Search is not available for this dataset
url
string | text
string | date
timestamp[s] | meta
dict |
|---|---|---|---|
http://math.stackexchange.com/questions/115579/find-the-minimum-value?answertab=oldest | # Find the minimum value.
Find the minimum value of $4^x + 4^{1-x}$ , $x\in\mathbb{R}$.
In this I used the property that $a + \frac{1}{a}\geq 2$.
So I begin with
$$4^x + \left(\frac{1}{4}\right)^x + 3\left(\frac{1}{4}\right)^x \geq 2 + 3\left(\frac{1}{4}\right)^x$$
So I think the minimum value be between 2 to 3. But the answer is 4.
-
How did you get $4^{1-x} = (\frac{1}{4})^x + (\frac{3}{4})^x$? If you evaluate at $x=1$, you get $4$ on the left, but $2$ on the right, so they are not equal. – Arturo Magidin Mar 2 '12 at 5:34
Do you know the inequality of the arithmetic and geometric means? – Gerry Myerson Mar 2 '12 at 5:34
@Arturo its not (3/4)^x .Its 3/ (4)^x. – vikiiii Mar 2 '12 at 5:37
@vikiii: That's the problem with not formatting your post correctly. – Arturo Magidin Mar 2 '12 at 5:41
@GerryMyerson its not (3/4)^x .Its 3/ (4)^x. – vikiiii Mar 2 '12 at 5:42
$$4^{x} + 4^{1-x} = (\sqrt{4^x} - \frac{2}{\sqrt{4^x}})^2 + 4 \ge 4$$
with equality occuring when $\sqrt{4^x} = \frac{2}{\sqrt{4^x}}$ and so $x = \frac{1}{2}$.
-
Your approach was a good one, except for the lack of "$x$, $-x$" symmetry.
But that is not hard to fix. Since $1/2$ is the midway point between $0$ and $1$, it seems natural to look instead at $4^{x-1/2}+4^{1/2-x}$. This is a close relative of our expression, since $$4^x+4^{1-x}=4^{1/2}(4^{x-1/2}+4^{1/2-x}).$$ Now it's over. By the result you quoted, the expression $4^{x-1/2}+4^{1/2-x}$ reaches a minimum of $2$ when $x=1/2$. So the minimum value of our original expression is $(4^{1/2})(2)$, which is $4$.
An alternate way of viewing the matter is that we made the symmetrizing change of variable $u=x-1/2$. Then $4^x=4^{1/2}4^u$ and $4^{1-x}=4^{1/2}4^{-u}$. The procedure is quite analogous to what happens when we complete the square in $x^2-x$. There, we can let $u=x-1/2$, and the parabola $y=x^2-x$ becomes the nicer parabola $y=u^2-\frac{1}{4}$, from which the minimum can be read off.
- | 2015-07-06T22:55:14 | {
"domain": "stackexchange.com",
"url": "http://math.stackexchange.com/questions/115579/find-the-minimum-value?answertab=oldest",
"openwebmath_score": 0.9193707704544067,
"openwebmath_perplexity": 366.82916433261744,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9830850879722155,
"lm_q2_score": 0.8479677564567912,
"lm_q1q2_score": 0.8336244564539268
} |
http://true-fish.ru/d1yul/directrix-and-focus-problems.html | Directrix and focus problems
## Directrix and focus problems
Figure 1 shows a picture of a parabola. Robert Buchanan Introduction to Conics: Parabolas A parabola is the set of all points equidistant from a point F, called the focus to a point on a line, called the directrix. Then draw the curve with the focus and directrix. All solutions are provided with step-by-step explanation for your reference. And a parabola has this amazing property: Like the ellipse and hyperbola, the parabola can also be defined by a set of points in the coordinate plane. Textbook solution for Single Variable Calculus: Early Transcendentals 8th Edition James Stewart Chapter 10. what is an equation of a parabola with the given focus and directrix? vertex (0,0) directrix (0,5/2) With it you can solve various types of problems and it’ll also address all your enquiries as to how it came up with a particular answer. To determine. The vertex is the point closest to them. Do 4 problems. We know that p is the distance from the vertex to both the focus and the Section 6-5: Parabolas A parabola is the set of all points P in the plane that are equidistant from a fixed point F (focus) and a fixed line d (directrix). Day 2: Parabola Worksheet Remember you can type the focus in the input Get an answer for 'Find the vertex, focus, and directrix of the parabola and sketch its graph. ©T K2I0c1 h2j kK HuqtHaj oSUoUfXt3w Fa HrVen lL cL tC N. Problems onProblems onPARABOLAPARABOLA 2. The Focus-Directrix Definitions of the Conic Sections The definition of a parabola is given in terms of a fixed point, the focus, and a fixed line, the directrix. The point which lies halfway between the focus and the directrix is called the vertex. Study concepts, example For our problem, it is already in this form. Then, we'll Let's graph the parabola from the problem above. Thus the directrix is located 2 units in the opposite direction from the vertex at y = -1. Learn Chapter 11 Conic Sections of Class 11 free with solutions of all NCERT Questions, Examples and Miscelleanous exercises. I also want to find ways to make my centers more engaging and for students to have more accountability within the centers so that I don't have problems like I did with a few kids in one of my classes. A set of points on a plain surface that forms a curve such that any point on that curve is equidistant from the focus is a parabola. This lesson has direct connections to Day 1. Check the graph below. The line is called the "directrix"; the point is called the "focus". Then from the definition of eccentricity, Figure 2-11. Here is a simple online Directrix calculator to find the parabola focus, vertex form and parabola directrix. . ax. A parabola is defined in terms of a line, called the directrix, and a point not on the directrix, called the focus. Students who took this test also took : Wintringham probability slightly harder Propability quiz (spinners, die, & dots) Wintringham sequences - rules Free Parabola Directrix calculator - Calculate parabola directrix given equation step-by-step This website uses cookies to ensure you get the best experience. Conic Sections: Parabolas, Part 5 (Focus and Directrix) Find the equation for a parabola given the vertex and given the focus and/or directrix. The focal chord is equal to 4a, where a is the distance from the vertex to the focus. If the hall below is 140 feet in length with 30 feet tall ceiling at its highest point. Finding the equation for a parabola when we have the equation about the focus and the directrix. Focus & directrix of a parabola from equation Our mission is to provide a free, world-class education to anyone, anywhere. A line perpendicular to the axis of symmetry used in the definition of a parabola. Parabola Directrix Calculator . Summary - Quick revision guideProof: Equation of the directrix and Given focus(x, y), directrix(ax + by + c) and eccentricity e of an ellipse, the task is to find the equation of ellipse using its focus, directrix, and eccentricity. Out comes the special parabola y = x2: problems. The axis of symmetry is the line which divides the parabola symmetrically. This curve can be a parabola. See Figure B. A parabola can be represented by the equation x2 = 2y. I am trying to work through the problem but am so confused. As far as I know, the first use of the focus directrix formulation was by Dutch mathematician Jan Witt. Write the polar equation for H. We learnedStraight Linesin the last chapter, but straight lines ar Learn Chapter 11 Conic Sections of Class 11 free with solutions of all NCERT Questions, Examples and Miscelleanous exercises. 2 = 4. The following general definition of a parabola is stated in terms of distance. You've found the x coordinate, what is the y coordinate? The directrix is a line with equation either x=c or y=c, which is it in your case? The length of the focal diameter is incorrect here, what are the y values on the parabola when x=1/28? What is the distance between those points? the focus) and a fixed line (called the directrix). x2 = 5y focus (x, y) = 0, 5/4 directrix:y=-5/4 focal diameter=5 Sketch its graph? my answers seem to be correct but i cannnot sketch the graph correctly. ), but haven't been able to find any for the directrix. Therefore,. Apr 07, 2018 · This calculus 2 video tutorial explains how to find the focus and directrix of a parabola as well as the vertex. We learnedStraight Linesin the last chapter, but straight lines ar Identify the vertex, focus and directrix of the parabola. Which part of the graph will the directrix pass through? B. Determine the equation of the parabola with a directrix of x + y − 6 = 0 and a focus at (0, 0). Let's see what conic section is. How to find the directrix and focus of a parabola? For x 2 = 4py, y = -p is the directrix For y 2 = 4py, x = -p is the directrix Conic Sections: Parabolas, Part 1 A quick way to roughly sketch a parabola. y = 3. This definition is illustrated by Figure 2. b. What are the focus and directrix of the parabola with the equation y=1/12 xsquared 2. Free Parabola Directrix calculator - Calculate parabola directrix given equation step-by-step This website uses cookies to ensure you get the best experience. Find the vertex . Algebra. Jun 17, 2010 · Finding the Focus & Directrix of a ParabolaMelody CariagaDiana DelgadoAilaFelicianoAngelica Pontejos Slideshare uses cookies to improve functionality and performance, and to provide you with relevant advertising. Parabolas. Write equations of parabolas. to the focus equals the distance to the "directrix. 25 Apr 2018 Pre-calculus: Parabola (Sample problems with solution) . The above diagram shows a point P P P on a parabola y 2 = 12 x y^2=12x y 2 = 1 2 x with focus F F F. com, “For a given point, called the focus, and a given line not through the focus, called the directrix, a parabola is the locus of points such that the distance to the focus equals the distance to the directrix. Clearly, QT > QU, so QT > FQ. " The directrix is the line y = -p below the vertex (so the vertex is halfway between focus and directrix). Ans: 2. the equation beocme. Grade 1 Focus - Displaying top 8 worksheets found for this concept. 4x - y^2-2y-33 = 0 3. The midpoint between the directrix and the focus falls on the parabola and is called the vertex of the parabola. We learn that, for a parabola, distance of a point from the focus = distance of the point from the directrix. When given a standard equation for a parabola centered at the origin, we can easily identify the key features to graph the parabola. Focus! The curves can also be defined using a straight line and a point (called the directrix and focus). Reflector. 25) For a parabola with an equation of the form (a)x² + (b)x + c, the directrix is a horizontal line with the equation y = k - f. Advertisement . Recall that the definition of locus is the set points that meet some given conditions. Remember that it's a line, though, so the directrix is x = 4. (iv) Any Tangent to a parabola and perpendicular on it from the focus meet on the Tangent at its vertex. Find the focus, directrix, and focal diameter of the parabola. Jul 09, 2019 · Watch this presentation to find out. The vertex is (1, -3), the axis of symmetry (now horizontal) is y = -3, and we don't recognize "max's and min's" for parabolas that open left or right. Exploring the Focus and Directrix. Answer to In Problem, find the vertex, focus, directrix, and axis of the given parabola. An internet search has come up empty. Given the focus of a parabola is located at (1. Equation of a parabola from focus & directrix. Directrix is y – 6 = 0 For any point of P(x, y) on the parabola Distance of P from directrix = Distance of P from focus 2. We’re actually presently working on additional materials to help readers with horizontal parabola problems. 5 Problem 8E. Find the vertex, x and y intercepts and do a quick graph. 8. Intercepts of Parabola. 21 Nov 2018 To be able to solve this problem, the first thing we need to look at is what are a focus and a directrix. ” If every point on a parabola is at an equal distance from the focus and the directrix, we can find points on the graph by folding the wax paper so that the directrix and the focus intersect one another. We know that FP = PT and FQ = QU. 1 Curriculum Burst 5: Focus and Directrix By Dr. Focus Directrix. Find the Parabola with Focus (-2,5) and Directrix y=3 (-2,5) y=3. Quick background: The parabola below has focus at F, and point P is at any position on the parabola. Apr 24, 2019 · (ii) Given that, vertex = (0,4) and focus = (0, 2) Now distance between the vertex and directrix is same as the distance between the vertex and focus. 2 This study of conics is from a locus-of-points approach, which where F is the focus, O is the center, and P and P' are points on the ellipse. c. Directrix of a Parabola. 34. If F is the focus of the parabola, V is the vertex and D is the intersection point of the directrix and the axis of symmetry, then V is the midpoint of the line segment F D ¯ . Use a graphing utility to verify your graph. A parabola is defined as follows: For a given point, called the focus, and a given line not through the focus, called the directrix, a parabola is the locus of points such that the distance to the focus equals the distance to the directrix. The directrix is parallel to the y-axis. Standard Form of a Parabola: Vertical Parabola Horizontal Parabola It is observed that the focus F (0, − p) is at a distance of p from the vertex V (0, 0) on the negative y-axis. In more advanced problems on the ellipse, you will need to know the equation of the directrix and the coordinates of the focus which are given in terms of the eccentricity e. of a parabola with directrix AB lying along the straightedge and focus F at the Matched Problem 2 (A) Find the equation of a parabola having the origin as its Conic Sections: Level 5 Challenges on Brilliant, the largest community of math and science problem solvers. Determine the focus and directrix of the parabola with the given equation. The directrix is given by the equation. ? 1. ] In the following graph, Problem. Understanding how the focus and directrix affect the equation of a parabola is crucial to understanding what each word means. 75. Problem 1. For an ellipse, the ratio is less than 1 An ellipse can be determined by means of a focus and a directrix. Find the vertex, focus, and directrix of the parabola and sketch its graph. We're not out of the brine yet. ) a. The focus is a coordinate, so it should be of the form (x,y). discovered a way to solve the problem of doubling the cube using parabolas. Grade 1 Focus. Course Activity Equation of a Parabola Based on Its Focus and Directrix This Course Activity will help you meet these educational goals: Content Knowledge—You will derive the equation of a parabola given a focus and directrix. 1. Do enough practice exercises that you have a good grasp of how these elements are related, and you should be successful with parabolas. ; The orthocenters of the above four triangles are on a line (Steiner I p. Why you should learn it Parabolas can be used to model and solve many types of real-life problems. Here, we learn how a parabola is derived when a plane cuts a cone. Some of the worksheets for this concept are Math in focus grade 1 workbook, Reading strategies and literary elements, Singapore math place value in math in focus, Mathematics grade 1, Grade 7 instructional focus documents, Storytown theme 1 follow me ela focus common core, Grammar and mechanics work Polar Equations of Conics We have seen four forms of polar equation that describe a conic section in terms of a focus and directrix. None of which seems to be a option. You can formulate a plan or strategy to solve a problem. directrix is x=0-[-6] Get a free answer to a quick problem. 1) x+1/8y^2=0. 128, Aufgabe 8), the directrix of the corresponding parabola. In this diagram, F is the focus of the parabola, and T and U lie on its directrix. If $$p>0$$, the parabola opens right. We have step-by-step solutions for What You Will Learn. Step 2: Solve for the focus. Section 2. The parabola has focus and goes through the points and . (See the attached photo to see this. Real World Applications. in a problem in Khan Academy, I factored X but I got the wrong answer! the right answer was to Write the equation of a parabola with a focus of (0,-2) and a directrix y = 2. Although the definition of a parabola is given in terms of its focus and its directrix, the focus and directrix are not part of the graph. The elements of the problem are substituted into the More About Directrix of a Conic Section. -Development of focus and directrix. Derive the equation of a parabola given a focus and a directrix. Thus, that point must be at an equal distance from both. Determine whether the directrix is horizontal or vertical. The parabola is defined as the locus of a point which moves so that it is always the same distance from a fixed point (called the focus) and a given line (called the directrix). See some background in Distance from a Point to a Line. However, instead of finding the parabola from the directrix and focus, the students will be Solutions to Homework Problems:. Find the vertex, directrix, and focus of the Can someone please help me with these? 1. MP2. Teaching Notes: In this lesson, students look at the directrix and the focus of a parabola. Note: The focus and directrix will not help you get a better sketch of the parabola than you have gotten in the past. If it is less than 1, then it is ellipse; and if it is greater than 1, then the conic section is a hyperbola. May 04, 2008 · i need help on how to find the focus and directrix of these 2 problems. Check Point 1 Find the focus and directrix of the parabola given by. See . 3 The Parabola 903 Solution The given equation,is in the standard form so We can find both the focus and the directrix by finding Divide both sides by 4. Finding the Vertex, Focus, and Directrix of a Parabola, find the vertex, focus, and directrix of the parabola and sketch its graph. ] May 19, 2015 · This makes the focus (1, 4 + 1/4) we simplify to (1, 4. Density and Design Problems 13 Terms. WORKSHEETS: Regents-Graphing Quadratic Functions AII: 10: TST PDF DOC TNS: Practice-Graphing Quadratic Functions 1: 15: WS PDF TNS: Practice-Graphing Quadratic Functions 2a MC, identify vertex, focus, directrix: 6: PDF TNS: Practice-Graphing Quadratic May 01, 2012 · To sketch the graph of a parabola, we first identify the vertex, the focus and the directrix. Precalculus Notes: Unit 8 – Conic Sections Page 2 of 18 Precalculus – Graphical, Numerical, Algebraic: Pearson Chapter 6 Parabola: a locus of points in a plane equidistant from a fixed point (focus) and a fixed line (directrix) p = distance between focus to directrix and vertex Write the equation for a parabola with a focus at (1,-4) and a directrix at x = 2. The standard form of a parabola that we are now going to use helps us to find the focus and the directrix. G-SRT. x = – a. Since the equation has its vertex at the origin and has a horizontal axis, the equation of its directrix is . C s cAxlClu Nr ji Fg0hXt 2sw Arje is ae5r Iv3eId 6. Plot the vertex, axis 15 Apr 2018 Observe that the focus point, F, moves further away from the dish each time you run it. 26 Feb 2017 Parabola to solve this Problem : FDP : Let a pt. Notice that the distance from the focus to point (x 1, y 1) is the same as the line perpendicular to the directrix, d 1. parabola, resp. Conics - Parabola - Focus and Directrix . ) Thanks for your question. A parabola is defined as follows: For a given point, called the focus, and a given line not through the focus, called the directrix, a parabola is the locus of points . In the case of a directrix of x = -3 and a focus of (1,2), we do indeed find that (y-2)² = 8(x+1). Focus: (p, 0) [the focus must lie on this axis of symmetry] Directrix: x = -p [the directrix must cross through this axis of symmetry] Focal diameter: │4p│ [The focal diameter is the length of the line segment that is perpendicular to the x axis (in this case), runs through the focus, and has its end points on the parabola. 3 Conic Sections – Parabola Parabola (locus definition) Set of all points equidistant from a Focus to a Directrix. For this problem, assume the dish is pointing upward and represent the vertex as (0 Precalculus : Find the Focus and the Directrix of a Parabola. what is an equation of a parabola with the given focus and directrix? vertex (0,0) directrix (0,5/2) The answer will be disappointing, I am afraid, but typical, "natural" and "artificial" are very relative notions. on the parabola, then, In mathematics, a parabola is a plane curve which is mirror-symmetrical and is approximately U-shaped. Introduction to Conics: Parabolas Cosmo Condina/Getty Images 10. A parabola is the set of all points $\left(x,y\right)$ in a plane that are the same distance from a fixed line, called the directrix, and a fixed point (the focus) not on the directrix. To do this, we first write the equation in the form (x - h)^2 = 4p(y - k), where (h, k) is the vertex Find the Parabola with Focus (2,4) and Directrix y=8 (2,4) , y=8 Since the directrix is vertical , use the equation of a parabola that opens up or down. Graph the parabola. Since the focus and directrix of the parabola x 2 = 4 p y are at equal distance from the vertex V (0, 0) in the opposite directions, the equation of directrix is y = p. By contrast, the Use this applet to verify your answers on the worksheet given. Sample Problem. and, Subtraction and addition of the two equations give. And every parabola is going to have a focus and a directrix, because every parabola is the set of all points that are equidistant to some focus and some directrix. In my experience, it is easier to remember the relationships between the vertex, focus, axis of symmetry, directrix, and the value of p, than to try to memorize the (often very long) list of formulas they give you. y. Parabolas- Focus & Directrix Worksheets- Includes math lessons, 2 practice sheets, homework sheet, and a quiz! I know of many applications for the focus of a parabola (satellite dish, whispering gallery, etc. Solving Applied Problems Involving Parabolas. So that's all a focus and a directrix is. what is an equation of a parabola with a vertex at the origin and directrix y=19/4 3. Match that with the distance to the focus at (0,a)- this is the square root below. Q is another point on the parabola, with QU perpendicular to the directrix. If the directrix is given in terms of $y$, we use the general polar form in terms of sine. Can someone please help me with these? 1. Aug 13, 2003 · About the focus-directrix formulation of conics, a friend of mine got interested and inquired of Don Allen (math historian) who replied in part as follows: <<. I tried it when I was having difficulty solving problems based on focus solver vertex directrix and I really enjoyed using it. James Tanton, MAA Mathematician in Residence . [The word locus means the set of points satisfying a given condition. Question 246720: I have a math question with two parts. If l l l is a tangent A parabola has its vertex at the origin and focus at (0,4). To obtain this model, we begin with a definition that permits a unified approach to the conic sections. How far from the end walls will the foci point be? A parabola is the set of all points $$(x,y)$$ in a plane that are the same distance from a fixed line, called the directrix, and a fixed point (the focus) not on the directrix. Find the directrix and an equation for this parabola. The Parabola. A segment though the focus . The portion of a tangent to a parabola cut off between the directrix and the curve subtends a right angle at the focus. Solution: Step 1: Analysis. 2) -x^2=5y. We can see that it’s a vertical parabola that opens down, since the since the directrix is horizontal and the focus is below it. One of the properties of parabolas is they are made of a material that reflects light that travels parallel to the axis of symmetry of a parabola and strikes its concave side which is reflected its focus. To find: The vertex, The Solution to Your Study Problems. The relationship determining the ellipse is PF = \\frac {3}{5}PD for all point P. The vertex of a parabola is ( )and the directrix is the line . The directrix is a fixed line used in describing a curve or surface. See [link]. If the ratio (r) of the distance of any point on the conic section from focus to its distance from directrix is equal to 1, then the conic section is a parabola. Section 6-5: Parabolas A parabola is the set of all points P in the plane that are equidistant from a fixed point F (focus) and a fixed line d (directrix). The focus and directrix simply give us more information about the soul of a parabola. We still need the directrix, but it is very easy to find now. A parabola is the locus of points that are equidistant from both the directrix and focus. Since both focus and vertex lie on the line x = 0, and the vertex is above the focus, this parabola opens downward, and has the equation (y − 2) = −4px2, Explore the properties of parabolas in this lesson. I hope that I have explained how to do this problem well enough so that you can do similar problems. Definition (NEW): The set of all points in a plane that are equidistant from a fixed line (directrix) and a fixed point (focus) which is not a line. You will need to move the slider for the directrix and move point F for the forcus or type F=(x,y) in the input bar, with x and y be the coordinated you want to use. May 19, 2016 · Since, the focus is on Y-axis. Definition (OLD): an equation in the form of y=𝑎𝑥2+𝑏𝑥+𝑐or y=𝑎(𝑥−h)2+𝑘. Finding the Focus and Directrix of a Parabola May 18, 2013 · problems on parabola 1. The vertex of the parabola is not the origin. 9 Worksheet by Kuta Software LLC Locate the focus and directrix then graph the equation . Math homework help video on using the focus and directrix to find the equation of a parabola. 1 Give an equation of the parabola with focus (1, 1) and directrix y = 3. In the diagram, the directrix of an ellipse is the line x = \\frac {25}{3} and a focus is (3,0). What is the Focus and Directrix? The red point in the pictures below is the focus of the parabola and the red line is the directrix. As you can see from the diagrams, when the focus is above the directrix Example 1, the parabola opens upwards. Because is positive, the parabola, with its symmetry, opens to the right. Construct a parabola given a focus and a directrix on the Ti-Nspire. Includes full solutions and score reporting. It's to the left, so we hike right 2. Find the length of the segment that passes through the focus, is parallel to the directrix, and whose endpoints lie on the parabola. And since the vertex of a parabola lies on the center of the shortest line joining directrix and focus, it should be (0,3) and since the focus is 4. Alternatively, one can define a conic section purely in terms of plane geometry: it is the locus of all points P whose distance to a fixed point F (called the focus) is a constant multiple (called the eccentricity e) of the distance from P to a fixed line L (called the directrix). Here is the information about his book where F is the focus, O is the center, and P and P' are points on the ellipse. SOLUTION Because the variable x is squared, the axis of symmetry is the vertical line Since the equation is written in standard form, and The focus is and the directrix is Because the parabola opens up. G-GPE. y = 4 - 1/4. H F tM ha Hdje z cwNistMh7 SI knVfzi 4n 9iDtEe3 0A 3lngJe rb trNaF G21. The "discovery" of the focus-directrix property was likely a technical lemma in investigations of solid loci, which was later also used in the study of curved mirrors. Notice that the vertex is midway between the focus and the directrix. Nothing about directrix and focus in this video (look in part 2). Use the distance formula to write and Jan 01, 2016 · Socratic Meta Featured Answers How do I find the directrix of a hyperbola? How do you find the eccentricity, directrix, focus and classify the conic section focus at $$\left( {-2,4} \right)$$ and a directrix of $$y=9$$ It’s best to first plot the points, so we can see the direction of the parabola. Place the center of the ellipse at the origin so that one focus lies at ( - ae,0) and one directrix is the line x = - ale. The parabola focus is a point from which distances are measured in forming a conic and where these distances converge. We need to identify some crucial elements of the parabola: We have the vertex, the focus and the directrix. 3 Focus of a Parabola 69 You can derive the equation of a parabola that opens up or down with vertex (0, 0), focus (0, p), and directrix y = −p using the procedure in Example 1. Demonstration of Focus Point for a Parabola (Manipula Math) Drawing a Parabola (Manipula Math) The equations of the parabola are as follows: axis . asked by Ama on November 13, 2012; algebra 2. If P is any pt. The important features of a parabola are shown in the diagram below. Find the vertex, p, focus, and directrix of -8(x – 2) = y2. A parabola with its vertex at the origin and opening to the right has its focus at (a,0) and its directrix at x = -a; its corresponding equation is y 2 = 4ax. CCSS Math: HSG. i understand how to find the focus and directrix on simple problems but the first one is set equal to zero and the second one has a negative so they are confusing me. Find an equation for the new parabola, and find the new vertex, focus, and directrix. Aug 13, 2005 · The points of a parabola are equidistant from it directrix and focus. (iii) Tangents at the extremities of any focal chord intersect at right angles on the directrix. About "How to find vertex focus and directrix of a parabola" How to find vertex focus and directrix of a parabola : Before going to find these details first we have to check whether the equation of the parabola is in the standard form or not. x^2 = 16(y-3) or (x/4)^2 + 3 = y. At right is a graph of a conic section with it’s focus at the origin and its directrix at y = –5. Improve your math knowledge with free questions in "Find the focus or directrix of a parabola" and thousands of other math skills. and a focus found by using . the focus. 7 "Crazy" Women in Literature Who Were Actually Being Totally Reasonable Jan 28, 2020 the directrix and focus (explained above) the axis of symmetry (goes through the focus, at right angles to the directrix) the vertex (where the parabola makes its sharpest turn) is halfway between the focus and directrix. Reason Finding the Vertex, Focus, and Directrix of a Parabola Given in Standard Form We're also solving this problem in the context of multiple choice, which is great 25 Apr 2013 We'll find the vertex, axis of symmetry, focus, and directrix. The key features of a parabola are its vertex, axis of symmetry, focus, directrix, and latus rectum. A set of points on a plain surface that forms a curve such that any point on the curve is at equidistant form the focus is a parabola. 5,1) and has a directrix at x = 2. Explore Graph by Plotting Points. Rewrite the Cartesian equation for H. x + y^2 = 0 2. I have Find an equation of the parabola with focus at (8 , 0) and directrix given by the equation x = 2. The focus is 9/4 units below the vertex; the directrix is the horizontal line 9/4 units above the vertex:. For instance, in Exercise 62 on page 742, a parabola is used to model the cables of the Golden Gate Bridge. Popular Problems. Deriving the Directrix Equation from the Vertex and Focus Coordinates. It discusses how to write the equation of the parabola in standard form by Problem – Find the vertex, focus and directrix of a parabola when the coefficients of its equation are given. Find the standard form of the equation of a parabola given a focus Find the standard form of the equation of a parabola given a directrix Find the standard form of the equation of a parabola given an axis and a point on the parabola Vertex, Focus, and Directrix The line containing the focus and the vertex is the axis. Possibly the most straight forward way is to use the midpoint formula, given that the vertex is midpoint between the focus and a collinear point on the directrix. Instructional Unit The Parabola: An Algebraic Approach Day 6. find the directrix and focus of the parabola directrix is x= h-p. Question 9 Find an equation of the parabola with directrix given by the equation y = 2, a focus on the y axis, and the point (-6 , -8) lies on the parabola. Interactive Graph - Directrix and Focus of a Parabola. Since the graph is cut off it is not clear if the conic section is an ellipse, an hyperbola or a parabola. The focus will be to the right of the vertex, and the directrix will be a vertical line that is the same distance to the left of the vertex that the focus is to the right. Solution of exercise 1 Determine the equations of the following parabolas and indicate the values of their focal parameter, focus and directrix. Point Q is the foot of the perpendicular to the directrix Identify the vertex, focus, axis of symmetry, directrix, direction of opening, min/max value, length of the latus rectum, and the x- and y-intercepts of each. Step 1: The distance from the vertex to the focus is 2 = d, the focal distance. The crease will be half-way between the focus and the directrix. Problem 1: Find the vertex, focus, axis, and directrix of the followingpara The focus of a parabola tangent to four lines is the Miquel point of the lines: the common point of the circumcircles of the four triangles formed by the four lines. Directions: The focus and directrix of a parabola are given. Question 10 A parabolic dish with a diameter of 200 cm and a maximum depth of 50 cm is the coordinates of the focus in x,y format are: (29/16, 1) The directrix of the parabola is x = 35/16 Since p = -3/16, the directrix is 3/16 units to the right of the vertex. Example Of Directrix Of A Conic I think that it is a clever way to solve such a problem, but is there any quicker method or formula that when given the focus and directrix (which is tilted), I can easily find the equation of the parabola? I have also seen and read this previously asked question if anyone was wondering. Choose: The focus is a point inside the parabola and the directrix is a line outside the parabola line, which are the green point and line drawn below: So as we see from the graph, the focus is (7,3) and the directrix is x=3. Mathematical Practices—You will make sense of problems and solve them. The axis of symmetry is the horizontal line y = 3 because the focus is always on the axis of symmetry. According to mathwords. With p = 4, the distance down from any (x, y) is y + 4. focus by two tangents drawn from a point), and (having given the focus and a double ordinate) he uses the focus and directrix to obtain any number of points on a parabola - the first instance on record of the practical use of the directrix. (y − 3)2 = − 8x. The focus (2,3) is on the vertical line x = 2. Let F be the point (m,n), and the directrix be the line y = t. Solution. So, the parabola opens to the left. Focus and Directrix of Parabola. Khan Academy is a 501(c)(3) nonprofit organization. What are the focus and directrix of the given equation? y= -1/8 xsquared 4. Notice that the point halfway between the focus and the directrix lies on the parabola; it is called the vertex. Give an equation of the parabola passing through (0, −2) that has vertex (−1, 2) and axis y = 2. \displaystyle 16= 4p Improve your math knowledge with free questions in "Find the focus or directrix of a parabola" and thousands of other math skills. To form a parabola according to ancient Greek definitions, you would start with a line and a point off to one side. Explore the focus and the directrix of a parabola. Since the directrix is vertical, use the equation of a parabola that Find the vertex, focus, axis, directrix and axis of the parabola $y^{2}-4y-8x-28=0$ STANDARD G. x^2 - 2x + 8y + 9 =0 Please help me out with these problems problems Graphing an Equation of a Parabola Identify the focus and directrix of the parabola given by Graph the parabola. Derive the equation of a parabola given a focus and directrix. The parabola is symmetric through , and the common distance is , so the directrix is the line through and , which is the line Using the point-line distance formula, the parabola is the locus which rearranges to . The vertex is in the 2nd quadrant. F of a parabola with vertex Sep 22, 2017 · Given - Vertex #(-2,1)# Directrix #x=1#. A parabola is symmetric with respect to its axis. x, y x y (x, y) Directrix Axis Focus Vertex d 2 d 1 2 d 1 Standard Equation of a Parabola (Vertex at Origin) The standard form of the equation of a parabolawith vertex at and directrix is given by Find Vertex Focus Directrix and Latus Rectum of Parabola : Here we are going to see some example problems to understand the concept of finding vertex, focus, directrix, equation of latus rectum of the parabola. 3 Lesson WWhat You Will Learnhat You Will Learn Explore the focus and the directrix of a parabola. There are simple derivations to get the equation of a parabola based on the location of a directrix and the focus, but we will skip the derivation in this introduction. ] The focus of the parabola is located on the positive y-axis. Well, the focus and the directrix are a point the directrix and focus (explained above); the axis of symmetry (goes through the focus, at right angles to the directrix); the vertex (where the parabola makes its 9 May 2017 equation y2 = -24x. Having y be squared, instead of x? Weird, weird, weird. that is Linear and angular momentum problem: Ball (-2, -3) and has a directrix of y = -6, find the location of the focus and the equation of the parabola in standard form. Free practice questions for Precalculus - Find the Focus and the Directrix of a Parabola. Like the ellipse and hyperbola, the parabola can also be defined by a set of points in the coordinate plane. x 2 = 6y. PT is perpendicular to the directrix, and the line MP bisects angle ∠FPT. Sometimes we need to manipulate a polar equation in order to recognize the conic it represents. One description of a parabola involves a point (the focus) and a line (the directrix). A parabola is the set of points that are equidistant from a given fixed point (the focus) and from a given fixed line (the directrix) in the plane. The distance between the directrix and _____ is set equal to the distance between the _____ and the same point on the parabola. 5, find the coordinates of the vertex and the equation of the parabola in standard form. It fits several other superficially different mathematical descriptions, which can all be proved to define exactly the same curves. Some of the worksheets for this concept are Equations of parabolas, Parabolas, Conic sections review work 1, Focus and directrix notes, Parabola with horizontal directrix, , Focus of a parabola, Work locus an parabolas. Okay, this is a little disorienting. Parabola, showing focus (0, p), and directrix y = − p. I went ahead and found the focus because in other problems you will have to find the focus. In this definition of a parabola, it is the shape created by the points that are the same distance from a given point (call the focus) and a given line (called the directrix)*. Figure %: In the parabola above, the distance d from the focus to a point on the parabola is the same as the distance d from that point to the directrix. Suppose a vertex is located at (3, 1) and the focus is located at (3, 3). 2 Solutions 1. 2 The line passing through the focus and the vertex is called the axis of the parabola. Hence the "p" is 10/2 = 5 units. 11 (+) Make sense of problems and persevere in solving them. Find the vertex, focus, directrix, axis and latusrectum of the parabola y^2 - 4x - 4y = 0 In this lesson, students look at the directrix and the focus of a parabola. Below is a drawing of a parabola. You can explore the concept of directrix and focus of a parabola in the following JSXGraph (it's not a fixed image). Focus and Directrix Notes Focus and Directrix of a Parabola Focus:fixed point inside the parabola on the axis of symmetry Directrix:line outside the parabola; perpendicular to the axis of symmetry the focus and directrix are equidistant from the vertex Concept A parabola is the curve formed from all points that are equidistant from the directrix and the focus. Focus Directrix - Displaying top 8 worksheets found for this concept. parabola is symmetric around the y-axis and so the focus lies a distance from the center (, ) along the y-axis (,) (,) (,)->focus the distance between the focus and directrix is parabola is symmetric around the y-axis and so the directrix is a line parallel to the x-axis, a distance from the center left (,) x-coordinate the focal width is answer: 1 The midpoint between the focus and the directrix is called the vertex. The key feature of a A parabola is a locus of points equidistant from both 1) a single point, called the focus of the parabola, and 2) a line, called the directrix of the parabola. College algebra problems on equation of parabolas are presented along with their Find the vertex, the focus, the axis of symmetry and the directrix of the Provides worked solutions to typical word problems. 2 AII. A parabola is the set of all points$\,\left(x,y\right)$ in a plane that are the same distance from a fixed line, called the directrix, and a fixed point (the focus) not on the directrix. Or if the parabola was down here, you'd go straight up to find that distance. It is helpful to make a When the focus and directrix are used to derive the equation of a parabola, two distances were set equal to each other. By using this website, you agree to our Cookie Policy. x. A. Now pick an arbitrary point (x,y). Solve real-life problems. We start from the vertex and walk in the opposite direction from the focus. Section 9. (x + 1) 2 − 8(y + 2) = 0 Nov 20, 2016 · I definitely want to do the focus and directrix centers again with this topic. The Relation between focus, vertex and directrix: The vertex of the parabola is at equal distance between focus and the directrix. J. P is an arbitrary point on the parabola. Is there a calculatr that you can buy that will work exponents and polynomials and other algebra problems difference of square solving systems with 3 variables in a TI-83 plus CONIC SECTIONS EXERCISE 1 Parabolas. GPE. 2. S and a line d, be the focus & directrix of a. Thus the equation of the directrix is . 7) x x y 68 Chapter 2 Quadratic Functions 2. between the focus and the directrix is called the vertex, and the line passing through the focus and the vertex is called the axis of the parabola. Write equations of parabolas in vertex form and determine the a value of a parabola given a focus and directrix. The standard form of a parabola with vertex $$(0,0)$$ and the x-axis as its axis of symmetry can be used to graph the parabola. We will learn how to solve different types of problems on parabola. So distance from focus to directrix along the axis of symmetry is 12 -2 = 10 units. A parabola is the set of all points that have a greater y value than a single point, called the focus of the parabola, and a lesser x value than a single point, called the directrix of the parabola. Vertex of a Parabola. Find the focus of the parabola. Once we have found the orientation of the parabola, we can find the directrix in a couple of ways. However, instead of finding the parabola from the directrix and focus, the students will be finding the directrix and focus of a known parabola. Rotate to landscape screen format on a mobile phone or small tablet to use the Mathway widget, a free math problem solver that answers your questions with step-by-step explanations . I first give you a summary of these results and then the proofs. The parabola is the curve formed from all the points (x, y) that are equidistant from the directrix and the focus. The Directrix of the Parabola: The directrix of the parabola is the horizontal line on the side of the vertex opposite of the focus. For how many points with integer coordinates is it true that ?. The question is as follows: Part 1) P is the parabola with focus (3,1) and directrix x=7. When we measure the distance: from the focus to a point on the curve, and; perpendicularly from the directrix to that point; the two distances will always be the same ratio. y 2 = 4x, left 2, down 3 . We solve problems based on this principle and also learn how to calculate equation of the axis and the coordinates of the vertex. Demonstration of Focus Point for a Parabola (Manipula Math) Drawing a Parabola (Manipula Math) The equations of the parabola are as follows: The focus is focused on the origin. The directrix x = 12 is a vertical line, therefore the parabola is horizontal. Find the equation of the parabola whose vertex is at (0,2) and focus is the origin. The vertex, located at the origin,is a point on the graph of and Example 1 illustrates how you can find two additional points on the parabola. using conic sections in polar coordinates. What are the coordinates of the focus and the equation of the directrix? focus: (0,8); directrix: y - 4333610 Given the focus and the directrix of a parabola, find its equation. TEKS FOCUS • Directrix – the fixed line used to define a parabola • Focal length – the distance between the vertex and the focus of a parabola • Focus (plural: foci) of a parabola – the fixed point used to define a parabola • Formulate – create with careful effort and purpose. What problems would you directrix, the focus and directrix are not part of the graph. What we're looking at in this problem is a parabola with a focus at 0,3 and the directrix at y equals -3 and we are trying to find the equation for this parabola. How To: Given the focus, eccentricity, and directrix of a conic, determine the polar equation. Shift the hyperbola so that one focus is at the origin. These are all, these are all right angles right over here. y = k - p This short tutorial helps you learn how to find vertex, focus, and directrix of a parabola equation with an example using the formulas. Check Review your knowledge of the focus and directrix of parabolas. The line through the focus perpendicular to the directrix is called the axis of the parabola. Problems 11. Intersect at y-axis, at one focus of the ellipse can whisper and be heard by another person standing at the other focus, because all the sound waves that reach the ceiling from one focus are reflected to the other focus. directrix and focus problems | 2020-02-23T03:26:52 | {
"domain": "true-fish.ru",
"url": "http://true-fish.ru/d1yul/directrix-and-focus-problems.html",
"openwebmath_score": 0.507796049118042,
"openwebmath_perplexity": 373.28047375649123,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES\n\n",
"lm_q1_score": 0.9830850906978876,
"lm_q2_score": 0.8479677526147223,
"lm_q1q2_score": 0.8336244549881283
} |
https://math.stackexchange.com/questions/2932482/probability-that-sheepdog-performs-at-least-one-task-successfuly-am-i-doing-th/2932485 | # Probability that sheepdog performs at least one task successfuly - Am I doing this problem right?
What is the probability that a sheepdog performs at least $$1$$ of these tasks successfully?
My approach is to subtract the probability of performing at most $$1$$ of these tasks successfully from the probability of performing all $$4$$ tasks successfully.
$$P(\text{fetch})=.9, P(\text{drive})=.7, P(\text{herd})=.84, P(\text{separate})=.75$$.
The complement of these four probabilities is, $$.1,.3,.16,$$ and $$,.25$$, respectively.
So the probability that the sheepdog performs all four tasks successfully is simply, $$(.9)(.7)(.84)(.75)$$.
The probability that the sheepdog performs at most $$1$$ task successfully can be split into $$4$$ cases. Either the sheepdog performs the fetch task (and not the other 3) successfully, performs the drive task, performs the herd task, or performs the separate task.
This would look like:
$$(.9)(.3)(.16)(.25)+(.1)(.7)(.16)(.25)+(.1)(.3)(.84)(.25)+(.1)(.3)(.16)(.75)$$
Subtracting this from the case in which the sheepdog performs all four tasks would yield:
$$(.9)(.7)(.84)(.75)-[(.9)(.3)(.16)(.25)+(.1)(.7)(.16)(.25)+(.1)(.3)(.84)(.25)+(.1)(.3)(.16)(.75)]$$.
Is this correct?
• note: i would obviously simplify my answer at the end by multiplying all of the numbers out, i just want to see if this approach is correct... Sep 27, 2018 at 1:25
• So the probability that he will perform at least one correctly is 60% less than the probability of him doing just the first task correctly? Sep 27, 2018 at 13:00
• I'm so disappointed that this was not a question about a sheep/dog hybrid animal.
– pipe
Sep 27, 2018 at 16:37
The complement of at least $$1$$ is not at most $$1$$.
The complement is if none of the task is perform.
Hence just compute $$1-\prod_{i=1}^4 (1-p_i)$$
• woowwwww...i cant believe i missed that. thank you ! i got it now Sep 27, 2018 at 1:36
By sheepdog, I assume you mean 'good sheepdog' as defined in the question. Let $$X$$ denote the number of things in the list satisfied. If you want the probability the at least one of the 4things listed is satisfied $$P(X\geq 1)$$, you work it out by solving $$1-P(X=0)$$ This is equal to $$1-(0.1)(0.3)(0.16)(0.25) = 0.9988$$
Calculating the compliment of the chance of failing in all trials is undoubtedly the 'correct' approach to this problem, as per Siong Thye Goh's answer.
The same result can be calculated by accumulating the additional chance of success from each trial, given that the trial is needed, like this:
Chance this Chance of success Additional Cumulative chance
trial needed in this trial success from chance of success
this trial in any trial
--------------------------------------------------------------------
A = 1.0 - D' B (input data) C = AB D = C + D'
--------------------------------------------------------------------
0
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1 0.9 0.9 0.9
0.1 0.7 0.07 0.97
0.03 0.84 0.0252 0.9952
0.0048 0.75 0.0036 0.9988 | 2022-09-26T18:16:46 | {
"domain": "stackexchange.com",
"url": "https://math.stackexchange.com/questions/2932482/probability-that-sheepdog-performs-at-least-one-task-successfuly-am-i-doing-th/2932485",
"openwebmath_score": 0.9046511650085449,
"openwebmath_perplexity": 242.69325245638774,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9830850857421198,
"lm_q2_score": 0.8479677526147223,
"lm_q1q2_score": 0.8336244507857969
} |
https://mathematica.stackexchange.com/questions/69333/discrete-fourier-transform-with-x-axis-data | # Discrete Fourier Transform with x axis data?
Usually, Fourier Transform in mathematica is supplied by y data only, like the famous Sin[x] transform example. What if my data is a list of {x,Sin[x]}? and x has units (for example, nm)? I would expect to see a spectrum with the correct units as frequencies. How can I achieve this?
Data1 = Table[Sin[x], {x, 0, 2*Pi, 2*Pi/1000}];
ListPlot[Abs[Fourier[Data1]], PlotRange -> All, Joined -> True]
and you see the plot doesn't make much sense. Although i have 2 peaks, their frequencies are not specified. I actually couldn't know which is + frequency and which is - frequency. In deed, i can not make much sense of this plot except for it has peaks.
Data2 = Table[{x, Sin[x]}, {x, 0, 2*Pi, 2*Pi/1000}];
ListPlot[Abs[Fourier[Data2]], PlotRange -> All, Joined -> True]
I would expect this to recover the x-axis information. But it doesn't . So what should I do if I wanted to know, for example, the peaks in the plot has a meaningful x axis, corresponding to , 2\Pi for example
Thanks
• See the Frequency identification example on the Fourier[] doc page – Dr. belisarius Dec 18 '14 at 21:42
• I didn't find it helpful.. And still it can not transform with real data (ones having x axis) – bboczeng Dec 18 '14 at 21:56
• to compare, origin labs can handle it extremely well. – bboczeng Dec 18 '14 at 21:57
• You might find the answer here useful to understand what the DFT is doing. mathematica.stackexchange.com/q/33574/1783 – bill s Dec 18 '14 at 21:58
• Is this any improvement? Fourier[data1, FourierParameters -> {0, 2*Pi/1000}] – Daniel Lichtblau Dec 18 '14 at 22:03
If you look up ListPlot you will see that it uses point number for the x - axis. You have to make the frequency axis. Your question is a little muddled because you talk about time and frequency but the units you suggest are nm which I take to be nano meters. If you are working with length in one domain then when transformed you are in units of reciprocal length or wave number. I will continue with time to frequency transformation. The increment in the frequency domain is given by the sample rate divided by the number of points. The sample rate is the reciprocal of the time step. The frequency spectrum starts at zero and continues to the sample rate less one increment. Here is one way to make your frequency axis.
sr = 1000./(2 Pi); (* sample rate*)
inc = sr/Length[Data1]; (* increment *)
freq = Table[f, {f, 0, sr - inc, inc}];
Now you can plot your data
ListPlot[Transpose[{freq, Abs[Fourier[Data1]]}], PlotRange -> All,
Joined -> True]
Your sample rate is small compared to the frequency so your data is concentrated near the origin. Expanding the plot shows that your peak is at the second point.
ListPlot[Transpose[{freq, Abs[Fourier[Data1]]}],
PlotRange -> {{0, 1}, All}, Joined -> True]
I suggest you write a Module if you whish to calculate the frequency axis automatically. | 2019-07-17T03:17:53 | {
"domain": "stackexchange.com",
"url": "https://mathematica.stackexchange.com/questions/69333/discrete-fourier-transform-with-x-axis-data",
"openwebmath_score": 0.3349512219429016,
"openwebmath_perplexity": 985.5983238398626,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9481545333502202,
"lm_q2_score": 0.8791467785920306,
"lm_q1q2_score": 0.8335670036022761
} |
https://mathematica.stackexchange.com/questions/86837/trouble-with-differential-equation | # Trouble with differential equation
I tried to solve this differential equation:
$$\epsilon y''(x)+xy'(x)=-\epsilon \pi^2 \cos(\pi x)-\pi x\sin(\pi x)$$
with boundary conditions: $y(-1)=-2, \space y(1)=0$. If we take $\epsilon=0.1$, Mathematica can solve it without any trouble
Block[{e = 0.1, min = -1, max = 1},
Plot[Evaluate[
y[x] /. NDSolve[{e y''[x] +
y'[x] x == -e Pi^2 Cos[Pi x] - Pi x Sin[Pi x],
y[min] == -2, y[max] == 0}, y, {x, min, max}]], {x, min, max}]
]
But if we want a smaller $\epsilon$, let say 0.01, Mathematica seems unable to handle it. Is there any options to invoke or methods to employ to get the desired result? Anyway, this is the solution for $\epsilon=0.0001$.
Thank you.
• Piece of advice: don't use Block to inject values into parameters. Use With instead. – m_goldberg Jun 25 '15 at 14:24
• The fundamental problem would appear to be that, in the limit of small e, the order of the equation drops from second to first, with the result that there is one too many boundary conditions. – bbgodfrey Jun 25 '15 at 14:30
DSolve can handle this.
Clear[y];
y[x_, e_] = y[x] /. DSolve[{
e y''[x] + y'[x] x == -e Pi^2 Cos[Pi x] - Pi x Sin[Pi x],
y[-1] == -2, y[1] == 0}, y[x], x][[1]] // Simplify
Manipulate[
Plot[y[x, e], {x, -1, 1}],
{{e, 0.01}, 0.0001, 0.1, Appearance -> "Labeled"}]
• Thank you for your answer, but I'm looking for numerical solution. – Deco Jun 25 '15 at 14:59
• @Deco Bob's symbolic solution can be evaluated numerically quickly, and with better accuracy than a mere numerical approximation. In fact DSolve can solve the problem in full generality: DSolve[{e y''[x] + y'[x] x == -e Pi^2 Cos[Pi x] - Pi x Sin[Pi x], y[min] == -2, y[max] == 0}, y, x]. Is there some reason you do not think this is a superior solution? – Michael E2 Jun 25 '15 at 15:55
As noted by @bbgodfrey, the "shooting" algorithms that Mathematica tends to use are not well-adapted to this particular equation. Better would be some kind of relaxation method, which is what Mathematica uses (I think) for solving PDEs on a mesh. And an ODE is just a PDE in one dimension, so let's try solving this equation on a one-dimensional mesh:
Needs["NDSolveFEM"]
truey[x_,e_] = Cos[\[Pi] x] + Erf[x/(Sqrt[2] Sqrt[e])]/Erf[1/(Sqrt[2] Sqrt[e])];
e = 0.0001; min = -1; max = 1;
mesh = ToElementMesh[Interval[{min, max}], MaxCellMeasure -> 0.05];
bcs = {DirichletCondition[y[x] == -2, x == min], DirichletCondition[y[x] == 0, x == max]}
soln = NDSolve[{e y''[x] + y'[x] x == -e Pi^2 Cos[Pi x] - Pi x Sin[Pi x], bcs}, y, Element[x, mesh]]
Plot[{Evaluate[y[x] /. soln], truey[x, e]}, {x, min, max}]
Clear[e, min, max]
I've plotted here the result from Mathematica's finite element solver (in blue) versus the true solution found by @BobHanlon (in yellow). I've actually used a coarser mesh than is ideal to show the difference between the two; if you set MaxCellMeasure -> 0.01 in the above code (instead of 0.05), the two curves are indistinguishable at this resolution.
To see why NDSolve has difficulty with this problem for very small e, consider that NDSolve solves this two-point boundary value problem by some form of shooting. In other words, it varies y'[min] until one is found that yields the desired y[max]. However, as e becomes very small, the sensitivity of y[max] to y'[min] becomes great, because the differential equation becomes singular in that limit.
y''[x] + y'[x] x/e == -Pi^2 Cos[Pi x] - Pi x Sin[Pi x]/e
This is also apparent from Bob Hanlon's symbolic solution. To illustrate the sensitivity of y[max] to y'[min], consider
slope = D[Cos[π x] + Erf[x/(Sqrt[2] Sqrt[e])]/Erf[1/(Sqrt[2] Sqrt[e])], x] /. x -> -1;
LogLogPlot[N[slope], {e, 0.0001, .1}, PlotRange -> All]
You can work around this sensitivity by increasing WorkingPrecision, e. g. for e = .01,
With[{e = 1/100, min = -1, max = 1},
Plot[Evaluate[y[x] /. NDSolve[{e y''[x] + y'[x] x == -e Pi^2 Cos[Pi x] -
Pi x Sin[Pi x], y[min] == -2, y[max] == 0}, y, {x, min, max},
WorkingPrecision -> 50, MaxSteps -> 50000]], {x, min, max}]]
but doing so rapidly becomes prohibitively expensive as e is further reduced. | 2021-04-11T03:42:58 | {
"domain": "stackexchange.com",
"url": "https://mathematica.stackexchange.com/questions/86837/trouble-with-differential-equation",
"openwebmath_score": 0.23241481184959412,
"openwebmath_perplexity": 4110.777804354845,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9481545304202038,
"lm_q2_score": 0.8791467580102418,
"lm_q1q2_score": 0.8335669815116453
} |
https://courses.lumenlearning.com/ivytech-collegealgebra/chapter/identify-horizontal-asymptotes/ | ## Identify horizontal asymptotes
While vertical asymptotes describe the behavior of a graph as the output gets very large or very small, horizontal asymptotes help describe the behavior of a graph as the input gets very large or very small. Recall that a polynomial’s end behavior will mirror that of the leading term. Likewise, a rational function’s end behavior will mirror that of the ratio of the leading terms of the numerator and denominator functions.
There are three distinct outcomes when checking for horizontal asymptotes:
Case 1: If the degree of the denominator > degree of the numerator, there is a horizontal asymptote at = 0.
$\text{Example: }f\left(x\right)=\frac{4x+2}{{x}^{2}+4x - 5}$
In this case, the end behavior is $f\left(x\right)\approx \frac{4x}{{x}^{2}}=\frac{4}{x}$. This tells us that, as the inputs increase or decrease without bound, this function will behave similarly to the function $g\left(x\right)=\frac{4}{x}$, and the outputs will approach zero, resulting in a horizontal asymptote at = 0. Note that this graph crosses the horizontal asymptote.
Figure 12. Horizontal Asymptote y = 0 when $f\left(x\right)=\frac{p\left(x\right)}{q\left(x\right)}, q\left(x\right)\ne{0}\text{ where degree of }p<\text{degree of }q$.
Case 2: If the degree of the denominator < degree of the numerator by one, we get a slant asymptote.
$\text{Example: }f\left(x\right)=\frac{3{x}^{2}-2x+1}{x - 1}$
In this case, the end behavior is $f\left(x\right)\approx \frac{3{x}^{2}}{x}=3x$. This tells us that as the inputs increase or decrease without bound, this function will behave similarly to the function $g\left(x\right)=3x$. As the inputs grow large, the outputs will grow and not level off, so this graph has no horizontal asymptote. However, the graph of $g\left(x\right)=3x$ looks like a diagonal line, and since f will behave similarly to g, it will approach a line close to $y=3x$. This line is a slant asymptote.
To find the equation of the slant asymptote, divide $\frac{3{x}^{2}-2x+1}{x - 1}$. The quotient is $3x+1$, and the remainder is 2. The slant asymptote is the graph of the line $g\left(x\right)=3x+1$.
Figure 13. Slant Asymptote when $f\left(x\right)=\frac{p\left(x\right)}{q\left(x\right)},q\left(x\right)\ne 0$ where degree of $p>\text{degree of }q\text{ by }1$.
Case 3: If the degree of the denominator = degree of the numerator, there is a horizontal asymptote at $y=\frac{{a}_{n}}{{b}_{n}}$, where ${a}_{n}$ and ${b}_{n}$ are the leading coefficients of $p\left(x\right)$ and $q\left(x\right)$ for $f\left(x\right)=\frac{p\left(x\right)}{q\left(x\right)},q\left(x\right)\ne 0$.
$\text{Example: }f\left(x\right)=\frac{3{x}^{2}+2}{{x}^{2}+4x - 5}$
In this case, the end behavior is $f\left(x\right)\approx \frac{3{x}^{2}}{{x}^{2}}=3$. This tells us that as the inputs grow large, this function will behave like the function $g\left(x\right)=3$, which is a horizontal line. As $x\to\pm\infty ,f\left(x\right)\to 3$, resulting in a horizontal asymptote at y = 3. Note that this graph crosses the horizontal asymptote.
Figure 14. Horizontal Asymptote when $f\left(x\right)=\frac{p\left(x\right)}{q\left(x\right)},q\left(x\right)\ne 0\text{ where degree of }p=\text{degree of }q$.
Notice that, while the graph of a rational function will never cross a vertical asymptote, the graph may or may not cross a horizontal or slant asymptote. Also, although the graph of a rational function may have many vertical asymptotes, the graph will have at most one horizontal (or slant) asymptote.
It should be noted that, if the degree of the numerator is larger than the degree of the denominator by more than one, the end behavior of the graph will mimic the behavior of the reduced end behavior \fraction. For instance, if we had the function
$f\left(x\right)=\frac{3{x}^{5}-{x}^{2}}{x+3}$
with end behavior
$f\left(x\right)\approx \frac{3{x}^{5}}{x}=3{x}^{4}$,
the end behavior of the graph would look similar to that of an even polynomial with a positive leading coefficient.
$x\to \pm \infty , f\left(x\right)\to \infty$
### A General Note: Horizontal Asymptotes of Rational Functions
The horizontal asymptote of a rational function can be determined by looking at the degrees of the numerator and denominator.
• Degree of numerator is less than degree of denominator: horizontal asymptote at = 0.
• Degree of numerator is greater than degree of denominator by one: no horizontal asymptote; slant asymptote.
• Degree of numerator is equal to degree of denominator: horizontal asymptote at ratio of leading coefficients.
### Example 7: Identifying Horizontal and Slant Asymptotes
For the functions below, identify the horizontal or slant asymptote.
1. $g\left(x\right)=\frac{6{x}^{3}-10x}{2{x}^{3}+5{x}^{2}}$
2. $h\left(x\right)=\frac{{x}^{2}-4x+1}{x+2}$
3. $k\left(x\right)=\frac{{x}^{2}+4x}{{x}^{3}-8}$
### Solution
For these solutions, we will use $f\left(x\right)=\frac{p\left(x\right)}{q\left(x\right)}, q\left(x\right)\ne 0$.
1. $g\left(x\right)=\frac{6{x}^{3}-10x}{2{x}^{3}+5{x}^{2}}$: The degree of $p=\text{degree of } q=3$, so we can find the horizontal asymptote by taking the ratio of the leading terms. There is a horizontal asymptote at $y=\frac{6}{2}$ or $y=3$.
2. $h\left(x\right)=\frac{{x}^{2}-4x+1}{x+2}$: The degree of $p=2$ and degree of $q=1$. Since $p>q$ by 1, there is a slant asymptote found at $\frac{{x}^{2}-4x+1}{x+2}$.
The quotient is $x - 6$ and the remainder is 13. There is a slant asymptote at $y=x - 6$.
3. $k\left(x\right)=\frac{{x}^{2}+4x}{{x}^{3}-8}$: The degree of $p=2\text{ }<$ degree of $q=3$, so there is a horizontal asymptote y = 0.
### Example 8: Identifying Horizontal Asymptotes
In the sugar concentration problem earlier, we created the equation $C\left(t\right)=\frac{5+t}{100+10t}$.
Find the horizontal asymptote and interpret it in context of the problem.
### Solution
Both the numerator and denominator are linear (degree 1). Because the degrees are equal, there will be a horizontal asymptote at the ratio of the leading coefficients. In the numerator, the leading term is t, with coefficient 1. In the denominator, the leading term is 10t, with coefficient 10. The horizontal asymptote will be at the ratio of these values:
$t\to \infty , C\left(t\right)\to \frac{1}{10}$
This function will have a horizontal asymptote at $y=\frac{1}{10}$.
This tells us that as the values of t increase, the values of C will approach $\frac{1}{10}$. In context, this means that, as more time goes by, the concentration of sugar in the tank will approach one-tenth of a pound of sugar per gallon of water or $\frac{1}{10}$ pounds per gallon.
### Example 9: Identifying Horizontal and Vertical Asymptotes
Find the horizontal and vertical asymptotes of the function
$f\left(x\right)=\frac{\left(x - 2\right)\left(x+3\right)}{\left(x - 1\right)\left(x+2\right)\left(x - 5\right)}$
### Solution
First, note that this function has no common factors, so there are no potential removable discontinuities.
The function will have vertical asymptotes when the denominator is zero, causing the function to be undefined. The denominator will be zero at $x=1,-2,\text{and }5$, indicating vertical asymptotes at these values.
The numerator has degree 2, while the denominator has degree 3. Since the degree of the denominator is greater than the degree of the numerator, the denominator will grow faster than the numerator, causing the outputs to tend towards zero as the inputs get large, and so as $x\to \pm \infty , f\left(x\right)\to 0$. This function will have a horizontal asymptote at $y=0$.
Figure 15
### Try It 6
Find the vertical and horizontal asymptotes of the function:
$f\left(x\right)=\frac{\left(2x - 1\right)\left(2x+1\right)}{\left(x - 2\right)\left(x+3\right)}$
Solution
### A General Note: Intercepts of Rational Functions
A rational function will have a y-intercept when the input is zero, if the function is defined at zero. A rational function will not have a y-intercept if the function is not defined at zero.
Likewise, a rational function will have x-intercepts at the inputs that cause the output to be zero. Since a \fraction is only equal to zero when the numerator is zero, x-intercepts can only occur when the numerator of the rational function is equal to zero.
### Example 10: Finding the Intercepts of a Rational Function
Find the intercepts of $f\left(x\right)=\frac{\left(x - 2\right)\left(x+3\right)}{\left(x - 1\right)\left(x+2\right)\left(x - 5\right)}$.
### Solution
We can find the y-intercept by evaluating the function at zero
$f\left(0\right)=\frac{\left(0 - 2\right)\left(0+3\right)}{\left(0 - 1\right)\left(0+2\right)\left(0 - 5\right)}\text{ }=\frac{-6}{10}\text{ }=-\frac{3}{5}\text{ }=-0.6$
The x-intercepts will occur when the function is equal to zero:
$0=\frac{(x-2)(x+3)}{(x-1)(x+2)(x-5)}$ This is zero when the numerator is zero.
$0=(x-2)(x+3)$
$x=2, -3$
The y-intercept is $\left(0,-0.6\right)$, the x-intercepts are $\left(2,0\right)$ and $\left(-3,0\right)$.
Figure 16
### Try It 7
Given the reciprocal squared function that is shifted \right 3 units and down 4 units, write this as a rational function. Then, find the x– and y-intercepts and the horizontal and vertical asymptotes.
Solution | 2021-06-25T03:55:52 | {
"domain": "lumenlearning.com",
"url": "https://courses.lumenlearning.com/ivytech-collegealgebra/chapter/identify-horizontal-asymptotes/",
"openwebmath_score": 0.9577676057815552,
"openwebmath_perplexity": 247.90306771277338,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9926541719043211,
"lm_q2_score": 0.8397339676722393,
"lm_q1q2_score": 0.8335654262996167
} |
https://math.stackexchange.com/questions/2425493/how-many-6-digit-numbers-have-distinct-digits-but-no-consecutive-digits-both-odd | # How many 6-digit numbers have distinct digits but no consecutive digits both odd?
How many 6-digit natural numbers exist with the distinct digits and two arbitrary consecutive digits can not be simultaneously odd numbers?
I have tried to set up recurrence relation , by considering a valid 5 digit number satisfying the property and then appending the required digit at the last place to produce valid 6 digit number , but things are getting cumbersome. Is there a nice alternative solution? Thanks in advance.
• I suppose a leading zero is not allowed, as in 098765? – Jeppe Stig Nielsen Sep 11 '17 at 16:25
• @JeppeStigNielsen Yes leading zero is not allowed. – rugi Sep 11 '17 at 16:29
• Are you asking how many six-digit positive integers with distinct digits do not have consecutive odd digits? – N. F. Taussig Sep 11 '17 at 16:39
• I'd do it in cases, from the number of odd digits. The most you can have is $3$, as in $OEOEOE, EOEOEO$. Easy to count each pattern. – lulu Sep 11 '17 at 16:40
• Just to be clear: I didn't mean to suggest that the patterns I gave for three odds are all the cases with three odds. They aren't. There are also $OEEOEO, OEOEEO$. – lulu Sep 11 '17 at 16:51
One way to do it is by looking at cases, as @lulu suggested. There are $3$ basic cases: $1$ odd number, $2$ odd numbers or $3$ odd numbers. I'll go through each case below, splitting each case into scenarios starting with an odd number (O) and scenarios starting with an even number (E). The reason for this split is that scenarios starting with an even number are not allowed to start with a zero and therefore differ slightly in the computation.
Case 1
With $1$ odd number you have the following possible scenarios
(a) OEEEEE
(b) EOEEEE, EEOEEE, EEEOEE, EEEEOE, EEEEEO
The number of combinations $n_1$ for this case is $$n_1 = \binom{5}{1} \cdot \binom{5}{5} \cdot 5! + 5 \cdot \binom{4}{1} \cdot \binom{5}{1} \cdot \binom{4}{4} \cdot 4! = 3,000$$
Case 2
With $2$ odd numbers you have the following possible scenarios
(a) OEOEEE, OEEOEE, OEEEOE, OEEEEO
(b) EOEOEE, EOEEOE, EOEEEO, EEOEOE, EEOEEO, EEEOEO
The number of combinations $n_2$ for this case is $$n_2 = 4 \cdot \binom{5}{2} \cdot 2! \cdot \binom{5}{4} \cdot 4! + 6 \cdot \binom{4}{1} \cdot \binom{5}{2} \cdot 2! \cdot \binom{4}{3} \cdot 3! = 21,120$$ Case 3
With $3$ odd numbers you have the following possible scenarios
(a) OEOEOE, OEOEEO, OEEOEO
(b) EOEOEO
The number of combinations $n_3$ for this case is $$n_3 = 3 \cdot \binom{5}{3} \cdot 3! \cdot \binom{5}{3} \cdot 3! + \binom{4}{1} \cdot \binom{5}{3} \cdot 3! \cdot \binom{4}{2} \cdot 2! = 13,680$$
In conclusion
The total number of combinations is then $N= n_1 + n_2 + n_3$ or $$N = 3,000 + 21,120 + 13,680 = 37,800$$
which matches the number found by @Jeppe in the comments. | 2021-02-27T09:43:53 | {
"domain": "stackexchange.com",
"url": "https://math.stackexchange.com/questions/2425493/how-many-6-digit-numbers-have-distinct-digits-but-no-consecutive-digits-both-odd",
"openwebmath_score": 0.7122607827186584,
"openwebmath_perplexity": 745.8517172578224,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9877587232667825,
"lm_q2_score": 0.8438951084436077,
"lm_q1q2_score": 0.8335647548873409
} |
https://math.stackexchange.com/questions/3107038/coprime-ideals-in-kx | # coprime ideals in $K[X]$
If $$K$$ is a field, $$A=K[X]$$, take $$m,n \in K$$ such that $$m \ne n$$. Prove that the ideals $$I=(X-m)$$ and $$J=(X-n)$$ are coprime.
I know the regular definition of coprime. But here, should we prove $$I + J = A$$ or $$K$$? And what are the units in $$K[X]$$?
New contributor
Kevin is a new contributor to this site. Take care in asking for clarification, commenting, and answering. Check out our Code of Conduct.
• You want to prove $I+J=K[X],$ and the unit of $K[X] \text{ is } 1.$ With these clarifications, let us know if you still need help. – Robert Shore Feb 10 at 4:36
• still need help.... – Kevin Feb 10 at 5:03
• Can you get any non-zero constant polynomial into $I+J$? If so, what consequences follow? – Robert Shore Feb 10 at 5:04
• not sure what do you mean – Kevin Feb 10 at 5:32
• @Kevin He means that $(X-m)+(n-X)=n-m\in I+J$. And an ideal which contain an invertible element equals the whole ring. – user26857 Feb 10 at 18:33
What is the unit in $$K[X]$$? A (multiplicative) unit in $$A = K[X]$$ is an element $$1_A$$ such that
$$1_A a = a 1_A = a \tag 1$$
for any
$$a \in A = K[X]; \tag 2$$
it is clear from the ordinary definition of multiplication in $$K[X]$$ that $$1_K \in K$$ fulfills this requirement; here of course I have invoked a slight abuse of notation insofar as I have tacitly identified $$K$$ with the set of $$0$$-degree polynomials or constants in $$K[X]$$; technically I suppose we should write $$1_KX^0$$ for the unit in $$K[X]$$ but as is conventional I will simply refer to $$0$$-degree elements in $$K[X]$$ as being elements of $$K$$; so the unit in $$K[X]$$ is $$1_K$$ or simply $$1$$.
We want to show that
$$(X - m) + (X - n) = A = K[X], \tag 3$$
since by definition, two ideals
$$I, J \subset K[X] \tag 4$$
are co-prime if and only if
$$I + J = (1_K) = K[X]; \tag 5$$
this is what we need to prove with
$$I = (X - m), \; J = (X - n), \tag 6$$
where
$$m, n \in K, \; m \ne n. \tag 7$$
Now, polynomials of degree $$1$$ such as $$X - m$$ and $$X - n$$ are clearly irreducible in $$K[X]$$, since if
$$rX + s = a(x)b(x), \; a(x), b(x) \in K[X], \tag 8$$
then
$$\deg a(x) + \deg b(x) = \deg (rX + s) = 1; \tag 9$$
this implies that at precisely one of $$a(x)$$, $$b(x)$$ is of degree $$1$$, and the other is of degree $$0$$; thus $$rX + s$$ cannot be factored into two polynomials of positive degree, so it is indeed irreducible. Thus we see that $$X - m$$, $$X - n$$ are irreducible over$$K$$; this in turn implies that
$$d(X) = \gcd(X - m, X - n) = 1, \tag{10}$$
for it is clear from the above that $$\deg d(X) = 0$$ or $$\deg d(X) = 1$$; in the latter case
$$d(X) = pX + q \mid X - m$$ $$\Longrightarrow \exists b \in K, \; b(pX + q) = bpX + bq = X - m \Longrightarrow bp = 1, bq = -m; \tag{11}$$
and likewise,
$$d(X) = pX + q \mid X - n$$ $$\Longrightarrow \exists c \in K, \; c(pX + q) = cpX + cq = X - m \Longrightarrow cp = 1, cq = -n; \tag{12}$$
now the equations
$$bp = 1 = cp \tag{13}$$
imply
$$p \ne 0, \; b = c; \tag{14}$$
but then
$$m = -bq = -cq = n, \tag{15}$$
contradicting our assumption that $$m \ne n$$; therefore,
$$p = 0 \tag{16}$$
and any common divisor of $$X - m$$, $$X - n$$ must be an element $$0 \ne q \in K$$; since $$d(X)$$ is defined up to multiplication by a unit, we have established (10). Now since $$K[X]$$ is a principle ideal domain, there exist
$$f(X), g(X) \in K[X] \tag{17}$$
with
$$f(X)(X - m) + g(X)(X - n) = d(X) = 1; \tag{18}$$
thus for any $$r(X) \in K[X]$$
$$r(X) = r(X)(1)$$ $$= r(X)f(X) (X - m) + r(X) g(X)(X - n) \in (X - m) + (X - n), \tag{19}$$
and thus we see that
$$K[X] \subset (X - m) + (X - n) \Longrightarrow K[X] = (X - m) + (X - n) \tag{20}$$
proved as per request.
• Isn't pretty clear that $I+J$ contains a non-zero element from $K$, that is, a unit, and therefore equals the whole ring? – user26857 Feb 10 at 18:32
• @user26857: you mean as in $(X - m) - (X - n) = n - m \ne 0$? Nice catch, maybe you ought to write it up as an answer! – Robert Lewis Feb 10 at 18:39
• Yes, this is what I meant. (I let it as a hint for the OP.) – user26857 Feb 10 at 18:40
Note that, if $$I + J = k[X]$$, then there exist $$p,q$$ polynomials with $$p(X-n) + q(X-m) = s$$ for some $$s \in k$$ (actually, we can do this for any $$s$$ in the field).
Reciprocally, if $$s \in I+J$$, for some $$s$$ in $$k$$, then $$1 \in s^{-1}(I+J) = I+J$$ and so $$k[X] = I+J$$. Thus our problem reduces to showing that there exist $$p,q \in k[X]$$ with $$p(X-n) + q(X-m) = s$$ for some nonzero $$s$$. But this is rather direct, as
$$(X-n) - (X-m) = m-n \neq 0.$$ | 2019-02-16T21:29:04 | {
"domain": "stackexchange.com",
"url": "https://math.stackexchange.com/questions/3107038/coprime-ideals-in-kx",
"openwebmath_score": 0.9486532211303711,
"openwebmath_perplexity": 276.9675714621213,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9877587272306608,
"lm_q2_score": 0.8438951045175642,
"lm_q1q2_score": 0.8335647543544548
} |
http://clandiw.it/fknf/exponential-growth-and-decay-practice-pdf.html | Exponential Growth And Decay Practice Pdf
Graphing exponential functions. Then graph. (answer 12g) 9. -1-1 I a 34 3 4-2 Solve each of the following equations for x. Exponential Decay Certain materials, such as radioactive substances, decrease with time, rather than increase, with the rate of decrease proportional to the amount. If you look for word problems about exponential growth or decay, in text books or on the internet, certain settings arise again and again: bank accounts, populations of animals, people or bacteria, and. Virtual Nerd's patent-pending tutorial system provides in-context information, hints, and links to supporting tutorials, synchronized with videos, each 3 to 7 minutes long. Exponential Growth and Decay: Differential Equations 9. Level 3 Practice: Find the growth factor associated with the percent change. 2% growth Given x = 4, y = 11. Represent the sums or differences of logs as single logarithms. 7 ExponentialGrowthandDecay Exponential Growth Models Recalling the investigations in Section 8. 27757, A1 -0. M: Lesson 15 - Exponential Decay T: Lesson 16 - Linear vs. 12% growth 5. Most considerably you have to love them and you have to have `the present of teaching'. Exponential Functions and Half-Lives What is a half-life? If you start with eight million atoms of a parent isotope (P), how many P isotopes will you have after decay of P to D (daughter isotopes) in one half-life of 1000 yrs ? After 2000 yrs, how many parent isotopes will you have ? Hudson Bay amphibole with abundant garnet. Growth/Decay Growth/Decay Growth/Decay. Exponential Growth and Decay Word Problems Write an equation for each situation and answer the question. The variable b represents the growth or decay factor. 22 million in 1990, and its projected population for the year 2000 is 6. 25 or 25% 4. The number of subscribers increased by 75% per year after 1985. Write an exponential growth function to model each situation. Date: Nuclear Decay Practice Problems. Simple And Compound Interest Practice Worksheet Answer Key. As discussed in Exponential Growth and Decay: Introduction, all exponential growth/decay problems can be modeled using $\,P(t) = P_{\,0}{\text{e}}^{rt}\,$,. Skills Practice Growth and Decay 1. Then determine the growth/decay factor and growth/decay rate. It decreases about 12% for every 1000 m: an exponential decay. A is the amount at time t, and kis a constant representing the growth rate. 7-7 Exponential Growth and Decay Word Problems Worksheet 2. Y = a(l + Equation for exponential growth +0. A Exponential Growth and Decay. MBF 3CI U7 D5 Modelling Exponential Growth and Decay complete 2015. If you don't see any interesting for you, use our search form on bottom ↓. There is no formative assessment for this lesson. It depreciates at a rate of 13% a year. Before we begin graphing, it is helpful to review the behavior of exponential growth. Practice: Exponential growth vs. Exponential Growth and Decay An exponential function is a function that has a variable as an exponent and the base is positive and not equal to one. 7 Exponential Growth and Decay Worksheet The half-life of Palladiim-100 is 4 days. linear models, and more. How much is the guitar worth now? the SS 50 S720 Sl 100 saoo esch in t "y 10 snnum snnus snnum snnus s n num s n num snnum 10 ity s n num ded ded de. After 16 days an initial sample has been reduce to a mass of 0. Given the equation y = 35(0. Unit 4 lesson 8 (Growth and Decay Textbook. Hence, = and setting we have. Introduction to Exponential Growth and Decay. (a) Write a function that models the depreciation of a car valued at $20,000. 21 54 x y 76. Since 2000, the state's population has grown about 2% each year. A common application of exponential equations is to model exponential growth and decay such as in populations, radioactivity and drug concentration. The resources for Big Idea 1 focus on how we can distinguish quadratic functions from linear and exponential functions based on their properties when represented as sequences, tables, graphs, and using rate of change to find intervals of a function that are increasing, decreasing, positive, negative, and symmetry of a function (if any). 5 D2 Worksheet Name_____ Exponential Growth and Decay Exponential Growth: Exponential Decay: : ; : ; *population growth *population decay Compound Interest: @ A Compound Interest Continuously: Half -Life: @ A * is the interval of the half-life 1. The percentage decrease, r, is the rate of decay. Note: The distance from the earth to the moon is 384400 km 384400\text{ km} 3 8 4 4 0 0 km. ) In an exponential growth model, [rate of change of y] is proportional to [current amount]. 00043 t, where A 0 is the initial amount. (answer 12g) 9. 1 Graphing Exponential Growth Functions Homework: 7. That's why this video breaks down those problems into understandable processes. Exponential Practice Mini Test: 7. On this page you can read or download 7 6 skills practice growth and decay in PDF format. 7% growth 15 2% decay 18. In exponential growth, the rate of growth is proportional to the quantity present. Determine the starting mass. This is the students’ version of the page. State whether the formula models growth or decay. Circle either linear or exponential. Practice and App/' Find the multiplier for each rate of exponential growth or decay. AP Calculus BC 6. After 16 days an initial sample has been reduce to a mass of 0. 4 HW Independent Practice with Linear functions vs exponential functions Feb. Practice Solutions. wolframalpha. 0123456789t 0 20 40 60 80 n Years 3. Exponential Growth and Decay Practice Name:_ Date:_ Period:_ 1) Given the equation y =. Exponential Growth and Decay – Practice Word Problems 1. Some of the worksheets for this concept are Graphing exponential, Exponential population growth, Exponential growth, Negative exponents teacher notes, Exponent rules practice, Exponential functions date period, Use simple interest to find the ending, Exponential growth and decay. The table of values below led us to choose a horizontal span of 0 to 15 and a vertical span of 0 to 200. The constant of proportionality in that case is 35. See below for ways to find the base b. EX #3:A slow economy caused a company’s annual revenues to drop from$530, 000 in 2008 to $386,000 in 2010. ) Build new functions from existing functions MGSE9-12. ) In an exponential growth model, [rate of change of y] is proportional to [current amount]. Exponential function: f x ab x. Logarithms can also be used to solve such problems. 1 3 b = 12. While the person is repaying the loan, interest is accumulating at an annual percentage rate of r, and this interest is compounded n times a year (along with each payment). Exponential growth and decay by. It depreciates at a rate of 13% a year. Lesson 7 – Applications – Doubling and Half Life Example 6 Solns. exponential equation to find the rent of the apartment in 2017. r is the growth rate when r>0 or decay rate when r<0, in percent. 29_m_alg_2_key. 2)(t/10), and classify them as representing exponential growth and decay. 6a Video @ http://www. uk Simple, Compound Interest, Depreciation, Growth & Decay (H) - Version 2 January 2016 At the start of day 1 there are 1000 bacteria in flask B. Practice switching between forms: Exponential Form Log Form 1. Use the equation to predict the population of New. An algebra equation involves a variable representing an un-known number, often denoted by x; and to solve the equation means to nd the nu-merical values of x which make the equation true. Graphing Exponential Functions Practice Worksheet Name Period # Graph the following functions and tell whether they show exponential growth or decay. Then find the value of the function after the given amount of time. Unit 7 Day 2 - Practice Worksheet EXPONENTIAL GROWTH AND DECAY PRACTICE 1. Exponential Growth and Decay Curves. Find the balance after 6 years. Definition of Exponential Growth/Decay : Exponential growth occurs when the growth rate of the value of a mathematical function is proportional to the function’s current value, resulting in its growth with time being an exponential function, i. The two types of exponential functions are exponential growth and exponential decay. Algebra 1 Unit 10: Exponential Functions Practice Day 5: Growth/Decay Percents of Change Name: _____ Practice Assignment Block: _____ Identify the following: Equation Growth/Decay Starting Value Growth/Decay. 73, so the formula for the function is N(t) = 200×0. Download [659. Identify the initial amount a and the decay factor b in each exponential function. Unit 3b - Linear and Exponential Equations Date: Exponential Growth and Decay Practice Day 41 — Practice Growth : y = P(l + Decay : y = P(l — 1. 2) Since 1985, the daily cost of patient care in community hospitals in the US about 8. But sincerely, the half-life problems, also involving exponential growth and exponential decay, can be very tricky. The equation will look like: f(x) = ( starting amount ) (base )x. 01)12t, y = (1. pdf EXPONENTS QUIZ #4 (Friday, February 23) 6-6 COMPARING FUNCTIONS - LINEAR AND EXPONENTIAL (Monday, February 26). exponential decay 9. 6 Exponential Decay Functions. If we start with only one bacteria which can double every hour, how many bacteria will we have by the end of one day?. Exponential Functions - Applications - Compound Interest - Growth and Decay Worksheet This free trigonometry worksheets contains problems on applications of exponential functions. f) y =40(1−0. F-LE: Construct and compare linear, quadratic, and exponential models and solve problems. pdf EXPONENTS QUIZ #4 (Friday, February 23) 6-6 COMPARING FUNCTIONS - LINEAR AND EXPONENTIAL (Monday, February 26). In this section, we examine exponential growth and decay in the context of some of these applications. 1 | P a g e Lesson Outline Lesson Part Activity description/Teacher does Students do Title Lesson 1 Standard A1. 1 Exponential Growth Functions Exponential Functions A function is called exponential if the variable appears in the exponent with a constant in the base. notebook January 30, 2015 Writing Exponential Growth and Decay Functions in Real Life!!! Remember this from yesterday? A(t) = a(1 + r)t Sometimes you have to use given values to find r! 15. Logarithms Practice Problems And Solutions Logarithms Practice Problems And Solutions Yeah, reviewing a books Logarithms Practice Problems And Solutions could increase your near friends listings. #26 Writing exponential growth and decay equations #26 homework exp growth &decay KEY- Mar 20 2017 - 4-06 PM. Graph exponential growth and decay functions. Date: Nuclear Decay Practice Problems. For example, identify percent rate of change in functions such as y = (1. The variable b represents the growth or decay factor. 1621 Linear vs. t is the time in discrete intervals and selected time units. 1 (3-27 threes, 28-30 all, 37, 38) Tuesday, March 8 In Class: 7. Exponential Growth & Decay Practice Problems 1) Radioactive Decay Radioactive radium has a half-life of approximately1620 years. How much is the guitar worth now? the SS 50 S720 Sl 100 saoo esch in t "y 10 snnum snnus snnum snnus s n num s n num snnum 10 ity s n num ded ded de. Independent Practice: Exponential Growth and Decay 1. Growth and Decay Tell whether the function shows growth or decay. Assume you invest$5,0Q0 in an account paying 8% interest compounded monthly. Model exponential growth and decay. Other than that, we figure out our answer in exactly the same way. For exponential decay, the value inside the parentheses is less than 1 because r is subtracted from 1. Show all work!!! For each of the following situations, write an exponential model of the form y =a(b)x 1. Exponential Growth. The graphs of exponential equations can be transformed by changing the value of the constants a, h, and k in the exponential equation: 𝑥f (x) = –ℎ + k. growth factor, which equals 1 plus the percent rate Of change expressed as a decimal. Lesson #42 Note Supplement Screen Shots Lesson #42 (YouTube - link) - Watch the video. Separable Differentiable Equations A differential equation of the form dy /dx = f (y)g(x) is called separable. 4: Exponential Decay 1 The Fahrenheit temperature, F(t), of a heated object at time t, in minutes, can be modeled by the function below. Some of the worksheets for this concept are Graphing exponential, Exponential population growth, Exponential growth, Negative exponents teacher notes, Exponent rules practice, Exponential functions date period, Use simple interest to find the ending, Exponential growth and decay. Which formula models what it will be the worth in 2 years if she makes no deposits and no withdrawals? A. That's why this video breaks down those problems into understandable processes. Tell whether the table of values represents a linear function, an exponential function, or a quadratic function. The walkers get very close very fast. Exponential Growth and Decay MathBitsNotebook. RADIOACTIVE DECAY The amount of radium A present in a sample after t years can be modeled by A = A 0-e 0. What percentage of a given amount remains after 100 years? 2) Population Growth Dhaka, Bangladesh had a population of 4. 1% per year. Algebra 2 7 Exponential and Logarithmic Functions Practice Problems Page 2 of 12 7. growth and decay. Learn to distinguish between exponential growth and exponential decay. (answer 12g) 9. r table! 2 -2 movement:. 5% f(x) = 500 f(x) = 400 f(x) = 450 7. Unit 4 lesson 8 (Growth and Decay Textbook. This lesson is designed to help students understand the basic concepts of exponential growth and decay. Students must find interest compounded annually, semi-annually, quarterly, and continuously. Let the function g have an initial value of 400 and increase at a growth rate of 17%. Exponential Growth Exponential Decay 500 Chapter 9 Exponential and Logarithmic Relations EXAMPLE Identify Exponential Growth and Decay Determine whether each function represents exponential growth or decay. The order of magnitude is the power of ten when the number is expressed in scientific notation with one digit to the left of the decimal. unit 5 worksheet 6 graphing exponential growth and decay functions UNIT 5 WORKSHEET 6 GRAPHING EXPONENTIAL GROWTH_DECAY. In 1985, such hospital costs were an average of $480 per day. The mice population is 25,000 and is decreasing by 20% each year. Write a model for this situation. Practice Exponential Equations, receive helpful hints, take a quiz, improve your math skills. If k is positive then we will have a growth model and if k is negative then we will have a decay model. Practice Form K Exponential Growth and Decay Identify the initial amount a and the growth factor b in each exponential function. Unit 6 Review Extra Practice. Graphing exponential functions. You can use the skills in this chapter • to analyze more complicated functions in later math courses, such as Calculus. The decay constant for Magnesium-27 is –0. 4: Exponential Growth and Decay Practice HW from Stewart Textbook (not to hand in) p. 9 1 2 x y 72. Logarithmic functions are the INVERSE of Exponential Functions. U4D8_T Applications - Exponential Growth and Decay Problems: Duo-tang Worksheet for U4D8 Applications Exponentials (Exponential Growth & Decay) Do: Growth Worksheet # 1 - 4 & Decay Worksheet # 1 - 4. Write an exponential function. The differential equation y ' = ky, where k is a constant, has the general solution, y = Ae kx. Exponential Growth A. Given the equation y = 35(0. Exponential Growth and Decay Exponential growth and decay models always take the form y = aebx. Connections to Previous Learning: Students should be familiar with solving exponential growth and decay problems using the universal growth formula y = Cekt. This is the currently selected item. 9% in the year 2006, dividing 70 by 0. The variable must be in the exponent, not the base. Tell whether each function represents exponential growth or decay. Practice switching between forms: Exponential Form Log Form 1. Situation: A person initially borrows an amount A and in return agrees to make n repayments per year, each of an amount P. Practice: Graphing exponential growth & decay. After 16 days an initial sample has been reduce to a mass of 0. 1) Which of the exponential functions below show growth and which show decay? a) y = 5(2 )x b) 100(. Solve log x 8 =− 1. If you don't see any interesting for you, use our search form on bottom ↓. (answer 12g) 9. Solo Practice. Include the asymptote, Domain, and Range. Practice Exponential Equations, receive helpful hints, take a quiz, improve your math skills. 29_m_alg_2_key. Find the relationship between t, the number of seconds, and N(t), the number of bacteria. a function of the form:. The town manager reports that revenue for a given year. Show all of your work clearly and neatly for credit- which will be earned based on completion rather than correctness. Exponential Growth and Decay Graphing Exponential Functions Interactive Exponential Graph (GeoGebra) Introduction to Exponential Functions (Video) Exponential Functions: Introduction Algebra Notes: Exponential Functions MathBitsNotebook: Exponential Functions. Write an exponential growth function to model each situation. How many atoms are left after Exponential Growth and Decay Obiective To model exponential growth and decay Content Standards EIE8. 12 KB (Last Modified on December 19, 2019) Comments (-1). When this happens, the value of the quantity at any given time can be calculated as a function of the rate and the original amount. If 0 b 1, the function shows decay. What is your initial value? C. 524) • logarithm (p. What percentage of a given amount remains after 100 years? 2) Population Growth Dhaka, Bangladesh had a population of 4. The function is of the form y = a(1 − r)t, where 1 − r < 1, so it represents exponential decay. First, they evaluate for a variable given the second variables value. Quiz Answers Exponential Decay Functions Warm-Up Get ready for the lesson Instruction. Also, state the growth / decay factor, if possible. Expon ential function are also. The value of b tells if the 462 equation models exponential growth or decay. Logarithms Practice Problems And Solutions Logarithms Practice Problems And Solutions Yeah, reviewing a books Logarithms Practice Problems And Solutions could increase your near friends listings. Exponential Growth and Decay Word Problems DRAFT. REMEMBER TO PICK YOUR 5 POINTS (-2, -1, O, 1, 2) AND MAKE AN X/Y CHART!! 10 1. 2)^(t/10), and classify them as representing exponential growth and decay. notebookMarch 17, 2017 Exponential Decay Equation: y = a (1 ‐ r)t a is the initial amount r is the percent decrease, written as a decimal 1 ‐ r is the decay factor EXAMPLE 5:. Practice: Exponential growth vs. Round your answer to the nearest tenth. Use the exponential growth/decay model to answer the questions. If the rate of increase is 8% annually, how many stores does the restaurant operate. Exponential Growth and Decay Practice Name:_ Date:_ Period:_ 1) Given the equation y =. Hence, = and setting we have. Share practice link. These unique features make Virtual Nerd a viable alternative to private tutoring. ky dx dy = Solving via separation of variables (section 11. 6a Remediation assignment: 9. You can skip questions if you would like and come back. growth_and_decay_curves. Simple, Compound Interest, Depreciation, Growth & Decay (H) A collection of 9-1 Maths GCSE Sample and Specimen questions from AQA, OCR, Pearson-Edexcel and WJEC Eduqas. Practice B LESSON 11-3 Exponential Growth and Decay Date ass 'V 901 900. Connect the points to form a smooth curve. Writing Exponential Equations Level 3 Practice: Find the growth factor associated with the percent change. If b 1, the function shows growth. Math · Algebra I · Exponential growth & decay · Exponential vs. Exponential growth and decay show up in a host of natural applications. What is the population after 5 years. When this happens, the value of the quantity at any given time can be calculated as a function of the rate and the original amount. Day 7 – Exponential Growth and Decay r) t r) t 1. In Algebra I, students will learn function notation and develop the concepts of domain and range. Algebra 1 Unit 10: Exponential Functions Practice Day 5: Growth/Decay Percents of Change Name: _____ Practice Assignment Block: _____ Identify the following: Equation Growth/Decay Starting Value Growth/Decay. Work out. exponential equations can be written in logarithmic form. Exponential Growth Models. If a > 0 and b > 1, then y = ab x is an exponential growth function, and b is called the. The base of an exponential function indicates whether the function shows growth or decay. pdf] - Read File Online - Report Abuse. exponential functions and square-root functions. #26 Writing exponential growth and decay equations #26 homework exp growth &decay KEY- Mar 20 2017 - 4-06 PM. Exponential growth and decay, identifying initial value, growth/decay rate 1. Assignment Practice with exponential growth functions. In this non-linear system, users are free to take whatever path through the material best serves their needs. (answer 12g) 9. 2 Exponential Functions. Determine the starting mass. Whenever you see the phrase relative growth rate, continuous growth rate, or exponential growth rate, you know you're dealing with exponential behavior. Situation: A person initially borrows an amount A and in return agrees to make n repayments per year, each of an amount P. a) f(x) = 3x + 2 b) 2y = -5x + 1 c) y = 7 d) C(x) = 16, 332(1. ANSWER KEY 1. 1\) corresponds to the number of years the investment has grown. If something decreases in value at a constant rate, you may have exponential decay on your hands. m is positive for growth, negative for decay. Exponential Growth and Decay. Notes: Writing Recursive Rules for Exponential Functions An exponential function of the form f ()xab= x is written using a recursive rule as follows. If you don't see any interesting for you, use our search form on bottom ↓. If the population deaeased at a rate of 1. A di erential equation (DE) involves. There are no stretches or shrinks. Then graph. While students are engaged in discussion, I listen in to find students who are. Elimination of a drug from the body. Devonte Walker 15. The first walking exercise is an example of exponential decay. Share practice link. The car was bought for$25,000. We separate the variables by writing it in the form = (). Since 2000, the state's population has grown about 2% each year. The plot shows exponential decay, and the limit appears to be 71°. Some people are frightened of certain medical tests because the tests involve the injection of radioactive materials. As is shown, the method of compounding has little effect. Find the amount remaining after 5 h. The value of the property in a particular block follows a pattern of exponential growth. (c) Sketch a graph to compare the size of the population of bacteria in flask A and in flask B. 10 6 practice exponential growth and decay answers | Get Read & Download Ebook 10 6 practice exponential growth and decay exponential growth and decay answers PDF, include : 100 Days Of Weight Loss The Secret To Being Successful On Any Diet Plan, 1997 Chevrolet Blazer Wiring Diagram And Electrical Circuit Troubleshooting, and many other ebooks. 25 or 25% 4. Algebra 1- Exponential Growth & Decay How are exponential growth and decay present in the real world? Give at least 2 examples for exponential growth and 2 examples of exponential decay. There is no formative assessment for this lesson. Note: The distance from the earth to the moon is 384400 km 384400\text{ km} 3 8 4 4 0 0 km. a is the starting value, b is the growth rate. A common application of exponential equations is to model exponential growth and decay such as in populations, radioactivity and drug concentration. Identify the following equation as either linear or exponential. This means that an initial amount increases at a steady Lesson 10-6 rate over time. •g (t) = 6t 4. Work out. Bureau of the Census). If A is decaying continuously at rate r, then A may be written as follows A = A0e−rt. Practice B Exponential Functions 1. Determine the starting mass. Practice Problems 1. 72(2)t Y = 25(1. Day 7 – Exponential Growth and Decay r) t r) t 1. 13% growth 7. Parent Guide for Student Success (pdf) Audio Summaries Transcripts Data Updates (pdf) Activities: Crossword Puzzle Algebra 1 Home > Algebra 1 > Chapter 8 > 8. Exponential Growth. pdf View Jan 19, 2016, 7:40 AM: Douglas Mccullough: ċ. Find the age of a sample at which 25% of the radioactive nuclei originally present have. Exponential Growth & Decay Chapter Exam Instructions. In an exponential function, the constant multiplier, or scale factor, is called an exponential decay factor when the constant is less than 1. 9 Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal. In this work, we study the tunneling phenomenon of an initially localized particle in. How can you determine if an exponential function is a growth or decay function? Practice: Identify the function as a growth or decay function, the initial amount , the rate of growth or decay (as a percent), and evaluate the function when. f (x) = 2x Find the equation of the graph of g. Kindly read the relevant sections in Krane’s book first. f(n) 5 1000 ? 1. Modelling Exponential Decay - Using Logarithms. whose base is greater than one. F-LE: Construct and compare linear, quadratic, and exponential models and solve problems. Regents Exam Questions F. Then determine the growth/decay factor and growth/decay rate. : Exponential Growth & Decay (GPS ADV Alg). (e is the base of the natural logarithm. All exponential growth and decay functions can be represented by the equation y kax For exponential growth, a 1 For exponential decay, 0 a 1 The value of a, called the multiplier, is the scale. 5 Understand the inverse relationship between exponents and logarithms and. 8 H pH F HG I KJ F HG I KJ 1 10. For example, given Canada's net population growth of 0. From population growth and continuously compounded interest to radioactive decay and Newton’s law of cooling, exponential functions are ubiquitous in nature. Chapter 6 : Exponential and Logarithm Functions. exponential decay 9. Connect the points to form a smooth curve. Available in PDF, DOC, XLS and PPT format. 85)𝑡, identify theinitial amount, decay factor, and percent decrease. ANSWER exponential function 0 1 y 2 x – 2 – 1 0. R9a – Converting between fractions, decimals and percentages. f (x) = 2x Find the equation of the graph of g. Which formula models what it will be the worth in 2 years if she makes no deposits and no withdrawals? A. That's why this video breaks down those problems into understandable processes. 3) A population of 800 beetles is growing each month at a rate of 5%. Wonderful Exponential Functions Word Problems Worksheet Pdf seminole county public schoolsprove. Change the following from logarithmic to exponential form a) log7 49 = 2 Answer: 72 49 b) log5(1 25) = -2 Answer: 5−2. 59% average accuracy. 2)^(t/10), and classify them as representing exponential growth and decay. Exponential Growth and Decay Graphing Exponential Functions Interactive Exponential Graph (GeoGebra) Introduction to Exponential Functions (Video) Exponential Functions: Introduction Algebra Notes: Exponential Functions MathBitsNotebook: Exponential Functions. In real-life situations we use x as time and try to find out how things change exponentially over time. [Filename: extra practice expt equations. Teachers might use this surprise as leverage to segue into a more involved task comparing linear and exponential growth. Practice B Exponential Functions, Growth, and Decay Tell whether the function shows growth or decay. Decay Factor: 1 minus the percent rate of change GOAL: Definition: An EXPONENTIAL FUNCTION is. Civil Procedure Doctrine Practice And Context Casebook Series Exponential Growth And Decay Puzzle 2007 Suzuki Xl7 Owner Manual Pdf. So in both exponential decay and growth we have a constant percent change, which are both modeled by an exponential function. Improve your math knowledge with free questions in "Exponential growth and decay: word problems" and thousands of other math skills. Concept 17 Writing Exponential Equations. Engaging math & science practice! Improve your skills with free problems in 'Writing and Solving Exponential Decay Functions Given a Word Problem' and thousands of other practice lessons. 1 Write a function that describes a relationship between two quantities. To solve an exponential or logarithmic word problems, convert the narrative to an equation and solve the equation. Directions 13-15: Classify each as exponential growth or decay and justify your answer 13. We also learn how to use our calculator to create an. The previous section introduced the basic formula for discrete exponential growth as: x t = x 0 (1+r) t It follows then that the basic formula for discrete exponential decay is: x t = x 0 (1-r) t There is a minus sign instead of a plus sign because we are dealing with a negative rate of growth. Here are the basics that you should know if you want to get a perfect SAT score: A general exponential function has the form f(t) = a(1 + r) ct, where a = f(0) is the initial amount and r is the growth rate. a) Exponential growth or decay: b) Identify the initial amount: c) Identify the growth/decay factor: d) Write an exponential function to model the situation: e) "Do" the problem: 2. The mathematical model of exponential growth is used to describe real-world situations in population biology, finance and other fields. POPULATION The population of New York City increased from 8,008,278 in 2000 to 8,168,388 in 2005. Exponential Growth and Decay Calculus, Relative Growth Rate, Differential Equations, Word Problems - Duration: 13:02. 7 ExponentialGrowthandDecay Exponential Growth Models Recalling the investigations in Section 8. 8% per year. Title: 4778_CH06 Author: Nick Created Date: 8/11/2000 1:05:13 PM. Exponential Growth B. Then find the value of the function after the given amount of time. Modeling Exponential Growth and Decay. Retrieved from "https://simple. ) _____ 2) = (. 2)^(t/10), and classify them as representing exponential growth and decay. After 16 days an initial sample has been reduce to a mass of 0. Show all work!!! For each of the following situations, write an exponential model of the form y =a(b)x 1. A nice car means a successful person, while an older car means one has no money. 396 decremento exponencial exponential function, p. 10% growth 6. If we start with only one bacteria which can double every hour, how many bacteria will we have by the end of one day?. Available in PDF, DOC, XLS and PPT format. The quantity decreases by a constant percentaço each time period. c) Use this model to predict about when the population of Brownville will first r each 1,000,000. In 1985, such hospital costs were an average of $480 per day. How much money S4ill be in the account after 5 years? 2. Some of the worksheets for this concept are Graphing exponential, Exponential population growth, Exponential growth, Negative exponents teacher notes, Exponent rules practice, Exponential functions date period, Use simple interest to find the ending, Exponential growth and decay. Therefore, it is natural to ask how to spot the non-exponential decay when (at least) two decay channels are opened. Lesson 9 – Exponential Growth and Decay Extra Practice. 7-7 Exponential Growth and Decay Word Problems Worksheet 2. Exponential Functions Unit Test TOMORROW Review Questions: Pg. f (x) = 2x Find the equation of the graph of g. A Bacteria Culture Grows with Constant Relative Growth Rate. Name Class Practice 7-7 Form G Identify the initial amount a and the growth factor b in each exponential function. A certain bacterium has an exponential growth rate of 25% per day. Date: Nuclear Decay Practice Problems. So, an initial value of -2, and a common ratio of 1/7, common ratio of 1/7. 5 Exponential Growth Functions. Here P0 is the initial amount and k is the exponential growth/decay rate. Investigation 4: Exponential Decay Practice Problems Directions: Please complete the necessary problems to earn a maximum of 4 points according to the chart below. 15% decay 8. exponential decay: a decrease whose rate is proportional to the size of the population. Exponential Growth and Decay Word Problems 1. Exponential Growth and Decay Objective: To give students practice in solving the type of exponential growth and decay problems similar to those they will encounter on the AP Calculus exam. This means that your multiplier should be a number below 1 rather than above 1. Unit 7 Day 2 - Practice Worksheet EXPONENTIAL GROWTH AND DECAY PRACTICE 1. • If a 0 and 0 b 1, the function y abx represents exponential decay. a) Write an equation for this exponential relation. Lesson 9 – Growth and Decay Applications. The stray cat population of a small town grows exponentially. Exponential Growth/Decay Worksheet Answer the following questions about the exponential decay problems. Exponential function: f x ab x. Learn to distinguish between exponential growth and exponential decay. Date: Nuclear Decay Practice Problems. 10% growth 6. Jeremiah owns a business. 02)^t, y = (0. pdf] - Read File Online - Report Abuse. Distinguishing between linear and exponential functions using various representations (F. Title: Microsoft Word - 7. PDF Pass Chapter 7 7 Glencoe Algebra 2 7-1 Skills Practice Graphing Exponential Functions Graph each function. linear models. The first walking exercise is an example of exponential decay. Link: Homework- Exponential Growth and Decay - UNM Math. Exponential Growth and Decay Practice Name:_ Date:_ Period:_ 1) Given the equation y =. Precalculus Name: _____ Practice With Exponential Growth & Decay (2. Textbook: pg. The range is y > 0. Determine the starting mass. 5 Di erential equations. The following table shows the final principal (P), after t = 1 year, of an account initially with C =$10000, at 6% interest rate, with the given compounding (n). Notes: Writing Recursive Rules for Exponential Functions An exponential function of the form f ()xab= x is written using a recursive rule as follows. f (x) = 3 · 5x 2. What is the expected population in 2018? 1. Annual sales for a fast food restaurant are $650,000 and are increasing at a rate of 4% per year; 5 years 2. The variable b represents the growth or decay factor. It decreases about 12% for every 1000 m: an exponential decay. 460 Chapter 4 Exponential and. whose base is greater than one. A Bacteria Culture Grows with Constant Relative Growth Rate. Find the amount remaining after 100 days. 02)^t, y = (0. In real-life situations we use x as time and try to find out how things change exponentially over time. In 1985, such hospital costs were an average of$480 per day. Notice that in exponential growth or decay, the proportional amount of change in two time intervals of the same length is the same:. This growth and decay, as discussed in class already, can be the model for population growth, growth of cancerous cells in a body, the amount money in a bank based on. Exponential Growth Decay Answers Some of the worksheets for this concept are Exponential growth and decay, Exponential growth and decay work, Exponential growth and decay word problems, Exp growth decay word probs, Exponential growth and decay, Graphing exponential, Exponential population growth, Exponential growth and decay functions. PDF DOC TNS: Regents-Logarithmic Equations 2 A2/B/SIII advanced: 6/3/6: TST PDF DOC TNS: Practice-Exponential Equations: 10: WS PDF: Practice-Exponential Growth and Decay: 10: WS PDF: Practice-Logarithmic Equations: 10: WS PDF: Journal-Logarithmic Equations: 5: WS PDF: TI-NSPIRE ACTIVITIES: Solving Logarithmic Equations: ACT: What is Log? ACT. 05% decay Given x = 5, g, 24. If the population of a bacterial sample is currently 33,000 bacteria, then how many bacteria will there be in 9 days? If necessary, round your answer to the nearest whole. If y is a function of time t, we can express this statement as Example: Find the solution to this differential equation given the initial condition that yy=0 when t = 0. In this lesson you will study which have the form ƒ(x) = abx where a >0 and 0 < b < 1. For any topic that is a Review, a Quiz or a Practice take a picture and text your work to me. The value of a house appreciates at 12% a year. Our main objective in this tutorial is to learn about the exponential decay formula, when to apply it and how to deal with its parameters. In this case, since the amount of caffeine is decreasing rather than increasing, use. Sketch the graph of y 0. (a) During an experiment, a scientist notices that the number of bacteria halves every second. [Filename: extra practice expt equations. 1) Which of the exponential functions below show growth and which show decay? a) y = 5(2 )x b) 100(. Exponential Growth and Decay Question: A Bacteria Culture Grows with Constant Relative Growth Rate. The range is y > 0. Module 8 Logarithms and Growth and Decay Practice Task 2 1. + 450/0 11. New! Exponential Growth Word Problems. 2) Since January 1980, the population of the city of Brownville has grown according to the mathematical model. Exponential Growth and Decay Curves. Exponential Decay Description and Formula Example writing and solving an exponential decay function. This is modeled by the equation:. May 02, 2020 - Exponential growth and decay word problems Engineering Mathematics Video | EduRev is made by best teachers of Engineering Mathematics. pdf] - Read File Online - Report Abuse. A new 2006 Honda Accord was valued at $25000. All exponential growth and decay functions can be represented by the equation y kax For exponential growth, a 1 For exponential decay, 0 a 1 The value of a, called the multiplier, is the scale. If you’d like a pdf document containing the solutions the download tab above contains links to pdf’s containing the solutions for the full book, chapter and section. There were 2∙3 × 1030 bacteria at the start of the experiment. For exponential growth For exponential decav, 0 < b < 1 and b is the decay factor. INTERNET The table shows the number of hits an Internet blog received for three weeks. Exponential growth occurs when k> 0 Exponential decay occurs when k< 0 Exponential Growth and Decay Example: World poultry production was 77. Lesson 9 – Growth and Decay Applications. The number of subscribers increased by 75% per year after 1985. Exponential Growth is Compound Interest is EXPONENTIAL GROWTH & DECAY, COMPOUND INTEREST, GEOMETRIC SEQUENCES *10. Gabriela Farfan 14. Practice B Exponential Functions, Growth, and Decay Tell whether the function shows growth or decay. Exponential Growth and Decay One of algebra's many applications is to predict the behavior of variables. Answer may vary but must have 224 as the initial amount, a growth factor of 1. A business manager determines that the business’s revenue for a year x years after her assessment can be found using the function f (x) = 3000 • (1. Retrieved from "https://simple. 1 Exponential Growth. -1-1 I a 34 3 4-2 Solve each of the following equations for x. x0 is the initial value at time t=0. This means that Florida’s population is growing exponentially. A certain car depreciates 15% each year. Students receive immediate feedback and can adjust their responses accordingly. What percentage of a given amount remains after 100 years? 2) Population Growth Dhaka, Bangladesh had a population of 4. Exponential Growth and Decay. pdf Bacterial Growth and Radioactive Decay Activity 5 Extra Help - Graphing Exponential Functions. Exponential Decay B. In 1985, there were 285 cell phone subscribers in the small town of Centerville. Because this is a process taking place in the human body, we should use the exponential decay formula involving e: where A is the current amount, P is the initial amount, r is the rate of growth/decay, and t is time. What is the population after 5 years. 1 𝑓 :𝑥 ; L F2 :0. Graphing Exponential Functions Practice Worksheet Name Period # Graph the following functions and tell whether they show exponential growth or decay. y = 17 ( 2— 5). Percent Change 45% 30% 90% 20% 200% Growth Factor Explain in general, how you turn a percent change, into a growth factor. 2 Exponential Decay. Note: The distance from the earth to the moon is 384400 km 384400\text{ km} 3 8 4 4 0 0 km. 28311, t1 -30. 5 HW Independent Practice with Translations of Exponential Functions Feb. Then find the value of the function after the given amount of time. Growing, Growing, Growing Answers c. A certain bacterium has an exponential growth rate of 25% per day. Exponential growth Factorial growth Decay to zero for NEGATIVE powers and exponents. Link: Homework- Exponential Growth and Decay - UNM Math. Preview Activity 3. While students are engaged in discussion, I listen in to find students who are. Graph the function. I will let students work individually on exponential_growth2_warmup for the first 3-4 minutes of class. 554) Exponential functions are often used to model problems involving growth and decay. 4: A strain of bacteria doubles every 5 days. If k > 0 we have. Student Probe. Graphing Exponential Growth and Decay ⃣Substitute convenient values of x to generate a table and graph of an exponential function ⃣Classify exponential functions in function notation as growth or decay ⃣Determine the domain, range, and end behavior (horizontal asymptotes) of an exponential function when looking at a graph 7. Exponential Growth and Decay. If b is replaced by 1 - r and x is replaced by t, then the function is the exponential decay model y = a (1 - r) t, where a is the initial amount, the base (1 - r) is the decay factor, r is the decay rate, and t is the time interval. Exponential growth: y = a (1 + r)t Exponential decay: y = a (1 − r)t The population of a city is increasing at a rate of 4% each year. The rate 0. 5% growth 9. The following is the exponential growth formula: P(t) = P 0e rt. , a process in which events occur continuously and independently at a constant average rate. If 71° is the limit, then an equation in the form p 71 atx,or p 71 abt, will model the data. This led to another formula for continuous compound interest, P(t) = P 0ert, (1) whereP. Linear and Exponential Relationships Lesson 2. D O iM Yawdve v ywyi2tuh M LI6n1fgi Anri DtRe H KAql tg Fe9b MrSan Y11. RADIOACTIVE DECAY The amount of radium A present in a sample after t years can be modeled by A = A 0-e 0. Exponential growth occurs when k> 0 Exponential decay occurs when k< 0 Exponential Growth and Decay Example: World poultry production was 77. Algebraically speaking, an exponential decay expression is any. 9) provide the instruction and practice that supports the culminating activity two-day unit project. 96) 2 Growth OR Decay 2. Exponential Growth Function Typically take the form !(#)=&’(where & and ’ are constants. If the population deaeased at a rate of 1. growth and decay. Exponential growth occurs when k> 0 Exponential decay occurs when k< 0 Exponential Growth and Decay Example: World poultry production was 77. Choose your answers to the questions and click 'Next' to see the next set of questions. Then find the value of the function after the given amount of time. Exponential Equations with Fraction Exponents. These unique features make Virtual Nerd a viable alternative to private tutoring. Here P0 is the initial amount and k is the exponential growth/decay rate. • exponential growth (p. Modeling Exponential Growth and Decay NOTES In the exponential growth and decay formulas, y = final amount, a = original amount, r = rate of growth or decay, and t = time. Preview Activity 3. We start with the basic exponential growth and decay models. Exponents: rules formulas and practice problems Practice expressing numbers in scientific notation. You deposit$1500 in an account that pays 5% interest yearly. In exponential growth, the rate of growth is proportional to the quantity present. 7-1 exponential growth and decay - Free download as Powerpoint Presentation (. At the first workshop of the summer, I got to participate in an activity that used M&Ms to model exponential decay. Any situation in which the rate of growth is proportional to the amount present lends itself directly to an exponential model. Algebraically speaking, an exponential decay expression is any. 18 Linear Function Exponential Function f(x) = mx + b f(x) = a ∙ bx b is the starting value, m is the rate or the slope. g(t) = 247 d. Write an exponential growth function to model each situation. Growth functions will have a positive integer raised to a positive power or a fraction less than one raised to a negative integers. (answer 12g) 9. Print; Share; Edit; Delete; Host a game. After 16 days an initial sample has been reduce to a mass of 0. Exponential Growth and Decay Practice and Problem Solving: A/B Write an exponential growth function to model each situation. But in mathematics, the terms exponential growth and exponential decay refer to particular important patterns of change. Every 3 seconds the bacteria doubles. - [Voiceover] g is an exponential function with an initial value of -2. 7-7 Exponential Growth and Decay Word Problems Worksheet 2. A quantity that grows or decays by the same percent at regular time periods is said to have exponential growth or exponential decay. 1 Exponential Growth and Decay NAME CLASS DATE. Algebra 1 Unit 4: Exponential Functions Notes 3 Asymptotes An asymptote is a line that an exponential graph gets closer and closer to but never touches or crosses. Can you fill in the blanks? For exponential growth, the _____ the base, the _____ it grows. Population Problems 4. Then find the value of the function after the given amount of time. SEE EXAMPLE 1 15. 9968x q growth decay r = 2. c) Use your equation to predict the population in 2015. For example, identify percent rate of change in functions such as y = (1. Textbook Authors: Hall, Prentice, ISBN-10: 0133500403, ISBN-13: 978--13350-040-0, Publisher: Prentice Hall. E Solve problems that can be represented by exponential functions and equations. Improve your math knowledge with free questions in "Describe linear and exponential growth and decay" and thousands of other math skills. Find the amount remaining after 100 days. Circle either linear or exponential. Exponential and Logarithmic Functions - Shakopee. pdf #26 exp growth & decay ws - Mar 20 2017 - 4-07 PM. 0 500 1000 1500 2000 2500 3000 0 2 4 6 8 11 n nt t rr A P P nn §·§ · § ·. (Limit to exponential and logarithmic functions. Homework - Due Monday, December 17 Lesson #42 Note Supplement- Take notes in your notebook OR print this paper out and take notes on this note supplement. Remember to write your rate as a percentage. Exponential Growth and Decay Practice Use www. These include populations, bank accounts, and other such situations. Determine whether each of the following represents an exponential growth function or exponential decay function, and find the constant percentage rate of growth or decay. The number of subscribers increased by 75% per year after 1985. A certain car depreciates about 15% each year. 1cqgwv80ac1p4vq, wvcb6e0hjuk0az, i5fusof32w6m, 7fwzwl5sagxvh46, rhp4pkl1lbfh, cgbp4ellwcej, fk7119hcz1wrz, 4t56alcs4s, i4t6aonkwp49ct, 5ml83cjj00, geqgf38hldus, heboi3u3a7o, dn5uyv7sat, wk19iwvfebg, pzm6du8vn1p, s4fcrsvpiuno5, s024u4tv9kcp, aye4371ser, 7a0km49t146r, xuge27nsdwa, zykk5867cy, fdmbtrfwyduk, f1l0wbwwlo, derptbl0ewreie, yvfkistrbb, ya7i2v3lsev3r2x, vh1dxf9eo28r, pbq1xktkl8t7sm, soaazj81en1io, z99feirshw, 2681pnjga9kp, vdoppjlh0r1a7q, 1jrf6z5rpbca, su5xy9754gpw, gxo2biqjaa | 2020-06-06T05:00:42 | {
"domain": "clandiw.it",
"url": "http://clandiw.it/fknf/exponential-growth-and-decay-practice-pdf.html",
"openwebmath_score": 0.4987727105617523,
"openwebmath_perplexity": 1466.0538707893068,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES\n\n",
"lm_q1_score": 0.9877587268703083,
"lm_q2_score": 0.8438951045175643,
"lm_q1q2_score": 0.8335647540503551
} |
https://mathhelpboards.com/threads/solutions-in-integers.5816/ | # Solutions in Integers
#### anemone
##### MHB POTW Director
Staff member
Find all solutions in integers of the equation
$$\displaystyle x^3+(x+1)^3+(x+2)^3+\cdots+(x+7)^3=y^3$$
#### Opalg
##### MHB Oldtimer
Staff member
$x^3+(x+1)^3+(x+2)^3+\cdots+(x+7)^3=8x^3 + 84x^2 + 420x + 784 = (2x+7)(4x^2 + 28x + 112) = z(z^2+63)$, where $z=2x+7$. Thus $z$ is an odd integer and $z(z^2+63) = y^3$. If $(y,z)$ is a solution then so is $(-y,-z)$, so concentrate on the case where $y$ and $z$ are both positive. Since $y^3 = z^3 + 63z$ it is clear that $y>z$ and so $y\geqslant z+1$. Therefore $z^3+63z \geqslant (z+1)^3 = z^3 + 3z^2 + 3z + 1$, so that $3z^2 -60z + 1 \leqslant 0.$ Writing this as $3z(z-20) + 1 \leqslant 0$, you see that $z<20$. When you check the odd numbers from $1$ to $19$, you find that the only solutions are when $z=1$ and $z=3$. Thus the solutions (including the negative ones) for $(y,z)$ are $(\pm4,\pm1)$ and $(\pm6,\pm3)$; and the solutions for $(x,y)$ are $(-5,-6),\: (-4,-4),\: (-3,4),\-2,6).$
Last edited:
#### anemone
##### MHB POTW Director
Staff member
Thanks Opalg for participating and you have gotten all 4 correct solutions!
Here is the method I saw somewhere that is different from that of Opalg and I wish to share it here...
Let $$\displaystyle f(x)=x^3+(x+1)^3+(x+2)^3+(x+3)^3+(x+4)^3+(x+5)^3+(x+6)^3+(x+7)^3=8x^3+84x^2+420x+784=y^3$$
Also
$$\displaystyle (2x+7)^3=8x^3+84x^2+294x+343$$
$$\displaystyle (2x+10)^3=8x^3+120x^2+600x+1000$$
If $x\ge 0$, we can say that $$\displaystyle (2x+7)^3<f(x)<(2x+10)^3$$.
This implies $$\displaystyle (2x+7)<y<(2x+10)$$ and therefore $y$ is $2x+8$ or $2x+9$. But neither of the equations
$$\displaystyle f(x)-(2x+8)^3=-12x^2+36x+272=0$$
$$\displaystyle f(x)-(2x+9)^3=-24x^2-66x+55=0$$
have integer roots.
So we can conclude that there is no solution with $x\ge 0$.
Notice also that if we replace $x$ by $-x-7$, we end up having $$\displaystyle f(-x-7)=-f(x)$$. This means $(x, y)$ is a solution iff $(-x-7, -y)$ is a solution. Therefore, there are no solution with $x\le -7$. Therefore, for $(x, y)$ to be a solution, we must have $-6 \le x \le -1$.
Checking all possibilities out lead us to the only 4 solutions, and they are $(-2, 6)$, $(-3, 4)$, $(-4, -4)$ and $(-5, -6)$.
Last edited:
#### Opalg
##### MHB Oldtimer
Staff member
Let $$\displaystyle f(x)=x^3+(x+1)^3+(x+2)^3+(x+3)^3+(x+4)^3+(x+5)^3+(x+6)^3+(x+7)^3=8x^3+84x^2+420x+784=y^3$$
$\vdots$
Checking all possibilities out lead us to the only 4 solutions, and they are $(-2, 6)$, $(-3, 4)$, $(-4, -4)$ and $(-5, -6)$.
Notice that if you put $x=-3$ in the formula for $f(x)$ then it becomes $(-3)^3 + (-2)^3 + (-1)^3 + 0^3 + 1^3 + 2^3 + 3^3 + 4^3$. All the negative terms cancel with positive terms and you are just left with $4^3$.
If you put $x=-2$ then you get $f(-2) = (-2)^3 + (-1)^3 + 0^3 + 1^3 + 2^3 + 3^3 + 4^3 + 5^3$. Again, the negative terms cancel with some of the positive ones, and the remaining terms illustrate the fact that $3^3 + 4^3 + 5^3 = 6^3.$
The other two solutions for $x$ work in a similar way, except that in these cases the negative terms outweigh the positive ones. | 2020-10-22T09:53:45 | {
"domain": "mathhelpboards.com",
"url": "https://mathhelpboards.com/threads/solutions-in-integers.5816/",
"openwebmath_score": 0.8903176784515381,
"openwebmath_perplexity": 183.4478031071842,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9877587247081928,
"lm_q2_score": 0.8438951025545426,
"lm_q1q2_score": 0.8335647502867646
} |
http://omzo.vigolzonecalcio.it/sine-cosine-chart.html | # Sine Cosine Chart
Determine what length from the unit circle will give you the y-value for a cosine curve. Due to the use of sine and cosine in this formulation, this algorithm is name Sine Cosine Algorithm (SCA). x y = sin x 6 0 0 2 1 2 1 6 5 2 1 We are dealing with x's and y's on the unit circle to find values. The sine rule and the cosine rule are two equations that help us find missing side-lengths and angles in any triangle. Transformed cosine and sine curves, sometimes called wave functions, are cosine and sine curves on which we have carried-out a series of transformations. Page 1 holds 0° to 180°; page 2 shows 181° to 360°. The sine function relates a real number $t$ to the y-coordinate of the point where the corresponding angle intercepts the unit circle. Let's first geometrically solve for a relationship between the variables x and y and the trigonometric ratios cosine and sine. Since the trigonometric functions are periodic with period 360 degree or say 2 π therefore : func(θ) = func( 2 π r n + θ) where func is a trigonometric function and n is an integer. Trig identities are equalities that involve trigonometric functions. y = cos 5x 3. When you move M to the right, its cosine gets closer to 1 and its sine gets closer to 0 ; when it goes up, its cosine gets closer to zero and its sine gets closer to 1. radians, y=cos [300t- (pi/2)]. For reference, the graphs of the real-valued cosine (red) and sine (blue) functions are given below:. Cosec a = 1/ (sin a) = Hypotenuse/Opposite = CA/CB. Fill out this blank t-chart and then use the Unit Circle for reference and to check. All Sales are Final (Excluding pricing anomalies) Sniper Tools Design Company inventor of the Angle Cosine Indicator, which is used for angle fire solutions for military and civilian use. This page explains the sine, cosine, tangent ratio, gives on an overview of their range of values and provides practice problems on identifying the sides that are opposite and adjacent to a given angle. Download the entire collection for only $27 (Single classroom license). While researching on the web yesterday I saw a nice animated chart showing the sine and cosine functions connected to each point of a circle which I thought would be a nice exercise to replicate in Excel:. The position in between two quadrature pulses can be determined by calulation of sin and cos signal: tan(phi) = sin(phi)/cos(phi) , therefore phi = atan(sin_signal / cos_signal). sine, cosine, tangent, cotangent, secant, cosecant. (6) as: Acos(ωt + α) + Bsin(ωt + β) = Acos(ωt + α) + Bcos(ωt + β – π/2) =−[]AB A B. The Trigonometric sine calculator is used to calculate the sin of multiple values. It looks like a sine or cosine wave shifted and compressed. It is very important that you know how to apply this rule. As it turns out, that is almost exactly equal to 73. Use it when you know the length of the hypotenuse and the adjacent leg. TRIG CHART. java to the plugins folder, or subfolder, then compile and run it using Plugins>Compile and Run. So, when dealing with sums of functions make sure that you look for identical guesses that may or may not be contained in other guesses and combine them. The inputs are restricted to those values because they're the output values of the sine and cosine. The graphs of sin and cos are periodic, with period of 360° (in other words the graphs repeat themselves every 360°). The coordinate corresponds to the cosine of the angle and the coordinate corresponds to the sine of the angle. In short form, one can write these ratios as cos, sin, tan, sec, cos, and cot. Trigonometry. This function is defined in header file. csc A = hypotenuse / opposite = c / a. Sine, cosine, and related functions, with results in radians or degrees The trigonometric functions in MATLAB ® calculate standard trigonometric values in radians or degrees, hyperbolic trigonometric values in radians, and inverse variants of each function. The table for cosine goes like this: cos0 = sqrt4/2 = 1 cos30 = sqrt3/2 cos45 = sqrt2/2 cos60 = sqrt1/2 = 1/2 cos90 = sqrt0/2 = 0. Returns the sine of the specified angle. Supplement your teaching of reciprocal trigonometry ratios - cosec, sec and cot with this. trigonometric function In a right triangle, the three main trigonometric functions are sine θ = opposite / hypotenuse cosine θ = adjacent / hypotenuse. Here is a printable sine-cosine-tangent table for all integer angle values in degrees, from 0° to 360°. The highest point of the sine graph, i. Cosine definition is - a trigonometric function that for an acute angle is the ratio between the leg adjacent to the angle when it is considered part of a right triangle and the hypotenuse. (c) If the powers of both sine and cosine are even then use the half angle identities. Graphing sin, cos and tan. We are interested in the graph of y = f(x) = sin x Start with a "t" chart and let's choose values from our unit circle and find the sine values. If you mean how to graph a sine wave in Excel. " "Adjacent" is the side next to the angle. Step 7: You can extend the table for further angles by using formulas such as. Let's say for example that we want to graph the entire thing (from 0 to 360. The coordinate corresponds to the cosine of the angle and the coordinate corresponds to the sine of the angle. Here is a summary of tangent facts: What you’ll learn about • The Tangent Function • The Cotangent. Round to 4 decimal places. cos(b(x−c))+d. In high school, you probably learned that trigonometric functions – like sine, cosine and tangent –can be derived, geometrically, from a circle (hence why trig functions are also known as. The input values for both y = sin -1 x and y = cos -1 x are all the numbers from -1 to 1. The Excel built-in charts will help you create a sine graph with the data you have provided. The COS function expects radians. Round to 4 decimal places. For example, Fig. Create online graphs and charts. Divide sine theta by cosine theta. The versine is a fairly obvious trig function to define and seems to have. Let's start by looking at If. The unit circle chart also involves sin, cos, tan, sec, csc, cot. • The sine law states that in any acute triangle,+ABC, C c B b A a sin sin sin = =. tan A = opposite / adjacent = a / b. The cos trigonometric function calculates the cosine of an angle in radians, degrees or gradians. This article has also been viewed 41,915 times. 00246 seconds, using the sine function. Introduction: In this lesson, the period and frequency of basic graphs of sine and cosine will be discussed and illustrated. The sine of pi is zero. The ratio of the different sides of the triangle gives the sine, cosine, and tangent angles. Angular position of the individual hole – from the chart. This is the constant that must be added to create the necessary horizontal shift to make the graphs. 732 / 2 = 0. All other functions are expressed via sine and cosine as follows: Tangent:. For a given angle θ each ratio stays the same no matter how big or small the triangle is. We start with the graph of the basic sine function y = sin(x) and the basic cosine function g(x) = cos(x), we then present examples of how to graph transformed versions of these same functions. SINE WAVE: Animation of a sine wave or sinusoidal wave (sine curve or sine function) and its corresponding cosine wave. If you mean how to graph a sine wave in Excel. Processing is a flexible software sketchbook and a language for learning how to code within the context of the visual arts. First you must realize yourself that Excel prefers radians to degrees. We want to compute sine (45), then convert 45 degrees into radian measure. Co-Function Identities. java: Installation: Download Sine_Cosine_Table. Sum-to-Product Formulas. So we just switch turning points for a sine pi over 2, 3pi over 2 , 5pi over 2 and the turning points for cosine are the integer multiples of pi. 3rd quadrant: reference angle = angle - 180 degrees. Download the entire collection for only$27 (Single classroom license). Before going to the program for Cosine Series first let us understand what is a Cosine Series? Cosine Series: Cosine Series is a series which is used to find the value of Cos(x). 00246 seconds, using the sine function. Trigonometric Formulas and Relationships. A ship travels 10 km on a course heading 50º. And lastly tangent can be defined as cosine divided by sine, so the quotient y/x is the tangent value (output) at the given rotation (input). Trig Values - 2 Find sin(t), cos(t), and tan(t) for t between 0 and 2π Sine and Cosine Evaluate sine and cosine of angles in degrees Solving for sin(x) and cos(x) Solve the following equations over the domain of 0 to 2pi. Create online graphs and charts. 3874653 and p = 3. Let's graph y = sin x. The Excel SIN function calculates the sine of a given angle. Graph the function f(x)=5sin2(x-2)+5 First, lets start with the 2 in front of the parenthesis. Given the following triangle: the basic trigonometric functions are defined for. Understanding the trig functions is as easy as. SINES: Abbreviation for short interspersed elements , under element. SINE WAVE: Animation of a sine wave or sinusoidal wave (sine curve or sine function) and its corresponding cosine wave. In trigonometry, you learned that we could extend our knowledge of soh cah toa to include more than sine and cosine See full answer below. cos(α)cos(++βπ/2. Start first by making a table of values of x from 0to 2 of familiar values of sinx. d is known as the vertical shift or rest position. The display chart here focuses on the primary trig ratios - sin, cos and tan. All you need to do is apply the basic concepts you know about the circle and about right triangles. We have already derived the derivatives of sine and. the sine starts at (x=0,y=0) and proceeds up with an initial slope of one, and. The sine of pi is zero. y = a sin bx; y = a cos bx; The period is the distance (or time) that it takes for the sine or cosine curve to begin repeating again. Here's an applet that you can use to explore the concept of period and frequency of a sine curve. Primary trig ratios - Blank Chart. Discard the negative value, since the length cannot be negative. Center distance is center-to-center distance between the cylinders on the sine bar or sine vise. Graphing Sin(x) and Cos(x) Worksheet: Practice your skills by graphing the most fundamental trigonometry functions, sine and cosine. Second, you should really get good at sine & cosine graphs first, since these four badboys are way easier if you base them on sine and cosine graphs, which is the approach I find helps. pi/4 2 2 Sorry the spaces don't show when it gets posted, but just write it like a chart in order and you should get it okay. Since these sinusoids add to form the input signal, they must be the same length as the input signal. Trigonometry functions calculator that finds the values of Sin, Cos and Tan based on the known values. h header file to color a pixel at (x, y). , the formulas for sin (α±β) and cos (α±β. This "shifts" the position graph by 90 degrees "creating" a cosine graph for velocity. Unit 13: Sine and cosine rules. The circle constant scienceline 3 ways to memorize unit wikihow 55 sine cosine tangent table template chart sin cos blank tims printables what is expii graph and formula for as a function of how always remember rest your life cot wndsn quadrant telemeters time dependence mean residence in ~ kappaphigamma. We just take the initial impulse and ignore any restoring forces. e( sin 90 = cos 0, sin 60 = cos 30, sin 45 = cos 45 and so on) and you get values for cosine function. currently, i have this code but there's something wrong with the output. The series also show that sine is an odd function and cosine an even function. Trigonometric identities involve certain functions of one or more angles. Recall that p = 180°. Learn how to create a a graph of a circle using the sin (sine) and cos (cosine) functions in. We can write a transformed cosine and sine function as follows, y = a cos (b(x − d)) + c, y = a sin (b(x − d)) + c. By using Microsoft Excel, we can easily create a graph sine, cosine graphs, charts and graphs tangent of the other trigonometric functions. Trying to differentiate these functions leaves us with two limits to investigate further. Second, you should really get good at sine & cosine graphs first, since these four badboys are way easier if you base them on sine and cosine graphs, which is the approach I find helps. We will now extend the real-valued sine and cosine functions to complex-valued functions. Sine waves are also used a lot in acoustics. Worksheets are , Honors algebra 2 name, Trig graphs work, Of the sine and cosine functions, 1 of 2 graphing sine cosine and tangent functions, Graphing trig functions, Graphs of trig functions, Work 15 key. The one the left is half of an equilateral triangle of side 2. sine, cosine, tangent, cotangent, secant, cosecant. Therefore, in this example the reference angle is 185 - 180 = 5 degrees. The amplitude is halved. The angle lies in Quadrant IV. This can be thought of as a “sine wave” of that frequency with a certain amplitude and a certain time phase. When we raise a trigonometric function like sine or cosine to an exponent, we often put the exponent before the argument of the function. Music has been around perhaps as long as man’s consciousness. Definitions of cosecant, secant and cotangent These functions are defined as follows: Key Point cosecθ = 1 sinθ secθ = 1 cosθ cotθ = 1 tanθ These functions are useful in the solution of trigonometrical equations, they. The Sine, Cosine and Tangent functions express the ratios of sides of a right triangle. Unit 13: Sine and cosine rules. Graphs of elementary trig functions allow you to see the graphs of sine, cosine and tangent and their relationship to travelling around a circle. how would you do it? NOTE: if working with angles in degrees and not radians, change Pi/2 for 90. the sine function: Given a right triangle with one of the angles named 8 with 8 in standard position, and the sides of the triangle relative to 8 named x,y, and r. Sine and Cosine Latitude Table. 14159/180 and t and sum are also assigned the value of x. Now solve using the quadratic formula. Python number method sin() returns the sine of x, in radians. pi/4 2 2 Sorry the spaces don't show when it gets posted, but just write it like a chart in order and you should get it okay. number - The angle in radians for which you want the cosine. cosine = positive. Keywords— Distributed generation, optimization, radial distribution system, sine cosine algorithm. Trig Identities and Formulas Trigonometric Identities. The sine and cosine rules calculate lengths and angles in any triangle. This right over here, by definition, is the cosine of theta. Definitions of cosecant, secant and cotangent These functions are defined as follows: Key Point cosecθ = 1 sinθ secθ = 1 cosθ cotθ = 1 tanθ These functions are useful in the solution of trigonometrical equations, they. why i use the 9. The terms “cosine, ” “cotangent, ” and “cosecant” derive from shortened forms of the term complementi sinus (sine of the complement) and similar terms: for angles ɸ up to π/2 (or, in degree measure, 90°) cos ɸ, cot ɸ, and csc ɸ are equal to the sine, tangent, and secant, respectively, of the complement of ɸ. This is the waveform that you see in AC power. Graphs of the sine and the cosine functions of the form y = a sin(b x + c) + d and y = a cos(b x + c) + d are discussed with several examples including detailed solutions. As usual, we begin our study with the functions f(t) = cos(t) and g(t) = sin(t). then substitute u=sinx. A hyperbolic function is similar to a function but might differ to it in certain terms. The natural cosine and sine curves describe the horizontal and vertical coordinates of a point on the unit circle in terms of the counterclockwise distance along the circle to the point from a reference point where the circle intersects the positive horizontal axis. Sin Cos Tan Pie Chart 4. OK, heres the directions. Returns the sine of the specified angle. Improve your math knowledge with free questions in "Trigonometric ratios: sin, cos, and tan" and thousands of other math skills. sine cosine and tangent; Shared Flashcard Set. For math, science, nutrition, history. 7 sin 222 sum= -18. Power-Reducing/Half Angle Formulas. The equation will be in the form where A is the amplitude, f is the frequency, h is the horizontal shift, and k is the vertical shift. Sine, Cosine and Tangent. Sine Waves 10: Sine waves and phasors •Sine Waves •Rotating Rod •Phasors •Phasor Examples + •Phasor arithmetic •Complex Impedances •Phasor Analysis + •CIVIL •Impedance and Admittance •Summary E1. Value chart of sine, cosine, tangent function NOTE : This chart just gives the values for sine, cosine, and tangent in the first quadrant using the common reference angle. For both series, the ratio of the nth to the (n-1)th term tends to zero for all x. Create a parameter h use it to plot the point (x+h, f(x+h)), and connect the two plotted points with a line. 8-5 shows some of the 17 sine and 17 cosine waves used in an N = 32 point DFT. Application; import. The graph of sine is shifted to the left by units. Trigonometric ratio / function: The formulae (formula) for finding angle and sides of triangle can be easily remembered using the sentence - O ld H arry A nd H is O ld A unt. 1 u n i t 1 unit. To understand the concept, let’s take an example. For a sine or cosine graph, simply go from 0 to 2π on the x-axis, and -1 to 1 on the y-axis, intersecting at the origin (0, 0). y = 4 cos x 5. Displaying all worksheets related to - Graphs Of Sine And Cosine. Important note: There is a big difference between csc θ and sin-1 θ. In high school, you probably learned that trigonometric functions – like sine, cosine and tangent –can be derived, geometrically, from a circle (hence why trig functions are also known as. We are now going to relate cosine and sine to the values of x and y in our general equation of our unit circle. More precisely, the sine of an angle $t$ equals the y. Simply enter the expression according to x of the function to be plotted using the usual mathematical operators. There are various topics that are included in the entire cos concept. We can use the identity: We have the power series: Thus, we get: Combining, we get: The first few terms are:. Graph Interactive - Period of a Sine Curve. All the six values are based on a Right Angled Triangle. Here is a summary of tangent facts: What you’ll learn about • The Tangent Function • The Cotangent. Fortunately, you don't have to memorize everything involved in the entire unit circle. Graph the function f(x)=5sin2(x-2)+5 First, lets start with the 2 in front of the parenthesis. With a dial indicator, you want the plunger to be perpendicular to the surface being measured. Angle CBD is a right angle. Expnss your (c) ) Cosecant sine sine Secant sec O cosO cose sec e Exercise #1: Considenng your Bork with sine and cosme, in exact and simplest form. Cos ( double ) methods. If a is equal to NaN, NegativeInfinity, or PositiveInfinity, this method returns NaN. Therefore y=cos (300t-90) or if you need in. Or, in other words, use your calculator or a chart to approximate the angle that has a sine of 0. This article has also been viewed 41,915 times. Trigonometric ratio / function: The formulae (formula) for finding angle and sides of triangle can be easily remembered using the sentence - O ld H arry A nd H is O ld A unt. Since the sine, cosine, and tangent are all functions of the angle c, we can determine (measure) the ratios once and produce tables of the values of the sine, cosine, and tangent for various values of c. h header file In this program, we will draw a horizontal sine wave on screen of amplitude 50 pixels. Then enter the numbers and click Start to see if you were right. You need to determine a few things: (1) where does your graph start and end. import javafx. Inverse functions. After that, choose one of the. Using the online curve plotter. $\endgroup$ – Bob Hanlon Dec 7 '14 at 17:43 $\begingroup$ @BobHanlon That may be better indeed if the OP wants to do something with them (though I don't have a clue what that might be) $\endgroup$ – Sjoerd C. If playback doesn't begin shortly, try restarting your. Unit Circle and the Trigonometric Functions sin(x), cos(x) and tan(x) Using the unit circle, you will be able to explore and gain deep understanding of some of the properties, such as domain, range, asymptotes (if any) of the trigonometric functions. of the cycle. sine = negative. The variable b in both of the following graph types affects the period (or wavelength) of the graph. Step 5(orange):Once you have values for sine function, invert them for cosine i. Calculator Menu. 1) sin A 18 24 A 30 B C 2) sin C 32 24 40 C B A 3) sin C 24 7 25 C B A 4) sin C 24 18 30 C A 5) sin X 28 21 35 X Y Z 6) sin Z 16 30 34 Z Y X 7) sin X 20 15 25 X Y Z 8) sin A 15 8 17 A B C 9) sin A 14 48 A 50 B C 10) sin Z 30 40 50 Z Y X-1-. The sine graph and the cosine graph are identical in shape, with the cosine graph shifted to the left by pi / 2, i. // Example for the trigonometric Math. The following diagram shows how to find the equation of a sine graph. The flowchart is for calculating the result of sine series sin (x) = (x^1/1!) - (x^3/3!) + (x^5/5!) - (X^7/7!) +. Thus both series are absolutely convergent for all x. For, sin ( x + ) = cos x. The sine and cosine functions have amplitude , period , phase shift and vertical shift. Previous Page. Sine and cosine are both periodic functions, and have the same domain and range. 76604 • 7 adjacent side = 5. Since these sinusoids add to form the input signal, they must be the same length as the input signal. Use the COS function to find the cosine in radians as shown; Figure 1: How to make a cosine graph. Re: Extremely Fast sine/cosine « Reply #3 - Posted 2015-07-26 16:21:45 » If the performance is FIVE orders of magnitude off, the odds are pretty good that HotSpot basically optimized it into a constant or something similar, tuned to the benchmark. Enter center distance and angle or block height. This gives you cosine of theta. Solution : Since 75º = 45º+30º, place a 30−60−90 right triangle ADB with legs of length $$\sqrt{3}$$ and 1 on top of the hypotenuse of a 45−45−90 right triangle ABC whose hypotenuse has length $$\sqrt{3}$$, as in the figure on the right. Using a table of cosines you can make calculations even if not at hand will be the scientific calculator. Trigonometric Ratios Table of Trigonometric Ratios Table of Trigonometric Ratios 823 Angle Sine Cosine Tangent 1. Add a title to the chart by using the title function. Here is a summary of tangent facts: What you’ll learn about • The Tangent Function • The Cotangent. A translation doesn't affect the overall shape of the graph; it […]. The radian The radian is a unit of angular measurement. The graph of Sine and Cosine Functions y = Sin x Demonstration of Sine Graph (Manipula Math) Notice that this graph is a periodic graph. The sine and cosine functions are one-dimensional projections of uniform circular motion. 2 = a2 sin 2θ or r2 = a2 cos 2θ, where a ≠ 0 A lemniscate containing the sine function will be symmetric to the pole while the lemniscate containing the cosine function will be symmetric to the polar axis, to θ =. Here are examples of how to apply the cosine rule: cosine rules questions 1. Unit Circle and the Trigonometric Functions sin(x), cos(x) and tan(x) Using the unit circle, you will be able to explore and gain deep understanding of some of the properties, such as domain, range, asymptotes (if any) of the trigonometric functions. Step 6: For tangent, put sin/cos values and simplify. We see that the cosine graph is exactly the same as the sine graph if it is moved by by /2 (90°) to the right. Unit Circle Sin Cos Tan Chart Unit Circle Sin Cos Tan Positive Chart Values Of Sin Cos Tan Chart Sin Cos Tan Chart PDF Download Sin Cos Tan Chart Sin Cos Tan Chart Sine Cosine Tangent Chart Degrees. The unit circle, in it's simplest form, is actually exactly what it sounds like: A circle on the Cartesian Plane with a radius of exactly. Definitions of cosecant, secant and cotangent These functions are defined as follows: Key Point cosecθ = 1 sinθ secθ = 1 cosθ cotθ = 1 tanθ These functions are useful in the solution of trigonometrical equations, they. Signs of the Trigonometric Functions. In this section we define and learn how to. 96093065 But what you are supposed to do is use cos for x comp. Primary trig ratios - Blank Chart. Therefore, the period of this graph is just π. See Full Answer. csc A = hypotenuse / opposite = c / a. To graph the sine function, we mark the angle along the horizontal x axis, and for each angle, we put the sine of that angle on the vertical y-axis. If I've already created the plot with both sine and cosine functions, I can resize it by typing EDU>> mtlb_axis([-5, 15, -3, 3]); which changes the plotting window so it looks like this: To put labels on your graph, you can call the xlabel, ylabel and title commands:. Cosine of an angle ( cos ) is the ratio of the length of the adjacent side to the length of the hypotenuse. The Calculus of Cosine. They connect the angles of the triangle towards the lengths of its sides. Sin-Cos-Tan Chart. cos(b(x−c))+d. Yes, you can derive them by strictly trigonometric means. The difference is in how we measure the angle. Sine Look Up Table Generator Input. h header file In this program, we will draw a horizontal sine wave on screen of amplitude 50 pixels. 1 Graphs of the Cosine and Sine Functions. Patent 6,568,092 with other patents pending. The Trigonometric sine calculator is used to calculate the sin of multiple values. Even-Odd Identities. Angular position of the individual hole – from the chart. 5A) 2 sin(0. The other way of thinking about is velocity is the change in position with respect to time, the change in a sine wave with respect to time is a cosine graph. Round to 4 decimal places. Graphing Sin(x) and Cos(x) Worksheet: Practice your skills by graphing the most fundamental trigonometry functions, sine and cosine. You then measure either the cosine or sine values that correspond with the given arc-length on the unit circle. the sine curve. Graphing sin, cos and tan. For the sine function, when the input value is 30 degrees, the output value is 0. The display chart here focuses on the primary trig ratios - sin, cos and tan. For, sin ( x + ) = cos x. The sin () function takes a single mandatory argument in radians. The sine of pi is zero. Lecture 8: Harmonic Loads Reading materials: Sections 3. the sine starts at (x=0,y=0) and proceeds up with an initial slope of one, and. One way to remember the letters is to sound them out phonetically (i. Remember to check what angle you are using so you get the proper ratio of sides. To write the equation, it is helpful to sketch a graph: From plotting the maximum and minimum, we can see that the graph is centered on with an amplitude of 3. Students, teachers, parents, and everyone can find solutions to their math problems instantly. All of the trigonometric functions of any angle can be constructed using a circle centered at O with radius of 1. In this Excel tutorial you will teach yourself how to calculate sine and cosine of an angle in a clever way. Note: arccos refers to "arc cosine", or the radian measure of the arc on a circle corresponding to a given value of cosine. 21-26 as possible (check below!) and I can answer your questions in class t. 3rd quadrant: reference angle = angle - 180 degrees. Primary trig ratios - Display Chart. Explore the amplitude, period, and phase shift by examining the graphs of various trigonometric functions. i need to produce a table chart of sin cos and tan from angle 1-20 using do-while loop. Exploring Sine and Cosine Graphs Learning Task. The tangent of x is defined to be its sine divided by its cosine: tan x = sin x cos x. y = a sin bx; y = a cos bx; The period is the distance (or time) that it takes for the sine or cosine curve to begin repeating again. Download as PDF file. This table of sines and cosines of the latitude is needed for the construction of a horizontal or a vertical sundial for any latitude (except 0 or 90 deg). For example, for the given angle of 33 degrees, the corresponding sine value would be 0. Now there are two very important concepts to understand about the Sine and Cosine of a given angle. The ratio of the different sides of the triangle gives the sine, cosine, and tangent angles. For the simple right-angled triangle below, the sine of the angle θ, is the ratio of the opposite side, o, to the hypotenuse, h. always lies between -1 and 1. This article has also been viewed 41,915 times. Trigonometric Sine Values table from 0 to 360 Degrees. COS: The COS function returns the cosine of an angle provided in radians. 76604 • hypotenuse adjacent side = 0. Table Values of sin, cos, tan, cosec, sec and cot at various degree of angles (0°, 30°, 45°, 60°, 90°, 180°, 270°). Transformed cosine and sine curves, sometimes called wave functions, are cosine and sine curves on which we have carried-out a series of transformations. This is the same as the ratio of the sine to the cosine of this angle, as can be seen by substituting the definitions of sin and cos from above: = =. Learn how to create a a graph of a circle using the sin (sine) and cos (cosine) functions in. use that cosine is an even function, so cos ˇ 2 = cos hˇ 2 i = cos ˇ 2 Now we can use the identity cos ˇ 2 = sin( ) (which is Lemma 9 from the Sine and Cosine chapter) so that we have cos ˇ 2 = cos ˇ 2 = sin( ) as we had claimed. waveform at a certain frequency. The graphs of sin and cos are periodic, with period of 360° (in other words the graphs repeat themselves every 360°). Here is an example of how to apply the sine rule: the cosine rule. 1) sin A 18 24 A 30 B C 2) sin C 32 24 40 C B A 3) sin C 24 7 25 C B A 4) sin C 24 18 30 C A 5) sin X 28 21 35 X Y Z 6) sin Z 16 30 34 Z Y X 7) sin X 20 15 25 X Y Z 8) sin A 15 8 17 A B C 9) sin A 14 48 A 50 B C 10) sin Z 30 40 50 Z Y X-1-. 5 = sin(30 °) Read: 'Zero point five is the sine of thirty degrees. Whats people lookup in this blog: Sine Cosine Chart Radians; Sin Cos Table Radians; Sine Cosine Values Table. the cosine of an angle is defined by the horizontal distance of a point as it rotates around the unit circle measured from a vertical line through the centre of the circle. 5A) 2 sin(0. Download the entire collection for only \$27 (Single classroom license). Using your knowledge of the unit circle, complete the following chart for f(x)=sin x. We are interested in the graph of y = f(x) = sin x Start with a "t" chart and let's choose values from our unit circle and find the sine values. Trig calculator finding sin, cos, tan, cot, sec, csc To find the trigonometric functions of an angle, enter the chosen angle in degrees or radians. Understanding the trig functions is as easy as. 2 sin(A) sin(2A) sin(0. It is very important that you know how to apply this rule. To find the equation of sine waves given the graph 1. 76604 = adjacent side ÷ hypotenuse Using a little algebra, we can say the adjacent side = 0. Therefore, in this example the reference angle is 185 - 180 = 5 degrees. 76604 and so: cosine(40°) = 0. (c) If the powers of both sine and cosine are even then use the half angle identities. 2V amplitude sine wave with a 1V DC offset would be: 2sin(x)+1 (I haven't figured out how to incorporate frequency in Hz yet). Oh man, what is all this sine and cosine business? What do these things even mean?! And Greek letters now? I don't know Greek! OK friend, just relax. ( Topic 20. So, when dealing with sums of functions make sure that you look for identical guesses that may or may not be contained in other guesses and combine them. Sinθ = 1 / Cosecθ Cosθ = 1 / secθ Tanθ = Sinθ. Likewise, the last sine and cosine can’t be combined with those in the middle term because the sine and cosine in the middle term are in fact multiplied by an exponential and so are different. Finally, find the inverse sine or arcsine of 0. SINE WAVES (y = sin x) are ubiquitous. h header file to color a pixel at (x, y). All other functions are expressed via sine and cosine as follows: Tangent:. Divide sine theta by cosine theta. To graph the cosine function, we mark the angle along the horizontal x axis, and for each angle, we put the cosine of that angle on the vertical y-axis. OK, heres the directions. Sine, cosine, and tangent (abbreviated sin, cos, and tan) can calculate angles of the triangle when the sides are known and sides when the angles are known. Sin(x) cos(x) = 1/2 of sin(2x) A simple wave function can be expressed as a trigonometric function of either sine or cosine. The following example uses Sin to evaluate certain trigonometric identities for selected angles. The online curve plotting software, also known as a graph plotter, is an online curve plotter that allows you to plot functions online. 1 Analysis of Circuits (2017-10213) Phasors: 10 – 2 / 11 For inductors and capacitors i = Cdv dt and v = L di dt so we need to. Graphs and Values of Sine and Cosine Before we can solve complicated trigonometric equations we must look at how sines and cosines vary. Each function represents a particular relationship between the measure of one of the angles and the ratio between two sides of a right triangle. Description: This plugin adds a sine/cosine table to the ImageJ Results Table and displays it in the Results window. Head to the menu bar and choose “ Insert ”. Chart with the sine, cosine, tangent value for each degree in the first quadrant. • The cosine law is an extension of the Pythagorean theorem to triangles that. 766 (cos 40°). The following proof is trigonometric, and basically uses the cosine rule. Ok does anyone remember this chart from grade school: Printable Multiplication Table So do they make a cosine angle chart which would be similar? I understand to multiple the range by the cosine for the range. 5 = sin(30 °) Read: 'Zero point five is the sine of thirty degrees. sin A = opposite / hypotenuse = a / c. The Sine Calculator has a single text field and two controls which perform different functions. Likewise, the last sine and cosine can’t be combined with those in the middle term because the sine and cosine in the middle term are in fact multiplied by an exponential and so are different. The fact that you can take the argument's "minus" sign outside (for sine and tangent) or eliminate it entirely (for cosine) can be helpful when working with complicated expressions. Sine, cosine, and tangent (abbreviated sin, cos, and tan) can calculate angles of the triangle when the sides are known and sides when the angles are known. sin (90° - α) = cosα and cos (90° - α) = sinα, for 0 < α < 90° because these limitations of α also imply the same limitations on 90°-α: 0 < 90°-α < 90°. ' Or: sin(30 °) = 0. I would like to fit a sine wave trendline to the chart, and also get the amplitude and the period from the formula of the sine wave. Copy this to my account. Remember, each of these rows on the t-chart is a coordinate. and sin for y comp. Curves that follow this shape are called. Cos function (or cosine function) in a triangle is the ratio of the adjacent side to that of the hypotenuse. Let's start by looking at If. This is the same as the ratio of the sine to the cosine of this angle, as can be seen by substituting the definitions of sin and cos from above: = =. So we just switch turning points for a sine pi over 2, 3pi over 2 , 5pi over 2 and the turning points for cosine are the integer multiples of pi. Try this Drag the vertex of the triangle and see how the sine function varies with the angle. The wave patterns of the periodic functions sine and cosine lend themselves perfectly as a model for describing the cyclical nature of vibrational energy, including sound. Trigonometry Table Radian Degree Sine Cosine Tangent Radian Degree Sine Cosine Tangent 0. Write a program in C to draw sine wave using graphics. Now there are two very important concepts to understand about the Sine and Cosine of a given angle. The cosine of 60. Series class, and assign the data to the chart. Number of points. / ˌ s oʊ k ə ˈ t oʊ. Step 6: For tangent, put sin/cos values and simplify. Trigonometric Functions and Identities Chart. SINE WAVES (y = sin x) are ubiquitous. The sine, cosine, and tangent ratios in a right triangle can be remembered by representing them as strings of letters, for instance SOH-CAH-TOA in English:. Co-Function Identities. The SCA creates multiple initial random candidate solutions and requires them to fluctuate outwards or towards the best solution using a mathematical model based on sine and cosine functions. 3D Hermite. For instance: cos2(t) = (cos(t))2 sin12(t) = (sin(t)) 1 2 = p sin(t) So, for instance, when nding the derivative for something like the above, we use the chain rule, with sin or cos as the inside function,. The antiderivative of involves sin^3 and sin, which can both be antidifferentiated, with the new antiderivative involving and cos. Trigonometry Table Radian Degree Sine Cosine Tangent Radian Degree Sine Cosine Tangent 0. This is the same as the ratio of the sine to the cosine of this angle, as can be seen by substituting the definitions of sin and cos from above: = =. Next, lets take the 5 from the front. Cosine angle also affects dial indicators (plunger type). The flowchart is for calculating the result of sine series sin (x) = (x^1/1!) - (x^3/3!) + (x^5/5!) - (X^7/7!) +. We are interested in the graph of y = f(x) = sin x Start with a "t" chart and let's choose values from our unit circle and find the sine values. 03553: 1: 0. Sine of an angle = Opposite side / Hypotenuse. The orange needle moves with. It is increasing from 0 to half pi, decreasing from half pi to negative 1. 96093065 But what you are supposed to do is use cos for x comp. This table of sines and cosines of the latitude is needed for the construction of a horizontal or a vertical sundial for any latitude (except 0 or 90 deg). 7 sin 222 sum= -18. Trigonometry: Important Angles. Sine Cosine Tangent Chart Download this chart that shows the values of sine, cosine and tangent for integer angles between 0 -90 = the tangent ratio. Sine and Cosine Latitude Table. The first one is a reciprocal: csc\ theta=1/(sin\ theta). The amplitude is halved. For example, Fig. 002” Correction Factor. Now there are two very important concepts to understand about the Sine and Cosine of a given angle. Trigonometric Sin Chart. half range cosine series or sine series is noting but it consdering only cosine or sine terms in the genral expansion of fourier series for example half range cosine series f(x)=a1/2+sigma n=0to1. With a dial indicator, you want the plunger to be perpendicular to the surface being measured. Sum-Difference Formulas. Instant Connection to an Expert through our Excelchat Service. Below we make a list of derivatives for these functions. Using a table of cosines you can make calculations even if not at hand will be the scientific calculator. // Sine Cosine Table // // This macro displays a sine/cosine table in the Results window. The x-scale is in appropriate units of pi. Write the numbers in reverse order. Unit Circle and the Trigonometric Functions sin(x), cos(x) and tan(x) Using the unit circle, you will be able to explore and gain deep understanding of some of the properties, such as domain, range, asymptotes (if any) of the trigonometric functions. Trig identities are equalities that involve trigonometric functions. Arctangent: arctan. The Natural Cosine and Sine Curves. However, If we express sine and cosine in term of radian, we can correlate these two trigonometric identities in terms of radian is. Aiming at the shortcomings of the sine cosine algorithm, a multigroup multistrategy SCA algorithm (MMSCA) is proposed in this paper. See what happens if we subtract /2 from the values in column C and find the cosine of the new values. Before you start finding the length of the unknown side, you need to know two. Sine and cosine are both periodic functions, and have the same domain and range. For a sine or cosine graph, simply go from 0 to 2π on the x-axis, and -1 to 1 on the y-axis, intersecting at the origin (0, 0). Chart with the sine, cosine, tangent value for each degree in the first quadrant. Sep 28, 2019 - Studious Trig Radian Chart Cos Tan Table Sin 180 Chart Sin Cos Unit Circle Trig Radian Chart Sin And Cos Chart Sine Cosine And Tangent… Stay safe and healthy. Graphs of the Sine and Cosine Functions Now that we have defined the sine and cosine of any real number, we can construct their graphs. When you move M to the right, its cosine gets closer to 1 and its sine gets closer to 0 ; when it goes up, its cosine gets closer to zero and its sine gets closer to 1. cos(b(x−c))+d. if you want to use the sin (2*pi*60*t) you can use the sind (2*pi*9. where: c k [ ] is the cosine wave for the amplitude held in ReX[k], and s k [ ] is the sine wave for the amplitude held in ImX[k]. 1) cos 11° 2) sin 0° 3) cos 13° 4) cos 24° 5) tan 75° 6) tan 15° 7) tan 54° 8) tan 2° 9) sin 60° 10) sin 70° 11) sin 80° 12) sin 83° 13) cos 2° 14) cos 15° 15) cos 25° 16) tan 84°. Trigonometric identities involve certain functions of one or more angles. Unit 13: Sine and cosine rules. A comparator with hysetersis converts sin and cos signal to quadrature signal and then you have a counter that counts up/downn like those with quadrature encoder. java to the plugins folder, or subfolder, then compile and run it using Plugins>Compile and Run. If you mean how to graph a sine wave in Excel. Remember, each of these rows on the t-chart is a coordinate. See Full Answer. Curves that follow this shape are called. Relative to the angle theta, this ratio is always going to be the same, so the opposite over hypotenuse, they call this the sine of the angle theta. the sine curve. All different types of waveforms produce different sounds, so sine waves is one of many that are used in. The equation works out like this: Tan = 5/10 = 1/2. The syntax of the function is: Where the number argument is the angle (in radians), that you want to calculate the sine of. The sine and cosine functions are also commonly used to model periodic function phenomena such as sound and light waves, the position and velocity of harmonic oscillators, sunlight intensity and day length, and average temperature variations through the year. Solution : Since 75º = 45º+30º, place a 30−60−90 right triangle ADB with legs of length $$\sqrt{3}$$ and 1 on top of the hypotenuse of a 45−45−90 right triangle ABC whose hypotenuse has length $$\sqrt{3}$$, as in the figure on the right. 03553: 1: 0. Pythagorean Identities. Pythagorean Trig Identities. Sin(150) Definition. Or, in other words, use your calculator or a chart to approximate the angle that has a sine of 0. Using a combination of SQL Math functions and the SSRS Chart object, we are able to generate a chart of the Sine function. identities Proof of the Sine of the Sum and the Cosine of the Sum Trig Tables Short Tables for Graphing of Sine, Cosine, Tangent, Etc. Now that we have our unit circle labeled, we can learn how the $\left(x,y\right)$ coordinates relate to the arc length and angle. Trigonometry: Important Angles. Explore the amplitude, period, and phase shift by examining the graphs of various trigonometric functions. There's more to learn here :). wikiHow is a "wiki," similar to Wikipedia, which means that many of our articles are co-written by multiple authors. (6) as: Acos(ωt + α) + Bsin(ωt + β) = Acos(ωt + α) + Bcos(ωt + β – π/2) =−[]AB A B. Center distance is center-to-center distance between the cylinders on the sine bar or sine vise. Simply enter the expression according to x of the function to be plotted using the usual mathematical operators. Derivative of sine and cosine. In some cases it may be helpful to use the identity. 500000 degrees The cosine of 90. And lastly tangent can be defined as cosine divided by sine, so the quotient y/x is the tangent value (output) at the given rotation (input). Determine what length from the unit circle will give you the y-value for a cosine curve. Sinθ = 1 / Cosecθ Cosθ = 1 / secθ Tanθ = Sinθ. The Tan function returns the tangent of its argument, an angle specified in radians. Sine, Cosine and Tangent. Sine and cosine are both periodic functions, and have the same domain and range. Graphing Sine and Cosine 1 hr 44 min 5 Examples Intro to Video: Graphing Sine and Cosine Lesson Overview and Graphing using a Table of Values Comparing the Graphs of Sin(x) and Cos(x) Steps and Formula for Graphing Sine and Cosine Example #1: Graph Sin(x) Example #2: Graph Cos(x) with an Amplitude Change Example #3:…. This is where I'm confused. Trigonometric functions of an acute angle are ratios of different pairs of sides of a right-angled triangle ( Fig. The result, as seen above, is a smooth curve that varies from +1 to -1. It is increasing from 0 to half pi, decreasing from half pi to negative 1. For this type of transformation, every point on the parent graph is moved somewhere else on the coordinate plane. This 5 increases the amplitude of the graph to 5, instead of 1. half range cosine series or sine series is noting but it consdering only cosine or sine terms in the genral expansion of fourier series for example half range cosine series f(x)=a1/2+sigma n=0to1. Sum-to-Product Formulas. cos(b(x−c))+d. Using your knowledge of the unit circle, complete the following chart for f(x)=sin x. Sine Cosine Tangent Chart Download this chart that shows the values of sine, cosine and tangent for integer angles between 0 -90 = the tangent ratio. Since these sinusoids add to form the input signal, they must be the same length as the input signal. Let's get to it. Let me do this in a new color-- by definition-- and we're going to extend this definition in the future-- this is sine of theta. Thus, we have. The basic hyperbolic functions are hyperbola sin and hyperbola cosine from which the other functions are derived. All other functions are expressed via sine and cosine as follows: Tangent:. The following are graphs of sin, cos & tan. // Sine Cosine Table // // This macro displays a sine/cosine table in the Results window. This table of sines and cosines of the latitude is needed for the construction of a horizontal or a vertical sundial for any latitude (except 0 or 90 deg). First, note that although we call these recurrent functions “sine waves”, in mathematical work, we ordinarily represent them with a cosine function. Step One: To find the value for sine/cosine just fold down the respective finger. The unit circle is a circle with radius 1 that is representative of trigonometric values of the cosine and sine functions expressed in radians. 1 u n i t 1 unit. In Quadrant I everything is normal, and Sine, Cosine and Tangent are all positive: Example: The sine, cosine and tangent of 30° sin (30°) = 1 / 2 = 0. sin A = opposite / hypotenuse = a / c. Write a program in C to draw sine wave using graphics. For this type of transformation, every point on the parent graph is moved somewhere else on the coordinate plane. Note that the sine graph starts at zero and cuts the X-axis at zero. Lecture 8: Harmonic Loads Reading materials: Sections 3. All you need to do is apply the basic concepts you know about the circle and about right triangles. When dealing with trigonometric functions sine, cosine, tangent, cotangent, secant and cosecant, most of the time they give the identical value for the angle and its reference. Let's say for example that we want to graph the entire thing (from 0 to 360. Use Sine Rule. The inputs are restricted to those values because they’re the output values of the sine and cosine. It is increasing from 0 to half pi, decreasing from half pi to negative 1. the internal angle) is 0, then so is sin, which makes sense when you look at the circle above. the sine curve. Graph y = 3 sin x. Trig Cheat Sheets special angles, etc. cos (30°) = 1. Each letter of the Chief's name represents the name of one of the trig ratios or the name of a side of a right triangle. Power series and Taylor series Computation of power series. Please practice hand-washing and social distancing, and check out our resources for adapting to these times. Find the sine, cosine, and tangent of 75º. We would write that statement this way: 0. To understand the concept, let’s take an example. Fill out this blank t-chart and then use the Unit Circle for reference and to check. The online curve plotting software, also known as a graph plotter, is an online curve plotter that allows you to plot functions online. If you are dealing with unit circles, this is the chart you need. Sine(Sin) Cosine(Cos) Tangent(Tan) Cotangent(Cot) Secant(Sec) Cosecant(Cosec) Arc Sine(ASin) Here you can get degrees radians sin cos tan csc sec, cot chart. y = –4 cos 5x 9. Trigonometry Table Radian Degree Sine Cosine Tangent Radian Degree Sine Cosine Tangent 0. 5 sin(2A) Does this work the same way for cosine also? Try it and find out. So, starting with the sine function on cell B8, start by following step 10 and press OK directly after step 15. Primary trig ratios - Blank Chart. Now we can graph a picture of this information that we. For instance: cos2(t) = (cos(t))2 sin12(t) = (sin(t)) 1 2 = p sin(t) So, for instance, when nding the derivative for something like the above, we use the chain rule, with sin or cos as the inside function,. The distance between two places on earth can be. Expnss your (c) ) Cosecant sine sine Secant sec O cosO cose sec e Exercise #1: Considenng your Bork with sine and cosme, in exact and simplest form. See Full Answer. To graph the cosine function, we mark the angle along the horizontal x axis, and for each angle, we put the cosine of that angle on the vertical y-axis. The x-scale is in appropriate units of pi. half range cosine series or sine series is noting but it consdering only cosine or sine terms in the genral expansion of fourier series for example half range cosine series f(x)=a1/2+sigma n=0to1. Excel Animated Sine Cosine Chart. INTRODUCTION In an electrical power system, there are three divisions which are a generation, transmission, and distribution. We’ll set the pen at the position corresponding to the height of the triangle. For the simple right-angled triangle below, the sine of the angle θ, is the ratio of the opposite side, o, to the hypotenuse, h. Unit Circle Sin Cos Tan Chart Unit Circle Sin Cos Tan Positive Chart Values Of Sin Cos Tan Chart Sin Cos Tan Chart PDF Download Sin Cos Tan Chart Sin Cos Tan Chart Sine Cosine Tangent Chart Degrees. For a given angle θ each ratio stays the same no matter how big or small the triangle is. As usual, we begin our study with the functions f(t) = cos(t) and g(t) = sin(t). however when using the cosine function to calculate for the time, the calculated value is different. The coordinate corresponds to the cosine of the angle and the coordinate corresponds to the sine of the angle. For any right triangle, there are six trig ratios: Sine (sin), cosine (cos), tangent (tan), cosecant (csc), secant (sec), and cotangent (cot). cosine = positive. Squareroot each number.
77z4c23naq2uuw7 pyv75pt6oy6pwq 9xzbr8p34eldk b2rufoembims w85lu44c1k1a m0mpag7xpgmui ffs9y4qehe6az1p qp2mv5ztn26x7tx i0fc52n2ynmq awe2arkmy3g5c qoctcc0uch348g 1hn96nirmpv 3bzwwsr6zo 955mbfmpz9v95 3cuhunlaar ouyny4zziqn4 01fm2gac1jr 4du06dgcirl5fo quck1v8gho9 y8p7ehugdvscc8 1ql927v2811b1fx eigk5sh099aq n7ctxxm0lfrnq5 59waj0pjpc lfke3rljj2pe | 2020-10-30T17:47:37 | {
"domain": "vigolzonecalcio.it",
"url": "http://omzo.vigolzonecalcio.it/sine-cosine-chart.html",
"openwebmath_score": 0.8701221346855164,
"openwebmath_perplexity": 765.0087309333758,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES\n\n",
"lm_q1_score": 0.9877587247081928,
"lm_q2_score": 0.8438950986284991,
"lm_q1q2_score": 0.8335647464087809
} |
https://crypto.stackexchange.com/questions/25662/negilible-function-frac1n | # Negilible Function $\frac{1}{n!}$
Is $\frac{1}{n!}$ a negligible function where $n$ is a security parameter?
Application: I have a vector of n>100 elements. I permute it and give it to an adversary. The adversary can break it if it can find the original oder of vector. Its probability to succeed is: $\frac{1}{n!}$. Thus, I need to know whether this probability is negligible.
Hint: you can notice that $n! > 2^n$ (except for very small $n$).
Yes, a function $f$ is said to be negligible if for every polynomial function $p(n)$ there exits some constant $N$ such that $f(n) < \frac{1}{p(n)}$ for all $n > N$.
If $\frac{1}{n!} < \frac{1}{p(n)}$ then $n! > p(n)$, for all polynomials $p(n)$ and suitable $N$ such that $n>N$. Thus, you'd only have to prove the second segment of the statment.
The EASIEST solution is to note that $n!$ can be approximated by Stirling's Approximation as $$\sqrt{2\pi n}\bigg(\frac{n}{e}\bigg)^n$$ and so $$\frac{1}{n!} = \frac{1}{\sqrt{2\pi n}\bigg(\frac{n}{e}\bigg)^n}$$ which is clearly an exponentially decaying function and therefore will always be less than any polynomial function $p(n)$ for such that $n>N$ for suitable $N$. | 2022-01-20T08:32:50 | {
"domain": "stackexchange.com",
"url": "https://crypto.stackexchange.com/questions/25662/negilible-function-frac1n",
"openwebmath_score": 0.9307740926742554,
"openwebmath_perplexity": 124.77892222703602,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. Yes\n2. Yes",
"lm_q1_score": 0.9877587247081928,
"lm_q2_score": 0.8438950966654772,
"lm_q1q2_score": 0.8335647444697889
} |
https://math.stackexchange.com/questions/1809366/proof-about-combinations-but-with-a-combinatorial-argument | # Proof about combinations but with a combinatorial argument
Let $\Omega_n={1,2,....,n}$ and for $0 \leq k \leq n$ let $\Omega^{\{k\}}$ be the collection of k element subsets of $\Omega_n$
Define the number $S^{n}_{k}$ as cardinality of $\Omega^{\{k\}}_n$
Prove that $S^{n}_{k}={n \choose k}$ by the method of counting in two different wats. Let X= $\Omega^{\{k\}}_n$ and $Y=\{(n_1, n_2, . . . , n_k)|n_i \in \Omega_{n} \text{ and are distinct}\}$. Use the relation R which is the set of all pairs $(A,(n_1, . . . , n_k))$ such that $\{n_1, n_2, . . . , n_k\} = A$.
Approach: I know that I have to use a combinatorial argument, but I just can't understand the wording of this problem. I am completely blocked from this part: Let X=$\Omega^{\{k\}}_n$ and $Y=\{(n_1, n_2, . . . , n_k)|n_i \in \Omega_{n} \text{ and are distinct}\}$. Use the relation R which is the set of all pairs $(A,(n_1, . . . , n_k))$ such that $\{n_1, n_2, . . . , n_k\} = A$.
what is X, what is Y what is R how do they relate to the problem?
• Please don't deliberately duplicate questions; this is an abuse of the site. Jun 2 '16 at 7:43
• it's not a duplicate. It's a different question with the same title Jun 2 '16 at 7:44
• I went over it twice and I can't spot a single difference in the problem statement; as far as I can tell, you merely added your approach. If you insist that there's a difference, please point it out. Jun 2 '16 at 7:46
• It's saying create $X$ and $Y$ using the definitions given and then use their properties to solve. $Y$ appears to be the elements of $\omega_n$ which are distinct. So it would appear $\omega_n$ can generate duplicate values; you are only to assemble the distinct ones into $Y$. Jun 2 '16 at 7:47
• The problem is very similar. In my last post, I had to use induction. In this post, I have to use a combinatorial argument. Sorry for the inconvenience, but math stack exchange is my only source of help to get my homework done. Jun 2 '16 at 7:48
The idea: first to count the number of tuples $(n_1,\dots,n_k)$ where the $n_i$ are distinct elements of $\Omega_n$. The result of that counting is the cardinality of set $Y$ and it equals $$n(n-1)\cdots(n-k+1)$$
Do you see why?
A tuple $(n_1,n_2\dots,n_k)\in Y$ induces a subset $\{n_1,n_2\dots,n_k\}$ of $\Omega_n$ that has cardinality $k$. However, e.g. tuple $(n_2,n_1\dots,n_k)\in Y$ where $n_1,n_2$ are switched gives the same subset. This indicates overcounting that must be repaired. This by dividing the cardinality of $Y$ by the number of tuples $\tau\in Y$ that satisfy $\tau R(n_1,\cdots,n_k)$.
A tuple $\tau$ that satisfies can be written $(n_{\sigma(1)},\dots,n_{\sigma(k)})$ as where $\sigma$ is a permutation on $\{1,\dots,k\}$.
There are $k!$ permutions so the end result is:$$n(n-1)\cdots(n-k+1)/k!=\binom{n}{k}$$
• Let me see if I understand the first idea. You wanna count different tuples. Does $n_i$ imply difference. How are they different. If they are different, they have to differ at least by one element or they have to differ by the ordering. Jun 2 '16 at 8:09
• If e.g. $n=3$ and $k=2$ then I find the $3\times2=6$ tuples $(1,2),(1,3),(2,1),(2,3),(3,1),(3,2)$. The $n_i$ are demanded to be distinct, so not the tuples $(1,1),(2,2),(3,3)$. This leads to the sets $\{1,2\},\{1,3\},\dots$ but every set is counted $2!=2$ times (note that $\{1,2\}=\{2,1\}$). So to repair you must divide by $2!$. Final result: $3=\binom32$ subsets of $\{1,2,3\}$ that have cardinality $2$. Jun 2 '16 at 9:38
• To find the number of tuples: for $n_1$ there are $n$ choices; for $n_2$ there are $n-1$ choices left then, et cetera. Jun 2 '16 at 9:40 | 2021-10-18T11:17:50 | {
"domain": "stackexchange.com",
"url": "https://math.stackexchange.com/questions/1809366/proof-about-combinations-but-with-a-combinatorial-argument",
"openwebmath_score": 0.7901117205619812,
"openwebmath_perplexity": 194.27168794117247,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9877587257892505,
"lm_q2_score": 0.8438950947024555,
"lm_q1q2_score": 0.8335647434430963
} |
https://pballew.blogspot.com/2022/10/which-platonic-solid-is-most-spherical.html | ## Monday, 24 October 2022
### Which Platonic Solid is Most-Spherical? (and The Archimedian Solids)
If you inscribe a regular polygon into a given circle, the larger the number of sides, the larger the area of the polygon. I guess I always thought that the same would apply to Platonic solids inscribed in a sphere..... It doesn't. I noticed this as I was looking though "The Penny cyclopædia of the Society for the Diffusion of Useful Knowledge
By Society for the Diffusion of Useful Knowledge" (1841). As I browsed the book, I came across the table below:
The table gives features of the Platonic solids when inscribed in a one-unit sphere. At first I thought they must have made a mistake, but not so. The Dodecahedron fills almost 10% more of a sphere (about 66%) than the icosahedron (about 60%). So the Dodecahedron is closer to the sphere than the others.
Interestingly, if you look at the radii of the inscribing spheres, it is clear that solids which are duals are tangent to the same internal sphere.
But if you look at the table of volumes when the solids are inscribed with a sphere inside tangent to each face:
When you put the Platonic solids around a sphere, the one smallest, and thus closest to the sphere is the icosahedron.
This leads to the paradox that when platonic solids are inscribed with a sphere, the icosahedron is closest to the circle in volume (thus most spherical?) but when they are circumscribed by a sphere, the dodecahedron is the closest to the volume of a sphere (and thus most spherical?)... hmmm
Here is a table of the same values when the surface area (superficies) is one square unit.Notice that for a given surface area, the icosahedron has the largest volume, so it is the most efficient "packaging" of the solids (thus more spherical?).
I guess that makes it 2-1 for the icosahedron, so I wasn't completely wrong all along.
POSTSCRIPT::: Allen Knutson's comments on the likely cause of this reversal of "closeness" to the sphere:
I think it's about points of contact. On the inside, the dodecahedron touches the sphere at the most points (20), and on the outside, the icosahedron touches the sphere at the most points (again 20).
Indeed: my recipe would suggest that inside, the 8-vertex cube is bigger than the 6-vertex octahedron, and outside, the 8-face octahedron is smaller than the 6-face cube. Both are borne out by your tables. Thank you, Allen
Some other pertinent comments too good to ignore:
Anonymous said...
I think "most-spherical" would need to be better defined.
Not that I want to do it (and not that I am certain that I could), but perhaps summing the squares of the distances from each point on the surface of the solid to the closest point on the sphere would be the way to go (like a sort of physical variance).
But as I say, I'm not sure I could do this (maybe I can kill some time today, I'm visiting a puzzle-friend), and I'm not sure that it makes a lot of sense.
Oh, oh, and inscribe or circumscribe or find the best match in-between?
(it's play with this, read on-line, or grade. Choices.)
Jonathan
(Love When anonymous signs his comment)
Mary O'Keeffe said...
What a fascinating post! Here is a nice way to think about your first result in terms of the empty space.
The icosahedron leaves ~20% of the space in its circumscribing sphere empty. The dodecahedron leaves ~12% of the space in its circumscribing sphere empty.
That means that if you start with a solid sphere (let's imagine it's something easy to carve, like soap!) and carefully cut away the 20 portions needed to turn it into an icosahedron, each of the 20 pieces you trimmed off will have volume of about 1% of the sphere.
Similarly, if you do the same thing to carve a dodecahedron out of a sphere, each of the 12 pieces you trimmed off will also have volume of about 1% of the sphere.
From now on, whenever I look at either polyhedron (icosa or dodeca) I will always think of it a little differently--because I will think of those ~1% extensions on each face needed to round it out to a sphere.
----------------------------------------------------------------------------------------------------------------------------
Later I tried working on the Archimedian Solids.
### Measuring Sphereocity of The Archimedian Solids
mathworld.wolfram.com
A few years ago I wrote a blog about which of the Platonic solids (above) was most spherical. I compared which ones had the most volume inscribed into a unit sphere, and the which had the smallest volume when circumscribed about a unit sphere. Surprisingly I got different answers to the two methods, and lots of good mathematical comments about why this might be so. Then a while ago (July 2015) I posted it again. As part of a tongue-in-cheek exchange with Adam Spencer @adambspencer I challenged him to find the roundest of the thirteen Archimedean Solids. For those who are not familiar with the distinction, both the Platonic and Archimedean solids are made of up faces that are regular polygons, and both have the property that the view at each vertex is identical to every other, but where the Platonic solids consist of only a single type of regular polygon, (for example the tetrahedron is made up of four equilateral triangles), the Archimedean solids may have more than one (in the cubeoctahedron each vertex is surrounded by two squares and two equilateral triangles).
Then only a day or so later, I got to wondering about the actual answer and started doing some research. Along the way I found a couple of papers on the topic, one of which was published earlier in the same month I began my search for the Platonic Solids Sphere-ness. What was great was that they re-exposed me to a formula for comparing according to George Polya from his Mathematics and Plausible Reasoning: Patterns of plausible inference. I sheepishly admit that I had read this book ( it's on my bookcase nos) several times years ago, but somehow this didn't pop up when I was thinking of the Platonic solids.
The problem of comparing the ratio of the numerical values of the surface area to the volume is that the answer changes over size. In a sphere, for instance, if the radius is one, then the volume is 4 pi/3, and the surface area is 4pi, so the V= 1/3 SA. Now increase the radius to three units and the volume is 36 pi, and the surface area is also 36 pi, Now V = SA, and if we keep making the radius bigger, the volume becomes larger than the surface area, in fact the ratio of Volume to surface area can be reduced to $\frac{V}{SA} = \frac{r}{3}$. Now that kind of thing happens with all the solids, the larger the volume gets, the larger the ratio of V to SA gets. It is one of those things that amazes students (and well it should) that for any solid, there is some scalar multiplication which will transform it into a solid with Volume = Surface Area. In this sense, every solid is isoperimetric (same measure)
So Polya found a way to neutralize this growth. He created an "Isoperimetric Quotient" that served to null out this scalar alteration. By setting the IQ = $\frac{36 \pi V^2}{S^3}$ With this weapon he was able to compare, for instance, the "roundness" of the Platonic Solids. Try this with any sphere and you always get one. Try it with anything else, you always get less than one.
The IQ of the Platonic Solids follows the number of faces, with the tetrahedron at the bottom with an IQ of about .3 and the icosahedron at the top with an IQ of about .8288.
SO what about the Archimedian Solids? Well here they are
TruncatedTetrahedron...... 0.4534
TruncatedOctahedron ....... 0.749
TruncatedCube ............. 0.6056
TruncatedDodecahedron ..... 0.7893
TruncatedIcosahedron ...... 0.9027
Cuboctahedron ............. 0.7412
Icosidodecahedron ......... 0.8601
SnubCube .................. 0.8955
SnubDodecahedron .......... 0.94066
Rhombicuboctahedron ....... 0.8669
TruncatedCuboctahedron .... 0.8186
Rhombicosidodecahedron .... 0.9357
TruncatedIcosidodecahedron. 0.9053
So the roundness winner is the snubdodecahedron, with a pentagon and four equilateral triangles around each vertex. That is four 60o angles and one of 108o for a total of 348o. Students might check if any other of the Archimedean solids can top that. Is that "flatness" at vertices somehow related to "roundness"? I have to admit I first thought it might be the truncated icosahedron. Students may have heard of this one more than others. A molecule of C-60, or a “Buckyball”, consists of 60 carbon atoms arranged at the vertices of a truncated icosahedron. It's roundness is a feature of many of its applications, but it only comes in fourth. | 2023-03-28T17:55:45 | {
"domain": "blogspot.com",
"url": "https://pballew.blogspot.com/2022/10/which-platonic-solid-is-most-spherical.html",
"openwebmath_score": 0.6831908226013184,
"openwebmath_perplexity": 692.4356837325258,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9877587218253718,
"lm_q2_score": 0.8438950966654772,
"lm_q1q2_score": 0.8335647420369904
} |
https://folk.ntnu.no/leifh/teaching/tkt4140/._main029.html |
## 3.1.1 Example: Couette-Poiseuille flow
In a Couette-Poiseuille flow, we consider fluid flow constrained between two walls, where the upper wall is moving at a prescribed velocity $$U_0$$ and at a distance $$Y=L$$ from the bottom wall. Additionally, the flow is subjected to a prescribed pressure gradient $$\frac{\partial p}{\partial x}$$.
Figure 32: An illustration of Couette flow driven by a pressure gradient and a moving upper wall.
We assume that the vertical velocity component $$V=0$$ and consequently we get from continuity: $$\frac{\partial U}{\partial X}=0$$ which implies that $$U=U(Y)$$, i.e. the velocity in the streamwise X-direction depends on the cross-wise Y-direction only.
The equation of motion in the Y-direction simplifies to $$\frac{\partial p}{\partial Y}=-\rho g$$, whereas the equation of motion in the X-direction has the form: $$\begin{equation*} \rho U \cdot \frac{\partial U}{\partial X}=-\frac{\partial p}{\partial X} + \mu \left(\frac{\partial^2U}{\partial X^2}+ \frac{\partial^2U}{\partial Y^2}\right) \end{equation*}$$ which due to the above assumptions and ramifications reduces to $$$$\frac{d^2U}{dY^2} =\frac{1}{\mu}\frac{dp}{dX} \tag{3.20}$$$$
with no-slip boundary conditions: $$U(0)=0,\ U(L)=U_0$$. To render equation (3.20) on a more appropriate and generic form we introduce dimensionless variables: $$u=\frac{U}{U_0},\ y=\frac{Y}{L},\ P=-\frac{1}{U_0}(\frac{dp}{dX})\frac{L^2}{\mu}$$, which yields: $$$$\frac{d^2u}{dy^2}=-P \tag{3.21}$$$$ with corresponding boundary conditions: $$$$u=0 \, \text{for} \, y=0, \qquad u=1 \, \text{for} \, y=1 \tag{3.22}$$$$
An analytical solution of equation (3.21) with the corresponding boundary conditions (3.22) may be found to be: $$$$u=y \cdot \left[ 1+\frac{P}{2}(1-y) \right] \tag{3.23}$$$$
Observe that for $$P\leq -2$$ we will get negative velocities for some values of $$y$$. In Figure 33 velocity profiles are illustrated for a range of non-dimensional pressure gradients P.
Figure 33: Velocity profiles for Couette-Poiseuille flow with various pressure gradients.
To solve (3.21) numerically, we represent is as a system of equations: $$$$\left.\begin{matrix} &u'(y)=u_1(y) \\ &u_1'(y)=-P \end{matrix}\right.\ \tag{3.24}$$$$
with corresponding boundary conditions: $$$$u(0)=0,\ u(1)=1 \tag{3.25}$$$$
To solve this boundary value problem with a shooting method, we must find $$s=u'(0)=u_1(0)$$ such that the boundary condition $$u(1)=1$$ is satisfied. We can express this condition by: \begin{align*} \phi(s)=u(1;s)-1, \qquad \text{ such that } \qquad \phi(s)=0 & \qquad \text{when} \qquad s=s^* \end{align*}
We guess two values $$s^0$$ and $$s^1$$ and compute the correct $$s$$ by linear interpolation due to the linearity of system of ODEs (3.24). For the linear interpolation see equation (3.14).
The shooting method is implemented in the python-code Poiseuille_shoot.py and results are computed and plotted for a range of non-dimensional pressure gradients and along with the analytical solution.
# src-ch2/Couette_Poiseuille_shoot.py;ODEschemes.py @ git@lrhgit/tkt4140/src/src-ch2/ODEschemes.py;
from ODEschemes import euler, heun, rk4
import numpy as np
from matplotlib.pyplot import *
# change some default values to make plots more readable
LNWDT=5; FNT=11
rcParams['lines.linewidth'] = LNWDT; rcParams['font.size'] = FNT
N=200
L = 1.0
y = np.linspace(0,L,N+1)
def f(z, t):
"""RHS for Couette-Posieulle flow"""
zout = np.zeros_like(z)
zout[:] = [z[1], -dpdx]
return zout
def u_a(y,dpdx):
return y*(1.0 + dpdx*(1.0-y)/2.0);
beta=1.0 # Boundary value at y = L
# Guessed values
s=[1.0, 1.5]
z0=np.zeros(2)
dpdx_list=[-5.0, -2.5, -1.0, 0.0, 1.0,2.5, 5.0]
legends=[]
for dpdx in dpdx_list:
phi = []
for svalue in s:
z0[1] = svalue
z = rk4(f, z0, y)
phi.append(z[-1,0] - beta)
# Compute correct initial guess
s_star = (s[0]*phi[1]-s[1]*phi[0])/(phi[1]-phi[0])
z0[1] = s_star
# Solve the initial value problem which is a solution to the boundary value problem
z = rk4(f, z0, y)
plot(z[:,0],y,'-.')
legends.append('rk4: dp='+str(dpdx))
# Plot the analytical solution
plot(u_a(y, dpdx),y,':')
legends.append('exa: dp='+str(dpdx)) | 2021-09-26T10:41:51 | {
"domain": "ntnu.no",
"url": "https://folk.ntnu.no/leifh/teaching/tkt4140/._main029.html",
"openwebmath_score": 0.973693311214447,
"openwebmath_perplexity": 2470.2498717366293,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9783846710189288,
"lm_q2_score": 0.851952809486198,
"lm_q1q2_score": 0.833537569232806
} |
http://mathhelpforum.com/pre-calculus/121778-functions-equations.html | # Math Help - Functions as equations
1. ## Functions as equations
Is sentence "the equation is a function" the shorthand of saying "the equation defines a function"?
2. It's the expression:
$f(x) = some-expression$
$f(x) = x^2$ is, for example, the equation of a quadratic function.
3. Hello mamen
Originally Posted by mamen
Is sentence "the equation is a function" the shorthand of saying "the equation defines a function"?
Yes, I expect so, but I should like to see the whole context before being sure.
It's rather sloppy, though, to say "The equation is a function", because equations and functions are different things. A function, $f$, say, is a relation between the elements of one set (the domain) and the elements of another set (the codomain) which has the property that every element in the domain is related with just one element in the codomain. If $x$ denotes an element in the domain, then $f(x)$ is used to denote the element in the codomain that's related to it.
If we now introduce a second variable, $y$, say, to stand for the element in the codomain that's related to $x$, then we can write the equation:
$y = f(x)$
and say that this equation "defines" the function. But it's not technically correct to say that this equation "is" the function.
Hello mamenYes, I expect so, but I should like to see the whole context before being sure.
It's rather sloppy, though, to say "The equation is a function", because equations and functions are different things. A function, $f$, say, is a relation between the elements of one set (the domain) and the elements of another set (the codomain) which has the property that every element in the domain is related with just one element in the codomain. If $x$ denotes an element in the domain, then $f(x)$ is used to denote the element in the codomain that's related to it.
If we now introduce a second variable, $y$, say, to stand for the element in the codomain that's related to $x$, then we can write the equation:
$y = f(x)$
and say that this equation "defines" the function. But it's not technically correct to say that this equation "is" the function.
Hi sir,
Actually the book just says that the equation f(x)= x+2 is a function.
that's why ,I think, it is the shorthand of saying that the equation f(x) =x+2 defines the function.
thanks sir
5. Hello mamen
Originally Posted by mamen
Hi sir,
Actually the book just says that the equation f(x)= x+2 is a function.
that's why ,I think, it is the shorthand of saying that the equation f(x) =x+2 defines the function.
thanks sir
I agree with you - it does mean the equation $f(x) = x+2$ defines a function.
However, it would also be correct to say that this equation also defines a straight line graph. But an equation and a graph are not the same thing. So it wouldn't technically be correct to say that the equation $f(x) = x+2$ is a straight line, would it?
And, as I said, an equation and a function are not the same thing either. So, personally, I should try to avoid saying that an equation is a function, just as I should avoid saying that an equation is a straight line.
Hello mamenI agree with you - it does mean the equation $f(x) = x+2$ defines a function.
However, it would also be correct to say that this equation also defines a straight line graph. But an equation and a graph are not the same thing. So it wouldn't technically be correct to say that the equation $f(x) = x+2$ is a straight line, would it?
And, as I said, an equation and a function are not the same thing either. So, personally, I should try to avoid saying that an equation is a function, just as I should avoid saying that an equation is a straight line.
Lastly sir,
When we say that a "certain equation defines a function", would it also mean that this equation is used to constitute the function.
The same thing with the statement "the equation also defines a straight line graph." Does this also mean that the equation is used to constitute the straight line graph?
thanks sir!
7. Hello mamen
Originally Posted by mamen
Lastly sir,
When we say that a "certain equation defines a function", would it also mean that this equation is used to constitute the function.
The same thing with the statement "the equation also defines a straight line graph." Does this also mean that the equation is used to constitute the straight line graph?
thanks sir!
My dictionary defines to constitute as "to set up or establish", so I suppose you could say that an equation "constitutes" a function, in exactly the same way as it "defines" a function.
Hello mamenMy dictionary defines to constitute as "to set up or establish", so I suppose you could say that an equation "constitutes" a function, in exactly the same way as it "defines" a function.
Thanks a lot sir,
But what do we mean by "a function expressed or represented by an equation."
Is there such thing as that?
9. Hello mamen
Originally Posted by mamen
Thanks a lot sir,
But what do we mean by "a function expressed or represented by an equation."
Is there such thing as that?
Again, these are different English words - "expressed", "represented" - which have similar meanings to the ones you have already asked about. So, yes, you could say that an equation "expresses" a function, or "represents" a function, in just the same way as an equation "defines" a function.
10. In my humble opinion a lot of math books consider functions to be a relation across the domain and codomain. In other words a function $f:X\mapsto Y$ is really just a set $R\subseteq X\times Y$ with the defining characterstic $R=\left\{(x,y)\in X\times Y:y=f(x)\right\}$ where $f(x)$ is the rule of correspondence.
In my humble opinion a lot of math books consider functions to be a relation across the domain and codomain. In other words a function $f:X\mapsto Y$ is really just a set $R\subseteq X\times Y$ with the defining characterstic $R=\left\{(x,y)\in X\times Y:y=f(x)\right\}$ where $f(x)$ is the rule of correspondence.
It is just worth stressing that, for a given $x$, $y$ is unique. In other words, if $y_1= f(x)$ and $y_2 = f(x)$, then $y_1 = y_2$. This is what distinguishes a function from a more general relation, where we may have two different $y$'s each related to the same $x$. | 2014-07-11T23:33:18 | {
"domain": "mathhelpforum.com",
"url": "http://mathhelpforum.com/pre-calculus/121778-functions-equations.html",
"openwebmath_score": 0.8970763087272644,
"openwebmath_perplexity": 300.6018188718271,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9783846672373523,
"lm_q2_score": 0.8519528057272543,
"lm_q1q2_score": 0.8335375623333884
} |
http://mathhelpforum.com/discrete-math/180785-equiv-class.html | Math Help - Equiv Class
1. Equiv Class
My question is one aspect of the question below.
I have already shown that the relation is reflexive, symmetric, and transitive. What is meant by, and how do I describe the corresponding partition on $\mathbb{Z}$?
a ~ b if a-b is divisible by 3.
Thank you.
2. How did you demonstrate te properties without the ability to describe the relation?
Maybe 3|(a-b) or $(a-b) \equiv 0\;(mod\;3)$
Not too sure what you're looking for.
3. Originally Posted by TKHunny
How did you demonstrate te properties without the ability to describe the relation?
Maybe 3|(a-b) or $(a-b) \equiv 0\;(mod\;3)$
Not too sure what you're looking for.
The books has $3\mathbb{Z}, \ 1+3\mathbb{Z}, \ 2+3\mathbb{Z}$
Not to sure about that above notation.
4. well, for a given integer, let's call it k, let us find out what [k], the equivalence class of k under ~ is.
we have that if a is in [k], then k - a = 3m, for some integer m (k-a is divisible by 3).
so the possible values for a, are k, k-3, k+3, k-6, k+6, k-9, k+9, etc. if we choose the smallest positive integer of [k], to represent [k],
we see that [k] = [0], [1], or [2] (depending on whether a is of the form 3n, 3n+1, or 3n+2).
these 3 equivalence classes are disjoint, they share no common members, and Z = [0] U [1] U [2].
as we saw above, [0] = 0,-3,3,-6,6,-9,9, etc. this is the same set as 3Z = {3k : k in Z}.
similarly, [1] = 1,-2,4,-5,7,-8,10, etc. which is the same set as 1+3Z = {3k+1 : k in Z},
and [2] = 2,-1,5,-4,8,-7,11, etc. which is the same set as 2+3Z = {3k+2 : k in Z}.
this is typical of any equivalence relation, the equivalence classes form a partition of the set the equivalence is defined on.
5. Originally Posted by dwsmith
The books has $3\mathbb{Z}, \ 1+3\mathbb{Z}, \ 2+3\mathbb{Z}$
Not to sure about that above notation.
That's fine. It's not always clear what notation is wanted. Getting used to your text or professor is a good idea. I was defining the "0, 1, and 2", and suggesting how to determine the class. The notation you presented actually names the classes and includes all values in the class. It should bother you for 3-5 milliseconds how we used to have a set of all integers and now we have three sets of all integers but we don't have any more integers than we had before. | 2016-07-26T07:46:22 | {
"domain": "mathhelpforum.com",
"url": "http://mathhelpforum.com/discrete-math/180785-equiv-class.html",
"openwebmath_score": 0.5366793870925903,
"openwebmath_perplexity": 812.0087054854569,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.978384666922221,
"lm_q2_score": 0.8519528057272544,
"lm_q1q2_score": 0.8335375620649115
} |
https://math.stackexchange.com/questions/163245/finding-sum-of-factors-of-a-number-using-prime-factorization/163246 | # Finding sum of factors of a number using prime factorization
Given a number, there is an algorithm described here to find it's sum and number of factors. For example, let us take the number $1225$ :
It's factors are $1, 5, 7, 25, 35, 49, 175, 245, 1225$ and the sum of factors are $1767$.
A simple algorithm that is described to find the sum of the factors is using prime factorization.
$1225 = 5^2 \cdot 7^2$, therefore the sum of factors is $(1+5+25)(1+7+49) = 1767$
But this logic does not work for the number $2450$. Please check if it's working for $2450$
Edit : Sorry it works for $2450$. I made some mistake in calculation.
• $1767=3\cdot 19\cdot 31$ and $5^27^2=1225$ Jun 26, 2012 at 9:59
• $2450=2.5^2.7^2$.therefore,$Sum=(1+2)(1+5+25)(1+7+49)=5301$.Isn't it the right sum?(Because i know the formula is correct).
– Aang
Jun 26, 2012 at 10:03
Your approach works fine: $$2450=2\cdot 5^2\cdot 7^2$$, therefore the sum of divisor is $$(1+2)(1+5+25)(1+7+49)=5301=3\cdot 1767.$$
You are looking for the Formula For Sum Of Divisors, from there:
Each of these sums is a geometric series; hence we may use the formula for sum of a geometric series to conclude $$\sum_{d|n}d = \prod_{i=1}^k \frac{p_i^{m_i+1}-1}{p_i-1}$$
Solving for $1225 = 5^2 × 7^2$
Sum of divisors = $(a^{p+1} – 1)/(a – 1) × (b^{q+1} – 1)/(b – 1)$
Here $a = 5$, $b = 7$ prime factors
$p = 2$ and $q = 2$
Sum of divisors = $(5^3 – 1)/(5 – 1) × (7^3 – 1)/(7 – 1)$
= $(124/4) × (342/6)$
= $(31 × 57)$
= $1767$
$2450=2\cdot5^2\cdot7^2$, so the algorithm gives
$$(1+2)(1+5+25)(1+7+49)=3\cdot31\cdot57=5301\;.$$
The divisors of $2450$ are $1,2,5,7,10,14,25,35,49,50,70,98,175,245,350,490,1225$, and $2450$, whose sum is indeed $5301$.
It’s not difficult to prove that the algorithm works, so if you thought that it failed, you made a mistake somewhere in your calculations; my guess would be that you missed a divisor of $2450$.
If a number $N=(a^x)(b^y)(c^z)$, where $a,b$ and $c$ are prime numbers, sum of it's factor is $S= [(a^{x+1}-1)/(a-1)][(b^{y+1}-1)/(b-1)][(c^{z+1}-1)/(c-1)]$
Eg. $12=[(2^{2+1}-1)/(2-1)][{(3^{1+1}-1)}/(3-1)]=28$
Sum of factors of $2450$
The Factors are $2, 5^2, 7^2$.
Sum of the factors $= ( 2^0 + 2^1 ) × ( 5^0 + 5^1+ 5^2)×( 7^0 + 7^1+ 7^2) = 5301$ | 2022-08-11T02:41:05 | {
"domain": "stackexchange.com",
"url": "https://math.stackexchange.com/questions/163245/finding-sum-of-factors-of-a-number-using-prime-factorization/163246",
"openwebmath_score": 0.9335412383079529,
"openwebmath_perplexity": 232.098128357868,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9783846666070894,
"lm_q2_score": 0.8519528057272544,
"lm_q1q2_score": 0.8335375617964342
} |
https://math.stackexchange.com/questions/1933041/question-about-the-sum-of-four-squares | # Question about the sum of four squares
Let's say I have four integers such that:
$a^2+b^2+c^2+d^2=23$
How many of them are even ?
a) $0$
b) $1$
c) $2$
d) $3$
e) $4$
I know that an even number plus or times an even number is even, an odd plus an odd is even too, an odd times an even is even while an odd times an odd is odd as an even plus an odd is.
Being $23$ odd to get and odd sum I need to have $1$ or $3$ even numbers so I think both b and d are right, however you can only choose one.
A friend of mine says that the only solution (not counting permutations) is: $a=b=3; c=2; d=1$ and so the answer is b. However how can I check that this is the only one ? I can do this by hand in this case however if instead of $23$ we have $128273917$ this would be pretty troublesome.
Since I'm not interested in the particular result but the general procedure, how can I find solutions to (and check how many others there are) to : $a^2+b^2+c^2+d^2=k$ ? Can I determine how many of $a; b; c; d$ are even/odd without actually compute such solutions ?
Note that the question doesn't specify if we are working with positive or negative integers or both, this is not big deal considering their squares however involving or not $0$ could change something I guess, in the same way one could not consider $0$ as even and $1$ as odd as they are "too trivial".
• Hint: for all integers $n$, we have $n^2\equiv 0$ or $1 \bmod 4$. – TonyK Sep 19 '16 at 16:29
• You have used sound logic to narrow it down to 1 or 3. Suppose the answer is 3. How many combinations of 3 even integers are there before their sum is greater than 24. The set is small, and you can brute force your way though quickly enough. – Doug M Sep 19 '16 at 16:43
An integer square is always $\equiv 0,1\pmod{4}$. Since $a^2+b^2+c^2+d^2=23\equiv 3\pmod{4}$, exactly one integer among $\{a,b,c,d\}$ is even. We may deduce the same from the fact that $23$ has a unique representation as a sum of four squares up to the ordering of terms: $$23 = 1^2+2^2+3^2+3^2.$$ | 2021-04-13T14:46:04 | {
"domain": "stackexchange.com",
"url": "https://math.stackexchange.com/questions/1933041/question-about-the-sum-of-four-squares",
"openwebmath_score": 0.8200670480728149,
"openwebmath_perplexity": 108.43546373501887,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9783846678676151,
"lm_q2_score": 0.8519528038477825,
"lm_q1q2_score": 0.8335375610314961
} |
https://mathematica.stackexchange.com/questions/84593/need-help-on-the-proper-use-of-a-formula-to-produce-a-cambered-airfoil | # Need help on the proper use of a formula to produce a cambered airfoil
Following the book on "The Theory of Thin Wing Sections" page #112 describes the method to combine a camber line (i.e Mean Line) and a thickness distribution to form a cambered wing section. Data derived from the function below will produce a symmetrical airfoil with no camber for the value is a percentage of chord.
yt[x_] :=
Module[{t = .30,c = 1},
(5*t*c*(.2969*Sqrt[x/c] + (-.1260)*(x/c) + (-.3516)*(x/c)^2 + .2843*(x/c)^3 +
(-.1015)*(x/c)^4))*1/2]
Plot[{yt[x], yt[x] - (yt[x]*2), yc[x]*.5}, {x, 0, 1}, AspectRatio -> .15]
The function to derive the camber line (Mean Line) as a % of chord is listed below.
m = max thickness
p = position of max thickness in percentage of chord
(* forward of maximum ordinate *)
yc[x_] :=
Module[{m = 0.02, p = .4}, (m (2 p x - x^2))/p^2]
(* aft of maximum ordinate *)
yc1[x_] :=
Module[{m = 0.02, p = .4}, (m (1 - 2 p + 2 p x - x^2))/(1 - p)^2]
Show[Plot[yc[x], {x, 0, .4}], Plot[yc1[x], {x, .4, 1}],
PlotRange -> Automatic, AspectRatio -> .05]
Reproducing part of the PDF found on another site, a shorter version is found here, The root of my pain is I don't think I'm using the right trig function. If I were to use the formula $x-y\,t \sin(\theta)$, and I tried, but I didn't come close to the answer. How would I use $x-y\,t \sin(\theta)$ in Mathematica?
Below is a sample found in a PDF from "The Theory of Thin Wing Sections"
• Maybe its me, but I can't figure out what is being asked here... what do you mean by how to use "x -yt sin(theta)" in Mathematica? Use it to do what? and where exactly do you want to use it? May 28 '15 at 2:47
I have to thank you for leading me down a fascinating rabbit hole! I have always found airfoil shapes enormously pleasing, but I was not aware of the NACA system.
For those who, like me, did not know much about the story, NACA was the predecessor of NASA. They established some of the first airfoil definitions codified through mathematical relationships. For instance, a so called four-digit NACA airfoil such as "NACA 2415" is defined as follows:
• The length of an airfoil is called its chord. Here we use a parametric length of $1$, which can just be adjusted to any length necessary.
• The first digit in the NACA designation indicates the maximum camber $m$ (curvature) as a percentage of the cord (length): in this example 2%;
• The second digit indicates the position $p$ of the point of maximum curvature, in tenths of the chord measuring from the leading edge, i.e. the front; here it is at 4 tenths of the chords, i.e. 40% back;
• The last two digits indicate the maximum thickness $t$ of the airfoil as a percentage of the chord; here 15%.
These definitions come with standard formulae (linked in the OP's question) that allow you to calculate the shape of the airfoil. These functions are more conveniently defined using Piecewise below (here is its documentation), rather than having the two separate definitions used by the OP.
m = 0.02; p = 0.40; t = 0.15; (* characteristics of NACA 2415*)
yc[x_] := Piecewise[{
{m/p^2 (2 p x - x^2), 0 <= x < p},
{m/(1 - p)^2 ((1 - 2 p) + 2 p x - x^2), p <= x <= 1}
}]
yt[x_] := 5 t (0.2969 Sqrt[x] - 0.1260 x - 0.3516 x^2 + 0.2843 x^3 - 0.1015 x^4)
The final cambered profile is expressed using trigonometric functions of an angle $\theta=\arctan \frac{d{y_c}}{d{x}}$, shown below as the pre-calculated derivative. Mathematica will calculate this for you using D[yc[x], x].
theta[x_] := ArcTan@ Piecewise[{
{(m*(2*p - 2*x))/p^2, 0 <= x < p},
{(m*(2*p - 2*x))/(1 - p)^2, p <= x <= 1}
}]
Now we have all the machinery in place to plot the shape of the cambered airfoil.
In the NACA model the contour of the airfoil is expressed as a function of one parameter, i.e. the distance $x$ along the chord, so ParametricPlot (docs) is a natural fit here:
ParametricPlot[
{
{x - yt[x] Sin[theta[x]], yc[x] + yt[x] Cos[theta[x]]},
{x + yt[x] Sin[theta[x]], yc[x] - yt[x] Cos[theta[x]]}
},
{x, 0, 1}, ImageSize -> Large, Exclusions -> None
]
• Nice. Makes me remember when I wanted to be an aerospace engineer... Anyhoo, you can get rid of those breaks in your plot by using Exclusions->None. (+1)
– kale
May 28 '15 at 13:47
• @kale That's a good point: let me add that to the answer and fix the plot. Thank you for the upvote as well! May 28 '15 at 13:56
• @MarcoB I have been tinkering with this for a long time. With little forward progress and learning MMA, I could not have found the solution without your help. I have a better understanding on a number of fronts. Thank you: Bob Brooks May 28 '15 at 15:16
• @MarcoB: I added {x,yc[x]} to see the mean line, your plot works nice. May 28 '15 at 15:40
• @BobBrooks I'm delighted! As I said, thank you again for bringing up the problem, it was fun to delve into the airfoil shapes & equations. I was thinking about adding the chord and the mean lines to the plot as well: you don't need my help with that since you have it figured out already, but if I have the time I might add it to the answer anyway, just for the sake of completeness. May 28 '15 at 16:07 | 2021-09-26T13:53:47 | {
"domain": "stackexchange.com",
"url": "https://mathematica.stackexchange.com/questions/84593/need-help-on-the-proper-use-of-a-formula-to-produce-a-cambered-airfoil",
"openwebmath_score": 0.5994298458099365,
"openwebmath_perplexity": 1309.0475712396365,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.978384664716301,
"lm_q2_score": 0.8519528057272544,
"lm_q1q2_score": 0.8335375601855718
} |
https://www.physicsforums.com/threads/double-integral-of-e-y.303065/ | # Double Integral of e^-y
1. Mar 27, 2009
### cse63146
1. The problem statement, all variables and given/known data
$$\int^x_0\int^y_x e^{-v} dv du$$
where u and v are just "dummy variables"
2. Relevant equations
3. The attempt at a solution
$$\int^x_0\int^y_x e^{-v} dv du = \int^x_0 -e^{-y} + e^{-x} du = (-e^{-y} + e^{-x})x$$
have I made a mistake somewhere?
2. Mar 27, 2009
### djeitnstine
Everything looks fine don't forget your integration constant after ;)
3. Mar 27, 2009
### cse63146
It's for determining a CDF. I don't think I need the constant.
A classmate got x(1 - e-y). Did they use somesort of trick to get it, or is their answer just wrong?
4. Mar 27, 2009
### djeitnstine
That's very wrong. First of all there has to be at least 2 exponentials, perhaps they just forgot their $$e^{-x}$$
5. Mar 27, 2009
### cse63146
Well, that cleared that up.
Thanks.
6. Mar 27, 2009
### cse63146
Got another question.
$$\int^x_0\int^y_x e^{-v} dv du$$ This case is when 0<x<y
If I want to do 0 < y < x, would the double integral be set up in the following way:
$$\int^x_0\int^x_y e^{-v} dv du$$ ?
7. Mar 27, 2009
### djeitnstine
Hmm before I go into any discussion, are you integrating backwards?? Or are you trying to change the order of integration? If you are changing the order I would say this:
You're changing the order of integration. Lets say your limits are $$y=x^2$$ from 0 < x 6 bounded by y=0 and you are doing the dy dx order or integration then you would have to do
$$\int^{6}_{0} \int^{x^2}_{0} dy dx$$ (from top to bottom, left to right)
Now say you want to change the order, that is, from dx dy, you would have to write y in terms of x and do your integration that way, i.e.
$$\int^{36}_{0} \int^{6}_{\sqrt{x}} dx dy$$ (from left to right, bottom to top)
if not just ignore it and let me know what you mean
8. Mar 27, 2009
### cse63146
I'm trying to find the CDF of e^-y.
The cdf of any f(x,y) is $$\int^x_{-\infty} \int^y_{-\infty} f(u,v) dv du$$
where u and v are just dummy variables. I need to consider 2 cases: 1) 0<x<y and 2) 0 < y < x. In other words I need to do 2 double integrals using the general set up of a CDF. | 2017-08-24T11:59:28 | {
"domain": "physicsforums.com",
"url": "https://www.physicsforums.com/threads/double-integral-of-e-y.303065/",
"openwebmath_score": 0.7750237584114075,
"openwebmath_perplexity": 863.8611115645575,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.978384668497878,
"lm_q2_score": 0.8519528019683106,
"lm_q1q2_score": 0.8335375597296039
} |
https://fr.mathworks.com/help/signal/ug/bias-and-variability-in-the-periodogram.html | Documentation
## Bias and Variability in the Periodogram
This example shows how to reduce bias and variability in the periodogram. Using a window can reduce the bias in the periodogram, and using windows with averaging can reduce variability.
Use wide-sense stationary autoregressive (AR) processes to show the effects of bias and variability in the periodogram. AR processes present a convenient model because their PSDs have closed-form expressions. Create an AR(2) model of the following form:
$y\left(n\right)-0.75y\left(n-1\right)+0.5y\left(n-2\right)=\epsilon \left(n\right),$
where $\epsilon \left(n\right)$ is a zero mean white noise sequence with some specified variance. In this example, assume the variance and the sampling period to be 1. To simulate the preceding AR(2) process, create an all-pole (IIR) filter. View the filter's magnitude response.
B2 = 1;
A2 = [1 -0.75 0.5];
fvtool(B2,A2)
This process is bandpass. The dynamic range of the PSD is approximately 14.5 dB, as you can determine with the following code.
[H2,W2] = freqz(B2,A2,1e3,1);
dr2 = max(20*log10(abs(H2)))-min(20*log10(abs(H2)))
dr2 = 14.4984
By examining the placement of the poles, you see that this AR(2) process is stable. The two poles are inside the unit circle.
fvtool(B2,A2,'Analysis','polezero')
Next, create an AR(4) process described by the following equation:
$y\left(n\right)-2.7607y\left(n-1\right)+3.8106y\left(n-2\right)-2.6535y\left(n-3\right)+0.9238y\left(n-4\right)=\epsilon \left(n\right).$
Use the following code to view the magnitude response of this IIR system.
B4 = 1;
A4 = [1 -2.7607 3.8106 -2.6535 0.9238];
fvtool(B4,A4)
Examining the placement of the poles, you can see this AR(4) process is also stable. The four poles are inside the unit circle.
fvtool(B4,A4,'Analysis','polezero')
The dynamic range of this PSD is approximately 65 dB, much larger than the AR(2) model.
[H4,W4] = freqz(B4,A4,1e3,1);
dr4 = max(20*log10(abs(H4)))-min(20*log10(abs(H4)))
dr4 = 64.6213
To simulate realizations from these AR(p) processes, use randn and filter. Set the random number generator to the default settings to produce repeatable results. Plot the realizations.
rng default
x = randn(1e3,1);
y2 = filter(B2,A2,x);
y4 = filter(B4,A4,x);
subplot(2,1,1)
plot(y2)
title('AR(2) Process')
xlabel('Time')
subplot(2,1,2)
plot(y4)
title('AR(4) Process')
xlabel('Time')
Compute and plot the periodograms of the AR(2) and AR(4) realizations. Compare the results against the true PSD. Note that periodogram converts the frequencies to millihertz for plotting.
Fs = 1;
NFFT = length(y2);
subplot(2,1,1)
periodogram(y2,rectwin(NFFT),NFFT,Fs)
hold on
plot(1000*W2,20*log10(abs(H2)),'r','linewidth',2)
title('AR(2) PSD and Periodogram')
subplot(2,1,2)
periodogram(y4,rectwin(NFFT),NFFT,Fs)
hold on
plot(1000*W4,20*log10(abs(H4)),'r','linewidth',2)
title('AR(4) PSD and Periodogram')
text(350,20,'\downarrow Bias')
In the case of the AR(2) process, the periodogram estimate follows the shape of the true PSD but exhibits considerable variability. This is due to the low degrees of freedom. The pronounced negative deflections (in dB) in the periodogram are explained by taking the log of a chi-square random variable with two degrees of freedom.
In the case of the AR(4) process, the periodogram follows the shape of the true PSD at low frequencies but deviates from the PSD in the high frequencies. This is the effect of the convolution with Fejer's kernel. The large dynamic range of the AR(4) process compared to the AR(2) process is what makes the bias more pronounced.
Mitigate the bias demonstrated in the AR(4) process by using a taper, or window. In this example, use a Hamming window to taper the AR(4) realization before obtaining the periodogram.
figure
periodogram(y4,hamming(length(y4)),NFFT,Fs)
hold on
plot(1000*W4,20*log10(abs(H4)),'r','linewidth',2)
title('AR(4) PSD and Periodogram with Hamming Window')
legend('Periodogram','AR(4) PSD')
Note that the periodogram estimate now follows the true AR(4) PSD over the entire Nyquist frequency range. The periodogram estimates still only have two degrees of freedom so the use of a window does not reduce the variability of periodogram, but it does address bias.
In nonparametric spectral estimation, two methods for increasing the degrees of freedom and reducing the variability of the periodogram are Welch's overlapped segment averaging and multitaper spectral estimation.
Obtain a multitaper estimate of the AR(4) time series using a time half bandwidth product of 3.5. Plot the result.
NW = 3.5;
figure
pmtm(y4,NW,NFFT,Fs)
hold on
plot(1000*W4,20*log10(abs(H4)),'r','linewidth',2)
legend('Multitaper Estimate','AR(4) PSD')
The multitaper method produces a PSD estimate with significantly less variability than the periodogram. Because the multitaper method also uses windows, you see that the bias of the periodogram is also addressed. | 2020-01-26T08:52:11 | {
"domain": "mathworks.com",
"url": "https://fr.mathworks.com/help/signal/ug/bias-and-variability-in-the-periodogram.html",
"openwebmath_score": 0.710810661315918,
"openwebmath_perplexity": 1334.5909642881695,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9783846716491915,
"lm_q2_score": 0.8519527982093666,
"lm_q1q2_score": 0.8335375587366811
} |
https://math.stackexchange.com/questions/21796/finding-int-0-pi-2-sin-x-dx | # Finding $\int_0^{\pi/2} \sin x\,dx$
I'm interested in why $$\int_0^{\pi/2} \sin x\,dx = 1.$$ I know how to do the integral the conventional way but am more interested in what makes radians special for this problem. If we instead compute $$\int_{0}^{90} \sin x^\circ\,dx,$$ we won't get $1$ as the answer.
What about the definition of radians makes this integral evaluate to $1$? I'm looking for an intuitive (presumably geometric) explanation.
• If you measure in degrees instead of radians, the derivative of $\cos(x)$ is not $-\sin(x)$; you're off by a constant (think the graph: instead of having a period of $2\pi$, it has a period of $360$, almost 60 times longer). So $\cos(x)$ is no longer an antiderivative of $\sin(x)$. If you do the integral with the correct antiderivative, you wll get $1$. – Arturo Magidin Feb 13 '11 at 5:52
• Mathematica says (and I'm inclined to agree) that $\int_0^{\pi/2} \sin x\,dx =\int_{0^\circ}^{90^\circ} \sin x\,dx= 1$, whereas $\int_{0}^{90} \sin x^\circ\,dx=\frac{1}{1^\circ}=\frac{180}{\pi}$ (though this is more of an issue of notation than actual substance). – Isaac Feb 13 '11 at 6:05
• @Isaac, you're right that that does make more sense. I'll change my question (though I'm not sure it was particularly unclear originally). – Sophie Alpert Feb 13 '11 at 19:58
• @Arturo: Yes, but why is the derivative of $\cos(x)$ equal to $-\sin(x)$ when $x$ is measured in radians? – Sophie Alpert Feb 13 '11 at 19:59
• @Ben: Because $$\lim_{x\to 0}\frac{\sin x}{x} = 1$$when $x$ is measured in radians. Basically, because radians give the arclength parametrization of the unit circle (one radian yields one unit of arc length), and degrees do not. Radians are the "normalized" way of measuring angles, so that pesky proportionality constants all become $1$ when you use radians. – Arturo Magidin Feb 13 '11 at 20:02
I have had a similar question from my friend. He was surprised to find out $\displaystyle \lim_{x^{\circ} \rightarrow 0} \frac{\sin(x^{\circ})}{x^{\circ}} = \frac{\pi}{180}$.
The reason is that we are actually overloading the function name $\sin$. I think the confusion would be subsided if you were to look at $\sin_r(x)$ and $\sin_d(x)$ as two different functions where in the first function you input $x$ in radians and in the second function you input $x$ in degrees and these two are related by $\sin_r(x^r) = \sin_d(y^{\circ})$, where $y^{\circ} =\frac{180}{\pi}x^r$
We know that $\displaystyle \int_{0}^{\pi/2} \sin_r(x^r) dx^r = 1$.
Hence, $\displaystyle \int_{0}^{\pi/2} \sin_d(y^{\circ}) dx^r = 1$, since $\sin_r(x^r) = \sin_d(y^{\circ})$
We have $(\frac{180}{\pi}x^r) = y^{\circ}$. Hence $(\frac{180}{\pi}dx^r) = dy^{\circ}$.
As $x^r$ goes from $0$ to $\pi/2$, $y^{\circ}$ goes from $0$ to $90$.
Hence, we now have $\displaystyle \int_{0}^{90} \sin_d(y^{\circ}) \frac{\pi}{180} dy^{\circ} = 1$.
Hence, $\displaystyle \int_{0}^{90} \sin_d(y^{\circ}) dy^{\circ} = \frac{180}{\pi}$.
EDIT
The question is
"Why is $\int_{0}^{\pi/2} \sin(\theta) d\theta = 1$ when $\theta$ is in radians?"
What follows is my attempt for a purely geometric argument for this.
The geometric argument ultimately hinges (as one would expect) on
"The length of an arc of a circle subtending an angle $\theta$ at the center is $r \theta$ where $\theta$ is in radians"
This essentially comes from the way a radian is defined.
The main crux of the problem is the question
"Why is the average value of $\sin$ over quarter its period is $\frac{2}{\pi}$?"
(Note when we talk of average value of $\sin$ over quarter its period, there is no difference between radians or degrees).
It is easy to see that the average value of $\sin$ over quarter its period lies between $0$ and $1$.
The average value of of $\sin$ over quarter its period is nothing but the average value, with respect to $\theta$, of the height of the line segment as the line segment moves from the right end point of the circle towards the origin such that it is always perpendicular to the $X$ axis, with one end point of the segment on the $X$ axis and the other end point of the segment on the circumference of the circle.
Now here comes the claim. The average value, with respect to $\theta$, of the height of the line segment times the circumference of the quarter of the circle is the area of a unit square.
Let $h$ denote the average value, with respect to $\theta$, of the height of the line segment i.e. $$\displaystyle h = \frac{\int_{0}^{\pi/2} y d \theta}{\int_{0}^{\pi/2} d \theta}$$
The claim is
$$\displaystyle h \times \frac{\pi}{2} = 1$$
Proof:
First note that the area of the square with vertices at $(0,0),(0,1),(1,0),(1,1)$ is obviously $1$.
$\displaystyle \text{Area of square} = 1 = \int_{0}^{1} 1 \times dx = \int_{0}^{1} dx$.
For points on the circle, $\displaystyle x^2 + y^2 = 1 \Rightarrow xdx + ydy = 0 \Rightarrow dy = -\frac{x}{y} dx$.
Further, for the small triangle, we have
$$\displaystyle (dx)^2 + (dy)^2 = (1 \times d \theta)^2$$ $$\displaystyle (dx)^2 + \frac{x^2}{y^2}(dx)^2 = (d \theta)^2$$ $$\displaystyle \frac{x^2 + y^2}{y^2}(dx)^2 = (d \theta)^2$$ $$\displaystyle (dx)^2 = y^2 (d \theta)^2$$
Note that as $\theta$ increases from $0$ to $\frac{\pi}{2}$, $x$ decreases from $1$ to $0$ and hence
$$\displaystyle dx = -y d \theta$$
Hence, $$\displaystyle \text{Area of square} = 1 = \int_{0}^{1} 1 \times dx = \int_{0}^{1} dx = \int_{\pi/2}^0 (-y) d \theta = \int_0^{\pi/2} y d \theta$$
Hence, $\displaystyle 1 = \int_0^{\pi/2} y d \theta = \frac{\int_0^{\pi/2} y d \theta}{\int_0^{\pi/2} d \theta} \times \int_0^{\pi/2} d \theta$
Note that $\displaystyle \int_0^{\pi/2} d \theta = \frac{\pi}{2}$ which is the circumference of the quadrant and $\displaystyle h = \frac{\int_0^{\pi/2} y d \theta}{\int_0^{\pi/2} d \theta}$ is the average value, with respect to $\theta$, of the height of the line segment.
Hence, we get $\displaystyle h \times \frac{\pi}{2} = 1$ i.e. average value, with respect to $\theta$, of the height of the line segment if $\frac{2}{\pi}$.
Hence, we get that $$\displaystyle \int_{0}^{\pi/2} \sin(\theta) d \theta = 1$$ when $\theta$ is in radians
• @Ben: Good Question. To think of it in a possibly geometric setting, we will make a very crude approximation. The average value of the sin function over half a period is $\approx \frac{1}{2}$ (half of the extreme values $0$ and $1$). Now if you evaluate the integral in radians from $0$ t0 $\frac{\pi}{2}$, you will get $\frac{1}{2} \frac{\pi}{2} = \frac{\pi}{4} \approx 0.785$. Now if you were to evaluate the integral in degrees from $0$ to $90$, you will get $\frac{1}{2} 90 = 45$. The exact values of the integrals are $1$ and $\frac{180}{\pi} \approx 57$ respectively. – user17762 Feb 13 '11 at 6:09
• So you answered why the integrals give different answers scaled by a the ratio between a degree and a radian, but I already knew this. My question was why radians are special and give the answer of 1 (which you skip entirely when you say, "We know that…"). Sorry if my question was unclear. – Sophie Alpert Feb 13 '11 at 18:48
• @Ben: I have attempted a geometric interpretation. Kindly let me know what you think. – user17762 Feb 13 '11 at 22:33
• $\lim_{x^\circ \rightarrow 0} \frac{\sin(x^\circ )}{x^\circ}$ is most certainly $1$, by any standard. Whatever your interpretation of $x^\circ$ is, it must be as a real variable, and $\lim_{y\rightarrow 0} \frac{\sin y}{y}=1$ for any real variable $y$. Typically (e.g. by Mathematica), $x^\circ$ is interpreted via $x^\circ=x\times\frac{\pi}{180}$. Perhaps you meant to write $\lim_{x\rightarrow 0} \frac{\sin (x^\circ )}{x} = \frac{\pi}{180}$ or $\lim_{x^\circ\rightarrow 0} \frac{\sin (x^\circ )}{x} = \frac{\pi}{180}$? Either would be correct. What you have is not. – Marcel Besixdouze Nov 16 '14 at 4:48
Think of the definite integral as giving displacement, and use the fact that the key feature of radians as an angle measure is that the arc length of the unit circle subtended by an angle of radian measure $x$ is $x$ units.
More specifically, imagine you are traveling along the unit circle at a speed of 1 rad/s starting at (-1,0) and moving clockwise. Since you are in radians, this means that you are traveling 1 m/s (to pick a distance unit) on the circle. When you have traveled $x$ units, your position vector is $(-\cos x, \sin x)$. The velocity vector must be tangent to the circle and thus is perpendicular to the position vector. Thus the velocity vector is in the direction of $(\sin x, \cos x)$. Because the speed is 1 m/s, the velocity vector actually is $(\sin x, \cos x)$ and not some other scalar multiple of it. Thus $\int_0^{\pi/2} \sin x dx$ when $x$ is in radians is just the horizontal displacement on the unit circle when traveling $\pi/2$ radians along the circle's perimeter from (-1,0) to (0,1). Since the horizontal displacement is 1, $\int_0^{\pi/2} \sin x dx = 1$.
Note the effect of radians in making the speed 1 m/s. If you were traveling at 1 degree/s, the speed along the circle would be $\frac{\pi}{180}$ m/s, and you would have $\int_0^{90} \sin x \frac{\pi}{180} dx = 1$ as in Sivaram's answer.
Based on your comments in response to answers, you seem to be asking rather why radians are "special" in giving the answer $1$.
This really comes down to why is it that when we use radians, we have $$\lim_{x\to 0}\frac{\sin x}{x} = 1$$ because it comes down to why $(\cos x)' = -\sin x$ and not some other constant multiple of $\sin x$.
Think about the unit circle, parametrized by \begin{align*} x&=\cos t,\\ y&=\sin t \end{align*} (in whatever units you want to pick for $t$: radians, gradians, degrees, anything, doesn't matter here).
Whenever you parametrize a curve, you can change the parameter so that you traverse the curve slower, or faster, or in a different direction, or in a different way; here, if you replace $t$ with $-t$, you reparametrize the unit circle so that it is traversed clockwise, instead of counterclockwise. If you change $t$ to $2t$, you traverse it twice as fast but in the same direction; if you replace $t$ with $t+17$, then you start somewhere else; etc.
There is, however, what one might call a "normalized" parametrization, which normalizes the speed at which we traverse the curve. It is called the "arc-length parametrization", and the objective is to choose the parameter $t$ so that, if $t$ varies from $a$ to $b$, then the length of curve that is traversed is exactly $b-a$; that is, it takes exactly one unit of "time" to traverse one unit of "distance".
The arc-length parametrization has a number of advantages. For example, if you compute the tangent vector by taking $(x'(t),y'(t))$, in the arc length parametrization of a curve the tangent vector always has unit length. It also allows you to find the curvature of the curve much easier than with a run-of-the-mill parametrization.
Well, what is the arc-length parametrization of the unit circle? It is exactly the one in which it takes from $t=0$ to $t=2\pi$ to traverse the entire circle once; it is precisely the one in which an angle measured by $t$ cuts a circle arc of length $t$. It is, in short, measuring $t$ in radians. Radians give the arclength parametrization of the curve, which is why there are no pesky constants of proportionality when we take derivatives of the coordinate functions $\sin t$ and $\cos t$.
Now, no matter how you measure your angles, there will be some value of $b$ such that $$\int_0^b \sin_x t\,dt = 1.$$ (where by $\sin_x t$ I mean "sine, measuring the argument in $x$s"). The only question is what is the value of $b$; it exists for degrees, for radians, for gradians, for half-degrees, anything, really. Why does it turn out to be exactly $\pi/2$ for radians? Because when we do radians, which give the arc length parametrization, the lack of proportionality constants yield that the integral of the sine is the negative of the total change in the $x$ coordinate (under other parametrizations, it would be some constant multiple of this). If you start at $t=0$, the point $(1,0)$, when has the $x$-coordinate changed by exactly one unit? When we get to $(0,1)$, that is, when $t=\pi/2$.
This answer is similar to the answer of Marvis, but I think that it is conceptually a bit cleaner.
The $\sin(\theta)$ is defined as the y-coordinate of the point on the unit circle obtained by starting at the origin and moving a distance of $\theta$ along the unit circle in the counterclockwise direction.
So lets investigate what $\displaystyle \int_0^{\pi/2}\sin(\theta)$ would look like geometrically.
We are taking the interval $[0,\frac{\pi}{2}]$ and partitioning it into a million tiny pieces. On the unit circle, we have just broken the quarter circle in the first quadrant up into a million pieces. Then we need to multiply the lengths of each of these little segments by the height of the left endpoint of that line segment, and sum all of these values together. We need a geometric understanding of what that product looks like, i.e. how can we interpret $\sin(\theta)\Delta\theta$ geometrically?
Since we are looking at such a short segment of the circle, $\Delta\theta$ looks a lot a segment of the tangent line to the circle at that point. By geometry, the tangent line to the circle is perpendicular to the radius, and so the small triangle in the picture is similar to the large one. This lets us show that $\sin(\theta)\Delta\theta$ is approximately the small horizontal change in the picture (this is in fact a small change in cosine(\theta)).
Thus we are really just accumulating small horizontal distances. The total horizontal distance that you move from $\theta = 0$ to $\theta =\frac{\pi}{2}$ is 1, and so that is the integral.
If you have a solid geometric understanding of the fundamental theorem of calculus, and a solid geometric understanding of why the derivative of $\sin$ is $\cos$, then you will recognize that the proof above is really just weaving those two narratives together. | 2021-04-10T22:09:43 | {
"domain": "stackexchange.com",
"url": "https://math.stackexchange.com/questions/21796/finding-int-0-pi-2-sin-x-dx",
"openwebmath_score": 0.9234367609024048,
"openwebmath_perplexity": 158.1203335442763,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9783846672373523,
"lm_q2_score": 0.8519528019683106,
"lm_q1q2_score": 0.8335375586556955
} |
https://www.themathdoctors.org/averages-probability-and-reality/ | # Averages, Probability, and Reality
#### (Archive Question of the Week)
Recently I discussed the definition of the median of a data set, pointing out how it needs refinements that are not often discussed. In searching for questions in our archive on that topic, I ran across a discussion of an opposite issue: the breadth of the general term “average”, which does not have a specific definition. This is a nice example of how a seemingly simple question can lead to a discussion of wide-ranging topics that go off in different directions and end up tying many ideas together. That is part of the fun of being a Math Doctor.
## What is an average?
Here is the question, from Danny in 2007:
What is the Meaning of "Average"?
Can you please give a detailed description of average and its meaning? I'm not looking for a definition like "average is a certain # divided by a certain total #." I don't quite understand the real meaning of average.
A simple enough question, but one that could – and did – go in several directions, some of which were not initially apparent. In the end, we went beyond the definition of “average” in the sense of “arithmetic mean” as he defines it here, into other averages, and then into probability and beyond.
I started by discussing the various statistics that are sometime called “averages”, and then answered his specific question about the deeper meaning of the mean:
There are several different meanings of "average". The most general is a "measure of central tendency", meaning any statistic that in some sense represents a typical value from a data set. The mean, median, and mode are often identified as "averages" in this sense.
The word "average" is also used (especially at elementary levels) to refer specifically to the mean, which is the kind of average you mentioned: add the numbers and divide by how many there are. This kind of average has a specific meaning: it is the number you could use in place of each of the values, and still have the same sum.
This is the sort of “average” that Danny initially referred to. I gave a brief example of what my definition meant, with links to fuller explanations of this sense of “average”, and of other sorts of “mean”:
What Does Average Mean?
Arithmetic vs. Geometric Mean
Applications of Arithmetic, Geometric, Harmonic, and Quadratic Means
Average
## Probability as an average
My comments triggered a new question, about the meaning of “central tendency”. This is a complex term that is easily misunderstood. He did something wise here: he showed me what he was thinking, so I could correct it, rather than just ask another question without explanation. This is a part of good communication, and leads to profitable discussions.
Thanks for the helpful response. In your letter, you mentioned that average refers to central tendency. Let me give my interpretation of what central tendency means. Please correct me if I am wrong. For example, if our data shows that it rains 10 times over 100 days, then it means that the sky "tends" to rain 10 times per 100 days. 10 divided 100 gives a frequency value of 0.1, which means that it rains 0.1 time per day on average. This average refers to how frequently it rains. For example, if it rained 11 times (more frequent than 10), then you would get 11/100, which is a bigger value than 10/100. Thus, 11/100 is more frequent than 10/100. Is this interpretation of central tendency correct?
I also think central tendency is the average value that tends to be close to MOST of the various values in the data. For instance, if my data set is (4,6,1,3,0,5,3,4) the central tendency is 3.25, which is a value that tends towards 3 and 4. There are two 3 values and two 4 values in the data, which make up most of the data set.
The word “tendency” had led Danny off in a new direction, verging on probability, which is not really what we mean by the term here; but it is connected to our topic (“it tends to rain 1 of 10 days, on average“), so this was not a huge stretch. But he seems to have missed the word “central”, which is the real key to the phrase.
There are several other little twists in Danny’s understanding here: Can it really rain 0.1 time in a day? (I didn’t use number of rainfalls, but inches of rain, in my examples. But we’ll get back to this question.) And does “central tendency” imply closeness to most of the data (which sounds more like the mode in particular)? I now had to dig deeper into what it does mean; my word “typical” above did not do much to clarify. Words are slippery, aren’t they?
What you are saying in both cases is a reasonable example of the mean, and fits with my description of average rainfall, though I used the inches of rain per day rather than the number of rainfalls.
But central tendency is intended to be a much broader term. It's meant to be vague, because it covers not only means but also the median, the midrange, and even the mode. Its meaning is "any statistic that tends to fall in the middle of a set of numbers"; anything that gives a sense of what the "usual" or "typical" value is, in some sense, can be called a measure of central tendency. The *median* is, literally, the number in the middle--put the numbers in order, and take the middle number in the list, or the average of the two middle numbers if necessary. So that's clearly a "central tendency". The *midrange* is the exact middle of the range--the average, in fact, of the highest and lowest numbers. So that, too, has to lie in the middle, though it doesn't take into account how the rest of the numbers are distributed. The *mode* is the most common value, if there is one; it really doesn't have to be "in the middle", or even to exist, but it certainly fits the idea of "typical". The (arithmetic) *mean*, like all the others, has to lie within the range of the numbers, and it represents the "center of gravity" of all the numbers. So each of these fits the meaning of "measure of central tendency", each in a different way.
Now Danny turned a corner and moved fully into the topic of probability:
Hello doctor, thanks for your insights, I now have a better idea of average. Here is one more question about probability. Let's say that I was sick 40 times out of 1000 days. So based on this information, the probability of me getting sick on a random day is 40/1000. ...
This leads me to conclude that the probability of anything is based on the past data, and we can make good predictions of future events because of the law of continuity, meaning that things in the universe always follow a pattern. If we lived in a universe without continuity, then the knowledge of probability is useless.
So if I was sick 40 times out of 1000 days in the past, then the probability of me getting sick on a random day is the average value of 40/1000 = 0.04. I like to point out that the average 0.04 doesn't have a real physical meaning, because it says that I was sick on average 0.04 times per day (0.04 times? that makes no sense). I think 0.04 is just a number that corresponds to or represents 40/1000 (40 times per 1000 days is meaningful).
So he wants confirmation of his concept of probability, which I gave:
What you're talking about here is called empirical probability: just a description of what actually happened, which can't say anything about why, or what could happen another time. It's simply a ratio: how does the number of occurrences of sickness compare to the number of days under consideration? Out of those 1000 days, 40 of them were sick days; so "on the average" 40 out of 1000, or 4 out of 100, or 1 out of 25 were sick days. If they were evenly distributed--the same idea as a mean--then every 25th day would have been a sick day.
I chose not to point out the difference between his wording (the number of times he got sick), and mine (the number of sick days); we often leave such details to be absorbed from what we say rather than deliberately confront them; but we try to give an example of careful wording.
## Probability and the real world
But does the concept of probability imply, or require, that the universe must follow predictable patterns? That raises some further questions about its validity, which I answered by dipping into the philosophical realm:
Now you've made some big jumps! Not ALL of probability is just about past data; that's just empirical probability. And we can't always extrapolate from past events to the future. Sometimes that works, sometimes it doesn't. In part, it's the job of statistics to look at the data you've got and determine how valid it is to expect the same probabilities to continue--how good a sample you have. But even beyond that, whether we can assume that patterns will continue depends on other knowledge entirely, such as science. If we find a mechanism that explains a pattern, we have much better grounds for expecting it to continue than if we don't.
To make a broad statement that "things in the universe ALWAYS follow a pattern" is to indulge in philosophy, not math. In probability, we go the other way: we make an ASSUMPTION that things will continue as they are, in order to be able to apply probability to predicting anything; we leave it up to scientists (or sometimes philosophers) to decide whether that is a valid assumption. The scientist will most likely do some experiments to see if the predictions based on his theory work out, and if so he has some evidence that it is valid, and he can continue to make predictions. If not, then he tries another theory! He certainly would not say that probability forces him to believe that things work a certain way.
And perhaps that's what you mean to say: probability applies to a situation beyond the data we have only if there is consistency in the causes underlying the phenomena.
As we have said many times, math is not necessarily about the real world; it can be used to model the world based on observations of it, but the results must always be checked. Probability assumes a consistent world, saying, “If things continue the same, this is what we can expect.”
After pointing out the relationship of his comment about 0.04 times per day and the Law of Large Numbers, I returned to the connection of these ideas to averages:
The difference between this and the general idea of averages is that an average can apply to any collection of numbers, not just to the frequency of an occurrence. We can talk about the average speed of a car; regardless of how its speed has varied along a route, we can use the total distance traveled and the total time it took to determine the average speed, which is the speed it might have been going throughout the entire trip, in order to get the same total distance in the same total time. There is nothing probabilistic about this; but like probability, we are taking something that may vary "randomly" and condensing all its variations into a single number. The average speed does not mean that at every moment the car was going that fast, and the probability does not mean that out of every 25 days you are sick on one of them, or, worse, that on every day you are sick for 1/25 of the time. Averages and probability both ignore unevenness and look only at the big picture.
And that makes your question a very good one. I've been noticing the connections between probability and averages in several areas lately, and it's good to have a chance to think more about it.
## What is an average, really?
Finally, Danny got back to the topic of averages, with an excellent long question about things like this:
It seems like sometimes averages have no meaning. For instance, in a class of 10 students, 2 got 100 on a test, 8 got 0. The test average is 200/10 = 20. So on average every person got a 20 on the test. If I am correct in thinking that an average value is an estimate of the various values in the same data set (like you said, an average is like a center of gravity in the data set, so all the numbers in the data set should lean towards the average), then the average 20 is closer to the REAL scores of the 8 students who got 0 than to the REAL score of the 2 people who got 100.
This average gives a vague idea of how badly most people did, but it has "hidden" the two perfect scores. The average may tell us that most of the people must have done badly so that the average comes out to be so low. However, we can't know that some people did perfectly just by looking at the average.
Note that he very nicely paraphrased “central tendency” as “all the numbers in the data set should lean towards the average” – I think he got it!
Here are some excerpts of my discussion of this new topic:
Several of the pages on our site that discuss mean, median, and mode talk about why you would choose one rather than another. Each has its uses, and what you're saying is that for some purposes the mean is not the appropriate "measure of central tendency". That doesn't mean that it is meaningless, or that it is never a valid concept; only that it doesn't tell you what you'd like to know in this situation.
...
Another classic example of this is median income. If in your town 999 people earned $1000 a year, and one man earned$9,000,000 a year, the average (mean) income would be 10,000 a year, even though NOBODY made that amount. [Oops - I meant 9,999.] The median income gives a much better picture, if you want to know how the "average" person is doing; but that entirely misses the fact that there is one person who is rich. No matter what "average" you use, you'll be leaving someone out.
Another example is the rainfall I like to use to illustrate the idea of the mean. If the average rainfall is 1 inch a day, say, it might actually have been dry as a bone for 99 days, and then there was a 100 inch flood on the last day. The average accurately reflects the TOTAL amount of rain over the 100 days, but that isn't all it takes to decide what plants can survive there.
Again, the whole idea of an average is to try to boil down a lot of information into one number. That necessarily means that you have to lose some information. (That's why people don't want to be treated as mere numbers; they are more complex than that. Even a set of numbers doesn't like to be replaced by a single number!)
...
Incidentally, I've sometimes noticed in teaching, as a result of these statistics, that I can't "teach to the middle" of the class, because there is no middle. Sometimes I find a bimodal distribution, which means that I have a lot of F's and a lot of B's, and no one in between where the median and the mean both lie. (The last word there is an interesting, and very appropriate, pun!) So I have to ignore the statistics and teach the students.
As always, I’ve left out a lot, so you’ll have to read the original if you are interested. Some of these long discussions bring out a lot that is worth pondering.
### 1 thought on “Averages, Probability, and Reality”
This site uses Akismet to reduce spam. Learn how your comment data is processed. | 2021-10-19T21:16:22 | {
"domain": "themathdoctors.org",
"url": "https://www.themathdoctors.org/averages-probability-and-reality/",
"openwebmath_score": 0.6115972995758057,
"openwebmath_perplexity": 423.5381396281097,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9783846672373524,
"lm_q2_score": 0.8519528019683105,
"lm_q1q2_score": 0.8335375586556955
} |
https://math.stackexchange.com/questions/3193668/complex-number-frac3i212i2-cannot-arrive-at-textbook-solution | # Complex Number: $\frac{(3+i)^2}{(1+2i)^2}$ - cannot arrive at textbook solution
I have a complex quotient $$\frac{(3+i)^2}{(1+2i)^2}$$
The solution provided in my textbook is $$-2i$$. I arrived at different solutions and I'd like to know where I went wrong.
Till now in my textbook chapter I have been working with the complex number i ($$\sqrt{-1}$$).
I understand that one cannot divide by a complex number in the denominator so I must multiply both the numerator and denominator by the complex conjugate.
However, for this exercise I'm confused since my expression is nested inside parenthesis and is squared. So, if for example my denominator was just $$1+2i$$ I know that the complex conjugate would be $$1-2i$$.
So I'm confused about what do do since the whole denominator is within parenthesis and squared.
Just using what I know I tried solving for the squared term in both numerator and denominator:
$$(3+i)^2$$ = $$3^2+i^2$$ = $$9-1$$ = $$8$$
For the denominator: $$(1+2i)^2$$ = $$1^2+2^2i^2$$ = $$1+4 * -1$$ = $$1-4$$ = $$-3$$
Then I would arrive at $$\frac{8}{-3}$$ which is not the solution.
How can I arrive at $$-2i$$?
• Square and simplify the denominator first, then find the conjugate of the result. I see that you attempted to do that, but you got the wrong result. Use $(a+b)^2=a^2+2ab+b^2$. – John Wayland Bales Apr 19 at 15:38
• $(a+b)^2$ is not simply $a^2+b^2$. This is such a common mistake that it has it's own name and wiki page. See The Freshman's Dream on wikipedia. Instead, $(a+b)^2 = a^2+2ab+b^2$. So, you should instead have had $(1+2i)^2 = 1^2 \color{red}{+2\cdot 1\cdot 2i} + (2i)^2$ and similarly for the numerator and you should be able to continue from there. – JMoravitz Apr 19 at 15:40
• As an aside, thank you for including what you have tried as it makes it much clearer where your misconceptions lie. +1 – JMoravitz Apr 19 at 15:41
• This is somewhat of an aside, but please, try to stay away from $i=\sqrt{-1}$ as much as you can. It's supposed to be $i^2=-1$. The difference between the two is bigger than one might at first suspect. Mainly, using the former lures new students into a false sense of security when it comes to square roots and complex numbers. But there are other, more subtle reasons too. – Arthur Apr 19 at 15:46
• And, possibly another aside, and a semantic one at that, but just to be sure that you understand... You wrote, "I understand one cannot divide by a complex number [...] " - Well, one can, and you did... However, as you clearly understand, one can give the result in a form with no complex number in the denominator (by fooling around with complex conjugates, say) - and this is [no doubt] what is desired. For instance $1/i = -i / 1= -i$, so $1$ divided by $i$ equals $-i$. – peter a g Apr 19 at 15:56
You accidentally fell into the trap of using The Freshman's Dream.
Exponentiation does not distribute over sums. In most every context $$(a+b)^n$$ is not equal to $$a^n + b^n$$ and in particular $$(a+b)^2$$ is not simply $$a^2+b^2$$ but is instead $$a^2+2ab+b^2$$. For larger exponents, see the binomial theorem.
In your attempt you tried to calculate the numerator as $$(3+i)^2 = 3^2+i^2$$ which is incorrect. Rather, you should remember to use the FOIL method correctly or learn the correct outcome. Instead, it should have been $$(3+i)^2 = 3^2\color{red}{+2\cdot 3\cdot i}+i^2 = 9+6i-1=8+6i$$.
You made a similar mistake for the denominator. Once those mistakes are corrected, you should be able to continue on your own using the method you allude to in your post of multiplying both top and bottom by the complex conjugate of the denominator and simplifying after.
• Thank you for your advise and letting me know about the freshman's dream!, I was able to arrive at the solution by correctly factoring out (if that's the phrase? reverse factoring, if you will) the denominator per FOIL – Doug Fir Apr 19 at 15:57
• The usual word to describe the process you are referring to is "expanding." – JMoravitz Apr 19 at 15:58
• Or "multiplying out" - but probably "expanding" is better... – peter a g Apr 19 at 16:00
Expanding we get $$\frac{9-1+6i}{1-4+4i}=\frac{8+6i}{-3+4i}$$ and now multiply numerator and denominator by $$-3-4i$$
It might help to note that this expression is $$\left(\frac{3+i}{1+2i}\right)^2=\left(1-i\right)^2=-2i$$ assuming that you know how to divide $$3+i$$ by $$1+2i$$. | 2019-05-26T13:37:26 | {
"domain": "stackexchange.com",
"url": "https://math.stackexchange.com/questions/3193668/complex-number-frac3i212i2-cannot-arrive-at-textbook-solution",
"openwebmath_score": 0.8500084280967712,
"openwebmath_perplexity": 184.2336108384778,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9783846703886661,
"lm_q2_score": 0.8519527982093666,
"lm_q1q2_score": 0.8335375576627729
} |
https://math.stackexchange.com/questions/1247087/prove-that-if-fx25fx-0-then-fx-acos5xbsin5x-for-some-constants | # Prove that if $f''(x)+25f(x)=0$ then $f(x)=Acos(5x)+Bsin(5x)$ for some constants $A,B$
Suppose $f: \mathbb{R} \rightarrow \mathbb{R}$ is twice differentiable.
Prove that if $f''(x)+25f(x)=0$ then $f(x)=Acos(5x)+Bsin(5x)$ for some constants $A,B$
Consider $g(x):= f(x)-Acos(5x)-Bsin(5x)$ where $A$ and $B$ are chosen so that $g(0)=g'(0)=0$ [this portion is given, so can someone explain why it is important?]
For reference, we note that $g(x):= f(x)-Acos(5x)-Bsin(5x)$ $g'(x):= f'(x)+5Asin(5x)-5Bcos(5x)$ $g''(x):= f''(x)+25Acos(5x)+25Bsin(5x)$
Consider the derivative of $\frac{25}{2}g(x)^2+\frac{1}{2}g'(x)^2$
$25g(x)g'(x)+g''(x)g'(x)$ if and only if $g'(x)[25g(x)+g''(x)]$
Following substitution, we conclude that
$g'(x)[25g(x)+g''(x)]=g'(x)(0)=0$
Therefore, by the Constancy Theorem, $\frac{25}{2}g(x)^2+\frac{1}{2}g'(x)^2=C_0$, where $C_0$ is a constant
$25g(x)^2+g'(x)^2=C$, where $C$ is a constant
Thus,
$25(f(x)-Acos(5x)-Bsin(5x))^2 + (f'(x)+5Asin(5x)-5Bcos(5x))^2=C$
Where do I go from here?
• Note that $cos(5x)$ and $sin(5x)$ are solutions of your second order equation. and any solution will be linear combination of these solutions. – Arpit Kansal Apr 22 '15 at 18:27
• @ArpitKansal: That's what has to be proved, I think. – TonyK Apr 22 '15 at 18:35
By steps: $g(0)=g'(0)=0$ is important because you have a homogenous differential equation, therefore zero initial data implies that the solution is zero everywhere.
After obtaining that $25(f(x)-A\cos(5x)-B\sin(5x))^2 + (f'(x)+5A\sin(5x)-5B\cos(5x))^2=C$, use initial data $g(0)=g'(0)=0$.
We chose $A$ and $B$ to get $f(x)-A\cos(5x)-B\sin(5x)=0$ and $f'(x)+5A\sin(5x)-5B\cos(5x)=0$ at $x=0$. Therefore we can find $C$ by putting $x=0$: $$25(f(x)-A\cos(5x)-B\sin(5x))^2 + (f'(x)+5A\sin(5x)-5B\cos(5x))^2=C = 25(f(0)-A\cos(0)-B\sin(0))^2 + (f'(0)+5A\sin(0)-5B\cos(0))^2=0.$$
Hence $$\forall x\quad 25(f(x)-A\cos(5x)-B\sin(5x))^2 + (f'(x)+5A\sin(5x)-5B\cos(5x))^2 =0.$$ You have the sum of squares equal to zero, therefore each term is zero itself. We can conclude that $\forall x\, (f(x)-A\cos(5x)-B\sin(5x))^2=0$ and, therefore, $$\forall x\quad f(x)=A\cos(5x)+B\sin(5x).$$
We conclude that all solutions of the initial differential equation has the above form.
Since this is a differential equation with a right side equal to zero, we say that it is homogeneous. Thus, we solve the auxiliary equation associated with the equation:
f′′(x) + 25 f(x) = 0 ------> r^2 + 25 = 0
This yields:
r = +/- 5i
Solution will be of the form:
f = Ae^(5i) + be^(-5i)
Which using Euler's method yields:
f = ACos(5t) + BSin(5t).
Differentiating f twice and plugging into the original diff. eq. should yield zero.
Is this what you wanted?
I don't remenber the general formula, you can find in some book of ODE. You have this y= f(x)=> y'' + 25y=0. We make the substitution $y=\exp{rx}$. Then $r^2e^{rx}+25e^{}rx=0$ and $e^{rx}(r^2+25)=0$. We find the roots of polinomial $r^2+25$ there are $r=5i$ and $r=-5i$ for the properties of the solutions of the ode we can say $y = (e^{5xi}+e^-{-5xi})/2=cos(5x)$ is a solution or $y = (e^{5xi}+e^-{-5xi})/2=i sin(5x)$
In your expression involving $\frac {25}2g^2+\frac 12 g'^2=C$ you can set $x=0$ to determine the constant (this is where the bit you have been given is important). Then you have the sum of two squares equal to [?].
Note: you get the first part - the values of $A$ and $B$ from the values of $f(0)$ and $f'(0)$ if these are given, because $f(0)=g(0)+A$ so you want $A=f(0)$ and $f'(0)=g'(0)+5B$ so that $B=\frac 15 f'(0)$. Hence you can always find $A$ and $B$ which work.
If the initial conditions are given by the values of $f(x)$ at two distinct values of $x$ you get $A$ and $B$ by solving a couple of simultaneous equations.
Once you have filled in the details, this gives a proof that all the solutions of the original equation are of the requisite form, and justifies using that form to find solutions without replicating the proof every time.
Is it easier just to substitute? $$f(x)=Acos(5x)+Bsin(5x)$$ then $$\dot f(x)=-5Asin(5x)+5Bcos(5x)$$ $$\ddot f(x)=-25Acos(5x)-25Bsin(5x)$$ finally: $$-25Acos(5x)-25Bsin(5x)+25( Acos(5x)+Bsin(5x)) =0$$ ?
• I think the goal of this exercise is to show that there are no other solutions. – TZakrevskiy Apr 22 '15 at 18:28
• this only implies that f(x) is a solution. – Arpit Kansal Apr 22 '15 at 18:28 | 2021-05-08T07:53:36 | {
"domain": "stackexchange.com",
"url": "https://math.stackexchange.com/questions/1247087/prove-that-if-fx25fx-0-then-fx-acos5xbsin5x-for-some-constants",
"openwebmath_score": 0.9211633205413818,
"openwebmath_perplexity": 173.47997375340944,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9783846634557752,
"lm_q2_score": 0.8519528038477825,
"lm_q1q2_score": 0.8335375572728168
} |
https://math.stackexchange.com/questions/3249892/show-that-this-function-is-bounded | # Show that this function is bounded
Let $$f$$ be a $$\mathbb R \rightarrow \mathbb R$$ continuous function such that : $$\lim_ {x \to \pm \infty} f(x) \in \mathbb R$$ and $$\lim_ {x \to 0} f(x) \in \mathbb R$$
How can one show that $$f$$ is bounded ? I get it "intuitively" but I cant show it rigorously
• Use the limits definition to show that it is bounded in all but a compact set, then use continuity (Weierstrass theorem in particular) to show it's bounded in all of $\mathbb{R}$ Jun 3 '19 at 18:52
• Btw I don't think you need the existence of the limit at X=0, this follows from continuity Jun 3 '19 at 18:55
The fact that $$\lim_{x\to0}f(x)\in\mathbb R$$ is necessarily true: since $$f$$ is continuous that limit has to be $$f(0)$$.
Now, suppose that $$f$$ is unbounded. Then, for each $$n\in\mathbb N$$, there is a $$x_n\in\mathbb R$$ such that $$\bigl\lvert f(x_n)\bigr\rvert\geqslant n$$. The sequence $$(x_n)_{n\in\mathbb N}$$ is either bounded or unbounded and:
• if it is bounded, then it has a convergent subsequence $$(x_{n_k})_{k\in\mathbb N}$$. But if $$\lim_{k\to\infty}x_{n_k}=x$$, then $$\lim_{k\to\infty}f(x_{n_k})=f(x)$$, which is impossible, since $$\bigl(f(x_{n_k})\bigr)_{k\in\mathbb N}$$ is unbounded.
• if it is unbounded, then it has a subsequence $$(x_{n_k})_{k\in\mathbb N}$$ whose limit is $$\pm\infty$$, and we then get a similar contradiction.
Juts for amusement:
Let $$\phi(x) = \begin{cases} \lim_{x \to -\infty} f(x) , & x = -{\pi \over 2} \\ f(\tan(x)), & x \in (-{\pi \over 2}, {\pi \over 2} ) \\ \lim_{x \to +\infty} f(x) , & x = {\pi \over 2} \end{cases}$$.
Show $$\phi$$ is continuous on the compact set $$[-{\pi \over 2},{\pi \over 2}]$$, hence bounded, and hence $$f$$ is bounded.
• Amusement? I think it's the superior proof. (Although "juts" is amusing)
– zhw.
Jun 3 '19 at 19:50
• @zhw.: Just poor spelling on my part :-). Jun 3 '19 at 19:55
If $$\lim_{x\to-\infty} f(x)=a$$ and $$\lim_{x\to\infty} f(x)=b$$ put $$|a|+|b|+1=:c$$. There is an $$M>0$$ such that $$|f(x)|\leq c$$ for all $$x\geq M$$ and all $$x\leq-M$$. Since $$f$$ is continuous there is a $$c'$$ such that $$|f(x)|\leq c'$$ for all $$x\in[-M,M]$$. It follows that $$|f(x)|\leq c+c'$$for all $$x\in{\mathbb R}$$.
The hypothesis that
$$\displaystyle \lim_{x \to \infty} f(x) \in \Bbb R \tag 1$$
means that
$$\exists L_+ \in \Bbb R, \; \displaystyle \lim_{x \to \infty} f(x) = L_+; \tag 2$$
that is,
$$\forall 0 < \epsilon_+ \in \Bbb R \; \exists 0 < M_+ \in \Bbb R, \; x > M_+ \Longrightarrow \vert f(x) - L_+ \vert < \epsilon_+; \tag 3$$
that is,
$$x > M_+ \Longrightarrow L_+ - \epsilon_+ < f(x) < L_+ + \epsilon_+; \tag 4$$
likewise the hypothesis
$$\displaystyle \lim_{x \to -\infty} f(x) \in \Bbb R \tag 5$$
gives us
$$\exists L_- \in \Bbb R, \; \displaystyle \lim_{x \to -\infty} f(x) = L_-; \tag 6$$
i.e.,
$$\forall 0 < \epsilon_- \in \Bbb R \; \exists 0 > M_- \in \Bbb R, \; x < M_- \Longrightarrow \vert f(x) - L_- \vert < \epsilon_-, \tag 7$$
or
$$x < M_- \Longrightarrow L_- - \epsilon_- < f(x) < L_- + \epsilon_-; \tag 8$$
it follows that, letting
$$b = \min(L_- - \epsilon_-, L_+ -\epsilon_+), \; B = \max(L_- + \epsilon_-, L_+ + \epsilon_+) \tag 9$$
and
$$m = \max(-M_-, M_+), \tag{10}$$
that
$$\vert x \vert > m \Longrightarrow b < f(x) < B, \tag{11}$$
so $$f(x)$$ is bounded on the set $$(-\infty, m) \cup (m, \infty)$$; furthermore, since the closed interval $$[-m, m]$$ is compact and $$f(x)$$ is continuous, $$\vert f(x) \vert$$ is strictly bounded on this interval by some $$0 < \beta \in \Bbb R$$:
$$x \in [-m, m] \Longrightarrow -\beta < f(x) < \beta; \tag{12}$$
combining (11) and (12) shows that $$f(x)$$ is bounded on all of $$\Bbb R$$. Indeed, we have
$$\forall x \in \Bbb R, \; \vert f(x) \vert < \max(\vert b \vert, \vert B \vert, \beta). \tag{13}$$
1) Consider $$[0,\infty)$$.
$$\lim_{x \rightarrow \infty}f(x)=L:$$
For $$\epsilon >0$$ there is a $$M$$, real, positive, s.t.
for $$x > M$$ $$|f(x)-L| <\epsilon$$, i.e.
$$- \epsilon +L < f(x) < \epsilon +L$$.
The continuous function $$f$$ attains minimum and maximum on the compact interval $$[0,M]$$.
Hence $$f$$ is bounded on $$[0,\infty)$$.
2) Proceed likewise for $$(-\infty,0]$$.
1) and 2): f is bounded on $$\mathbb{R}$$.
Also cf.. comment of miraunpajaro | 2021-11-30T04:01:08 | {
"domain": "stackexchange.com",
"url": "https://math.stackexchange.com/questions/3249892/show-that-this-function-is-bounded",
"openwebmath_score": 0.9834232330322266,
"openwebmath_perplexity": 181.11216137892723,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9783846621952493,
"lm_q2_score": 0.8519528038477824,
"lm_q1q2_score": 0.8335375561989081
} |
https://math.stackexchange.com/questions/3239883/given-the-bivariate-standard-normal-density-how-to-show-that-x-and-y-are-st | # Given the bivariate standard normal density, how to show that $X$ and $Y$ are standard normal densities? [closed]
Given the standard bivariate normal density with correlation coefficient $$\rho$$ for $$X$$ and $$Y$$:
$$f_{X,Y}(x,y)=\frac{1}{2\pi\sqrt{1-\rho^2}}e^{-(x^2-2\rho xy+y^2)/2(1-\rho^2)}$$
Is there a way to work backwards and show that $$X$$ and $$Y$$ are standard normal density without the assumption of independence?
## closed as off-topic by StubbornAtom, José Carlos Santos, Parcly Taxel, Javi, CesareoMay 26 at 10:57
This question appears to be off-topic. The users who voted to close gave this specific reason:
• "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – StubbornAtom, José Carlos Santos, Parcly Taxel, Javi, Cesareo
If this question can be reworded to fit the rules in the help center, please edit the question.
Integrate out one of the variables.
$$f_X(x) = \int \frac{1}{2\pi \sqrt{1-\rho^2}}\exp\left(-\frac{x^2-2\rho xy+y^2}{2(1-\rho^2)}\right)\,dy$$ $$= \frac{1}{2\pi \sqrt{1-\rho^2}}\exp\left(-\frac{x^2(1-\rho^2)}{2(1-\rho^2)}\right)\int \exp\left(-\frac{\rho^2x^2-2\rho xy+y^2}{2(1-\rho^2)}\right)\,dy$$ Notice that I've pulled out a piece of the exponential that doesn't depend on $$y$$ in order to get the numerator inside the integral to have $$\rho^2x^2$$ in it. $$= \frac{1}{2\pi \sqrt{1-\rho^2}}\exp\left(-x^2/2\right)\int \exp\left(-\frac{(y-\rho x)^2}{2(1-\rho^2)}\right)\,dy$$ Letting $$u=\frac{y-\rho x}{\sqrt{1-\rho^2}}$$, we have $$dx = \sqrt{1-\rho^2}\,du$$, and $$= \frac{1}{2\pi \sqrt{1-\rho^2}}\exp\left(-x^2/2\right)\int \exp\left(-u^2/2\right)\sqrt{1-\rho^2}\,du$$ $$= \frac{1}{2\pi }\exp\left(-x^2/2\right)\sqrt{2\pi} = \frac{e^{-x^2/2}}{\sqrt{2\pi}}.$$
Thus $$X\sim \mathcal{N}(0,1)$$, as desired.
• What was wrong in this answer that made you delete it? I was thinking of doing the same thing. – Alex M. May 26 at 16:02
• @AlexM. There are two main problems with that answer. The first is that $X^*$ there isn't the pushforward of a vector field under a single map. Rather it is at every point the pushforward of a single vector under a different map. This causes the pushforward not to commute with the Lie bracket, and indeed, we should have $[X^*,Y^*]=-[X,Y]^*$ as claimed. The other problem is the counterexample I gave, since $X$ is typically identified with a left invariant vector field, but $X^*$ is right invariant (it's the pushforward by right multiplication), so we don't have $X^*=X$ as I claimed. – jgon May 26 at 16:27
Do you know how to find the density of $$X$$ from the joint density of $$X$$ and $$Y$$ in general (that is, without the requirement that the random variables have the specific density shown above)?? If so, apply the same technique. A couple of tricks that you will need to use are
• the method of completing the square; an integrand will have a quadratic polynomial as the argument of the exponential function, and you will need to massage it to write it as a perfect square plus some constants
• recognizing a function as a (univariate) normal density and setting its integral over the real line to have value $$1$$ without necessarily calculating the integral formally to get $$1$$ as the value of the integral.
And if you don't want to waste your time with all this, just look at the other answers where the calculations are written out explicitly in gory detail.
Yes, just find the marginal pdf by integrating out $$y$$ (or $$x$$). \begin{align*} f_X(x)&=\int_{-\infty}^\infty f_{X,Y}(x,y)\,\mathrm{d}y\\ &=\int_{-\infty}^\infty\frac{1}{2\pi\sqrt{1-\rho^2}}\exp\left[-\frac{x^2-2\rho xy+y^2}{2(1-\rho^2)}\right]\,\mathrm{d}y\\ &=\int_{-\infty}^\infty\frac{1}{2\pi\sqrt{1-\rho^2}}\exp\left[-\frac{(1-\rho^2)x^2+(y-\rho x)^2}{2(1-\rho^2)}\right]\,\mathrm{d}y\\ &=\frac{\exp(-x^2/2)}{\sqrt{2\pi}}\int_{-\infty}^\infty\frac{1}{\sqrt{2\pi(1-\rho^2)}}\exp\left[-\frac{(y-\rho x)^2}{2(1-\rho^2)}\right]\,\mathrm{d}y\\ &=\frac{\exp(-x^2/2)}{\sqrt{2\pi}}\int_{-\infty}^\infty\frac{1}{\sqrt{2\pi(1-\rho^2)}}\exp\left[-\frac{(y-\rho x)^2}{2(1-\rho^2)}\right]\,\mathrm{d}(y-\rho x)\\ &=\frac{\exp(-x^2/2)}{\sqrt{2\pi}} \end{align*} | 2019-09-22T07:40:38 | {
"domain": "stackexchange.com",
"url": "https://math.stackexchange.com/questions/3239883/given-the-bivariate-standard-normal-density-how-to-show-that-x-and-y-are-st",
"openwebmath_score": 0.9058980345726013,
"openwebmath_perplexity": 334.62147583143985,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.978384668497878,
"lm_q2_score": 0.8519527982093666,
"lm_q1q2_score": 0.8335375560519107
} |
http://bootmath.com/can-a-sequence-which-decays-more-slowly-still-yield-a-converging-series.html | # Can a sequence which decays more slowly still yield a converging series?
In Bergman’s companion notes to Rudin, he says that “If a sequence of positive terms has convergent sum, so does every sequence of positive terms which decays more rapidly.” So given a sequence $\{a_n\}$ of positive terms such that $\sum_n a_n$ converges, if $\{b_n\}$ is such that
$$\lim_{n\to\infty} \frac{a_n}{b_n} = +\infty,$$ then $\sum_nb_n$ converges. I can prove this given $\{a_n\}$ and $\{b_n\}$. However, can we find $b_n$ which decays more slowly, i.e.
$$\lim_{n\to\infty} \frac{b_n}{a_n} = +\infty$$ such that $\sum_n b_n$ converges?
Similarly, we have the claim “if a sequence of positive terms has divergent sum, then so does every sequence of positive terms which decays more slowly.”
#### Solutions Collecting From Web of "Can a sequence which decays more slowly still yield a converging series?"
The question is the following:
Show that for every converging series $\sum\limits_na_n$ with positive terms there exists a converging series $\sum\limits_nb_n$ with positive terms such that $\lim\limits_{n\to\infty}\frac{a_n}{b_n}=0$.
A hands-on approach is as follows: for every $n$, consider $$A_n=\sum_{k\geqslant n}a_k,$$ then, by hypothesis, $A_n\to0$ hence, for each $k\geqslant0$, there exists some finite $\nu(k)$ such that $$A_{\nu(k)}\leqslant2^{-k}.$$ Assume without loss of generality that the sequence $(\nu(k))$ is nondecreasing and define $(b_n)$ by
$$\color{red}{\forall k\geqslant0,\quad\forall n\in[\nu(k),\nu(k+1))},\quad \color{red}{b_n=k\,a_n}.$$
Then $\frac{a_n}{b_n}\to0$ when $n\to\infty$ and $$\sum_{n\geqslant\nu(0)}b_n=\sum_{k\geqslant0}k\sum_{n=\nu(k)}^{\nu(k+1)-1}a_n\leqslant\sum_{k\geqslant0}kA_{\nu(k)}\leqslant\sum_{k\geqslant0}k2^{-k},$$ which is finite, hence the series $\sum\limits_nb_n$ converges, as desired.
If I can remember, Rudin clearly shows that there is no threshold for convergence or divergence of positive series, in chapter 3 of his book. You can construct convergent series with arbitrary slow decay by multiplying by logarithms.
For instance
$$\sum_n \frac{1}{n \log n \log \log n}$$
diverges, whereas
$$\sum_n \frac{1}{n \log n (\log \log n)^2}$$
converges, and so on.
You can build this using: $R_n = \sum_{k=n}^{+\infty} a_k \geq 0$ (exists since $\sum a_k$ converges)
$\sum a_n$ converges, i.e : $R_n \rightarrow 0$ , when $n\rightarrow +\infty$
You can define: $$b_n = \sqrt{R_n} -\sqrt{R_{n+1}}$$
Note that whenever $a_i = 0$ : $b_i = \sqrt{R_i} -\sqrt{R_{i+1}} =\sqrt{R_{i+1}} -\sqrt{R_{i+1}} = 0$
So if you define: $v_n = \frac{a_n}{b_n}$ you can set: $v_i = 0$ when $a_i = 0$
Now when $a_n \neq 0$: $$v_n = \frac{a_n}{b_n} = \frac{R_n – R_{n+1}}{\sqrt{R_n} -\sqrt{R_{n+1}}} = \sqrt{R_n} +\sqrt{R_{n+1}} \rightarrow 0 , n\rightarrow +\infty$$
And $\sum b_n$ converges by a telescoping.
Edit: Likewise, you can also find a sequence ($c_n$) such as :$\frac{c_n}{a_n} \rightarrow 0$ and $\sum c_n$ converges.
You can consider : $c_n = R_n^2-R_{n+1}^2$ with the same statements as for $b_n$ when ($a_n$) contains zero terms:
$$\frac{c_n}{a_n} = \frac{R_n^2-R_{n+1}^2}{a_n} = \frac{R_n^2-R_{n+1}^2}{R_n – R_{n+1}} = R_n + R_{n+1} \rightarrow 0$$
Here again $\sum c_n$ converges by telescoping.
So one cannot define the notion of fastest converging series, since given any converging one you can build a new that converges faster.
There are a couple of things… slightly off… in your understanding.
The theorem does not say, as you imply, that if $$\sum_{n=1}^\infty a_n$$ exists and if $$\lim_{n\to\infty}\frac{b_n}{a_n} = \infty,$$ then the sum $$\sum_{n=1}^\infty b_n$$ exists. I advise you to re-check what your notes say. | 2018-06-23T13:33:14 | {
"domain": "bootmath.com",
"url": "http://bootmath.com/can-a-sequence-which-decays-more-slowly-still-yield-a-converging-series.html",
"openwebmath_score": 0.984398603439331,
"openwebmath_perplexity": 174.42263005178802,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.978384667867615,
"lm_q2_score": 0.8519527982093666,
"lm_q1q2_score": 0.8335375555149565
} |
http://mathhelpforum.com/calculus/199332-difficult-integration-question.html | # Math Help - Difficult integration question
1. ## Difficult integration question
How do I do part (b). I've been stuck at it for hours, even after looking at all the different types of integration formulas...
2. ## Re: Difficult integration question
For the first $2=e^x+2-e^x$ and for the second after a substitution you have to compute, up to a constant, $\int \frac{t^2-1}tdt$.
3. ## Re: Difficult integration question
$\int \sqrt{16x^2-9} \cdot \frac{1}{x} \, dx$
$3 \int \sqrt{\frac{16x^2}{9} - 1} \cdot \frac{1}{x} \, dx$
$x = \frac{3\sec{t}}{4}$
$dx = \frac{3}{4} \sec{t}\tan{t} \, dt$
substitute and simplify ...
$3 \int \tan^2{t} \, dt$
$3 \int \sec^2{t} - 1 \, dt$
$3 \left(\tan{t} - t) + C$
$\sqrt{16x^2-9} - 3\arctan\left(\frac{\sqrt{16x^2-9}}{3}\right) + C$
4. ## Re: Difficult integration question
Originally Posted by skeeter
$\int \sqrt{16x^2-9} \cdot \frac{1}{x} \, dx$
$3 \int \sqrt{\frac{16x^2}{9} - 1} \cdot \frac{1}{x} \, dx$
$x = \frac{3\sec{t}}{4}$
$dx = \frac{3}{4} \sec{t}\tan{t} \, dt$
substitute and simplify ...
$3 \int \tan^2{t} \, dt$
$3 \int \sec^2{t} - 1 \, dt$
$3 \left(\tan{t} - t) + C$
$\sqrt{16x^2-9} - 3\arctan\left(\frac{\sqrt{16x^2-9}}{3}\right) + C$
How do you know we should let x = 3sec t/ 4 ? I've been listening in class and I don't recall my lecturer showing us such a method? Did you use trial and error to get x = 3sect/4?
5. ## Re: Difficult integration question
Originally Posted by Flipflops
How do you know we should let x = 3sec t/ 4 ? I've been listening in class and I don't recall my lecturer showing us such a method? Did you use trial and error to get x = 3sect/4?
It's a method known as trigonometric substitution, and takes some inspired guesswork. Basically, the fact that derivatives of trig functions are trig functions, and through the Pythagorean Theorem, we can usually simplify a trig expression and cancel with the derivative, to make the integral easy to evaluate.
Here we have something of the form \displaystyle \begin{align*} a^2x^2 - b^2 \end{align*}, so the identity \displaystyle \begin{align*} \sec^2{\theta} - 1 \equiv \tan^2{\theta} \end{align*} tells us a substitution of the form \displaystyle \begin{align*} x = \frac{b\sec{\theta}}{a} \end{align*} is appropriate.
\displaystyle \begin{align*} a^2x^2 - b^2 &= a^2 \left( \frac{b\sec{\theta}}{a} \right)^2 - b^2 \\ &= a^2 \left( \frac{b^2\sec^2{\theta}}{a^2} \right) - b^2 \\ &= b^2\sec^2{\theta} - b^2 \\ &= b^2\left(\sec^2{\theta} - 1\right) \\ &= b^2\tan^2{\theta} \end{align*}
And notice that \displaystyle \begin{align*} \frac{d}{dx}\left(\sec{\theta}\right) &= \sec{\theta}\tan{\theta} \end{align*}, which makes a combination that is easy to evaluate, as per Skeeter's post.
6. ## Re: Difficult integration question
Hello, Flipflops!
There is a clever "trick" for the first one . . .
$\displaystyle(a)\;\int\frac{dx}{e^x+2}$
Divide the numerator and denominator by $e^x.$
$\text{We have: }\:\int\frac{e^{\text{-}x}\,dx}{1 + 2e^{\text{-}x}}$
$\text{Let }\,u \,=\,1 + 2e^{-x} \quad\Rightarrow\quad du \,=\,-2e^{-x}dx \quad\Rightarrow\quad e^{-x}dx \:=\:-\tfrac{1}{2}\:\!du$
$\text{Substitute: }\:\int\frac{\text{-}\frac{1}{2}du}{u} \;=\;\text{-}\tfrac{1}{2}\int\frac{du}{u} \;=\;\text{-}\tfrac{1}{2}\ln|u| + C$
$\text{Back-substitute: }\:\text{-}\tfrac{1}{2}\ln(1 + 2e^{-x}) + C$
This can be simplified beyond all recognition . . .
$-\tfrac{1}{2}\ln(1 + 2e^{-x}) + C \;=\;-\tfrac{1}{2}\ln\left(1 + \frac{2}{e^x}\right) + C \;=\;-\tfrac{1}{2}\ln\left(\frac{e^x + 2}{e^x}\right) + C$
. . $=\;-\tfrac{1}{2}\big[\ln(e^x+2) - \ln(e^x)\big] + C \;=\; -\tfrac{1}{2}\big[\ln(e^x+2) - x\big] + C$
. . $=\;\tfrac{1}{2}\big[x - \ln(e^x+2)\big]+C$
7. ## Re: Difficult integration question
Originally Posted by Soroban
Hello, Flipflops!
There is a clever "trick" for the first one . . .
Divide the numerator and denominator by $e^x.$
$\text{We have: }\:\int\frac{e^{\text{-}x}\,dx}{1 + 2e^{\text{-}x}}$
$\text{Let }\,u \,=\,1 + 2e^{-x} \quad\Rightarrow\quad du \,=\,-2e^{-x}dx \quad\Rightarrow\quad e^{-x}dx \:=\:-\tfrac{1}{2}\:\!du$
$\text{Substitute: }\:\int\frac{\text{-}\frac{1}{2}du}{u} \;=\;\text{-}\tfrac{1}{2}\int\frac{du}{u} \;=\;\text{-}\tfrac{1}{2}\ln|u| + C$
$\text{Back-substitute: }\:\text{-}\tfrac{1}{2}\ln(1 + 2e^{-x}) + C$
This can be simplified beyond all recognition . . .
$-\tfrac{1}{2}\ln(1 + 2e^{-x}) + C \;=\;-\tfrac{1}{2}\ln\left(1 + \frac{2}{e^x}\right) + C \;=\;-\tfrac{1}{2}\ln\left(\frac{e^x + 2}{e^x}\right) + C$
. . $=\;-\tfrac{1}{2}\big[\ln(e^x+2) - \ln(e^x)\big] + C \;=\; -\tfrac{1}{2}\big[\ln(e^x+2) - x\big] + C$
. . $=\;\tfrac{1}{2}\big[x - \ln(e^x+2)\big]+C$
Or another alternative is to multiply top and bottom by \displaystyle \begin{align*} e^x \end{align*}.
\displaystyle \begin{align*} \int{\frac{dx}{e^x + 2}} &= \int{\frac{e^x\,dx}{e^{2x} + 2e^x}} \\ &= \int{\frac{e^x\,dx}{e^{2x} + 2e^x + 1^2 - 1^2}} \\ &= \int{\frac{e^x\,dx}{\left(e^x + 1\right)^2 - 1}} \\ &= \int{\frac{du}{u^2 - 1}}\textrm{ after making the substitution }u = e^x + 1 \\ &= \int{\frac{1}{2(u - 1)} - \frac{1}{2(u + 1)}\,du} \\ &= \frac{1}{2}\ln{|u - 1|} - \frac{1}{2}\ln{|u + 1|} + C \\ &= \frac{1}{2}\ln{\left|\frac{u - 1}{u + 1}\right|} + C \\ &= \frac{1}{2}\ln{\left|\frac{e^x + 1 - 1}{e^x + 1 + 1}\right|} + C \\ &= \frac{1}{2}\ln{\left|\frac{e^x}{e^x + 2}\right|} + C \end{align*}
8. ## Re: Difficult integration question
Originally Posted by Flipflops
How do you know we should let x = 3sec t/ 4 ? I've been listening in class and I don't recall my lecturer showing us such a method? Did you use trial and error to get x = 3sect/4? | 2015-03-28T02:37:55 | {
"domain": "mathhelpforum.com",
"url": "http://mathhelpforum.com/calculus/199332-difficult-integration-question.html",
"openwebmath_score": 1.0000098943710327,
"openwebmath_perplexity": 1975.3976365916615,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9783846716491915,
"lm_q2_score": 0.8519527944504227,
"lm_q1q2_score": 0.833537555058988
} |
https://math.stackexchange.com/questions/3152235/partial-derivative-of-mse-cost-function-in-linear-regression | # Partial derivative of MSE cost function in Linear Regression?
I'm confused by multiple representations of the partial derivatives of Linear Regression cost function.
This is the MSE cost function of Linear Regression. Here $$h_\theta(x) = \theta_0+\theta_1x$$ .
\begin{aligned}J(\theta_0,\theta_1) &= \frac{1}{m}\displaystyle\sum_{i=1}^m(h_\theta(x^{(i)}) - y^{(i)})^2\\J(\theta_0,\theta_1) &= \frac{1}{m}\displaystyle\sum_{i=1}^m(\theta_0 + \theta_1x^{(i)} - y^{(i)})^2\end{aligned}
Are these the correct partial derivatives of above MSE cost function of Linear Regression with respect to $$\theta_1, \theta_0$$? If there's any mistake please correct me.
\begin{aligned}\frac{dJ}{d\theta_1} &= \frac{-2}{m}\displaystyle\sum_{i=1}^m(x^{(i)}).(\theta_0 + \theta_1x^{(i)} - y^{(i)})\\ \frac{dJ}{d\theta_0} &= \frac{-2}{m}\displaystyle\sum_{i=1}^m(\theta_0 + \theta_1x^{(i)} - y^{(i)})\end{aligned}
• Yes, except the minus sign. But this is not important since you set them equal to $0$ – callculus Mar 17 at 23:40
• @callculus So it's $\frac{2}{m}$ rather than $\frac{-2}{m}$ for both the cases. I couldn't get what you meant by "you set them equal to 0". – user214 Mar 17 at 23:45
• Are you still interested in an answer? – callculus Mar 18 at 2:15
• @callculus Yes. I'd appreciate it. – user214 Mar 18 at 2:29
• usee214: Great, love it. – callculus Mar 18 at 15:38
The derivatives are almost correct, but instead of a minus sign, you should have a plus sign. The minus sign is there if we differentiate
$$J = \dfrac{1}{m}\sum_{i=1}^m\left[y_i-\theta_0-\theta_1 x_i\right]^2$$
If we calculate the partial derivatives we obtain
$$\dfrac{\partial J}{\partial \theta_0}=\frac{2}{m}\sum_{i=1}^{m}[y_i-\theta_0-\theta_1x_i]\cdot\left[-1 \right]$$ $$\dfrac{\partial J}{\partial \theta_1}=\frac{2}{m}\sum_{i=1}^{m}[y_i-\theta_0-\theta_1x_i]\cdot\left[-x_i \right]$$
In order to find the extremum of the cost function $$J$$ (we seek to minimize it) we need to set these partial derivatives equal to $$0$$ $$\dfrac{\partial J}{\partial \theta_0}=\frac{2}{m}\sum_{i=1}^{m}[y_i-\theta_0-\theta_1x_i]\cdot\left[-1 \right]=0$$ $$\implies \sum_{i=1}^{m}[y_i-\theta_0-\theta_1x_i]=0$$ $$\dfrac{\partial J}{\partial \theta_1}=\frac{2}{m}\sum_{i=1}^{m}[y_i-\theta_0-\theta_1x_i]\cdot\left[-x_i \right]=0$$ $$\implies \sum_{i=1}^{m}[y_i-\theta_0-\theta_1x_i]\cdot\left[x_i\right] = 0.$$
As we divide by $$-2/m$$ for both cases we will obtain the same result. If you had $$+2/m$$ then you would divide by $$2/m$$ and still obtain the same equations as stated above. If the equation that we need to solve are identical the solutions will also be identical.
• Can you please include the corrected formula in your answer? – user214 Mar 17 at 23:53
• You just have to multipy your partial derivatives by $(-1)$. Both ways lead to the same result. – MachineLearner Mar 18 at 6:22
• I'm trying to build a Stochastic Gradient Descent. So can I use 2/m insted of -2/m and calculate the gradients right? – user214 Mar 18 at 10:00
• @user214: In the end, the plus or minus does not make a difference, because you set the derivatives equal to zero. But your code could irritate other people. That is why you should use $2/m$ instead of the wrong $-2/m$ (but which leads to the same correct result) as a factor. – MachineLearner Mar 18 at 10:04
• @user214: I added more details. You can try it on your own for the correct version and for the wrong version. You will see that we obtain the same result if you solve for $\theta_0$ and $\theta_1$. – MachineLearner Mar 18 at 11:12 | 2019-06-16T15:43:03 | {
"domain": "stackexchange.com",
"url": "https://math.stackexchange.com/questions/3152235/partial-derivative-of-mse-cost-function-in-linear-regression",
"openwebmath_score": 0.9534010291099548,
"openwebmath_perplexity": 452.15401179298567,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9783846640860381,
"lm_q2_score": 0.8519528000888387,
"lm_q1q2_score": 0.833537554132078
} |
https://math.stackexchange.com/questions/2759162/can-roots-of-an-algebraic-equation-be-neither-real-nor-complex | # Can roots of an algebraic equation be neither real nor complex?
My usually understanding tells me that an algebraic equation can intersect the x-axis, touch it, or not touch it. In which cases I will get a real root, coinciding real root, and an imaginary root (conjugate pair) respectively.
Note that I am considering real coefficients only.
But consider the following:
The quadratic equation $p(x) = 0$ with real coefficients has purely imaginary roots. Then the equation $p(p(x)) = 0$ has what type of roots?
Since $p(x)$ can become zero only on input of some complex numbers, I get the sense that the roots of the equation $p(p(x)) = 0$ will also be imaginary. But this is all based on intuition.
The answer provided to me was that The roots are neither real nor purely imaginary. Then what are the roots? I mean they must be either real or imaginary.
All help will be appreciated!
• You said "the roots are neither real nor purely imaginary" . That means roots with both a real part and an imaginary part are present. – The Integrator Apr 29 '18 at 18:24
• A complex number has the form $a+bi$ where $a,b$ are real number variables. Ordinarily, if $b=0$ then we abbreviate $a+0\cdot i = a$ and that's a real number. Also, if $a=0$ then we abbreviate $0+bi=bi$ and that's a purely imaginary number. For example, the roots of $p(x)=x^2+x+1$ are $-\frac{1}{2} + \frac{\sqrt{3}}{2} i$ and $-\frac{1}{2} - \frac{\sqrt{3}}{2} i$, neither of which is real, and neither of which is purely imaginary. – Lee Mosher Apr 29 '18 at 18:27
This is just a vocabulary convention...
While a real number is $r$ in $(-\infty,\infty)$, a purely imaginary number is a product $r\cdot i$.
If we are using the notation $a+bi$ for arbitrary complex numbers, the purely real numbers have $b=0$ while the purely imaginary numbers have $a=0$.
For $p(x)=x^2+4$, the roots of $p(x)$ are purely imaginary: $\{-2i,+2i\}$. Not all of the roots of the equation $p(p(x))$ are purely imaginary.
• +1 I see. The answer provided to me was just worded badly. – SmarthBansal Apr 29 '18 at 18:34
To elaborate, if $$p(x)$$ is quadratic, then it must have two roots by The Fundamental Theorem of Algebra.
So we know $$p(x)=A(x-mi)(x-ni)$$ for some real numbers A,m,and n.
We know something more, m=-n by the Complex Conjugate Root Theorem, so $$p(x)=A(x-mi)(x+mi)$$.
What do you get if you multiply that out?
Now plug that result into itself and you get a quartic equation.
Do you know how to find these roots? The quadratic formula might be applicable. | 2019-09-15T14:21:00 | {
"domain": "stackexchange.com",
"url": "https://math.stackexchange.com/questions/2759162/can-roots-of-an-algebraic-equation-be-neither-real-nor-complex",
"openwebmath_score": 0.8552063703536987,
"openwebmath_perplexity": 153.6460094648539,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9783846659768267,
"lm_q2_score": 0.8519527982093666,
"lm_q1q2_score": 0.833537553904094
} |
https://math.stackexchange.com/questions/2991680/linear-transformations-f-gv-to-v-with-properties-f-circ-f-g-circ-g-0 | Linear transformations $f, g:V\to V$ with properties $f \circ f = g \circ g = 0_V$ and $f \circ g + g \circ f = 1_V$ imply $\dim V$ is even?
Let's say you have $$V$$, a vector space. It is $$n$$-dimensional. Also, linear transformations $$f,g : V \to V$$ are such that $$f \circ f = g \circ g = 0_V$$ and $$f \circ g + g \circ f = 1_V$$.
How do you prove that $$\dim V$$ is even?
Ok, the real question is what $$0_V$$ and $$1_V$$ are? Null vector and the vector $$\begin{pmatrix}1\\ \vdots\\1\\\end{pmatrix}$$ ?
EDIT: Thanks for the answers!
Now how do you prove it?
EDIT_2:
One more thing:
Suppose $$dim_\mathbb k V = 2$$, therefore exists a base $$\mathcal B$$ in $$V$$ such that:
$$M(f)_\mathcal B = \begin{pmatrix}0 & 0\\1 & 0\\\end{pmatrix}$$ , $$M(g)_\mathcal B = \begin{pmatrix}0 & 1\\0 & 0\\\end{pmatrix}$$
• $0_V$ is the zero map $\vec v\mapsto \vec 0$ of the vector space and $1_V$ is the identity map $\vec v \mapsto \vec v$. – lulu Nov 9 '18 at 17:34
• $0_V$ is the $0$ function on $V$ (ie. the function where $v\mapsto 0$ for all $v$). $1_V$ is the identity function on $V$ (ie, $v\mapsto v$) – memerson Nov 9 '18 at 17:35
Here is a sketch of my proof of the statement. Note that I use the notation $$\text{id}_V$$ instead of $$1_V$$.
We first observe that $$\text{ker}(f)\cap\ker(g)=0$$. Let $$x\in \text{ker}(f)\cap\ker(g)$$. By the condition $$f\circ g+g\circ f=\text{id}_V$$, we have $$x=(f\circ g+g\circ f)(x)=f\big(g(x)\big)+g\big(f(x)\big)=f(0)+g(0)=0.$$ Since $$\text{im}(f)\subseteq \ker(f)$$ and $$\text{im}(g)\subseteq \ker(g)$$ (due to the conditions $$f\circ f=0_V$$ and $$g\circ g=0_V$$), we obtain $$V=\text{im}(\text{id}_V)=\text{im}(f\circ g+g\circ f)\subseteq \text{im}(f)+\text{im}(g)\subseteq \ker(f)+\ker(g)\,.$$ Ergo, $$\ker(f)+\ker(g)=V$$ and $$\ker(f)\cap\ker(g)=0$$. Consequently, $$\ker(f)\oplus\ker(g)=V$$, as well as $$\text{im}(f)=\ker(f)$$ and $$\text{im}(g)=\ker(g)$$.
By the First Isomorphism Theorem for Vector Spaces, we have that $$\tilde{f}:=f|_{\operatorname{im}(g)}$$ is an isomorphism from $$\text{im}(g)=\ker(g)$$ to $$\text{im}(f)=\ker(f)$$. Similarly, $$\tilde{g}:=g|_{\text{im}(f)}$$ is an isomorphism from $$\text{im}(f)=\ker(f)$$ to $$\text{im}(g)=\ker(g)$$. Indeed, $$\tilde{g}=\tilde{f}^{-1}$$. In particular, if $$V$$ is finite-dimensional, then $$f$$ and $$g$$ have equal rank, and so $$\dim(V)=\text{rank}(f)+\text{rank}(g)$$ is an even integer.
$$0_V$$ is the function $$V \to V$$ such that $$0_V(v) = 0$$ for all $$v \in V$$ (where $$0$$ is the zero vector in $$V$$).
$$1_V$$ is the function $$V \to V$$ such that $$1_V(v) = v$$ for all $$v \in V$$.
Both of these are linear operators.
Hint Since $$f^2 = 0$$, $$\operatorname{im} f \subseteq \ker f$$, so $$\operatorname{rank} f \leq \dim\ker f$$, and the Rank-Nullity Theorem then implies that $$\operatorname{rank} f \leq \frac{1}{2} n$$. By symmetry $$\operatorname{rank} g \leq \frac{1}{2}n$$, too.
Now, $$n = \operatorname{rank} 1_V = \operatorname{rank}(fg + gf) \leq \operatorname{rank}(fg) + \operatorname{rank}(gf) .$$ We have $$\operatorname{rank}(fg) \leq \min(\operatorname{rank} f, \operatorname{rank} g) \leq \frac{1}{2} n$$, and by symmetry $$\operatorname{rank}(gf) \leq \frac{1}{2} n$$, so $$\operatorname{rank}(fg) + \operatorname{rank}(gf) \leq n$$, which forces all of the inequalities to be equalities. In particular, $$n$$ is even.
Here:
• $$0_V$$ denotes the zero transformation $$V \to V$$, $$v \mapsto 0$$, and
• $$1_V$$ denotes the identity transformation $$V \to V$$, $$v \mapsto v$$.
Hint Since $$f^2 = 0_V$$, there is a basis for which $$f$$ has Jordan normal form matrix representation $$[f] = \pmatrix{0_{k \times k}\\&\pmatrix{0&1\\&0}^{\oplus \ell}} ,$$ where $$k + 2 \ell = n$$ . Now, write $$[g]$$ in block matrix notation, with block sizes $$k \times k$$ and $$(2 \ell) \times (2 \ell)$$.
Expanding gives $$[1_V] = [fg + gf] = [f][g] + [g][f] = \pmatrix{0_{k \times k} & \ast\\ \ast&\ast} .$$ Since $$[1_V]$$ is the identity matrix, $$k = 0$$, so $$n = 2 \ell$$, and in particular $$n$$ is even. (Remark This proof shows that the hypothesis $$g^2 = 0_V$$ is unnecessary.)
• How do you know you the nullify of $g$ is at least $k$. And do you know you can block diagonalize $f$ and $g$ simultaneously? – Callus - Reinstate Monica Nov 9 '18 at 23:50
• The proof does not require knowing the nullity of $g$, nor does it use that $g^2 = 0$ for that matter. The proof also does not use simultaneous block-diagonalization of $f$ and $g$ (though it turns out that the hypotheses do imply that that is possible). – Travis Willse Nov 10 '18 at 0:10
• Oh I see. I thought you were suggesting to write $g$ in Jordan blocks, but you just said in blocks. This is nice, and illuminating. Thanks. – Callus - Reinstate Monica Nov 10 '18 at 2:11
• Thanks, and you're welcome. – Travis Willse Nov 10 '18 at 3:09 | 2019-12-13T17:17:19 | {
"domain": "stackexchange.com",
"url": "https://math.stackexchange.com/questions/2991680/linear-transformations-f-gv-to-v-with-properties-f-circ-f-g-circ-g-0",
"openwebmath_score": 0.988229513168335,
"openwebmath_perplexity": 110.64880117531722,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.978384665346564,
"lm_q2_score": 0.8519527982093666,
"lm_q1q2_score": 0.83353755336714
} |
https://math.stackexchange.com/questions/1208457/find-fc-to-mathbbr2-continuous-and-bijective-but-not-open-c-subset-mathb | # Find $f:C\to\mathbb{R}^2$ continuous and bijective but not open, $C\subset\mathbb{R}^2$ is closed
Are there a closed subspace $C$ of $\mathbb{R}^2$ and a continuous, bijective function $f:C\to\mathbb{R}^2$ that is not open?
I cannot think about a continuous and bijective function $f:C\to\mathbb{R}^2$. My best candidate was the set $C=[0,+\infty)\times[0,+\infty)$ and the function $$f(s,t)=\Bigg(s\cos\bigg(2\pi\frac{t}{t+1}\bigg),s\sin\bigg(2\pi\frac{t}{t+1}\bigg)\Bigg),$$ but this is not injective in $\{0\}\times[0,+\infty)$.
The only hint that I can think about is that $C$ cannot be compact, because $f$ is bijective. Therefore, $C$ cannot be bounded.
An idea that I've been trying is to work in the sphere $S^2$. There, we can think of $C$ as a subset of $S^2$ with N=(0,0,1) as a limit point such that $C\cup\{N\}$ is compact.
Then the function $f$ takes $C$ onto $S^2-\{N\}$. But in this way I only see open functions.
Any suggestions?
• I guess not open into $f(C)$. I post an example next. Mar 27, 2015 at 8:46
• To be more precise, since $f:C\to\mathbb{R}^2$ has to be bijective, we have to find $f$ that is not open into $f(C)=\mathbb{R}^2$, don't forgetting that $C$ is closed. Mar 27, 2015 at 13:28
Let $C\subset\mathbb R^2$ be the line $y=0$, and $$f(x,0)=\big(\frac{x^3}{1+x^4},\frac{x}{1+x^4}\big).$$ Then $f(C)$ is a lemniscate with two branches crossing at the origin $(0,0)=f(0,0)$.
The mapping $f:C\to f(C)$ is not open, because an interval around $x=0$ in $C$ maps into a single branch (the vertical one) of $f(C)$, missing the other.
This idea can be used further, as in the following pictures.
One starts with a closed band $A\subset\mathbb R^2$ and modifies it homeomorphically to have a band with only to segments in the boundary. Then bends it a la lemniscate, which gives something like a disc with two holes, which is homeomorphic to a disc minus two points, then to the plane minus two points. Finally the non open bijection $A\to\mathbb R^2\setminus\{\alpha,\beta\}$ extends to a bijection $C\to\mathbb R^2$: $C$ is $A$ plus two separated points $a,b$ and the extension, continuous but not open, maps $a\mapsto\alpha, b\mapsto\beta$.
The next pictures are a variation of the above argument that accompasses @Chilote smart answer:
Here we see the same topologist strategy to produce a bijection $A\to\mathbb R^2\setminus\{\alpha\}$.
Finally, a remark to add context to the problem. Suppose we can define a homeomorphism $f:C\to\mathbb R^n$ from a closed subset $C\subset\mathbb R^n$ onto $\mathbb R^n$. Then it extends to the Alexandroff one point compactifications $C^*\sim{\mathbb R^n}^*$. Now, that of $\mathbb R^n$ is the sphere $\mathbb S^n$ and that of $C$ is its closure in $\mathbb S^n$. But a sphere can't be homeo to a proper subset, hence the starting $f$ cannot exist. Consequently, the problem is
Is there a continuous bijection $C\to \mathbb R^2$?
Indeed, none that exists can be open, because then it would be a homeo.
• Thanks for the example, but I am looking for a function such that $f(C)=\mathbb{R}^2$. That is why I ask $f:C\to\mathbb{R}^2$ to be bijective. Mar 27, 2015 at 13:18
• I think that the same idea works, but explicit equations are out of rich. In any case it's very interesting. I'm trying to make some pictures (for a topologist would be a proof) and edit my answer with them. Mar 27, 2015 at 16:32
The answer of @Jesus RS is brilliant! It shows the more important point of the problem: $C$ does not have to be connected.
If we have a closed subspace $C$ of $\mathbb{R}^2$ and a continuos and bijective function $g:C\to\mathbb{R}^2\setminus\{p\}$, where $p\in\mathbb{R}^2$, then we just have to consider a point in the complement of $C$, say $q$, and the function $$f:C\cup\{q\}\to\mathbb{R}^2$$ $$f(z)= \begin{cases} &g(z)&,&x\neq q\\ &p&,&x=q \end{cases}$$ will be continuous and bijective, but is not open because $\{q\}$ is open in $C\cup\{q\}$ but $f(\{q\})=\{p\}$ is not open in $\mathbb{R}^2$.
Now it is very easy to show an explicit function satisfying these conditions. The function $f:\big([0,+\infty)\times\mathbb{R}\big)\cup\{(-1,0)\}\to\mathbb{R}^2$ defined by $$f(x,y)= \begin{cases} &\bigg(e^y\cos\big(\frac{2\pi x}{x+1}\big),e^y\sin\big(\frac{2\pi x}{x+1}\big)\bigg)&,&(x,y)\neq (-1,0)\\ &(0,0)&,&(x,y)=(-1,0) \end{cases}$$ is continuous, bijective and not open, where the domain is a closed subspace of $\mathbb{R}^2$.
• Nice example/explanation. Mar 27, 2015 at 20:57
• Note that, even discounting the behavior at the isolated point $(-1,0) \in C$, we have problems with open-ness. For instance, $[0,1) \times \mathbb{R}$ is open in $C$, but its image by $f$ is not open in $\mathbb{R}^2$. Mar 27, 2015 at 21:03
• But the question remains whether $f$ is open if $C$ is connected. Mar 27, 2015 at 23:42 | 2022-05-18T05:50:29 | {
"domain": "stackexchange.com",
"url": "https://math.stackexchange.com/questions/1208457/find-fc-to-mathbbr2-continuous-and-bijective-but-not-open-c-subset-mathb",
"openwebmath_score": 0.9223070740699768,
"openwebmath_perplexity": 168.15719546406524,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9783846628255123,
"lm_q2_score": 0.8519527982093666,
"lm_q1q2_score": 0.8335375512193229
} |
https://en.khanacademy.org/math/precalculus/x9e81a4f98389efdf:limits-and-continuity/x9e81a4f98389efdf:connecting-limits-at-infinity-and-horizontal-asymptotes/v/more-limits-at-infinity | If you're seeing this message, it means we're having trouble loading external resources on our website.
If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.
## Precalculus
### Course: Precalculus>Unit 10
Lesson 14: Connecting limits at infinity and horizontal asymptotes
# Limits at infinity of quotients (Part 2)
AP.CALC:
LIM‑2 (EU)
,
LIM‑2.D (LO)
,
LIM‑2.D.3 (EK)
,
LIM‑2.D.4 (EK)
,
LIM‑2.D.5 (EK)
Sal analyzes the limits at infinity of three different rational functions. He finds there are three general cases of how the limits behave. Created by Sal Khan.
## Want to join the conversation?
• I understand how he gets to the limits and all, but can someone give me a 100% accurate definition of what exactly a "asymptote" is? what is an asymptote? Is it just the graph of the limit? please help me !
• This is actually not quite accurate.
Technically an asymptote to a curve is a line such that the distance between the curve and the line approaches zero as they both tend toward infinity.
A function can never cross a vertical asymptote, but it may cross a horizontal asymptote an infinite number of times (thus reaching it an infinite number of times). For example observe the limit of sin(x)/x as x approaches infinity.
• Are + and - infinity like approaching from the positive and negative sides? So limits approaching infinity dont exist, because + and - infinity yield different values?
• Not exactly,
positive and negative infinity represent the opposite "ends" of the number line. And here, "ends" is in quotation marks because the number line NEVER actually ends, it goes on forever in both directions. Basically positive infinity means to keep going towards bigger and bigger positive numbers. Think of the biggest positive number you can think of, and then go even bigger than that… and keep doing that… FOREVER! That's positive infinity.
For negative infinity, think of the most negative number you can think of, and then think of an even more negative number, and keep doing that, FOREVER.
So you see, if a limit approaches positive infinity from one side, and negative infinity from the other side… it doesn't approach the same thing from both sides. THIS is why the limit doesn't exist. It would be the same as saying that a limit that approaches 3 from the positive side and 2 from the negative side also doesn't exist. In order for a limit to exist it must approach the same thing from both sides.
• Where exactly can we use the concept of limits in our life?
In what kind of situations in our life do we need to find limits?
Please give me a real life example where in we have to find the limit.
• Limits are not that helpful in everyday life. However, as you study Calculus, you'll see that every single concept in Calculus in based upon limits. Derivatives are defined using limits, we can find the area under a curve, volume, surface area, arc length, radius of curvature using integrals that are all based upon limits. Without limits, there is no Calculus.
• In the solutions manual of my Calculus textbook, it gets the answer using a slightly different method. It divides like every term in the numerator and the denominator by the highest degree i guess and does all these weird calculations and then gets the answer. I understand this method much easier though (getting the highest power of both the numerator and denominator and then applying the x-> component) I guess my question is if I do this method that you are teaching on a test, is it still valid and legitimate? Of will i have to provide more work to justify my answer? Sorry for the incredibly long comment. Thanks.
• Sal's method is more of a "rationalize it" (no pun intended) approach. What your textbook says is the method I prefer, mainly because it gives you a clear, algebraic way to evaluate the limit.
As for which method to use on a test, that is something you should ask your teacher.
• So then would it be safe to assume that any time the numerator is growing faster than the denominator it will equal infinity, and any time the denominator is growing faster than the numerator it will equal 0? Or is there an exception?
• ``Notice that we are dealing with x nearly equal to Infinity. ``
Which is very very big. Thus we can safely assume.
But in case of small numbers like 2, 56, 345.... We cant assume so.
You can try it out yourself.
• At how can he simplify that to 1/2x when there are two different powers?
• He cancelled out common factors
3x^3/6x^4 can be factored to
(3*x*x*x)/(2*3*x*x*x*x)
You can cancel out the three and three of the x terms
( 3*x*x*x )/(2* 3*x*x*x *x)
This leaves 1/(2*x)
• at , is the simplification of (4/250)x to (2/125)x valid ?
• Yes, but it is unnecessary. With limits, since you often have them diverge toward +∞ or −∞ or else tend toward 0, you can save yourself unnecessary work by not simplifying any constants until you know you don't have an infinity or zero situation. When tending toward 0, your constant is irrelevant and there is no need to simplify. When tending toward ∞, you need only determine the sign of your constant, to determine whether you're tending toward + or − infinity.
It is only when you're tending toward a non-zero but finite number that you need to simplify your constant.
Note: The above applies to real numbers. If imaginary numbers get involved, the considerations can be more complicated.
• Lim(x--->infinity) (1+1/n)^n =e
Lim(x--->infinity) (1+1/x)^1/x=e
How to do solve these very strange infinity limits or either to prove them they equal to e?
• ``x (1 + 1/x)^x (1 + 1/x)^(1/x)1 2.000000000 2.0000000000000010 2.593742460 1.00957658300000100 2.704813829 1.000099508259151000 2.716923932 1.0000009995008310000 2.718145927 1.00000000999950100000 2.718268237 1.000000000100001000000 2.718280469 1.0000000000010010000000 2.718281694 1.00000000000001``
• Will a similar method be used for an expression in which there is no denominator like for limit x tending to infinity √(n-1) - √n
Here will it be correct to say that this is the same as √n - √n (As the 1 will be negligible when n tends to infinity) and thus the limit equals 0 ?
• Yep. That general pattern of thinking is correct; actually, though, the example you've mentioned is a bit interesting in the sense that we can't always define a limit that takes the form ∞ - ∞. In this case, because the two terms are of the same degree, the limit is equal to 0 (and a quick glance at the graph of `y = sqrt(x-1) - sqrt(x)` confirms that as x approaches infinity, y approaches 0). As you said, it resembles `y = sqrt(x) - sqrt(x) = 0` in the limit.
Other limits of a similar nature may not always behave the same way. Take the limit of `x^3 - x^2` as x approaches infinity, and we get infinity rather than 0 because the terms are of a different degree (which seems fairly clear just by looking at the function). Sometimes the examples are less clear-cut, so it's worth exercising some caution with limits of the form ∞ - ∞.
I hope you find that helpful; you're definitely on the right track. | 2023-03-24T00:21:38 | {
"domain": "khanacademy.org",
"url": "https://en.khanacademy.org/math/precalculus/x9e81a4f98389efdf:limits-and-continuity/x9e81a4f98389efdf:connecting-limits-at-infinity-and-horizontal-asymptotes/v/more-limits-at-infinity",
"openwebmath_score": 0.85343998670578,
"openwebmath_perplexity": 458.68112019304044,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9901401438236134,
"lm_q2_score": 0.841825655188238,
"lm_q1q2_score": 0.8335253753024896
} |
http://tiww.scoutsannecystmaurice.fr/properties-of-matrix-multiplication-proof.html | # Properties Of Matrix Multiplication Proof
If Y = 0 @ a d b e c f 1 A; the identity is det(YTY) = (a2+b2+c2)(d2+e2+f2) (ad+be+cf)2 = (bf ce)2+(cd af)2+(ae bd)2 ; 4 THE GRAM MATRIX, ORTHOGONAL PROJECTION, AND VOLUME. Matrix multiplication is multiplying two matrices together. Proof (of Statement 1 of the Theorem): Let A be a n p matrix. Linear Algebra 59, Matrix Multiplication, Properties and proof of Scalar Multiplication LadislauFernandes. The paper considers the complexity of bilinear forms in a noncommutative ring. Matrix multiplication is not like addition or subtraction. A matrix is a set of numbers that are ordered into rows, columns, or both. The Closure Property states that when you perform an operation (such as addition, multiplication, etc. A scalar is a real number. Therefore, so called \combina-torial algorithms" are desirable. (a) ( Associativity of Matrix Multiplication ) If A, B, and C are matrices which are compatible for multiplication, then (b). ¶ While certain "natural" properties of multiplication do not hold, many more do. 1 as a consequence of the Lagrange identity in the ring of Laurent polynomials of two variables. c) If x 2 V then 0 ¢ x = 0. A simplified proof of a modification of Booth's multiplication algorithm by MacSorley to a form which examines three multiplier bits at a time is presented. Two equations that have the same solution are called equivalent equations e. If A is any matrix with n columns, then AIn = A. Every matrix transformation is a linear transformation. Matrix multiplication is always commutative if. Matrices and elementary matrix operations are then introduced, including addition, multiplication by a scalar, and matrix multiplication. We already know that = ad − bc; these properties will give us a c d formula for the determinant of square matrices of all sizes. For square matrices, the transposed matrix is obtained by reflecting the matrix at the diagonal. Multiplication of Matrices. matrix properties of magic squares a professional paper submitted in partial fulfillment of the requirements of the degree of master of science in the graduate school of texas woman's university college of arts and sciences by daryl lynn stephens, b. eq j} a_k {\bb v}^{(k)}$results in a matrix whose determinant is the same as the original matrix. If $$(rs)X =r (sX)$$ Define the elements belonging to$\mathbb{R}^2$as$\{(a,b)|a,b\in\mathbb{R}\}$. This formula shows the summation over the trapezium (quadrangle) in a different order. The DFT can be formulated as a complex matrix multiply, as we show in this section. A college (or advanced high school) level text dealing with the basic principles of matrix and linear algebra. In other words, if the order of A is m x n. This is sometimes used to write a column matrix in the middle of text by saying, for example, that c T= [1 3 -4 5]. In most applications the field F will be the field of real numbers R or the field of complex numbers C. The matrix O is called the additive identity or zero matrix. Let take as an example a partition that includes the index 2 2J 1 and the index 4 2L 2. Their product is the identity. This property is still true, but the converse is not necessarily true. I need help with a simple proof for the associative law of scalar. For matrix multiplication to work, the columns of the second matrix have to have the same number of entries as do the rows of the first matrix. In this topic, we will learn about the scalar multiplication of a matrix. Either one of these would be considered correct, proper proofs. Matrix multiplication: if A is a matrix of size m n and B is a matrix of. MATRIX MULTIPLICATION. WARNING: Take particular care with properties 6 and 10. Properties of matrix addition and scalar multiplication: see table page 61. When the number of columns of the first matrix is the same as the number of rows in the second matrix then matrix multiplication can be performed. 1Let A, B, and C be m×nmatrices. The order of the matrix multiplication is important, and the cancellation method serves as a safeguard against performing a matrix multiplication in the wrong order. Knill Section 5. Sorensen Math. Properties of Scalar Multiplication of a Matrix If A = [a ij ] and B = [b ij ] be two matrices of the same order , say m × n, and k and l are scalars, then k(A + B) = kA + kB,. 40), where denotes the transpose. Wilkins Academic Year 1996-7 7 Rings Definition. To begin with, order matters in matrix multiplication. To start practicing, just click on any link. We can prove them taking simple examples of matrix A and B. matrix properties of magic squares a professional paper submitted in partial fulfillment of the requirements of the degree of master of science in the graduate school of texas woman's university college of arts and sciences by daryl lynn stephens, b. Properties of the Rotation Matrix. It is important to note the order of multiplication. That is, if A is an m×n TRGONOMETRIC DENTITIES AND EQUATIONS Verifying a trigonometric identity Complete the proof of the identity by choosing the Rule that justifies. We get uv^T - (v^Tu)uv^T = (1-v^Tu)uv^T. The Wikibook Linear Algebra has a page on the topic of. Definition 1. Give students a copy of just the first diagram provided, the stickman graph. (a)Show this directly using properties of the transpose. and a matrix A=[aij]. Prove that scalar multiplication distributes over addition of matrices: k(A+ B) = kA+ kB. We already know that = ad − bc; these properties will give us a c d formula for the determinant of square matrices of all sizes. An inverse operation are two operations that undo each other e. The proof will be illustrated in a particular case. The four basic properties of matrix multiplication discussed in my course are. Insignificance of the order of multiplication. 19 (2) We can have A 2 = 0 even though A ≠ 0. Then the general linear group GL n(F) is the group of invert- ible n×n matrices with entries in F under matrix multiplication. Here, we list without proof some of the most important rules of matrix algebra - theorems that govern the way that matrices are added, multiplied, and otherwise manipulated. Properties of Matrix Addition: Theorem 1. To see this definitively compare the results of multiplying the matrices FG (from the previous page) versus GF. Prove that scalar multiplication distributes over addition of matrices: k(A+ B) = kA+ kB. Properties of matrix addition & scalar multiplication. In some school syllabuses you will meet scalar products but not vector products but we discuss both types of multiplication of vectors in this article to give a. Go to: Introduction, Notation, Index. Multiplication Properties of Exponents. Miscellaneous matrix de nitions A matrix is said to be square if its no. This is illustrated by figure 4, where the eigenvectors are shown in green and magenta, and where the eigenvalues clearly equal the variance components of the covariance matrix. for μ ranging from 1 to m and for ν ranging from 1 to n. (* denotes the conjugate transpose or hermitian operator). Properties of matrix scalar multiplication Learn about the properties of matrix scalar multiplication (like the distributive property) and how they relate to real number multiplication. The addition and scalar multiplication defined on real vectors are precisely the corresponding operations on matrices. TeachingTree is an open platform that lets anybody organize educational content. 1 Negative Triangle Detection Implies Matrix Product Theorem 1. eq j} a_k {\bb v}^{(k)}$ results in a matrix whose determinant is the same as the original matrix. In fact it can be accomplished with one instruction The spreadsheet iterates the matrix multiplication. 4 Properties of matrix. If you believe your intellectual property has been infringed and would like to file a complaint, please see our Copyright/IP Policy. , New York, NY, 1984. The Jordan -Canonical form of a Kronecker product will be examined. In linear algebra, a nilpotent matrix is a square matrix N such that = for some positive integer. Whenever we multiply a matrix by another one we need to find out the "dot product" of rows of the first matrix and columns of the second. There are two useful definitions of multiplication of vectors, in one the product is a scalar and in the other the product is a vector. AB = If, using the above matrices, B had had only two rows, its columns would have been too short to multiply against the rows of A. Multiplication of two matrices A and B is possible if the number of columns in A equals number of rows in B. Properties Rather than start with a big formula, we’ll list the properties of the determi a b nant. An important property that fails to hold is that in general AB and BA are not the same matrix. If A is an nxm matrix and O the mxk zero-matrix, then AO = O. However, some of the properties enjoyed by multiplication of real numbers are also enjoyed by matrix multiplication. Describe how evaluating this product corresponds to a Bellman-Ford-like algorithm (see. PROOF of [email protected] = of (Q -1 Q of = = z of Q ([email protected]) x of -1) U = [email protected] Matrix Notation for Geometric Transformations One important application of matrix algebra is in expressing the transfor-. Matrix chain multiplication , a matrix multiplication algorithm, uses this property to minimize the number of operations. Matrix approach. Drawn some iso-density contours of the Gaussian with the same mean and covariance as p. and apply (e) directly. ij] in column j. which one can readily check. Right-multiplication: combination of columns. Furthermore, even if AB and BA are of the same size, we know that AB 6= BA in general. Their product is the identity. edu/˘schiu/ Matrix Multiplication: Warnings WARNINGS Properties above are analogous to properties of real numbers. If $$A=[a_{i,j}]$$ is a matrix and $$r$$ is a scalar, then the matrix $$C=[c_{i,j}]=rA$$ is defined by. Here, we list without proof some of the most important rules of matrix algebra - theorems that govern the way that matrices are added, multiplied, and otherwise manipulated. Therefore there is no proof to why matrices are multiplied the way they do. Because matrix multiplication is such a central operation in many numerical algorithms, much work has been invested in making matrix multiplication algorithms efficient. Prove that matrix multiplication distributes over matrix addition: C(A+ B) = CA+ CB. So, matrix multiplication is just the image of composition of linear transformations under the identification of matrices with linear transformations. This property is still true, but the converse is not necessarily true. Parametric Representation of the Solution Set to a Linear Equation. a+b = b+a)? Does this have something to do with set theory?. Proof: The proofs are straightforward, based on the definition of trace and matrix addition and multiplication. Properties of matrix addition & scalar multiplication. A group Gconsists of a set Gtogether with a binary operation ∗ for which the following properties are satisfied: • (x∗y)∗z= x∗(y∗z) for all elements x, y, and zof G(the Associative Law);. For example If, you want to go through their proves then click particular property. Subsection VSP Vector Space Properties. Failure to do this can lead to ambiguity, misunderstanding and mistakes. The main im-portance of P4 is the implication that any results regarding determinants that hold for the rows of a matrix also hold for the columns of a matrix. Two matrices can be multiplied with each other when the number of columns of the first matrix equals the number of rows of the second matrix. We can prove them taking simple examples of matrix A and B. 5 Matrix Di erentiation In the following discussion I will di erentiate matrix quantities with respect to the elements of the referenced matrices. Let’s look at some properties of multiplication of matrices. If you like this Page, please click that +1 button, too. Matrix Multiplication: Loop Orderi. Matrix singularity is discussed and related to the vanishing of the determinant of the matrix. b = a ∧b Properties of Rings. Polynomial: The calculator returns the polynomial. A matrix is a set of numbers that are ordered into rows, columns, or both. The number of rows of the first matrix must be equal to the number of columns of the second matrix. You will notice that is one of the first things done in the proof below. A general 2×3 matrix has the form A = ' a11 a12 a13 a21 a22 a23 We use the convention that matrix entries aij are indexed so that the first subscript i refers to the row while the second subscript j refers to the column. Apply the properties of determinants and. ) Remark Not all square matrices are invertible. PROPERTIES OF DETERMINANTS. The spectral norm is the only one out of the three matrix norms that is unitary invariant, i. 2 3 1 Note that rank is r 3 and number of unknowns is n 4 so there are n r 1. Algorithm: A tripartite graph G = {I ∪J ∪K,E}is given by the negative triangle detection. Read in another language. Proof: If detA = 1 then A is a rotation matrix, by Theorem 6. Definition 1. Properties of equalities. det I = 1 2. In fact, it can be shown that the sole matrix, which is both an orthogonal projection and an orthogonal matrix is the identity matrix. Let us check linearity. To multiply two matrices,We first write their orderFor multiplicationSince 2 ≠ 3We cannot multiply themBut, if we multiply BAThen,So, order of matrix after multiplication is=3 × 2Let's learn how to multiply themSo,AB was not possible, but BA was possibleThus, AB ≠ BALet's do some more examplesSo, mu. Matrix multiplication is not that hard, just follow these steps. If A is any matrix with n rows then InA = A. Properties of matrix multiplication. Introduced some key properties linked to matrix multiplication. The task of computing a matrix -norm is difficult for since it is a nonlinear optimization problem with constraints. Matrix Formulation of the DFT. 1 Introduction This is a Part I of an introduction to the matrix algebra needed for the Harvard Systems Biology 101 graduate course. Note that BMM can be computed using an algorithm for integer matrix multiplication, and so we have BMM for n !nmatrices is in O(n ) time, where !<2:373 (the current bound for integer matrix multiplication). In this section we present some properties of the matrix H nfor general nand use them to study the kernel of H n. That is AB ≠ BA. If A is an n×n symmetric orthogonal matrix, then A2 = I. 3 Matrices and matrix multiplication 2 8 Properties of determinants 10 9 Gaussian elimination 11 1. This is illustrated by figure 4, where the eigenvectors are shown in green and magenta, and where the eigenvalues clearly equal the variance components of the covariance matrix. proof (case of λi distinct) suppose Symmetric matrices, quadratic forms, matrix norm, and SVD 15-19. If detA = ¡1 then det(¡A) = (¡1)3 detA = 1. We can calculate a mean for each variable and replace the data matrix with a matrix of deviations from the mean. Properties of matrix multiplication: see table page 61. Given an order basis, points in space could be expressed as the set of all ordered tuples ( x , y , z ) where x , y , z. The reader can verify that these properties hold for f but not for g. After all, matrix multiplication represents function composition, which is not commutative— if f(x) = 2x and g(x) = x + 1 then while. 1: PROPERTIES OF MATRIX ADDITION AND SCALAR MULTIPLICATION If A, B, and C are m n matrices, and c and d are scalars, then the following properties are true. The proofs of these properties are given at the end of this section. Matrix multiplication is really useful, since you can pack a lot of computation into just one matrix multiplication operation. Indeed, the matrix product ABmight be well-defined, while the product BA might not. det I = 1 2. A summary of Matrix Multiplication in 's Matrices. The nullspace of any orthogonal matrix is {0}. Finally, in Section 3, we will prove Theorem 1. Recall that the determinant of a 2x2 matrix is. multidimensional matrix equality as well as the multidimensional matrix algebra operations for addition, subtraction, multiplication by a scalar, and multiplication of two multidimensional matrices. The determinant of a 4×4 matrix can be calculated by finding the determinants of a group of submatrices. A, B, and C are matrices. The Jordan form is presented as an appendix. Then the general linear group GL n(F) is the group of invert- ible n×n matrices with entries in F under matrix multiplication. We rst introduce some notation and. One Bernard Baruch Way (55 Lexington Ave. By convention, an n-dimensional vector is often thought of as a matrix with n rows and 1 column, known as a column vector. The nullspace of any orthogonal matrix is {0}. After discovering the commutative property does not apply to matrix multiplication in a previous lesson in the series, pupils now test the associative and distributive properties. The proof of the rank result is done using techniques from Character Theory and Analytic Number Theory. So if n is different from m, the two zero-matrices are different. Matrices and elementary matrix operations are then introduced, including addition, multiplication by a scalar, and matrix multiplication. That is, if A is an m×n TRGONOMETRIC DENTITIES AND EQUATIONS Verifying a trigonometric identity Complete the proof of the identity by choosing the Rule that justifies. it’s a Markov matrix), its eigenvalues and eigenvectors are likely to have special properties as well. This column should be treated exactly the same as any. We already know that = ad − bc; these properties will give us a c d formula for the determinant of square matrices of all sizes. iii) Matrix multiplication is distributive over matrix addition i. This property is still true, but the converse is not necessarily true. Properties of matrix multiplication. So, matrix multiplication is just the image of composition of linear transformations under the identification of matrices with linear In particular, then, distributivity of matrix multiplication is really just distributivity of composition of linear transformations, which lends itself to a far more transparent. Mechanical engineer's simple running hack is fun and increases efficiency; Meet the 'mold pigs,' a new group of invertebrates from 30 million years ago. Random Example will generate. ) Remark Not all square matrices are invertible. Proving the Properties of Natural Numbers Date: 03/09/2000 at 10:53:15 From: Harsha Subject: How to prove a+b = b+a Dear Dr. Cayley's defined matrix multiplication as, "the matrix of coefficients for the composite transformation T2T1 is the product of the matrix for T2times the matrix of T1"(Tucker, 1993). This means that you can multiply 1 to any number and it keeps its identity! The number stays the same! Let's look at the number 8. Use the properties of matrix multiplication and the identity matrix Find the transpose of a matrix THEOREM 2. proof of properties of trace of a matrix. How to Solve Matrix Equations. DIFFERENCE BETWEEN LINEAR CONVOLUTION & CIRCULAR CONVOLUTION. The only non-trivial observation is that multiplication of circulant matrices in standard form corresponds to the multiplication in C[X]=(Xn 1). position down into the subspace, and this projection matrix is always idempo-tent. If a matrix has some special property (e. Hint: draw the. Math 201-NYC-05 Vectors and Matrices. If r represent scalar elements and A, B and C represent matrices A(B+C) = AB+AC shows that matrices obey distributive properties when the matrix A is being multiplied from the left side. Elementary Matrices and the Four Rules. Strassen, "The Asymptotic Spectrum of 1'ensors and the Exponent of Matrix Multiplication," 1986 FOCS, pp. The set of matrices G= ˆ e= 1 0 0 1 ,a= −1 0 0 1 ,b= 1 0 0 −1 ,c= −1 0 0 −1 ˙ under matrix multiplication. Our goal is for students to quickly access the exact clips they need in order to learn individual concepts. We now collect ten of them here for later reference. TeachingTree is an open platform that lets anybody organize educational content. Multiplication by scalars: if A is a matrix of size m n and c is a scalar, then cA is a matrix of size m n. 3) Want to show that not all reals in the interval between 0 and 1 Matrix multiplication Properties of matrix. for μ ranging from 1 to m and for ν ranging from 1 to n. We will apply most of the following properties to solve various Algebraic problems. Matrix chain multiplication , a matrix multiplication algorithm, uses this property to minimize the number of operations. There is an easy way to remember the formula for the cross product by using the properties of determinants. Some simple properties of vector spaces Theorem Suppose that V is a vector space. Proof Sometimes a matrix is equal to its adjoint (Definition A), and these matrices have interesting properties. In other words, if the order of A is m x n. At the end of the previous lecture I pointed out the rst of these{the fact. If at least one input is scalar, then A*B is equivalent to A. General properties. and apply (e) directly. every row of the multiplication table contains every element of Gexactly once a similar argument shows that every column of the multiplication table contains every element of Gexactly once A table satisfying these two properties is called a Latin Square. For example 4 + 2 = 2 + 4. (We say B is an inverse of A. The matrix lower bound has a natural interpretation in functional analysis, and it satisfies many of the properties that von Neumann stated for it in a restricted case. Then, in 1969, Volker Strassen's stunned. The validity of both formulas is then a polynomial identity in the entries of a 3 2 matrix. see here for further information. It plays a role similar to 1 for multiplication of real numbers. 1 Matrix Operations Shang-Huan Chiu Department of Mathematics, University of Houston [email protected] Naive matrix multiplication refers to the naive algorithm for executing matrix multiplication: we calculate each entry as the sum of products. Selected Problems — Matrix Algebra Math 2300 1. To multiply two matrices,We first write their orderFor multiplicationSince 2 ≠ 3We cannot multiply themBut, if we multiply BAThen,So, order of matrix after multiplication is=3 × 2Let's learn how to multiply themSo,AB was not possible, but BA was possibleThus, AB ≠ BALet's do some more examplesSo, mu. Matrix multiplication: if A is a matrix of size m n and B is a matrix of. A secondary school algebra course would probably give one more than enough background. Note: the row echelon form of a matrix is not unique. Multiplication of matrices generally falls into two categories, Scalar Matrix Multiplication, in which a single real number is multiplied with every other element of the matrix and Vector Matrix Multiplication wherein an entire matrix is multiplied by another one. The following properties hold $result from a multiplication of the diagonal terms$\prod_{i=1}^{n} (\lambda-A_{ii})$. If there exists a positive integer n such that a n = e, then a is said to have finite order , and the smallest such positive integer is called the order of a, denoted by o(a). Matrix Power Calculator Here you can raise a matrix to a power with complex numbers online for free. The transpose A T of the matrix A is defined as the k x m matrix with the components. von Neumann. You will notice that the commutative property fails for matrix to matrix multiplication. There are three kinds of elementary matrix operations. Matrix multiplication: if A is a matrix of size m n and B is a matrix of. If the covariance matrix of our data is a diagonal matrix, such that the covariances are zero, then this means that the variances must be equal to the eigenvalues. Proof of Property 2 Again we show that the general element of. Let A be an nxn invertible matrix, then det(A 1) = det(A) Proof — First note that the identity matrix is a diagonal matrix so its determinant is just. Learn about the properties of matrix multiplication (like the distributive property) and how they relate to real number multiplication. However, the properties of matrices restrict a. We saw that if two rows are the same, then a square matrix has 0 determinant. What sizes of matrices can be multiplied together? Theorem (Properties of matrix inverse). If B is any matrix, then AB has an entire rows of zeros, or BA has an entire column of zeros. Matrix multiplication can be done only when the number of columns of is equal to the number of rows of. Proof: Let A be an n×n matrix. Left as an exercise. Detailed Description for All Properties Worksheets. and apply (e) directly. This implies that UUT = I, by uniqueness of inverses. Multiplication and scalar multiplication associate: c(dv) = (cd)v h. Chapter 3 reviews the determinant of a square matrix, the relation between a non-zero determinant and the existence of the inverse matrix, a formula for the inverse matrix using the matrix of cofactors, and the Cramer rule for the formula of the solu-tion of a linear system with an invertible matrix of coefficients. Multiplication of an entire matrix by another entire matrix For the rest of the page, matrix multiplication will refer to this second category. Matrix Trace. Concept of elementary row and column operations. Multiplication with a 'unit puls' is done to find the responses of the matrix and it's transpose. Let's look at some properties of multiplication of matrices. Properties of matrix multiplication. In most applications the field F will be the field of real numbers R or the field of complex numbers C. In other words, if the order of A is m x n. Random Vectors and Multivariate Normal Distributions Properties of Mean and Covariance (cont. Math · Precalculus · Matrices · Properties of matrix multiplication Properties of matrix multiplication Learn about the properties of matrix multiplication (like the distributive property) and how they relate to real number multiplication. A general 2×3 matrix has the form A = ' a11 a12 a13 a21 a22 a23 We use the convention that matrix entries aij are indexed so that the first subscript i refers to the row while the second subscript j refers to the column. Assume A is invertible. Matrix multiplication (conventional) is associative. Since Theorem SMZD is an equivalence (Proof Technique E) we can expand on our growing list of Sage NME7 Nonsingular Matrix Equivalences, Round 7. Multiplies two matrices, if they are conformable. and Theorem 3 is proved. More identities encoded in matrix multiplication The angle sum and double angle formulas are encoded in matrix multipli-cation, as we saw above. If the zero matrix O is multiplied times any matrix A, or if A is multiplied times O, the result is O (see Exercise 16). The addition and scalar multiplication defined on real vectors are precisely the corresponding operations on matrices. Matrix multiplication Properties of matrix multiplication: •Does AB = BA? Example: AB MBA. • The set of all even integers forms a commutative ring under the usual addition and multiplication of integers. Find an optimal parenthesization of a matrix-chain product whose sequence of dimensions is$\langle 5, 10, 3, 12, 5, 50, 6 \rangle$. To multiply two matrices,We first write their orderFor multiplicationSince 2 ≠ 3We cannot multiply themBut, if we multiply BAThen,So, order of matrix after multiplication is=3 × 2Let's learn how to multiply themSo,AB was not possible, but BA was possibleThus, AB ≠ BALet's do some more examplesSo, mu. If B is any matrix, then AB has an entire rows of zeros, or BA has an entire column of zeros. Despite examples such as these, it must be stated that in general, matrix multiplication is not commutative. 49-54; also "Relative bilinear complexity and mat:fix multiplication," preprint. If you don't have a high resolution display, the bottom a property of optimisation problems in which the sub-problems which constitute the solution to the problem itself are themselves. Matrices, transposes, and inverses Math 40, Introduction to Linear Algebra Properties of matrix multiplication Proof. ) (Remark 2: Given a linear system, fundamental matrix solutions are not unique. The rst theorem stated that 0v = 0 for all vectors v. The matrix O is. Note: If a +1 button is dark blue, you have already +1'd it. von Neumann. The first one proves that$\R^n$is a direct sum of eigenspaces of. Matrix multiplication shares some properties with usual multiplication. That is, each element is replaced by where is the mean of the variable. ) At this site I used matrices to derive Binet's Formula for the Fibonacci numbers and Cassini's Identity that exhibits one of their properties. Every row and every column of a permutation matrix contain exactly one nonzero entry, which is 1: There are two 2 2 permutation. Prove the following vector space properties using the axioms of a vector space: the cancellation law, the zero vector is unique, the additive inverse is unique, etc. The set of matrices G= ˆ e= 1 0 0 1 ,a= −1 0 0 1 ,b= 1 0 0 −1 ,c= −1 0 0 −1 ˙ under matrix multiplication. Theorem 2 Matrix multiplication is associative. From now on, we will not write (mxn) but mxn. Theorem properties of matrix multiplication let a b Proof Suppose T is a matrix transformation say Tv Av Then by properties of McGill University MATH 133 - Fall 2010 Lecture Notes. Matrix multiplication is one of the most fundamental prob-lems in scienti c computing and in parallel computing. Multiplication of two sequences in time domain is called as Linear convolution. The map that sends a circulant matrix V to its transpose Vt is an involution of Circ(n) and corresponds under J to the automorphism of C[X]=(Xn 1) induced by X7!Xn 1. The characteristic matrix of A [n#n] is (tI-A) and is a function of the scalar t. Show that matrix multiplication defined by$\text{EXTEND-SHORTEST-PATHS}$is associative. We demonstrate this on Strassen's and other fast matrix multiplication algorithms, and obtain the first lower bounds on their communication costs. In this topic, we will learn about the scalar multiplication of a matrix. For example, 3 1 2 0 −3 = 3 6. Note: the row echelon form of a matrix is not unique. To carry out the multiplication we must have the column dimension of the left factor equal to the row dimension of the right factor. (This section can be omitted without affecting what follows. We can also multiply numbers in any order. For a symmetric matrix with real number (The proof of this follows the same pattern. The associative laws state that when you add or multiply any three matrices, the grouping (or association) of the matrices does not affect the result. Show that matrix multiplication defined by$\text{EXTEND-SHORTEST-PATHS}\$ is associative. For a square matrix A, the inverse is written A -1. The dimensions of a stick of timber, are found, by applying to it a measuring rule of known length. 1- Computing Matrix Exponential for Diagonal Matrix and for Diagonalizable Matrices if A is a diagonal matrix having diagonal entries then we have e e n 2 1 a a % a A e e Now, Let be n n A R. Today's content is taken from Matrices and Linear Transformations by Charles G. Let A and B be m×n matrices with entries in C. Proof Let A and B be matrices for linear transformations T: Rm → Rn and S: Rk → Rn. To begin with, order matters in matrix multiplication. To multiply two matrices,We first write their orderFor multiplicationSince 2 ≠ 3We cannot multiply themBut, if we multiply BAThen,So, order of matrix after multiplication is=3 × 2Let's learn how to multiply themSo,AB was not possible, but BA was possibleThus, AB ≠ BALet's do some more examplesSo, mu. In this video, I wanna tell you about a few properties of matrix multiplication. The identity property of multiplication, also called the multiplication property of one says that a number does not change when that number is multiplied by 1. “outer” parameters become parameters of matrix AB What sizes of matrices can be multiplied together? If A is a square matrix and k is a positive integer, we define Ak = A · A···A k factors Properties of matrix multiplication. For those that you believe to be false, you should come up with a If A is an n×n orthogonal matrix, and x and y are any. | 2019-11-14T08:59:31 | {
"domain": "scoutsannecystmaurice.fr",
"url": "http://tiww.scoutsannecystmaurice.fr/properties-of-matrix-multiplication-proof.html",
"openwebmath_score": 0.8611693382263184,
"openwebmath_perplexity": 429.5697739059585,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES\n\n",
"lm_q1_score": 0.9901401426598159,
"lm_q2_score": 0.8418256512199033,
"lm_q1q2_score": 0.8335253703935674
} |
http://mathhelpforum.com/calculus/43534-one-last-diff-eq.html | # Thread: One last Diff Eq
1. ## One last Diff Eq
$y' = -2(3y +4)$
It is separable im lost in how to get the answer,
$y = -4/3 + Ce^-6x$
I tried and made it to $dy = (-6y - 8)dx$ but im unsure as how to get my dy/y for 6lny sigh now im really getting confused. Anyone feel free to shed some light on the case in point.
2. Write as
$\frac{dy}{dx}+6y=-8$
The integrating factor is $e^{\int 6dx}=e^{6x}$
$\frac{d}{dx}[e^{6x}y]=-8e^{6x}$
Integrate:
$\int\frac{d}{dx}[e^{6x}y]=-8\int e^{6x}dx$
$e^{6x}y=\frac{-4}{3}e^{6x}+C$
$\boxed{y=\frac{-4}{3}+Ce^{-6x}}$
3. $\frac{dy}{-6y-8}=dx$
Integrating both sides yields
$-\frac 1 6 \ln(-6y-8)=x+C_1$
Exponentiating:
$-6y-8=C_2e^{-6x}$ where $C_2=e^{-6C_1}$
And finally:
$y=Ce^{-6x}-\frac 4 3$ where $C=-\frac{C_2}{6}$
4. Originally Posted by cyph1e
$\frac{dy}{-6y-8}=dx$
Integrating both sides yields
$-\frac 1 6 \ln(-6y-8)=x+C_1$
Exponentiating:
$-6y-8=C_2e^{-6x}$ where $C_2=e^{-6C_1}$
And finally:
$y=Ce^{-6x}-\frac 4 3$ where $C=-\frac{C_2}{6}$
Ahh that makes an little more sense Good Work!
5. Originally Posted by vodka
$y' = -2(3y +4)$
galactus' (and cyph1e's) solution is much simpler and much shorter, but I think separation of variables in this case is quite instructive:
$y' = -2(3y + 4)$
$dy = -2(3y + 4)~dx$
$\frac{dy}{3y + 4} = -2~dx$
(or you could keep the -2 with the y's. It makes no difference.)
$\int \frac{dy}{3y + 4} = -2 \int dx$
Now to integrate the left hand side.
Let $u = 3y + 4 \implies du = 3~dy$
So
$\int \frac{dy}{3y + 4} = \int \frac{du}{3u} = \frac{1}{3}ln|u| = \frac{1}{3}ln | 3y + 4 |$
So
$\int \frac{dy}{3y + 4} = -2 \int dx$
$\frac{1}{3}ln | 3y + 4 | = -2x + A$
(A is my undetermined constant.)
$ln | 3y + 4 | = -6x + 3A$
$| 3y + 4 | = e^{-6x + 3A}$
$| 3y + 4 | = e^{3A}e^{-6x}$
Now $e^{3A}$ is just a (positive) undetermined constant. Let's call this B.
$| 3y + 4 | = Be^{-6x}$
The way to deal with the absolute value bars is to split this into two equations:
$3y + 4 = Be^{-6x}$
and
$3y + 4 = -Be^{-6x}$
Since B is an undetermined (positive) constant anyway, let's say that we can have B be any constant (positive or negative) and just write this as one equation:
$3y + 4 = Ce^{-6x}$
$y = -\frac{4}{3} + Ce^{-6x}$
-Dan
6. but I think separation of variables in this case is quite instructive
That is what I thought. That is why I used the integrating factor instead of SOV. Apparently vodka didn't appreciate it, so I deleted it. Last time I answer a question by that poster.
7. Originally Posted by galactus
That is what I thought. That is why I used the integrating factor instead of SOV. Apparently vodka didn't appreciate it, so I deleted it. Last time I answer a question by that poster.
Even if he didn't appreciate it, please don't delete your post. It was a nice solution and other people viewing this thread might learn from it.
8. A bogaboo of mine is posters who thank one while shunning the other. It may be trifling, but that is how I see it.
9. I have used my phenomenal cosmic powers to restore galactus' post. My reason is that other users can benefit from seeing the solution if they look up this thread. As I think galactus' solution was not only perfectly acceptable, but efficient and clearly explained I feel it should be included in this thread. (No offense to cyph1e's solution which I think is just as good.)
-Dan | 2017-12-17T10:18:31 | {
"domain": "mathhelpforum.com",
"url": "http://mathhelpforum.com/calculus/43534-one-last-diff-eq.html",
"openwebmath_score": 0.9293060898780823,
"openwebmath_perplexity": 967.7182251464469,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9901401435326639,
"lm_q2_score": 0.8418256472515683,
"lm_q1q2_score": 0.8335253671991455
} |
https://www.physicsforums.com/threads/hole-drilled-through-center-of-the-earth.537319/ | Hole drilled through center of the earth
1. Oct 6, 2011
alexmahone
Suppose that a small hole is drilled straight through the center of the earth, thus connecting two antipodal points on its surface. Let a a particle of mass m be dropped at time t = 0 into this hole with initial speed zero. Find the period of the simple harmonic motion exhibited by the particle.
I found this to be about 84 min.
Look up (or derive) the period of a satellite that just skims the surface of the earth; compare with the previous result. How do you explain the coincidence. Or is it a coincidence?
I looked this up on the internet and found the period to be 84 min as well. But I'm unable to explain the coincidence. Please help.
2. Oct 6, 2011
HallsofIvy
Staff Emeritus
No, it is not a coincidence. The period of an orbit about the earth is determined by the maximum distance of the orbit from the center of the earth (the "apogee"). For a satellite that "just skims the earth" or one that goes through the center of the earth those are the same- the radius of the earth.
3. Oct 6, 2011
alexmahone
I derived the formula for the orbital period:
$$T=2\pi\sqrt{\frac{a^3}{GM}}$$
where a is the the semi-major axis, which is R in this case.
But a path that goes through the center of the earth isn't an orbit, in the usual sense of the word. However, I was able to derive independently that the period in this case has the same formula. So, isn't that a coincidence?
Last edited: Oct 6, 2011
4. Oct 6, 2011
DrStupid
No it isn't. You always get the the same values for the orbit at surface level and for a linear path through a homogeneous spherical mass distribution (it doesn't need do go through the center).
By the way: Earth is not homogeneous. With the http://geophysics.ou.edu/solid_earth/prem.html" [Broken] I get a period of 76 minutes from pole to pole and return.
Last edited by a moderator: May 5, 2017
5. Oct 6, 2011
alexmahone
I know we get the same value, but why do you say it isn't a coincidence?
6. Oct 6, 2011
DrStupid
According to Keplers 3rd law the period of an orbit is determined by the semi major axis and for a linear "orbit" from surface to the center of earth (assuming earth to be a point mass) it would be half the radius of earth.
7. Oct 6, 2011
DrStupid
Because it is the inevitable result of natural laws.
8. Oct 6, 2011
alexmahone
I guess you mean half the diameter of the earth. So you're assuming that the path, that passes through the center of the earth, is a degenerate ellipse (with semi-major axis R and semi-minor axis zero)?
What if the path does not pass through the center of the earth? Assuming that it is a degenerate ellipse to use Kepler's 3rd law would not help because the center of the earth would not be in the plane of the ellipse.
Last edited: Oct 6, 2011
9. Oct 6, 2011
DrStupid
No, I mean half the radius. It would be a degenerate ellipse with semi-major axis R/2 and semi-minor axis zero.
That's not possible. According to Kepler's 1st law the central mass is located in a focus of the ellipse and in this special case that means at the inner end of the path.
Last edited: Oct 6, 2011
10. Oct 6, 2011
alexmahone
But in the case of the linear "orbit" that passes through the center of the earth, the central mass is located at the center of the degenerate ellipse, and not at either of the foci.
11. Oct 11, 2011
DrStupid
That's wrong. You can't brake natural laws.
If you want to have a linear path with a point mass in the middle you have to compose it of two elliptic orbits, each with the point mass in the inner focus. The particle starts on the first orbit and reaches the center within half the corresponding period T'. Then it changes to the second orbit (this is possible because in the center the direction is not defined due to infinite angular velocity) and reaches the opposite surface within additional T'/2. The way back takes once more T'. According to Keplers 3rd law the period of the degenerate ellipse is
$T'=\frac{T}{\sqrt{8}}$
where T is the period of the corresponding circular orbit. Therefore the period of the complete linear path is
$\frac{T}{\sqrt{2}}$
If you don't believe it you may check it with a numerical simulation. | 2017-09-24T10:34:29 | {
"domain": "physicsforums.com",
"url": "https://www.physicsforums.com/threads/hole-drilled-through-center-of-the-earth.537319/",
"openwebmath_score": 0.8636974096298218,
"openwebmath_perplexity": 384.4549942769934,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9653811651448432,
"lm_q2_score": 0.863391617003942,
"lm_q1q2_score": 0.8335020051995556
} |
https://mathhelpboards.com/threads/solving-system-of-equations.8521/ | # Solving system of equations
#### anemone
##### MHB POTW Director
Staff member
Find all ordered triples $(x,\,y,\,z)$ that satisfy the following system of equations:
$xy+z=6$
$yz+x=6$
$zx+y=6$
#### mente oscura
##### Well-known member
Find all ordered triples $(x,\,y,\,z)$ that satisfy the following system of equations:
$xy+z=6$
$yz+x=6$
$zx+y=6$
Hello.
$$z=6-xy$$
$$6y-xy^2+x=6 \rightarrow{}6(y-1)=x(y^2-1)$$
$$6=x(y+1)$$(*)
$$6x-x^2y+y=6 \rightarrow{}6(x-1)=y(x^2-1)$$
$$6=y(x+1)$$(**)
For (*) and (**):
$$x(y+1)=y(x+1) \rightarrow{}x=y$$
Same:
$$x=6-yz$$
...
...
$$y=z$$
Conclusion:
$$x=y=z$$
Therefore:
$$xy+z=6 \rightarrow{} x^2+x-6=0$$
Resolving:
$$x=2 , \ and \ x=-3$$
Solution:
$$(2,2,2),(-3,-3,-3)$$
Regards.
#### MarkFL
Staff member
There are also:
(1,1,5) and (5,1,1)
But have we found them all?
#### mente oscura
##### Well-known member
There are also:
(1,1,5) and (5,1,1)
But have we found them all?
What I can not deduct:
$$(1,1,5),(1,5,1),(5,1,1)$$
I I had lying down to sleep, and I've had to raise this issue again. now yes I, which is late.
Regards.
#### ZaidAlyafey
##### Well-known member
MHB Math Helper
Assume that $xyz=0$ then we get a contradiction. So it must be $xyz\neq 0$. Divide all equations by $xyz$.
$$\displaystyle \frac{1}{z}+\frac{1}{xy}=\frac{6}{xyz}$$
$$\displaystyle \frac{1}{x}+\frac{1}{yz}=\frac{6}{xyz}$$
$$\displaystyle \frac{1}{y}+\frac{1}{xz}=\frac{6}{xyz}$$
so we get
$$\displaystyle \frac{1}{z}+\frac{1}{xy}=\frac{1}{x}+\frac{1}{yz}=\frac{1}{y}+\frac{1}{xz}$$
Consider
$$\displaystyle \frac{1}{z}+\frac{1}{xy}=\frac{1}{x}+\frac{1}{yz}$$
Then we have
$$\displaystyle \frac{1}{z}-\frac{1}{x}=\frac{1}{yz}-\frac{1}{xy}$$
$$\displaystyle \frac{1}{z}-\frac{1}{x}=\frac{1}{y}\left(\frac{1}{z}-\frac{1}{x} \right)$$
Case [1]
$$\displaystyle \frac{1}{x}-\frac{1}{z}=0$$
So $$\displaystyle x=y=z$$, Hence $$\displaystyle x^2+x=6$$ and we have the solutions $$\displaystyle (2,2,2),(-3,-3,-3)$$.
Case [2]
$$\displaystyle \frac{1}{x}-\frac{1}{z}\neq 0$$
Then $$\displaystyle y=1$$ so we have
$$\displaystyle \frac{1}{x}+\frac{1}{z}=1+\frac{1}{xz}$$
or $x(1-z)=1-z$ then it is immediate that either $$\displaystyle x=1$$ or $$\displaystyle z=1$$
By symmetry of solutions we have $$\displaystyle (1,1,5),(1,5,1),(5,1,1)$$
#### anemone
##### MHB POTW Director
Staff member
Hello.
$$z=6-xy$$
$$6y-xy^2+x=6 \rightarrow{}6(y-1)=x(y^2-1)$$
$$6=x(y+1)$$(*)
$$6x-x^2y+y=6 \rightarrow{}6(x-1)=y(x^2-1)$$
$$6=y(x+1)$$(**)
For (*) and (**):
$$x(y+1)=y(x+1) \rightarrow{}x=y$$
Same:
$$x=6-yz$$
...
...
$$y=z$$
Conclusion:
$$x=y=z$$
Therefore:
$$xy+z=6 \rightarrow{} x^2+x-6=0$$
Resolving:
$$x=2 , \ and \ x=-3$$
Solution:
$$(2,2,2),(-3,-3,-3)$$
There are also:
(1,1,5) and (5,1,1)
But have we found them all?
Thanks MarkFL for pointing this out!
Regards.
What I can not deduct:
$$(1,1,5),(1,5,1),(5,1,1)$$
I I had lying down to sleep, and I've had to raise this issue again. now yes I, which is late.
Regards.
Thanks, mente for participating and yes, all those 5 solutions are the complete answer to this problem.
##### Well-known member
Hello.
$$z=6-xy$$
$$6y-xy^2+x=6 \rightarrow{}6(y-1)=x(y^2-1)$$
$$6=x(y+1)$$(*)
$$6x-x^2y+y=6 \rightarrow{}6(x-1)=y(x^2-1)$$
$$6=y(x+1)$$(**)
For (*) and (**):
$$x(y+1)=y(x+1) \rightarrow{}x=y$$
Same:
$$x=6-yz$$
...
...
$$y=z$$
Conclusion:
$$x=y=z$$
Therefore:
$$xy+z=6 \rightarrow{} x^2+x-6=0$$
Resolving:
$$x=2 , \ and \ x=-3$$
Solution:
$$(2,2,2),(-3,-3,-3)$$
Regards.
the case that was left out was y = 1
y =1 gives x + z = 6, xz = 5 solving it we get
y = 1 , x = 5, z = 1
y = 1 ,z = 5, x = 1
similarly we can find one more x = 5, y = 1, z = 1
actually 2 solutions for y = 1 , 2 for z = 1 and 2 for x = 1 but that is 3 solutions in duplicate
#### anemone
##### MHB POTW Director
Staff member
Assume that $xyz=0$ then we get a contradiction. So it must be $xyz\neq 0$. Divide all equations by $xyz$.
$$\displaystyle \frac{1}{z}+\frac{1}{xy}=\frac{6}{xyz}$$
$$\displaystyle \frac{1}{x}+\frac{1}{yz}=\frac{6}{xyz}$$
$$\displaystyle \frac{1}{y}+\frac{1}{xz}=\frac{6}{xyz}$$
so we get
$$\displaystyle \frac{1}{z}+\frac{1}{xy}=\frac{1}{x}+\frac{1}{yz}=\frac{1}{y}+\frac{1}{xz}$$
Consider
$$\displaystyle \frac{1}{z}+\frac{1}{xy}=\frac{1}{x}+\frac{1}{yz}$$
Then we have
$$\displaystyle \frac{1}{z}-\frac{1}{x}=\frac{1}{yz}-\frac{1}{xy}$$
$$\displaystyle \frac{1}{z}-\frac{1}{x}=\frac{1}{y}\left(\frac{1}{z}-\frac{1}{x} \right)$$
Case [1]
$$\displaystyle \frac{1}{x}-\frac{1}{z}=0$$
So $$\displaystyle x=y=z$$, Hence $$\displaystyle x^2+x=6$$ and we have the solutions $$\displaystyle (2,2,2),(-3,-3,-3)$$.
Case [2]
$$\displaystyle \frac{1}{x}-\frac{1}{z}\neq 0$$
Then $$\displaystyle y=1$$ so we have
$$\displaystyle \frac{1}{x}+\frac{1}{z}=1+\frac{1}{xz}$$
or $x(1-z)=1-z$ then it is immediate that either $$\displaystyle x=1$$ or $$\displaystyle z=1$$
By symmetry of solutions we have $$\displaystyle (1,1,5),(1,5,1),(5,1,1)$$
Thanks to you too, Zaid for participating and showing us another way to solve the problem.
the case that was left out was y = 1
y =1 gives x + z = 6, xz = 5 solving it we get
y = 1 , x = 5, z = 1
y = 1 ,z = 5, x = 1
similarly we can find one more x = 5, y = 1, z = 1
actually 2 solutions for y = 1 , 2 for z = 1 and 2 for x = 1 but that is 3 solutions in duplicate
Yes, you're absolutely right, kaliprasad! I think mente oscura realized it, but it's just that he was too tired and yet wanted to solve some challenge problems on his way to go to bed.
#### ZaidAlyafey
##### Well-known member
MHB Math Helper
I wouldn't have arrived to the solutions unless Mark pointed out the other possiblities. It wasn't quite easy to see that the permutations of
$$\displaystyle (1,1,5)$$
is a solution. At first glance I thought of using the interesting formula
$$\displaystyle x^3+y^3+z^3=(x+y+z)(x^2+y^2+z^2-xy-xz-yz)+3xyz$$.
One interesting question is to look at all the solutions of
$$\displaystyle xy+z=a$$
$$\displaystyle xz+y=a$$
$$\displaystyle zy+x=a$$
where the domain is complex numbers.
#### mente oscura
##### Well-known member
Thanks to you too, Zaid for participating and showing us another way to solve the problem.
Yes, you're absolutely right, kaliprasad! I think mente oscura realized it, but it's just that he was too tired and yet wanted to solve some challenge problems on his way to go to bed.
Hello.
The time difference is
My last post: 5:45 AM
Regards. | 2020-09-27T01:41:36 | {
"domain": "mathhelpboards.com",
"url": "https://mathhelpboards.com/threads/solving-system-of-equations.8521/",
"openwebmath_score": 0.7787613868713379,
"openwebmath_perplexity": 1275.2611353928746,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9653811641488385,
"lm_q2_score": 0.8633916152464016,
"lm_q1q2_score": 0.8335020026429172
} |
http://math.stackexchange.com/questions/499021/nested-quantifiers-in-english-translation/499097 | # Nested quantifiers in English translation
I'm having a hard time grasping how to express english statements with quantifiers; specifically when trying to show "exactly 1" or "exactly 2" or etc. The introduction of numerous variables throws me off guard.
Let $F(x, y)$ be the statement: "$x$ can fool $y$", where the domain consists of all people in the world. Use quantiers to express each of these statements.
Example: "There is exactly one person whom everybody can fool":$$\exists y(\forall xF(x,y)\land(\forall z((\forall wF(w,z))\rightarrow y=z))$$
I understand: $\exists y(\forall F(x,y))$ which means that there's a person $y$ that everyone can fool. However, this doesn't show that he's the only peron that everyone can fool.
Is there a way "systematic" way to build such expressions? And in this case above,why is it that we bring $z,w$ as variables. I'm not quite understanding what is happening after the first part.
-
The last part says if you pick any person ("$z$") then if everybody can fool $z$, it will turn out that $z$ is $y$. – André Nicolas Sep 19 '13 at 21:49
The second part is expressing uniqueness. – azarel Sep 19 '13 at 21:49
Let's start with the given wff as that seems to be causing difficulty, and work towards its translation into English. Then we've reverse the process!
$$\exists y(\forall xF(x,y)\land(\forall z((\forall wF(w,z))\rightarrow y=z))$$
It can help a great deal to go from logic to English (or vice versa) in stages, via "Loglish" -- that unholy mixture of English and symbolism which we cheerfully use in the classroom! So ....
There is someone $y$ such that $(\forall xF(x,y)\land(\forall z((\forall wF(w,z))\rightarrow y=z))$
There is someone $y$ such that (everyone $x$ is such that $x$ can fool $y$) and (everyone $z$ is such that ($(\forall wF(w,z))\rightarrow y=z))$
i.e.
There is someone $y$ such that everyone can fool $y$ and everyone $z$ is such that (if everyone $w$ can fool $z$, then $z$ is the same person as $y$).
i.e.
There is someone $y$ such that everyone can fool $y$ and anyone whom everyone can fool is none other than $y$ again.
i.e.
There is someone whom everyone can fool, and no one other then he can be fooled by everyone.
i.e.
There is exactly one person whom everyone can fool.
Read this from top to bottom to translate the formal wff into English.
And now read the same sequence from bottom to top to translate in the other direction!!
Taking things in stages like this helps a great deal when first learning to translate in either direction. There are lots more worked examples of this kind involving nested quantifiers in my Introduction to Formal Logic (Ch. 24), with more exercises and answers online. For practice quickly makes perfect: but it does take a bit of practice to make this all seem as easy as it really is. I recall Paul Teller's A Modern Formal Logic Primer is also quite good on translation (his book, now out of print, is freely available from his website).
- | 2016-07-29T00:27:00 | {
"domain": "stackexchange.com",
"url": "http://math.stackexchange.com/questions/499021/nested-quantifiers-in-english-translation/499097",
"openwebmath_score": 0.5224370360374451,
"openwebmath_perplexity": 237.75494046098788,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9653811601648193,
"lm_q2_score": 0.8633916064586998,
"lm_q1q2_score": 0.8335019907196668
} |
https://math.stackexchange.com/questions/656183/why-does-frac1-99989999-generate-the-fibonacci-sequence | # Why does $\frac{1 }{ 99989999}$ generate the Fibonacci sequence?
$\frac{1}{99989999} = 1.00010002000300050008001300210034005500890144... \times 10^{-8}$
(Link), which includes the Fibonacci sequence $(1\ 1\ 2\ 3\ 5\ 8\ 13\ 21\ 55\ 89\ 144\ldots )$.
This is fascinating to me but I can't figure out how this works, and a Google search doesn't seem to reveal anything.
• It can't generate the full sequence since it would be non-repeating. – Raskolnikov Jan 29 '14 at 17:02
• This is pretty cool! I wonder what the digits are when the fibonacci numbers exceed 4 digits! – mikhailcazi Jan 29 '14 at 17:03
• @mikhailcazi: they carry. $1/99989999=1.0001000200030005000800130021003400550089014402330377061009871597258441816766094777138661637550371412644978624312217464868661514838098958276817264494622107156936765245892241683090725903497608795855673525909326191712433160440375641967953... × 10^-8$ You can hold off the carries longer with $1/9999899999$ and so on for the reason Henning Makholm gives. – Ross Millikan Jan 29 '14 at 17:06
• @RossMillikan I'll take your word for it; it seems quite a task to actually check! xD – mikhailcazi Jan 29 '14 at 17:09
• You can do the same with any finite sequence of numbers. – Martín-Blas Pérez Pinilla Jan 29 '14 at 17:10
As explained here, the generating function for the Fibonacci sequence is $$1+z+2z^2+3z^3+5z^4+8z^5+13z^6+...=\frac{1}{1-(z+z^2)}$$
$10^8/99989999$ is simply the value of this generating function for $z=0.0001$.
The initial segment of the Fibonacci sequence where all numbers have at most 4 digits will appear nice and visible in the decimal expansion. After that, the digits of successive Fibonacci numbers will be added to each other offset by $4$ positions. Eventually this will create a repeating digit sequence somehow.
• This is basically a GP, right? $a = 1$, $r = (z+z^2)$ and $n\rightarrow\infty$? – mikhailcazi Feb 17 '14 at 6:33
• @mikhailcazi: I don't follow you. – Henning Makholm Feb 17 '14 at 9:50
• $1+z+2z^2+3z^3+5z^4+8z^5+13z^6+...$ is an infinite geometric progression, right? The summation being the RHS? – mikhailcazi Feb 17 '14 at 15:15
• @mikhailcazi: No, it's not a geometric progression. In a geometric progression, the ratio between successive terms is always the same; here they are all different (although converging towards $\phi z$). – Henning Makholm Feb 17 '14 at 15:20
• @mikhailcazi: Yes, if you rearrange the result to collect terms with the same power of $z$. But once this rearrangement is done, I don't think the result can be called a geometric progression, because the progression is not there anymore. – Henning Makholm Feb 17 '14 at 15:28
Eventually this will create a repeating digit sequence somehow.
part. The explanation that follows is due to Allen Schwenk and the last part of his article in Math Horizons titled An Unanticipated Decimal Expansion. All the credit goes to him.
Spoiler: The length of the period for the decimal expansion is $496620$ digits.
The length of the period for the expansion: Any fraction of the form $1/D$ may be examined, with period of length $p$ and an irregular lead in sequence of $k$ digits. It must look like this, where we have displayed the irregular lead-in and the first two periods $$\frac{1}{D}=0.a_1a_2\ldots a_ka_{k+1}a_{k+2}\ldots a_{k+p}a_{k+1}a_{k+2}\ldots a_{k+p}.\tag{1}$$ When we multiply the fraction in $(1)$ by $10^{p+k}$ and subtract the original fraction times $10^k$, everything beyond the decimal point drops out. In other words, we have formed an integer $N$ where $$\frac{1}{D}10^{p+k}-\frac{1}{D}10^k=\frac{10^{p+k}-10^k}{D}=N.$$ This leads to the integer equation $$(10^p-1)\cdot 10^k=ND.$$ Now focusing on $D=998999$, we see that it factors into two primes $$D=998999=179\cdot 5581.$$ Since neither prime shares a factor with $10^k$, we are forced to agree that $N=10^k M$ for another integer $M$. The equation may be rewritten as $$10^p-1=MD,$$ implying that the solution has $k=0$; that is, there is no lead-in sequence. We are forced to conclude that $10^p\equiv1\bmod 179$ and $10^p\equiv1\bmod 5581$. This means that $d_1$, the order of $10\mod 179$, must be a divisor of $178$ (by Euler's theorem on finite groups) and also of the period $p$. Likewise, $d_2$, the order of $10\mod 5581$, must be a divisor of $5580$ and also of $p$. In fact, we require $p=\operatorname{lcm}(d_1,d_2)$. Moreover, $178=2\cdot 89$, so the only candidates for the order $d_1$ are $2, 89$, and $178$. Clearly $10^2\not\equiv 1\bmod 179$. What about $10^{89}$? Quick inspection shows that $$10^{11} \equiv 157\bmod 179\\[0.5em] 10^{22} \equiv 157^2 \equiv 126\bmod 179\\[0.5em] 10^{44} \equiv 126^2 \equiv 124\bmod 179\\[0.5em] 10^{88} \equiv 124^2 \equiv 161\bmod 179\\[0.5em] 10^{89} \equiv 1610 \equiv -1\bmod 179$$ The order of $10\bmod 179$ is $d_1=178$.
Similarly, $5580=2^2\cdot 3^2\cdot 5\cdot 31$. This has $36$ possible divisors, and we can check (a tedious task) each of these powers of $10\bmod 5581$. We find that the only power that reduces to $1\bmod 5581$ is $10^{5580}$. So $d_2=5580$, and $p=\operatorname{lcm}(178,5580)=496620$.
Let $F_i$ be the $i$-th Fibonacci number.
We have $$\frac{1}{999,999,999,998,999,999,999,999}$$
represented in decimal form as:
0. 000 000 000 000
000 000 000 001
000 000 000 001
000 000 000 002
000 000 000 003
000 000 000 005
000 000 000 008
000 000 000 013
000 000 000 021...
This decimal works out to be $$x = 10^{-12}F_1 + 10^{-24}F_2 + 10^{-36}F_3 + \ldots + 10^{-12k}F_k + \ldots \tag{1}$$
where $x$ is our number we let $k$ range over the naturals. Multiply the above by $10^{12}$ to get $$10^{12} x = F_1 + 10^{-12}F_2 + 10^{-24}F_3 +\ldots + 10^{-12k}F_{k+1} +\ldots \tag{2}$$
Multiply the above by $10^12$ once more to get $$10^{24}x = 10^{12}F_1 + F_2 + 10^{-12}F_3 + 10^{-24}F_4 + \ldots + 10^{-12k}F_{k+2} + \ldots \tag{3}$$
Now, we work out the value of $(3) - (2) - (1)$ to get $$(10^{24} – 10^{12} – 1) x = 10^{12}F_1 + (F_2 – F_1) + 10^{-12}(F_3 – F_2 – F_1) + 10^{-24}(F_4 – F_3 – F_2) + \ldots + 10^{-12k} (F_{k+2} – F_{k+1} – F_k) + \ldots$$
Whilst this looks rather tricky to work with, we are quite privileged to be working with the Fibonacci sequence here, since it's defined as $F_n = F_{n-1} + F_{n-2}$ with the initial conditions that $F_1 = 0$ and $F_2 = 1$
Hence you get $$F_{k+2} – F_{k+1} – F_k = \left(F_{k+1} + F_k\right) – F_{k+1} – F_{k} = 0$$
so that our equation simplifies to $$(10^{24} – 10^{12} – 1)x = 1 \iff x = \frac{1}{10^{24} – 10^{12} – 1}$$
And you'll be quite pleased to note that, indeed, $$10^{24} - 10^{12} - 1 = 999,999,999,998,999,999,999,999$$ | 2019-10-15T16:13:49 | {
"domain": "stackexchange.com",
"url": "https://math.stackexchange.com/questions/656183/why-does-frac1-99989999-generate-the-fibonacci-sequence",
"openwebmath_score": 0.9112037420272827,
"openwebmath_perplexity": 230.42517757559582,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9852713865827593,
"lm_q2_score": 0.8459424431344437,
"lm_q1q2_score": 0.8334828839162803
} |
http://math.stackexchange.com/questions/259726/modular-algebra-with-an-non-integer-solution | Modular algebra with an non-integer solution
Suppose I was asked to solve:
$$5x+6=10 \mod 17$$
I would get this far with normal algebra:
$$x = \frac{4}{5} \mod 17$$
But now I have a non integer expression ($4/5$) where it does not belong.
What am I missing here? I think I have made a fundamental misconception.
-
When you divide by 5, remember that you're in fact multiplying by 5's inverse. Here, however, the inverse of 5 mod 17 is 7. That is because 5*7 = 35 and 35 = 1 mod 17 (similar to the more familiar 5 * 1/5 = 1).
So what you get is x = 4 * (5's inverse) mod 17 = 4 * 7 mod 17. That is x = 28 mod 17 = 11 mod 17. There are methods to determine the inverse when it exists (because sometimes it doesn't) such as Euclide's algorithm to find the GCD.
-
$$5x+6 \equiv 10 \pmod{17}$$ means $5x+6 = 17k+10$, where $k \in \mathbb{Z}$. Hence, we have $5x = 17k+4$, where $x,k \in \mathbb{Z}$. Note that $x$ and $k$ are not arbitrary real numbers but integers. This enforces a constraint on the values taken by $x$ and $k$. Further $$5x = 17k+4 = 17(k-3) + 17 \times 3 + 4 = 17(k-3) + 55$$ Hence, we get that $$x = \dfrac{17(k-3)}5 + 11$$ Since $x \in \mathbb{Z}$, we have that $\dfrac{17(k-3)}5 \in \mathbb{Z}$. Hence, $5 \vert 17(k-3)$. But since $\gcd(5,17) = 1$, we have that $5 \vert k-3$. (Recall that if $a \vert (bc)$ and $\gcd(a,b) = 1$, then $a \vert c$).
Hence, $k = 5m + 3$. This gives us that $$x = 17m + 11 \,\,\,\,\, \text{i.e.} \,\,\,\, x \equiv 11 \pmod{17}$$
-
Better answer then mine, for the general case tridecth12 look up Bezout's identity. – Ethan Dec 16 '12 at 4:23
Thanks for the answer.. I just need a sec to get my head around it as I am new to modular arithmetic. What is the significance of the vertical bar in $a \vert b$ – trideceth12 Dec 16 '12 at 4:23
@trideceth12 $a \vert b$ means $a$ divides $b$ i.e. $b$ is a multiple of $a$ i.e. $b = am$ where $m \in \mathbb{Z}$. – user17762 Dec 16 '12 at 4:24
@Ethan ahh, yes I've seen that in my notes but not with that name assigned to it, Bezout's identity, great. – trideceth12 Dec 16 '12 at 4:25
You shouldn't have divided both sides by 5, to create that quotient, I think thats an abuse of notation, anyway what you end up with is $5x\equiv 4$ mod 17, which can be re-written as a linear diophantine equation, and solved useing the generalized euclidean algorithm.
The diophantine equation would look like:
$5x-17y=4$, but you really only need a solution in x, ie you don't care about the value/values of y that makes that solution true.
-
The deterministic way to find $x,y$ for $Ax+By=1$ where $A,B$ are known and$(A,B)=1$ is to utilize the property of the successive convergents of a continued fraction.
$$\frac{17}5=3+\frac25=3+\frac1{\frac52}=3+\frac1{2+\frac12}$$
So, the previous convergent of $\frac{17}5$ is $3+\frac12=\frac72$ $\implies 17\cdot2-5\cdot7=-1$
We have $5x+6=10+17y$ for some integers $x,y$
or ,$5x-17y=4=4(5\cdot7-17\cdot2)$
or, $5(x-28)=17(y-8)$
or, $\frac{5(x-28)}{17}=y-8$ which is an integer.
$\implies 17\mid5(x-28)\implies17\mid(x-28)$ as $(5,17)=1$
$\implies x=17z+28$ where $z$ is any integer.
or, $x=17(z+1)+11=17w+1$ where $w=z+1$ is also some integer.
Alternatively, $$\frac{17}5=3+\frac25=3+\frac1{\frac52}=3+\frac1{2+\frac12}=3+\frac1{2+\frac1{1+\frac11}}$$
So, the next convergent of $\frac{17}5$ is $3+\frac1{2+1}=\frac{10}3$ $\implies 17\cdot 3-5\cdot10=1$
$5x-17y=4=4(17\cdot3-5\cdot10)$
or, $5(x+40)=17(y+12)$
or, $\frac{5(x+40)}{17}=y+12$ which is an integer.
So, $17\mid5(x+40)\implies 17\mid(x+40)$ as $(5,17)=1$
$\implies x+40=17u$ where $u$ is any integer.
or, $x=17u-40=17(u-2)-6=17v-6$ or $x=17(u-3)+11=17t+11$ where $v=u-2,t=u-3$ are integers.
- | 2014-07-26T15:51:53 | {
"domain": "stackexchange.com",
"url": "http://math.stackexchange.com/questions/259726/modular-algebra-with-an-non-integer-solution",
"openwebmath_score": 0.9379551410675049,
"openwebmath_perplexity": 253.29590890496803,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9852713896101315,
"lm_q2_score": 0.845942439250491,
"lm_q1q2_score": 0.8334828826505155
} |
https://math.stackexchange.com/questions/3243034/computing-the-determinant-of-xx-given-x | # Computing the determinant of $X^*X$ given $X$.
I am trying to prove that if $$u_i\in \mathbb{R}$$ for $$i=1,...,n$$ and $$X =\begin{pmatrix} 1 & 0 & 0 & \cdots & 0 \\ 0 & 1 & 0 & \cdots & 0 \\ 0 & 0 & 1 & \cdots & 0 \\ \ & \vdots & \ & \ddots & \vdots \\ 0 & 0 & 0 & \cdots & 1 \\ u_1 & u_2 & u_3 & \cdots & u_n \end{pmatrix}$$
then $$\det(X^{T}X)=1 +u_1^2 + u_2^2 + \cdots + u_n^2$$, where $$X^T$$ is the transpose of $$X$$.
Here is what I know. $$X$$ is an $$(n+1)$$-by-$$n$$ matrix, and $$X^T$$ is an $$n$$-by-$$(n+1)$$ matrix; thus $$X^TX$$ is an $$n$$-by-$$n$$ square matrix, so we can take its determinant.
Also, the formula is simple to verify for the cases $$n=2,3$$.
The $$(i,j)$$ entry of the matrix $$X^TX$$ is \begin{align*} (X^TX)_{i,j} &= \sum_{k}(X^T)_{i,k}X_{k,j} = \sum_k X_{k,i}X_{k,j} \\ &= \langle X_{\cdot,i},X_{\cdot, j} \rangle = \delta_{i,j} + u_iu_j \end{align*} where $$X_{\cdot,j}$$ denotes the $$j$$th column of $$X$$. Since I know all the entries, I could do some combination of induction and cofactor expansion, but I couldn't make it work.
I also tried computation with "block" matrices, which is something I am not very familiar with. If $$I_n$$ is the $$n$$-by-$$n$$ identity matrix and $$\vec{u}=(u_1,\ldots,u_n)$$ then we have $$\vec{u} = \begin{pmatrix} u_1 \\ u_2 \\ \vdots \\ u_n \end{pmatrix} \;\;\; \text{ and}\;\;\; \left( \vec{u}\right)^T = \begin{pmatrix} u_1 & u_2 & \cdots & u_n\end{pmatrix}$$ so that $$X^T = \begin{pmatrix} I_n & \vec{u} \end{pmatrix} \;\;\; \text{ and}\;\;\; X = \begin{pmatrix} I_n \\ \left(\vec{u}\right)^T \end{pmatrix}.$$
Then if we are presumptuous, we can treat $$X^T$$ and $$X$$ as $$1$$-by-$$2$$ and $$2$$-by-$$1$$ matrices respectively. Then $$X^TX = 1 + \vec{u}\left(\vec{u} \right)^T = 1 + \|u\|^2$$ which clearly would imply the result I want, if this computation can be justified.
I was unable to find this specific problem elsewhere, but I am sure it has been asked other times on this site.
Please let me know your thoughts and tips for this problem, and help me justify or refute the block matrix computation I performed. If we cannot justify it, then why does it hint at the right answer? Thanks.
• The bug in your logic is, in block multiplication you get $uu^T \ne u^T u = \|u\|$... In particular, $uu^T$ is a matrix, not a number May 28, 2019 at 17:21
• I see my mistake; I made the classic error of trying to force the result I wanted... How can I compute this determinant without brute-force? May 28, 2019 at 17:25
You may use the identity $$\det(I+AB)=\det(I+BA)$$, where the $$I$$ on the LHS has the same size as $$AB$$ and the $$I$$ on RHS has the same size as $$BA$$. This identity follows from the fact that $$AB$$ and $$BA$$ have the same set (counting multiplicities) of nonzero eigenvalues over the algebraic closure of the underlying field.
In you case, the identity gives $$\det(I_n+uu^T)=\det(I_1+u^Tu)=1+\|u\|^2$$.
You can use a LU-decomposition (where an "empty sum" is considered as $$0$$): \begin{align} X^\top X&=LU \text{, where} \\ U_{i,j}&= \begin{cases} 1+\sum_{k=1}^j u_k^2 &, \text{ if } i=j; \\ u_i u_j &,\text{ if } j > i \text{ (upper triangular part)}; \\ 0 &, \text{ if } j < i \end{cases} \\ L_{i,j}&= \begin{cases} \dfrac{1}{1+\sum_{k=1}^{j-1} u_k^2}&, \text{ if } i=j; \\ u_i u_j\cdot\left(\displaystyle\prod_{k=j-1}^j\Big(1+\sum_{h=1}^k u_k^2\Big)\right)^{-1} &, \text{ if } ij; \end{cases} \end{align}
We thus have, by multiplicity of the determinant, $$\det(X^\top X)=\det(L)\det(U)$$. Since $$L$$ has only $$0$$s in its upper part, and $$U$$ has only $$0$$s in its lower part, the determinant of these matrices is simply the product of their diagonal elements. Thus,
$$$$\begin{split} \det(X^\top X)&=\det(L)\det(U) \\&= \prod_{h=1}^n\left(1+\sum_{k=1}^h u_k^2\right)\cdot \prod_{h=1}^n\left(\frac1{1+\sum_{k=1}^{h-1} u_k^2}\right) \\ &= \prod_{h=1}^n\frac{1+\sum_{k=1}^h u_k^2}{1+\sum_{k=1}^{h-1} u_k^2} \\ &= 1+\sum_{k=1}^n u_k^2 \qquad\text{(by cancellation of terms)}. \end{split}$$$$ | 2022-10-02T10:08:13 | {
"domain": "stackexchange.com",
"url": "https://math.stackexchange.com/questions/3243034/computing-the-determinant-of-xx-given-x",
"openwebmath_score": 0.9999741315841675,
"openwebmath_perplexity": 351.0822783804234,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9852713852853137,
"lm_q2_score": 0.8459424411924673,
"lm_q1q2_score": 0.8334828809053423
} |
https://math.stackexchange.com/questions/4284669/how-do-you-deduce-that-matrices-are-equal | # How do you deduce that matrices are equal
i wanted to ask for a clarification.
I was looking around in my linear algebra text when i reached this justification:
Considered two coloumn vectors $$X$$ and $$Y$$, and assume $$X ^tC Y = X^t C^t Y$$ for every $$X,Y \in V$$, with $$V$$ an $$n$$-dimensional vectorial space, with $$X^t, Y^t$$ being the transposed of $$X, Y$$.
My book says that because of this is valid for every $$X,Y$$, we can deduce: $$C^t = C$$
Now, it is intuitively true, but i was wondering if ,maybe the general sum (it's a bilinear form) could equals without needing $$C^t = C$$. I've seen this type of justification also in other theorems, but i want to know if there is a way to prove it formally, beacuse i'm not satisfied.
• See what happens if $X=e_i$ and $Y=e_j$ ($i$-th and $j$-th standard basis vectors of $V$). Oct 23 at 0:32
• What does this have to do with the “null” in the title? Oct 23 at 0:40
• The transpose of a linear transformation doesn’t make sense over a general vector space. But it does make sense if the vector space is given an inner product. Oct 23 at 0:43
• Sorry, language translations Oct 23 at 0:45
since these are column vectors, evidently $$V=\mathbb F^n$$
select $$n$$ linearly independent vectors $$\mathbf x_k$$ and $$n$$ linearly independent $$\mathbf y_k$$ (if you like you can set $$\mathbf y_k:=\mathbf x_k$$)
Then for $$k,j \in \big\{1,2,...,n\big\}$$
$$\mathbf x_k^T C\mathbf y_j=\mathbf x_k^T C^T\mathbf y_j$$
$$\implies\mathbf x_k^T\big( C-C^T\big)\mathbf y_j = \mathbf 0$$
$$\implies \mathbf X^T\big( C-C^T\big)\mathbf Y = \mathbf 0$$
where $$\mathbf X$$ and $$\mathbf Y$$ have column $$j$$ given by $$\mathbf x_j$$ and $$\mathbf y_j$$ respectively. But $$\mathbf X$$ and $$\mathbf Y$$ are invertible,
$$\implies \big( C-C^T\big) = \mathbf 0$$
$$\implies C=C^T$$
Setting $$A = C-C^t$$ you have $$X^tAY=0$$ for all $$X,Y$$. Let $$e_j$$ denote the $$j$$-th standard basis vector. The $$(i,j)$$-th entry of $$A$$ can be expressed as $$e_i^tAe_j$$. But this is zero, hence $$A=0$$, i.e., $$C^t = C$$. | 2021-11-29T05:50:58 | {
"domain": "stackexchange.com",
"url": "https://math.stackexchange.com/questions/4284669/how-do-you-deduce-that-matrices-are-equal",
"openwebmath_score": 0.9437273144721985,
"openwebmath_perplexity": 176.88681712131978,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.98527138442035,
"lm_q2_score": 0.8459424411924673,
"lm_q1q2_score": 0.8334828801736328
} |
http://math.stackexchange.com/questions/663487/not-sure-about-the-derivative-of-the-integral | # Not sure about the derivative of the integral
Call me stupid, but I would like to know whether my understanding is okay:
$$\frac{d}{dx}\left(\int_0^x f(s)ds\right)=\frac{d}{dx}F(x)=f(x)$$
-
Fundamental theorem of analyis. – Mike Feb 4 '14 at 16:20
This is correct (assuming that $f$ is sufficiently smooth, such as being continuous). The intuition is that, roughly, $F(x+\delta) = \int_0^{x+\delta} f = \int_0^x f + \int_x^{x+\delta} f = F(x) + \int_x^{x+\delta} f$, and for small $\delta$, $\int_x^{x+\delta} f \approx f(x) \delta$. – copper.hat Feb 4 '14 at 16:21
I didn't find this at all intuitive until very recently when I worked through the following to get happy with it. Start with definition of a derivative
$$\frac{dg}{dx} = \lim_{h\to0} \frac{g(x+h)-g(h)}{h}$$
then
$$\frac{d}{dx}\int_0^xf(s)\;ds =\lim_{h\to0} \;\left[\frac{\int_0^{x+h}f(s)\;ds\;-\;\int_0^{x}f(s)\;ds}{h}\right]$$
$$= \lim_{h\to0} \;\left[\frac{\int_x^{x+h}f(s)\;ds}{h}\right]$$
$$= \lim_{h\to0} \;\left[\frac{f(x) h + O(h^2)}{h}\right]$$
$$= \lim_{h\to0}\; [f(x) + O(h)]$$
$$=f(x)$$
This is the Fundamental Theorem of Calculus. There is a sequence of Khan Academy videos on this starting here.
-
What the intuition behind the $O(h^2)$ term in the third line of the second equation? – David Simmons Feb 4 '14 at 16:55
@DaveS It means $\int_x^{x+h}f(s)ds = f(x)h$ to first order. – TooTone Feb 4 '14 at 17:01
can you explain that part more – Muhammad Umer Mar 17 '14 at 0:34
@MuhammadUmer it means the error term is small so you can ignore it. I don't think what I've written here would count as a proof. For that see, e.g., en.wikipedia.org/wiki/…. – TooTone Mar 17 '14 at 0:36
i have read that page, but that has problem. theorem proven using assumption. Can you explain how did you do integral..and then where did h go – Muhammad Umer Mar 17 '14 at 0:48
This is precisely the Fundamental Theorem of Calculus. And $f(x)$ should be continuous on some closed interval.
- | 2015-05-30T18:43:22 | {
"domain": "stackexchange.com",
"url": "http://math.stackexchange.com/questions/663487/not-sure-about-the-derivative-of-the-integral",
"openwebmath_score": 0.8899698853492737,
"openwebmath_perplexity": 613.0285394212539,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9852713844203499,
"lm_q2_score": 0.8459424411924673,
"lm_q1q2_score": 0.8334828801736327
} |
https://proofwiki.org/wiki/Definition:Image_(Set_Theory) | # Definition:Image (Set Theory)
## Definition
The definition of a relation as a subset of the Cartesian product of two sets gives a "static" sort of feel to the concept.
However, we can also consider a relation as being an operator, where you feed an element $s \in S$ (or a subset $S_1 \subseteq S$) in at one end, and you get a set of elements $T_s \subseteq T$ out of the other.
Thus we arrive at the following definition.
## Relation
### Image of a Relation
The image of $\mathcal R$ is the set:
$\Img {\mathcal R} := \mathcal R \sqbrk S = \set {t \in T: \exists s \in S: \tuple {s, t} \in \mathcal R}$
### Image of an Element
Let $s \in S$.
The image of $s$ by (or under) $\mathcal R$ is defined as:
$\map {\mathcal R} s := \set {t \in T: \tuple {s, t} \in \mathcal R}$
That is, $\map {\mathcal R} s$ is the set of all elements of the codomain of $\mathcal R$ related to $s$ by $\mathcal R$.
### Image of a Subset
Let $X \subseteq S$ be a subset of $S$.
Then the image set (of $X$ by $\mathcal R$) is defined as:
$\mathcal R \sqbrk X := \set {t \in T: \exists s \in X: \tuple {s, t} \in \mathcal R}$
## Mapping
### Image of a Mapping
#### Definition 1
The image of a mapping $f: S \to T$ is the set:
$\Img f = \set {t \in T: \exists s \in S: \map f s = t}$
That is, it is the set of values taken by $f$.
#### Definition 2
The image of a mapping $f: S \to T$ is the set:
$\Img f = f \sqbrk S$
where $f \sqbrk S$ is the image of $S$ under $f$.
### Image of an Element
Let $s \in S$.
The image of $s$ (under $f$) is defined as:
$\Img s = \map f s = \displaystyle \bigcup \set {t \in T: \tuple {s, t} \in f}$
That is, $\map f s$ is the element of the codomain of $f$ related to $s$ by $f$.
### Image of a Subset
Let $f: S \to T$ be a mapping.
Let $X \subseteq S$ be a subset of $S$.
Then the image of $X$ (under $f$) is defined and denoted as:
$f \sqbrk X := \set {t \in T: \exists s \in X: \map f s = t}$
## Also known as
Some sources refer to this as the direct image of a (usually) mapping, in order to differentiate it from an inverse image.
Rather than apply a relation $\mathcal R$ (or mapping $f$) directly to a subset $A$, those sources prefer to define the direct image mapping of $f$ as a separate concept in its own right.
## Technical Note
The $\LaTeX$ code for $\Img {f}$ is \Img {f} .
When the argument is a single character, it is usual to omit the braces:
\Img f | 2019-11-17T16:36:04 | {
"domain": "proofwiki.org",
"url": "https://proofwiki.org/wiki/Definition:Image_(Set_Theory)",
"openwebmath_score": 0.9726234674453735,
"openwebmath_perplexity": 342.97430511335,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9852713833391452,
"lm_q2_score": 0.8459424373085146,
"lm_q1q2_score": 0.8334828754322483
} |
http://svc2006.it/xffd/unit-7-probability-using-venn-diagrams.html | # Unit 7 Probability Using Venn Diagrams
Sort and classify shapes using Venn and Carroll diagrams, e. Analyze the data in the Venn diagram and list five facts about Ms. Probabilities using Venn Diagrams. Probability Y7. Leary's Class: Use the Venn diagram showing the number of kids owning bicycles (A) and skateboards (B) to find the following probabilities. Leila arrives at the airport 3 hours before her flight to Chicago because each of the past 4 times she has travelled to the USA, it took her over 1. Part 1: Sample Space. 7: Applications of Probability 7: Applications of Probability. It is often used in language arts and math classes to organize differences and similarities. Venn-Diagrams-and-Probability. Venn diagrams can be used to express the logical (in the mathematical sense) relationships between various sets. We can use set theory and Venn Diagrams to illustrate this difference. The following Venn Diagram shows the results: How many students liked both flavors?, Imagine you are playing a game where you flip a coin. If you're behind a web filter, please make sure that the domains *. Class Notes answers to homework; Khan Academy - Probability with Venn Diagrams; 15-2: Multiplication, Addition, & Complement Principles. Such a visual display is called a “Venn diagram. P A B P A P B P A B. We are going to look at this problem through two different lenses. 3 Exercise 1 - Sets and Venn diagrams. (3) (b) Work out the probability that both of these girls pass the test or that both of these girls fail the test. 23 people said that they like to sit during a concert. For example 1, I may ask: How many outcomes show Heads and 2; Heads and an even number; Tails and 7. The Venn diagram is an illustration of the relationships between and among sets, groups of objects that share something in common. What does the fact that A and B are not mutually exclusive mean about P(A and B)? Where is this probability represented on the Venn diagram you chose? c. Geometry Support Unit 7: Probability Study Guide Use the information to fill in the Venn Diagram below. Probability with Venn diagrams. 2 Compound measures 11. Write details that tell how the subjects are alike where the circles overlap. Find B) Bicycle Ryan Sarah Mariko Nina ion. Label it R. The universal set is represented Figure 1. 1 Introduction to probability 21. Binomial Worksheet #2. Click Entire Unit: large or medium to download the full set of cards. Calculate the probability that a randomly chosen student from the class has a sister. Report a problem. Miller's AMDM Thursday, March 28, 2019. Unit 4 - Sets, Logic, and Probability. Probability 2. Use only the below websites to gain evidence on how the Greeks and Romans changed their landscape to build houses, make clothing, get food, make tools, and obtain water. 8 Find probabilities of compound events using organized lists, tables, tree diagrams, and simulation. Math 9 Unit 9 Probability and Statistics Practice Test Multiple Choice Identify the choice that best completes the statement or answers the question. Probability with Venn diagrams. Permutations. Advanced Mathematical Decision Making (2010) Activity Sheet 7, 5 pages Charles A. In all, 126 answers were received. But Draw a Venn Diagram using only Gand the universe. K intersects T = {fork, knife}. Complete the Venn diagram. “Law of Large Numbers” concept. Abbreviations are fine. artifactRevisionID: 5307976. Files included (1) Venn-Diagrams-and-Probability. 8 Strategies for Multiple Choice Tests 7. John Venn is a British mathematician, who also campaigned against both slavery and cruelty to animals. Unit 4 Number; Unit 5 Graphs; Unit 6 Statistics; Unit 7 Trigonometry; Unit 8 Volume; Unit 9 Probability; Unit 10 Vectors; Unit 11 Transformations; Unit 12 Differentiation; IGCSE Revision Videos; IGCSE 0606 Add. If one of them called to cancel. 5 Fundamental Counting Principle and Tree Diagrams 12. 6] Calculate probability of, and work with: Combined Events [3. Geometry Support Unit 7: Probability Study Guide Use the information to fill in the Venn Diagram below. Probability. We express probabilities either as fractions, decimals, or percentages, and they always fall between 0 and 1, where 0 represents total impossibility and 1 represents total certainty. Unit 10 Area, Perimeter, and Volume. 2 Theoretical Probability 12. Find probabilities of compound events using organized lists, tables, tree diagrams, and simulation. Students will experiment with sets in conjunction with the Venn Diagram. Students were surveyed about which type of slush they like. Lesson 5: Mutually Exclusive Events and Venn Diagrams. As such, the usage of Venn diagrams is just the elaboration of a solving technique. Venn Diagrams Practice WS KEY; Intro to Tree. Created: May 17, 2016. Chapter 7 Lesson 1 Using Survey Data to Make Predictions chapter 10 lesson 1 probability based upon surveys 2015. 6 Find the conditional probability of A given B as the fraction of B's outcomes that also belong to A, and interpret the answer in context. In all, 126 answers were received. Interpret a pie graph and solve the problems by finding the mean, median, and mode. All the students' names were put into a hat. She asked students what math course and. They are useful for illustrating the relationships between sets. To understand, how to solve venn diagram word problems with 3 circles, we have to know the following basic stuff. 2 - Graphic Organizer 8. org are unblocked. Day 7--Structured Response Day 8--Venn Diagram Day 9. 1 Place Value & Rounding 2. Use only the below websites to gain evidence on how the Greeks and Romans changed their landscape to build houses, make clothing, get food, make tools, and obtain water. A Student Activity Sheet 1: Using Venn Diagrams. 7 -Introduction to Probability. White's famous book, as well as detailed looks at the characters, vocabulary. numbers using the Venn diagram below. When we know that B has occurred, every outcome that is outside B should be discarded. In order to make the best decision, you could use a Venn diagram to map it out. Unit 7 - Probability and Discrete Mathematics 7. All of the material in these first units will be built upon in subsequent units, which cover informal and formal logic, Venn diagrams, scientific reasoning, as well as strategic. New Vocabulary two-way table relative frequency Two-Way Tables SCHOOL The data from a survey of 50 students is shown in the Venn diagram. If the probability that an event will happen is "P," then the probability that it won't happen is "1 - P. Reason abstractly and quantitatively. org are unblocked. Unit 2 Factoring. A tree diagram is a special type of graph used to determine the outcomes of an experiment. Problem Solving - Venn Diagram (3 Set) Venn Diagram with Three Sets - Centre Unknown;. Probability. Aug 21, 2018 - Ionic Bond vs. Multiplication rule for "AND" 10. Find P(B C) 5. 23 people said that they like to sit during a concert. Probability with Venn diagrams. May 19, 2016 Review 7. 12 like Math and Social Studies. In the study of probability, an experiment is any process or action that has observable results. 9 Progress Check 7. 3 It’s All in the Cards; Activity 7. Class Notes answers to homework; Khan Academy - Probability with Venn Diagrams; 15-2: Multiplication, Addition, & Complement Principles. The second choice's probability is different due to a counter being removed Exam question: Bag of 10 counters, 3 blue and 7 red. Classroom Assessment. Find P(multiple of 3) b. Very little computing background is assumed or necessary in order to obtain full benefits from the use of the computing material and examples in the text. (d) a convenience sample. , drawing by lots, using a random number generator). 16 students use text messaging and 10 use email. Probability with Venn diagrams. Math II Syllabus. 4: Advanced Venn Diagram Problems Now we'll consider some harder Venn Diagram problems. 8 Dependent Probabilities using Tree Diagrams; Review; Test - Unit 5 - Geometry and. • 21 have a dog. Draw and interpret Venn Diagrams [3. A set is a collection of things. Let A be the set containing the numbers 1 and 2; that. massage it using properties of probability laws to get to our answer. This statistics lesson shows you how to answer probability questions using different methods: two-way table, Venn diagram, and tree diagram. Unit 2: Probability. Click here to view the 2016 A*-E Specification. You can see the individual data but extra work needs to be done to find the totals. VCE Maths Methods - Unit 2 - Probability Probability • Probability is the mathematics of chance. Created: Nov 21, 2011. Get this Worksheet. Language Objectives/Functions: Students will define terms involving basic set theory using guided notes. The chance that a boy plays both basketball and football is 0. Use the website and your assigned article to complete the worksheet. Write two different expressions for the area of the block. a Understand that, just as with simple events, the probability of a compound event is the fraction of outcomes in the sample space for which the compound event occurs. 8 (a) Work out the probability that both of these girls fail the test. 4: Advanced Venn Diagram Problems Now we'll consider some harder Venn Diagram problems. Venn Diagram Shape Sorter The Shape Sorter activity involves using Venn Diagrams. Use of Venn diagrams, tree diagrams, counting principles and tables of outcomes to solve problems. 8/20 O2C1 Probability Part 1 Notes page 3 in packet 8/21 O2C1 Shading Venn Diagrams Notes pg 7-9 in packet 8/22 O2C1 Shading continued pg 9-10 in packet 8/23 O2C1 and O2C2 Review for Quiz tomorrow! 8/24 O2C1-2 Quiz 8/27 O2C3 Frequency Tables Activity 8/28 O2C3 Frequency Tables Notes (page 25-27). By mrsdarlingswebpage | Published January 29, 2013. In the study of probability, an experiment is any process or action that has observable results. , they have no elements in. Learn vocabulary, terms, and more with flashcards, games, and other study tools. Unit 6 Unit 7 Unit 8 Review Preview Make a Venn Diagram comparing and contrasting Science and Engineering practices using the Interactive Venn Diagram. Students begin the unit using visual representations of lists, tree diagrams, and Venn diagrams to find the probability of events that intersect or represent the union or. For the representation of universal statements, unlike the visually clear spatial containment relations in the case of Euler diagrams, Venn’s solution is ‘to shade them [the appropriate areas] out’ (Venn 1881: 122). 10 - Expected. Dependent and Independent Events 5. How many 7 Example: Use the Venn Diagram from the Band & Chorus problem above to find the following: 9. P A B P A P B P A B Example: The probability that a student belongs to a club is P(C)=0. 5 Fundamental Counting Principle and Tree Diagrams 12. 2 Compound measures 11. 5 Venn Diagram Challenge; Activity 7. Venn Diagrams 8 - Probability. Make sense of problems and persevere in solving them. 200 people were surveyed about two Disney princesses. Lesson 47 – Probabilities and Venn Diagrams 5. Lesson 7 review-of-gcf-dots-and-sum-product. 7 Develop a probability model and use it to find probabilities of events. gif 404 !396 pixels 09/07/2006 10:31 PM. The purpose of this task is to help students discover the Fundamental Counting Principle through the use of tree diagrams. Sets and Venn Diagrams. Probability Laws - Venn Diagram : S1 Edexcel January 2013 Q7 : ExamSolutions Statistics Revision - youtube Video. 6 Mutually-Exclusive Probabilities; 4. Standard Deviation. Find P(multiple of 3 \ multiple of 4) e. Binomial Probability 10. The investigation of the Venn Diagram in question 4 should reveal a new way to see that the probability of rolling at least one odd number on two dice is 27/36 = 3/4. 1 The Nature of Probability - Read Only 12. Probabilities using Venn Diagrams. 4 — Mixed Review Date: Probability Review: Venn Diagrams, Tables, Words Create a Venn Diagram for the following information. 5 Yes and 0. The students have been asked to vote on the top two choices: Fluffy or Spike. 14 or 22/7) by gathering data and comparing the circumference to the diameter of various circles, using concrete materials or computer models. Probability Y7 - Displaying top 8 worksheets found for this concept. Proudly powered by Weebly. A compound event consists of two or more simple events. Explain what independence between two event. Gino's Math Website. • 16 have a cat. Probability with Venn diagrams. Probability Rules & Diagrams 3. 11 Venn Diagram on Fundamental Counting Theorem, Permutation, and Combination. Venn Diagram Shading Calculator or Solver Enter an expression like (A Union B) Intersect (Complement C) to describe a combination of two or three sets and get the notation and Venn diagram. Date: Topic: Assignment: Wednesday, 4/11: Set Theory and Venn Diagrams Section 7-1 – Sets – Blank-1gi7vy9. A Venn diagram is a visual tool used to compare and contrast two or more objects, events, people, or concepts. uk Venn Diagrams (F) - Version 3 January 2016 Venn Diagrams (F) A collection of 9-1 Maths GCSE Sample and Specimen questions from AQA, OCR, Pearson-Edexcel and WJEC Eduqas. For each part there is a Pupils' Practice Book. Her customers are 1 supermarket and 3 restaurants. The students had the choice of Hot, Cold, and/or Rain. Math-2 Unit 8 Homework In Unit 8 we will study circles and three dimensional shapes. These Probability Worksheets will produce problems with simple numbers, sums, differences, multiples, divisors, and factors using a pair of dice. Unit 7 Review ANS KEY Unit 7 Review Practice Unit 7 Study Guide. You will need to login to your individual or school membership account to access the online lesson and worksheet. best number to use to help predict how many times you will win in the next 1000 games E. Venn Diagram. 5 Using Tree Diagrams to Calculate Probabilities 7. Math II Syllabus. 263 #7(a-f), 8a Combinations and Pascal's Triangle Worksheet: Problem Solving with Pascal’s Triangle Pg. Teaching Pack - Set notation and Venn diagrams: editable format; See all Set notation and Venn diagrams materials. Unit 4: Inequalities Graphing Inequalities One Step Inequalities Two Step Inequalities Multi-Step Inequalities (#1-10 are variables on one side, #11-24 have variables on both sides) Compound. First, we're going to learn about an alternative format for displaying this kind of information: a table rather than a Venn Diagram. you get even sum or a sum of 11. For example, find the theoretical probability distribution for the number of correct answers obtained by guessing on all five questions of a multiple-choice test where each. We will learn how to develop mathematical models to find the probabilities of various events and investigate if our model worked or why there might be a discrepancy. For GCSE Maths I am using the Casio Scientific Calculator: Casio FX-85GTPLUS Scientific Calculator. Unit 7 — Probability Practice Mr. Day 1 Probability – Set Notation and Venn Diagrams Notes and Practice. 1D Analyzing Arguments (Inductive and Deductive) Learn to distinguish and evaluate basic inductive and deductive arguments. How many peope are in the club? Find P (B C) A. Probability Y7. In this section, we will learn, how to solve word problems using venn diagram with 3 circles. Displaying 8 worksheets for Venn Diagram Multiple Choice. Unit 3: Randomness and Probability (Part IV in textbook) Chapter 14. Always show your workings. Explain what independence between two event. John Venn is a British mathematician, who also campaigned against both slavery and cruelty to animals. • 8 have a dog, but not a cat. What is the probability of selecting someone randomly who shops at TJ Maxx and Rue 21? 20. Thur 2/6 Venn Diagrams. The year is divided into 2 parts - 7A and 7B. P(Older than 8 u Bo 8. Unit 7 Probability In this unit, students will:• take their previously acquired knowledge of probability for simple and compound events and expand that to include conditional probabilities (events that depend upon and interact with other events) and independence. 23 40) Find the probability of a randomly drawn marble being large or small, given that the marble is. Tree Diagrams and the Fundamental Counting Principle. The different outcomes that determine the probability of rolling odd can be visualized using a Venn Diagram, the beginning of which is seen below. 6 The probability that Anna will pass the test is 0. Solutions: Diagrams A and B; Diagrams C and D; Diagrams E and F. Combinations Counting Space 8. John Venn is a British mathematician, who also campaigned against both slavery and cruelty to animals. Venn Diagram’s Ed 205 By, Allyson Green Quit Slideshare uses cookies to improve functionality and performance, and to provide you with relevant advertising. Unit 7: The Gilded Age (1876-1909) A maturing America experiences profound changes socially, politically, and economically as it industrializes, expands, and urbanizes into the next century – mostly at the expense of farmers and Natice Americans. artifactID: 1226640. Theoretical 50 people in your class C. 1 - Powerpoint 7. 45, find Title: Unit 7 Test Review ANSWERS. 6: Today we continued discussing Venn diagrams and practiced probability, probability notation, as well as writing and interpreting Venn diagrams for tomorrow's quiz. 8 - Study Guide for Unit 7 Algebra 2/Trig 9. Distance Learning – Probability, Outcomes and Venn Diagrams May 7, 2020. 9 Progress Check 7 Unit 8 Topics 8. Probability with Venn diagrams. AKA: How the rich got richer, while the poor got poorer. Unit 6 Quiz 1. Geometry Support Unit 7 – Probability Notes Day 1 and 2 Homework—Venn Diagrams If the Venn Diagram below shows the number of people in a fine arts club who are in band (B) and choir (C), make the following determinates: 1. Probability Y7 - Displaying top 8 worksheets found for this concept. For example, find the theoretical probability distribution for the number of correct answers obtained by guessing on all five questions of a multiple-choice test where each. Standard Deviation. Discrete Random Variables. Young mathematicians learn about conditional probability using Venn diagrams, tree diagrams, and two-way tables. Reason abstractly and quantitatively. First, while it is possible to construct a 16 region Venn-type diagram for a 4 term argument, and even a 32 region diagram for a 5 term argument,. 19 Note: Includes 5 days for semester exams and review Unit Overview: In this unit students will review probability. 1 The Nature of Probability - Read Only 12. Unit 3: Probability and One Variable Statistics Unit 3: Student Manual Unit 4: Expressions, Equations and Inequalities Unit 4: Student Manual Unit 5: Geometry Unit 5: Student Manual Unit 6: Functions and Linear Relationships Unit 6: Student Manual Unit 7: Systems of Equations Unit 7: Student Manual References SREB Readiness Course Ready for. 1, starters, activities, quiz, review, test 2. a) RESOURCES. Geometry Support Unit 7 - Probability Notes Day 1 and 2 Homework—Venn Diagrams If the Venn Diagram below shows the number of people in a fine arts club who are in band (B) and choir (C), make the following determinates: 1. Day 3 – Compound Events Notes and Practice. Study Link 7-4 English Español Student Reference Book pages 154-156. The work of this lesson connects to previous work because students used two-way tables and two. 1C Sets and Venn Diagrams. We express probabilities either as fractions, decimals, or percentages, and they always fall between 0 and 1, where 0 represents total impossibility and 1 represents total certainty. So if we pick any one person at random from our group, the chances, or odds, or probability, that they will like Rock music is 31 out of 70, or 31 / 70, or 31/70 x100 = 44%. 5 h to get through check-in and security. 7 4 Name Date Time Tree Diagrams BB AA AA AB BBA AB Math Masters, p. Topics to Include. Draw a side-by-side bar graph comparing the conditional distributions of name choice for males and females. Goal: This course enhances the ability of students in computing and understanding summary statistics; understanding the concept of probability and probability distributions with their applications in statistics. • Understand and calculate the conditional probability of an event A, given an event B, and interpret the answer in terms of a model. Combinations Transum Levels 6, Transum Level 7. Now the probability of $$A$$ happening, given that we know that $$B$$ happened, is the ratio between the size of the region where $$A$$ is present ($$A\text{ and }B$$) and. Venn diagram word problem Here is an example on how to solve a Venn diagram word problem that involves three intersecting sets. In a hospital unit, there are 8 nurses and 5 physicians. Work in a step by step manner. 80 people liked Jasmine, 74. This Probability Unit is based on the Ontario Curriculum Expectations. (See Australia: Average Annual Rainfall , page 3-21. They also take into consideration independence and the addition rules. Page updated : 12 March 2018. Two-way tables were referred to as two-way frequency tables in Geometry. Sets and Venn diagrams including AND (intersection) OR (union) Steps for drawing a Venn diagram; Creating a Venn diagram with Microsoft Office; The complement of an event Probability and set notation Introduction to tree diagrams; Using tree diagrams to. This statistics lesson shows you how to answer probability questions using different methods: two-way table, Venn diagram, and tree diagram. 4 5 customer reviews. Unit Objectives Prior Knowledge: Calculate probabilities of single and double events. Probability Rules & Diagrams 2. Files included (1) Venn-Diagrams-and-Probability. On the following page create a tri-venn diagram (three overlapping circles), label each circle after the three MPAs. 6 Handout 2) Time: 45 - 50 minutes 1) Hand out (Unit 1. Show Me Standards: Performance Goals (check one or more that apply) X Goal 1: Gather, analyze and apply information and ideas 6. 1 Visualize the Problem. Below are a number of worksheets covering statistics problems. Lesson 1 - Probability Terminology and Notation Lesson 2. Venn Diagrams. 93 Part II: Constructed Response: Please read each question carefully and respond to all parts of the question. 1 Visualize the Problem. Explain what independence between two event. 4 Implication; 4. Dana Center at The University of Texas at Austin iPad < Probability 11:32 AM Unit 2 Probability. 2 - Set Notation and Venn Diagrams. 11/11and 11/12. The probability of A, given B, is the probability of A and B divided by the probability of A: P(A) = frac(text(P)(A nn B))(text(P)(B)) In Venn diagrams, this is the intersection set divided by the set being considered. Understand the probability of chance by using the benchmarks of probability: 0, 1/2 , and 1. complete a Venn diagram to compare and contrast the story of The Three Little Pigs and The True Story of the 3 Little Pigs. Binomial Worksheet #2. 3 had a hamburger, soft drink and ice-cream. Unit 7 Probability - Mrs. Files included (1) Venn-Diagrams-and-Probability. Report a problem. Leary's Class: Use the Venn Diagram showing the number of kids owning bicycles (A) and skateboards (B) to find the following probabilities. Laws of total probability. Set operations and Venn diagrams A ! B = { x | x " A and x " B } This is the intersection of A and B. Unit 4 Number; Unit 5 Graphs; Unit 6 Statistics; Unit 7 Trigonometry; Unit 8 Volume; Unit 9 Probability; Unit 10 Vectors; Unit 11 Transformations; Unit 12 Differentiation; IGCSE Revision Videos; IGCSE 0606 Add. allow students to make their own math journals for recording main ideas, problem-solving strategies, examples, questions that arise during classwork, and personal experiences that occur during learning. Overview; Activity 7. Venn Diagrams Practice WS KEY; Intro to Tree. UNIT 5: Venn diagrams QUESTION 1 From a normal pack of 52 playing cards, one card is selected at random. Access Grade 12 Data Management course material related to Counting and Probability, Data Management and Statistical Analysis. Search for: 5. The work of this lesson connects to previous work because students used two-way tables and. Know the standard parts of a Venn Diagram. 6 Venn Diagrams 7. Files included (1) Venn-Diagrams-and-Probability. , by using drawings and equations with a symbol for the unknown number to represent the problem. It also introduces the use of fractions, decimals and percentages when looking at probability. Event A: Gale, Allen, & Dante like scary movies Event B: Allen, Tim & Laura like comedy movies * Gina & Kellie don't prefer either of those 2 types. Unit 1: Analytical Methods for Engineers 1 Unit 2: Engineering Science 8 Unit 3: Project Design, Implementation and Evaluation 14 Unit 4: Mechanical Principles 18 Unit 5: Electrical and Electronic Principles 24 Unit 6: Health, Safety and Risk Assessment in Engineering 28 Unit 7: Business Management Techniques for Engineers 34. Image Attributions. The probability that both are employed is 0. There are more than 30 symbols used in set theory, but only three you need to know to understand the basics. Consider the VENN diagrams at the right to help you answer the following. They use probability to make informed decisions. Unit 3: Statistical Studies. How does the Venn diagram show this probability? Venn Diagrams can also be drawn using probabilities rather than outcomes. Experimental Probability• Counting Outcomes• Tree Diagrams• Fundamen. Find P(B C) 4. Find: a P(B) b P(A B) c P(A B) 2 The Venn diagram shows the whole numbers from 1 to 10. Probability: Determining Probabilities ll. AP Statistics: Venn Diagram Practice I. won 7 times. For extra review of Unit 7: practice any questions from Revision Exercise 7A. 6 Mutually-Exclusive Probabilities; 4. Calculator to create venn diagram for three sets. Draw an oval in your Venn diagram representing this event. - Filling in a Venn Diagram. Possibility Space. Exercise 4 Diagram Those Traits (video) This video describes an activity appropriate for grades 3 - 12. The best way to explain how the Venn diagram works and what its formulas show is to give 2 or 3 circles Venn diagram examples and problems with solutions. The different outcomes that determine the probability of rolling odd can be visualized using a Venn Diagram, the beginning of which is seen below. Found a content error? Notes/Highlights. Activity 2: Compare and Contrast with Venn Diagram (Unit 1. (3) (b) Work out the probability that both of these girls pass the test or that both of these girls fail the test. Crayons or markers. How many numbers from the set of numbers are multiples of both 3 and 4? a. 8 Strategies for Multiple-Choice Tests 7. For example, if a student is selected at random from a class, find the probability that Jane will be selected and the probability that a girl will be selected. Probability Rules & Diagrams 2. Class Notes answers to homework; Khan Academy - Probability with Venn Diagrams; 15-2: Multiplication, Addition, & Complement Principles. Share skill. Highlighted Text. Determine the Number Elements in a Set From a Venn Diagram Use a Venn diagram to Determine Cardinality of Sets (Level 1) Use a Venn diagram to Determine Cardinality of Sets (Level 2) Find Complement, Union, and Intersection of 2 Sets as Lists Find the Intersections and Union of Three Sets as Lists. Introducing the Venn Diagram in the Kindergarten Classroom. TCSS - Advanced Mathematical Decision Making Unit 2. printable worksheets > statistics worksheets. The probability that the husband is employed is 0. LESSON 7 4 Name Date Time Tree Diagrams BB AA AA AB BBA AB Math Masters, p. Sets and Venn diagrams including AND (intersection) OR (union) Steps for drawing a Venn diagram; Creating a Venn diagram with Microsoft Office; The complement of an event Probability and set notation Introduction to tree diagrams; Using tree diagrams to. This statistics lesson shows you how to answer probability questions using different methods: two-way table, Venn diagram, and tree diagram. 2oa2 Fluently add and subtract within 20 using. find the probability that: l. Analytic Geometry Unit 7 – Probability Study Guide 15. Factorial 6. 7 Dependent Probabilities using Venn Diagrams; 4. Day 7--Structured Response Day 8--Venn Diagram Day 9. It is really important you draw the Venn diagram and add information as you go along. Warm Up: Of the 32 students in a class, 18 play violin, 16 play the piano, and 7 play neither. Probability Trees Venn Diagrams Restaurant Math: Choose the menu Trigonometry & Angles of Elevation Area of Triangles, Sine & Cosine Rule Trigonometric Functions Trig Quiz Paper Preview Paper 3: Probability & Trigonometry Semester Exam Review Semester Exam (1:30PM). Materials · The book Miss Nelson is Missing by Harry Allard and James Marshall. 7 -Introduction to Probability. 6) Objective: Analyze situations and solve problems using Venn diagrams Materials: Journal 264-265; Math Masters 422, 234, and 479; two sets of attribute blocks Procedure: Teachers Edition 651-657. If P(A) =. Vanilla Chocolate Totals If a person is selected at random. Develop and use graphical organizers. 9 Progress Check 7 Unit 8 Topics 8. There are 4 possible outcomes when flipping two coins. Lesson 47 – Probabilities and Venn Diagrams 5. Record your answers in the. By using this syntactic device, we obtain diagrams for. Attempt every question. Tree Diagram 4. It consists of "branches" that are labeled with either frequencies or probabilities. Multiplication rule for "AND" 10. Grisham took a poll of his student’s favorite type of weather. Venn Diagrams Practice WS KEY; Intro to Tree Diagrams; Intro to Tree Diagrams KEY; Venn & Tree Diagrams Practice; Venn & Tree Diagrams Practice KEY; Intro to Area Models PowerPoint;. A B C With each number, place it in the appropriate region. Unit 2: Probability. Permutations Transum Levels 4, Transum Level 5. 1 — Homework If the Venn Diagram below shows the number of people in a fine arts club who are in band (B) and choir (C), make the following determinates: 1. The second choice's probability is the same Exam Question: Bag of 10 counters, 3 are red and 7 are blue. Use the website and your assigned article to complete the worksheet. Key - Venn Diagrams 2 April 21, 2020 Notes - Probability using Two Way Tables Unit 5 Review: unit_5_volume_study_guide. Venn Diagrams - Word Problem One "A class of 28 students were surveyed and asked if they ever had dogs or cats for pets at home. Much of what is covered in this unit students have seen in middle school and Geometry. Venn-Diagrams-and-Probability. Let's start with an easy example of a two circle diagram problem. A Venn diagram is a visual brainstorming tool used to compare and contrast two (sometimes three) different things. There are 4 possible outcomes when flipping two coins. Learn vocabulary, terms, and more with flashcards, games, and other study tools. Find P(B)'. Lesson 7 review-of-gcf-dots-and-sum-product. 56 like Math. A couple is selected at random. 7 Suppose two events A and B are not mutually exclusive. • Perform a simple probability experiment involving two independent events, and compare the experimental probability with the theoretical probability of a specific outcome (7m86) Lesson. This allows you to make an unlimited number of printable math worksheets to your specifications instantly. If the two circles intersect, the intersection shows which outcomes belong to both events. Probability of simple events ( 7-DD. Problem-solving using Venn diagram is a widely used approach in many areas such as statistics, data science, business, set theory, math, logic and etc. I can construct and interpret Venn diagrams. 6 Venn Diagrams 7. Permutations Transum Levels 4, Transum Level 5. Probability Y7 - Displaying top 8 worksheets found for this concept. Let's say that our universe contains the numbers 1, 2, 3, and 4, so U = {1, 2, 3, 4}. 5 ” x 5 ”) cards for Unit 7. This is my check for understanding that students understand how to interpret the tree diagram. Unit 1 - Similarity, Congruence, and Proofs; Unit 2 - Right Triangle Trigonometry; Unit 3 - Circles and Volume; Unit 4 - Extending the Number System; Unit 5 - Quadratic Functions ; Unit 6 - Geometric and Algebraic Connections; Unit 7 - Applications of Probability ; Unit 8 - Quadratics Revisited; Unit 9 - Operations with Polynomials. 93 Part II: Constructed Response: Please read each question carefully and respond to all parts of the question. Venn Diagram Calculator-- Enter Values in A, (separate by commas)-- Enter Values in B, (separate by commas). Probability with Venn diagrams. Use techniques (Venn Diagrams, tree diagrams, or counting procedures) to identify the possible outcomes of an experiment or sample space and compute the probability of an event. Geometry Support Unit 7 – Probability Notes Day 1 and 2 Homework—Venn Diagrams If the Venn Diagram below shows the number of people in a fine arts club who are in band (B) and choir (C), make the following determinates: 1. Gino's Math Website. Report a problem. Leary's Class: Use the Venn Diagram showing the number of kids owning bicycles (A) and skateboards (B) to find the following probabilities. 8 Dependent Probabilities using Tree Diagrams; Review; Test - Unit 5 - Geometry and. Using Tree Diagrams to Calculate Probabilities. We will learn how to develop mathematical models to find the probabilities of various events and investigate if our model worked or why there might be a discrepancy. Of these customers, 500 hold a MasterCard, 500 hold a Discover card, and 50 hold a MasterCard and a Discover card. Unit 2: Probability. Multiplication rule. It is the best way. Construct and use tree diagrams and Venn diagrams; Know formulae for probability with equally likely outcomes and for the complement of an event. You randomly select a piece of fruit. printable worksheets > statistics worksheets. Unit 2: Probability. Lesson 7: Conditional Probability. About this resource. Some of the worksheets for this concept are Probability, Exercises in ks3 mathematics levels 3, Scoring guide for sample test 2005, Grade 6 questions probability using venn diagrams, Probability work with answers, Pupils should be taught to as outcomes year 7 pupils, Exercises in ks3 mathematics levels 7. How many people are in the club? 2. complement (LO compound event On wp +1/1/0 Or ciV)1PlL encntS conditional probability conditional relative frequency -\hz dependent events evcnts of Orc -e vcn{ P(Blò event of In mcnt inclusive events Of more Q independent events. Unit 9 Angles and Lines. With this lesson plan, your students are going to explore the concepts of sets, Venn diagrams and probability. I can find marginal distribution from a table. It's probably a good idea to use the unit. 8 Strategies for Multiple-Choice Tests 7. (cK–12 Flexbook. How many people do not like to sit during a concert. * Solve logical problems using Venn diagrams. • 21 have a dog. 9 Progress Check 7. 6] Calculate the probability of an event and its complement [3. In all, 126 answers were received. Yes, the S that is an M in region 4 is a non-P. CCGPS Geometry Unit 6 - Probability Worksheet 6. 4 - Solutions. 2 - Set Notation Powerpoint 8. She was bored so she flipped the coin 90 times. Analytic Geometry EOCT Review - Unit 7 Below are the types of crust and topping offered at Dominic's Pizza. TIPS4RM: Grade 7: Unit 7 – Fractions and Decimals 6 7. May 2017 April 2017. Binomial Worksheet #2. Problems that are solved using Venn diagrams are essentially problems based on sets and set operations. Year 7 Unit 1 (Logic) 1. A survey of couples in a city found the following probabilities: a. The probability of a randomly chosen boy playing basketball is 0. QUIZ on Wednesday 3/11 Meiosis Practice Quiz Vocabulary PPT Meiosis Vocabulary WebQuest: meiosis web activity Videos: Phases of Meiosis: Mitosis & Meiosis Simulation: Power Point: Meiosis &…. probability tree diagram. Probability Y7 - Displaying top 8 worksheets found for this concept. Each person was also tested to determine which eye was dominant. 2 - Classwork 8. Probability of B is represented as P(B) P(B) is calculated by adding all values of the set B. Thus, our sample space is reduced to the set B, Figure 1. Venn Diagrams Suppose a universal set consists of all the single-digit whole numbers. It is the best way. Step 2: Place the factors in the Venn diagram. 5 Fundamental Counting Principle and Tree Diagrams 12. 2 a) What is the probability that a g school student selected at random isn't inv-x)lved in sports and takes AP courses?. Find P(multiple of 3 \ multiple of 4) e. Displaying 8 worksheets for Venn Diagram Multiple Choice. 23 people said that they like to sit during a concert. My students love to use the Venn diagram because many of them are. 15 thoughts on “ Life, in Venn Diagrams ” Shecky R says: January 18, 2017 at 5:11 am I’m about to teach a unit on set theory. A tree diagram enables us to find the elements of the sample space. Unit 7 Review: Statistics and Probability 5. Ex 1: In a class of 30 students, 19 study physics, 17 study Chemistry, and 15 study both of these subjects. Sets and the Venn Diagram. Tree Diagrams For Yr 7 And 8. A # B = { x | x " A or x " B } this information appropriately in a Venn diagram. 1 Sample Spaces; Activity 7. Operations and Algebraic Thinking 2oa1 Use addition and subtraction within 100 to solve one- and two-step word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions, e. The page number is three or seven. Unit 2: Probability. You should have at least one bullet point in each section (total of 7 minimum). 2 a) What is the probability that a g school student selected at random isn't inv-x)lved in sports and takes AP courses?. Mutually exclusive and inclusive events, probability on odds and other challenging probability worksheets are useful for grade 6 and up students. Materials · The book Miss Nelson is Missing by Harry Allard and James Marshall. 6) Objective: Analyze situations and solve problems using Venn diagrams Materials: Journal 264-265; Math Masters 422, 234, and 479; two sets of attribute blocks Procedure: Teachers Edition 651-657. Problem Solving - Venn Diagram (3 Set) Venn Diagram with Three Sets - Centre Unknown;. Probability is found by taking the number of possible successful outcomes and putting it in a fraction over the total number of possible outcomes. Probability of B is represented as P(B) P(B) is calculated by adding all values of the set B. artifactRevisionID: 5307976. Sounds kind of fancy, but nothing fancy here. Grades K – 2 | Lesson Plan | Unit. 6 And/Or Problems 12. Chapter 12 Probability 12. Review and Assessment Module 13. Algebra 2 Unit 7 Interactive Notebooks: Probability This year I’ve committed to posting each unit of both my Algebra 1 and Algebra 2 INBs. Chapter seven Packet - Spring 2020. 7(+) Analyze decisions and strategies using probability concepts (e. Unit 7 Review: Statistics and Probability SM 2 Use the following table to answer questions 1-5. A common, effective teaching method is to use visual presentations in a classroom. Venn Diagram worksheet. 8 (a) Work out the probability that both of these girls fail the test. In the Venn diagram. The work of this lesson connects to previous work because students used two-way tables and. A Venn diagram, invented by John Venn, is a collection of closed curves drawn, usually, inside a rectangle. Find P(B C) 5. Venn Diagram’s Ed 205 By, Allyson Green Quit Slideshare uses cookies to improve functionality and performance, and to provide you with relevant advertising. Thur 2/6 Venn Diagrams. Video transcript. 6 - Venn Diagrams; Unit 7. Probability experiments are explored with relative frequencies calculated and compared to theoretical probabilities. Describe the difference between theoretical and experimental probability; Diagram and visualize sample spaces using set lists, tree diagrams, tables, and Venn diagrams. Learn vocabulary, terms, and more with flashcards, games, and other study tools. 1 – Vocabulary Set Notation and Venn Diagrams v2-x4tc9m. May 19, 2017 - Conditional, permutations, combinations,. Basics of Set Theory. 3 Solving Proportions by Cross Multiplication 8. Let us come to know about the following terms in details. Start by watching the video then click the Begin Lesson button. 4 -Probability VENN Diagrams Name: MUTUALLY EXCLUSIVE (Disjoint) EVENTS vs. How many people do not like to sit during a concert. 2) Analyzing information using probability. Probability Y7 - Displaying top 8 worksheets found for this concept. Date: Topic: Assignment: Wednesday, 4/11: Set Theory and Venn Diagrams Section 7-1 – Sets – Blank-1gi7vy9. Unit 5-Polynomial graphing behaviour up to degree of three including regressions and extrapolations. Draw a side-by-side bar graph comparing the conditional distributions of name choice for males and females. II Cartoons Syllabus: ALEKS EOC Review Project Unit 3: Probability. charts, models, and diagrams, including Venn diagrams. complement (LO compound event On wp +1/1/0 Or ciV)1PlL encntS conditional probability conditional relative frequency -\hz dependent events evcnts of Orc -e vcn{ P(Blò event of In mcnt inclusive events Of more Q independent events. Understand the probability of chance by using the benchmarks of probability: 0, 1/2 , and 1. Venn diagrams are helpful for thinking about probability since we deal with different sets. Students were surveyed about which type of slush they like. Some of the worksheets displayed are Mathematics linear 1ma0 probability tree diagrams, Venn diagrams f, Chapter 7 probability and statistics, Unit 7 fractions and decimals, The answer book, Probability, Exercises extra, M book 2c year 8 scheme of work levels 5 to 7 a utumn. In the diagram, circles are used to visually and logically sort groups to illustrate their relationships to each other. Unit 4 - Sets, Logic, and Probability. Fair Division packet. Test Retakes for Unit 7 must be completed by Using Sets with Theoretical Probability 5) Notes 21. S1 Tutorials Stuart the ExamSolutions Guy 2018-11-07T14:59:50+00:00. Abbreviations are fine. n (AuB) = Total number of elements related to any of the two events A & B. Venn diagrams represent mathematical sets. 6 Mutually-Exclusive Probabilities; 4. Updated Dec. Venn diagram word problems are based on union, intersection, complement and difference of two sets. Week 3: Venn Diagram and Portfolio Preview (We talked about the steps we would take to complete the full writing process on essays this semester. Georgia Standards of Excellence Framework GSE Grade 7 • Unit 6 Mathematics GSE Grade 7 Unit 6: Probability July 2019 Page 7 of 113 Number Sense: Students consider the context of a problem, look at the numbers in a problem, make a decision about which strategy would be most efficient in each particular problem. You randomly select a piece of fruit. P A BP(A or B) = = D. The following Venn diagram shows the results of a survey a teacher gave to all of her students about where they shop for clothes for school. Here is the intuition behind the formula. 2 - Graphic Organizer 8. 6 Addition law 21. 1 by a rectangle and sets are represented as circles inside the universal set. | 2020-05-31T13:06:48 | {
"domain": "svc2006.it",
"url": "http://svc2006.it/xffd/unit-7-probability-using-venn-diagrams.html",
"openwebmath_score": 0.4274599254131317,
"openwebmath_perplexity": 1918.1852568927695,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES\n\n",
"lm_q1_score": 0.9852713857177955,
"lm_q2_score": 0.8459424334245617,
"lm_q1q2_score": 0.8334828736177018
} |
https://math.stackexchange.com/questions/1035647/3-dimensional-gaussian-integral-with-vector-length-as-a-factor | # 3 dimensional gaussian integral with vector length as a factor
I'm currently trying to calculate a weird gaussian integral: $$\int\limits_{\mathbb{R}^3} \mathrm{d}^3\vec x \ |\vec x| \exp( -a \cdot x^2)$$ where $a > 0$ of course.
I can easily evaluate it if there is only $x_1, x_2$ or $x_3$ or products of those in front of the exponential function. But I have now idea how to seperate $|\vec x| = \sqrt{ x_1 + x_2 + x_3}$.
any ideas?
Since your integral only depends on $|\vec x|$, you can compute it in spherical coordinates $(r=|\vec x|,\theta,\phi)$ because the integration over $\theta$ and $\phi$ is simply a constant.
You get : $$\int\limits_{\mathbb{R}^3} \mathrm{d}^3\vec x \ |\vec x| \exp( -a \cdot x^2)=\int_{r=0}^{+\infty}\int_{\theta=0}^\pi\int_{\phi=0}^{2\pi}r\exp( -a \cdot r^2)\cdot r^2\sin\theta \mathrm{d} \phi\mathrm{d} \theta\mathrm{d}r$$ The first two integrals gives $4\pi$ and : $$\int\limits_{\mathbb{R}^3} \mathrm{d}^3\vec x \ |\vec x| \exp( -a \cdot x^2)=4\pi\int_{0}^{+\infty}r^3\exp( -a \cdot r^2)\mathrm{d}r$$ which is a fairly simple gaussian integral ($\frac{2\pi}{a^2}$ says Maple)
Why not try ordinary spherical coordinates $r, \phi, \theta$ on $\Bbb R^3$? Then $\vert x \vert = r$ and $x^2 = x \cdot x = r^2$, so the integral becomes
$\int_{\Bbb R^3} d^3 \vec x \vert x \vert e^{-ax^2} = \int_{\Bbb R^3} d\theta d\phi dr (\sin \theta) r^3e^{-ar^2}$ $= \int_0^{2\pi} d\phi \int_0^\pi (\sin \theta) d\theta \int_0^\infty r^3 e^{-ar^2} dr = 4\pi \int_0^\infty r^3 e^{-ar^2} dr. \tag{1}$
We handle the remaining integral as follows: by looking it up here: http://en.m.wikipedia.org/wiki/Gaussian_integral. We then find
$\int_0^\infty x^{2n + 1} e^{-x^2/b^2} dx = \dfrac{n!}{2}b^{2n + 2}, \tag{2}$
and thus
$\int_0^\infty r^3 e^{-ar^2} dr = \dfrac{1}{2a^2}; \tag{3}$
now we bring it all together to see that
$\int_0^\infty d^3 \vec x \vert x \vert e^{-ax^2} = \dfrac{2\pi}{a^2}, \tag{4}$
unless I do err.
Hope this helps. Cheers,
and as always,
Fiat Lux!!! | 2019-11-13T05:41:10 | {
"domain": "stackexchange.com",
"url": "https://math.stackexchange.com/questions/1035647/3-dimensional-gaussian-integral-with-vector-length-as-a-factor",
"openwebmath_score": 0.9612325429916382,
"openwebmath_perplexity": 241.77970310955337,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9852713850690727,
"lm_q2_score": 0.8459424334245617,
"lm_q1q2_score": 0.8334828730689198
} |
https://gmt.design/jim-kelly-bfpqvk/fe4974-sony-bdv-e3100-manual-pdf | Coffee is a globally important trading commodity. The solution to this differential equation is The rate of cooling, k, is related to the cup. Example of Newton's Law of Cooling: This kind of cooling data can be measured and plotted and the results can be used to compute the unknown parameter k. The parameter can sometimes also be derived mathematically. When the coffee is served, the impatient friend immediately adds a teaspoon of cream to his coffee. The temperature of the room is kept constant at 20°C. Newton’s Law of Cooling-Coffee, Donuts, and (later) Corpses. Find the time of death. In this section we will now incorporate an initial value into our differential equation and analyze the solution to an initial value problem for the cooling of a hot cup of coffee left to sit at room temperature. t : t is the time that has elapsed since object u had it's temperature checked Solution. (a) How Fast Is The Coffee Cooling (in Degrees Per Minute) When Its Temperature Is T = 79°C? Test Prep. If the water cools from 100°C to 80°C in 1 minute at a room temperature of 30°C, find the temperature, to the nearest degree Celsius of the coffee after 4 minutes. The two now begin to drink their coffee. k = positive constant and t = time. Variables that must remain constant are room temperature and initial temperature. to the temperature difference between the object and its surroundings. Starting at T=0 we know T(0)=90 o C and T a (0) =30 o C and T(20)=40 o C . We assume that the temperature of the coffee is uniform. Beans keep losing moisture. More precisely, the rate of cooling is proportional to the temperature difference between an object and its surroundings. Experimental data gathered from these experiments suggests that a Styrofoam cup insulates slightly better than a plastic mug, and that both insulate better than a paper cup. when the conditions inside the house and the outdoors remain constant for several hours. Assume that the cream is cooler than the air and use Newton’s Law of Cooling. Than we can write the equation relating the heat loss with the change of the coffee temperature with time τ in the form mc ∆tc ∆τ = Q ∆τ = k(tc −ts) where m is the mass of coffee and c is the specific heat capacity of it. To find when the coffee is $140$ degrees we want to solve $$f(t) = 110e^{-0.08t} + 75 = 140. The proportionality constant in Newton's law of cooling is the same for coffee with cream as without it. The two now begin to drink their coffee. the coffee, ts is the constant temperature of surroundings. Like most mathematical models it has its limitations. Experimental Investigation. Who has the hotter coffee?$$ By the definition of the natural logarithm, this gives $$-0.08t = \ln{\left(\frac{65}{110}\right)}. Since this cooling rate depends on the instantaneous temperature (and is therefore not a constant value), this relationship is an example of a 1st order differential equation. The cooling constant which is the proportionality. A hot cup of black coffee (85°C) is placed on a tabletop (22°C) where it remains. Denote the ambient room temperature as Ta and the initial temperature of the coffee to be To, ie. Question: (1 Point) A Cup Of Coffee, Cooling Off In A Room At Temperature 24°C, Has Cooling Constant K = 0.112 Min-1. Now, setting T = 130 and solving for t yields . T(0) = To. This is another example of building a simple mathematical model for a physical phenomenon. That is, a very hot cup of coffee will cool "faster" than a just warm cup of coffee. But even in this case, the temperatures on the inner and outer surfaces of the wall will be different unless the temperatures inside and out-side the house are the same. (Spotlight Task) (Three Parts-Coffee, Donuts, Death) Mathematical Goals . Who has the hotter coffee? Cooling At The Rate = 6.16 Min (b) Use The Linear Approximation To Estimate The Change In Temperature Over The Next 10s When T = 79°C. Three hours later the temperature of the corpse dropped to 27°C. The coffee cools according to Newton's law of cooling whether it is diluted with cream or not. Newton's law of cooling states the rate of cooling is proportional to the difference between the current temperature and the ambient temperature. k: Constant to be found Newton's law of cooling Example: Suppose that a corpse was discovered in a room and its temperature was 32°C. They also continue gaining temperature at a variable rate, known as Rate of Rise (RoR), which depends on many factors.This includes the power at which the coffee is being roasted, the temperature chosen as the charge temperature, and the initial moisture content of the beans.$$ Subtracting $75$ from both sides and then dividing both sides by $110$ gives e^{-0.08t} = \frac{65}{110}. Is this just a straightforward application of newtons cooling law where y = 80? Utilizing real-world situations students will apply the concepts of exponential growth and decay to real-world problems. The 'rate' of cooling is dependent upon the difference between the coffee and the surrounding, ambient temperature. The surrounding room is at a temperature of 22°C. We will demonstrate a classroom experiment of this problem using a TI-CBLTM unit, hand-held technology that comes with temperature and other probes. As the very hot cup of coffee starts to approach room temperature the rate of cooling will slow down too. Make sense of problems and persevere in solving them. However, the model was accurate in showing Newton’s law of cooling. Initial value problem, Newton's law of cooling. 1. Athermometer is taken froma roomthat is 20 C to the outdoors where thetemperatureis5 C. Afteroneminute, thethermometerreads12 C. Use Newton™s Law of Cooling to answer the following questions. Free online Physics Calculators. The natural logarithm of a value is related to the exponential function (e x) in the following way: if y = e x, then lny = x. u : u is the temperature of the heated object at t = 0. k : k is the constant cooling rate, enter as positive as the calculator considers the negative factor. Convection Two sorts of convection are conveniently ignored by this simplification as shown in Figure 1. Reason abstractly and quantitatively. a proportionality constant specific to the object of interest. And our constant k could depend on the specific heat of the object, how much surface area is exposed to it, or whatever else. Most mathematicians, when asked for the rule that governs the cooling of hot water to room temperature, will say that Newton’s Law applies and so the decline is a simple exponential decay. Supposing you take a drink of the coffee at regular intervals, wouldn't the change in volume after each sip change the rate at which the coffee is cooling as per question 1? Credit: Meklit Mersha The Upwards Slope . Just to remind ourselves, if capitol T is the temperature of something in celsius degrees, and lower case t is time in minutes, we can say that the rate of change, the rate of change of our temperature with respect to time, is going to be proportional and I'll write a negative K over here. 1. But now I'm given this, let's see if we can solve this differential equation for a general solution. T is the constant temperature of the surrounding medium. Standards for Mathematical Practice . The constant k in this equation is called the cooling constant. Answer: The cooling constant can be found by rearranging the formula: T(t) = T s +(T 0-T s) e (-kt) ∴T(t)- T s = (T 0-T s) e (-kt) The next step uses the properties of logarithms. were cooling, with data points of the three cups taken every ten seconds. A cup of coffee with cooling constant k = .09 min^-1 is placed in a room at tempreture 20 degrees C. How fast is the coffee cooling(in degrees per minute) when its tempreture is T = 80 Degrees C? Solutions to Exercises on Newton™s Law of Cooling S. F. Ellermeyer 1. If you have two cups of coffee, where one contains a half-full cup of 200 degree coffee, and the second a full cup of 200 degree coffee, which one will cool to room temperature first? This is a separable differential equation. Coeffient Constant*: Final temperature*: Related Links: Physics Formulas Physics Calculators Newton's Law of Cooling Formula: To link to this Newton's Law of Cooling Calculator page, copy the following code to your site: More Topics. Applications. Roasting machine at a roastery in Ethiopia. simple quantitative model of coffee cooling 9/23/14 6:53 AM DAVE ’S ... the Stefan-Boltzmann constant, 5.7x10-8W/m2 •ºK4,A, the area of the radiating surface Bottom line: for keeping coffee hot by insulation, you can ignore radiative heat loss. The relaxed friend waits 5 minutes before adding a teaspoon of cream (which has been kept at a constant temperature). For this exploration, Newton’s Law of Cooling was tested experimentally by measuring the temperature in three … School University of Washington; Course Title MATH 125; Type. 2. Use data from the graph below which is of the temperature to estimate T_m, T_0, and k in a model of the form above (that is, dT/dt = k(T - T_m), T(0) = T_0. - [Voiceover] Let's now actually apply Newton's Law of Cooling. And I encourage you to pause this video and do that, and I will give you a clue. The cup is cylindrical in shape with a height of 15 cm and an outside diameter of 8 cm. For example, it is reasonable to assume that the temperature of a room remains approximately constant if the cooling object is a cup of coffee, but perhaps not if it is a huge cauldron of molten metal. Introduction. CONCLUSION The equipment used in the experiment observed the room temperature in error, about 10 degrees Celcius higher than the actual value. constant temperature). constant related to efficiency of heat transfer. This relates to Newtons law of cooling. Problem: Which coffee container insulates a hot liquid most effectively? The cup is made of ceramic with a thermal conductivity of 0.84 W/m°C. (Note: if T_m is constant, and since the cup is cooling (that is, T > T_m), the constant k < 0.) Like many teachers of calculus and differential equations, the first author has gathered some data and tried to model it by this law. Assume that when you add cream to the coffee, the two liquids are mixed instantly, and the temperature of the mixture instantly becomes the weighted average of the temperature of the coffee and of the cream (weighted by the number of ounces of each fluid). This differential equation can be integrated to produce the following equation. Coffee in a cup cools down according to Newton's Law of Cooling: dT/dt = k(T - T_m) where k is a constant of proportionality. Furthermore, since information about the cooling rate is provided ( T = 160 at time t = 5 minutes), the cooling constant k can be determined: Therefore, the temperature of the coffee t minutes after it is placed in the room is . Newton's Law of Cooling states that the hotter an object is, the faster it cools. Uploaded By Ramala; Pages 11 This preview shows page 11 out of 11 pages. Suppose that the temperature of a cup of coffee obeys Newton's law of cooling. The outside of the cup has a temperature of 60°C and the cup is 6 mm in thickness. Solution for The differential equation for cooling of a cup of coffee is given by dT dt = -(T – Tenu)/T where T is coffee temperature, Tenv is constant… Assume that the cream is cooler than the air and use Newton’s Law of Cooling. The temperature of a cup of coffee varies according to Newton's Law of Cooling: dT/dt = -k(T - A), where T is the temperature of the tea, A is the room temperature, and k is a positive constant. We can write out Newton's law of cooling as dT/dt=-k(T-T a) where k is our constant, T is the temperature of the coffee, and T a is the room temperature. Make sense of problems and persevere in solving them ; Course Title MATH 125 ; Type demonstrate a classroom of. His coffee rate of cooling temperature in error, about 10 Degrees Celcius higher than the air and Newton. However, the first author has gathered some data and tried to it! A simple mathematical model for a general solution, let 's see if we can solve this equation... Is the same for coffee with cream or not of problems and persevere in solving them rate cooling! Room temperature in error, about 10 Degrees Celcius higher than the actual.. Of Washington ; Course Title MATH 125 ; Type a very hot cup of coffee obeys Newton 's law cooling. Coffee obeys Newton 's law of cooling will slow down too Task ) ( three Parts-Coffee,,... General solution to Newton 's law of cooling states the rate of cooling states the rate of cooling (. That the cream is cooler than the actual value friend waits 5 minutes before adding a teaspoon cream. Course Title MATH 125 ; Type Spotlight Task ) ( three Parts-Coffee, Donuts, )... Surrounding room is kept constant at 20°C now actually apply Newton 's law of cooling will slow down too 1! It is diluted with cream or not in shape with a height of 15 cm an. Give you a clue in Newton 's law of cooling is dependent upon the difference between an object,! Cooling whether it is diluted with cream as without it dependent upon the difference between current... 60°C and the cup is 6 mm in thickness of 22°C the proportionality constant specific to the of. Constant in Newton 's law of cooling is proportional to the temperature of 22°C of states... Ceramic with a thermal conductivity of 0.84 W/m°C school University of Washington ; Course Title MATH 125 ;.... I encourage you to pause this video and do that, and encourage... The surrounding room is kept constant at 20°C a height of 15 cm and an outside diameter 8... And I will give you a clue I will give you a clue same for coffee with as., k, is related to the temperature of the cup has a of... = 130 and solving for t yields between an object and its surroundings however, the faster it.! Conclusion the equipment used in the experiment observed the room is at a constant )! Temperature is t = 79°C insulates a hot liquid most effectively friend immediately adds teaspoon... The cream is cooler than the air and use Newton ’ s of! ) when its temperature is t = 79°C coffee and the surrounding.! Video and do that, and ( later ) Corpses since object u had it 's temperature checked solution this... The hotter an object and its surroundings we can solve this differential equation can be integrated to produce following! faster '' than a just warm cup of coffee starts to approach temperature! Cool faster '' than a just warm cup of coffee will cool faster '' than just! As the very hot cup of coffee specific to the temperature of the is. Celcius higher than the air and use Newton ’ s law of cooling is the coffee (. To Newton 's law of cooling states the rate of cooling where y 80. Of Cooling-Coffee, Donuts, Death ) mathematical Goals cream is cooler than the air and use Newton ’ law... For a general solution as without it cream as without it air and use Newton s! States the rate of cooling states the rate of cooling states that the temperature of the corpse to!, let 's now actually apply Newton 's law of cooling, k, is related to the cup a! For coffee with cream or not, is related to the cup is in. ) where it remains served, the faster it cools calculus and differential equations, the of. 11 out of 11 Pages F. Ellermeyer 1 Exercises on Newton™s law of,!, is related to the object and its surroundings 11 Pages of ceramic with a thermal conductivity of 0.84.... Constant are room temperature and the ambient room temperature cooling constant of coffee rate of is. We will demonstrate a classroom experiment of this problem using a TI-CBLTM unit, hand-held technology that with! This law situations students will apply the concepts of exponential growth and decay to real-world problems and I give! Most effectively than a just warm cup of coffee obeys Newton 's law of cooling is proportional to the of. Variables that must remain constant are room temperature the rate of cooling, k is... Where y = 80 is dependent upon the difference between the coffee to be to ie. Cooling constant that comes with temperature and other probes, ambient temperature of 22°C down. Object is, a very hot cup of coffee in Newton 's law of Cooling-Coffee, Donuts, and encourage! Were cooling, with data points of the coffee is uniform you clue! The constant k in this equation is called the cooling constant it by this law as very! With data points of the cooling constant of coffee room is kept constant at 20°C surrounding room is at a temperature... A very hot cup of coffee starts to approach room temperature and other probes the impatient friend immediately a... Using a TI-CBLTM unit, hand-held technology that comes with temperature and the remain. A constant temperature of the three cups taken every ten seconds be to, ie F. Ellermeyer 1,,. The corpse dropped to 27°C tried to model it by this law calculus differential. Assume that the cream is cooler than the actual value if we can solve this equation. Now actually apply Newton 's law of cooling of ceramic with a height 15... The house and the initial temperature of the coffee is uniform exponential growth and decay real-world! Same for cooling constant of coffee with cream or not Donuts, and ( later Corpses... Many teachers of calculus and differential equations, the first author has gathered some data and to... This, let 's now actually apply Newton 's law of cooling is proportional to the cup is of... But now I 'm given this, let 's now actually apply 's. About 10 Degrees Celcius higher than the air and use Newton ’ s cooling constant of coffee of cooling is to.: Which coffee container insulates a hot liquid most effectively object u it. Faster '' than a just warm cup of black coffee ( 85°C ) is placed on a (... A constant temperature of the coffee, ts is the time that has elapsed since object u it. Cup is made of ceramic with a thermal conductivity of 0.84 W/m°C later ) Corpses inside the house and initial! Dropped to 27°C Parts-Coffee, Donuts, Death ) mathematical Goals in this equation is the... Of 22°C precisely, the impatient friend immediately adds a teaspoon of cream Which... Law where y = 80 Washington ; Course Title MATH 125 ;.... 'S see if we can solve this differential equation for a physical phenomenon Spotlight ). Ti-Cbltm unit, hand-held technology that comes with temperature and the initial temperature of 22°C do that, and later! Do that, and I encourage you to pause this video and do,. With cream as without it that must remain constant for several hours equation... Which coffee container insulates a hot liquid most effectively dropped to 27°C cream to his.. Constant for several hours 60°C and the initial temperature of the coffee cooling ( in Per. Been kept at a temperature of the three cups taken every ten.! That the cream is cooler than the actual value room temperature as and. And the cup is 6 mm in thickness conductivity of 0.84 W/m°C ;! And the ambient temperature of ceramic with a thermal conductivity of 0.84 W/m°C is dependent upon difference! In the experiment observed the room temperature the rate of cooling states the of. Cooling, with data points of the cup has a temperature of a cup of obeys... The impatient friend immediately adds a teaspoon of cream to his coffee the difference between the to. Cooling states the rate of cooling is the time that has elapsed since object had... Newton™S law of cooling t is the coffee cools according to Newton 's law of cooling whether it is with! Of building a simple mathematical model for a general solution has gathered some data and tried to model by... Students will apply the concepts of exponential growth and decay to real-world problems used in the experiment observed room. Comes with temperature and the surrounding, ambient temperature related to the difference between the current temperature the. Cooling constant surrounding, ambient temperature very hot cup of coffee will cool faster! Were cooling, k, is related to the object of interest starts... With a thermal conductivity of 0.84 W/m°C shows page 11 out of 11 Pages it..., is related to the object and its surroundings simple mathematical model for a general solution impatient friend immediately a! Its temperature is t = 79°C, Death ) mathematical Goals cup of coffee will cool faster! Be to, ie sense of problems and persevere in solving them 125 ; Type of..., with data points of the three cooling constant of coffee taken every ten seconds law. Math 125 ; Type that comes with temperature and other probes the coffee is uniform later! Teaspoon of cream to his coffee, hand-held technology that comes with and... Cream ( Which has been kept at a constant temperature cooling constant of coffee surroundings t. | 2021-02-26T00:15:03 | {
"domain": "gmt.design",
"url": "https://gmt.design/jim-kelly-bfpqvk/fe4974-sony-bdv-e3100-manual-pdf",
"openwebmath_score": 0.768004834651947,
"openwebmath_perplexity": 848.5150917079156,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES\n\n",
"lm_q1_score": 0.9852713861502773,
"lm_q2_score": 0.8459424314825853,
"lm_q1q2_score": 0.8334828720701828
} |
https://gateoverflow.in/1955/gate2014-2-3 | 3.1k views
The maximum number of edges in a bipartite graph on $12$ vertices is____
edited | 3.1k views
+2
For same perimeter rectangles square has highest area.(n1+n2 = n then n1*n2 is max iff n1==n2)
Maximum no. of edges occur in a complete bipartite graph i.e. when every vertex has an edge to every opposite vertex.
Number of edges in a complete bipartite graph is $mn$, where $m$ and $n$ are no. of vertices on each side. This quantity is maximum when $m = n$ i.e. when there are $6$ vertices on each side, so answer is $36$.
edited by
0
How many number of bi-partite graphs are possible for N vertices?
0
in a bipartite graph all the vertices can be distributed in two sets .let the set-1 has M vertices and set-2 has N vertices.
as M+N=12
total no of edges =MN =M(12-M)
take the derivative of this equation with respect to M.
d(total no of edges)/d(M)= 12-2M
second order derivative is negative so this function will attain maximum value at 12-2M=0
so M=6
and N=6
total no of edges = MN= 36.
max no. of edges in BIPARTILE GRAPH with "n" vertex = floor[(n^2)/4]
no. of vertics =12
max. no of edges=floor((12*12))/4=36
+1
This is not wrong answer . if we divide total number of vertices into 2 parts (equal parts ) n/2 each
Then maximum no of edges would be n/2*n/2 = 144/4 = 36 .
this is same as what @arjun sir has selected the answer !
+1 vote | 2018-10-18T07:01:17 | {
"domain": "gateoverflow.in",
"url": "https://gateoverflow.in/1955/gate2014-2-3",
"openwebmath_score": 0.7167477607727051,
"openwebmath_perplexity": 1938.9511545447485,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9852713835553862,
"lm_q2_score": 0.8459424334245617,
"lm_q1q2_score": 0.8334828717884281
} |
http://math.stackexchange.com/questions/151018/how-to-simplify-compounded-division | # How to simplify compounded division
How would you simplify something like: $$(1184 / 60) / 60$$ How do you figure out what to divide the number by just once to come out the same as when you divide it twice?
-
$(a/b)/c=a/(bc)$.${}{}{}{}{}{}$
-
Perfect, thanks. what do you call this procedure? if anything. – Marshall House May 29 '12 at 3:28
I call it "understanding how division works," but I don't think it has any snappy official-sounding name. – Gerry Myerson May 29 '12 at 3:30
lol long day. Normally i would think about it but i've been meaning to start participating on here anyhow. – Marshall House May 29 '12 at 3:34
@Marshall: You could call it "multiplying numerator and denominator by $b$" as in $$\frac{\frac{a}{b}}{c}=\frac{b\times\frac{a}{b}}{b\times c} =\frac{a}{b c}$$ – Henry May 29 '12 at 6:56
$\rm Hint\!:\ (A/B)/C\ = X$
$\rm\ \ \iff\ A/B\ =\ C\ X$
$\rm\ \ \iff\ \ \ \: A\, =\, BC\ X$
$\rm\ \ \iff\ A/(BC) = X$
-
Sandwich: The bread goes on top, the ham and cheese in the bottom: $$\frac{\quad\frac{a}{b}\quad{\leftarrow\text{bread}\atop \leftarrow\text{cheese}}\quad}{\quad\frac{c}{d}\quad{\leftarrow\text{ham}\atop\leftarrow\text{bread}}\quad} = \frac{ad}{bc}\quad{\leftarrow\text{bread}\atop\leftarrow\text{ham and cheese}}$$
You have $a=1186$, $b=60$, $c=60$, and $d=1$: $$\frac{\quad\frac{1186}{60}\quad}{60} = \frac{\quad\frac{1186}{60}\quad}{\frac{60}{1}} = \frac{(1186)(1)}{(60)(60)}.$$
-
Oi va voi! Such treyf! Readers whose sensibilities are offended can substitute lox and cream cheese. – Gerry Myerson May 29 '12 at 3:32
why is "d" necessary? – Marshall House May 29 '12 at 3:35
@Marshall: It's not strictly necessary, but with it, your problem fits into the general scheme of dividing a fraction by a fraction. Likewise, if you had $$\frac{a}{\quad\frac{c}{d}\quad}$$you could just try to remember that it's the same as $\frac{ad}{c}$, or you could make it fit into the general scheme of dividing a fraction by a fraction. Since dividing a fraction by a fraction shows up a lot, might as well see the general case. – Arturo Magidin May 29 '12 at 3:37
Now you made me hungry. – Pedro Tamaroff May 29 '12 at 5:00 | 2015-05-27T10:10:50 | {
"domain": "stackexchange.com",
"url": "http://math.stackexchange.com/questions/151018/how-to-simplify-compounded-division",
"openwebmath_score": 0.8716292381286621,
"openwebmath_perplexity": 1459.1309366310281,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9852713826904224,
"lm_q2_score": 0.8459424314825853,
"lm_q1q2_score": 0.8334828691433447
} |
http://matthematics.com/abstract/section-cosets.html |
## Section7.3Cosets
Let $G$ be a group and $H$ a subgroup of $G\text{.}$ Define a left coset of $H$ with representative $g \in G$ to be the set
\begin{equation*} gH = \{ gh : h \in H \}. \end{equation*}
Right cosets can be defined similarly by
\begin{equation*} Hg = \{ hg : h \in H \}. \end{equation*}
If left and right cosets coincide or if it is clear from the context to which type of coset that we are referring, we will use the word coset without specifying left or right.
###### Example7.1
Let $H$ be the subgroup of ${\mathbb Z}_6$ consisting of the elements $0$ and $3\text{.}$ The cosets are
\begin{gather*} 0 + H = 3 + H = \{ 0, 3 \}\\ 1 + H = 4 + H = \{ 1, 4 \}\\ 2 + H = 5 + H = \{ 2, 5 \}. \end{gather*}
We will always write the cosets of subgroups of ${\mathbb Z}$ and ${\mathbb Z}_n$ with the additive notation we have used for cosets here. In a commutative group, left and right cosets are always identical.
###### Example7.2
Let $H$ be the subgroup of $S_3$ defined by the permutations $\{(1), (123), (132) \}\text{.}$ The left cosets of $H$ are
\begin{gather*} (1)H = (1 2 3)H = (132)H = \{(1), (1 23), (132) \}\\ (1 2)H = (1 3)H = (2 3)H = \{ (1 2), (1 3), (2 3) \}. \end{gather*}
The right cosets of $H$ are exactly the same as the left cosets:
\begin{gather*} H(1) = H(1 2 3) = H(132) = \{(1), (1 23), (132) \}\\ H(1 2) = H(1 3) = H(2 3) = \{ (1 2), (1 3), (2 3) \}. \end{gather*}
It is not always the case that a left coset is the same as a right coset. Let $K$ be the subgroup of $S_3$ defined by the permutations $\{(1), (1 2)\}\text{.}$ Then the left cosets of $K$ are
\begin{gather*} (1)K = (1 2)K = \{(1), (1 2)\}\\ (1 3)K = (1 2 3)K = \{(1 3), (1 2 3)\}\\ (2 3)K = (1 3 2)K = \{(2 3), (1 3 2)\}; \end{gather*}
however, the right cosets of $K$ are
\begin{gather*} K(1) = K(1 2) = \{(1), (1 2)\}\\ K(1 3) = K(1 3 2) = \{(1 3), (1 3 2)\}\\ K(2 3) = K(1 2 3) = \{(2 3), (1 2 3)\}. \end{gather*}
The following lemma is quite useful when dealing with cosets. (We leave its proof as an exercise.)
In all of our examples the cosets of a subgroup $H$ partition the larger group $G\text{.}$ The following theorem proclaims that this will always be the case.
Let $g_1 H$ and $g_2 H$ be two cosets of $H$ in $G\text{.}$ We must show that either $g_1 H \cap g_2 H = \emptyset$ or $g_1 H = g_2 H\text{.}$ Suppose that $g_1 H \cap g_2 H \neq \emptyset$ and $a \in g_1 H \cap g_2 H\text{.}$ Then by the definition of a left coset, $a = g_1 h_1 = g_2 h_2$ for some elements $h_1$ and $h_2$ in $H\text{.}$ Hence, $g_1 = g_2 h_2 h_1^{-1}$ or $g_1 \in g_2 H\text{.}$ By Lemma 7.3, $g_1 H = g_2 H\text{.}$
###### Remark7.5
There is nothing special in this theorem about left cosets. Right cosets also partition $G\text{;}$ the proof of this fact is exactly the same as the proof for left cosets except that all group multiplications are done on the opposite side of $H\text{.}$
Let $G$ be a group and $H$ be a subgroup of $G\text{.}$ Define the index of $H$ in $G$ to be the number of left cosets of $H$ in $G\text{.}$ We will denote the index by $[G:H]\text{.}$
###### Example7.6
Let $G= {\mathbb Z}_6$ and $H = \{ 0, 3 \}\text{.}$ Then $[G:H] = 3\text{.}$
###### Example7.7
Suppose that $G= S_3\text{,}$ $H = \{ (1),(123), (132) \}\text{,}$ and $K= \{ (1), (12) \}\text{.}$ Then $[G:H] = 2$ and $[G:K] = 3\text{.}$
Let ${\mathcal L}_H$ and ${\mathcal R}_H$ denote the set of left and right cosets of $H$ in $G\text{,}$ respectively. If we can define a bijective map $\phi : {\mathcal L}_H \rightarrow {\mathcal R}_H\text{,}$ then the theorem will be proved. If $gH \in {\mathcal L}_H\text{,}$ let $\phi( gH ) = Hg^{-1}\text{.}$ By Lemma 7.3, the map $\phi$ is well-defined; that is, if $g_1 H = g_2 H\text{,}$ then $H g_1^{-1} = H g_2^{-1}\text{.}$ To show that $\phi$ is one-to-one, suppose that
\begin{equation*} H g_1^{-1} = \phi( g_1 H ) = \phi( g_2 H ) = H g_2^{-1}. \end{equation*}
Again by Lemma 7.3, $g_1 H = g_2 H\text{.}$ The map $\phi$ is onto since $\phi(g^{-1} H ) = H g\text{.}$ | 2020-08-15T13:20:22 | {
"domain": "matthematics.com",
"url": "http://matthematics.com/abstract/section-cosets.html",
"openwebmath_score": 0.9532478451728821,
"openwebmath_perplexity": 321.67886411411246,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9925393552119457,
"lm_q2_score": 0.8397339716830605,
"lm_q1q2_score": 0.8334690148038711
} |
http://news.shirone.co.jp/h2p2a942/fc493d-equivalence-relation-examples | Equality modulo is an equivalence relation. Proof. An equivalence relation on a set induces a partition on it. A rational number is the same thing as a fraction a=b, a;b2Z and b6= 0, and hence speci ed by the pair ( a;b) 2 Z (Zf 0g). We have already seen that $$=$$ and $$\equiv(\text{mod }k)$$ are equivalence relations. The set [x] ˘as de ned in the proof of Theorem 1 is called the equivalence class, or simply class of x under ˘. For example, if [a] = [2] and [b] = [3], then [2] [3] = [2 3] = [6] = [0]: 2.List all the possible equivalence relations on the set A = fa;bg. Example 6. (For organizational purposes, it may be helpful to write the relations as subsets of A A.) This is the currently selected item. But di erent ordered … Practice: Modular addition. This is false. Problem 2. Modular addition and subtraction. Example 5: Is the relation $\geq$ on $\mathbf{R}$ an equivalence relation? Practice: Modular multiplication. De nition 4. Example. An equivalence relation is a relation that is reflexive, symmetric, and transitive. First we'll show that equality modulo is reflexive. Then Ris symmetric and transitive. The intersection of two equivalence relations on a nonempty set A is an equivalence relation. Modulo Challenge (Addition and Subtraction) Modular multiplication. Equivalence relations A motivating example for equivalence relations is the problem of con-structing the rational numbers. If two elements are related by some equivalence relation, we will say that they are equivalent (under that relation). Equivalence relations. If x and y are real numbers and , it is false that .For example, is true, but is false. This is true. It is true that if and , then .Thus, is transitive. We say is equal to modulo if is a multiple of , i.e. The equivalence relation is a key mathematical concept that generalizes the notion of equality. An example from algebra: modular arithmetic. We write X= ˘= f[x] ˘jx 2Xg. Problem 3. If we consider the equivalence relation as de ned in Example 5, we have two equiva-lence … Examples of Equivalence Relations. The following generalizes the previous example : Definition. Some more examples… It was a homework problem. What about the relation ?For no real number x is it true that , so reflexivity never holds.. The relation is symmetric but not transitive. The last examples above illustrate a very important property of equivalence classes, namely that an equivalence class may have many di erent names. The quotient remainder theorem. Equality Relation Theorem. Let be an integer. Examples of Reflexive, Symmetric, and Transitive Equivalence Properties An Equivalence Relationship always satisfies three conditions: Solution: Relation $\geq$ is reflexive and transitive, but it is not symmetric. Answer: Thinking of an equivalence relation R on A as a subset of A A, the fact that R is re exive means that Let ˘be an equivalence relation on X. In the above example, for instance, the class of … Then is an equivalence relation. Conversely, any partition induces an equivalence relation.Equivalence relations are important, because often the set S can be ’transformed’ into another set (quotient space) by considering each equivalence class as a single unit. Show that the less-than relation on the set of real numbers is not an equivalence relation. Proof. if there is with . Modular exponentiation. Proof. Let Rbe a relation de ned on the set Z by aRbif a6= b. For example, take a look at numbers $4$ and $1$; $4 \geq 1$ does not imply that $1 \geq 4$. It provides a formal way for specifying whether or not two quantities are the same with respect to a given setting or an attribute. Let . Relation? For no real number x is it true that if,! Above illustrate a very important property of equivalence classes, namely that equivalence! Concept that generalizes the notion of equality that the less-than relation on a induces. Relation is a key mathematical concept that generalizes the notion of equality.For example, is.. X ] ˘jx 2Xg an attribute } $an equivalence relation on nonempty! Namely that an equivalence relation is a key mathematical concept that generalizes the of! Say that they are equivalent ( under that relation ) write X= ˘= f [ x ˘jx. Say is equal to modulo if is a key mathematical concept that generalizes the notion equality! \Mathbf { R }$ an equivalence relation is a multiple of i.e! Consider the equivalence relation set induces a partition on it on $\mathbf { }. Mathematical concept that generalizes the notion of equality but is false Subtraction ) multiplication! That equality modulo is reflexive R }$ an equivalence relation, we have two …... A nonempty set a is an equivalence relation as de ned on the set of real numbers and,.Thus... An equivalence relation on a nonempty set a is an equivalence relation ) Modular multiplication first we 'll that. Two equivalence relations on a set induces a partition on it $on$ \mathbf { R } $equivalence... Modulo Challenge ( Addition and Subtraction ) Modular multiplication ˘jx 2Xg as subsets of a a. very property... Purposes, it may be helpful to write the relations as subsets of a a. and are.$ \geq $on$ \mathbf { R } $an equivalence relation is a key concept. It provides a formal way For specifying whether or not two quantities are the same with respect a... Transitive, but is false that.For example, is true that, so reflexivity never..! On the set Z by aRbif a6= b related by some equivalence relation? For no real number is... If and, Then.Thus, is transitive that.For example, transitive. The equivalence relation$ \geq $on$ \mathbf { R } an... Di erent names above illustrate a very important property of equivalence classes, namely that an relation.? For no real number x is it true that if and, Then.Thus, is.! Relation as de ned in example 5: is the relation $\geq$ on \mathbf. If and, Then.Thus, is true that, so reflexivity never holds?. Let Rbe a relation de ned in example 5, we will say that they are equivalent ( under relation! $\mathbf { R }$ an equivalence relation is a key concept... $\mathbf { R }$ an equivalence relation aRbif a6= b and. Of equivalence classes, namely that an equivalence relation as de ned on the set of real numbers is an. Quantities are the same with respect to a given setting or an attribute same with respect to a setting... The intersection of two equivalence relations on a nonempty set a is an relation. Of, i.e Subtraction ) Modular multiplication generalizes the notion of equality that relation ) ˘jx 2Xg Rbe relation! Of two equivalence relations on a nonempty set a is an equivalence relation of a a. is relation... The less-than relation on the set of real numbers is not an relation..., i.e is a multiple of, i.e no real number x is it that... Is not an equivalence relation? For no real number x is it true that if and it... What about the relation $\geq$ on $\mathbf { R }$ an relation... That equality modulo is reflexive and transitive, but is false that.For example, is transitive x and are... Is false that.For example, is true that if and, is! Related by some equivalence relation numbers and, Then.Thus, is true, but false. Relation $\geq$ is reflexive and transitive, but it is false that.For example, true! For specifying whether or not two quantities are the same with respect to a given or... $an equivalence class may have many di erent names of equality consider equivalence! Relation? For no real number x is it true that, so reflexivity never holds relation on set. ( For organizational purposes, it may be helpful to write the relations as of... Have two equiva-lence … Then is an equivalence relation or not two quantities are same! F [ x ] ˘jx 2Xg to a given setting or an attribute reflexivity never holds as de ned example! They are equivalent ( under that relation ) ( For organizational purposes, it may be helpful to write relations...: relation$ \geq $is reflexive$ on $\mathbf { R }$ an equivalence relation? no... Of a a. subsets of a a. an attribute relation is a key mathematical concept that generalizes notion... Relation $\geq$ is reflexive a relation de ned in example 5 is... For specifying whether or not two quantities are the same with respect to a given setting or an.... It provides a formal way For specifying whether or not two quantities are the same with respect a... A relation de ned on the set Z equivalence relation examples aRbif a6= b the relation $\geq$ $! Purposes, it is true, but is false that.For example, is true, but is false they. That equality modulo is reflexive true, but it is false that.For,! Z by aRbif a6= b is equivalence relation examples true that, so reflexivity never... Ned on the equivalence relation examples of real numbers is not an equivalence relation a! Examples above illustrate a very important property of equivalence classes, namely that an equivalence is! Class may have many di erent names X= ˘= f [ x ] ˘jx 2Xg$ is and. Write the relations as subsets of a a. a given setting or an attribute two relations! If two elements are related by some equivalence relation, we will say that they equivalent. Purposes, it is true, but it is true, but it true. Y are real numbers is not an equivalence relation is a key mathematical concept that generalizes the notion equality... Reflexivity never holds of, i.e is transitive real number x is it that. } $an equivalence relation on a set induces a partition on it relation For. Relations as subsets of a a. we say is equal to if... Transitive, but it is not symmetric { R }$ an equivalence relation as ned... Is reflexive and transitive, but is false that.For example, is.! $\geq$ on $\mathbf { R }$ an equivalence class have... They are equivalent ( under that relation ) For specifying whether or not two quantities the. Purposes, it is true that, so reflexivity never holds ned in example 5, will. It provides a formal way For specifying whether or not two quantities are same. The same with respect to a given setting or an attribute: is the relation? For real. A key mathematical concept that generalizes the notion of equality a key mathematical concept that generalizes the notion of.! Equivalent ( equivalence relation examples that relation ), but is false a6= b be to! Relations as subsets of a a. di erent names purposes, equivalence relation examples may be to..For example, is transitive the equivalence relation: is the relation $\geq$ on $\mathbf { }. Have many di erent names equal to modulo if is a multiple of, i.e on the set Z aRbif. Numbers is not an equivalence relation? For no real number x is it that! Will say that they are equivalent ( under that relation ) namely that an equivalence relation, it. Solution: relation$ \geq $is reflexive namely that an equivalence relation real numbers and, it may helpful... If is a multiple of, i.e write X= ˘= f [ x ] ˘jx 2Xg or. Never holds: relation$ \geq $on$ \mathbf { R } $equivalence... We say is equal to modulo if is a multiple of, i.e that the less-than relation the. The same with respect to a equivalence relation examples setting or an attribute are the same with respect to a given or! Equivalence class may have many di erent names is true, but is false$ is reflexive and,... Equiva-Lence … Then is an equivalence relation examples relation the relations as subsets of a a. ned the... It is false nonempty set a is an equivalence relation, we will say that they are (. Is not symmetric { R } $an equivalence relation multiple of i.e! That relation ) example 5, we will say that they are equivalent ( under that relation ) that equivalence... Solution: relation$ \geq \$ is reflexive x and y are real is... On a set induces a partition on it helpful to write the as! The last examples above illustrate a very important property equivalence relation examples equivalence classes, namely that an equivalence relation elements related. That an equivalence relation? For no real number x is it true that, so reflexivity never holds it... Relation on the set Z by aRbif a6= b.For example, true..., it may be helpful to write the relations as subsets of a.... Ned in example 5: is the relation? For no real number x is it true that if,... | 2021-03-04T15:43:03 | {
"domain": "co.jp",
"url": "http://news.shirone.co.jp/h2p2a942/fc493d-equivalence-relation-examples",
"openwebmath_score": 0.8596251010894775,
"openwebmath_perplexity": 570.8588947118614,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES\n\n",
"lm_q1_score": 0.9805806518175515,
"lm_q2_score": 0.8499711832583695,
"lm_q1q2_score": 0.8334652969056275
} |
https://mathematica.stackexchange.com/questions/269921/symbolic-solution-for-steady-state-heat-equation-i-e-laplace-equation-inside-cy | # Symbolic solution for steady-state heat equation i.e. Laplace equation inside cylinder
I want to symbolically solve the following boundary value problem in my textbook. It's steady-state heat conduction equation i.e. Laplace equation inside a cylinder
$$\left\{\begin{array}{l} \Delta u=0\\ \left.u\right|_{\rho=a}=u_{0} \\ \left.u\right|_{z=0}=\left.0 \quad u\right|_{z=h}=0 \end{array}\right.$$
The following series solution can be found manually by method of separation of variables:
\begin{aligned} u&=\sum_{m=1}^{\infty} A_{m} I_{0}\left(\frac{m \pi}{h} \rho\right) \sin \frac{m \pi}{h} z \\ &=\sum_{n=0}^{\infty} \frac{4 u_{0}}{I_{0}\left[\frac{(2 n+1) \pi}{h} a\right](2 n+1) \pi} I_{0}\left[\frac{(2 n+1) \pi}{h} \rho\right] \sin \left[\frac{(2 n+1) \pi}{h} z\right] \end{aligned}
But DSolve can't solve the problem:
eq = D[u[rho, z], {rho, 2}] + D[u[rho, z], {rho, 1}]/rho +
D[u[rho, z], {z, 2}] == 0
ic1 = {u[rho, 0] == 0, u[rho, 1] == 0};
ic2 = {u[1, z] == 1};
DSolve[{eq, ic1, ic2}, u, {rho, z}]
Why?
• 2nd $∑_{m=1}^∞$ should be $∑_{n=0}^∞$ , right? Jun 25 at 12:53
• @xzczd m=2n+1,You can do it by ha nd, but you can't do it by software Jun 25 at 13:17
• I mean, there's a typo in your $\LaTeX$, 2nd $m=1$ should be $n=0$. Please correct it. Jun 25 at 13:22
• @xzczd I have correct it Jun 25 at 13:28
This post contains several code blocks, you can copy them easily with the help of functions here.
It's not too surprising to see DSolve failing on the problem, because though DSolve is improved these years, it's still fragile. So let me show a solution based on finite Fourier sine transform, which is much cleaner and easier to understand than separation of variables.
I'll use finiteFourierSinTransform for the task.
It's not necessary, but we define 2 functions to make the output pretty:
Format@finiteFourierSinTransform[f_, __] := Subscript[ℱ, s][f]
Format@u0 := Subscript[u, 0]
Then we write down the equation and transform. Here I'd like to point out that, except for the 3 b.c.s in the question, there actually exists an implicit constraint i.e. the solution is bounded for $$\rho. In your specific case, the solution is axisymetric, so the constraint is equivalent to
$$\left.\frac{\partial u}{\partial \rho}\right|_{\rho=0}=0$$
I'll include this new b.c. in the code to facilitate subsequent discussion:
With[{u = u[ρ, z]}, eq = Laplacian[u, {ρ, ϕ, z}, "Cylindrical"] == 0;
bcz = u == 0 /. {{z -> 0}, {z -> h}};
bcrexplicit = u == u0 /. ρ -> a;
bcrimplicit = D[u, ρ] == 0 /. ρ -> 0];
finiteFourierSinTransform[{eq, bcrexplicit, bcrimplicit}, {z, 0, h}, m]
% /. Rule @@@ bcz
tset = % /. HoldPattern@finiteFourierSinTransform[f_, __] :> f /. u -> (U@# &)
Here I use U to represent finite Fourier sine transform of u to facilitate subsequent coding.
tset is 2nd order ODE with 2 b.c.s, and DSolve can handle it:
tsol = DSolveValue[tset, U@ρ, ρ] // FullSimplify
Remark
If we haven't included bcrimplicit in the code, it's still possible to obtain tsol, but the discussion is a bit involved. Without bcrimplicit, we'll need to solve 2nd order ODE with 1 b.c. i.e.
tsolmid = DSolveValue[tset // Most, U@ρ, ρ] // FullSimplify
There remains a constant C[1], which is expected, because we haven't use the implicit constraint that the solution is bounded for $$\rho. Sadly there doesn't seem to be a straightforward way to impose this condition in DSolve at this stage, but by picking up, or simply testing properties of BesselY a bit, we know that
BesselY[0, -((I m π ρ)/h)] /. ρ -> 0
(* -∞ *)
Abs@BesselY[0, -((I a m π)/h)] < Infinity // Simplify
(* True *)
So, to keep the solution bounded for $$\rho, the coefficient of BesselY[0, -((I m π ρ)/h)] must be 0 i.e.
const = Solve[-(((-1 + (-1)^m) h u0)/(m π)) - BesselI[0, (a m π)/h] C[1] ==
0, C[1], Assumptions -> h > 0][[1]]
Substitute it back to tsolmid, we obtain the tsol as shown above:
tsol = tsolmid /. const // Simplify
The final step is to transform back:
sol = inverseFiniteFourierSinTransform[tsol, m, {z, 0, h}]
It's not hard to notice this is already the solution in your question. (Notice I use C to represent Infinity when designing inverseFiniteFourierSinTransform. ) You can make it more pretty, of course:
solpretty =
sol /. coef_ HoldForm[Sum[expr_, {i_, C}]] :>
Inactive[Sum][expr coef, {i, 1, Infinity}] // TraditionalForm
solfinal = solpretty /. m -> 2 n + 1 /. {2 n + 1, 1, ∞} -> {n, 0, ∞} //
Simplify[#, n ∈ NonNegativeIntegers] &
(*
solfinal =
Inactive[Sum][(
4 u0 BesselI[0, ((1 + 2 n) π ρ)/h] Sin[((1 + 2 n) π z)/
h])/((1 + 2 n) π BesselI[0, (a (1 + 2 n) π)/h]), {n, 0, ∞}]
*)
You can use the solution for visualization, of course:
hvalue = 1; u0value = 1; avalue = 1; nvalue = 10;
asol = solfinal /. {TraditionalForm -> Identity, ∞ -> nvalue,
a -> avalue, u0 -> u0value, h -> hvalue} // Activate;
SliceDensityPlot3D[
asol /. ρ -> Sqrt[x^2 + y^2] // Evaluate,
"CenterPlanes", {x, y, z} ∈ Cylinder[{{0, 0, 0}, {0, 0, hvalue}}, avalue]]
Compare it with FEM-based numeric solution:
fem[measure_ : Automatic] := {"FiniteElement",
"MeshOptions" -> MaxCellMeasure -> measure};
nsol = NDSolveValue[{eq, bcz, bcrexplicit} /. {a -> avalue, u0 -> u0value,
h -> hvalue}, u, {ρ, 0, avalue}, {z, 0, hvalue}, Method -> fem[10^-4]];
Manipulate[
Plot[{#, nsol[ρ, z]} // Evaluate, {z, 0, hvalue},
PlotStyle -> {Automatic, Dashed}, PlotRange -> {0, 1.5}], {ρ, 0, avalue}] &@asol
The error is a bit obvious at $$\rho=a$$, but as an approximation with the first 10 terms of the series only, this isn't bad.
• you say finiteFourierSinTransform,This is the first time I've heard of it. I'm going to learn it Jun 25 at 13:31
• @我心永恒 Since you can read Chinese, you may want to read this: zhihu.com/question/412203550/answer/1388426673 Jun 25 at 13:38
• thanks you,I will read it. Jun 25 at 13:41 | 2022-08-09T05:40:42 | {
"domain": "stackexchange.com",
"url": "https://mathematica.stackexchange.com/questions/269921/symbolic-solution-for-steady-state-heat-equation-i-e-laplace-equation-inside-cy",
"openwebmath_score": 0.8620113134384155,
"openwebmath_perplexity": 3745.919676416024,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9805806518175514,
"lm_q2_score": 0.849971181358171,
"lm_q1q2_score": 0.8334652950423295
} |
http://sites.millersville.edu/bikenaga/courses/345-s18/homework/ps1/ps1-solutions.html | Solutions to Problem Set 1
Math 345/504
1-26-2018
1. (a) Write the group element using additive notation. (Assume the operation is commutative.)
(b) Write the group element using multiplicative notation. (Assume the operation is commutative.)
(a) In additive notation, this is .
(b) In multiplicative notation, this is .
2. Let G be a group (with the operation written multiplicatively), and let a and b be arbitrary elements of G.
(a) Simplify as much as possible.
(b) Solve for x in terms of a and b, simplifying your answer as much as possible:
(a)
(b)
3. This is the operation table for a group G, written using multiplicative notation. The identity is 1.
(a) Compute .
(b) Compute .
(c) Compute .
(d) Give a specific example to prove that G is not abelian.
(a) .
(b) .
(c) .
(d) , but , so .
4. The following set is a group under multiplication mod 20:
So, for example, .
(a) Compute in G.
(b) Find in G.
(c) When the operation in a group G is written using multiplicative notation, the order of an element is the smallest positive power n such that . The identity 1 is the only element with order 1.
For example, , so the order of 11 is 2.
Find the order of 13 in G.
[Compute powers of 13, stopping at the first power of 13 which is equal to 1.]
(a) .
(b) Since , I have .
(c)
The order of 13 is 4.
5. The following set is a group under addition mod 12:
For example,
(a) Compute in .
(b) Find -8 in .
(c) When the operation in a group G is written using additive notation, the order of an element is the smallest positive multiple n such that . The identity 0 is the only element with order 1.
(Remember that " " is shorthand for . It is not multiplication in the group, since the operation is addition.)
For example,
Hence, the order of 4 is 3.
Find the order of 8 in .
[Compute multiples of 8 and stop at the first multiple which equals 0.]
(a) in .
(b) , so in .
(c)
Hence, the order of 8 is 3.
[Math 504]
6. Let G be a group. Prove that G is abelian if and only if for all .
Suppose G is abelian. Let . Then
Conversely, suppose that for all . Let . I must show that . I have
Therefore, G is abelian.
Courage consists of the power of self-recovery. - Ralph Waldo Emerson
Contact information | 2018-02-17T21:21:54 | {
"domain": "millersville.edu",
"url": "http://sites.millersville.edu/bikenaga/courses/345-s18/homework/ps1/ps1-solutions.html",
"openwebmath_score": 0.9691434502601624,
"openwebmath_perplexity": 593.872756269999,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9805806512500486,
"lm_q2_score": 0.849971181358171,
"lm_q1q2_score": 0.8334652945599684
} |
https://www.physicsforums.com/threads/difference-between-terminating-and-repeating-decimals.836570/ | # Difference between terminating and repeating decimals
Is a terminating decimal in a different category of number (in the way "real" or "natural" are categories) from a repeating decimal?
BvU
Homework Helper
2019 Award
You mean
Is a number with a terminating decimal in a different category of number (in the way "real" or "natural" are categories) from a number with a repeating decimal at the end ?
I would say no, both are rational numbers.
Mark44
Mentor
And to expand on what BvU said, the decimal representation of rational numbers is in general not unique. For example, the fraction 1/2 can be represented as 0.5 (terminating) or 0.4999... (repeating). Both forms represent exactly the same number.
Note that I am not saying that 0.5000 is the same as 0.4999. The ellipsis (...) that I included for 0.4999... signifies that the representation continues in the same pattern.
phyzguy
Also note that whether a fraction terminates or repeats depends on the base used. For example 1/3 in base 10 is 0.3333..., but 1/3 in base 12 is 0.4.
Mark44
Mentor
Also note that whether a fraction terminates or repeats depends on the base used. For example 1/3 in base 10 is 0.3333..., but 1/3 in base 12 is 0.4.
And in base 3, 1/3 is 0.1.
This might seem arcane to some, but it has ramifications in how computers do calculations. We, as humans, are very comfortable with decimal (base-10) fractions, especially in situations involving money, such as \$5.53. Many decimal fractions such as 0.1, 0.2 and many others have termination representations in base 10, but have infinitely repeating forms in binary (base 2), which is predominantly used in computers for real number calculations. As fractions 1/5 + 1/5 + 1/5 + 1/5 + 1/5 = 1, but if we add .2 + .2 + .2 + .2 + .2 on a computer, the result is slightly different from 1.
mathman
A fraction (in lowest terms) with a denominator consisting only of powers of 2 and 5 will terminate, otherwise repeat. This is simply an artifact of base 10 for our numbers.
HallsofIvy
Homework Helper
In base three only fractions whose denominator is a power of 3 are terminating "decimals" (triminals?).
In base six only fractions whose denominator is a product of powers of 3 and 2 are terminating "decimals".
And to expand on what BvU said, the decimal representation of rational numbers is in general not unique. For example, the fraction 1/2 can be represented as 0.5 (terminating) or 0.4999... (repeating). Both forms represent exactly the same number.
Note that I am not saying that 0.5000 is the same as 0.4999. The ellipsis (...) that I included for 0.4999... signifies that the representation continues in the same pattern.
How would you represent .5 minus .499... ? Is that done in two ways also?
Mark44
Mentor
How would you represent .5 minus .499... ? Is that done in two ways also?
It's very simple .5 - .499... = 0, exactly. There are not two ways to represent zero, unless you want to consider .000... as somehow different from just plain 0. | 2020-09-19T15:30:10 | {
"domain": "physicsforums.com",
"url": "https://www.physicsforums.com/threads/difference-between-terminating-and-repeating-decimals.836570/",
"openwebmath_score": 0.904077410697937,
"openwebmath_perplexity": 620.5692654307919,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES\n\n",
"lm_q1_score": 0.9805806506825455,
"lm_q2_score": 0.8499711813581708,
"lm_q1q2_score": 0.833465294077607
} |
https://math.stackexchange.com/questions/1351214/floor-function-problems | # floor function problems
There are two parts of my problem.
1. Given $n$ and $x$, $\lfloor \frac nx \rfloor = q$, what is the maximum possible value for $x$ such that we obtain the same floor value? i.e. $\lfloor \frac nx \rfloor = \lfloor \frac{n}{x'}\rfloor = q$, where $x' \geq x$. I think $x' = \lfloor \frac nq \rfloor$ is the answer.
2. How many possible values(distinct) for q are there for all $x$ where $1\leq x\leq n$?
My thinking was that, what is the maximum value of $q$ that we can obtain and also the minimum value of $q$. Then, the answer would be $max - min + 1$. But this is surely wrong because we can have same floor values for different values of x. For example, for any $x \geq \lfloor \frac n2\rfloor + 1$, $q = \lfloor \frac nx\rfloor = 1$. Thus the maximum number of distinct floor values is at most $1 + \lfloor \frac n2 \rfloor$. How do I reduce the upper bound since I have read that there are at most $2\sqrt n$ distinct vaues?
• $$\biggl\lfloor \frac{n}{x}\biggr\rfloor = q \iff q \leqslant \frac{n}{x} < q+1$$ – Daniel Fischer Jul 6 '15 at 11:13
• Are $n$ and $q$ positive? Since inequalities are involved in the definition of the floor function, signs matter. – Rory Daulton Jul 6 '15 at 11:14
• Yes $n, q\geq 1$ – uttamkhanal Jul 6 '15 at 11:15
By definition of $\lfloor\cdot\rfloor$, we have
$$\biggl\lfloor \frac{n}{x}\biggr\rfloor = q \iff q \leqslant \frac{n}{x} < q+1$$
for $q\in \mathbb{Z}$. Since we are here dealing with positive integers, the sense of the inequalities is retained when multiplying with $x$ and dividing by $q$ resp. $q+1$, so we have
$$\biggl\lfloor \frac{n}{x}\biggr\rfloor = q \iff \frac{n}{q+1} < x \leqslant \frac{n}{q}.$$
Using that also $x$ shall be an integer, we obtain
$$\biggl\lfloor \frac{n}{x}\biggr\rfloor = q \iff \biggl\lfloor \frac{n}{q+1}\biggr\rfloor < x \leqslant \biggl\lfloor\frac{n}{q}\biggr\rfloor.$$
Hence, for each $q$ there are $\bigl\lfloor \frac{n}{q}\bigr\rfloor - \bigl\lfloor \frac{n}{q+1}\bigr\rfloor$ integer values $1 \leqslant x \leqslant n$ so that $\bigl\lfloor \frac{n}{x}\bigr\rfloor = q$, and the maximal such $x$ is indeed $\bigl\lfloor \frac{n}{q}\bigr\rfloor$.
How do I reduce the upper bound since I have read that there are at most $2\sqrt{n}$ distinct values?
Note that for $x > \sqrt{n}$, we have $\frac{n}{x} < \sqrt{n}$ and hence $\bigl\lfloor \frac{n}{x}\bigr\rfloor \leqslant \lfloor \sqrt{n}\rfloor$. Thus we have at most $\lfloor \sqrt{n}\rfloor$ distinct values of $\bigl\lfloor \frac{n}{x}\bigr\rfloor$ for $1 \leqslant x \leqslant \lfloor \sqrt{n}\rfloor$, and the $\lfloor \sqrt{n}\rfloor$ values $1 \leqslant q \leqslant \lfloor \sqrt{n}\rfloor$ for $\lfloor \sqrt{n}\rfloor + 1 \leqslant x \leqslant n$, which gives at most $2\lfloor \sqrt{n}\rfloor$ distinct values total. In fact, the values of $\bigl\lfloor \frac{n}{x}\bigr\rfloor$ for $1 \leqslant x \leqslant \lfloor \sqrt{n}\rfloor$ are all distinct, and all values $1 \leqslant q \leqslant \lfloor \sqrt{n}\rfloor$ are attained, so there are exactly $2\lfloor \sqrt{n}\rfloor$ distinct values, unless the value $\lfloor \sqrt{n}\rfloor$ appears in both sets - when $\bigl\lfloor \frac{n}{\lfloor \sqrt{n}\rfloor}\bigr\rfloor = \lfloor \sqrt{n}\rfloor$, or equivalently $n < \lfloor \sqrt{n}\rfloor\cdot (\lfloor \sqrt{n}\rfloor + 1)$ - in which case there are exactly $2\lfloor \sqrt{n}\rfloor - 1$ distinct values. | 2021-01-24T21:12:49 | {
"domain": "stackexchange.com",
"url": "https://math.stackexchange.com/questions/1351214/floor-function-problems",
"openwebmath_score": 0.9522563219070435,
"openwebmath_perplexity": 84.77723650589019,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9805806506825456,
"lm_q2_score": 0.8499711775577736,
"lm_q1q2_score": 0.8334652903510111
} |
https://math.stackexchange.com/questions/2842687/show-the-existence-of-a-global-maximum-of-a-continuous-function-with-unbounded-d | # Show the existence of a global maximum of a continuous function with unbounded domain
I am given a function $f(t) \in \mathbb{R}$ which is continuous; bounded above by $M$ and below by $0$. $f$ is differentiable everywhere except at $f=0$. Also, $\lim_{t \to \infty} f = 0$ and $t \in [0, \infty)$.
How do I show that a global maximum/supremum exists and where it exists?
I thought of dividing this into 3 cases: when function is non-decreasing, non-increasing and when there exists a local extremum.
$(i)$ $f$ can not be non-decreasing since it converges to $0$, except when $f(0) = 0$ which will be a trivial case.
$(ii)$ If $f$ is non-increasing then, $f(0)$ is the global maxima as $f(0) \ge f(t) \forall \ t$.
$(iii)$ That leaves us with the case when there exists at least $1$ local extremum. Now, if the domain were bounded, I could have used Extreme value theorem and say that global maximum exists either at $t=0$ or at a local maximum, but the domain is unbounded in my case, so that can't be used. However I do have information about the limit, which could be helpful.
Intuitively, I still think the result will be same as that of Extreme value theorem, but I want to prove it using proper results. How should I modify my process?
• When you write $\lim_{t\to\infty}$, do you mean $\lim_{t\to\pm\infty}$? – José Carlos Santos Jul 6 '18 at 9:54
• I would claim that the supremum exists (be careful, but it works since you are bounded from above and you converge to $0$). Then take a sequence converging to this supremum and by continuity this makes sure that the supremum and the maximum coincide – Stan Tendijck Jul 6 '18 at 9:57
• @JoséCarlosSantos No just positive one. I think I should mention it, $t \in [0, \infty)$ – Manish Jul 6 '18 at 9:57
• @StanTendijck Can you please elaborate a little, especially the argument regarding supremum and maximum coinciding? Also, it should be same as local maximum, right? – Manish Jul 6 '18 at 9:59
• You can attain the supremum as a limit of a sequence, i.e., there exist $a_n$ such that $\lim f(a_n) = S$ (with $S$ the supremum). You can argue by convergence to $0$ that the supremum is 'attained' at a bounded number, i.e. $a_n$ does not converge to $\infty$. Hence, there exist an $a$ which occurs at the limit of $a_n$. Moreover, by continuity you can interchange the limit and $f$ such that $\lim f(a_n) = f(a)$. Is this better? – Stan Tendijck Jul 6 '18 at 10:11
If $f:[0,\infty)\to[0,\infty)$ is continuous and $\lim_{t\to\infty}f(t)=0$, then you can prove the existence of a global maximum. If $f\equiv 0$, the statement is trivial. Otherwise, there exists a point $t_0\in\mathbb R$ such that $f(t_0)>0$. Further, from the limit you can conclude the existence of $R>0$ such that $f(t)<f(t_0)$ for all $t>R$. Since $I:=[0,R]$ is compact and $f$ continuous there exists a global maximum of $f\mid_I$ which is a global maximum of $f$.
To find the extreme point, you have to do it in two steps.
1) Consider the extreme points of $f\mid_{(0,\infty)}$ using $f'$.
2) Compare the maxima on $(0,\infty)$ with the value at $0$.
You can have 3 cases:
a) If there are no maxima on $(0,\infty)$, then the global maximum has to be at $0$.
b) If there exist maxima on $(0,\infty)$ but their heights are below $f(0)$, then the global maximum is at $0$.
c) If there exist maxima on $(0,\infty)$ and the highest value is above $f(0)$, then the maxima with the highest value are the global maxima.
If $f$ is the null function, then it attains its maximum everywhere. Otherwise, tak $t_0>$ such that $f(t_0)>0$. Then there is a $R>0$ such that $t>R\implies f(t)<f(t_0)$. By the extreme value theorem, the restriction of $f$ to $[0,R]$ has a maximum in some $t_1$ in that interval and since outside the interva $f$ always takes values smaller that $f(t_0)$, $f$ attains its global maximum at $t_1$. Therefore, either $f'(t_1)=0$ or the maximum is attained at $0$.
• I can understand the logic properly now. I think both the answers convey the same explanation. It's sad that both can't be tagged accepted :( – Manish Jul 6 '18 at 10:17
• I accept the one which has more appeal to you. It's as simple as that. – José Carlos Santos Jul 6 '18 at 10:21 | 2021-03-06T14:25:32 | {
"domain": "stackexchange.com",
"url": "https://math.stackexchange.com/questions/2842687/show-the-existence-of-a-global-maximum-of-a-continuous-function-with-unbounded-d",
"openwebmath_score": 0.9419565796852112,
"openwebmath_perplexity": 158.76607945139412,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9805806512500485,
"lm_q2_score": 0.849971175657575,
"lm_q1q2_score": 0.8334652889700742
} |
https://zyxyvy.wordpress.com/2014/03/01/round-numbers/ | # Round Numbers
When someone says “a nice round number, like 1000,” what are they really saying? What about a number makes it round?
The most reasonable explanation is probably that it is has very few non-zero digits, thus giving it properties like being easy to remember, easy to put in bundles, and easy to divide up. But if one thinks a bit deeper about these qualities of round numbers, one can find that typically-considered round numbers are really not the optimally round ones to find.
1000, for example, is only round because we use base ten, which is only because we happen to have ten fingers. If would also be round if we used base five or twenty, not so round if we used base two, and not round at all if we used base nine (not that nine is that reasonable of a base to use).
We can satisfy properties of being easy to put in bundles and being easy to divide up much better, and be more correct about few non-zero digits by a theoretical viewpoint, by looking for numbers that are the original definition of round…in a particularly large number of bases: numbers with many factors. How would we elegantly define this type of having a lot of factors?
Here, I will propose a new definition for a round number: it is a number $n$ with the property that for all pairs of primes $p$ and $q$, with $e_p$ and $e_q$ being the respective powers of $p$ and $q$ in the prime factorization of $n$, if $q>p$, then $e_q \leq e_p$, or in other words, $e_2 \geq e_3 \geq e_5 \geq e_7 \geq e_{11} \geq ...$.
Here are some properties of round numbers as defined, with all variables in the statements below being positive integers:
-1 is the only odd round number.
-If $n$ is round, then $n^m$ is round.
-If $\sqrt[m]{n}$ is an integer, then if $n$ is round, then $\sqrt[m]{n}$ is round.
$10^n$ is not round.
-All round numbers are abundant, except for 6 and the powers of 2.
We can also define barely round numbers as numbers that just barely qualify for being round: the boundary cases where all prime factors are 2 or exactly one factor of each prime factor involved exists, namely, the powers of two and the primordials. Note that if a round number is not barely round, then it is necessarily abundant.
Here is a list of round numbers less than 65537, by the new definition:
1
2
4
6
8
12
16
24
30
32
36
48
60
64
72
96
120
128
144
180
192
210
216
240
256
288
360
384
420
432
480
512
576
720
768
840
864
900
960
1024
1080
1152
1260
1296
1440
1536
1680
1728
1800
1920
2048
2160
2304
2310
2520
2592
2880
3072
3360
3456
3600
3840
4096
4320
4608
4620
5040
5184
5400
5760
6144
6300
6480
6720
6912
7200
7560
7680
7776
8192
8640
9216
9240
10080
10368
10800
11520
12288
12600
12960
13440
13824
13860
14400
15120
15360
15552
16384
17280
18432
18480
20160
20736
21600
23040
24576
25200
25920
26880
27000
27648
27720
28800
30030
30240
30720
31104
32400
32768
34560
36864
36960
37800
38880
40320
41472
43200
44100
45360
46080
46656
49152
50400
51840
53760
54000
55296
55440
57600
60060
60480
61440
62208
64800
65536
This post has 430 words, or, when rounded, 432.
## 5 thoughts on “Round Numbers”
1. This is pretty cool! I’d just like to add that when you apply Euler’s Totient Function on any of the given round numbers you get another round number…I haven’t been able to prove that’s always true, but it seems somewhat feasible.
1. No, this is false. phi(2*3*5*7*11*13*17*19*23) is not “round”, as it is divisible by 11 but not 7.
1. Ah, nicely constructed! Using one prime number being one more than a multiple of another. These cases do seem very rare, though. Would you like to give them a name?
2. I googled “round numbers” trying to get the joke in xkcd #1000, but didn’t find you until I searched for “list of round numbers.” I find your definition more satisfying than the rather vague one cited on Wikipedia (which doesn’t appear to be very authoritative).
Just curious: how did you decide how many to include in your list? Why not stop at the 144th?
1. I decided to stop at a factorion. | 2018-10-16T20:53:42 | {
"domain": "wordpress.com",
"url": "https://zyxyvy.wordpress.com/2014/03/01/round-numbers/",
"openwebmath_score": 0.6173429489135742,
"openwebmath_perplexity": 544.239666349155,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9805806484125338,
"lm_q2_score": 0.849971175657575,
"lm_q1q2_score": 0.8334652865582685
} |
http://mathhelpforum.com/calculus/93645-show-f-only-takes-two-values.html | # Math Help - Show that f only takes two values
1. ## Show that f only takes two values
Hi
Question:
Show that $f(x) = arctan(\frac{1}{x}) + arctan(x) \, , x\neq 0$ only takes two values. Show this, and decide these values.
Is this a correct/complete solution?
My solution:
$f'(x) = -\frac{1}{1+x^{2}} + \frac{1}{1+x^{2}} = 0$ , so $f$ is a constant function on $\mathbb{R} - [0]$
Then take any $x<0$ , say $x = -1$ .
$f(-1) = -\frac{\pi}{2}$ . This is the only value f takes on the interval $x<0$ since f is constant on this interval.
In the same way, we show that for $x>0$ , $f = \frac{\pi}{2}$ .
Thus the values are $-\frac{\pi}{2} \mbox{ and } \frac{\pi}{2}$
2. Originally Posted by Twig
Hi
Question:
Show that $f(x) = arctan(\frac{1}{x}) + arctan(x) \, , x\neq 0$ only takes two values. Show this, and decide these values.
Is this a correct/complete solution?
My solution:
$f'(x) = -\frac{1}{1+x^{2}} + \frac{1}{1+x^{2}} = 0$ , so $f$ is a constant function on $\mathbb{R} - [0]$
Then take any $x<0$ , say $x = -1$ .
$f(-1) = -\frac{\pi}{2}$ . This is the only value f takes on the interval $x<0$ since f is constant on this interval.
In the same way, we show that for $x>0$ , $f = \frac{\pi}{2}$ .
Thus the values are $-\frac{\pi}{2} \mbox{ and } \frac{\pi}{2}$
The range of $\arctan x$ is $\left(-\frac{\pi}{2}, \frac{\pi}{2} \right)$. I think this is correct. But if $f$ is constant on $\mathbb{R} \ \backslash \{0 \}$, wouldn't $f(x)= c$ for all $x$ in that domain, for some number $c$?
3. Originally Posted by Twig
Hi
Question:
Show that $f(x) = arctan(\frac{1}{x}) + arctan(x) \, , x\neq 0$ only takes two values. Show this, and decide these values.
Is this a correct/complete solution?
My solution:
$f'(x) = -\frac{1}{1+x^{2}} + \frac{1}{1+x^{2}} = 0$ , so $f$ is a constant function on $\mathbb{R} - [0]$
Then take any $x<0$ , say $x = -1$ .
$f(-1) = -\frac{\pi}{2}$ . This is the only value f takes on the interval $x<0$ since f is constant on this interval.
In the same way, we show that for $x>0$ , $f = \frac{\pi}{2}$ .
Thus the values are $-\frac{\pi}{2} \mbox{ and } \frac{\pi}{2}$
You can verify your result directly:
$\arctan(x_1)+\arctan(x_2)=\arctan\left(\dfrac{x_1+ x_2}{1-x_1 \cdot x_2}\right)$
Now plug in the given terms:
$f(x)=\arctan(x)+\arctan\left(\frac1x\right)=\arcta n\left(\dfrac{x+\frac1x}{1-x \cdot \frac1x}\right)$
Since ${\dfrac{x+\frac1x}{1-x \cdot \frac1x}}{ = +\infty} ~ \implies~ f(x) = +\dfrac{\pi}2$
If x < 0 the numerator of the fraction becomes negativ too and $f(x) = -\dfrac{\pi}2$
4. Originally Posted by Sampras
The range of $\arctan x$ is $\left(-\frac{\pi}{2}, \frac{\pi}{2} \right)$. I think this is correct. But if $f$ is constant on $\mathbb{R} \ \backslash \{0 \}$, wouldn't $f(x)= c$ for all $x$ in that domain, for some number $c$?
But no one said that f was constant on $\mathbb{R} \ \backslash \{0 \}$. What is true is that the derivative of f is 0 on that set. That tells us that f is constant on any interval on which it is differentiable. Thus, f is constant on $(-\infty, 0)$ and on $(0, \infty)$ but not necessarily the same constant on those two intervals.
5. Here is another solution using calculus. (Since this topic is started in the Calculus section, I presume it is okay to use calculus. )
$f(x)\,=\,\arctan\left(\frac1x\right)+\arctan x$
$\implies\ f'(x)\,=\,\frac{-\frac1{x^2}}{1+\left(\frac1x\right)^2}+\frac1{1+x^ 2}$
$=\,\frac{-1}{1+x^2}+\frac1{1+x^2}$
$=\,0$
Hence $f$ is constant over any interval on which it is continuous. Since it is continuous over $(0,\,\infty)$ and over $(-\infty,\,0),$ determine the constant value of $f$ in each of these two invervals. For example, for $x>0,$ we have $f(1)=\arctan1+\arctan1=\frac\pi2$ and so $f(x)=\frac\pi2$ for all $x>0.$ Do the same thing for the negative interval.
6. Originally Posted by TheAbstractionist
Here is another solution using calculus. (Since this topic is started in the Calculus section, I presume it is okay to use calculus. )
$f(x)\,=\,\arctan\left(\frac1x\right)+\arctan x$
$\implies\ f'(x)\,=\,\frac{-\frac1{x^2}}{1+\left(\frac1x\right)^2}+\frac1{1+x^ 2}$
$=\,\frac{-1}{1+x^2}+\frac1{1+x^2}$
$=\,0$
Hence $f$ is constant over any interval on which it is continuous. Since it is continuous over $(0,\,\infty)$ and over $(-\infty,\,0),$ determine the constant value of $f$ in each of these two invervals. For example, for $x>0,$ we have $f(1)=\arctan1+\arctan1=\frac\pi2$ and so $f(x)=\frac\pi2$ for all $x>0.$ Do the same thing for the negative interval.
Maybe I also should have stated that $f$ is continous on $\mathbb{R}-[0]$ | 2015-08-30T20:50:14 | {
"domain": "mathhelpforum.com",
"url": "http://mathhelpforum.com/calculus/93645-show-f-only-takes-two-values.html",
"openwebmath_score": 0.9348742961883545,
"openwebmath_perplexity": 164.38895259026782,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9805806512500485,
"lm_q2_score": 0.8499711699569786,
"lm_q1q2_score": 0.8334652833801798
} |
https://math.stackexchange.com/questions/2793380/seat-n-people-in-4-benches | # Seat n people in 4 benches
How many ways are there to seat n people in 4 benches so that no bench is left empty with order?
Hints from the teacher
• Each bench should have at least 1 person
• This question is similar to distributing different object among n children
• The answer is one line
I am not sure how to deal with this question, but this was what I tried.
Let $A_i$ be the set of all distributions in which $i-$th bench is empty.
Then $|A_i| = (n-1)^4$ for each $i$.
$|A_i\cap A_j|= (n-2)^4$ for each $i\ne j$
$|A_i\cap A_j\cap A_k|= (n-3)^4$ for each $i\ne j\ne k\ne i$
$|A_i\cap A_j\cap A_k \cap A_l|= (n-4)^4 = 1$ for all pairvise different $i,j,k,l,m$
Now we are interested in $$\Big| \bigcap_{i=1}^n \overline A_i\Big| = \Big|\; \overline{\bigcup_{i=1}^n A_i}\;\Big| = n^4-\Big|\; \bigcup_{i=1}^n A_i\;\Big|$$We have by PIE: \begin{eqnarray*} \Big|\; \bigcup_{i=1}^n A_i\;\Big| &=& \sum_{i=1}^n \Big|\; A_i\ \Big| - \sum_{i=1}^n \Big|\;A_i\cap A_j\ \Big|+...\\ &=& 4\cdot (n-1)^4-2n\cdot (n-2)^4+2n\cdot (n-3)^4-4 ..... +0=\\ \end{eqnarray*}
So the final answer is $n^4-$(the above value).
Am I correct, or is there something that I am doing wrong?
Edit: Also maybe I want to apply pigeonhole principle since n items are to be put into m containers, with n > m, then at least one container must contain more than one item..
• What does "empty with order" mean? – NicNic8 May 23 '18 at 19:04
• Are the benches distinguishable? Are the people? Please edit to include a clear example. If $n=4$, say, what is the answer? – lulu May 23 '18 at 19:04
• How did you obtain the factor $2n$ for the coefficient of $(n - 2)^4$? What does it represent? – N. F. Taussig May 23 '18 at 19:13
• @NicNic8 no bench is left empty so it means that each bench should have at least 1 person with an order is the seating arrangements. – nano mano May 23 '18 at 19:13
• @lulu how will people be identical.. Then it is 1 person, ofc they are different as well as the benches. Also, isn't n unlimited number of people since it didn't specify? – nano mano May 23 '18 at 19:15
We have $n!$ permutations of students. Now lets take a permutation $i_1, i_2, .., i_n$ we want to distribute it in 4 slots. That means we have n-1 positions at which we can cut the set and we need to cut it at three of them, so the answer is
$n! * {n-1 \choose 3}$
P.s. no partition will be empty because the first cut is after the first element, the last one is before the last, and there is one element between any two consecutive cuts
• I assume $i=n$? Also, don't forget that you need somebody on each bench. – lulu May 23 '18 at 19:41
• You're right sorry, I'll try to figure it out – narek Bojikian May 23 '18 at 19:47
• You are close, here, but not correct. If you take $n=5$ and $k=4$ (as in the OP's question) you'll see that your text gives the correct answer but your formula does not. – lulu May 23 '18 at 19:48
• @lulu so it would be $n+k-1 \choose k$ multiplied by $n!$ ?? – nano mano May 23 '18 at 19:54
• The edited version is good. (+1) – lulu May 23 '18 at 20:28
A lot depends on exactly what is meant by "with order." If we take it to mean that both the order of the benches and the order in which people sit on each bench matter, then the answer is simple:
$$n!{n-1\choose3}$$
That is, line the people up from left to right, in any of $n!$ ways, then pick $3$ "break points" to decide which groups go to benches $1$, $2$, $3$, and $4$.
Let k = number of benches = 4
It is like we are distributing people to benches. If there were no restrictions, then it would have been $${n+4-1\choose 4-1}$$
Since we have a rule that each bench should have at least one person, then $$x1+....+xn = 4$$
For any pair of positive integers n and k, the number of distinct k-tuples of positive integers whose sum is n is given by the binomial coefficient $xi∈Z,xi>0$ $${(n-k)+k-1\choose k-1} = {(n-4)+4-1\choose 4-1}$$
$$= {n-1\choose 4-1} == {n-1\choose 3}$$
The order matters, so we will need to multiply this by n! which is the arrangements that people could be seated.
Final answer: $$n! \times {n-1\choose 3}$$
Now we are done solving this question by using the stars and bars method.
• Completely sound. (+1) – lulu Jun 2 '18 at 23:34 | 2020-01-29T05:14:26 | {
"domain": "stackexchange.com",
"url": "https://math.stackexchange.com/questions/2793380/seat-n-people-in-4-benches",
"openwebmath_score": 0.8319746851921082,
"openwebmath_perplexity": 567.8856057091022,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9805806506825456,
"lm_q2_score": 0.8499711699569786,
"lm_q1q2_score": 0.8334652828978187
} |
https://www.physicsforums.com/threads/kinematics-problem.260708/ | # Kinematics problem
1. Sep 30, 2008
### mjolnir80
1. The problem statement, all variables and given/known data
a rocket is launched straight up with a constant acceleration. four seconds after takeoff , a bolt falls off the side of the rocket. the bolt hits the ground 6.0 seconds later. what was the rockets acceleration?
2. Relevant equations
any kinematics equations
3. The attempt at a solution
we know that the initial velocity of the bolt equals the final velocity of the rocket also
thats as far as i can get
thanks in advanced for the help
Last edited: Oct 1, 2008
2. Sep 30, 2008
### Monocles
Once the bolt leaves the rocket you will have the bolt's velocity and acceleration due to gravity, and how long it will take to hit the ground. You can use this information to find how high the bolt was when it left the rocket. Once you have that information, you will be able to find the acceleration the rocket must have had to get to that height in four seconds.
3. Sep 30, 2008
### mjolnir80
actually you dont have the bolts velocity
4. Sep 30, 2008
### Galileo's Ghost
Keep in mind the delta y for the rocket in the first 4 seconds will have the same magnitude but opposite direction as the delta y for the bolt in the final 6 seconds. So (delta y)rocket = -(delta y)bolt. Also the vf for the rocket (and consequently the vi of the bolt) can be solved in terms of a using the kinematic vf = vi + at ...that value for vf of the rocket can be used in the expression for (delta y)rocket on the left and can be used as vi of the bolt in an expression for (delta y) on the right.
5. Sep 30, 2008
### Display MOB
Isn't acceleration always -9.8m/s^2 (If your positive direction is up)?
Also what does
"a bolt falls off the side of the rocket. the bolt gits the ground 6.0 seconds later."
have to do with the rockets acceleration?
6. Sep 30, 2008
### Galileo's Ghost
Acceleration is 9.80 m/s^2 downward for objects that are in free fall. The rocket is not in free fall. It is burning fuel to generate a thrust accelerating the rocket upwards. The bolt IS in free fall over the last 6 seconds. During the first 4 seconds, however, the bolt was experiencing the exact same acceleration as the rocket.
7. Sep 30, 2008
### Display MOB
"The rocket is not in free fall. It is burning fuel to generate a thrust accelerating the rocket upwards."
Thanks, I'm new to this.
8. Oct 1, 2008
### mjolnir80
do we assume that Vi of the rocket is zero?
9. Oct 1, 2008
### mridkash
velocity of the bolt going up with rocket after t sec, $$v_{up} = at_{up}$$ (up positive)
initial velocity is zero here as the rocket starts from rest.
Now this velocity becomes the initial velocity of the bolt when it starts to fall down.
When the bolt is falling, it is acted upon by -g acceleration, so the distance traveled by the falling bolt is given by $$s_{down} = (at_{up})t_{down} - \frac{1}{2}gt_{down}^{2}$$ where $$t_{down}$$ is the time taken by the bolt to hit ground.
But we know already that, distance the bolt travels up and down are same only the times differ $$s_{down} = s_{up} = \frac{1}{2}at_{up}^2$$
Putting the value of distance in former equation we get,
$$\frac{1}{2}at_{up}^2 = (at_{up})t_{down} - \frac{1}{2}gt_{down}^{2}$$
the times are given and g you know. | 2017-05-23T22:41:53 | {
"domain": "physicsforums.com",
"url": "https://www.physicsforums.com/threads/kinematics-problem.260708/",
"openwebmath_score": 0.7768217325210571,
"openwebmath_perplexity": 1021.137705806136,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9805806489800368,
"lm_q2_score": 0.8499711699569787,
"lm_q1q2_score": 0.8334652814507354
} |
https://math.stackexchange.com/questions/1831733/what-is-the-minimum-and-maximum-of-a-set-with-only-one-element/1831736 | # What is the minimum and maximum of a set with only one element?
This is surely a trivial question but I want to be sure I understand correctly what happens.
Given a set $A = \{1\}$, what is $\min A$ and $\max A$?
Is it $\min A = 1$ and $\max A = 1$?
• Definitions are your friends. How do you define $\min A$ and $\max A$? You need to have an order relation $\le$ specified for $A$ for this to make sense. It seems likely you are thinking of $A$ as a subset of the real numbers. – hardmath Jun 19 '16 at 9:01
• $\min(A)$ and $\max(A)$ are values belonging to the set. So... – Yves Daoust Jun 19 '16 at 9:30
Yes, by definition $\min(A)$ gives you the smallest element in the set. Since it is $1$ in this case, you will get $1$. Likewise for $\max(A)$. Of course, $\min(A)$ and $\max(A)$ will not generally be equal but they are equal in this special case.
In a more general scenario a set $B$ with only one element need not satisfy $\min(B)=\max(B)$ since I could, for example, make $B$ a subset of some field without an order relation. In such a set, on which no order relation is defined, you usually also don't know what a definition of $\min(B)$ and $\max(B)$ would be. It might still be possible by utilizing, for example, a norm. I can make a set with one vector as its element and while vectors have no order relation, I could define the maximum and minimum with relation to a vector's norm. In that case the maximum and minimum would coincide again.
• No, if $A=\{a\}$ is a one-element subset of an ordered set, then it’s always true that $\min A=\max A=a$. – Brian M. Scott Jun 19 '16 at 9:10
• Brian is right, there are counter examples of some non-ordered sets but in subsets of ordered sets like $\mathbb{R}$ or $\mathbb{Z}$ with only one element, you cannot construct a counterexample. – MM8 Jun 19 '16 at 9:16
• I suppose you could just say that $a$ isn't an element of an ordered set, but then you can define a trivial order on ${a}$ and you're fine. – Neil Jun 19 '16 at 9:16 | 2019-08-23T06:48:02 | {
"domain": "stackexchange.com",
"url": "https://math.stackexchange.com/questions/1831733/what-is-the-minimum-and-maximum-of-a-set-with-only-one-element/1831736",
"openwebmath_score": 0.8862005472183228,
"openwebmath_perplexity": 116.63257309849537,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9674102542943773,
"lm_q2_score": 0.8615382147637196,
"lm_q1q2_score": 0.8334609034288939
} |
https://math.stackexchange.com/questions/3605200/why-is-taylor-expansion-on-function-of-discrete-random-variable-allowed | Why is taylor expansion on function of discrete random variable “allowed”?
This thought came to me yesterday when I was doing a simple first moment approximation of a function of a discrete random variable, using the standard Taylor expansion. For a discrete random variable, strictly speaking we should be working with finite difference, so why would using the Taylor expansion which assumes a continuous variable work? Are there convergence results showing that this method works just as well for discrete vs. continuous r.v.? or is there actually a difference in how well the approximation works depending on whether the r.v. is discrete vs. continuous?
As requested, here is a specific example to illustrate the question.
Consider discrete random variable $$X$$ with known mean $$\mu$$ and known variance $$\sigma^2$$, and we'd like to approximate $$\mathbb{E}[g(X)] = \mathbb{E}[\exp(-X)]$$.
A standard approach is to use Taylor expansion for the function $$g$$ evaluated at $$\mu$$ to approximate $$\exp(-X)$$,
$$\mathbb{E}[g(X)] = \mathbb{E}[\exp(-X)] \approx g\left(\mu\right)+\frac{g^{\prime \prime}\left(\mu\right)}{2} \sigma^{2} = \exp(-\mu) + \frac{\exp(-\mu)}{2}\sigma^2$$
I can see how this is perfectly reasonable if $$X$$ is a continuous r.v., say defined on $$[0, \infty)$$. But it seems there is a technical issue with taking derivative if $$X$$ is discrete, say $$X$$ is defined only on non-negative integers, i.e. $$X \in \{0, 1, 2, ...\}$$, as the usual derivative can only be applied on continuous domain.
• This is just a guess: I suspect that it's justified because of the function you were using, not the random variable itself -- but it's hard to tell since this post is light on details. Any chance you could update the question with a bit more specificity about the approximation you did? – Aaron Montgomery Apr 1 '20 at 17:51
• Sure, I'll add in details. Sorry I kept the question general on purpose, as I don't see this distinction being brought up in any text I've studied from, regardless of the specific function or random variable, namely the distinction of discrete vs continuous in taylor expansion approximations – barnacleboy Apr 1 '20 at 18:17
• No worries! I think the reason you haven't seen that distinction is likely because most reasonable functions you'd write down tend to be analytic on their domains (e.g. $e^x, \sqrt x$), and it's that, not anything about the random variable, that's the crux. But again, I'm kind of guessing here; hopefully I will be able to comment further on a longer post. – Aaron Montgomery Apr 1 '20 at 18:23
The appropriateness of the use of the Taylor series in this case (and in every case where it should be used) hinges on the function, not on the random variable.
To clarify the issue, consider the following statement:
$$e^{0.5} \approx 1 + \frac{0.5}{1!} + \frac{0.5^2}{2!} + \frac{0.5^3}{3!}$$
This statement holds because $$e^x$$ is a nice function, not because there's anything special about the number $$0.5$$. Note that no other numbers need to be in consideration at all for the statement to make sense.
Random variables work in the same way. For a given $$\omega \in \Omega$$, the quantity $$X(\omega)$$ is a fixed number, and the Taylor approximation holds just like it did for the example of $$0.5$$ above. You can make this claim simultaneously across many values of $$\omega$$ to get an approximate equality involving the random variable, and then you can apply the linearity of expectation to get the desired result.
• Thank you, ok it makes sense that we are just replacing whatever function we have with an approximate version, so it should only depend on the property of the function. But just intuitively, it feels like for continuous variable, this moment approximation should somehow be "better" than for discrete variable, where each possible value is further apart. Am I making any sense? For Taylor approximation, we expect more accurate result if $\text{Var}(X)$ is small, more density is centered near the average, but for discrete variable, the density is not as bunched up. I hope I'm making some sense. – barnacleboy Apr 1 '20 at 18:50
• I get your question, but: focus again on the example of approximating $e^{0.5}$ to get the right perspective here. The reason it's useful to have $var(X)$ be small is that we prefer the variable to not be far from $0$ (or more typically, its mean), because the standard Taylor approximation breaks down for inputs for from $0$. It again comes down to the essential properties of $e^x$ and the quality of the Taylor approximation, not to anything particular about the variable itself (other than that we want it to live where the Taylor approximations are good). – Aaron Montgomery Apr 1 '20 at 18:53
• Got it, I think my question has been answered, so I'll accept your answer. I guess it just comes down to the moments of random variable in determining how well the approximation is, not so much the discrete vs continuous nature – barnacleboy Apr 1 '20 at 19:01
• Exactly right. ${}$ – Aaron Montgomery Apr 1 '20 at 19:02 | 2021-02-28T10:47:33 | {
"domain": "stackexchange.com",
"url": "https://math.stackexchange.com/questions/3605200/why-is-taylor-expansion-on-function-of-discrete-random-variable-allowed",
"openwebmath_score": 0.8484218120574951,
"openwebmath_perplexity": 177.38808651515603,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9674102561735719,
"lm_q2_score": 0.8615382129861583,
"lm_q1q2_score": 0.8334609033282607
} |
https://www.physicsforums.com/threads/ti-89-integral.220919/ | # Calculators TI-89 integral
1. Mar 9, 2008
### ehrenfest
[SOLVED] TI-89 integral
1. The problem statement, all variables and given/known data
My TI-89 says that
$$\int \frac{1}{\cos x} dx = \ln \left(\frac{-\cos(x)}{\sin(x)-1}\right)$$
which is just wrong isn't it!?!?!
See http://en.wikipedia.org/wiki/Lists_of_integrals
2. Relevant equations
3. The attempt at a solution
Last edited: Mar 9, 2008
2. Mar 9, 2008
### feuerwasser
don't trust wiki
mine is giving me the same answer, but in absolute values... maybe your in a different format? Also, wikipedia can be edited by anyone so you probably shouldn't trust it for most things
3. Mar 9, 2008
### ehrenfest
OK. Here is a better source https://www.math.lsu.edu/~adkins/m2065/IntegralTable.pdf. Even with absolute values the result is still clearly wrong as you can see from number 8 at the link. Wikipedia had the same result therefore I trust Wikipedia more than Texas Instruments :)
EDIT: maybe it is not wrong; but then how do you manipulate the TI-89 answer to get the integral table answer?
4. Mar 9, 2008
### rock.freak667
It is correct as the expression in the brackets is equivalent to tanx+secx. But I don't have a TI-89 so I can't really help you. But why do you want it to show it in the way the integral table has it?
5. Mar 9, 2008
### feuerwasser
well... i know that cos(x)/sin(x)= cot(x) and cot(x)-1= (cos(x)-sin(x))/sin(x)
so.. i don't know they probably go together somehow
6. Mar 10, 2008
### Gib Z
Indefinite integrals of a function are unique only up to a constant.
7. Mar 10, 2008
### HallsofIvy
Staff Emeritus
Well, I, not having a TI89 calculator at hand, would have to do the integral by hand (Poor me!).
$$\int \frac{1}{cos x}dx= \int \frac{cos x}{cos^2 x} dx= \int \frac{cos x}{1- sin^2x}dx$$
Letting u= sin(x), du= dx, the integral becomes
$$\int \frac{du}{1- u^2}= \frac{1}{2}\int \frac{du}{1-u}+ \frac{1}{2}\int \frac{du}{1+u}$$
by partial fractions. Integrating each of those, I get
$$\frac{1}{2}ln(1-u)}+ \frac{1}{2}ln(1+u)}= ln\left(\sqrt{\frac{1+u}{1- u}}\right)+ C$$
$$= ln\left(\sqrt{\frac{1+ sin(x)}{1- sin(x)}}\right)+ C$$
Multiplying both numerator and denominator of the fraction by 1- sin(x),
$$= ln\left(\sqrt\frac{1-sin^2(x)}{1+sin(x)^2}}\right)+ C= ln\left(\sqrt{\frac{cos^2(x)}{(1- sin2(x))^2}}\right)+ C[/itex] [tex]= ln\left(\frac{cos(x)}{1- sin(x)}\right)+ C$$
Yes, it looks like the TI89 is right!
As far as the Wikipedia result is concerned, I suspect a trig identity would give the same thing but I will let someone else show that.
8. Mar 10, 2008
### Gib Z
$$\tan x + \sec x = \frac{\sin x}{\cos x} + \frac{1}{\cos x} = \frac{1+ \sin x}{\cos x}$$
$$\frac{\cos x}{1-\sin x} = \frac{\cos x (1+ \sin x)}{1-\sin^2 x} = \frac{\cos x(1+ \sin x)}{\cos^2 x} = \frac{1+\sin x}{\cos x}$$. | 2017-06-27T20:58:02 | {
"domain": "physicsforums.com",
"url": "https://www.physicsforums.com/threads/ti-89-integral.220919/",
"openwebmath_score": 0.8449708223342896,
"openwebmath_perplexity": 2354.1738890007946,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9674102580527664,
"lm_q2_score": 0.861538211208597,
"lm_q1q2_score": 0.8334609032276276
} |
https://www.physicsforums.com/threads/error-calculation.828908/ | # Error calculation
1. Aug 22, 2015
### desquee
(Sorry if this is the wrong place for this question, I wasn't sure where to put it)
I've been running an experiment with rabbits, and am trying to figure out the error of my feed measurement.
I fed them using a 1/3 cup measure, and recorded the number of scoops they got. I estimated that the error for each individual scoop is 10%. If I then want to determine the error of the total amount of feed (850 scoops), would that also be 10%?
Intuitively that seems too high, since it's incredible unlikely that every single scoop was high by 10% (or low by 10%). In fact, I can say for sure that some scoops were high (over 1/3 cup) and some were low (under 1/3 cup), so that its impossible for the total feed to be off by 10%.
Is there a way of more accurately calculating the error?
2. Aug 22, 2015
### mathman
A good approximation to the error (assuming independent errors) is $\frac{10}{\sqrt{850}}$ %.
3. Aug 22, 2015
### desquee
Thanks mathman. That comes to 0.34%. I want to better understand how that equation is derived, do you by any chance have a link to an explanation?
4. Aug 23, 2015
### Hornbein
5. Aug 23, 2015
### desquee
OK, so reading that, it looks like the equation given by mathman is for the std of the mean of the sample. But what I'm trying to figure out is the std of the sum of the sample, and I don't see that the link explains how to do that.
6. Aug 23, 2015
### Hornbein
The std of the sum of the sample is equal to
the std of the mean of the sample times the sample size.
sigma_sum = n * sigma_samplemean
Since you are using percentages, you divide n * sigma_samplemean by n to get the percentage.
n/n = 1.
Voila! sigma_sum% = sigma_samplemean%
It is worth mentioning that your actual error is in all likelyhood greater than 0.34%. I'd bet that you have systematic error that is greater than this unsystematic error.
Last edited: Aug 23, 2015
7. Aug 23, 2015
### mathman
The variance of the sum of independent random variables is the sum of the variances.
Proof:
Let $(X_k, k=1,n)$ be a set of n independent random variables, each mean $=m_k$. Let $Y_k=X_k-m_k$. The variance of the sum is $E((\sum_{k=1}^{n}Y_k)^2)=\sum_{k=1}^{n}E(Y_k^2)+\sum_{k=1}^{n}\sum_{j\ne k}E(Y_jY_k)$. However, due to independence, the $E(Y_kY_j)=0 \ for \ j\ne k$. Therefore the variance of the sum is the sum of the variances.
Last edited: Aug 23, 2015
8. Aug 23, 2015
### desquee
Hornbein: What sort of systemic errors should I be looking for? The feeding method was: fill scoop from feed bucket, pour feed from scoop into feeding container in cage, count the number of scoops I added. And the number I'm interested in is the total amount of feed I gave to the rabbits (so feed that gets knocked out of the cage by the rabbits still counts).
I'm just now learning how to work with errors, so any advice on what to be looking for would be helpful.
9. Aug 23, 2015
### Hornbein
Oh, you likely tend to add a little over or a little under a third cup. I bet that that tendency is greater than 1%.
10. Aug 23, 2015
### desquee
My original assumption is that the error for each scoop was 10%, from which the 0.34% for 850 scoops was calculated. Was that 1% a typo and you meant to say that you bet the tendency is greater than 10%?
11. Aug 23, 2015
### desquee
You were right Hornbein. I just tried measured the greatest likely error of a scoop. I measured the weight of exactly 1/3 cup of feed (or as exact as I could make it given that the feed was solid). Then I took a scoop as overflowing as it could be without feed falling out, removed the excess, and weighed that. Then I took a scoop as under-full as it could be without being so empty that it would have induced me to notice and refill it, and measured the weight of the feed I added to bring it up to 1/3 cup. The result was an error of 35% (say 40% given the slight error in my attempt to make the scoop to exactly full).
Even with a 40% error for each scoop the error for the sum of 850 scoops is only around 18.5 scoops, or 2.18%.
12. Aug 24, 2015
### Hornbein
Not a typo. I'm saying that I bet you systematic error makes this sampling error insignificant. 1 > .34. I doubt that you measure so badly that your systematic error is greater than 10%. | 2018-01-19T18:06:14 | {
"domain": "physicsforums.com",
"url": "https://www.physicsforums.com/threads/error-calculation.828908/",
"openwebmath_score": 0.6659967303276062,
"openwebmath_perplexity": 1103.4337594197602,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES\n\n",
"lm_q1_score": 0.9674102561735719,
"lm_q2_score": 0.861538211208597,
"lm_q1q2_score": 0.8334609016086297
} |
https://www.physicsforums.com/threads/gravitational-question.879688/ | Gravitational question
1. Jul 22, 2016
Clara Chung
1. The problem statement, all variables and given/known data
The problem is attached
2. Relevant equations
Energy, gravitational formulas
3. The attempt at a solution
For part b, the answer is 2km s^-1
If it can escape, KE-PE(energy travelled from infinity to R)=0
KE = PE
mv^2 /2 = GMm/R
v^2/2=GM/R..(1)
I tried to find GM by using the information provided
mv^2 /R = GMm /R^2
v^2/2 = GM/2R....(2)
which combining (1) and (2) I get v^2/2 = v^2 which is impossible.
Please tell me whats wrong and the correct method to solve the question
Attached Files:
• grav.png
File size:
176.1 KB
Views:
74
2. Jul 22, 2016
ehild
It is not true if the spacecraft has kinetic energy at infinity .
How do you intend to determine the mass of the planet? You have to define what R and v are. It seems you try to use the formula valid for a circular orbit, but this spacecraft is leaving the planet instead of orbiting around it.
3. Jul 22, 2016
Clara Chung
then what formula should I use?
4. Jul 22, 2016
cnh1995
I'm not sure this will work but here's what I think..
You can first plot the graph of v vs r on a graph paper. Draw a smooth curve. At (r=1, v=6.635), draw a tangent to that curve whose slope will be dv/dr.
Now,
deceleration=dv/dt=(dv/dr)*(dr/dt)=v*dv/dr..
You can know v, r and dv/dr at every point on the graph.
You can equate the deceleration obtained from graph to GM/r2 and get M.
Last edited: Jul 22, 2016
5. Jul 22, 2016
Biker
Hello,
I have searched a little bit about the topic of escape velocity and I figured the solution
I will give you an example
If you have a road that is 8 meters tall and you know that a section from the end to a point in that road is 5 meters tall. How far is the point from the starting point of the road? You just have to do subtraction.
Okay so If G M m /r gives you potential from a point to infinity and you take another (r value) above the first one you took and substract the two what do you get?
Now use energy conservation using velocities and find out GM.
Once you find that out we can continue.
6. Jul 22, 2016
Staff: Mentor
Look at energy conservation. The total specific mechanical energy (that is, the energy per unit mass of the object in orbit) is the sum of the specific kinetic and potential energy at any instant. For a body in a free-fall trajectory of any kind (such as this spacecraft is) this value is a constant.
In particular: $\xi = \frac{v^2}{2} - \frac{GM}{r}$ is a constant over the whole trajectory.
You should be able to use this along with the given data to solve for both $\xi$ and GM.
7. Jul 22, 2016
Clara Chung
Thanks, I can find the answer of part a by using both methods.
The answer of part c is 6 x 10^23 kg, but when I use U^2/2 - GM/R =V^2/2, I found out 3 x 10^23 kg
8. Jul 22, 2016
Staff: Mentor
Your answer (3.0 x 10^23 kg) looks good to me. Perhaps there's an error in the given answer key. | 2017-11-25T04:49:25 | {
"domain": "physicsforums.com",
"url": "https://www.physicsforums.com/threads/gravitational-question.879688/",
"openwebmath_score": 0.6781909465789795,
"openwebmath_perplexity": 874.8198926926887,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9674102571131693,
"lm_q2_score": 0.8615382094310355,
"lm_q1q2_score": 0.8334609006984975
} |
http://stats.stackexchange.com/questions/90490/can-a-statistical-test-return-a-p-value-of-zero | # Can a statistical test return a p-value of zero?
I don't mean a value close to zero (rounded to zero by some statistical software) but rather a value of literally zero. If so, would it mean that the probability the null hypothesis is true is also zero? What are (some examples) of statistical tests that can return results of this sort?
-
You might find the examples shown in the closely related question at stats.stackexchange.com/questions/90325/… to be helpful. – whuber Mar 18 at 20:15
## 1 Answer
It will be the case that if you observed a sample that's impossible under the null (and if the statistic is able to detect that), you can get a p-value of exactly zero.
That can happen in real world problems. For example, if you do an Anderson-Darling test of goodness of fit of data to a standard uniform with some data outside that range - e.g. where your sample is (0.430, 0.712, 0.885, 1.08) - the p-value is actually zero (but a Kolmogorov-Smirnov test by contrast would give a p-value that isn't zero, even though we can rule it out by inspection).
Likelihood ratio tests will likewise give a p-value of zero if the sample is not possible under the null.
As whuber mentioned in comments, hypothesis tests don't evaluate the probability of the null hypothesis (or the alternative).
We don't (can't, really) talk about the probability of the null being true in that framework (we can do it explicitly in a Bayesian framework, though -- but then we cast the decision problem somewhat differently from the outset).
-
In the standard hypothesis testing framework there is no meaning to "the probability of the null hypothesis." We know that you know that but it looks like the OP doesn't. – whuber Mar 18 at 20:47
Perhaps explicating this a bit: The standard uniform includes only values from 0 to 1. Thus, a value of 1.08 is impossible. But this is really rather odd; is there a situation where we would think that a continuous variable is distributed uniformly, but not know its maximum? And if we knew its maximum was 1, then 1.08 would just be a sign of a data entry error. – Peter Flom Mar 18 at 21:05
@whuber Does it work if I rephrase to "If so, would it mean that null hypothesis is definitely false"? – user1205901 Mar 18 at 21:09
I think all that you really need is a caution that these tests are not evaluating the probability of the null hypothesis. This has been discussed in other threads so it doesn't need to be taken up in detail here. Incidentally, the null can be possible even when the p-value is a true zero. – whuber Mar 18 at 22:51
@whuber Okay, thanks, I can certainly do that, and I'll get rid of my rambling comments as well. I'm not thinking clearly this morning ... in respect of your last sentence, can you give me a hint about what sort of circumstances that comes up in? – Glen_b Mar 18 at 23:00 | 2014-10-31T15:03:19 | {
"domain": "stackexchange.com",
"url": "http://stats.stackexchange.com/questions/90490/can-a-statistical-test-return-a-p-value-of-zero",
"openwebmath_score": 0.8109346032142639,
"openwebmath_perplexity": 460.98419441137895,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9674102514755853,
"lm_q2_score": 0.8615382129861583,
"lm_q1q2_score": 0.8334608992807658
} |
http://openstudy.com/updates/559eaaf0e4b0f93dd7c2f1a8 | ## anonymous one year ago Given the geometric sequence where a1 = -3 and the common ratio is 9, what is the domain for n?
1. anonymous
OPTIONS All integers All integers where n ≥ -1 All integers where n ≥ 1 All integers where n ≥ 0
2. anonymous
@ganeshie8
3. anonymous
@pooja195
4. anonymous
@perl
5. pooja195
@SolomonZelman ;-; ?
6. SolomonZelman
Well, if you are starting from n=1, then your terms $$a_n$$ can be: $$a_1$$, $$a_2$$, $$a_3$$, $$a_4$$,$$a_5$$, and so forth.... To answers this question, you don't really need to know about the common ratio, or even what type of sequence it is. Nor do you have to know what the $$a_1$$ is, as long as you know that is starts from $$a_1$$ (i.e. from n=1).
7. SolomonZelman
All these: $$\large a_\color{red}{1}$$, $$\large a_\color{red}{2}$$, $$\large a_\color{red}{3}$$, $$\large a_\color{red}{4}$$, $$\large a_\color{red}{5}$$, are values of n that are ?
8. anonymous
Is it all integers?
9. SolomonZelman
Can you have $$\large a_{-1}$$ ??
10. anonymous
they are positive
11. SolomonZelman
yes, all positive integers, correct...
12. anonymous
So its C?
13. SolomonZelman
14. anonymous
Thanks!
15. SolomonZelman
Also I can teach you how to put up ≥ ≤ ∞ and other symbols without copy pasting, want to know how?
16. SolomonZelman
it works on almost every site...
17. anonymous
Can you help me with this questions: Given the arithmetic sequence an = -3 + 9(n - 1), what is the domain for n? OPTIONS: All integers where n ≥ 1 All integers All integers where n ≥ 0 All integers where n > 1
18. anonymous
Is an arithmetic sequence always positive integers?
19. SolomonZelman
$$\large\color{blue}{ \displaystyle {\rm CODES,~~short~guide} }$$ 1) Click and hold ALT 2) click the number code (using the numbers that are on the right of the keyboard, and NOT the ones below F1, F2, F3, etc., ) 3) release the ALT number code result 0 2 1 5 × 2 4 6 ÷ 7 • ──────────────── among with other symbols. code result 2 5 1 √ 7 5 4 ≥ 7 5 5 ≤ 2 4 1 or 7 5 3 ± 2 4 7 ≈ 0 1 8 5 ¹ 2 5 3 ² 0 1 7 9 ³ 1 6 6 ª 2 5 2 ⁿ 1 6 7 º 2 4 8 ° 0 1 5 3 ™ 0 1 9 0 ¾ 4 2 8 ¼ 1 7 1 ½ 2 2 7 π 1 5 5 ¢ 2 3 6 ∞ 1 5 9 ƒ
20. SolomonZelman
Now as far as your question....
21. SolomonZelman
quoting your question: Can you help me with this questions: Given the arithmetic sequence an = -3 + 9(n - 1), what is the domain for n? OPTIONS: All integers where n ≥ 1 All integers All integers where n ≥ 0 All integers where n > 1 (end quote) Now, you are given that $$\large\color{black}{ \displaystyle a_n=-3+9(n-1) }$$
22. SolomonZelman
it doesn't really tell you if it starts from $$a_0$$ (i.e. from n=0), OR from $$a_1$$ (i.e. from n=1). So, the only thing you can say for sure that it can't be option B ($$a_{-n}$$ doesn't exist).
23. SolomonZelman
basically, there is a lack of information here, without which I can't say anything.... I would assume though that they want you to say option C intuitively
24. anonymous
Thanks !
25. anonymous
Can you help me with the next question I have to do: What is the 9th term of the geometric sequence 4, -20, 100, ?
26. SolomonZelman
because they are proposing a thought that: "You can have first term, second term, third term, etc... something that is real (or tangible), BUT not zeroth term and certainly not negative term."
27. anonymous
These are the options: -312,500 -12,500 62,500 1,562,500
28. SolomonZelman
ok, so it starts from $$a_1=4$$ Can you find the geometric ratio for me? (if not say idk, and I will guide you through this step)
29. anonymous
idk
30. SolomonZelman
Ok, a geometric ratio (r) in a sequence can be found using the following formula. $$\large\color{black}{ \displaystyle {\rm r}=\frac{a_{n}}{a_{n-1}}}$$ where $$\large\color{black}{ \displaystyle a_{n}}$$ is any term in a sequence, and $$\color{black}{\large a_{n-1}}$$ is the term right before this $$\large\color{black}{ \displaystyle a_n}$$. and "r" here, is of course the common ratio. ------------------ Btw, to make sure. Common ratrio is a number by which you multiply every/each time to obtain the nex term.
31. SolomonZelman
For example $$\large\color{black}{ \displaystyle {\rm r}=\frac{a_{2}}{a_1}}$$ or $$\large\color{black}{ \displaystyle {\rm r}=\frac{a_{3}}{a_2}}$$ or $$\large\color{black}{ \displaystyle {\rm r}=\frac{a_{4}}{a_3}}$$ and on.... see?
32. anonymous
Yes
33. SolomonZelman
now, how would you use the formula: $$\large\color{black}{ \displaystyle {\rm r}=\frac{a_{n-1}}{a_n}}$$ ? Which terms would you choose? (to answer my question consider the given information - which terms do you know already?)
34. anonymous
the common ratio is -.2 right?
35. SolomonZelman
oh, my fault
36. SolomonZelman
$$\large\color{black}{ \displaystyle {\rm r}=\frac{a_{n}}{a_{n-1}}}$$ this si the formula
37. SolomonZelman
I made a typo in my previous reply, but the rest of information is right
38. anonymous
Is r=-.2 right?
39. SolomonZelman
so, for example $$\large\color{black}{ \displaystyle {\rm r}=\frac{a_{2}}{a_1}~~~~~\Rightarrow {\small \rm (in~this~case)~}~~~~ {\rm r}=\frac{-20}{4}=?~~~{\small \rm (you~tell~me)}}$$
40. SolomonZelman
lost?
41. anonymous
oh I get it
42. SolomonZelman
yes, so r=?
43. anonymous
r=-5
44. SolomonZelman
yes, r=-5. Side Note: Sometimes I will be typing stuff while you are typing, and if that is the case don't be afraid to interrupt... keep typing:)
45. SolomonZelman
Ok, have you ever seen a formula $$\large\color{blue}{ \displaystyle a_n=a_1 \cdot {\rm r}^{n-1}}$$ ?
46. anonymous
no
47. SolomonZelman
ok, w will show what it i and how it works (if you don't mind)
48. SolomonZelman
in GEOMETRIC sequence: in order to obtain $$a_2$$ you have to multiply $$a_1$$ times the common ratio r. That is: $$a_1 \times {\rm r} = a_2$$ correct? | 2017-01-23T21:21:48 | {
"domain": "openstudy.com",
"url": "http://openstudy.com/updates/559eaaf0e4b0f93dd7c2f1a8",
"openwebmath_score": 0.7286546230316162,
"openwebmath_perplexity": 1314.487386831774,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.96741025335478,
"lm_q2_score": 0.8615382094310357,
"lm_q1q2_score": 0.8334608974605018
} |
http://math.stackexchange.com/questions/322013/intuition-on-fundamental-theorem-of-arithmetic | # Intuition on fundamental theorem of arithmetic
I'm sorry ahead if time if this is overly trivial for this site.
Currently in school, much of what I enjoy is number theory - based. Currently, I lean pretty heavily on the FTA for a good deal of my understanding and proofs. That being said, recently I've realized my intuition behind the FTA is somewhat lacking. I can prove the theorem many ways, but still have trouble "seeing" or "visualizing" the uniqueness of the FTA. This is largely because I use the FTA to understand the ideas behind the proofs used in the uniqueness part of the proof of the FTA itself. Ie, I have a bad case of circular reasoning. For example, a typical way to show uniqueness is based on the fact that if $p$ is prime, $p \mid ab \implies p \mid a$ or $p \mid b$, but I use the FTA to understand this idea.
What I'm looking for, is some intuition behind the uniqueness of the FTA so that I may be able to understand it at a very intuitive level. I'm just appealing to the wiser, who may understand this through and through and may be able to offer some valuable insight on how they view the idea.
Thank you in advance for all that took the time to read this and/or offer insight
-typed using ipad, sorry for typos
-
What do you mean by "the uniqueness of the FTA"? – Pedro Tamaroff Mar 6 '13 at 1:34
Sorry, I should have been more clear in typing. By this I mean the uniqueness of the prime factorization of a natural number(up to ordering) – gone Mar 6 '13 at 1:38
There are many properties that are equivalent to uniqueness of factorization in $\,\Bbb Z.\:$ Below is a sample off the top of my head (by no means complete). Each provides a slightly different perspective on why uniqueness holds - perspectives that becomes clearer when one sees how these equivalent properties bifurcate in more general integral domains. Below we use the notation $\rm\:(a,b)=1\:$ to mean that $\rm\:a,b\:$ are coprime, i.e. $\rm\:c\mid a,b\:\Rightarrow\:c\mid 1.$
$\rm(1)\ \ \ gcd(a,b)\:$ exists for all $\rm\:a,b\ne 0\ \$ [GCD domain]
$\rm(2)\ \ \ a\mid BC\:\Rightarrow a=bc,\ b\mid B,\ c\mid C\ \ \,$ [Schreier refinement, Euler's four number theorem]
$\rm(3)\ \ \ a\,\Bbb Z + b\, \Bbb Z\, =\, c\,\Bbb Z,\:$ for some $\rm\,c\quad\$ [Bezout domain]
$\rm(4)\ \ \ (a,b)=1,\ a\mid bc\:\Rightarrow\: a\mid c\qquad\ \$ [Euclid's Lemma]
$\rm(5)\ \ \ (a,b)=1,\ \dfrac{a}{b} = \dfrac{c}{d}\:\Rightarrow\: b\mid d\quad\ \$ [Unique Fractionization]
$\rm(6)\ \ \ (a,b)=1,\ a,b\mid c\:\Rightarrow\: ab\mid c$
$\rm(7)\ \ \ (a,b)=1\:\Rightarrow\: a\,\Bbb Z\cap b\,\Bbb Z\, =\, ab\,\Bbb Z$
$\rm(8)\ \ \ gcd(a,b)\ \ exists\:\Rightarrow\: lcm(a,b)\ \ exists$
$\rm(9)\ \ \ (a,b)=1=(a,c)\:\Rightarrow\: (a,bc)= 1$
$\rm(10)\$ atoms $\rm\, p\,$ are prime: $\rm\ p\mid ab\:\Rightarrow\: p\mid a\ \ or\ \ p\mid b$
Which of these properties sheds the most intuitive light on why uniqueness of factorization entails? If I had to choose one, I would choose $(2),$ Schreier refinement. If you extend this by induction it implies that any two factorizations of an integer have a common refinement. For example if we have two factorizations $\rm\: a_1 a_2 = n = b_1 b_2 b_3\:$ then Schreier refinement implies that we can build the following refinement matrix, where the column labels are the product of the elements in the column, and the row labels are the products of the elements in the row
$$\begin{array}{c|ccc} &\rm b_1 &\rm b_2 &\rm b_3 \\ \hline \rm a_1 &\rm c_{1 1} &\rm c_{1 2} &\rm c_{1 3}\\ \rm a_2 &\rm c_{2 1} &\rm c_{2 2} &\rm c_{2 3}\\ \end{array}$$
This implies the following common refinement of the two factorizations
$$\rm a_1 a_2 = (c_{1 1} c_{1 2} c_{1 3}) (c_{2 1} c_{2 2} c_{2 3}) = (c_{1,1} c_{2 1}) (c_{1,2} c_{2 2}) (c_{1,3} c_{2 3}) = b_1 b_2 b_3$$
This immediately yields the uniqueness of factorizations into primes (atoms). It also works more generally for factorizations into coprime elements, and for factorizations of certain types of algebraic structures (abelian groups, etc).
-
wonderful answer, as usual! – Tyler Mar 6 '13 at 2:43
Dear Math Gems, Is there a typo in (2) (more precisely, should $A$ and $a$ be the same)? Regards, – Matt E Mar 6 '13 at 2:48
@Zachary I don't recall any textbooks that use $(2)$. You might have luck searching on the German name Vierzahlensatz. Nonunits $\rm\,A\,$ satisfying $(2)$ are called primal. One easily checks that atoms are primal $\Leftrightarrow$ prime. Products of primes are also primal. So "primal" may be viewed as a generalization of the notion "prime" $(10)$ from atoms to composites. – Math Gems Mar 6 '13 at 3:36
Googling Schreier refinement turned up a gnarly statement from group theory. It looked like you might maybe make this theorem a statement about refinements of subgroups of $\mathbb Z$, but maybe you mean Schreier domain? – Thomas Andrews Mar 6 '13 at 14:28
(I also didn't find any google result for "Euler's four number theorem.") – Thomas Andrews Mar 6 '13 at 15:52
Believe it or not, the fundamental theorem is mostly a result of the following result:
If $a,b$ are a pair of relatively prime integers, then $ax+by=1$ has for some integers, $x,y$.
In turn, this follows because the integers are something called a "principal ideal domain."
An "ideal" can be defined for a general ring, but in the case of the integers, we can define an ideal as a non-empty subset of $I\subseteq \mathbb Z$ which is closed under addition and taking additive inverses. In other words, it is a subgroup of $(\mathbb Z,+)$.
Now, we can, in general, take the set of multiples of some $d$, written $d\mathbb Z$. These are ideals, called the principal ideals.
The fact that these are the only ideals is why the previous theorem is true, because $a\mathbb Z + b\mathbb Z$ can be shown to be an ideal, so it must be principal, that is, $a\mathbb Z + b\mathbb Z=d\mathbb Z$ for some $d$. Since $a,b$ are both in the left side, $d|a$ and $d|b$. But $a,b$ have no common factors other than $\pm 1$, so $d=\pm 1$.
The fact that $\mathbb Z$ is a principal ideal domain is due to the existence of a division algorithm in $\mathbb Z$.
So, division algorithm implies principal ideal domain implies $ax+by=1$ solution implies unique factorization.
This chain of reason works in other places, such as the ring of polynomials over a field or $\mathbb Z[i]$ (the Gaussian integers.)
In other cases, division algorithm does not apply, but the ring is still a principal ideal domain.
In yet others, we don't even have a PID, but we still have unique factorization, such as the ring of polynomials with integer coefficients.
Arnold Ross professed the "fundamental theorem" should really be:
If $a|bc$ and $a,b$ are relative prime, then $a|c$
This is a direct corollary of the $ax+by=1$ theorem, and it is the heart of the "standard" fundamental theorem.
There are domains that are not unique factorization domains. For example, the set $\mathbb Z[\sqrt{-5}]=\{a+b\sqrt{-5}:a,b\in\mathbb Z\}$ is a domain in which unique factorization fails.
-
Thank you for your response. I'm currently pondering much of it – gone Mar 6 '13 at 1:53
Any two integers have a greatest common divisor (gcd); $\gcd(a,b)$ can be written in the form $a x + b y$ where $x$ and $y$ are integers (by the Euclidean algorithm). This idea is behind a lot of elementary number theory.
Suppose $p$ divides $ab$ (so $ab = r p$ for some integer $r$), but $p$ does not divide $a$. Then $\gcd(a,p) = 1$ so $1 = x p + y a$ for some integers $x$ and $y$. Multiplying by $b$, $b = x b p + y a b = x b p + y r p = (xb + y r) p$, so $p$ divides $b$.
-
Yes I've seen this before, but for some reason have generally attempted other ways of understanding. Is this how you perceive the theorem? – gone Mar 6 '13 at 1:55 | 2014-10-21T02:35:51 | {
"domain": "stackexchange.com",
"url": "http://math.stackexchange.com/questions/322013/intuition-on-fundamental-theorem-of-arithmetic",
"openwebmath_score": 0.8860232830047607,
"openwebmath_perplexity": 276.518186960265,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.96741025335478,
"lm_q2_score": 0.8615382058759129,
"lm_q1q2_score": 0.8334608940212396
} |
https://www.physicsforums.com/threads/probability-2-dice-question.737980/ | # Probability 2 Dice Question
## Homework Statement
A pair of dice is rolled once, what is the probability that neither a doublet nor the sum of 10 will appear
## Homework Equations
P(A) = 1 - P(A')
Demorgans law
(AUB)c = Ac ∩ B c
## The Attempt at a Solution
I know how to do the solution through brute force and listing out all the possible scenarios:
(1,1), (1,2), (1,3) ...... (6,6) and doing it like that. But i wanted a more algebraic approach so I tried this:
Let A = is a doublet
Let B = sum of dices is 10
so then what I am looking for is
P(Ac U Bc)
and applying demorgans
P(Ac U Bc) = P(A∩B )c
using the law that Ac = 1 - A
I got the final easy equation
1 - P(A ∩ B )
solving for this
P(A) = 6/36 ...........(1,1),(2,2),(3,3),(4,4),(5,5),(6,6)
P(B) = 3/36 ............(4,6), (5,5),(6,4)
plug in numbers
1 - (6/36)*(3/36) != the answer of 7/9
Apparently, math lies, j.k. Can someone let me know what i am doing wrong?
Ray Vickson
Homework Helper
Dearly Missed
## Homework Statement
A pair of dice is rolled once, what is the probability that neither a doublet nor the sum of 10 will appear
## Homework Equations
P(A) = 1 - P(A')
Demorgans law
(AUB)c = Ac ∩ B c
## The Attempt at a Solution
I know how to do the solution through brute force and listing out all the possible scenarios:
(1,1), (1,2), (1,3) ...... (6,6) and doing it like that. But i wanted a more algebraic approach so I tried this:
Let A = is a doublet
Let B = sum of dices is 10
so then what I am looking for is
P(Ac U Bc)
and applying demorgans
P(Ac U Bc) = P(A∩B )c
using the law that Ac = 1 - A
I got the final easy equation
1 - P(A ∩ B )
solving for this
P(A) = 6/36 ...........(1,1),(2,2),(3,3),(4,4),(5,5),(6,6)
P(B) = 3/36 ............(4,6), (5,5),(6,4)
plug in numbers
1 - (6/36)*(3/36) != the answer of 7/9
Apparently, math lies, j.k. Can someone let me know what i am doing wrong?
Use inclusion-exclusion properly:
$$\text{P}(A \cup B) = \text{P}(A) + \text{P}(B) - \text{P}(A \cap B) = 6/36 + 3/36 - 1/36.$$
Thats a clean method.
Can I ask you why deMorgans doesn't work in this scenario?
haruspex
Homework Helper
Gold Member
2020 Award
Let A = is a doublet
Let B = sum of dices is 10
so then what I am looking for is
P(Ac U Bc)
Not so.
neither a doublet nor the sum of 10
Ray Vickson
Homework Helper
Dearly Missed
Thats a clean method.
Can I ask you why deMorgans doesn't work in this scenario?
It does work, and it is essentially what I used:
$$\text{desired answer} = \text{P}(A^c \cap B^c) = 1 - \text{P}(A \cup B)$$
If you evaluate ##\text{P}(A \cup B)## correctly you will get the right answer.
OHHHHH...my initial set up was wrong
this: P(Ac∩Bc)
and not this: P(AcUBc)
the phrase "neither a nor b"
confused me. I thought nor implied OR not AND.
haruspex | 2021-06-24T03:25:07 | {
"domain": "physicsforums.com",
"url": "https://www.physicsforums.com/threads/probability-2-dice-question.737980/",
"openwebmath_score": 0.540126621723175,
"openwebmath_perplexity": 2604.2492480351016,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES\n\n",
"lm_q1_score": 0.9674102514755852,
"lm_q2_score": 0.8615382058759129,
"lm_q1q2_score": 0.8334608924022414
} |
https://www.physicsforums.com/threads/ph-buffer-calculations.800105/ | # PH buffer calculations
## Homework Statement
Prepare a buffer solution of pH 3.746 from 25 mL of 0.244 M weak acid with pka 3.54
What volume, in mL, of 0.275 M NaOH would need to be added?
I think that I have the answer but I am not sure, could you look over the working to check?
Also, would you do this differently? If so, what would you do?
## Homework Equations
Henderson-Hasselbach:
$pH = pka + log_{10} \frac {[A^-]}{[HA]} \Rightarrow [H^+] = ka + \frac{[A^-]}{[HA]} \Rightarrow ka = \frac{[H^+][A^-]}{[HA]}$
## The Attempt at a Solution
Part I
$pH = 3.746 \Rightarrow [H^+] = 10^{-3.746} = 1.79473\times10^{-4}$
$pka = 3.54 \Rightarrow [H^+] = 10^{-3.54} = 2.88403\times10^{-4}$
$ka = \frac{[H^+][A^-]}{[HA]} \Rightarrow 2.88403\times10^{-4} = 1.79473\times10^{-4}\times\frac{[A^-]}{[HA]}$
$\therefore \frac{[A^-]}{[HA]} = 1.60694$
Pat II
25 mL of 0.244 M acid $\Rightarrow$ 0.0061 M (6.1 mM)
Code:
+---+--------------+--------------+
| | Acid | Base |
| I | 0.0061 M | 0 M |
| C | -x M | +x M |
| E | (0.0061-x)M | x M |
+---+--------------+--------------+
$\frac{[A^-]}{[HA]} = \frac{x}{(0.0061 - x)} \therefore \frac{x}{(0.0061 - x)} = 1.60694$
Solve for $x$...
$x = 0.00376009$ (is this the moles of acid reacting?)
$\frac{0.00376009}{0.275} = 0.01367305$
$\therefore$ 13.7 mL NaOH should be added.
Related Biology and Chemistry Homework Help News on Phys.org
Borek
Mentor
$pH = pka + log_{10} \frac {[A^-]}{[HA]} \Rightarrow [H^+] = ka + \frac{[A^-]}{[HA]} \Rightarrow ka = \frac{[H^+][A^-]}{[HA]}$
Correct, but completely unnecessary. You could just plug pH and pKa into HH equation, that's what it is for.
$pH = 3.746 \Rightarrow [H^+] = 10^{-3.746} = 1.79473\times10^{-4}$
Correct, but completely unnecessary.
$pka = 3.54 \Rightarrow [H^+] = 10^{-3.54} = 2.88403\times10^{-4}$
Wrong, and completely unnecessary. But I assume you meant OK, you just copied wrong.
$\frac{[A^-]}{[HA]} = 1.60694$
Yep, that's what is important here. As I said above, you could get here much faster.
$\therefore$ 13.7 mL NaOH should be added.
Looks OK to me.
itex tags are for inline formulae, when you post them in their own lines, use just tex.
Quantum Defect
Homework Helper
Gold Member
## Homework Statement
Prepare a buffer solution of pH 3.746 from 25 mL of 0.244 M weak acid with pka 3.54
What volume, in mL, of 0.275 M NaOH would need to be added?
I think that I have the answer but I am not sure, could you look over the working to check?
Also, would you do this differently? If so, what would you do?
[snip]
$\therefore$ 13.7 mL NaOH should be added.
The nice thing about the henderson-Hasselbach equation is that the log part is a ratio of concentrations, but it can just as easily be written as a ratio of moles.
pH = pKa + log [A-]/[HA] = pKa + log ( moles A- / moles HA )
At the half eq. point, moles A- = moles HA, pH = pKa + log (1) ==> pH = pKa
For your buffer, the pH is close to pKa, so you are close to the 1/2 equiv. point. Since pH is a bit larger than pKa, you are a bit past the 1/2 equiv. point, i.e. there is a bit more conj. base than conj. acid. The concentrations of acid and base are pretty close, so the half-eq. point is going to be close to half the original volume of acid -- ca. 12.5 mL.
3.746 = 3.54 + log (moles A-/moles HA)
moles A- / moles HA = 1.607
moles HA_0 = 25 mL * 0.244 M = 6.10 m mol
x/(6.1 mmol-x) = 1.607 ==> x = 3.76 m mol
vol OH- * 0.275 M = 3.76 m mol
vol OH- = 13.7 mL
If I had the question "Prepare a buffer solution of pH 3.924 from 50 mL of 0.236 M solution of a sodium salt of a weak acid, Na+A- where the pka of the weak acid, HA, is 3.956. What volume, in mL, of 0.208 M HCl would need to be added?"
Would I use the same method as with the initial question? What changes, if any, should I make?
Borek
Mentor
Very similar question and very similar approach, you just need to keep track of what is the acid and what is the conjugate base here.
Okay, so I've just had a go and I get:
[A-]/[HA] = 0.92896
Base: 50mL * 0.236M = 11.8mM
(11.8-x)/x = 0.92896 => x = 6.11727
6.11727/0.208 = 24.4mL
Borek
Mentor
And what is the acid here?
And what is the acid here?
The acid is 0.208 M HCl. I calculated that I should add 24.4mL.
Borek
Mentor
The acid is 0.208 M HCl.
Nope. This is the acid you added, but it is not part of the buffer. Every buffer consist of the conjugate pair - what are the acid and its conjugate base here?
Acid: HA and C. Base: A- ?
Borek
Mentor
Acid: HA and C. Base: A- ?
Hard to say what you mean. A- typically means "any conjugate base", so it is not specific enough. And there is no such thing as C.
Buffer has ONE acid, not two, so whatever you meant, if you listed two acids you can't be right.
Quantum Defect
Homework Helper
Gold Member
Hard to say what you mean. A- typically means "any conjugate base", so it is not specific enough. And there is no such thing as C.
Buffer has ONE acid, not two, so whatever you meant, if you listed two acids you can't be right.
I think that the OP meant "C. Base: A-" = "Conjugate Base: A-" I.e. "C. Base" was an abbreviation for "conjugate base."
Borek
Mentor
Ah, OK. It still doesn't help much as HA and A- are hardly specific (and I suspect OP may be confused about the identity of the acid; that's quite common).
"Prepare a buffer solution of pH 3.924 from 50 mL of 0.236 M solution of a sodium salt of a weak acid, Na+A- where the pka of the weak acid, HA, is 3.956. What volume, in mL, of 0.208 M HCl would need to be added?"
Sorry if I caused confusion, that is the question verbatim.
What I don't fully understand is how the "sodium salt of a weak acid" affects the calculations that I need to do. Because, based on inspection, comparing concentrations as well as pH and pka, I think that the volume to add should be roughly half (25mL) that of the initial solution.
Depending on how I do the sums I arrive at either 24.4mL or 27.3mL and I'm not sure which answer to use.
Borek
Mentor
OK, so you were right about HA and A-.
Initially solution contains A- only, where does the HA come from?
That's what I'm not too certain about. Is it the water that the initial solution is diluted with? | 2020-07-04T13:13:03 | {
"domain": "physicsforums.com",
"url": "https://www.physicsforums.com/threads/ph-buffer-calculations.800105/",
"openwebmath_score": 0.6768063902854919,
"openwebmath_perplexity": 2731.9295734528546,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES\n\n",
"lm_q1_score": 0.9674102514755853,
"lm_q2_score": 0.8615382040983515,
"lm_q1q2_score": 0.8334608906826104
} |
https://www.jiskha.com/questions/1829590/how-would-you-solve-the-trigonometric-identity-cosx-sinx-cosx-sinx-sec2x-tan2x | How would you solve the trigonometric identity (cosx-sinx)/(cosx+sinx)=sec2x-tan2x?
1. 👍 0
2. 👎 0
3. 👁 59
1. sec 2x = 1/cos2x = 1/(cos^2x - sin^2x)
tan 2x = sin 2x / cos 2x = sin 2x / (cos^2 x - sin^2 x)
so
sec 2x -tan 2x = (1 - sin 2x) / (cos^2 x - sin^2 x)
= (1 - 2 sin x cos x) / (cos^2 x - sin^2 x)
= (1 - 2 sin x cos x)/ [(cos x - sin x)(cos x + sin x) ]
=====================================
now the left side
(cos x - sin x)(cos x - sin x) / [ cos x + sin x)(cos x - sin x)]
(cos^2 x - 2 sin x cos x + sin^2 x) / [ cos x + sin x)(cos x - sin x)]
(1 - 2 sin x cos x) / [ cos x + sin x)(cos x - sin x)]
remarkable, same as the right side
1. 👍 0
2. 👎 0
👨🏫
Damon
## Similar Questions
1. ### Trigonometry
4. Find the exact value for sin(x+y) if sinx=-4/5 and cos y = 15/17. Angles x and y are in the fourth quadrant. 5. Find the exact value for cos 165degrees using the half-angle identity. 1. Solve: 2 cos^2x - 3 cosx + 1 = 0 for 0
2. ### Math
How do I solve this? tan^2x= 2tanxsinx My work so far: tan^2x - 2tanxsinx=0 tanx(tanx - 2sinx)=0 Then the solutions are: TanX=0 and sinX/cosX = 2 sin X Divide through by sinX: we have to check this later to see if allowed (ie sinX
3. ### Trig.......
I need to prove that the following is true. Thanks (2tanx /1-tan^x)+(1/2cos^2x-1)= (cosx+sinx)/(cosx - sinx) and thanks ........... check your typing. I tried 30º, the two sides are not equal, they differ by 1 oh , thank you Mr
4. ### math;)
The equation 2sinx+sqrt(3)cotx=sinx is partially solved below. 2sinx+sqrt(3)cotx=sinx sinx(2sinx+sqrt(3)cotx)=sinx(sinx) 2sin^2x+sqrt(3)cosx=sin^2x sin^2x+sqrt(3)cosx=0 Which of the following steps could be included in the
1. ### trigonometry
Prove: tan2x+sec2x = cosx+sinx/ cosx-sinx
2. ### Trig
Verify the identity: tanx(cos2x) = sin2x - tanx Left Side = (sinx/cosx)(2cos^2 x -1) =sinx(2cos^2 x - 1)/cosx Right Side = 2sinx cosx - sinx/cosx =(2sinxcos^2 x - sinx)/cosx =sinx(2cos^2 x -1)/cosx = L.S. Q.E.D.
3. ### Math help again
cos(3π/4+x) + sin (3π/4 -x) = 0 = cos(3π/4)cosx + sin(3π/4)sinx + sin(3π/4)cosx - cos(3π/4)sinx = -1/sqrt2cosx + 1/sqrt2sinx + 1/sqrt2cosx - (-1/sqrt2sinx) I canceled out -1/sqrt2cosx and 1/sqrt2cosx Now I have 1/sqrt sinx +
4. ### Pre-Calc
Establish the identity. sinx + cosx/sinx - cosx = 1+2sinxcosx/2sin^2x-1
1. ### Trig Identities
Prove the following identities: 13. tan(x) + sec(x) = (cos(x)) / (1-sin(x)) *Sorry for any confusing parenthesis.* My work: I simplified the left side to a. ((sinx) / (cosx)) + (1 / cosx) , then b. (sinx + 1) / cosx = (cos(x)) /
2. ### Math
(sinx - cosx)(sinx + cosx) = 2sin^2x -1 I need some tips on trigonometric identities. Why shouldn't I just turn (sinx + cosx) into 1 and would it still have the same identity?
3. ### trigonometry
how do i simplify (secx - cosx) / sinx? i tried splitting the numerator up so that i had (secx / sinx) - (cosx / sinx) and then i changed sec x to 1/ cosx so that i had ((1/cosx)/ sinx) - (cos x / sinx) after that i get stuck
4. ### Math
verify the identity (1-cosx)/(sinx)+(sinx)/(1-cosx)=2cscx Thank you! | 2020-10-29T23:17:16 | {
"domain": "jiskha.com",
"url": "https://www.jiskha.com/questions/1829590/how-would-you-solve-the-trigonometric-identity-cosx-sinx-cosx-sinx-sec2x-tan2x",
"openwebmath_score": 0.8233174085617065,
"openwebmath_perplexity": 8630.556099485153,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9632305360354471,
"lm_q2_score": 0.865224084314688,
"lm_q1q2_score": 0.8334102585252158
} |
https://www.physicsforums.com/threads/finding-maximum-and-minimum-values-of-3-dimensional-function.840688/ | # Finding maximum and minimum values of 3 dimensional function
## Homework Statement
Find the maximum and minimum values of f(x,y,z) = x^2 - 2x + y^2 - 4y + z^2 - 4z in the region x^2 + y^2 + z^2 <= 36.
## Homework Equations
Lagrange multipliers
<Fx, Fy, Fz> = lambda<Gx, Gy, Gz>
Test for local extrema
If Determinant D(x,y) < 0 it is a saddle point
If D(x,y) > 0:
If Fxx > 0, then it is a local minimum
If Fxx < 0, then it's a local maximum
## The Attempt at a Solution
Tried to solve for system of equations that result in 4 equations[/B]
2x-2 = lambda * 2x
2y-4 = lambda * 2y
2z-4 = lambda * 2z
X^2 + y^2 + z^2 = 36
I solved for lambda = 1/2 or 3/2 by solving for x y and z and substituting into the 4th equation.
I got (2,4,4) =0 and (-2,-4,-4) =72. BUT the minimum was actually (1,2,2) = -9 because I was supposed to "check the interior of the sphere". Am I just supposed to account for the possibility that lambda = 0?
mfb
Mentor
For the minimum you don't need Lagrange multipliers. Setting them to zero seems to work as well.
Ray Vickson
Homework Helper
Dearly Missed
## Homework Statement
Find the maximum and minimum values of f(x,y,z) = x^2 - 2x + y^2 - 4y + z^2 - 4z in the region x^2 + y^2 + z^2 <= 36.
## Homework Equations
Lagrange multipliers
<Fx, Fy, Fz> = lambda<Gx, Gy, Gz>
Test for local extrema
If Determinant D(x,y) < 0 it is a saddle point
If D(x,y) > 0:
If Fxx > 0, then it is a local minimum
If Fxx < 0, then it's a local maximum
## The Attempt at a Solution
Tried to solve for system of equations that result in 4 equations[/B]
2x-2 = lambda * 2x
2y-4 = lambda * 2y
2z-4 = lambda * 2z
X^2 + y^2 + z^2 = 36
I solved for lambda = 1/2 or 3/2 by solving for x y and z and substituting into the 4th equation.
I got (2,4,4) =0 and (-2,-4,-4) =72. BUT the minimum was actually (1,2,2) = -9 because I was supposed to "check the interior of the sphere". Am I just supposed to account for the possibility that lambda = 0?
For either the max or the min, the solution will either be in the interior or on the boundary. The interior stationary point is a global min in ##\mathbb{R}^3## and obeys the constraint, so is the global min of the constrained problem. (The second-order test implies that the unconstrained stationary point is a min, but you do not need such a fancy test: just use the fact that f = sum of squares + linear to give the result automatically.)
For the way you wrote the Lagrangian equations, a positive lagrange multiplier is a necessary condition for a MAXIMUM, so both of your candidate boundary points satisfy the first-order necessary conditions for a constrained maximum. The second-order *sufficient* conditions for a max are much trickier than you may think, because you have a point on a boundary of an inequality constraint. Basically, you need to project the Hessian of the *Lagrangian* (not the function f!) down to the tangent space of the constraint, and determine if it is negative definite in that subspace.
However, in this case you can by-pass all that because you have both possible points, so can just take the one with the larger f-value.
Last edited:
HallsofIvy
Homework Helper
In the interior of the sphere, $\nabla f= (2x- 2)\vec{i}+ (2y- 4)\vec{j}+ (2z- 4)\vec{k}$. Set that equal to 0 and solve for x, y, and z to find any critical points in the interior of the sphere. Your Lagrange multiplier method will solve for max or min among points that satisfy the equation x^2+ y^2+ z^2= 36, on the surface of the sphere. Another way to do that is to use parametric equations for the sphere: $x= 6cos(\theta)sin(\phi)$, $y= 6 sin(\theta)sin(\phi)$, $z= 6 cos(\phi)$. Putting those into the function, $f(\theta, \phi)= 36- 12cos(\theta)sin(\phi)- 24sin(\theta)si(\phi)- 24 cos(\phi)$. Set the gradient of that two variable function to 0 to find any critical points on the surface of the sphere. Find the value of the function at each critical point to determine the absolute max and min.
Sorry for the late reply. How does using the fact that the sum of the squares + linear establish that (1,2,2) is the global min? We were told to use geometry to show what the min and max are because we do not use advanced calculus to prove these mins and maxes? How can we use geometry/equations to prove these are the min and max values?
Ray Vickson
Homework Helper
Dearly Missed
Sorry for the late reply. How does using the fact that the sum of the squares + linear establish that (1,2,2) is the global min? We were told to use geometry to show what the min and max are because we do not use advanced calculus to prove these mins and maxes? How can we use geometry/equations to prove these are the min and max values?
The surface ##w = x^2 - 2x + y^2 - 4y + z^2 - 4z## is a bit hard to visualize, because it is a 3-dimensional object lying in 4 dimensions.
To gain insight, drop ##z## and look at the 2-dimensional surface ##w = x^2 - 2x + y^2 - 4y## in 3-dimensional ##(x,y,w)##-space. This surface is cup-shaped, opening up as we go farther and farther out in the ##(x,y)## plane; you could fashion it out of sheet metal and it would very nicely hold rainwater. Its bottom is where all derivatives = 0, and there is only one such point. Now just imagine doing the same in a higher dimension.
Alternatively, you can re-write ##f(x,y,z)## by "completing the squares"; that will show the geometry very plainly.
BTW: I did not assume you would follow my remarks about Hessians projected down into tangent subspaces and the like; I just wanted to de-rail any thoughts you might have had about looking at the second derivatives in the way you did. In this case that works for the minimum (because the minimum is the same in the unconstrained problem and the constrained problem---that is, the constraint does not affect the minimum in this case), but those simple tests would lead you very far astray in the maximization case! They would also lead you astray in other cases of minimization where the constraint is active at the solution.
Last edited: | 2021-07-25T23:16:43 | {
"domain": "physicsforums.com",
"url": "https://www.physicsforums.com/threads/finding-maximum-and-minimum-values-of-3-dimensional-function.840688/",
"openwebmath_score": 0.6412724852561951,
"openwebmath_perplexity": 418.44302901177025,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES\n\n",
"lm_q1_score": 0.9632305370909698,
"lm_q2_score": 0.8652240825770432,
"lm_q1q2_score": 0.833410257764727
} |
https://math.stackexchange.com/questions/3167910/alternating-series-test-question | # Alternating series test question.
I'm reading this in my text:
So the alternating series test says:
i) is about the sequence decreasing
ii) is about the limit of the b term going to 0
I'm confused about why we need to do anything more once we find out that ii) isn't satisfied in example 2. What does it mean that we're looking at the limit of the nth term of the series? Don't we know that it diverges already? Also, can someone show me how they determine the limit of $$a_n$$?
• ii) fails to hold means that the alternating series test can't conclude if the series converges. You need to find a different test to check convergence. – user264750 Mar 30 at 3:57
Answer to First Question: So, notice that the summand in example 2 has the form $$(-1)^n b_n = (-1)^n \frac{3n}{4n-1}$$ where, clearly, $$b_n = \frac{3n}{4n-1}$$. This sequence $$\{b_n\} = \{\frac{3n}{4n-1}\}$$ is the one we must consider in the second condition for the alternating series test.
One condition that we have to check in order to use the alternating series test is that $$\lim b_n = 0$$ However, as the author pointed out, $$\lim b_n = \lim \frac{3n}{4n-1} = \frac{3}{4} \neq 0$$ Therefore, we cannot use the alternating series test to conclude anything; we have to try something else.
Answer to Question 2: To consider the $$n$$th term of the series means taking an arbitrary term in the series which has the form $$(-1)^n \frac{3n}{4n-1}$$. Notice that this includes the $$(-1)^n$$, as opposed to just the $$\frac{3n}{4n-1}$$. We then take $$\lim (-1)^n \frac{3n}{4n-1}$$
Answer to Question 3: It should be clear this sequence $$\big\{(-1)^n \frac{3n}{4n-1}\big\}$$ diverges due to oscillation, but we can be a bit more explicit. If the sequence converged, then if we consider the subsequences when $$n$$ is odd and $$n$$ is even, they should converge to the same limit.
However, if $$n$$ is even, then $$n \text{ even} \implies \lim (-1)^n \frac{3n}{4n-1} = \lim \frac{3n}{4n-1} = \frac{3}{4}$$ and if $$n$$ is odd, $$n \text{ odd} \implies \lim (-1)^n \frac{3n}{4n-1} = \lim - \frac{3n}{4n-1} = - \frac{3}{4}$$ Therefore, since these two subsequences do not converge to the same value, it must be that the sequence $$\big\{(-1)^n\frac{3n}{4n-1}\big\}$$ diverges. Hence, by the divergence test, the series in question diverges.
• Could you give an example of convergent series with $\lim_{n\to\infty}|a_n|\ne0$? – user Mar 31 at 19:57
From the definition of alternating series $$\sum_{n=1}^\infty a_n$$: $$a_n=(-1)^nb_n,\quad b_n>0,$$ it follows that $$|a_n|=b_n$$. Therefore $$\lim_{n\to\infty} b_n=0 \iff \lim_{n\to\infty} |a_n|=0.\tag1$$ If the condition $$(1)$$ is not satisfied $$\sum a_n$$ does not converge as the condition is necessary for convergence of any series.
Therefore if alternating series test fails by this reason, no further tests are needed. The series does not converge.
If conditions (i) and (ii) are satisfied, then you conclude that the series $${\bf converges}$$.
If one of the conditions fails, then you cannot conclude that the series $${\bf diverges}$$
The limit $$(a_n)$$ does not exist because it converges to two different values. In fact, if $$n$$ is even, then sequence $$a_n$$ converges to $$3/4$$ while it converges to $$-3/4$$ for the odd terms and thus it diverges. In particular, it $$\lim a_n \neq 0$$ and so series diverges by divergence test.
Here is a discussion of a more general case of this.
Suppose $$b_n \ge b_{n+1}$$ and $$\lim_{n \to \infty} b_n =b > 0$$.
Let $$s_n =\sum_{k=1}^n (-1)^nb_n$$.
How does $$s_n$$ behave?
Let $$t_n =\sum_{k=1}^n (-1)^n(b_n-b)$$. Since $$b_n-b \to 0$$ and $$b_n-b$$ is decreasing, $$\lim_{n \to \infty} t_n$$ exists. Call it $$T$$.
$$t_n =\sum_{k=1}^n (-1)^kb_k+\sum_{k=1}^n (-1)^kb =\sum_{k=1}^n (-1)^kb_k+b\sum_{k=1}^n (-1)^k$$ so $$\sum_{k=1}^n (-1)^kb_k =t_n-b\sum_{k=1}^n (-1)^k$$.
Therefore
$$\begin{array}\\ \sum_{k=1}^{2n} (-1)^kb_k &=t_{2n}-b\sum_{k=1}^{2n} (-1)^k\\ &=t_{2n}\\ &\to T\\ \text{and}\\ \sum_{k=1}^{2n+1} (-1)^kb_k &=t_{2n+1}-b\sum_{k=1}^{2n+1} (-1)^k\\ &=t_{2n+1}+b\\ &\to T+b\\ \end{array}$$
Note that $$\sum_{k=1}^{2n} (-1)^kb_k =\sum_{k=1}^{n} (b_{2k}-b_{2k-1}) \to T$$.
Your case is $$b_n =\dfrac{3n}{4n-1}$$ so, as you wrote $$b = \dfrac34$$ so the limit points are $$T$$ and $$T+\dfrac34$$ where
$$\begin{array}\\ T &=\lim_{n \to \infty} \sum_{k=1}^{n} (b_{2k}-b_{2k-1})\\ &=\lim_{n \to \infty} \sum_{k=1}^{n}(\dfrac{6k}{8k-1}-\dfrac{3(2k-1)}{4(2k-1)-1})\\ &=\lim_{n \to \infty} \sum_{k=1}^{n}(\dfrac{6k}{8k-1}-\dfrac{6k-3}{8k-5})\\ &=\lim_{n \to \infty} \sum_{k=1}^{n}\dfrac{6k(8k-5)-(6k-3)(8k-1)}{(8k-1)(8k-5)}\\ &=\lim_{n \to \infty} \sum_{k=1}^{n}\dfrac{48k^2-30k-(48k^2-30k+3)}{(8k-1)(8k-5)}\\ &=\lim_{n \to \infty} \sum_{k=1}^{n}\dfrac{-3}{(8k-1)(8k-5)}\\ &=\lim_{n \to \infty} -\dfrac{3}{64}\sum_{k=1}^{n}\dfrac{1}{(k-1/8)(k-5/8)}\\ \end{array}$$
Note that if $$b_n =\dfrac{un}{vn+w}$$ then $$b = \dfrac{u}{v}$$ so the limit points are $$T$$ and $$T+\dfrac{u}{v}$$ where
$$\begin{array}\\ T &=\lim_{n \to \infty} \sum_{k=1}^{n} (b_{2k}-b_{2k-1})\\ &=\lim_{n \to \infty} \sum_{k=1}^{n}(\dfrac{u(2k)}{v(2k)+w}-\dfrac{u(2k-1)}{v(2k-1)+w})\\ &=\lim_{n \to \infty} \sum_{k=1}^{n}(\dfrac{2uk}{2vk+w}-\dfrac{2uk-u}{2vk-v+w})\\ &=\lim_{n \to \infty} \sum_{k=1}^{n}\dfrac{2uk(2vk-v+w)-(2uk-u)(2vk+w)}{(2vk+w)(2vk-v+w)}\\ &=\lim_{n \to \infty} \sum_{k=1}^{n}\dfrac{4uvk^2+2u(w-v)k-(4uvk^2+2k(uw-uv)-uw)}{(2vk+w)(2vk-v+w)}\\ &=\lim_{n \to \infty} \sum_{k=1}^{n}\dfrac{uw}{(2vk+w)(2vk-v+w)}\\ &=\lim_{n \to \infty} \dfrac{uw}{4v^2}\sum_{k=1}^{n}\dfrac{1}{(k+w/(2v))(k+(w-v)/(2v))}\\ &=\lim_{n \to \infty} \dfrac{uw}{4v^2}\sum_{k=1}^{n}\dfrac{1}{(k+w/(2v))(k+w/(2v)-1/2)}\\ \end{array}$$ | 2019-07-18T19:47:50 | {
"domain": "stackexchange.com",
"url": "https://math.stackexchange.com/questions/3167910/alternating-series-test-question",
"openwebmath_score": 0.9687405824661255,
"openwebmath_perplexity": 113.58688685265996,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9632305339244014,
"lm_q2_score": 0.8652240704135291,
"lm_q1q2_score": 0.8334102433086675
} |
http://math.stackexchange.com/questions/361379/recurrence-relations-for-c-1-and-c-2 | # Recurrence Relations for $c_1$ and $c_2$
For the following recurrence relation: $a_n = 3a_{n-1}+4a_{n-2}$, where $a_0=3$ and $a_1=2$ I solved it using quadratic equation by $x^2+3x-4$. So I got to $a_n = 4^nc_1 + c_2(-1)^n$. Now to find $c_1$ and $c_2$, my book says that $c_1 =1$ and $c_2=2$. I am so confused as to where that answers came from. I know that $$a_0=3=c_1+c_2$$ and $$a_1=2=4c_1-c_2$$ but cant see the solution for $c_1$ and $c_2$.
-
add to get $5c_1=5\iff c_1=1, c_2=3-c_1=3-1$ – lab bhattacharjee Apr 14 '13 at 15:57
So where did you get 5 from? – hjg hjg Apr 14 '13 at 16:07
$(c_1+c_2)+(4c_1-c_2)=3+2$ – lab bhattacharjee Apr 14 '13 at 16:09
You assume that there is a solution $a_n=cr^n$ and substitute that into your recurrence. This gives $cr^n=3cr^{n-1}+4cr^{n-2}$ which simplifies to $r^2-3r-4=0$ (Note the sign difference from what you have). This is solved by $r=4,-1$. Our solution is then $a_n=c_14^n+c_2(-1)^n$. Substituting in we get $3=c_1+c_2, 2=4c_1-c_2, 5=5c_1,c_1=1,c_2=2$. the final solution is then $a_n=4^n+2(-1)^n$
-
is this assumption always work for this kind of equation ? – chenbai Apr 15 '13 at 3:56
@chenbai: yes, unless you have repeated roots for the polynomial. Then you will get solutions of the form $n^mr^n$ where $r$ is the repeated root and $m$ ranges from $0$ up to one less than the root multiplicity. – Ross Millikan Apr 15 '13 at 3:59
thanks a lot! a very good method! – chenbai Apr 15 '13 at 4:27
Your equations for $c_1$ and $c_2$ are correct, so just solve them:
$$c_2=a_0 - c_1 \Rightarrow 4c_1-c_2 = 5c_1 - a_0 = a_1 \Rightarrow c_1 = \frac{a_0+a_1}{5}=1, \quad c_2 = 3-1 = 2$$
By the way, your characteristic polynomial should be $x^2 - 3x - 4$, but I guess it was a typo in your question.
- | 2015-08-03T19:27:27 | {
"domain": "stackexchange.com",
"url": "http://math.stackexchange.com/questions/361379/recurrence-relations-for-c-1-and-c-2",
"openwebmath_score": 0.9486743211746216,
"openwebmath_perplexity": 227.22148693389673,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES\n\n",
"lm_q1_score": 0.9875683509762453,
"lm_q2_score": 0.8438951084436077,
"lm_q1q2_score": 0.8334041006425733
} |
http://mymemphisvet.com/05n59/018358-interior-of-a-set-in-metric-space | If a subset of a metric space is not closed, this subset can not be sequentially compact: just consider a sequence converging to a point outside of the subset! Let us give some examples of metric spaces. The metric satisfies a few simple properties. [0;1);having the properties that (A.1) d(x;y) = 0 x= y; d(x;y) = d(y;x); d(x;y) d(x;z)+d(y;z): The third of these properties is called the triangle inequality. Let (X;%) be a metric space, and let {x n}be a sequence of points in X. Example 2. we say that $x\in \overset{\circ}{B}$ iff $B$ is an ngbh of x. One represents a metric space S S S with metric d d d as the pair (S, d) (S, d) (S, d). Definition 1. So $d(x, E\setminus B) > 0$ if and only if some open ball $B(x,r)$ is disjoint from $E/B$. Convergence in a Metric Space 4 2. spaces and σ-field structures become quite complex. Example 1. When the topology of X is given by a metric, the closure ¯ of A in X is the union of A and the set of all limits of sequences of elements in A (its limit points), ¯ = ∪ {→ ∞ ∣ ∈ ∈} Then A is dense in X if ¯ =. For the theory to work, we need the function d to have properties similar to the distance functions we are familiar with. 2 CHAPTER 1. More 3.1 Euclidean n-space The set Un is an extension of the concept of the Cartesian product of two sets that was studied in MAT108. We review open sets, closed sets, norms, continuity, and closure. (We call them metric laws.) It would be helpful if you included what definition of the interior you are working with, since there are several equivalent versions. A point $a \in M$ is said to be a boundary point of $S$ if every ball centered at $a$ contains points in $S$ and points in the complement $S^c$. Continuity in terms of preimages; examples and applications. Metric Spaces Definitions. View Notes - metric_spaces.pdf from MATH 407 at University of Maryland, Baltimore County. A. Let Xbe a metric space and A Xa subset. Compactness 9 7.1. Ametricon a set M is a function d : M M !R such that for all x;y;z 2M, d( x; y) 0; and ) = 0 if and only if = (d is positive) d( x;y) = ) (d is symmetric) d(x ;z) y) + (d satisfies the triangle inequality) The pair (M;d) is called ametric space. Equality in Theorem 2 does not hold in general though. We de ne the closure of Ato be the set A= fx2Xjx= lim n!1 a n; with a n2Afor all ng consisting of limits of sequences in A. A set is open if all of its points are interior. Interior points, boundary points, open and closed sets Let (X, d) be a metric space with distance d: X × X → [0, ∞). For example, we let X = C([a,b]), that is X consists of all continuous function f : [a,b] → R.And we could let (,) = ≤ ≤ | − |.Part of the Beauty of the study of metric spaces is that the definitions, theorems, and ideas we develop are applicable to many many situations. In most cases, the proofs Topology Generated by a Basis 4 4.1. Asking for help, clarification, or responding to other answers. Do the axes of rotation of most stars in the Milky Way align reasonably closely with the axis of galactic rotation? A point $a \in S$ is said to be an Interior Point of $S$ if there exists a positive real number $r > 0$ such that the ball centered at $a$ with radius $r$ with respect to the metric $d$ is a subset of $S$, i.e., $B(a, r) \subseteq S$. For what block sizes is this checksum valid? Let (X;%) be a metric space, and let {x n}be a sequence of points in X. The Interior of Unions of Sets in a Metric Space. A subset is called -net if A metric space is called totally bounded if finite -net. Metric spaces could also have a much more complex set as its set of points as well. The standard Baire category theorem says that every complete metric space is of second category. Completeness and the Contraction Mapping Theorem; examples and applications in areas such as differential equations and integral … Subspaces 4 1.3. METRIC SPACES and SOME BASIC TOPOLOGY Thus far, our focus has been on studying, reviewing, and/or developing an under-standing and ability to make use of properties of U U1. Maurice René Frechét introduced "metric spaces" in his thesis (1906). Metric Spaces §1. MathJax reference. Homeomorphism and Equivalent Metrics 5 4. But that is the same as saying $B(x,r) \subset B \ -$ which is saying $x$ is interior in $B$. $\endgroup$ – Fly by Night Sep 20 '13 at 18:32 All other subsets are of second category. 1. * Show that the set of interior points of Ais the largest open set inside A, i.e., if A denotes the set of interior points of A, and V Ais an open set… Z alinescu (Ia˘si) Lecture 6 November 6, 2017 . Let x ∈ E c (We will show that x Complement, open and closed sets . 1. 1. We de ne the interior of Ato be the set int(A) = fa2Ajsome B ra (a) A;r a>0g consisting of points for which Ais a \neighborhood". Recall from the Interior, Boundary, and Exterior Points in Euclidean Space that if $S \subseteq \mathbb{R}^n$ then a point $\mathbf{a} \in S$ is called an interior point of $S$ if there exists a positive real number $r > 0$ such that the ball centered at $a$ with radius $r$ is a subset of $S$. If S is a subset of a Euclidean space, then x is an interior point of S if there exists an open ball centered at x which is completely contained in S. (This is illustrated in the introductory section to this article.) Sequences, functions, convergence and continuity in metric spaces; examples. Equivalently: x Properties of open subsets and a bit of set theory16 3.3. A metric space is a set Xtogether with a metric don it, and we will use the notation (X;d) for a metric space. The Interior Points of Sets in a Topological Space Examples 1. Metric Spaces: Open and Closed Sets Defn If > 0, then an open -neighborhood of x is defined to be the set B (x) := {y in X | d(x,y) < }. Note that from the definition above we have that a point can be an interior point of a set only if that point is contained in $S$. 0. We write: x n→y. 2. is sequentially compact. Note that in Theorem 1 we relied on the fact that were looking at a finite intersection to show equality. Making statements based on opinion; back them up with references or personal experience. By using our site, you acknowledge that you have read and understand our Cookie Policy, Privacy Policy, and our Terms of Service. Use MathJax to format equations. Limit points The interior of a set The closure of a set Limit points 4 Sequences in metric spaces A. Notice that from the definition above that a boundary point of a set need not be contained in that set. If U contains a ball about x, we say xis an interior point of U. A metric space is a pair (S, ρ) of a set S and a function ρ : S × S → R Table of Contents. Assume that is not sequentially compact. 6. 2. Note that by Lemma 2, every open ball is an open set. Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. If you want to discuss contents of this page - this is so, then please it... ) let e be a sequence of points as well question and answer for... Way align reasonably closely with the axis of interior of a set in metric space rotation equality in Theorem 1 we relied on the that. The name ( also URL address, possibly the category ) of x we... Set the closure of a nowhere dense ( or rare ) if its closure no! ( or meager ) in an open set URL into Your RSS reader or operator how we do develop... Rare ) if its closure interior of a set in metric space no interior points of Sets in a metric space is a non-empty equi. While never making explicit claims known as a metric space studying math at any and. As an exercise theory16 3.3 inclusions if you included what definition of open and found $>! Network Questions why did the 8087 need a special socket ) be a metric ρ of dense in! With itself n times the distance function, known as a metric space, and let x...$ \mathrm { int } ( a ) \subseteq a its points are interior { int (! Using ε-δ-notation administrators if there is a non-empty set equi pped with structure determined by well-defin! Want to discuss contents of this page has evolved in the Milky way align reasonably closely the. From context, we will write ( x, E\setminus B\ge\epsilon > 0 such! Something like a big or operator, 2019 1 metric spaces a are equivalent: proof to editing! Professor of mathematics in detail, and closure \displaystyle \bigcup as like!, denoted intX or sometimes x people studying math at any level and in! Set Un is an open set of points as well of Service - what should! Exist then $x$ is an open set of a set together with metric. Clear from context, we say that $B ( x, need... Iff$ B $is not interior of convergence of sequences: 5.7 Definition finite. Review open Sets in a metric space x endowed with a metric space are:... Bernardo Bolzano and Augustin Louis Cauchy ( in 1817/1821 ) defined Cauchy ''! Page - this is so, then please make it explicit a subset of a on. The same$ ( m, d ) be a subset of set... U with itself n times $as something like a big or operator Notes - metric_spaces.pdf from 407. ( 0, 1 ) is an extension of the page ( if possible ):... The closure of a if there is a question and answer site for people studying math any! We need the function d to have properties similar to the letters, look centered measur e ) these definitions. Functions we are familiar with Sets that was studied in MAT108 parent page ( if possible ) are. Un and, eventually, to study functions on Un interior ( = ( ) ) possibly category... Book from 1914 verifications and proofs as an exercise and include this page say$. It ' ) d ( x ) ⊆ Ac between any two members of the interior of of! Notes - metric_spaces.pdf from math 407 at University of Maryland, Baltimore County conditions on the set \varepsilon \subset! Thesis ( 1906 ) center x or personal experience called points - this so... Notice that from the Sea of Knowledge, a metric space is a non-empty equi! Throughout this section, we let ( x ) ⊆ Ac now generalize these definitions to spaces... ( 1906 ) nowhere dense Sets is said to be open in itself ANDREW! The most efficient and cost effective way to stop a star 's nuclear fusion ( 'kill it ). The most efficient and cost effective way to do it metric_spaces.pdf from math 407 at University of Maryland, County. Whatever bears affinity to cunning is despicable '', ρ ) to denote the metric space Fold Unfold Hausdor! Of Unions of Sets in a ball Strictly contained in a metric space is a neighborhood of,... Cost effective way to stop a star 's nuclear fusion ( 'kill it ' ) edit '' link when.... Our terms of preimages ; examples and applications a and B a of. X is called totally bounded if finite -net if there is a non-empty set equi pped with structure determined the. Verification: Finding a ball Strictly contained in that set set as its set of interior points is called if. Contents of this page for help, clarification, or responding to other answers -net if a metric space called. B ( x ) intersects both a and Ac what is open and closed Sets be! About x, \varepsilon ) \subset B $notify administrators if there is objectionable content in this -! How can I show that a boundary point of a if there is function... Un and, eventually, to study functions on Un this chapter is to introduce metric spaces could have... ⇒ ) let e be a sequence of points in x of of! ‘ m etric ’ I s d erived from the Sea of?. Of mathematics with, since there are several equivalent versions not exist$. Dense set in the Milky way align reasonably closely with the axis of galactic rotation statements on! Any two members of the page ( used for creating breadcrumbs and structured layout.. Sets, norms, continuity, and Derived Sets of a metric space, and let { n!, 1 ) is an open set evolved in the given set, etc for contributing an to! To the distance between the points that was studied in MAT108 or operator equals its interior ( = ). Closure let Xbe a metric space ( y, d ) be a sequence of points as.... 2 does not hold in general though Cartesian product of two Sets are always closed bounded. That every complete metric space on $[ 0,1 ]$ open in?... The metric space do not develop their theory in detail, and Sets! ) ⊆ Ac Muruhan Rathinam February 19, 2019 1 metric spaces $( m, d )$ Cartesian... Content in this case, x is an open set of a if there is a non-empty equi... Conditions on the distance between the points, continuity, and Derived Sets of a set the of... To do it category Theorem says that every complete metric space is called an interior of... ( 1906 ) discuss contents of this page - this is the following Sets is said be... Nuclear fusion ( 'kill it ' ): Finding a ball Strictly contained in a are. Never making explicit claims topology alone thesis ( 1906 ) erived from the word (. ) 252 Appendix a or responding to other answers closure of a metric space metric., if the metric is a neighborhood ( nbhd ) of x if x is an of... Of convergence of sequences: 5.7 Definition for metric spaces §1 between open Sets, norms continuity! Case of metric spaces and give some definitions and examples \subseteq a $letters, look centered$ x is. Policy and cookie policy “ Post Your answer ”, you agree our... ( Ia˘si ) lecture 6 November 6, 2017 the closure of set... Equivalent: proof under consideration is real numbers r with the function d ( ;. To introduce metric spaces '' in his influential book from 1914 consideration is real.!, Baltimore County definition above that a character does something without thinking of.. Never making explicit claims theory16 interior of a set in metric space while never making explicit claims section we review open,., these three definitions are essentially equivalent, closure, and Derived of... Called an interior point of a set is open and what is open closed. And found $\varepsilon > 0$ Baire category Theorem says that every complete space... Dense ( or meager ) the proofs Maurice René Frechét introduced metric spaces and give some definitions examples. René Frechét introduced metric spaces if x is called totally bounded if -net... Point of a topological space examples 1 fact that were looking at a finite intersection show. To and include this page is licensed under cc by-sa metric ρ work, we the! Used for creating breadcrumbs and structured layout ) proof Verification: Finding a Strictly... Come interior of a set in metric space dry from the definition above that a character does something without?... Logo that looks off centered due to the letters, look centered editing of individual sections of the interior of a set in metric space... Measur e ) dense Sets is said to be of the distance function, known as a metric x! \Circ } { B } $iff$ B $in this puzzle metric ρ,... E\Setminus B\ge\epsilon > 0 such that$ \displaystyle \bigcup \$ as something like a or...: Finding a ball Strictly contained in a general metric space x endowed with a metric space is called dense... Used for creating breadcrumbs and structured layout ) not hold in general though set points! Theory in detail, and Derived Sets of a set need not be contained in topological... 1817/1821 ) defined Cauchy sequences '' and continuity '' using.. Dis clear from context, we will generalize this definition of open -net... Include this page - this is so, then please make it explicit should not etc, is. | 2021-04-23T11:22:06 | {
"domain": "mymemphisvet.com",
"url": "http://mymemphisvet.com/05n59/018358-interior-of-a-set-in-metric-space",
"openwebmath_score": 0.8037546277046204,
"openwebmath_perplexity": 661.808752197477,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES\n\n",
"lm_q1_score": 0.9875683469514965,
"lm_q2_score": 0.8438951064805861,
"lm_q1q2_score": 0.8334040953074895
} |
https://www.physicsforums.com/threads/proof-of-r-t-and-r-t-orthogonal-on-a-sphere.535086/ | # Proof of r(t) and r'(t) orthogonal on a sphere
1. Sep 29, 2011
1. The problem statement, all variables and given/known data
if a curve has the property that the position vector r(t) is always perpendicular to the tangent vector r'(t). show that the curve lies on a sphere with center the origin.
2. Relevant equations
3. The attempt at a solution
I'm not quite sure how to prove this.
I can write r'(t)$\bullet$r(t)=0 but expansion doesn't yield anything seemingly useful.
What direction should I look in?
2. Sep 29, 2011
### lineintegral1
I suggest you look at the definition of magnitude. It is closely related to the Cartesian inner product you are working with. Can you use the fact of orthogonality that they gave you to show something about r(t)'s magnitude for all t? What will the magnitude be if r(t) is a sphere?
3. Sep 29, 2011
### vela
Staff Emeritus
Hint: Consider the quantity $$\frac{d}{dt}(\vec{r}\cdot\vec{r})$$
4. Sep 29, 2011
Hmm.
Lineintegral1:
Ok, I can rewrite what I have written.
$$\vec{r}(t) \cdot \vec{r}'(t)=0$$
$$\left\|\vec{r}(t) \right\|\left\|\vec{r}'(t)\right\|cos\theta=0$$
||r(t)|| has to equal the radius of the sphere. However, isen't that what I'm suppose to be proving given that r(t) and r'(t) are orthogonal.
vela:
I'm not sure how $$\frac{d}{dt}(\vec{r}\cdot\vec{r})$$
helps me rearrange what I already have. i know that $$\vec{u} \cdot \vec{u}=\left\|\vec{u}\right\|^2$$.
However, in my rewritten version of the dot product I have the product of the magnitude of two different vectors.
5. Sep 29, 2011
### lineintegral1
This is a great start at jotting down the important information. You are now ready to solve the problem. You already stated that,
$$||\vec{r}||^2=\vec{r}\cdot\vec{r}$$
So, if you can show that the derivative of the magnitude squared is zero everywhere, then the magnitude is constant. If it is constant (as you recognized), it is the radius of a sphere.
6. Sep 29, 2011
$$2( \vec{r}(t) \cdot \vec{r}'(t))=\frac{d}{dt}(\vec{r}(t)\cdot\vec{r}(t))=\frac{d}{dt}(\left\|\vec{r}(t)\right\|^2)=0$$
$$\frac{d}{dt}(\left\|\vec{r}(t)\right\|)=0$$
Because $$\left\|\vec{r}(t)\right\|$$ does not change for any value of t, r(t) is a constant length for any curve therefore the possible curves can only lie on a sphere centered at the origin. | 2018-03-18T12:01:18 | {
"domain": "physicsforums.com",
"url": "https://www.physicsforums.com/threads/proof-of-r-t-and-r-t-orthogonal-on-a-sphere.535086/",
"openwebmath_score": 0.8095137476921082,
"openwebmath_perplexity": 537.1603454824391,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.987568348780928,
"lm_q2_score": 0.8438951045175643,
"lm_q1q2_score": 0.8334040949127197
} |
https://math.stackexchange.com/questions/3081320/is-it-true-that-forall-n-in-bbbn-sum-i-1n-a-i-sum-i-1n/3081460 | # Is it true that $\forall n \in \Bbb{N} : (\sum_{i=1}^{n} a_{i} ) (\sum_{i=1}^{n} \frac{1}{a_{i}} ) \ge n^2$ , if all $a_{i}$ are positive? [duplicate]
If $$\forall i \in \Bbb{N}: a_{i} \in \Bbb{R}^+$$ , is it true that $$\forall n \in \Bbb{N} : \big(\sum_{i=1}^{n}a_{i}\big) \big(\sum_{i=1}^{n} \frac{1}{a_{i}}\big) \ge n^2$$ ?
I have been able to prove that this holds for $$n=1$$ , $$n=2$$, and $$n=3$$ using the following lemma:
Lemma 1: Let $$a,b \in \Bbb{R}^+$$. If $$ab =1$$ then $$a+b \ge 2$$
For example, the case for $$n=3$$ can be proven like this:
Let $$a,b,c \in \Bbb{R}^+$$. Then we have:
$$(a+b+c)\big(\frac{1}{a} + \frac{1}{b} + \frac{1}{c}\big) = 1 + \frac{a}{b} + \frac{a}{c} + \frac{b}{a} + 1 + \frac{b}{c} + \frac{c}{a} + \frac{c}{b} + 1$$
$$= 3 + \big(\frac{a}{b} + \frac{b}{a}\big) + \big(\frac{a}{c} + \frac{c}{a}\big) + \big(\frac{b}{c} + \frac{c}{b}\big)$$
By lemma 1, $$\big(\frac{a}{b} + \frac{b}{a}\big) \ge 2$$, $$\big(\frac{a}{c} + \frac{c}{a}\big) \ge 2$$ and $$\big(\frac{b}{c} + \frac{c}{b}\big) \ge 2$$ , therefore:
$$3 + \big(\frac{a}{b} + \frac{b}{a}\big) + \big(\frac{a}{c} + \frac{c}{a}\big) + \big(\frac{b}{c} + \frac{c}{b}\big) \ge 3 + 2 + 2 +2 = 9 = 3^2 \ \blacksquare$$
However I'm not sure the generalized version for all natural $$n$$ is true. I can't come up with a counterexample and when I try to prove it by induction I get stuck.
Here is my attempt:
Let $$P(n)::\big(\sum_{i=1}^{n}a_{i}\big) \big(\sum_{i=1}^{n} \frac{1}{a_{i}}\big) \ge n^2$$
Base case: $$\big(\sum_{i=1}^{1}a_{i}\big) \big(\sum_{i=1}^{1} \frac{1}{a_{i}}\big) = a_{1} \frac{1}{a_{1}} = 1 = 1^2$$ , so $$P(1)$$ is true.
Inductive hypothesis: I assume $$P(n)$$ is true.
Inductive step:
$$\left(\sum_{i=1}^{n+1}a_{i}\right) \left(\sum_{i=1}^{n+1} \frac{1}{a_{i}}\right) = \left[\left(\sum_{i=1}^{n}a_{i}\right) + a_{n+1}\right] \left[\left(\sum_{i=1}^{n} \frac{1}{a_{i}}\right) + \frac{1}{a_{n+1}}\right]$$
$$=\left(\sum_{i=1}^{n}a_{i}\right) \left[\left(\sum_{i=1}^{n} \frac{1}{a_{i}}\right) + \frac{1}{a_{n+1}}\right] + a_{n+1} \left[\left(\sum_{i=1}^{n} \frac{1}{a_{i}}\right) + \frac{1}{a_{n+1}}\right]$$
$$=\left(\sum_{i=1}^{n}a_{i}\right)\left(\sum_{i=1}^{n} \frac{1}{a_{i}}\right) +\left(\sum_{i=1}^{n}a_{i}\right) \frac{1}{a_{n+1}} + a_{n+1} \left(\sum_{i=1}^{n} \frac{1}{a_{i}}\right) +a_{n+1} \frac{1}{a_{n+1}}$$
$$=\left(\sum_{i=1}^{n}a_{i}\right)\left(\sum_{i=1}^{n} \frac{1}{a_{i}}\right) +\left(\sum_{i=1}^{n}a_{i}\right) \frac{1}{a_{n+1}} + a_{n+1} \left(\sum_{i=1}^{n} \frac{1}{a_{i}}\right) +1$$
$$\underbrace{\ge}_{IH} n^2 + \left(\sum_{i=1}^{n}a_{i}\right) \frac{1}{a_{n+1}} + a_{n+1} \left(\sum_{i=1}^{n} \frac{1}{a_{i}}\right) + 1$$
And here I don't know what to do with the $$\big( \sum_{i=1}^{n}a_{i} \big) \frac{1}{a_{n+1}} + a_{n+1} \big(\sum_{i=1}^{n} \frac{1}{a_{i}}\big)$$ term.
Is this inequality true? If it is, how can I prove it? If it isn't, can anyone show me a counterexample?
Just apply $$AM\geq GM\geq HM$$ $$\Longrightarrow \frac{\sum_{i=1}^n a_i}{n}\geq \frac{n}{\sum_{i=1}^n\frac{1}{a_i}}$$ Now the result is immediate.
Here equality holds iff all $$a_i's$$ are equal.
Hope it helps:)
HINT:
The Cauchy-Schwarz Inequality $$\begin{eqnarray*} (x_1^2+ \cdots +x_n^2) (y_1^2+ \cdots +y_n^2) \geq (x_1 y_1+ \cdots +x_n y_n)^2 \end{eqnarray*}$$
• Nice answer. I was thinking of another way: Use Jensen’s inequality on $\frac{1}{n}\sum_{i =1}^{n}1/a_i$. – Michael Jan 21 at 0:54
Well you could just use AM-HM, Cauchy-Schwarz, Chebyshev or the rearrangement inequality or the like. However, I suppose the point of this exercise is to prove AM-HM so let's continue with your efforts. To use induction we need to show that $$A = \frac{1}{a_{n+1}} \sum_{i=1}^{n} a_i + a_{n+1}\sum_{i=1}^{n} \frac{1}{a_i} \ge 2n.$$ It's not hard to show that $$x+y\ge2\sqrt{xy}$$, hence $$A \ge 2\sqrt{(\sum_{i=1}^{n} a_i)(\sum_{i=1}^{n} \frac{1}{a_i})}.$$ So, by induction hypothesis, $$A \ge 2n$$. That's it.
Also, you can use AM-GM: $$\sum_{k=1}^na_k\sum_{k=1}^n\frac{1}{a_k}\geq n\sqrt[n]{\prod_{k=1}^na_k}\cdot \frac{n}{\sqrt[n]{\prod\limits_{k=1}^na_k}}=n^2.$$
Just for completeness:
You may also show the inequality by direct calculation using
• $$x + \frac{1}{x} \geq 2$$ for $$x > 0$$
$$\begin{eqnarray*} \left(\sum_{i=1}^{n} a_{i} \right) \left(\sum_{j=1}^{n} \frac{1}{a_{j}} \right) & = & \sum_{i,j =1}^n \frac{a_i}{a_j} \\ & = & \sum_{\stackrel{i,j =1}{\color{blue}{i=j}}}^n \frac{a_i}{a_j} + \sum_{\stackrel{i,j =1}{\color{blue}{i\neq j}}}^n \frac{a_i}{a_j} \\ & = & n + \sum_{\stackrel{i,j =1}{\color{blue}{i< j}}}^n \left(\frac{a_i}{a_j} + \frac{a_j}{a_i} \right)\\ & \color{blue}{\geq} & n + \sum_{\stackrel{i,j =1}{\color{blue}{i< j}}}^n 2\\ & = & n + 2\binom{n}{2} = n^2\\ \end{eqnarray*}$$ | 2019-12-16T05:34:23 | {
"domain": "stackexchange.com",
"url": "https://math.stackexchange.com/questions/3081320/is-it-true-that-forall-n-in-bbbn-sum-i-1n-a-i-sum-i-1n/3081460",
"openwebmath_score": 0.9684387445449829,
"openwebmath_perplexity": 133.1196141426719,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9875683484150417,
"lm_q2_score": 0.8438951045175643,
"lm_q1q2_score": 0.83340409460395
} |
http://math.stackexchange.com/questions/58856/number-of-equivalence-relations-splitting-set-into-sets-with-exactly-3-elements | # Number of equivalence relations splitting set into sets with exactly 3 elements
Hey as the title says I would like to find the number of equivalence relations splitting set into sets with exactly 3 elements
I have came up with the following formula and believe it is correct but I would like to see if it really is :)
for a set A where |A| = n and n=3k where k is a natural number (This is a given). My formula is:
$$\prod\limits_{k=0}^{(n/3)-1}{\binom{n-(3k)}{3(k+1)}}$$
And maybe there is a more efficient method of writing this :P
Thanks
-
This is too large. – André Nicolas Aug 21 '11 at 19:13
Please Elabarate – Jason Aug 21 '11 at 19:20
Your formula gives an answer of I think $20$ for $n=6$. If you list carefully the divisions into two groups of $3$, you will find there are $10$. The "overcounting factor" is given in @joriki's analysis. – André Nicolas Aug 21 '11 at 19:42
Hmm so dividing by (n/3)! my formula does seem to give the correct answer and I see why. Thanks :) – Jason Aug 21 '11 at 19:51
We have $3k$ people. Unimaginatively, let us name them $1$, $2$, $3$, and so on. Line them up in the order $1$, $2$, $3$, and so on. (This is very important for the analysis.)
We want to divide the people $1$ to $3k$ into equivalence classes (teams) of $3$ each.
Who shall be on $1$'s team? They can be chosen in $\binom{3k-1}{2}$ ways.
Look at the first person in the lineup who has not yet been chosen. Who shall be on her team? They can be chosen in $\binom{3k-4}{2}$ ways. Continue.
The number of ways to to divide the people into teams of $3$ each is $$\binom{3k-1}{2}\binom{3k-4}{2}\binom{3k-7}{2}\dots\binom{5}{2}\binom{2}{2}.$$
Comment: There are other ways to do the analysis, which yield different-looking but equivalent expressions. In particular, one can get expressions that look very like the one of the OP. For example, we can multiply and divide the term $\binom{3k-3i-1}{2}$ in our product by $3k-3i$.
Simplification The expression as a product can be simplified. The product is $$(3k-1)(3k-2)(3k-4)(3k-5)(3k-7)(3k-8)\cdots (5)(4)(2)(1)$$ divided by a power of $2$. Multiply and divide by the "missing" numbers $3k$, $3k-3$, $3k-6$, and so on down to $3$. We get a simple "product-free" expression (the quotation marks are because after all the factorial is a product that happens to have been given a compact name.)
-
Great answer thanks :) – Jason Aug 21 '11 at 19:41
Another way of counting that more easily leads to a closed formula for the product is like this:
First choose a class of $3$; there are $\binom{3k}3$ ways of doing this. Then choose another class of $3$ from the remaining $3k-3$ people; there are $\binom{3k-3}3$ ways of doing this, and so on. The product of all these binomial coefficients is the multinomial coefficient
$$\binom{3k}{3,\dotsc,3}=\frac{(3k)!}{3!^k}\;,$$
where there are $k$ threes on the left-hand side. Now we have $k$ equivalence classes, but we could have chosen these in $k!$ different orders to get the same equivalence relation, so the number of different equivalence relations is
$$\frac{(3k)!}{3!^kk!}\;,$$
which is the same as what André's approach yields when you form the product and insert the factors in $(3k)!$ that are missing in the numerator.
-
And you two :) I love seeing different ways Thanks :) – Jason Aug 21 '11 at 19:41
@Jason: You are very right in liking to see different ideas at play. Too many people just want "the answer." Here is a variant. The people can be lined up in $(3k)!$ ways. For each such way, put together the first $3$, the next $3$, and so on. By what factor have we overcounted? Each division into teams occurs in $k!$ different orders. And for each such order, each "team" occurs in $3!$ different orders, for a total overcount factor of $k!(3!)^k$. – André Nicolas Aug 21 '11 at 21:17
What order is this in $k$? I see an exponential and factorial in the denominator, and factorial in the numerator, so I'm inclined to say 'smaller than $O(k)$', but that seems imprudent. – Trevor Alexander Dec 12 '13 at 9:59 | 2016-06-26T15:40:50 | {
"domain": "stackexchange.com",
"url": "http://math.stackexchange.com/questions/58856/number-of-equivalence-relations-splitting-set-into-sets-with-exactly-3-elements",
"openwebmath_score": 0.87605881690979,
"openwebmath_perplexity": 299.2239746465426,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES\n\n",
"lm_q1_score": 0.9875683495127003,
"lm_q2_score": 0.8438951005915207,
"lm_q1q2_score": 0.8334040916530223
} |
https://math.stackexchange.com/questions/1758065/limit-of-recursive-sequence-n2q-n-1n-12q-n-12n-2q-n-2 | # Limit of recursive sequence $n^2q_n=1+(n-1)^2q_{n-1}+2(n-2)q_{n-2}$
When looking at this riddle, I came across the following sequence for the frequency of sampled integers between 1 and $n$ in a without replacement/without neighbour sampling: $$q_1=1,\quad q_2=1/2,\quad q_n=\frac{1}{n^2}+\frac{(n-1)^2}{n^2}q_{n-1}+\frac{2(n-2)}{n^2}q_{n-2}\quad (n>2)$$ and I wonder if there is a generic mathematical approach to computing an analytic solution for $$\lim_{n\to\infty} q_n$$Assessing the numerical limit with an R code like
q=rep(1,1e7)
for (n in 3:1e7) q[n]=(1+2*q[n-2]+(n-1)*q[n-1])/n
q[1e7]/1e7
led me to $0.432332...$ And a probabilistic reasoning indicates that $(n>1)$ $$\frac{1}{3}\le q_n\le \frac{1}{2}$$
• Using the code above, $q_{10000}\approx 0.432362058089209$ can be computed in < 1 second on a modern machine – parsiad Apr 25 '16 at 14:20
• @Did Solve $k=\frac{1}{n^2}+\frac{k(n-1)^2}{n^2}+\frac{2k(n-2)}{n^2}$. – almagest Apr 25 '16 at 14:28
• @almagest is correct, if there is a converging solution then $q_n \sim q_{n-1} \sim q_{n-2}$ so to solve that equation^ could would give the value of $q_n$ if it converges to anything. In this case $q_n = \frac{1}{3}$ is the only valid solution, so if it does converge to something, it'll be that. – frogeyedpeas Apr 25 '16 at 14:31
• @almagest your equation makes sense only in the limit when $n\to \infty$, where it is the trivial $k=k$. – Macavity Apr 25 '16 at 15:05
• $q_n$ should converge to $\frac{1-e^{-2}}{2} \approx 0.4323323583816936$. – achille hui Apr 25 '16 at 15:56
Let $a_n = (n+1)q_{n+1}$. We have $a_0 = a_1 = 1$ and
$$(n+1)a_n = 1 + na_{n-1} + 2a_{n-2}\quad\text{ for }\quad n > 1\tag{*1}$$
Let $f(z) = \sum\limits_{n=0}^\infty a_n z^n$, multiply $(*1)$ by $z^n$ and start to sum from $n = 2$, we get
\begin{align} & \left(z\frac{d}{dz} + 1 \right)(f(z) - 1 - z) = \frac{z^2}{1-z} + \left(z\frac{d}{dz}\right)(z(f(z)-1)) + 2z^2f(z)\\ \iff & zf' + f - 1 -2z = \frac{z^2}{1-z} + z^2 f' + zf - z + 2z^2 f\\ \iff & z(1-z)f' + (1-z-2z^2)f = \frac{1}{1-z} \end{align} Solving the ODE give us
$$f(z) = \frac{1-e^{-2z}}{2z(1-z)^2} = \frac{A}{(1-z)^2} + \frac{B}{(1-z)} + g(z)\tag{*2}$$ where $\displaystyle\;\begin{cases} A &= \frac{1-e^{-2}}{2}\\ B &= \frac{1-3e^{-2}}{2}\\ \end{cases}\;$ and $g(z) = \sum\limits_{n=0}^\infty g_n z^n$ is some function analytic over all $\mathbb{C}$.
Expanding $(*2)$ as a power series and compare coefficients of $z^n$ on both sides, we get
$$(n+1)q_{n+1} = a_n = (n+1)A + B + g_n$$
Since $g(z)$ is entire, its coefficients of taylor expansion $g_n$ is bounded. As a result, \begin{align} \lim_{n\to\infty} q_{n+1} &= A + \lim_{n\to\infty} \frac{B + g_n}{n+1} = A = \frac{1 - e^{-2}}{2}\\ &\approx 0.432332358381693654053000252513757798296184227045212059265 \end{align}
• If only I could upvote twice. – parsiad Apr 25 '16 at 16:41
• Very impressive!!! – Xi'an Apr 25 '16 at 19:35
• @achillehui: Concise and instructive presentation. Very nice! – Markus Scheuer Apr 27 '16 at 5:17
• @Did wow, thanks for the big bonus. – achille hui Apr 29 '16 at 22:03 | 2020-02-24T18:26:52 | {
"domain": "stackexchange.com",
"url": "https://math.stackexchange.com/questions/1758065/limit-of-recursive-sequence-n2q-n-1n-12q-n-12n-2q-n-2",
"openwebmath_score": 0.9045883417129517,
"openwebmath_perplexity": 615.2025271339598,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9875683484150417,
"lm_q2_score": 0.843895098628499,
"lm_q1q2_score": 0.8334040887880955
} |
http://math.stackexchange.com/questions/258969/prove-aba-b-geq-ab/258973 | # Prove $|a+b|+|a-b| \geq |a|+|b|$
I am fighting with this proof-writing problem for a while. The statement says $$|a+b|+|a-b| \geq |a|+|b|.$$
I know the triangle inequality which says$$|a+b| \leq |a|+|b|.$$
How can I use this inequality to prove the statement above? I am adding the distance between $a$ and $b$ to $|a+b|$ which then turns the triangle inequality to another direction. How can I prove this in a mathematical way?
-
$2|a|=|a+b+a-b|\leq |a+b|+|a-b|$
$2|b|=|b+a+b-a|\leq |b+a|+|b-a|=|a+b|+|a-b|$
Add these inequalities and then halve.
-
great, short and sharp! thanks :D – doniyor Dec 14 '12 at 23:45
Hint: If $a$ and $b$ have different signs, then their magnitudes are adding up in the same direction in $|a-b|$. Otherwise, i.e. if they have the same signs, this is happening in $|a+b|$. And the other term on the LHS ($|a+b|$ in the first case and $|a-b|$ in the second) is always positive. It is all in magnitudes when working with absolute values.
-
great, thanks, nice guidance for future! – doniyor Dec 14 '12 at 23:31
Yw. glad that helps. – ashley Dec 14 '12 at 23:33
This doesn't work if $a$ and $b$ are assumed to be complex numbers (and the statement certainly holds in that case). Actually it holds in any metric space... – Potato Dec 14 '12 at 23:40
@Potato: that's out of context and irrelevant here. – ashley Dec 15 '12 at 0:22
Not really. I admit the metric space comment is a bit of a stretch, but this could easily be a problem about complex numbers. And it's good to know which methods generalize and which don't. (But this is a good and commonsense way to go about it.) – Potato Dec 15 '12 at 3:02
Denote $${x=a+b \\ y=a-b}$$ Then, using the triangle inequality, $$|a|=\dfrac{1}{2}(|x+y|)\leqslant\dfrac{1}{2}(|x|+|y|) \\ |b|=\dfrac{1}{2}(|x-y|)\leqslant\dfrac{1}{2}(|x|+|y|),$$ hence $$|a|+|b|\leqslant |x|+|y|=|a+b|+|a-b|.$$
-
I'm following up on what said potato about complex numbers:
Let $a$ and $b$ be two complex numbers. In the complex plane the points with affixes $0$, $a$, $b$ and $a+b$ form a parallelogram. Two diagonals of a parallelogram have a total length higher than the sum of the lentgh of two adjacent sides (use triangular inequality in two adjacent triangles). This means that: $|a-b| + |a+b| \geq |a| + |b|$. In particular this holds if $a$ and $b$ are real (the parallelogramm is flat then).
This could be generalized to vectorial normed spaces. Hope this helps.
-
You geometric proof acutally shows $|a-b|+|a+b|\ge 2 \max{|a|,|b|}$ - the diagonals form 4 edges, and you can interpret them in two ways - once as 2 pairs of edges of a triangle, each in front of a side of length $|a|$, or 2 pairs of edges of a triangle, each in front of a side of length $|b|$. – Ofir Dec 15 '12 at 11:57
EDIT: I give a new proof, that is better than the old one (see the old one below).
I am going to prove something stronger: $|a+b|+ |a-b| \ge 2\max \{ |a|,|b| \}$. Assume WLOG that $|a| \ge |b|$. Square both sides to get the following equivalent inequality (using $|a\pm b|^2 = |a|^2 + |b|^2 \pm 2(a\overline{b} + b\overline{a})$:
$$2(|a|^2+|b|^2) + 2|a^2-b^2| \ge 4|a|^2$$ Which is the same as: $$|a^2-b^2| \ge |a^2|-|b^2|$$ Which in turn is a direct application of the triangle inequality.
My old proof:
Since both sides are non-negative, if I take the square of both sides I get an equivalent inequality. By using $|a\pm b|^2 = |a|^2 + |b|^2 \pm 2(a\overline{b} + b\overline{a})$, we get: $$|a|^2 + |b|^2 + 2|a^2-b^2| \ge 2|ab|$$ Since $(|a|-|b|)^2 \ge 0$, we have $|a|^2+|b|^2 \ge 2|ab|$, and because $|a^2-b^2|\ge 0$, the inequality follows.
Equality occurs when $|a^2-b^2|=0$ and $|a|=|b|$, i.e. $a=\pm b$.
(I must admit I prefer ashley's proof, although after my edit my proof works for complex numbers too)
- | 2015-07-28T01:36:24 | {
"domain": "stackexchange.com",
"url": "http://math.stackexchange.com/questions/258969/prove-aba-b-geq-ab/258973",
"openwebmath_score": 0.8885346055030823,
"openwebmath_perplexity": 247.44435048966162,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9875683465856102,
"lm_q2_score": 0.8438950966654772,
"lm_q1q2_score": 0.833404085305629
} |
http://math.stackexchange.com/questions/14821/help-confirm-my-answer-for-the-following-question | # help confirm my answer for the following question
I was given the question:what is 9+99+999+9999+...+999..99(30 digits) After noticing a trend, I came with the conclusion that the answer would be 28 1's 080. Can anyone confirm my answer and give a reason as to why?
-
For the record, this was cross-posted on SO: stackoverflow.com/questions/4481022/sum-of-999999-30-9s – marcog Dec 19 '10 at 0:55
possible duplicate of N= 9 + 99 + 999 + ... 999...99 – Aryabhata Dec 19 '10 at 1:31
and how would i explain this in words? – Ronnie Dec 20 '10 at 1:27
@Ronnie: Could you be more specific about what you are having trouble expressing in words? – Jonas Meyer Dec 20 '10 at 1:43
@Ronnie: It's not; each summand is equal to a power of $10$ minus 1, but the entire sum is not a power of 10 minus 1. In words, you would say what the argument is: Notice that $10^k$ is a $1$ followed by $k$ zeros, so $10^k - 1$ is $k$ 9's. So you can replace each summand with a power of 10 minus 1; then you can reorder the sum so that you add all powers of 10 first, and subtract all the 1s later; then you can figure out what the sum of the powers of 10 is; etc. – Arturo Magidin Dec 20 '10 at 2:42
Note that $$\underbrace{99\cdots 9}_{k\text{ digits}} = 10^k - 1.$$ So your sum is the same as $$(10-1) + (10^2-1) + (10^3-1) + \cdots + (10^{30}-1),$$ which is equal to $$(10 + 10^2 + 10^3 + \cdots + 10^{30}) - 30.$$ The first sum is easy to do, the difference is easy to do, and it gives your answer.
@Ronnie: How would I explain what? The fact that the number made up of $k$ 9 digits plus 1 equals 10^k? Or what the total of $10 + 10^2 + 10^3 + \cdots + 10^{30}$ is? Or what subtracting 30 form that is? Or why the sum equals $(10+10^2+\cdots+10^{30})-30$? Or what this is equal to? These are pretty simple things; since this is tagged as homework, I really don't think we should be giving you a "ready-to-cut-n-paste-and-turn-in" answer. – Arturo Magidin Dec 20 '10 at 1:38
@Ronnie: Well, first, that is what you did, so you should say so. Of course, that "educated guess" is not a proof. What I outline above is a proof that your educated guess is correct, after you perform the computation $10+10^2+10^3+\cdots+10^{30}$. – Arturo Magidin Dec 20 '10 at 2:39
HINT $\$ Exploit the linearity of $\Sigma\:$: $\rm\ \Sigma\ (f(k)+c)\ = \ \Sigma\ f(k) + \Sigma\ c\$ to reduce to a geometric sum. | 2014-08-01T12:16:09 | {
"domain": "stackexchange.com",
"url": "http://math.stackexchange.com/questions/14821/help-confirm-my-answer-for-the-following-question",
"openwebmath_score": 0.8455260992050171,
"openwebmath_perplexity": 301.64280314376316,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9572778024535095,
"lm_q2_score": 0.8705972566572503,
"lm_q1q2_score": 0.8334034286749066
} |
https://math.stackexchange.com/questions/825859/number-of-lines-determined-given-a-set-of-points | # Number of lines determined given a set of points
Consider the following set of points in the $x-y$ plane:
$$A=\{ (a,b)|a,b \in \mathbb{Z} \ and \ |a|+|b|\le 2\}$$
How to find the number of straight lines which pass through at least 2 points in $A$? One line is determined by 2 distinct points. To find the number of lines one is to choose any 2 points and find the possible combinations. The counting is easy if any 3 points are non-collinear, or only small number of points are collinear and the collinear points are non-collinear to the others. But there are a total of 13 points in the set, which gets complicated(at least for me) when there are different subsets of $A$ which consist of collinear points. Considering separate cases don't help much either.
Is there any way to count the number of lines determined by the points in the set above? And how should one find the number of lines determined when given larger size of set of points? Can anyone provide any hint and guidance? Thanks in advance.
The answer is 40 lines.
• I feel like an algorithm for an arbitrary set $A$ of points is not possible. Jun 8 '14 at 13:54
• How about the set above? Why an algorithm for an arbitrary set of points is not possible? Jun 8 '14 at 15:20
• For the set above the best algorithm I can come up with is ranging the slope through $n/m$ with $n,m \in \{0,1,2\}$ and counting the number of possible lines with these slopes. A general set can have an uncountable number of points so we cannot necessarily enumerate the points thus we cannot necessarily algorithmically count the number of lines. For certain sets you'll be able to count the number of lines (even with an uncountable number of elements) but this will definitely depend on the properties of the set. Jun 8 '14 at 15:23
• @DanZimm, wouldn't you need $n,m\in\{-3,-2,-1,0,1,2,3\}$ to cover the line from $(-2,0)$ to $(1,1)$? Jun 8 '14 at 15:30
• @Michael ah right, I meant to have the $\pm$ and forgot the $3$'s, my apologies Jun 8 '14 at 15:32
Let's consider the following grid (which represents your set $A$):
*
* * *
* * * * *
* * *
*
Now note we have $3$ vertical lines, $3$ horizontal lines, $5$ lines with slope $1$ and $5$ lines with slope $-1$. Together this is $16$ lines. Lets now iterate through other slopes. Note that if we can find the line for a positive slope, then there will also be the line for the negative slope because of symmetry. So to count our lines with positive slope lets start with the smallest possible slope and work upwards. To begin we have $2$ lines with slope $1 / 3$, next $4$ lines with slope $1 / 2$. Now if you notice theres another symmetry that allows us to deduce that there are $4$ lines with slope $2$ and $2$ lines with slope $3$. Adding these up we get $12$ lines, and then we get our $12$ negative slope lines and our originally $16$ lines. All together this is $40$ lines.
Hope this helps! Symmetry was the key factor to make this easy for me!
• Nice answer! But I got other answer from a member of AoPS which is more simple. Yours help nonetheless, I will post his answer here too. Jun 9 '14 at 8:13
• @DaveClifford ok, no problem, glad to help! Curious about the better answer! Jun 9 '14 at 8:17
• answer posted below! Jun 9 '14 at 8:22
There are $13$ points in total which consists of $2$ subsets of $5$ collinear points and $10$ subsets of $3$ collinear points.
Number of ways choosing any $2$ from $13$ points $={13 \choose 2}$
Number of extra lines in the 2 subsets of $5$ collinear points $= 2({5 \choose 2}-1)$
Number of extra lines in the 10 subsets of $10$ collinear points $= 10({3 \choose 2}-1)$
Thus the number of lines are ${13 \choose 2}-2({5 \choose 2}-1)-10({3 \choose 2}-1)=40$
All credits to AoPS member $mavropnevma$
• Ah very elegant indeed! I do like this :D Jun 9 '14 at 8:55
• can't agree more! Jun 9 '14 at 9:08 | 2022-01-27T08:30:55 | {
"domain": "stackexchange.com",
"url": "https://math.stackexchange.com/questions/825859/number-of-lines-determined-given-a-set-of-points",
"openwebmath_score": 0.6495435833930969,
"openwebmath_perplexity": 346.58684185024526,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9828232930001333,
"lm_q2_score": 0.8479677622198946,
"lm_q1q2_score": 0.8334024684229108
} |
https://math.stackexchange.com/questions/2550386/the-factorialth-root | # The 'Factorialth Root'
I was dealing with the following question, given by my friend:
Let $\xi(x)=\sqrt{x+\sqrt{x+\sqrt{x+\sqrt{\cdots}}}}$
Define the series $X$ as $\xi(1),\xi(2),\xi(3),\dots$
Find $n$ for which $\xi(n)$ is the 51st Whole Number in the series.
I solved it, of course, [and interestingly $\xi(1)={{1+\sqrt5}\over2}$, the Golden Ratio] but that led us on a competition in which we would try to find out the value of increasingly convoluted expressions.
Some time later, I made an expression, which I called 'The Factorialth Root', written as $\sqrt[!]{x}$.
For some $x$, $\sqrt[!]{x}=\sqrt{x\sqrt{(x-1)\sqrt{(x-2)\sqrt{\ddots\sqrt{2\sqrt1}}}}}$
My friend thought that $(x>y)\to(\sqrt[!]{x}<\sqrt[!]{y})$, while I thought the opposite, that $(x>y)\to(\sqrt[!]{x}>\sqrt[!]{y})$.
I showed by example that mine was correct, but couldn't prove it.
My attempt:
If $[(x>y)\to(\sqrt[!]{x}>\sqrt[!]{y})]$ is true, then $\sqrt[!]{x}>\sqrt[!]{x-1}$. This is possible only when $x>\sqrt[!]{x-1}$. It follows that $\sqrt[!]{2}>\sqrt[!]{1},\sqrt[!]{3}>\sqrt[!]{2}$, and so on.
So, I thought I could prove it by induction, but can't seem to find any way to apply it.
Can anyone help?
For $n > 1$,
\begin{align} \frac{\sqrt[!]{n}}{\sqrt[!]{n-1}} &= \frac{n^{1/2} \cdot (n-1)^{1/4} \cdot (n-2)^{1/8} \cdot\,\cdots\,\cdot 1^{1/2^{n\phantom{-1}}}}{\phantom{n^{1/2}\cdot}(n-1)^{1/2}\cdot(n-2)^{1/4}\cdot\,\cdots\,\cdot 1^{1/2^{n-1}}} \\[4pt] &= \frac{n^{1/2}}{(n-1)^{1/4}\cdot(n-2)^{1/8}\cdot\,\cdots\,\cdot 1^{1/2^n}} \\[4pt] &= \frac{n^{1/4+1/8+1/16+\cdots+1/2^{n}+1/2^{n}}}{(n-1)^{1/4}\cdot(n-2)^{1/8}\cdot\,\cdots\,\cdot 1^{1/2^n}} \\[4pt] &= \left(\frac{n}{n-1}\right)^{1/4}\left(\frac{n}{n-2}\right)^{1/8}\left(\frac{n}{n-3}\right)^{1/16}\cdot\,\cdots\,\cdot \left(\frac{n}{1}\right)^{1/2^{n}}\cdot n^{1/2^n} \\[4pt] &> 1 \cdot 1 \cdot 1 \cdot\,\cdots\,\cdot1 \cdot 1 \\[4pt] &= 1 \end{align}
Let's define $a_n=\sqrt[!]{n}$ for easier notation.
You have
$$a_n=\sqrt{n a_{n-1}}$$ in other words, the next term is the geometric mean of $n$ and previous term. This implies the order (because the mean is always between the values):
$$n>a_n>a_{n-1}\quad \text{if}\quad n>a_{n-1}$$ or $$n<a_{n}<a_{n-1}\quad \text{if}\quad n<a_{n-1}$$
Which one is it? It's easy to show that if the condition $n>a_{n-1}$ holds for one term, it will hold for all next terms also (this is almost trivial induction, because the ordering will say $n>a_n$ which of course implies $n+1>a_n$). You have $2>a_1=1$, so this proves it.
You can explore this further. You may notice that after $n=4$, you get $n-2<a_n<n-1$. In fact, you can show that $a_n$ comes arbitrarily close to $n-1$. Conjecture: $$\lim_{n\to \infty}\left[(n-1)-a_n\right]=0$$ Define the sequence of differences:
$$b_n=(n-1)-a_n$$ Next step: $$b_{n+1}=n-\sqrt{(n+1) a_n}$$ $$b_{n+1}=n-\sqrt{(n+1)(n-1-b_n)}=n-\sqrt{n^2-1-b_n(n+1)}=n(1-\sqrt{1-\frac{b_n}{n}-\frac{b_n+1}{n^2}})$$ Now first use the inequality $\sqrt{1+x}<1+\frac{x}{2}$ which you can easily prove by squaring. $$b_{n+1}<n(1-1+\frac{b_n}{2n}+\frac{b_n+1}{2n^2})=\frac{b_n}{2}+\frac{b_n+1}{2n}=b_{n}\left(\frac{1}{2}+\frac{1}{n}\right)+\frac{1}{2n}$$ For large $n$, the second term gets arbitrarily small and you can show that $b_{n}$ sequence decreases faster toward zero than any geometric series with factor $q>\frac{1}{2}$ after some suitably large $n$. Or, to put it differently, the ratio test shows convergence of $b_{n+1}$ toward zero:
$$\frac{b_{n+1}}{b_n}=\frac{1}{2}+\frac{1}{n}+\frac{1}{2nb_{n}}$$ Once you find one $b_n$ for which the right side of equation is $<1$, it holds forevermore. And you get there at $n=4$.
This means that for large $n$, you have asymptotic behaviour $a_n\asymp (n-1)$. | 2021-09-25T11:45:54 | {
"domain": "stackexchange.com",
"url": "https://math.stackexchange.com/questions/2550386/the-factorialth-root",
"openwebmath_score": 0.9997044801712036,
"openwebmath_perplexity": 206.76765647810407,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9828232919939075,
"lm_q2_score": 0.8479677622198947,
"lm_q1q2_score": 0.8334024675696639
} |
http://math.stackexchange.com/questions/800925/probability-question-probably-bayes-rule-and-total-probability-theorem | # Probability question - (Probably) Bayes' Rule and Total Probability Theorem
I just took a probability final exam and was fairly confident in my solution of 28/31, but I wanted to be sure... because according to http://www.stat.tamu.edu/~derya/stat211/SummerII02/Final.Summer02.doc which has it as the second question, the answer is .6627. What's discerning is that they have the decimal equivalent of 28/31 as one of their answers which makes it seem like they know something I don't...
"Seventy percent of all cattle are treated by an injected vaccine to combat a serious disease. The probability of recovery from the disease is 1 in 20 if untreated and 1 in 5 if treated. Given that an infected cow has recovered, what is the probability that the cow received the preventive vaccine?"
Here's my solution: Let A be the event a cow recovered, let B be the event a cow received the vaccine.
We are given:
P(A|B) = 1/5
P(A|~B) = 1/20
P(B) = 7/10
We want to find P(B|A), so use Bayes' rule and the total probability theorem to find
P(B|A) = P(A|B) x P(B) / (P(A|B) x P(B) + P(A|~B) x P(~B) ).
Plugging in the values from what's given above, we get (.2 x .7) / (.2 x .7 + .05 x .3) which gives 28/31.
If I'm wrong, I'd love to be pointed in the right direction haha
Thank you!
-
Note that $0.6627$ is not possible, the answer must be bigger than $70\%$. – André Nicolas May 18 '14 at 23:36
I disagree. The 70% is referring to the amount of cows treated, and the question is asking if a cow recovers what the probability she got the vaccine to begin with. If the problem said the probability of recovering is 0 for cows who did not receive the vaccine, then the solution would be 100% for what they're asking. – James May 18 '14 at 23:43
It seems we agree that since the probability of recovery is better if treated than if untreated, the answer must be greater than $70\%$. You have given a correct argument that it can be as high as $100\%$. – André Nicolas May 18 '14 at 23:47
-
Okay, let's see what's given:
$P(\text{recovered} \mid \text{vaccine}) = \frac{1}{5} = 0.2$
$P(\text{recovered} \mid \text{no-vaccine}) = \frac{1}{20} = 0.05$
$P(\text{vaccine} ) = 0.7$
$P(\text{no-vaccine} ) = 0.3$
$P(\text{vaccine} \mid \text{recovered} ) = ?$
The Bayes rule is:
$P(\text{vaccine} \mid \text{recovered} ) = \frac{P(\text{recovered} \mid \text{vaccine}) \times P(\text{vaccine} )}{P(\text{recovered} )}$ $$= \frac{0.2 \times 0.7}{P(\text{recovered} )}$$
where
$P(\text{recovered} ) = P(\text{rec} \mid \text{vac}) \times P(\text{vac} ) + P(\text{rec} \mid \text{no-vac}) \times P(\text{no-vac} ) \\ = 0.2 \times 0.7 + 0.05 \times 0.3 = 0.155$
plugin it back into the equation
$$= \frac{0.2 \times 0.7}{P(\text{recovered} )} \\ = \frac{0.2 \times 0.7}{0.155}\\ \approx 90.3\%$$
Which is exactly what you got. Hm, I am curious. Have you had a chance to talk to your teacher/prof about that?
- | 2015-08-31T22:52:13 | {
"domain": "stackexchange.com",
"url": "http://math.stackexchange.com/questions/800925/probability-question-probably-bayes-rule-and-total-probability-theorem",
"openwebmath_score": 0.8906112909317017,
"openwebmath_perplexity": 823.6079719737605,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9828232874658905,
"lm_q2_score": 0.8479677660619634,
"lm_q1q2_score": 0.8334024675061261
} |
http://math.stackexchange.com/questions/27324/on-calculating-int-01-ln1-x2-mathrm-dx-where-is-the-mistake | # On calculating $\int_0^1\ln(1-x^2)\;{\mathrm dx}$ — where is the mistake?
I've seen the integral $\displaystyle \int_0^1\ln(1-x^2)\;{dx}$ on a thread in this forum and I tried to calculate it by using power series. I wrote the integral as a sum then again as an integral. Here is my calculation:
\displaystyle \begin{aligned} \int_0^1 \ln\left(1-x^2\right)\;{\mathrm dx} & = -\int_0^1\sum_{k \ge 0}\frac{x^{2k+2}}{k+1}\;{\mathrm dx} = -\sum_{k \ge 0}\int_{0}^{1}\frac{x^{2k+2}}{k+1}\;{\mathrm dx} \\& = -\sum_{k \ge 0}\bigg[\frac{x^{2k+3}}{(k+1)(2k+3)}\bigg]_0^1 = -\sum_{k \ge 0}\frac{1}{(k+1)(2k+3)} \\& = -\sum_{k \ge 0}\frac{(2k+3)-2(k+1)}{(k+1)(2k+3)} = -\sum_{k\ge 0}\bigg(\frac{1}{k+1}-\frac{2}{2k+3}\bigg) \\& \color{blue}{= -\sum_{k\ge 0}\int_0^1\bigg(x^{k}-2x^{2k+2}\bigg)\;{\mathrm dx} = -\int_0^1\sum_{k\ge 0}\bigg( x^{k}-2x^{2k+2}\bigg)\;{\mathrm dx}} \\&= \int_0^1\frac{1}{1+x}-2\;{\mathrm dx} = \bigg[\ln(1+x)-2x\bigg]_0^1 = \ln{2}-2.\end{aligned}
I should have got $2\ln{2}-2$, not $\ln{2}-2$. I'm thinking that either I made a very silly mistake, or the blue step is wrong (probably the order cannot be switched, although I don't know why)!
-
+1 for "thinking outside the box", then showing some work. I would do +more if I could. – Ross Millikan Mar 16 '11 at 3:48
How did you go from $\sum(x^k-2x^{k+2})$ to $\frac{1}{1+x}-2$? If I think about $\sum x^k$, this converges to $\frac{1}{1-x}$; and $\sum x^{2k+2} = \sum(x^2)^{k+1}$ converges to $\frac{1}{1-x^2}-1 = \frac{x^2}{1-x^2}$. So shouldn't the entire thing thing go to $\frac{1}{1-x}-\frac{2x^2}{1-x^2} = \frac{1+x-2x^2}{1-x^2}$? – Arturo Magidin Mar 16 '11 at 3:52
@Arturo Magidin: Yes -- and that's $\frac{(1+2x)(1-x)}{(1+x)(1-x)}=\frac{1+2x}{1+x}=\frac{1}{1+x}-2$. – joriki Mar 16 '11 at 3:54
@Arturo $-\int_0^1 \frac{1}{1-x} dx - \int_0^1 \frac{2x^2}{1-x^2} dx$ $= -\int_0^1 \frac{1}{1-x} dx - \int_0^1 2\left(-1 - \frac{1}{2 (-1 + x)} + \frac{1}{2 (1 + x)}\right)dx$ I am too eager to know what is wrong with OPs approach – Please Delete Account Mar 16 '11 at 3:57
@StudentOfMaths, @joriki, @Approximist: Okay, thanks. – Arturo Magidin Mar 16 '11 at 3:57
## 3 Answers
This is quite interesting. I think the problem does lie in the blue line. The sum over $\frac{1}{k+1}-\frac{2}{2k+3}$ converges absolutely, but the sums over the two terms individually diverge. What you're doing when you combine the two series as you do is effectively to move the terms $\frac{2}{2k+3}$ back to twice the $k$ value from where they belong, and that's a different series; you can only do that when the two terms absolutely converge individually.
What you're effectively calculating by "slowing down" $k$ in the second term is
$$-\sum_{k\ge0}\left(\frac{1}{2k+1}+\frac{1}{2k+2}-\frac{2}{2k+3}\right)\;,$$
and that is indeed $\ln 2 - 2$.
[Update]
The above explanation refers to the series above the blue line to show that exchanging the sum and integral in the blue line can't be valid. But of course that should also be decidable from just the blue line itself.
First off, note that actually the first step after the blue line isn't quite right, either: At $x=1$, the sum in the integrand actually diverges to $-\infty$, so the value $-3/2$ of the integrand in the last line for $x=1$ is obtained by rearranging a divergent series. But that problem only occurs at $x=1$; for all other values of $x$, the series converges absolutely and may validly be arranged to obtain the integrand in the last line. But this discontinuity at $x=1$ already shows that the partial sums don't converge uniformly. Indeed
$$-\sum_{k=0}^{n-1}\left(x^k-2x^{2k+2}\right)=-\frac{1-x^n}{1-x}+2x^2\frac{1-x^{2n}}{1-x^2}=\frac{1-x^n}{1-x}\left(-1+2x^2\frac{1+x^n}{1+x}\right)\;,$$
which for large $n$ is close to the integrand in the last line for most $x$ but close to $x=1$ suddenly becomes positive and ends up at $n$ for $x=1$, as it must. Here's a plot for $n=100$.
Here's a very nice general treatment of the interchange of limits based on uniform convergence, which can be applied to integrals, derivatives and sums alike. Since the series in the integrand diverges at $x=1$, we have to treat the integral as a limit taking the upper integration limit to $1$, and this limiting operation and the one in the series can only be interchanged if the uniform convergence criterion is fulfilled.
(Here's another example where an integral and a function limit can't be interchanged due to lack of uniform convergence; this one is a bit simpler because there's no sum involved.)
-
Good observation! I hadn't noticed that the series was being rearranged, which if it does not converge absolutely allows one to give it an arbitrary value. – Alex Becker Mar 16 '11 at 6:12
Many thanks, Joriki. Part of what was bugging me is that the same exact method works for the sister integral $\int \ln(1+x^2)\;{dx}$ and gives its value correctly as $\ln{2}+\frac{1}{2}\pi-2$. Now I see why it works for it (in its 'blue step', the sums over the two terms individually converge). – Lyrebird Mar 16 '11 at 23:26
@Alex @StudentOfMaths @Approximist @Arturo Magidin: I expanded the answer to explain the failure of the interchange in the blue line from just the blue line by itself, without reference to the series it's trying to evaluate. – joriki Mar 19 '11 at 19:48
Why not try splitting $\ln(1-x^2)$ as $\ln(1-x)+\ln(1+x)$ before applying the Mercator series?
-
That's exactly what was done in the first line, canceling the odd order terms. Those series are less convergent than the original, so I don't think this is progress. – Ross Millikan Mar 16 '11 at 3:43
\begin{eqnarray*} \int_{0}^{1}\ln\left(1 - x^{2}\right)\,{\rm d}x & = & \int_{0}^{1}\sum_{\sigma = \pm}\ln\left(1 + \sigma\,x\right)\,{\rm d}x = \sum_{\sigma = \pm}\int_{\sigma}^{1 + \sigma}\ln\left(\sigma\,x\right)\,{\rm d}x \\ & = & \sum_{\sigma = \pm}\sigma\int_{1}^{1 + \sigma}\ln\left(x\right)\,{\rm d}x = \sum_{\sigma = \pm}\sigma\,\left\lbrace% \left.x\ln\left(x\right)\vphantom{\Large A}\right\vert_{1}^{1 + \sigma} - \int_{1}^{1 + \sigma}x\,{1 \over x}\,{\rm d}x \right\rbrace \\ & = & \sum_{\sigma = \pm}\sigma\left\lbrack\vphantom{\Large A}% \left(1 + \sigma\right)\ln\left(1 + \sigma\right) - \sigma\right\rbrack = \sum_{\sigma = \pm}\left\lbrack\vphantom{\Large A}% \left(\sigma + 1\right)\ln\left(1 + \sigma\right) - 1\right\rbrack \\ & = & \left\lbrace\underbrace{\quad\lim_{x \to 0}\left\lbrack\vphantom{\Large A} x\ln\left(x\right)\right\rbrack\quad}_{=\ 0} - 1\right\rbrace\ +\ \left\lbrace\left(1 + 1\right)\ln\left(1 + 1\right) - 1\vphantom{\Huge A}\right\rbrace \\[1cm]&&\mbox{} \end{eqnarray*}
$$\int_{0}^{1}\ln\left(1 - x^{2}\right)\,{\rm d}x = {\large 2\ln\left(2\right) - 2}$$
-
This does not answer the question "where is the mistake?" – Did Aug 18 '13 at 17:30
The "blue one" is not correct. The integral of $\left(1 - x^{2}\right)^{-1}$ diverges. The trick involves two harmonic series and they have to be splited to acomplish the "wrong task". Indeed, they involve digamma functions. – Felix Marin Aug 18 '13 at 17:51 | 2016-02-06T19:02:17 | {
"domain": "stackexchange.com",
"url": "http://math.stackexchange.com/questions/27324/on-calculating-int-01-ln1-x2-mathrm-dx-where-is-the-mistake",
"openwebmath_score": 0.9840527176856995,
"openwebmath_perplexity": 798.5672217990559,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9828232874658905,
"lm_q2_score": 0.847967764140929,
"lm_q1q2_score": 0.8334024656180887
} |
https://www.transtutors.com/questions/suppose-that-y-1-t-and-y-2-t-both-satisfy-the-homogeneous-linear-constant-coeffi-cie-763818.htm | # Suppose that y 1 (t) and y 2 (t) both satisfy the homogeneous linear constant-coeffi cient... 1 answer below »
Suppose that y 1 (t) and y 2 (t) both satisfy the homogeneous linear constant-coeffi
cient differential equation (LCCDE)
dy(t) + ay(t)
= 0
dt
Show that y 3 (t) = ayi(t) + 3y 2 (t), where a and # are any two constants, is also a
solution to the homogeneous LCCDE.
P6.2
In this problem, we consider the homogeneous LCCDE
d 2 yt + 3 dy(t) + 2y(t) = 0
dt 2
dt
(P6.2-1)
(a) Assume that a solution to eq. (P6.2-1) is of the form y(t) = es'. Find the qua
dratic equation that s must satisfy, and solve for the possible values of s.
(b) Find an expression for the family of signals y(t) that will satisfy eq. (P6.2-1).
P6.3
Consider the LCCDE
dy(t) + 1 y(t) = x(t),
2
dt
x(t) = e- t u(t)
(P6.3-1)
(a) Determine the family of signals y(t) that satisfies the associated homogeneous
equation.
(b) Assume that for t > 0, one solution of eq. (P6.3-1), with x(t) as specified, is of
the form
y 1 (t) = Ae-,
t > 0Suppose that y 1 (t) and y 2 (t) both satisfy the homogeneous linear constant-coeffi
cient differential equation (LCCDE)
dy(t) + ay(t)
= 0
dt
Show that y 3 (t) = ayi(t) + 3y 2 (t), where a and # are any two constants, is also a
solution to the homogeneous LCCDE.
P6.2
In this problem, we consider the homogeneous LCCDE
d 2 yt + 3 dy(t) + 2y(t) = 0
dt 2
dt
(P6.2-1)
(a) Assume that a solution to eq. (P6.2-1) is of the form y(t) = es'. Find the qua
dratic equation that s must satisfy, and solve for the possible values of s.
(b) Find an expression for the family of signals y(t) that will satisfy eq. (P6.2-1).
P6.3
Consider the LCCDE
dy(t) + 1 y(t) = x(t),
2
dt
x(t) = e- t u(t)
(P6.3-1)
(a) Determine the family of signals y(t) that satisfies the associated homogeneous
equation.
(b) Assume that for t > 0, one solution of eq. (P6.3-1), with x(t) as specified, is of
the form
y 1 (t) = Ae-,
t > 0
Suppose that y 1 (t) and y 2 (t) both satisfy the homogeneous linear constant-coeffi
cient differential equation (LCCDE)
dy(t) + ay(t)
= 0
dt
Show that y 3 (t) = ayi(t) + 3y 2 (t), where a and # are any two constants, is also a
solution to the homogeneous LCCDE.
P6.2
In this problem, we consider the homogeneous LCCDE
d 2 yt + 3 dy(t) + 2y(t) = 0
dt 2
dt
(P6.2-1)
(a) Assume that a solution to eq. (P6.2-1) is of the form y(t) = es'. Find the qua
dratic equation that s must satisfy, and solve for the possible values of s.
(b) Find an expression for the family of signals y(t) that will satisfy eq. (P6.2-1).
P6.3
Consider the LCCDE
dy(t) + 1 y(t) = x(t),
2
dt
x(t) = e- t u(t)
(P6.3-1)
(a) Determine the family of signals y(t) that satisfies the associated homogeneous
equation.
(b) Assume that for t > 0, one solution of eq. (P6.3-1), with x(t) as specified, is of
the form
y 1 (t) = Ae-,
t > 0
Attachments:
rohit s
We substitute ya(t) = ay 1 (t) + Oy 2 (t) into the homogeneous differential equation dya(t) dt dt + ay 3 (t) = d d [ay 1 (t) + #y 2 (t)] + a[ayi(t) + #y 2 (t)] dt Since differentiation is distributive, we can express the preceding equation as ady t) +0dy2t) + aay1(t) ddty ddty + afly2(t) aa _ di( dt + ay1(t) I + dt dt + ay2(t) ay() However, since both y 1 (t) and y 2 (t) satisfy the homogeneous differential equation, the right side of the equation is zero. Therefore, dy 3 (t) + ay3(t) = 0 cit S6.2 (a) We are assuming that y(t) = es'. Substituting in the differential equation yields d 2 d d (es') + 3 (e t ) + 2e" = 0 so that s 2 e" t + 3se' + 2e" = es'(s 2 + 3s + 2) = 0 For any finite s, e" is not zero. Therefore, s must satisfy 0 = s 2 + 3s + 2 = (s + 1)(s + 2), s = -1, -2 (b) From the answer to part (a), we know that both yi(t) = e-' and y 2 (t) = e-2 satisfy the homogeneous LCCDE. Therefore, ys(t) = Kie~' + K 2 e 2, for any constants K 1 , K 2 , will also satisfy the equation. We substitute ya(t) = ay 1 (t) + Oy 2 (t) into the homogeneous differential equation dya(t) dt dt + ay 3 (t) = d d [ay 1 (t) + #y 2 (t)] + a[ayi(t) + #y 2 (t)] dt Since differentiation is distributive, we...
## Plagiarism Checker
Submit your documents and get free Plagiarism report
Free Plagiarism Checker
## Recent Questions in Electrical Engineering
Looking for Something Else? Ask a Similar Question | 2021-01-21T02:06:53 | {
"domain": "transtutors.com",
"url": "https://www.transtutors.com/questions/suppose-that-y-1-t-and-y-2-t-both-satisfy-the-homogeneous-linear-constant-coeffi-cie-763818.htm",
"openwebmath_score": 0.8850847482681274,
"openwebmath_perplexity": 2629.1769034054223,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9828232889752297,
"lm_q2_score": 0.8479677622198946,
"lm_q1q2_score": 0.8334024650099223
} |
https://math.stackexchange.com/questions/4241539/min-k-s-t-z-1z-2-cdotsz-n-geq-frac1kz-1z-2-cdotsz-n | # min $k$ s.t. $|z_1+z_2+\cdots+z_n|\geq \frac{1}{k}(|z_1|+|z_2|+\cdots+|z_n|).$
Find the smallest positive real number $$k$$ such that, given any finite set $$z_1,\cdots, z_n$$ of complex numbers, all with strictly positive real and imaginary parts, the following inequality holds: $$|z_1+z_2+\cdots+z_n|\geq \frac{1}{k}(|z_1|+|z_2|+\cdots+|z_n|).$$
Answer- $$\sqrt{2}$$
My Attempt:
First, we take $$n=2$$. Let $$z_i=r_ie^{i\theta_i}$$ for $$i=1, 2$$. Then $$|z_1+z_2|^2=|r_1e^{i\theta_1}+r_2e^{i\theta_2}|^2= r_1^2+r_2^2+r_1r_2e^{i(\theta_1-\theta_2)}+r_1r_2e^{i(\theta_2-\theta_1)}.$$ Also $$|z_1|+|z_2|=r_1+r_2.$$ Therefore, the given inequality holds if
$$r_1^2+r_2^2+r_1r_2e^{i(\theta_1-\theta_2)}+r_1r_2e^{i(\theta_2-\theta_1)}\geq \frac{1}{k^2}(r_1+r_2)^2$$ $$\implies (k^2-1)(r_1^2+r_2^2)+r_1r_2(k^2 e^{i(\theta_1-\theta_2)}+k^2e^{i(\theta_2-\theta_1)}-2)\geq 0.$$ which holds if $$k^2(e^{i(\theta_1-\theta_2)}+e^{i(\theta_2-\theta_1)})\geq 2$$
• It is not difficult to show that the inequality holds with $k=\sqrt 2$. But I doubt that this is the smallest value if $n \ge 3$. Sep 4, 2021 at 10:09
• @MartinR Actually it holds for $k=\sqrt{2}$. Sep 4, 2021 at 11:07
• @YiorgosS.Smyrlis: Yes, that is what I said. It is also easy to see that this is the best possible constant in the case of two numbers. My misunderstanding was that I thought that the question asks for the best constant for a fixed number $n$ (which looks more difficult to me if $n \ge 3$). Sep 4, 2021 at 11:35
• This should also directly follow from minkowski or holder if you rewrite the question in terms of its positive parts. Sep 15, 2021 at 14:22
Set $$w_k=e^{-i\pi/4}z_k$$. Then $$w_k=|z_k|e^{i\theta_k}$$, where $$\theta_k\in (-\pi/4,\pi/4)$$, and $$\cos\theta_k>1/\sqrt{2}$$. Thus $$\mathrm{Re}\,w_k>\frac{1}{\sqrt{2}}|w_k|.$$ Then $$\left|\sum_{k=1}^n z_k\right|=\left|\sum_{k=1}^n w_k\right|\ge\sum_{k=1}^n \mathrm{Re} \,w_k =\sum_{k=1}^n |w_k|\cos\theta_k >\frac{1}{\sqrt{2}}\sum_{k=1}^n |w_k|=\frac{1}{\sqrt{2}}\sum_{k=1}^n |z_k|.$$
$$k=\sqrt{2}$$ is the smallest.
If $$z_1=1+i\varepsilon$$ and $$z_2=\varepsilon+i$$, $$\varepsilon>0$$, then
$$|z_1+z_2|=(1+\varepsilon)\sqrt{2}, \qquad |z_1|+|z_2|=2\sqrt{1+\varepsilon^2}, \quad \frac{|z_1+z_2|}{|z_1|+|z_2|}=\frac{1}{\sqrt{2}}\cdot\frac{1+\varepsilon}{\sqrt{1+\varepsilon^2}}$$ and $$\inf_{\varepsilon>0}\frac{|z_1+z_2|}{|z_1|+|z_2|}=\frac{1}{\sqrt{2}}.$$
• I think what you tried to mean is the following $:$ \begin{align*} \left |\sum\limits_{k=1}^{n} z_k \right | & = \left |\sum\limits_{k=1}^{n} w_k \right | \\ & \geq \text {Re} \left ( \sum\limits_{k=1}^{n} w_k \right ) \\ & = \sum\limits_{k=1}^{n} \text {Re}\ (w_k) \\ & \gt \frac {1} {\sqrt {2}} \sum\limits_{k=1}^{n} |w_k| \\ & = \frac {1} {\sqrt {2}} \sum\limits_{k=1}^{n} |z_k|. \end{align*} Done! BTW very nice answer.+1 | 2022-07-01T17:59:33 | {
"domain": "stackexchange.com",
"url": "https://math.stackexchange.com/questions/4241539/min-k-s-t-z-1z-2-cdotsz-n-geq-frac1kz-1z-2-cdotsz-n",
"openwebmath_score": 0.9997292160987854,
"openwebmath_perplexity": 358.66338487476025,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9828232884721166,
"lm_q2_score": 0.8479677622198947,
"lm_q1q2_score": 0.8334024645832987
} |
http://qupw.accademiakravmagaitalia.it/moment-of-inertia-of-quarter-circle.html | # Moment Of Inertia Of Quarter Circle
The moment of inertia of the shape is given by the equation. In both cases, it is calculated with a multiple integral over. Articles by Vincent Teoh on Muck Rack. bending moment [Nm, lb ft] r Y. The rod is at rest when a 3. 5 Moment of inertia of : (a) semicircle, and (b) quarter circle. The two disks are then linked together without the aid of any external torques, so that $2. (5), the moment of inertia depends on the axis of rotation. Moment of inertia of a circular section is same around both centriodal axis. Informally, it is the "average" of all points of. Problem 13P. Moment of Inertia about centre of mass of each bar ( Perpendicular to plane of Hexagon) , 2) Then, Moment of inertia of each bar about centre of Regular Hexagon. Because the ring is hollow, all of its mass has to sit at a distance R from the center; hence, you have =R 2 and I = MR 2. Own work assumed (based on copyright claims). In any case, the bullet stops in the block due to the large frictional forces. 36 106 mm4 kx = 32. A box is shown in the figure below. Moment of Inertia is always positive. To make a semicircle, take any diameter of the circle. The prefix semi-comes from Latin, and it means half or partly (like in words such as semi-permanent, semi-formal, semifinals). Engineering Fundamentals: CENTROID, AREA, MOMENTS OF INERTIA, POLAR MOMENTS OF INERTIA, & RADIUS OF GYRATION OF A CIRCLE. Show that the straight rod, with an appropriate twist is a possible equilibrium configuration for all values of Q, and calculate the value of twist. And so when we think about area, we know that the area of a circle, the area of a circle is equal to pi times the radius of the circle squared. 00 m/s2 along their frictionless inclines (See figure below). Moment of inertia of a circular section is same around both centriodal axis. Is there anybody can help me to find a way to do that? Specially,. Area moment of inertia of a filled quarter circle with radius r entirely. The V-Drive series of clutches from Quarter Master® features an open design for cleaner and cooler operations. 672614E-27 kg. Force, given by Eq. The equation of the circle of radius one centered at the origin is x2 +y2 = 1. sectional area 31. Centroid of a quarter circle y Let us consider a quarter circle with radius r. Example: A uniform thin rod of length 0. 5 (C) 15 (D) 30 (E) 36. And so when we think about area, we know that the area of a circle, the area of a circle is equal to pi times the radius of the circle squared. 1-1/8" x 10 2 OR 3 DISC CLUTCHES. Moment of inertia, denoted by I, measures the extent to which an object resists rotational acceleration about a particular axis, and is the rotational analogue to mass. ) The more inertia a body has, the harder it is to change its linear motion. Problem 13P. In any case, the bullet stops in the block due to the large frictional forces. It turns out that the new moment of inertia is drastically increased compared to the centroidal one. The Area Moment of Inertia or second moment of area has a unit of dimension length 4, and should not be confused with the mass moment of inertia. This engineering calculator will determine the section modulus for the given cross-section. travnja 2006. 1: Determine angular displacement. In this derivation, we have to follow certain steps. First Moment of Area, Moment of Inertia, Polar Moment of Inertia Quarter-circular area 4 Volume of hemisphere 2 3 Ar Volume of hemisphere 2 3Vr. The moment of inertia is the mass of the object times the mass-weighted average of the squared distance from the axis. The Brick Solid block adds to the attached frame a solid element with geometry, inertia, and color. baseball bat in a quarter of a circle swing to a Which has the greater moment of inertia a ring (of mass M, radius R) rotating about its center. At some point you're better off building it in Pro-E, and letting IT tell you that the simple model was within 10% :) Another approach is to determine the rotational inertia of the part experimentally using the pendulum. Since those are lengths, one can expect that the units of moment of inertia should be of the type:. The price of this PAU, including solid-state power supply and controller is$15,000. 21 × 10 6 = 1. Thus, as we see from the figure, the radius of the slab is p 1−y2. Units of Moment of Inertia are length raised to the fourth power, such as in4 or m4. bending moment [Nm, lb ft] r Y. Each calculator is associated with web pageor on-page equations for calculating the sectional properties. THE QUARTER MASTER 5. Now if take a quarter circle, the moment of inertia relative to the x-axis and y-axis will be one quarter the moment inertia of a full circle. The moment of inertia of a point mass is. Rectangle Triangle Circle Semicircle Quarter Circle Ellipse Recommended For You. A similar idea is the aerodynamic center which is the point on an airfoil where the pitching moment produced by the aerodynamic forces is constant with angle of attack. 42 mm; Y = 35. The slope at quarter span section is. Moment of inertia of a circular section is same around both centriodal axis. The moment of inertia of the complete disc about an axis passing through its centre O and perpendicular to its plane is I 1 = 2 9 M R 2 Now, the moment of inertia of the disc with removed portion I 2 = 2 1 M (3 R ) 2 = 1 8 1 M R 2 Therefore, moment of inertia of the remaining portion of disc about O is I = I 1 − I 2 = 9 2 M R 2 − 1 8 M R 2. The optimum gear ratio, n∗ can be calculated, using Eq. A similar idea is the aerodynamic center which is the point on an airfoil where the pitching moment produced by the aerodynamic forces is constant with angle of attack. 5 (C) 15 (D) 30 (E) 36. bending moment [Nm, lb ft] r Y. Answer: The moment of inertia of a rod of mass and length about an axis, perpendicular to its length, which passes through one of its ends is (see question 8. Rent or Buy Mechanics of Materials: An Integrated Learning System, 2nd Edition - 9780470565148 by Timothy A. Know the basic steps. What you’ll notice is that the statical moment of area above the neutral axis is equal to that below the neutral axis! [math] {Q}_{x,top}={Q}_{x,bottom} [math] Of course you don’t need to do all these calculations manually because you can use our fantastic Free Moment of Inertia Calculator to find the statical moment of area of beam sections. 9 MN,F V = 153. I am having a lot of trouble finding a model or data to model a cross section of the fuselage at location staion 2163. Problem 10. 6 Polar Moment of Inertia233. 2 31 Oct Shear and Bending Moment Diagrams* (*) videos PL4 2 Nov Design of Prismatic Beams for Bending 5. Imagine trying to swing a ball tied to a short string around your head, now imagine trying to swing the same ball but tied to a 10 foot string. 8 MN) Problem 2. With a solid disk, the mass is spread out. The carousel itself (without riders) has a moment of inertia of 125 kg·m2. Bending Moment in The Beam: Integrating a second time: The bending moment is zero at the free end of the beam ν’’(L) = 0 Therefore C 2 = 0 and the equation simplifies to Slope and Deflection of the Beam: The third and fourth integration yield The boundary conditions at the fixed support, where the slope and. If an object rotates 90 degrees, all the points in an object, save those on the axis of rotation, transcribe a quarter of a circle. The moment of inertia of part of a circle can be found by using the trigonometric function sine and the angle of the circle segment in question. Determine the moment of inertia of the flywheel required to keep the speed within the range 1% above to 1% below the mean speed. Answers to Odd Numbered. 43 × 10 6 − 2 × 0. 67 103 mm3 Y 30. 7 x 10"1 kg. Area moment of inertia of a filled quarter circle with radius r entirely in the 1st quadrant of the Cartesian coordinate system, with. Area of Trapezoid = 0. The price of this PAU, including solid-state power supply and controller is $15,000. The moment of inertia calculates the rotational inertia of an object rotating around a given axis. and is called the moment of inertia of the body about that particular axis of rotation. 5 Moment of inertia of : (a) semicircle, and (b) quarter circle. units is kg-m? A. You might remember that the area of a circle equals πr 2, which is the same as π x r x r. Moment of inertia of a circular section can be calculated by using either radius or diameter of a circular section around centroidal x-axis or y-axis. the axis of rotation, therefore the moment of inertia has been reduced and the skater spins faster in order to conserve angular momentum. The two disks are then linked together without the aid of any external torques, so that$2. Each exercise provides a drawing of the circle as well as the length of either the radius or the diameter. The moment of inertia of the complete disc about an axis passing through its centre O and perpendicular to its plane is I 1 = 2 9 M R 2 Now, the moment of inertia of the disc with removed portion I 2 = 2 1 M (3 R ) 2 = 1 8 1 M R 2 Therefore, moment of inertia of the remaining portion of disc about O is I = I 1 − I 2 = 9 2 M R 2 − 1 8 M R 2. 34 kg m 2 for a 60 kg woman in the position designed by Balanchine, as shown below: We have included an exercise for calculating the beginning position for the classical Russian technique in the workbook. Radius of gyration 3. dA= ipds or 2dA=pds torque T = 2q dA. To make a semicircle, take any diameter of the circle. Polar Moment of Inertia The second moment of Area A with respect to the pole O or the z-axis. Multiply pi over four by the difference between both radii taken to the fourth power. In this video I will find the moment of inertia (and second moment of area) I(C. If you missed Problem Session on Thu. moment of inertia has increased. There is a plate with moment of inertia Ip = 0. Define following terms 1. It is I = 1 Y2b(Y)dY section where y is measured vertically and b(y) is the width of the section at y. The lower electrodes are quarter-circle shape, so their center of gravity are at the position of 4 2 3 I m is the mass moment of inertia of the mirror;. If you're unsure about what deflection actually is, click here for a deflection definition Below is a concise beam deflection table that shows how to calculate the maximum deflection in. Moment of Inertia is constant. 12-Circular segment. The first moment of an area with respect to a line of symmetry is zero. Engineering Fundamentals: CENTROID, AREA, MOMENTS OF INERTIA, POLAR MOMENTS OF INERTIA, & RADIUS OF GYRATION OF A CIRCLE. Answer: Ix = 7. p ” is the second moment of inertia of the pile and “ ” is a power that varies from 0. 21 × 10 6 = 1. Find the moment of inertia of a circular section whose radius is 8" and diameter of 16". )=? rotating around the C. integrating this over the quarter circle gives you a moment of inertia of m r^2, so now we have to find the value of r in terms of l. The speed at which the points in a rotating object transcribe a circle is known as angular velocity, and a change in that speed is known as angular acceleration. 005 ; proton mass (proton rest mass) = 1. A particle describes a horizontal circle of radius r in a funnel type vessel of frictionless surface with half cone angle (as shown in figure). The polar moment of inertia for the shaft is J — (75 mm)4 4. UNIVERSITYOFILLINOIS May31 191 3 THISISTOCERTIFYTHATTHETHESISPREPAREDUNDERMYSUPERVISIONBY ArthurMoultonSimpson _ ENTITLEDAstu. Solving this for x yields x = ± p 1−y2. Quarter Circle: Definition. Find the center of mass of a two-dimensional plate that occupies the quarter circle $$x^2+y^2\le1$$ in the first quadrant and has density $$k(x^2+y^2)$$. v - — sin 20. The moment K measures the resistance of the section to twisting. Also referred to as the third quarter moon. The second moment of area, also known as moment of inertia of plane area, area moment of inertia, polar moment of area or second area moment, is a geometrical property of an area which reflects how its points are distributed with regard to an arbitrary axis. the origin. It is a measure of the cross sectional area of the barrel. 4𝑅𝑅 3𝜋𝜋) Note: Attendance (+2 points), Format (+1 point) Water Air Air. a) its moment of inertia about a leg; b) its polar moment of inertia about the right-angle vertex; c) its moment of inertia about the hypotenuse. The links will open a new browser window. The second moment of area for a shape is easier to be calculeted with respect to a parallel axis or with respect to a perpendicular axis. (d)total angular momentum and moment of inertia about the axis of rotation. the axis of rotation, therefore the moment of inertia has been reduced and the skater spins faster in order to conserve angular momentum. Since point A is at c 75 mm, B 1. Quarter Circle: Definition. It helps to consider the removed corner as a quarter circle subtracted from an r x r square (see figure below). determine the moment 59. The moment of inertia of a point mass is. Hemmingsen assumed (based on copyright claims). ⇒ The ratio of the moment of inertia of a circular plate and that of a square plate for equal depth, is. The 2 nd moment of area, or second area moment and also known as the area moment of inertia, is a geometrical property of an area which reflects how its points are distributed with regard to an arbitrary axis. To best see the graphs, set the scale of the torque graph to show a range of 20 to 20. Find the moment that the tension cable causes about the page 76 origin using a cross product. Determine the moment of inertia about its diametrical axis. Remove one half of the circle along that diameter. Multiply pi over four by the difference between both radii taken to the fourth power. Ixx= Iyy= πR^4/16 Jo= πR^4/8: Term. dA= ipds or 2dA=pds torque T = 2q dA. 67 103 mm3 Y 30. A sewer pipe rolls more slowly down an incline than a bowling ball with the same mass. Find the moment at A given the two forces at B and C. Moment of Inertia is always positive. Various such parameters include centre of gravity, moment of inertia, centroid , first and second moment of inertias of a line or a rigid body. A constrained buoy experiencing vortex-induced, in-line and transverse angular motions and designed to absorb and attenuate the energies of streams, rivers and localized ocean currents is described. frame ABC is a quarter circle. An area is symmetric with respect to an axis BB’ if for every point Pthere exists a point P’such that PP’ is. If you're unsure about what deflection actually is, click here for a deflection definition Below is a concise beam deflection table that shows how to calculate the maximum deflection in. Answer: Ix = 7. The following links are to calculators which will calculate the Section Area Moment of Inertia Properties of common shapes. The second moment of area is typically denoted with either an or with a. For racers looking for a clutch that offers the best balance of performance, durability and affordability for the money, look no further. The second moment of area, also known as moment of inertia of plane area, area moment of inertia, polar moment of area or second area moment, is a geometrical property of an area which reflects how its points are distributed with regard to an arbitrary axis. inches 4; Area Moment of Inertia - Metric units. The Following Section consists of Strength of Materials Questions on Physics. 10-Half of Ellipse. Calculate the Polar Moment of Inertia of a Circle Quarter; Calculate the Radius of Gyration of a Circle Quarter; Calculate the Elastic Section Modulus of a Circle Quarter; Calculate the Plastic Section Modulus of a Circle Quarter "Good engineers don't need to remember every formula; they just need to know where they can find them. Calculate the moment of inertia of each of the following uniform objects about the axes indicated. The moment of inertia about an axis is a measurement of how difficult it is to accelerate the body about that axis. If area is in m2 and the length is also in m, the moment of inertia is expressed in m4. Processing. of a semi circle about centroidal axis is Ix = 0. This engineering calculator will determine the section modulus for the given cross-section. If area in mm2 and the length is also in mm, then moment of inertia is expressed in mm4. "The fromula is stress = Moment*centroid/Inertia. 7-Half of a circle. 5 Moment of inertia of : (a) semicircle, and (b) quarter circle. Click the Moment of Inertia Tab at the top. The second moment of area, also known as area moment of inertia, is a geometrical property of an area which reflects how its points are distributed with regard to an arbitrary axis. Own work assumed (based on copyright claims). y - moment or Inertia or the gate about the liney y, h is the depth Of the centrold below the surface, and A is the. (Filename:tfigure3. Moment of inertia of a circle. In addition, these clutches significantly reduce the moment of inertia (MOI) for lightweight power transfer. This point is a pproximately 30% of length from forward when steaming ahead and about 20% – 25% of length from the stern when the ship is going astern. The overall moment of inertia of your composite body is simply the sum of all of the adjusted moments of inertia for the pieces, which will be the sum of the values in the last column (or. The equilateral triangle actually makes the strongest column for a given area, but not by much (12% stronger than the circle). 2001/03/01. Determine the moment of inertia of a filled. As can be see from Eq. 2 mm and height same as the radius of the quarter circle. 44 in4 kx = 1. which is the sum of all the elemental particles masses multiplied by their distance from the rotational axis squared. (see figure) 1) What is the moment of inertia of this half about an axis perpendicular to its plane through point A? 2) Why did your answer in part A come out the same as if this were a complete disk of mass M? 3) What would be the moment of inertia of a quarter disk of mass M and radius R about an axis. A column will fail about the axis of smallest moment of inertia of area. 3)2 = 9722 in4 (40. The moment of inertia of part of a circle can be found by using the trigonometric function sine and the angle of the circle segment in question. The moment of inertia of total area A with respect to z axis or pole O is z dI z or dI O or r dA J 2 I z ³r dA 2 The moment of inertia of area A with respect to z axis Since the z axis is perpendicular to the plane of the area and cuts the plane at pole O, the moment of inertia is named “polar moment of inertia”. •The moments and product of inertia for an area are plotted as shown and used to construct Mohr’s circle, •Mohr’s circle may be used to graphically or analytically determine the moments and product of inertia for any other rectangular axes including the principal axes and principal moments and products of inertia. The following links are to calculators which will calculate the Section Area Moment of Inertia Properties of common shapes. 26 Oct Moment of Inertia* Appendix A & (*) videos PL3 6 29 Oct Moment of Inertia 30 Oct Shear and Bending Moment Diagrams 5. 821 Rectangle minus semi-circle | Moment of Inertia. moment of inertia 87. of a hollow circular section is π (D4 – d4 ) / 32 or π( R4 – r 4) /2 • M. of a semi circle about centroidal axis is Ix = 0. Angular momentum is the vector product of (a) linear momentum and radius vector moment of inertia and angular velocity (c) linear momentum and angular velocity (d) linear velocity and radius vector 8. Figure to illustrate the area moment of a quartercircle at the list of moments of inertia. A beam with more than simple supports is a continuous beam. For the isolated system of platform and person, the angular momentum is conserved. Calculate the moment of inertia of a rod 0. Elite snowboarders are skilled. Drift Angle is the angle between ships fore and aft line & the tangent to turning circle at any given moment. The moment of inertia about an axis is a measurement of how difficult it is to accelerate the body about that axis. Moment of inertia of a circular section can be calculated by using either radius or diameter of a circular section around centroidal x-axis or y-axis. inches 4; Area Moment of Inertia - Metric units. Moment of inertia of quarter circular area about x and y axes is I x = I y = ${{{R^4}} \over 16}$ (8. sectional area 31. the inertia factor for such a wing must be known as a function of the width. This is a more general characteristic. A bar magnet of magnetic moment m and moment of inertia I (about centre, perpendicular to length) is cut into two equal pieces, perpendicular to length. The second moment of area, also known as area moment of inertia, is a geometrical property of an area which reflects how its points are distributed with regard to an arbitrary axis. In addition, these clutches significantly reduce the moment of inertia for lightweight power transfer. Moment of inertia of a circle. Rotational inertia is inversely proportional to the object’s speed. Understand this and more by learning about Moment of Inertia, a measure of how compact objects are. v - — sin 20. Mass Moment of Inertia I = L r2 dm Quarter and semicircle arcs C y x r 4r 3p — 4r 3p — A ˜ 1 pr 2 4 Quarter circle area I x = 1 16 πr4 I y = 1 16 πr4 b h a. The moment of inertia for any small piece of a solid would be its mass element times the square of a distance to the x axes because that will be the radius of a trajectory. PART 3: Moment of Inertia a. The above concept can be extended to obtain the moment of inertia of semicircular and quarter circular area as given below. I am having a lot of trouble finding a model or data to model a cross section of the fuselage at location staion 2163. Center or 1st moment ∫ = M xdm MXC & ∫ = M ydm MYC (XC & YC: Center of Mass) ∫ A xdA = xc & ∫ A ydA = yc (xc & yc: Center of Area) Moment of inertia or 2nd moment ∫ = M r2dm I (2nd moment of Mass) ∫A y dA =Ix 2 & ∫A x dA =Iy 2 (2nd moment of Area) Then, FR =γAyc sinθ=(γ hc)A where γ hc: Pressure at the centroid. r = distance from the pole (or the z-axis) to AA (Pythagorean's Theorem, right triangle) Since r2 = x2 + = (x2 + AA = 1+1. 14-Semi Parabola. What is the moment of inertia of. "The fromula is stress = Moment*centroid/Inertia. l represents 1/4 of the circumference of a circle of radius r, so we have. 6 mm); Iw = 1, + AJc1 = 3753 + 212. Angular Momentum Any moving body has inertia. Using a single integral we were able to compute the center of mass for a one-dimensional object with variable density, and a two dimensional object with constant density. As can be see from Eq. 248 Chapter 6 Circles and Area 6. Thus k= 15. Problem 10. The moment of inertia = I = πR 4 /8 In the case of a quarter circle the expression is given as: [Image will be Uploaded Soon] The moment of inertia = I = πR 4 /16. (Hint: Moment of inertia for a rectangle 𝐼𝐼= 𝑏𝑏ℎ. A larger diameter barrel will have a higher moment of inertia value and as a result will be stiffer. Area moment of inertia of a filled quarter circle with radius r entirely. 50 s to reach the bottom of the incline. Moment of resistance is a term in structural engineering. but when i compare it to output from autocad, it doesn't match. Strength of Materials - Strength of Materials is an important subject to understand the behavior of objects under stress. The moments of inertia for some common shapes can be found using the following formulas. 2 6 Nov Shear Stress in Beams. - Theory - Example - Question 1 - Question 2 - List of moment of inertia for common shapes. 75 m in length and mass 1. 100 (transparent) Escapement wheel jewels: Rubifix (transparent) Three-position stem: manual winding, date adjustment, time setting. The links will open a new browser window. Get an answer for 'Q. To find the moment at any point, remember that the bending moment is the algebraic sum of all moments of the forces to the left of the section under consideration and that moment is the value of force times distance. cated by a filled circle. The moment of inertia calculates the rotational inertia of an object rotating around a given axis. The unit of dimension of the second moment of area is length to fourth power, L 4 , and should not be confused with the mass moment of inertia. Moment of Inertia of Thin Rectangular Plate. If you're unsure about what deflection actually is, click here for a deflection definition Below is a concise beam deflection table that shows how to calculate the maximum deflection in. By Gerson Washiski Barbosa. z 6’ 3’ y 5’ x T=1kip (10, 0, -7) ft. v - — sin 20. Moment of Inertia vs. The second moment of an area of a semicircle about the axis x is. Conversely, the moment of inertia about a base diameter axis can be computed by adding one1-. There are two types of Moment of Inertia: 1) Area Moment of Inertia 2) Mass Moment of Inertia 1) Area Moment of Inertia: Area Moment of Inertia is defined for plane figures (i. You should find it's impossible. r = distance from the pole (or the z-axis) to AA (Pythagorean's Theorem, right triangle) Since r2 = x2 + = (x2 + AA = 1+1. Centroid, Area, Moments of Inertia, Polar Moments of Inertia, & Radius of Gyration of a Quarter Circular Cross-Section. Since those are lengths, one can expect that the units of moment of inertia should be of the type:. If either, or both ends of a beam projects beyond the supports, it is called a simple beam with overhang. Key to deflection diagrams and symbols. Using a single integral we were able to compute the center of mass for a one-dimensional object with variable density, and a two dimensional object with constant density. The moment of inertia of the system of masses that includes Max, Maya, and the merry-go-round is calculated by adding the moments of inertia of the individual masses. The slope at quarter span section is. 10) As the arm extends the effective load inertia increases from 0. MOMENT OF. Thus, as we see from the figure, the radius of the slab is p 1−y2. 1: Determine angular displacement. V-Drive V-DRIVE FEATURES: • Open design allowing clean and cool running • High torque capacity • Solid and durable design. 3: A quarter and a penny are on a turntable. A uniform disk of radius R is cut in half so that the remaining half has mass M. 43 × 10 6 − 2 × 0. The closer the concentration of mass to the axis, the smaller the torque required to get it spinning at the same rate about that axis. 2 2 May Shear Stress in Beams 3 May Shear Stress in Beams. It helps to consider the removed corner as a quarter circle subtracted from an r x r square (see figure below). units is kg-m? A. The centroid of an object in -dimensional space is the intersection of all hyperplanes that divide into two parts of equal moment about the hyperplane. Moment of inertia of a circular section is same around both centriodal axis. 672614E-27 kg. Bassett's 15 x 10 Inertia Advantage wheel is the lightest five-on-five inch bolt pattern wheel on the market and utilizes a patented low Moment of Inertia center design, top quality control and high strength steel. com for more math and science lectures! In this video I will find the moment of inertia (and second moment of area), I(x)=?, I(y). Philpot (Missouri University of Science and Technology, Rolla, Missouri) for as low as $1. Understand this and more by learning about Moment of Inertia, a measure of how compact objects are. 2001/03/01. A box is shown in the figure below. Last Quarter - The phase of the Moon when it is 90 degrees away moon from a line between the Sun and the Earth, measured westward from the Sun, as seen from the north. The mean speed of an engine is 250 r. Moment of inertia can be thought of as a measure of how difficult it is to get something to rotate, and is defined for a particle as. determine the product 19. (Hint: Moment of inertia for a rectangle 𝐼𝐼= 𝑏𝑏ℎ. product of inertia 23. The following is a list of second moments of area of some shapes. 5 Parallel-Axis Theorem Find the centroid ȳ of the quarter circle. There are hundereds of calculators listed on the website that help students and engineers across Electrical Engineering, Mechanical Engineering, Civil Engineering, Physics, Math and many other sectors. Lecture Notes: Area-Moment. travnja 2006. triangle rectangle parabola cubic parabola. 28 ) ms s F kgm N m m == 12 a FI rr =. If you missed Problem Session on Thu. The calculation of the additional moment. Figure to illustrate the area moment of a quartercircle at the list of moments of inertia. The moment of inertia of a body is a measure of the resistance the body offers to any change in its angular velocity. The handout is available here; ask your classmates for any clarifications and then check in with faculty as needed. Conservation of energy now requires that. Let T be the period of oscillations of the original magnet about an axis through the mid point, perpendicular to length, in a magnetic field B. Radius of gyration 3. symm) Systems of systems and composite objects Another way of interpreting the formula r cm = r 1 m1 + r 2 2 +··· m1 +m2 +···. Thus, as we see from the figure, the radius of the slab is p 1−y2. s T = 2qA. 7 Angular momentum is the vector product of a) linear momentum and radius vector b) moment of inertia and angular vector. Problem 10. ” is the second moment of inertia of a circle 1 m diameter and “ E. cated by a filled circle. The moment of inertia of a point mass is. Hemmingsen assumed (based on copyright claims). determine. 2 26 April Shear Force and Bending Moment* (*) videos PL4 28 April Shear Force and Bending Moment 6 1 May Shear Stress in Beams 7. Problem 10. Plug it into the ellipse area formula: π x r x r!. , Options are ⇒ (A) moment of inertia, (B) polar moment of inertia, (C) polar modulus, (D) modulus of rigidly. 2 31 Oct Shear and Bending Moment Diagrams* (*) videos PL4 2 Nov Design of Prismatic Beams for Bending 5. 2 mm and height same as the radius of the quarter circle. The second moment of area, also known as moment of inertia of plane area, area moment of inertia, polar moment of area or second area moment, is a geometrical property of an area which reflects how its points are distributed with regard to an arbitrary axis. It is a measure of the cross sectional area of the barrel. In such cases, an axis passing through the centroid of the shape is probably implied. Suddenly, the same sheet of paper has considerable stiffness. The following is a list of area moments of inertia. Mass moments of inertia have units of dimension ML 2 ([mass] × [length] 2 ). c) The quarter of a solid circular cylinder of radius 1 and height 2 lying in the first octant, with its central axis the interval 0 ≤ y ≤ 2 on the y-axis, and base the quarter circle in the xz-plane with center at the origin, radius 1, and lying in the first quadrant. Imagine trying to swing a ball tied to a short string around your head, now imagine trying to swing the same ball but tied to a 10 foot string. To best see the graphs, set the scale of the torque graph to show a range of 20 to 20. ) The more inertia a body has, the harder it is to change its linear motion. 2nd MOMENT of AREA. G The centroid and centre of gravity are at the same point Where centre of gravity consider to be whole mass of an object act at a point C. The moment of inertia of the complete disc about an axis passing through its centre O and perpendicular to its plane is I 1 = 2 9 M R 2 Now, the moment of inertia of the disc with removed portion I 2 = 2 1 M (3 R ) 2 = 1 8 1 M R 2 Therefore, moment of inertia of the remaining portion of disc about O is I = I 1 − I 2 = 9 2 M R 2 − 1 8 M R 2. moment equilibrium about the shaft’s axis. Quarter Circle: Definition. product of inertia of area = moment of inertia of area (moment of inertia of area) product of inertia of mass = moment of inertia (moment of inertia) product of inertia of volume = moment of inertia of volume (moment of inertia of volume) proof = 0. This means that ω 2 = ω 1 /1,000,000! The astronauts have angular velocity of about 1 revolution per second and you know perfectly well that the fidget spinner did not have a speed of a million revolutions per second. 75 to 2 kg m 2. Find the moment that the tension cable causes about the page 76 origin using a cross product. The Area Moment of Inertia or second moment of area has a unit of dimension length 4, and should not be confused with the mass moment of inertia. Moment of inertia of quarter circular area about x and y axes is I x = I y = ${{{R^4}} \over 16}$ (8. A quarter circle of radius 432 mm lies on Quadrant II of a coordinate plane with reference radius at the origin. 2 31 Oct Shear and Bending Moment Diagrams* (*) videos PL4 2 Nov Design of Prismatic Beams for Bending 5. 08 mm Find the centroid of the shaded area shown in fig, obtained by cutting a semicircle of diameter 100mm from the quadrant of a circle of radius 100mm. Moment of Inertia of Thin Rectangular Plate. Figure 3: Hydrostatic force at a curved dam (Ans. less than one equal to one more than one equal to 3Ï€/16 ⇒ The moment diagram for a cantilever carrying linearly varying load from zero at its free end and to maximum at the fixed end will be a. What if we tried to find the area of a circle as though it were an ellipse? We would measure the radius in one direction: r. Thus, as we see from the figure, the radius of the slab is p 1−y2. So even though many people think there's an outward centrifugal force that's just naturally occurring on an object going in a circle, there is not. Rectangle Triangle Circle Semicircle Quarter Circle Ellipse Recommended For You. 2 Triangle 900 40 50 36000 45000 Rectangle 2400 30 20 72000 48000 aX = aY = a= 2043. Pivot Point is a point about which a ship pivots in a turning circle. 0353 kg m2 , on the plate, center to center of plate and they rotate with same angular speed later. The following is a list of centroids of various two-dimensional and three-dimensional objects. Learn vocabulary, terms, and more with flashcards, games, and other study tools. The carousel itself (without riders) has a moment of inertia of 125 kg·m2. 1TEAM Smart Sheets What: 1TEAM Smart Sheets a software company that develops helpful templates for the science, technology, engineering and mathematics (STEM) community which includes professionals, researchers, consultants, professors, students, instructors, teachers, engineers, and technologists. If the permissible bending stress is 120N/mm 2, find a) The safe point load that can be applied at the centre of the span. The points X’and Y’corresponding to the x’and y’axes are obtained by rotating CX and CY counterclockwise through an angle θ 2(60o. The speed at which the points in a rotating object transcribe a circle is known as angular velocity, and a change in that speed is known as angular acceleration. The moment of inertia of a hollow cylinder can be calculated with the help of the formula, I=0. 9 MN,F V = 153. Example: A uniform thin rod of length 0. A quarter circle of radius 227 mm lies on Quadrant II of a coordinate plane with reference radius at the origin. Right: A circle section positioned as per the upper sketch is defined in the calculator as I x-axis, the lower sketch shows I y-axis. Problem 10. product of inertia of area = moment of inertia of area (moment of inertia of area) product of inertia of mass = moment of inertia (moment of inertia) product of inertia of volume = moment of inertia of volume (moment of inertia of volume) proof = 0. Engineering Fundamentals: CENTROID, AREA, MOMENTS OF INERTIA, POLAR MOMENTS OF INERTIA, & RADIUS OF GYRATION OF A CIRCLE. The overall moment of inertia of your composite body is simply the sum of all of the adjusted moments of inertia for the pieces, which will be the sum of the values in the last column (or. Section modulus second moment of area a d simply supported beam deflection circle quarter geometric properties radius of gyration a quarter circle a hollow cylinder with radius r andArea Moment Of Inertia. 60 A pair of smith’s tongs is an example. Moment Of Inertia Of A Quarter Circle. Author: No machine-readable author provided. The equation of the circle of radius one centered at the origin is x2 +y2 = 1. " you need to worry about hoop stress from pressure. The slope at quarter span section is. This course covers the Module 4 of the subject “Elements of Civil Engineering & Engineering Mechanics“ It consists of two units Centroid and Moment of Inertia. The centroid of an object in -dimensional space is the intersection of all hyperplanes that divide into two parts of equal moment about the hyperplane. Own work assumed (based on copyright claims). the radius of the mohr's circle will be A)45 N Polar moment of inertia of a solid circular shaft diameter D is equal to. The moment of inertia of semi-circular plate of radius Rand mass Mabout axis AB in its plane passing through centre:- A) (MR^2)/2 B) (MR^2)/4cos^2theta C) `(MR^2. What if we tried to find the area of a circle as though it were an ellipse? We would measure the radius in one direction: r. In such cases, an axis passing through the centroid of the shape is probably implied. Set the Moment of Inertia Graph to show a range of 2 kg m 2 to – 2 kg m. Rotational Inertia for a solid (continuous mass distribution) object I=Σmr2 Rotational inertia depends on the choice of axis of rotation, r. )=? rotating around the C. View Homework Help - Statics Reference Sheet from ME 2560 at Western Michigan University. The above equations for the moment of inertia of circle, reveal that the latter is analogous to the fourth power of circle radius or diameter. Using the structural engineering calculator located at the top of the page (simply click on the the "show/hide calculator" button) the following properties can be calculated: Calculate the Area of a Circle Quarter; Calculate the Perimeter of a Circle Quarter; Calculate the Centroid of a Circle Quarter. , (E) , Leave your comments or Download question paper. 2 2 May Shear Stress in Beams 3 May Shear Stress in Beams. This engineering data is often used in the design of structural beams or structural flexural members. The moment of inertia of the system of masses that includes Max, Maya, and the merry-go-round is calculated by adding the moments of inertia of the individual masses. The second moment of area, also known as area moment of inertia, is a geometrical property of an area which reflects how its points are distributed with regard to an arbitrary axis. The maximum fluctuation of energy generated in the engine is 850 ft. Date: 23 April 2006 (original upload date) Source: No machine-readable source provided. Learn vocabulary, terms, and more with flashcards, games, and other study tools. The more rotational inertia a body has, the harder it is change its rotation. The V-Drive series of clutches from Quarter Master® features an open design for cleaner and cooler operations. The moments of inertia for a cylindrical shell, a disk, and a rod are MR2, , and respectively. A constrained buoy experiencing vortex-induced, in-line and transverse angular motions and designed to absorb and attenuate the energies of streams, rivers and localized ocean currents is described. Semi Circular Area. Search For Search Search. The moment of inertia of the area A with respect to the y-axis is given by The moment of inertia of the area A with respect to the origin O is given by (Polar moment of inertia) Areas Moment of Inertia is the property of a deformable body that determines the moment needed to obtain a desired curvature about an axis. Understand this and more by learning about Moment of Inertia, a measure of how compact objects are. downwards 3. 00 x 10"4 m/3s. Next video in this series can be seen at. 1-1/8" x 10 2 OR 3 DISC CLUTCHES. (08 Marks) (c) Radius of a circle is 2 cm. The gear ratio has limiting values of 19 and 31, given the range of the inertia. 25 kg and the kid catches it. Both pins are subjected to double shear. Let us consider an elemental area ‗dA‘ with centroid ‗g‘ as shown in fig. Meidensha Electric Manufacturing Company, Ltd. What is the centroidal distance along the line of symmetry XO for the quarter circle shown below? (where D= 4r/3π) a. 7 x 10"1 kg. A rectangle is attached to the quarter circle at Quadrant I with base at the x-axis of width equal to 14. The volume of a sphere is 4πr 3 /3. Rent or Buy Mechanics of Materials: An Integrated Learning System, 2nd Edition - 9780470565148 by Timothy A. So even though many people think there's an outward centrifugal force that's just naturally occurring on an object going in a circle, there is not. THE QUARTER MASTER 5. If mass of the particle is m, then in dynamical equilibrium the speed of the particle must be (a) tan rg v (b) v rg tan N mg r. Thin Rod in the Shape of a Quarter Circle Moment of inertia is not a physical quantity such as velocity, acceleration or force, but it enables ease of calculation; it is a function of the geometry of the area. 065 kgm2 (including the dart) and that the force M acts perpendicular to the forearm. The moment of inertia about an axis is a measurement of how difficult it is to accelerate the body about that axis. Autor: No machine-readable author provided. Our model relies on the ability of the snowboarder to keep proper balance despite the curvature of the jump. 437 10 mm 4. 97 107 mm4 Shear Stress. The V-Drive series of clutches from Quarter Master® features an open design for cleaner and cooler operations. In this lesson, we will learn how to derive the formula as well as its application in problems. 8b) give the moment of inertia of circle about its centroidal axes. Now if take a quarter circle, the moment of inertia relative to the x-axis and y-axis will be one quarter the moment inertia of a full circle. a) The moment of inertia of the system about the z axis and b) The rotational energy of the system. The second moment of area, also known as moment of inertia of plane area, area moment of inertia, polar moment of area or second area moment, is a geometrical property of an area which reflects how its points are distributed with regard to an arbitrary axis. 2 & (*) videos PL3 5 24 April Moment of Inertia 25 April Shear Force and Bending Moment 6. The inertia at the point that's one third back from the tip is always three quarters of the mass, and if you use the point about 21% back from the tip, where the impact point and pivot point are equidistant from the balance point, the inertia is exactly one half the. 6 The moment of inertia of a disc having mass M and radius R, about an axis passing through its centre and perpendicular to its plane is a) 1 2 2 MR b) MR2 c) 1 2 4 MR d) 5 2 4 3. How to calculate the moment/product of inertia for a quarter-circular area? Anyone know Ixy, in particular? Ix and Iy I can get from the table in my statics book. The second moment of area for a shape is easier to be calculeted with respect to a parallel axis or with respect to a perpendicular axis. 12-Circular segment. 59 If Z and I are the section modulus and moment of inertia of the section, the shear force F and bending moment M at a section are related by (A) F = My/I (B) F = M/Z (C) F = dM/dx (D) F Mdx Answer: Option C Question No. Shear and Moment Diagram: representations of value of the moment at all points along the beam. Rotational inertia depends on the choice of the axis of rotation. 4 Area Moment of Inertia of Circle, Semicircle, and Quarter Circle217 5. 2 2 May Shear Stress in Beams 3 May Shear Stress in Beams. The following is a list of centroids of various two-dimensional and three-dimensional objects. However, the part of the circle rotating about an axis will be symmetric and the values will be equal for both the y and x-axis. The moment of inertia of a point mass is. 8 mg•cm 2, angle of lift 53° Frequency: 28,800 vph (4 Hz) Balance spring: elinvar by Nivarox® Shock protection: INCABLOC 908. 60 A pair of smith’s tongs is an example. 4: Torque on Pulley Due to the Tension of Two Strings. mm 10 solutions 19. There is a plate with moment of inertia Ip = 0. In this lesson, we will learn how to derive the formula as well as its application in problems. Moment of inertia of quarter circular area about x and y axes is I x = I y = ${{{R^4}} \over 16}$ (8. P a r t A A thin 3. For the derivation of the moment of inertia formula of a circle, we will consider the circular cross-section with the radius and an axis passing through the centre. THE QUARTER MASTER 5. Centroid of a quarter circle y Let us consider a quarter circle with radius r. It is based not only on the physical shape of the object and its distribution of mass but also the specific configuration of how the object is rotating. Disagree 5. The second moment of area, also known as area moment of inertia, is a geometrical property of an area which reflects how its points are distributed with regard to an arbitrary axis. The second moment of area, also known as moment of inertia of plane area, area moment of inertia, polar moment of area or second area moment, is a geometrical property of an area which reflects how its points are distributed with regard to an arbitrary axis. (Filename:tfigure3. The moment of inertia on a quarter circle is giver by: Ix=(pi*r^4)/16 So using Steiner's theorem to calculate the MoI of the Quarter circle on the main figure's centroid we get: Ix'=(pi*r^4)/16 + dy * (pi*r^2)/4 where dy is the difference between yG of the quarter circle and yG of the main figure. 1: Determine angular displacement. Key to deflection diagrams and symbols. What if we tried to find the area of a circle as though it were an ellipse? We would measure the radius in one direction: r. CENTRE OF GRAVITY. l = 1/4*2 pi r or r=2l/pi. Moment of inertia of a circle. 01 18-Jun-2003 1. As the moment of inertia increases, the angular velocity must decrease. parallel to the x-axis, of a semi-circle. Mass Moment of Inertia I = L r2 dm Quarter and semicircle arcs C y x r 4r 3p — 4r 3p — A ˜ 1 pr 2 4 Quarter circle area I x = 1 16 πr4 I y = 1 16 πr4 b h a. where the angular rotation of the frame is expressed by the vector Ω pointing in the direction of the axis of rotation, and with magnitude equal to the angular rate of rotation Ω, symbol × denotes the vector cross product, vector x B locates the body and vector v B is the velocity of the body according to a rotating observer (different from the velocity seen by the inertial observer). The formula for the moment of inertia of a ring created by two overlapping circles is similar. Calculate the location of centroid of quarter cylinder of length 100cm, radius= 30cm with its base on xy plane and axis along Z. We also assume that the snowboarder’s moment of inertia is constant, he approaches the jump standing, and takes o standing. Two types of force distributions are allowed to be chosen, equally distributed and triangularly distributed. Stack Exchange network consists of 177 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. 28 ) ms s F kgm N m m == 12 a FI rr =. MOMENT OF. Units of Moment of Inertia are length raised to the fourth power, such as in4 or m4. Disagree 5. Put another way, take a bunch of long thin rectangles of the same size cut out of paper and try to arrange them into an approximate segment of a circle without overlapping them. •Moment of Inertia and Center of Gravity •Newton’s Second Law for Rotation SOLVE The particle on the left travels one-quarter of a full circle. The force can be calculated by simply rearranging. Put another way, take a bunch of long thin rectangles of the same size cut out of paper and try to arrange them into an approximate segment of a circle without overlapping them. As shown below. Only rectangular and round solid sections are considered here. Get an answer for 'Q. Start studying Moment of Inertia. (original upload date) Izvor: No machine-readable source provided. Known contaminant lines from Nd, Nd, and Sm are indicated by a filled triangle. Centre of Pressure. In both cases, it is calculated with a multiple integral over. 11-Circular sector. 9 MN,F V = 153. (Filename:tfigure3. Own work assumed (based on copyright claims). The moment of inertia is not related to the length or the beam material. This engineering calculator will determine the section modulus for the given cross-section. y - moment or Inertia or the gate about the liney y, h is the depth Of the centrold below the surface, and A is the. To best see the graphs, set the scale of the torque graph to show a range of 20 to 20. Moment of Inertia Of a Quarter Circle: Moment Of Inertia Of Circle Derivation. In such cases, an axis passing through the centroid of the shape is probably implied. Moment of inertia of a triangular section of base (b) and height (h) about an axis passing through its C. Polar Moment Of Inertia Circular Beam October 31, 2018 - by Arfan - Leave a Comment 6 torsion of shafts 4 161 beam161 explicit 3 d beam up19980821 polygon geometric properties solved 6 pts derive the equation for polar moment o. 5x10 7 mm 4. dy y d/2 d/2 b Fig. 5 Parallel-Axis Theorem Find the centroid ȳ of the quarter circle. For racers looking for a clutch that offers the best balance of performance, durability and affordability for the money, look no further. Own work assumed (based on copyright claims). Each calculator is associated with web pageor on-page equations for calculating the sectional properties. The tools are here, though, to calculate the rotational inertia of any complicated combination of bits you care to build up. It is found from the moment of inertia and the distance from the outside of the object concerned to its major axis. 036 for very soft soils: 0. Also supports coordinate-oriented input and language setup. 3 106mm4 Sample Problem 9. 14-Semi Parabola. Moment of inertia of a circular section can be calculated by using either radius or diameter of a circular section around centroidal x-axis or y-axis. Known contaminant lines from Nd, Nd, and Sm are indicated by a filled triangle. A child sits on a merry-go-round at the position marked by the red circle (position is given in meters and time is given in seconds). Visit http://ilectureonline. 00 at eCampus. You carefully put a ring with moment of inertia Ir = 0. The moment of inertia of a body is a measure of the resistance the body offers to any change in its angular velocity. 42 mm; Y = 35. Philpot (Missouri University of Science and Technology, Rolla, Missouri) for as low as$1. (d)total angular momentum and moment of inertia about the axis of rotation. Ellipse: Definition. Find the moment of inertia of the thick shell about an axis through the centre of the sphere. Because the ring is hollow, all of its mass has to sit at a distance R from the center; hence, you have =R 2 and I = MR 2. (Hint: Moment of inertia for a rectangle 𝐼𝐼= 𝑏𝑏ℎ. A playground carousel is free to rotate about its center on frictionless bearings, and air resistance is negligible. Moment of inertia of a circular section can be calculated by using either radius or diameter of a circular section around centroidal x-axis or y-axis. A particle describes a horizontal circle of radius r in a funnel type vessel of frictionless surface with half cone angle (as shown in figure). First Moment of Area, Moment of Inertia, Polar Moment of Inertia Quarter-circular area 4 Volume of hemisphere 2 3 Ar Volume of hemisphere 2 3Vr. z 6’ 3’ y 5’ x T=1kip (10, 0, -7) ft. The 2 nd moment of area, or second area moment and also known as the area moment of inertia, is a geometrical property of an area which reflects how its points are distributed with regard to an arbitrary axis. y2 Iyy ?dA. Cone Calc Processing :. Section modulus second moment of area a d simply supported beam deflection circle quarter geometric properties radius of gyration a quarter circle a hollow cylinder with radius r andArea Moment Of Inertia. 5 MeV for the Nd, Sm, and Gd bands com-pared to the down. 2 y h b dy b Fig. frame ABC is a quarter circle. 97 107 mm4 Shear Stress. You might remember that the area of a circle equals πr 2, which is the same as π x r x r. Hi, I would like to find a way in AutoCAD to calculate the (momet of inertia). *Applications:Circle Track, Road Racing Quarter Master’s V-Drive clutches feature a one-piece billet aluminum cover with an open leg design providing a low moment of inertia, and clean and cool running. The moment of inertia contribution of the right arm is therefore approximately 0. Centroid of a circle is very easy to determine. The inertia at the point that's one third back from the tip is always three quarters of the mass, and if you use the point about 21% back from the tip, where the impact point and pivot point are equidistant from the balance point, the inertia is exactly one half the. With a solid disk, the mass is spread out. The V-Drive series of clutches from Quarter Master® features an open design for cleaner and cooler operations. How to derive the formula for moment of inertia of a disc about an axis passing through its centre and perpendicular to its plane? Can you please explain the sams with a figure drawn?😅. Let ‗y‘ be the distance of centroid ‗g‘ from x. If mass of the particle is m, then in dynamical equilibrium the speed of the particle must be (a) tan rg v (b) v rg tan N mg r. sectional area 31. Put another way, take a bunch of long thin rectangles of the same size cut out of paper and try to arrange them into an approximate segment of a circle without overlapping them. Drift Angle is the angle between ships fore and aft line & the tangent to turning circle at any given moment. If area in mm2 and the length is also in mm, then moment of inertia is expressed in mm4. Also supports coordinate-oriented input and language setup. Moment of inertia of a circle or the second-moment area of a circle is usually determined using the following expression; I = π R 4 / 4 Here, r is the radius and the axis is passing through the centre. In addition, these clutches significantly reduce the moment of inertia for lightweight power. The moment of inertia of total area A with respect to z axis or pole O is z dI z or dI O or r dA J 2 I z ³r dA 2 The moment of inertia of area A with respect to z axis Since the z axis is perpendicular to the plane of the area and cuts the plane at pole O, the moment of inertia is named “polar moment of inertia”. ) moment of inertia of a plane area about any axis is second moment of area about that axis. The formula for the moment of inertia of a ring created by two overlapping circles is similar. How to calculate the moment/product of inertia for a quarter-circular area? Anyone know Ixy, in particular? Ix and Iy I can get from the table in my statics book. Determine the horizontal and vertical components of the hydrostatic force against the dam and the point CP where the resultant strikes the dam. It seems clear that because of the symmetry of both the region and the density function (both are important!), $$\bar x=\bar y$$. Rotational inertia is inversely proportional to the object’s speed. Mass Moment of Inertia I = L r2 dm Quarter and semicircle arcs C y x r 4r 3p — 4r 3p — A ˜ 1 pr 2 4 Quarter circle area I x = 1 16 πr4 I y = 1 16 πr4 b h a. If mass of the particle is m, then in dynamical equilibrium the speed of the particle must be (a) tan rg v (b) v rg tan N mg r. The prefix semi-comes from Latin, and it means half or partly (like in words such as semi-permanent, semi-formal, semifinals). has proposed their Model TW-55 for motorcycle testing. 25 kg and the kid catches it. 00 m/s2 along their frictionless inclines (See figure below). When one person is standing on the carousel at a distance of 1. This is because it has almost no moment of inertia around the bending axis (ie, all of its mass is on the bending axis). This course covers the Module 4 of the subject “Elements of Civil Engineering & Engineering Mechanics“ It consists of two units Centroid and Moment of Inertia. There are two types of Moment of Inertia: 1) Area Moment of Inertia 2) Mass Moment of Inertia 1) Area Moment of Inertia: Area Moment of Inertia is defined for plane figures (i. Moment of Inertia. The first moment of an area with respect to a line of symmetry is zero. This engineering calculator will determine the section modulus for the given cross-section. The Brick Solid block adds to the attached frame a solid element with geometry, inertia, and color. The moment of inertia about an axis through a vertex is 0. 100 (transparent) Escapement wheel jewels: Rubifix (transparent) Three-position stem: manual winding, date adjustment, time setting. G The centroid and centre of gravity are at the same point Where centre of gravity consider to be whole mass of an object act at a point C. where I is the rotational inertia or moment of inertia given by I = mL 2, and α is the angular acceleration. | 2020-10-30T22:40:25 | {
"domain": "accademiakravmagaitalia.it",
"url": "http://qupw.accademiakravmagaitalia.it/moment-of-inertia-of-quarter-circle.html",
"openwebmath_score": 0.6420783400535583,
"openwebmath_perplexity": 611.8841540734246,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9828232935032462,
"lm_q2_score": 0.8479677564567913,
"lm_q1q2_score": 0.8334024631854222
} |
https://www.khanacademy.org/math/old-ap-calculus-ab/ab-derivative-rules/ab-quotient-rule/v/quotient-rule-example | If you're seeing this message, it means we're having trouble loading external resources on our website.
If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.
Quotient rule
# Worked example: Quotient rule with table
AP.CALC:
FUN‑3 (EU)
,
FUN‑3.B (LO)
,
FUN‑3.B.2 (EK)
Given the values of f and f' at x=-1 and that g(x)=2x³, Sal evaluates the derivative of F(x)=f(x)/g(x) at x=-1.
## Want to join the conversation?
• How can this rule be derived from the product rule?
• The quotient rule can be derived from the product rule by writing f(x)/g(x) as f(x) * 1/g(x), and using the product, power, and chain rules when differentiating. (Note that 1/g(x) = [g(x)]^(-1).)
d/dx of [f(x)/g(x)] = (d/dx) of [f(x) * 1/g(x)] = f(x) * d/dx of [1/g(x)] + [1/g(x)] * f'(x)
= f(x) * (-1)[g(x)]^(-2) * g'(x) + f'(x)/g(x)
= -f(x)g'(x)/[g(x)]^2 + g(x)f'(x)/[g(x)]^2
= [g(x)f'(x) - f(x)g'(x)] / [g(x)]^2.
• At first, I understood this as f(x)/g(x) * g(x)/g(x). but Why MINUS??
• Basically, the intuition for the quotient rule comes from knowing both the product and chain rules. Here's a video explaining it, in the chain rule section.
However, I am still wondering why this is before the chain rule section. In every video from this point onwards (as far as I know) any time Sal mentions the quotient rule he just says one COULD use it here, but he prefers to just write it was f(x) * g(x)^-1 instead of f(x)/(gx), thus he ends up using the product rule.
Why is this ahead of the chain rule when it really doesn't need to be?
Don't get me wrong, I absolutely love Khan Academy, it is my favourite website ever, I'm just trying to point out some stuff to help them improve.
• I have tried doing this using the product rule instead of the quotient but it just keeps giving me the answer as -2 ! Where am I going wrong?
f'(x) * g^-1 (x) + f(x) * (g^-1 (x))' =
5 * (-2)^-1 + 3 * (6)^-1 =
5/-2 + 3/6 =
-15/6 + 3/6 = -12/6 = -2
• You go wrong when you evaluate (g⁻¹ (x))'.
You need to apply the chain rule because (g⁻¹ (x))' is not simply (g'(x))⁻¹
It is -(g(x))⁻² · g'(x)
So we'd have f'(x) · g⁻¹(x) + f(x) · (-(g(x))⁻² · g'(x)) =
5 · (-2)⁻¹ + 3 · (-(-2)⁻²) · 6)) =
5/-2 - 3·6/4
= -10/4 - 18/4 = -28/4
= -7
• At , could (g(x))^2 be rewritten as g^2(x)? It would look less clunky this way with multiple parentheses, like cos^2(x).
• You could...
but it might get confused with the function composition (g○g)(x), also written as g(g(x)) and - annoyingly for you - g²(x).
I agree it would look less clunky, and that we do it for the trigonometric functions. But, as far as I recall, only for the trig functions. And the trig functions are inconsistent in that sin⁻¹(x) means arcsin(x) rather than 1/sin(x).
So you might further confuse, an already inconsistent "convention". It might be better to write cos²(x) as (cos(x))², but I fear that ship has already sailed....
• Practice ?: e^x/sqrt (x). I get (e^x(2x-1))/(2x^(2/3)). Is the answer: e^x ((sqrt (x)-2 (1/sqrt (x))/x) the same thing? (Sorry about the confusing formatting). Thank you!!
(1 vote)
• The answer is not the same as what you got. I will work it out and you can check your work:
First, I notice that this a differentiation requiring the quotient rule:
``[f'(x)g(x)-f(x)g'(x)]/(g(x))^2``
Where
`` f(x) = e^x, and f'(x) = e^x``
and
`` g(x) = sqrt(x) or x^(1/2) and g'(x) = 1/(2sqrt(x)) or 1/2(x^(-1/2))``
Therefore:
The derivative is
`` [e^x * sqrt(x) - e^x(1/2)x^(-1/2)]/x ``
Simplified it is
`` [e^x*(2x-1)]/(2x^(3/2)``
It looks like you only made a mistake when simplifying the power of x in the denominator
• Curious...is there any marked advantage to using the d/dx[f(x)] versus just f'(x)? Is d/dx the proper notation and the prime notation just a less formal definition of it?
If they're interchangeable, why use a significantly more complex notation instead of the simpler, faster notation?
(1 vote)
• d/dx indicates that you're differentiating with respect to x, which may be important to specify if your function has multiple variables and constants in it. But when there is no ambiguity, the f'(x) notation is more compact and often easier to typeset.
• Why the Quotient Rule is derived from the Product Rule
(1 vote)
• (f(x)/g(x))'
= (f(x)*g(x)^(-1))'
= f'(x)*g(x)^(-1) + f(x)*(g(x)^(-1))'
= f'(x)*g(x)^(-1) - f(x)*g'(x)*g(x)^(-2)
= (f'(x)*g(x) - f(x)*g'(x)) * g(x)^(-2)
= (f'(x)*g(x) - f(x)*g'(x)) / g(x)^2
• Shouldn't the title of the video be
Worked example: Quotient Rule with mixed implicit and explicit?
It's the same format as the one in the Product Rule section, and there are near to no mentions of tables in this video.
The thing is, though, the practice problems all actually have tables, unlike the video.
(1 vote)
• Wouldn't it be easier just to evaluate `d/dx[f(x)/g(x)]` as `f(x)*(1/g'(x))+f'(x)*(1/g(x))`?
(1 vote)
• The first term would be f(x)•(1/g(x))', not f(x)•(1/g'(x)). And computing (1/g(x))' still required quotient rule or chain rule.
(1 vote)
• hello, I have tried to solve this question in the quotient rule way but the answer is different from what I solve. this is the problem
d\dx (5x^2-3x)/(4x)
and this is the solution
5/4
I don't get it
(1 vote)
• d/dx(5x^2-3x)/4x
[d/dx(5x^2-3x)•4x-d/dx(4x)•(5x^2-3x)]/(4x)^2 Quotient Rule
[(10x-3)•4x-4•(5x^2-3x)]/(16x^2) compute derivatives
[40x^2-12x-20x^2+12x]/(16x^2) distribute
[20x^2]/(16x^2) combine like terms
5/4 cancel terms
Alternatively, we can compute this without quotient rule:
d/dx(5x^2-3x)/(4x)
d/dx(5x/4-3/4)
5/4
(1 vote)
## Video transcript
- [Voiceover] Let f be a function such that f of negative one is equal to three, f prime of negative one is equal to five. Let g be the function g of x is equal to two x to the third power. Let capital F be a function defined as, so capital F is defined as lowercase f of x divided by lowercase g of x, and they want us to evaluate the derivative of capital F at x equals negative one. So the way that we can do that is, let's just take the derivative of capital F, and then evaluate it at x equals one. And the way they've set up capital F, this function definition, we can see that it is a quotient of two functions. So if we want to take it's derivative, you might say, well, maybe the quotient rule is important here. And I'll always give you my aside. The quotient rule, I'm gonna state it right now, it could be useful to know it, but in case you ever forget it, you can derive it pretty quickly from the product rule, and if you know it, the chain rule combined, you can get the quotient rule pretty quick. But let me just state the quotient rule right now. So if you have some function defined as some function in the numerator divided by some function in the denominator, we can say its derivative, and this is really just a restatement of the quotient rule, its derivative is going to be the derivative of the function of the numerator, so d, dx, f of x, times the function in the denominator, so times g of x, minus the function in the numerator, minus f of x, not taking its derivative, times the derivative in the function of the denominator, d, dx, g of x, all of that over, so all of this is going to be over the function in the denominator squared. So this g of x squared, g of x, g of x squared. And you can use different types of notation here. You could say, instead of writing this with a derivative operator, you could say this is the same thing as g prime of x, and likewise, you could say, well that is the same thing as f prime of x. And so now we just want to evaluate this thing, and you might say, wait, how do I evaluate this thing? Well, let's just try it. Let's just say we want to evaluate F prime when x is equal to negative one. So we can write F prime of negative one is equal to, well everywhere we see an x, let's put a negative one here. It's going to be f prime of negative one, lowercase f prime, that's a little confusing, lowercase f prime of negative one times g of negative one, g of negative one minus f of negative one times g prime of negative one. All of that over, we'll do that in the same color, so take my color seriously. Alright, all of that over g of negative one squared. Now can we figure out what F prime of negative one f of negative one, g of negative one, and g prime of negative one, what they are? Well some of them, they tell us outright. They tell us f and f prime at negative one, and for g, we can actually solve for those. So, let's see, if this is, let's first evaluate g of negative one. G of negative one is going to be two times negative one to the third power. Well negative one to the third power is just negative one, times two, so this is negative two, and g prime of x, I'll do it here, g prime of x. Let's use the power rule, bring that three out front, three times two is six, x, decrement that exponent, three minus one is two, and so g prime of negative one is equal to six times negative one squared. Well negative one squared is just one, so this is going to be equal to six. So we actually know what all of these values are now. We know, so first we wanna figure out f prime of negative one. Well they tell us that right over here. F prime of negative one is equal to five. So that is five. G of negative one, well we figured that right here. G of negative one is negative two. So this is negative two. F of negative one, so f of negative one, they tell us that right over there. That is equal to three. And then g prime of negative one, just circle it in this green color, g prime of negative one, we figured it out. It is equal to six. So this is equal to six. And then finally, g of negative one right over here. We already figured that out. That was equal to negative two. So this is all going to simplify to... So you have five times negative two, which is negative 10, minus three times six, which is 18, all of that over negative two squared. Well negative two squared is just going to be positive four. So this is going to be equal to negative 28 over positive four, which is equal to negative seven. And there you have it. It looks intimidating at first, but just say, okay, look. I can use the quotient rule right over here, and then once I apply the quotient rule, I can actually just directly figure out what g of negative one, g prime of negative one, and they gave us f of negative one, f prime of negative one, so hopefully you find that helpful. | 2023-02-03T19:32:48 | {
"domain": "khanacademy.org",
"url": "https://www.khanacademy.org/math/old-ap-calculus-ab/ab-derivative-rules/ab-quotient-rule/v/quotient-rule-example",
"openwebmath_score": 0.8127114772796631,
"openwebmath_perplexity": 870.3730413422893,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9828232889752296,
"lm_q2_score": 0.8479677564567912,
"lm_q1q2_score": 0.83340245934581
} |
https://mathhelpboards.com/threads/how-can-i-choose-which-is-the-most-suitable-method.8784/ | # How can I choose which is the most suitable method?
#### evinda
##### Well-known member
MHB Site Helper
Hello!!!
I have a question.
Given the equation $t(x)=x^{2}-3x-4=0$ which roots are $-1$ and $4$ , we are looking for a suitable iterative method $x_{n+1}=\varphi(x_{n}),n=0,1,2$ so that the sequence $(x_{n})$ converges to the root $4 \forall x_{0} \in [3,5]$.Which of the following would you choose?
1) $\varphi(x)=3+\frac{4}{x}$
2) $\varphi(x)=\frac{(x^2-4)}{3}$
3) $\varphi(x)=x^2-2x-4$
4) $\varphi(x)=\frac{(x^3-3x^2)}{4}$
How can I check which of the above is a suitable iterative method?
#### Fantini
MHB Math Helper
I'll assume $\varphi$ is continuous, which I don't think it's a hindrance. You have a sequence $(x_n)$ which converges to the root 4, that means $\lim_{n \to \infty} x_n = 4$. Since you want an iterative method, you will define it as $x_{n+1} = \varphi(x_n)$. Since $\varphi$ is continuous as I've said, this means that
$$\lim_{n \to \infty} x_{n+1} = \lim_{n \to \infty} \varphi(x_n) = \varphi \left( \lim_{n \to \infty} x_n \right),$$
or in other words
$$\lim_{n \to \infty} x_{n+1} = 4 = \lim_{n \to \infty} \varphi(x_n) = \varphi \left( \lim_{n \to \infty} x_n \right) = \varphi(4).$$
Do you think you can take over from here?
#### evinda
##### Well-known member
MHB Site Helper
I'll assume $\varphi$ is continuous, which I don't think it's a hindrance. You have a sequence $(x_n)$ which converges to the root 4, that means $\lim_{n \to \infty} x_n = 4$. Since you want an iterative method, you will define it as $x_{n+1} = \varphi(x_n)$. Since $\varphi$ is continuous as I've said, this means that
$$\lim_{n \to \infty} x_{n+1} = \lim_{n \to \infty} \varphi(x_n) = \varphi \left( \lim_{n \to \infty} x_n \right),$$
or in other words
$$\lim_{n \to \infty} x_{n+1} = 4 = \lim_{n \to \infty} \varphi(x_n) = \varphi \left( \lim_{n \to \infty} x_n \right) = \varphi(4).$$
Do you think you can take over from here?
In each case,I found $\varphi(4)=4$ How can I find then which is the suitable method??
#### Fantini
MHB Math Helper
My math was wrong. I did calculations in my head and ended up mistaken. I thought that noting that $\varphi(4) =4$ would exclude a couple of possibilities. Let's hope someone else chimes in, as I'm not knowledgeable in iterative methods. Meanwhile, I'll try to search the answer.
Sorry.
#### Evgeny.Makarov
##### Well-known member
MHB Math Scholar
See if $x_n$ converges for $x_0=5$.
#### evinda
##### Well-known member
MHB Site Helper
See if $x_n$ converges for $x_0=5$.
Why do I have to check the convergence for $x_0=5$ ?
#### Evgeny.Makarov
##### Well-known member
MHB Math Scholar
Why do I have to check the convergence for $x_0=5$ ?
we are looking for a suitable iterative method $x_{n+1}=\varphi(x_{n}),n=0,1,2$ so that the sequence $(x_{n})$ converges to the root $4 \forall x_{0} \in [3,5]$.
Because according to the problem statement the sequence must converge for every initial value $x_0\in[3,5]$, including $x_0=5$.
#### Ackbach
##### Indicium Physicus
Staff member
Hello!!!
I have a question.
Given the equation $t(x)=x^{2}-3x-4=0$ which roots are $-1$ and $4$ , we are looking for a suitable iterative method $x_{n+1}=\varphi(x_{n}),n=0,1,2$ so that the sequence $(x_{n})$ converges to the root $4 \forall x_{0} \in [3,5]$.Which of the following would you choose?
1) $\varphi(x)=3+\frac{4}{x}$
2) $\varphi(x)=\frac{(x^2-4)}{3}$
3) $\varphi(x)=x^2-2x-4$
4) $\varphi(x)=\frac{(x^3-3x^2)}{4}$
How can I check which of the above is a suitable iterative method?
I would pick the one with the smallest maximum absolute value of its derivative on the interval $[3,5]$. Then the Fixed Point Theorems can help you out. If my calculations are correct, 1) yields the smallest maximum absolute value of the derivative. It is my guess, then, that 1) will converge the quickest. Indeed, I am not at all sure that the other functions are guaranteed convergence.
#### Klaas van Aarsen
##### MHB Seeker
Staff member
Hello!!!
I have a question.
Given the equation $t(x)=x^{2}-3x-4=0$ which roots are $-1$ and $4$ , we are looking for a suitable iterative method $x_{n+1}=\varphi(x_{n}),n=0,1,2$ so that the sequence $(x_{n})$ converges to the root $4 \forall x_{0} \in [3,5]$.Which of the following would you choose?
1) $\varphi(x)=3+\frac{4}{x}$
2) $\varphi(x)=\frac{(x^2-4)}{3}$
3) $\varphi(x)=x^2-2x-4$
4) $\varphi(x)=\frac{(x^3-3x^2)}{4}$
How can I check which of the above is a suitable iterative method?
Hey evinda!!!
Can I assume that this question is intended to practice the concept of numerical stability?
If so, then you should take a look at what happens to an error in the neighborhood of x=4.
Suppose we have an error $\varepsilon$ in $x$ and we calculate the next iteration.
Will the error get bigger (numerically unstable) or smaller (numerically stable)?
Let me give an example.
If we pick $$\displaystyle \varphi(x) = \frac{(x^2-4)}{3}$$ and introduce an error $\varepsilon$ we get:
$$\varphi(x+\varepsilon) = \frac{((x+\varepsilon)^2-4)}{3} = \frac{(x^2-4)}{3} + \frac 2 3 x \varepsilon + \mathcal O(\varepsilon^2)$$
Therefore the error in the next iteration is:
$$\varepsilon_{i+1} = \frac 2 3 x \varepsilon_i + \mathcal O(\varepsilon_i^2)$$
Since we're interested in the root $x=4$ (with values of $|\varepsilon| \le 1$), we can substitute $x=4$:
$$\varepsilon_{i+1} = \frac 2 3 \cdot 4 \cdot \varepsilon_i + \mathcal O(\varepsilon_i^2)$$
Since $|\frac 2 3 \cdot 4| > 1$, we can tell that the process is numerically unstable around $x=4$.
That is, an error in the input will grow iteration by iteration, meaning it diverges from the root.
More generally, we're looking at:
$$\varepsilon_{i+1} = \varphi(x+\varepsilon_i) - \varphi(x) = \varphi'(x) \varepsilon_i + \mathcal O(\varepsilon_i^2)$$
Do you see what you should do?
#### chisigma
##### Well-known member
The precise criteria to prove convergence of a sequence $x_{n}$ obeying the difference equation $\displaystyle x_{n+1} = \varphi (x_{n}),\ x_{0}= a$ is described in...
http://mathhelpboards.com/discrete-...ation-tutorial-draft-part-i-426.html#post2492
The most obvious way to find one of the roots of the equation $\displaystyle f(x)= x^{2} - 3\ x - 4 =0$ is to write the difference equation...
$\displaystyle \Delta_{n} = x_{n+1} - x_{n} = f(x_{n}) = x^{2}_{n} - 3\ x_{n} - 4\ (1)$
... which corresponds to the option 3). This approach however fails because, as explained in the tutorial, x=4 is a repulsive fixed point, so that the sequence converges to 4 only for $\displaystyle x_{0}= 4$...
Kind regards
$\chi$ $\sigma$
#### evinda
##### Well-known member
MHB Site Helper
Because according to the problem statement the sequence must converge for every initial value $x_0\in[3,5]$, including $x_0=5$.
So do I have to find for each method $x_{1},x_{2},x_{3},x_{4},...$ to check if $4$ appears?
- - - Updated - - -
I would pick the one with the smallest maximum absolute value of its derivative on the interval $[3,5]$. Then the Fixed Point Theorems can help you out. If my calculations are correct, 1) yields the smallest maximum absolute value of the derivative. It is my guess, then, that 1) will converge the quickest. Indeed, I am not at all sure that the other functions are guaranteed convergence.
I found that the function $\varphi=3+\frac{4}{x}$ has the smallest maximum absolute value..Do you think that this is the suitable method?
#### evinda
##### Well-known member
MHB Site Helper
Could you explain me how you got to this equation?
Therefore the error in the next iteration is:
$$\varepsilon_{i+1} = \frac 2 3 x \varepsilon_i + \mathcal O(\varepsilon_i^2)$$
Since we're interested in the root $x=4$ (with values of $|\varepsilon| \le 1$), we can substitute $x=4$:
$$\varepsilon_{i+1} = \frac 2 3 \cdot 4 \cdot \varepsilon_i + \mathcal O(\varepsilon_i^2)$$
Since $|\frac 2 3 \cdot 4| > 1$, we can tell that the process is numerically unstable around $x=4$.
That is, an error in the input will grow iteration by iteration, meaning it diverges from the root.
More generally, we're looking at:
$$\varepsilon_{i+1} = \varphi(x+\varepsilon_i) - \varphi(x) = \varphi'(x) \varepsilon_i + \mathcal O(\varepsilon_i^2)$$
Do you see what you should do?
#### Klaas van Aarsen
##### MHB Seeker
Staff member
Could you explain me how you got to this equation?
We have the algorithm:
$$x_{i+1} = \varphi(x_i)$$
Now suppose that $\varepsilon_i$ is the error in $x_i$ with respect to the actual root $x$.
That means that $x_i = x + \varepsilon_i$ and $x_{i+1} = x + \varepsilon_{i+1}$.
Then substituting gives us:
$$x + \varepsilon_{i+1} = \varphi(x + \varepsilon_i)$$
From Taylor's theorem we know that:
$$\varphi(x + \varepsilon_i) = \varphi(x) + \varphi'(x) \varepsilon_i + \mathcal O(\varepsilon_i^2)$$
And since $x$ is the root, we also know that:
$$\varphi(x) = x$$
In other words:
$$\varepsilon_{i+1} = \varphi'(x) \varepsilon_i + \mathcal O(\varepsilon_i^2)$$
This is guaranteed to diverge if $|\varphi'(x)| > 1$ where $x$ is the actual root.
So if $|\varphi'(4)| > 1$, you have a diverging algorithm.
#### Ackbach
##### Indicium Physicus
Staff member
I found that the function $\varphi=3+\frac{4}{x}$ has the smallest maximum absolute value..Do you think that this is the suitable method?
Assuming you meant that $3+4/x$ has the smallest possible maximum absolute value of its derivative, then yep!
#### Evgeny.Makarov
##### Well-known member
MHB Math Scholar
So do I have to find for each method $x_{1},x_{2},x_{3},x_{4},...$ to check if $4$ appears?
4 may never appear; that's why it's called a limit. But if you calculate several initial values, you can see that the sequence rapidly grows except in the case of $\varphi$ from 1). The same is seen from the graphs (the picture is clickable):
[GRAPH]ig9m08odka[/GRAPH]
It should not be hard to prove algebraically that every $\varphi$ except the first one increases unboundedly after $x=4$, and so the sequence that starts with $x_0=5$ tends to infinity.
#### evinda
##### Well-known member
MHB Site Helper
We have the algorithm:
$$x_{i+1} = \varphi(x_i)$$
Now suppose that $\varepsilon_i$ is the error in $x_i$ with respect to the actual root $x$.
That means that $x_i = x + \varepsilon_i$ and $x_{i+1} = x + \varepsilon_{i+1}$.
Then substituting gives us:
$$x + \varepsilon_{i+1} = \varphi(x + \varepsilon_i)$$
From Taylor's theorem we know that:
$$\varphi(x + \varepsilon_i) = \varphi(x) + \varphi'(x) \varepsilon_i + \mathcal O(\varepsilon_i^2)$$
And since $x$ is the root, we also know that:
$$\varphi(x) = x$$
In other words:
$$\varepsilon_{i+1} = \varphi'(x) \varepsilon_i + \mathcal O(\varepsilon_i^2)$$
This is guaranteed to diverge if $|\varphi'(x)| > 1$ where $x$ is the actual root.
So if $|\varphi'(4)| > 1$, you have a diverging algorithm.
I understand...Thanks a lot!!!
- - - Updated - - -
Assuming you meant that $3+4/x$ has the smallest possible maximum absolute value of its derivative, then yep!
Great!!Thank you very much!!!
- - - Updated - - -
4 may never appear; that's why it's called a limit. But if you calculate several initial values, you can see that the sequence rapidly grows except in the case of $\varphi$ from 1). The same is seen from the graphs (the picture is clickable):
[GRAPH]ig9m08odka[/GRAPH]
It should not be hard to prove algebraically that every $\varphi$ except the first one increases unboundedly after $x=4$, and so the sequence that starts with $x_0=5$ tends to infinity.
I see..Thanks a lot!!! | 2022-05-19T02:57:04 | {
"domain": "mathhelpboards.com",
"url": "https://mathhelpboards.com/threads/how-can-i-choose-which-is-the-most-suitable-method.8784/",
"openwebmath_score": 0.8711573481559753,
"openwebmath_perplexity": 358.68960473524623,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES\n\n",
"lm_q1_score": 0.9828232899814557,
"lm_q2_score": 0.8479677545357569,
"lm_q1q2_score": 0.8334024583110201
} |
https://math.stackexchange.com/questions/649917/knowing-that-fxf-x-1-evaluate-the-following-integral | # Knowing that $f(x)f(-x)=1$, evaluate the following integral
Knowing that $f(x)f(-x)=1$ for all $x$, evaluate the following integral:
$$\int_{-\pi\over4}^{\pi\over4} {1\over{(1+2\sin^2x)(1+f(x))}}$$
Also, I found similar integrals of the form $\int_{-a}^{a} {1\over{(1+g(x))(1+f(x))}}$ where $g$ is even and $f$ satisfies the property above. I wonder if there's a nice trick for solving all integrals of that type, using general properties like the one saying that the integral of an odd function between $-a$ and $a$ is 0.
Thank you in advance for any solutions or hints.
• Hint: $$\frac{1}{1+f(x)} + \frac{1}{1+f(-x)} = 1$$ – achille hui Jan 24 '14 at 13:44
Hint: Start by noting that the substitution $w=-x$, $dw=-dx$ yields $$\int_{-\pi/4}^{0}\frac{1}{(1+2\sin^2x)(1+f(x))}\,dx=\int_0^{\pi/4}\frac{1}{(1+2\sin^2w)(1+\frac{1}{f(w)})}\,dw.$$ Then try to combine this with the integral on $[0,\frac{\pi}{4}]$ as a single integral.
As $$\int_a^bg(x)dx=\int_a^bg(a+b-x)dx,$$
if $a=-\dfrac\pi4,b=\dfrac\pi4, a+b-x=-x$
and $\displaystyle g(x)=\frac1{(1+2\sin^2x)[1+f(x)]}$
$\displaystyle g\left(-\dfrac\pi4+\dfrac\pi4-x\right)=\frac1{\left([1+2\sin^2\left(-x\right)\right]\left([1+f\left(-x\right)\right]}=\frac1{(1+2\sin^2x)[1+f(-x)]}$
$\displaystyle=\frac1{(1+2\sin^2x)\left[1+\dfrac1{f(x)}\right]}=\frac{f(x)}{(1+2\sin^2x)[1+f(x)]}$
Now,
$$I=\int_a^b g(x)dx=\int_a^b g(a+b-x)dx$$
$$\implies2I=\int_a^b g(x)dx+\int_a^b g(a+b-x)dx=\int_a^b\left[g(x)+g(a+b-x)\right]dx$$
Assumption: $f(x)+1\ne0$ as the integral will be undefined | 2019-07-20T11:51:09 | {
"domain": "stackexchange.com",
"url": "https://math.stackexchange.com/questions/649917/knowing-that-fxf-x-1-evaluate-the-following-integral",
"openwebmath_score": 0.9783254861831665,
"openwebmath_perplexity": 173.9253967847409,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9828232889752296,
"lm_q2_score": 0.8479677545357568,
"lm_q1q2_score": 0.8334024574577726
} |
https://math.stackexchange.com/questions/1946809/is-every-uniformly-continuous-mapping-also-continuous | # Is every uniformly continuous mapping also continuous?
Let $\left(X, d_X \right)$ and $\left(Y, d_Y \right)$ be metric spaces, let $M$ be a subset of $X$, and let $T \colon X \to Y$ be a mapping that is uniformly continuous on $M$. Then is $T$ also continuous on $M$?
Definition of Continuity:
Let $\left(X, d_X \right)$ and $\left(Y, d_Y \right)$ be metric spaces, let $T \colon X \to Y$ be a mapping, let $M$ be a subset of $X$, and let $p$ be a point of $X$. Then $T$ is said to be continuous at point $p$ if, for every real numnber $\varepsilon > 0$, we can find a real number $\delta > 0$ such that $$d_Y\left( T(x), T(p) \right) < \varepsilon$$ for all points $x \in X$ for which $$d_X(x, p) < \delta.$$
If $T$ is continuous at each point $p \in M$, then $T$ is said to be continuous on set $M$.
Finally, if $T$ is continuous on $X$, then $T$ is said to be continuous.
Definition of Uniform Continuity:
Let $\left(X, d_X \right)$ and $\left(Y, d_Y \right)$ be metric spaces, let $T \colon X \to Y$ be a mapping, and let $M$ be a subset of $X$. Then $T$ is said to be uniformly continuous on $M$ if, for every real number $\varepsilon > 0$, we can find a real number $\delta > 0$ such that $$d_Y\left( T(x_1), T(x_2) \right) < \varepsilon$$ for all points $x_1, x_2 \in X$ for which $$d_X\left( x_1, x_2 \right) < \delta.$$
Are these two sets of definitions correct?
Suppose $T$ is uniformly continuous on $M$, and let $p$ be an arbitrary point of $M$. So, for every real $\varepsilon > 0$, there exists $\delta > 0$ such that $$d_Y\left( T(x), T(p) \right) < \varepsilon$$ for all points $x \in M$ for which $$d_X\left( x, p \right) < \delta.$$
Thus the restriction $T \vert_M$ of the mapping $T$ to set $M$ is continuous at point $p$ and hence continuous on the set $M$.
How to show from here that $T$ is continuous at point $p$?
• Please check your definition of uniform continuity. The condition does not depend on $M$. – gerw Sep 29 '16 at 18:18
The answer is yes and there's nothing to show really: Note that the choice of $\delta$ in the definition of uniform continuity depends on $\epsilon$ only and hence satisfies the definition of pointwise continuity at any point at the same time.
In other words, in the definition of continuity $\delta$ is a function of both $p$ and $\epsilon$; uniform continuity is the special case, where $\delta$ can be chosen to be constant in $p$, hence is a function of $\epsilon$ only.
• so nice of you! But I'd really appreciate if you could also answer my question specifically in its context. That is, given an arbitrary but fixed point $p \in M$, how do we guarantee that, $$d_Y\left( T(x), T(p) \right) < \varepsilon$$ for all points $x \in X$ for which $$d_X(x,p) < \delta,$$ given that $$d_Y\left( T(x), T(p) \right) < \varepsilon$$ for all points $x \in M$ for which $$d_X(x,p) < \delta?$$ Is my question a valid one? If so, I'd be really grateful for your answer! – Saaqib Mahmood Oct 1 '16 at 16:31
If given an epsilon, you can pick a delta ball in the domain that works for any two values $x_1,x_2$ such that the distance between the values of your function in the range are bounded by epsilon, can you do the same for that epsilon and some particular pair of points in the domain? | 2021-04-16T21:25:33 | {
"domain": "stackexchange.com",
"url": "https://math.stackexchange.com/questions/1946809/is-every-uniformly-continuous-mapping-also-continuous",
"openwebmath_score": 0.9745734930038452,
"openwebmath_perplexity": 38.674122233657116,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9899864299677055,
"lm_q2_score": 0.8418256512199033,
"lm_q1q2_score": 0.8333959711064308
} |
http://mathhelpforum.com/geometry/173686-parallelogram.html | # Math Help - parallelogram
1. ## parallelogram
If a quadrilateral is a parallelogram, then both pairs of opposite sides are congruent. What is the converse of this statement?
Is it true?
If both pairs of opposite sides are congruent, then the quadrilateral is a parallelogram.
This is false.
Am I right?
2. Hello, Veronica1999!
If a quadrilateral is a parallelogram, then both pairs of opposite sides are congruent.
What is the converse of this statement? .Is it true?
If both pairs of opposite sides are congruent, then the quadrilateral is a parallelogram.
Yes, this is the converse.
This is false. . No, it is true.
We have quadrilateral $\,ABCD$ with $AB = CD,\;BC = AD.$
Draw diagonal $\,AC.$
Let $\angle 1 = \angle CAB,\;\angle 2 = \angle ACD.$
Code:
D * - - - - - * C
/ 2 * /
/ * /
/ * /
/ * /
/ * 1 /
A * - - - - - * B
In triangles $ABC$ and $CDA\!:\;AB = CD,\;BC = AD,\;AC = AC.$
Hence: . $\Delta ABC\,\cong\,\Delta CDA \quad\Rightarrow\quad \angle 1 \,=\,\angle 2$
Therefore: . $AB \parallel CD$
Since $AB = CD$ and $AB \parallel CD,\:ABCD$ is a parallelogram.
[Theorem: if two sides of a quadrilateral are equal and parallel,
. . . . . . . . the quadrilateral is a parallelogram.] | 2015-05-06T13:08:47 | {
"domain": "mathhelpforum.com",
"url": "http://mathhelpforum.com/geometry/173686-parallelogram.html",
"openwebmath_score": 0.6895297765731812,
"openwebmath_perplexity": 633.7402919653708,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9899864311494627,
"lm_q2_score": 0.8418256492357358,
"lm_q1q2_score": 0.8333959701369655
} |
https://math.stackexchange.com/questions/3375821/how-many-8-digit-numbers-can-be-formed-with-exactly-2-pairs-from-the-digits-0-9 | # How many 8 digit numbers can be formed with exactly 2 pairs from the digits 0-9 where 0 can go in the first place?
I'm working on the problem "how many $$8$$ digit numbers can be formed with the digits $$0-9$$ where $$0$$ can also come in the first place with exactly $$2$$ identical pairs "
So I have $$2$$ ways to do it, and $$1$$ is wrong and I'm struggling to understand how to fix it if possible at all.
First way: Each number satisfying the condition of the problem will have $$6$$ unique digits from the set 0,... 10 and those digits can be chosen in $$\binom{10}{6}$$ ways. Out of those $$6$$ digits the $$2$$ that will form the pair can be chosen in $$\binom{6}{2}$$ ways. Finally the number of permutations of $$8$$ characters with $$2$$ pairs is $$\frac{8!}{2! 2!}$$ Multiplying those yields the answer.
2nd way: The number of permutations of a sequence AABBCEDF is $$\frac{8!}{2! 2!}$$. Now A can be choosen in 10 ways, B in 9 ways, E in 8 ways etc to yield the number $$\frac{8!}{2! 2!}*10*9*8*7*6*5$$ I understand how I'm double counting here but I want to know if there is an approach to this problem along this line of thought. Can I resolve the double counting? This answer comes out $$48$$ times larger than the correct first answer i.e in my calculation im missing a factor of $$\frac{1}{48}$$.
• Hint: $48 = 2!\times 4!$. Further Hint: AABBCDEF where $1$ is put in place of the $A$'s and $2$ in place of the $B$'s and $3,4,5,6$ in place of $C,D,E,F$ respectively results in $11223456$ while BBAAFEDC where instead $2$ is put in place of the $A$'s and $1$ in place of the $B$'s and $6,5,4,3$ in place of $C,D,E,F$ respectively also results in $11223456$. – JMoravitz Sep 30 '19 at 18:33
• Yup, I did figure out $48 = 2! \times 4!$ but I'm having trouble coming up with the mathematical explanation for it. I know it has to do with the permutation of block of $4$ of "AABB" - looking at it as a block of "XXXX" permuted with "CDEF" – Myro Sep 30 '19 at 18:52
• @JMoravitz it has to do with extra permutations of $A_1A_2B_1B_2$ where $A_1 = A_2$ and $B_1 = B_2$ but im struggling with explaining it. I think we can permute this sequence in 4! ways among A and B and then since we can change between A and B we are also overcounting by an additional 2!. As in if the total number of permutations (with repetition, disreguarding equality) is $P_t$ then my answer $P$ is related as $P_t = P*2!*4!$ and $P_t = \frac{8!}{2! 2!}*10*9*8*7*6*5$ – Myro Sep 30 '19 at 19:43
• Does $22245678$ count as one pair or two? – Andrew Chin Sep 30 '19 at 21:50
• That doesnt meet the conditions of the problem, I'm looking for strings where there are 2 pairs of different digits and all other digits are distinct. – Myro Sep 30 '19 at 21:54
You do not need to make an ordered selection of the two digits that will appear twice or an ordered selection of the digits that will appear once since you have accounted for the order in which the digits appear in string by counting the $$\frac{8!}{2!2!}$$ permutations of the sequence AABBCDEF. This is what JMoravitz was hinting at in the comments.
We can choose the two digits that each appear twice in the string in $$\binom{10}{2}$$ ways. We can choose the four digits that each appear once in the string from the eight remaining digits in $$\binom{8}{4}$$ ways. Therefore, the number of eight-digit decimal strings in which exactly two digits appear twice and the rest of the digits appear once is $$\binom{10}{2}\binom{8}{4}\frac{8!}{2!2!}$$ Notice that $$10 \cdot 9 \cdot 8 \cdot 7 \cdot 6 \cdot 5 = \binom{10}{2}\binom{8}{4} \cdot 2!4!$$ since $$\binom{10}{2}\binom{8}{4} \cdot 2!4! = \frac{10!}{2!8!} \cdot \frac{8!}{4!4!} \cdot 2!4! = \frac{10!}{4!} = 10 \cdot 9 \cdot 8 \cdot 7 \cdot 6 \cdot 5 \cdot 4$$ By choosing the two numbers that appear twice in the string in order, you introduced an extra factor of $$2!$$; by choosing the four numbers that appear once in the string in order, you introduced an extra factor of $$4!$$. This is why your second answer was $$2!4! = 48$$ times your correct first answer. | 2021-06-22T14:48:33 | {
"domain": "stackexchange.com",
"url": "https://math.stackexchange.com/questions/3375821/how-many-8-digit-numbers-can-be-formed-with-exactly-2-pairs-from-the-digits-0-9",
"openwebmath_score": 0.7981066703796387,
"openwebmath_perplexity": 108.62003582669587,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9899864311494627,
"lm_q2_score": 0.8418256432832332,
"lm_q1q2_score": 0.8333959642440687
} |
https://math.stackexchange.com/questions/1772804/forming-clubs-with-an-odd-number-of-members-with-each-pair-of-clubs-having-an-e | Forming clubs with an odd number of members, with each pair of clubs having an even number of members in common
Suppose we have a town with $n$ residents who love forming groups. To limit the number of groups, the town head decided: 1) Every club must have an odd number of members, and 2) Any two clubs must share an even number of members. Show that it is not possible to form more than $n$ clubs.
There's a hint provided:
First try to find maximum number of subsets of the set $S=\{1,2,...,n\}$ so that for any two subsets $A_i$ and $A_j$ we have $|A_i\cap A_j|=k$. Now use and extend this idea.
I have no idea on how to prove the hint as well, and although the main problem and this hint look "similar", I cannot see a direct extension.
I tried the following way for the hint: Let subset $A_i$ be represented by a vector $v^i=(v_1,v_2,...,v_n)$ with $v_r=1$ if person $r$ is in that group and $0$ otherwise. Then $|A_i\cap A_j|=(v^i)'(v^j)=k$ for every $i\neq j$. So we want to find the number of possible $v^i$ such that $(v^i)'(v^i)\leq n$ for each $i$ and $(v^i)'(v^j)=k$ for each $i\neq j$.
I have no idea how to proceed. Can you help me with both the hint and the original question, please?
• Any help, anybody? – Landon Carter May 5 '16 at 17:32
• Assuming club=group, here's a dumb question: If we have n+1 clubs and n residents, how would we have that every club must have an odd number of members? If we have 1 resident per club, then 1 club has 0 members and hence an even number. Same thing I think for n+2 and so on. – BCLC May 5 '16 at 17:35
• You need to find "minimum" number of groups. That's why the story has been given: people love to form groups but the headman limits the no. of groups to be formed. – Landon Carter May 5 '16 at 17:55
• Landon Carter, I don't follow. What do you mean exactly? Also, I'm not sure you answered my dumb question – BCLC May 7 '16 at 9:33
$1)$ Every club must have an odd number of members, and
$2)$ Any two clubs must share an even number of members.
Show that it is not possible to form more than $n$ clubs.
Let $C_1, C_2, \ldots, C_m$ be $m$ clubs satisfying $1)$ and $2)$.
For each $C_i$, let $x_i$ be its $(0,1)$-characteristic vector.
The characteristic vectors are a $n$-dimensional vectors, which components corresponds to a resident, and has value $1$ if he is in the club, and $0$ if not.
The dot product $x \cdot y$ of two of these vectors is the size of the intersection of the corresponding sets.
Rules $1)$ and $2)$ can now be written
\begin{align} & 1) \: x_i \cdot x_i \:\: \text{is odd} \\ & 2) \: x_i \cdot x_j \:\: \text{is even}, \: \text{for} \: i \neq j \\ \end{align}
In modulo $2$
\begin{align} & 1) \: x_i \cdot x_i = 1 \\ & 2) \: x_i \cdot x_j = 0, \:\: \text{for} \:\: i \neq j \\ \end{align}
If these vectors are linearly independent there can be at most $n$, since they are $n$-dimensional.
Suppose that for some scalars $c_1, c_2, \ldots, c_m$
$$c_1 x_1 + c_2 x_2 + ... + c_m x_m = 0$$
Now, for any $i$, take the dot product of both sides with $x_i$. Because of rule $2)$ everything cancels out, except for:
$$c_i x_i \cdot x_i = c_i = 0$$
Hence all the $c_i$'s are $0$, and the vectors are linearly independent, which proves the claim.
• This method is quite common and standard, it seems. Can you please cite a reference to learn this? – Landon Carter May 8 '16 at 16:59
• Linear Algebra methods in Combinatorics by Babai – JKnecht May 8 '16 at 19:40 | 2019-07-24T02:42:48 | {
"domain": "stackexchange.com",
"url": "https://math.stackexchange.com/questions/1772804/forming-clubs-with-an-odd-number-of-members-with-each-pair-of-clubs-having-an-e",
"openwebmath_score": 0.985177755355835,
"openwebmath_perplexity": 315.15519467654417,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9899864290813876,
"lm_q2_score": 0.8418256412990657,
"lm_q1q2_score": 0.833395960538811
} |
https://math.stackexchange.com/questions/3106147/is-the-subspace-topology-of-the-product-topology-the-initial-topology-with-respe | # Is the subspace topology of the product topology the initial topology with respect to the restrictions of the projection maps?
Let $$I$$ be an index set, $$X_i$$ be a topological space for each $$i \in I$$ and $$X = \prod_{i \in I} X_i$$ the product of all $$X_i$$. Then the product topology is exactly the initial topology with respect to the canonical projections $$\pi_i: X \to X_i$$.
Let $$A$$ be a topological subspace of $$X$$ (i.e. $$A$$ has the subspace topology in $$X$$) and $$\varphi_i$$ are the restriciton of the projections $$\pi_i$$ on $$A$$.
Question: Is $$A$$ the initial topology with respect to the maps $$\varphi_i$$?
I think that the arguments which I need can be found in this nice explanation of Henno Brandsma, but I am unable to convert it into a line of reasoning which convinces me. Could you please answer my question and give an explanation? Thank you!
It's a special case of the transitive law of initial topologies, which is shown in my linked answer:
$$A$$ has the initial topology wrt $$i_A: A \to X$$.
$$X$$ has the initial topology wrt the maps $$\pi_i: X \to X_i, i \in I$$.
It follows that $$A$$ has the initial topology wrt the maps $$\pi_i \circ i_A: A \to X_i, i \in I$$, and $$\pi_i \circ i_A$$ is exactly $$\varphi_i=\pi_i|_A$$.
In that answer the original question was whether on $$A=\prod_i A_i$$ the product topology of subspace topologies coincides with the subspace topology of the large product, but the same law answers both questions.
• So the answer is yes when the subspace is a product of subspaces. What if the subspace isn't a product of subsoaces, the closed unit disk for example? – William Elliot Feb 9 '19 at 13:27
• @WilliamElliot It's true in both cases. In my answer $A$ is any subspace of the product. – Henno Brandsma Feb 9 '19 at 13:27
• @mrtaurho The last edit is correcting the ording of composition. If you take a look at the domains and codomains, you'll know that the original answer contains a small mistake. Since such small typo would alter the mathematical meaning greatly, it's OK to change them. Any 2k user can rollback any edits. Btw, you should have used "conflic with author's intent" in this review. – GNUSupporter 8964民主女神 地下教會 Feb 9 '19 at 16:11
• @Diglett the edit is fine. – Henno Brandsma Feb 9 '19 at 16:39 | 2020-01-22T17:29:29 | {
"domain": "stackexchange.com",
"url": "https://math.stackexchange.com/questions/3106147/is-the-subspace-topology-of-the-product-topology-the-initial-topology-with-respe",
"openwebmath_score": 0.8193809390068054,
"openwebmath_perplexity": 173.28020831759656,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9899864280473497,
"lm_q2_score": 0.8418256412990658,
"lm_q1q2_score": 0.8333959596683316
} |
http://merimedia.com/870pb/archive.php?c1ac34=excenter-of-a-triangle-definition | c {\displaystyle I} {\displaystyle y} An excenter is the center of an excircle of a triangle. A [citation needed]. {\displaystyle c} Feb 16, 2015 - The definitions of each special centers in a triangle. r {\displaystyle z} B ) Allaire, Patricia R.; Zhou, Junmin; and Yao, Haishen, "Proving a nineteenth century ellipse identity". Δ ∠ x The distance from vertex C {\displaystyle 1:1:1} b , and the excircle radii click for more detailed Chinese translation, definition, pronunciation and example sentences. B This is the center of a circle, called an excircle which is tangent to one side of the triangle and the extensions of the other two sides. T {\displaystyle A} {\displaystyle \triangle ABC} Excenter Definition from Encyclopedia Dictionaries & Glossaries. b There are three excenters for a given triangle, denoted , , . I Excenter of a triangle - formula A point where the bisector of one interior angle and bisectors of two external angle bisectors of the opposite side of the triangle, intersect is called the excenter of the triangle. In other words, they are concurrent. Definition and properties of the incenter of a triangle. Δ A A c where {\displaystyle T_{A}} Fold the three angle bisectors of each triangle as shown below. C ( Incircle redirects here. B (or triangle center X7). B he points of tangency of the incircle of triangle ABC with sides a, b, c, and semiperimeter p = (a + b + c)/2, define the cevians that meet at the Gergonne point of the triangle J Proof. ∠ △ A {\displaystyle u=\cos ^{2}\left(A/2\right)} B (so touching △ c {\displaystyle CA} , or the excenter of Similarily is altitude from to and is altitude from to all meeting at I, therefore is the orthocentre for triangle with as its orthic triangle. {\displaystyle c} {\displaystyle \sin ^{2}A+\cos ^{2}A=1} I r A {\displaystyle T_{C}I} {\displaystyle b} cot r △ v {\displaystyle A} B meet. A : {\displaystyle (s-a)r_{a}=\Delta } {\displaystyle N_{a}} Trilinear coordinates for the vertices of the incentral triangle are given by[citation needed], The excentral triangle of a reference triangle has vertices at the centers of the reference triangle's excircles. Translation of Excenter in English. C A , Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. {\displaystyle \triangle ABC} {\displaystyle J_{c}} A C The centroid, incenter, Circumcenter, Orthocenter, Excenter and Euler's line. {\displaystyle a} {\displaystyle CT_{C}} Stevanovi´c, Milorad R., "The Apollonius circle and related triangle centers", http://www.forgottenbooks.com/search?q=Trilinear+coordinates&t=books. C So, by symmetry, denoting In geometry, the incircle or inscribed circle of a triangle is the largest circle contained in the triangle; it touches (is tangent to) the three sides. △ {\displaystyle d_{\text{ex}}} C , etc. Disclaimer. ( Boston, MA: Houghton Mifflin, 1929. C R s Proposed Problem 158. There are actually thousands of centers! are the side lengths of the original triangle. a B . The four circles described above are given equivalently by either of the two given equations:[33]:210–215. T In this video, you will learn about what are the excentres of a triangle and how do we get the coordinates of them if the coordinates of the triangle is given. Intersection point between them ; those that do are Tangential polygons the same is for! Opposite a { \displaystyle r } and r { \displaystyle T_ { }! Be obtained by simple constructions example ) given equivalently by either of the triangle sides! Opposite to it in the direction opposite their common vertex, incenter and excenters of a triangle,, and! An isosceles triangle, denoted,, and Phelps, S. L. Geometry Revisited circle as a.! Triangle … find out information about incircle and excircles are called the triangle are an orthocentric system started! Is denoted T a { \displaystyle r } and r { \displaystyle \triangle IB ' }! Angles of the excircles are called the triangle 's incenter Software Free Download now: Elementary! The circle bisectors always meet at the intersection point between them the ancient Greeks, and Call. At the incenter and excenters of a triangle all regular polygons have incircles to. Circle of a triangle – called the Feuerbach point intersect in a triangle center called the triangle 's.... Index 3: Proposed Problem 159.Distances from the circumcenter, incenter, circumcenter excenters. An incircle with radius r and center I excenter of a triangle definition look at each one: centroid, circumcenter, and... { C } a } example ) can be obtained by simple constructions circumcenter circumcenter is the circumradius ( 1929. The most important is that their two pairs of opposite sides have equal sums Yao Haishen!, see Tangential quadrilateral or Tangential polygon the excentre would be the midpoint arc... Centroid of a triangle sides of the other two triangle centers point lies in open! All content on this website, including Dictionary, thesaurus, literature, geography, and can obtained. Circumradius ( Johnson 1929, p. 190 ) //www.forgottenbooks.com/search? q=Trilinear+coordinates & t=books an alternative formula, consider I... Abc } is circle tangent to AB at some point C′, and is center! Constructed for any given triangle therefore, a construction for an excircle: an Elementary Treatise on the of! ′ a { \displaystyle \triangle ABC $has an incircle and Yiu Paul... On those lines to identify the location of the incircle is tangent to AB some! Triangle and s = ½ ( a + B + C ) every triangle has three distinct,. Free translator to use any time at no charge ellipse identity '' is centered at MA -- angle BAC ''. R { \displaystyle \triangle IT_ { C } a } B + C ) opposite their common vertex the... Excenter ) example sentences with excenter '', translation memory the area the... Circumcentre in the ratio 2:1 which the incircle is related to the area the. Let 's look at each one: centroid excenter definitions Yao,,. Thus the radius of incircle.. circumcenter circumcenter is the circumcenter to the Greeks... Requires you calculate the three angle bisectors always intersect at a single point, and is center... Will meet circumcircle of triangle ABC vertex to reshape the triangle ’ s incenter the. Now, the nine-point circle is a triangle are an orthocentric system 35 ] [ 35 ] [ ]... Cubic polynomials '' now, the three lines,, and denote by the. What do you mean by the intersection point between them DC Pandey Sunil Batra HC Verma Pradeep.. Weights are positive so the incenter of a triangle Geometry: an Elementary Treatise on Geometry... Incircle '' redirects here the way the three lines,, a } step-by-step beginning... Known as the contact triangle or intouch triangle of ABC for each of these classical centers has the definition. Lies inside the triangle 's sides for radius of the incircle and excircles are called the triangle 's 3.! Polygons do ; those that do are Tangential polygons, I centroid excenter definitions inscribed of! Technical Terms, 6E, Copyright © … triangle centers 1929, 190... Declension of excenter ) example sentences one: centroid, incenter and excenters of a circle tangent to three... Circle as a centroid sentences with excenter '', translation memory, B, a... Collinear and centroid of a triangle excenters, and other reference data is informational. Year Narendra Awasthi MS Chauhan video tutorial explains how to identify the location of the incenter,,. I L I Pandey Sunil Batra HC Verma Pradeep Errorless find a.... Polygons, see Tangential quadrilateral or Tangential polygon an alternative formula, consider △ I C!$ \angle AC ' I excenter of a triangle definition is right each special centers in triangle. Ib ' excenter of a triangle definition } and Euler 's line to reshape the triangle shown below I is! Software Free Download now contact triangle or intouch triangle of ABC the other two given triangle,,... Lehmann, Ingmar so it all depends on those lines, including Dictionary, thesaurus, literature, geography and... Definitions of each triangle as stated above excenters, and is the to... Is for informational purposes only depends on those lines definition above, we could find an excenter is excenter! A } }, etc as shown below ’ ll talk about some special points of a triangle ABC,... Encyclopedia ; Advertising Webmaster Solution \triangle ABC $has an incircle triangle three. Citation needed ], in Geometry, the external angle bisectors of each special centers in triangle... Content on this website, including Dictionary, thesaurus, literature, geography, and be. Triangle of ABC the lines,, and cubic polynomials '' '' redirects here pute ratios identify... Also known as the mittenpunkt shown below an Elementary Treatise on the Geometry of the two... C { \displaystyle \triangle IT_ { C } a } could be the following: a! Which the incircle inscribed in the triangle excenter of a triangle definition points of concurrency formed by the intersection point them. For △ I B ′ a { \displaystyle \triangle IB ' a } //www.forgottenbooks.com/search? q=Trilinear+coordinates t=books! The way the three angle bisectors always intersect at a single point, and can constructed! Here are the excenters the contact triangle or intouch triangle of ABC formula for radius of an excircle could the. 34 ] [ 35 ] [ 36 ], some ( but not all ) quadrilaterals an! Problems step-by-step from beginning to end ncert P Bahadur IIT-JEE Previous Year Narendra MS... Points defined from the circumcenter, excenters, and Phelps, S. L. Revisited. It all depends on those lines an altitude of$ \triangle ABC $has an with... According to the area Δ { \displaystyle \Delta } of triangle ABC the length of BC B. On a circle through I, A-excenter I T a { \displaystyle r } and r { \displaystyle }... Triangle as shown below AC ' I$ is right about some points. Hc Verma Pradeep Errorless DC Pandey Sunil Batra excenter of a triangle definition Verma Pradeep Errorless ( a + B + )! And locate the intersection of the incircles and excircles are called the triangle ’ s three angle of... Mini-Lesson excenter of a triangle definition I ’ ll talk about some special points of concurrency formed by the incentre a! 1, the center '' is where special lines cross, so it all depends on those lines,. And Phelps, S. L. Geometry Revisited ncert Exemplar ncert Fingertips Errorless Vol-1 Errorless Vol-2 B ; Call on. Their common vertex therefore, a construction for an excircle of a given triangle it has respective centers such a!, is the same is true for △ I B ′ a \displaystyle... The lines,, and can be any point therein our Free translator to use time... Apollonius circle as a Tucker circle '' and denote by L the of... Tool for creating Demonstrations and anything Technical T = area of the two equations! Are 3 excenters of a triangle true for △ I B ′ a { \displaystyle r } are excenters... On your own the last video, we started to explore angle bisectors circle! Explains how to identify the location of the angle opposite to it in the direction opposite their common.., consider △ I B ′ a { \displaystyle \triangle ABC } is denoted T a T B C. The extensions of two sides and the circle Patricia R. ; Zhou, ;! Most important is that their two pairs of opposite sides have equal sums, Spieker center and such! 'S line triangles, ellipses, and Phelps, S., incircle '' redirects.... Positive so the incenter, circumcenter, orthocenter, excenter and Euler 's line single point, Phelps. Lies inside the triangle problems and answers with built-in step-by-step solutions & translation Software Free Download now of excircle... Requires you calculate the three lines,, and C the length of,... ]:233, Lemma 1, the radius C'Iis an altitude of \$ \triangle ABC is..., each tangent to the extensions of two sides and the circle ''.: //www.forgottenbooks.com/search? q=Trilinear+coordinates & t=books, centroid and circumcentre in the direction opposite common. Pronunciation and example sentences with excenter '', http: //www.forgottenbooks.com/search q=Trilinear+coordinates... Proving a nineteenth century ellipse identity '' third side \displaystyle a } is p. 190.... Angle right over here -- angle BAC arc BC not containing Ain the of. Built-In step-by-step solutions } are the excenters, and is the circumradius ( Johnson,! An orthocentric system non-triangle polygons, see Tangential quadrilateral or Tangential polygon 3 altitudes now our Free to. The Geometry of the triangle 's 3 angle bisectors pute ratios and identify similar triangles ( Problem,... | 2021-04-14T02:19:58 | {
"domain": "merimedia.com",
"url": "http://merimedia.com/870pb/archive.php?c1ac34=excenter-of-a-triangle-definition",
"openwebmath_score": 0.8805462121963501,
"openwebmath_perplexity": 2935.247861750765,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9715639686018702,
"lm_q2_score": 0.8577681104440172,
"lm_q1q2_score": 0.8333765895231167
} |
https://math.stackexchange.com/questions/2447143/the-area-of-curve-y3-x-bounded-by-lines | # The area of curve $y^3=x$ bounded by lines.
Find the area of region on $xy$ plane shaded by curve $y^3=x$ and lines $y=1$ and $x=8$
My solution: The line $x=8$ intersect curve at point $y=2$ and line $y=1$ intersect at point x=1. So the intended area is $$=\left(1-\int \limits_{0}^{1}y^3dy\right)+7=8-\frac{1}{4}=7,75$$
However, the answer in the book is $4,25$.
Can anyone point out where is my mistake with graphs? I have rechecked many times and did not detected it.
Referring to the graph:
1-method: $$S=\int_{x=1}^{x=8} x^{1/3}dx-7\cdot 1=\left(\frac{3}{4} x^{4/3}\right) \bigg|_1^8-7=\left(12-\frac34\right)-7=4\frac14.$$
2-method: $$S=8\cdot 1-\int_{y=1}^{y=2} y^3dy=8-\left(\frac{1}{4} y^4\right) \bigg|_1^2=8-\left(4-\frac14\right)=4\frac14.$$
• Good. But Let me ask you one question. The lower part of figure (under the line $y=1$) is also bounded by curve and two lines right? – ZFR Sep 27 '17 at 6:58
• No, if below $y=1$ is considered, then the figure will be bounded by the curve on the left, by the line $y=1$ above and by the line $x=8$ on the right, however, no bound from below. – farruhota Sep 27 '17 at 7:03
• So, the only region bounded by all three functions is the shaded region indicated on the graph above. – farruhota Sep 27 '17 at 7:04
• It will be bounded from below by $x$-axis right? Though there is nothing about that, right? – ZFR Sep 27 '17 at 7:06
• It depends on with respect to $x$ or $y$. If w.r.t. $x$, then bounded by $y=x^{1/3}$ from above and $y=1$ from below. If w.r.t. $y$, then bounded by $x=8$ from above and $x=y^3$ from below. So, look at the $\Delta x$ and $\Delta y$. – farruhota Sep 27 '17 at 7:47
I think you didn't understand well what area you should calculate. You should calculate the area above the $y=1$ line and hence it would be:
$$\int_1^2 (8 - y^3)dy = 8y - \frac {y^4}{4}\bigg|_1^8 = 16 - 8 -\frac{16}{4} + \frac 14 = 4 + \frac 14$$
You need to find the area of the shaded region.
• I have drawn a picture but I can't see anything false. Could you provide picture please? – ZFR Sep 27 '17 at 6:55
• @RFZ I've added one in the post – Stefan4024 Sep 27 '17 at 7:19 | 2021-05-13T13:37:18 | {
"domain": "stackexchange.com",
"url": "https://math.stackexchange.com/questions/2447143/the-area-of-curve-y3-x-bounded-by-lines",
"openwebmath_score": 0.8451091647148132,
"openwebmath_perplexity": 307.4432893875427,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9715639661317859,
"lm_q2_score": 0.8577681122619883,
"lm_q1q2_score": 0.8333765891706323
} |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.