Search is not available for this dataset
url
string | text
string | date
timestamp[s] | meta
dict |
|---|---|---|---|
https://www.physicsforums.com/threads/the-sum-of-series.108928/ | # The sum of series
1. Feb 1, 2006
### heaven eye
a series such as :-
∑ from n = 1 ,until m for n
can be found with the law of the sum of arithmetical series:-
[m(2a+(m-1)d)]/2
where :-
m= the number of terms
a= the first term in the series
d= the basic arithmetical
in the pervious example :-
a=1 , d=1 , m=m
when we solve the last information in the law we find :-
∑ from n = 1 ,until m for n = m(m+1)/2
for example :-
1+2+3+4+..........+18+19+20 = 20(20+1)/2 = 210
my question is :-
∑ from n = 1 ,until m for n^2 ???
it isn't an artithmetical or even a geometrical series then :-
what kind of series is it ?
and how could they find that :-
∑ from n = 1 ,until m for n^2 = [m(m+1)(2m+1)]/6 ???
for example :-
(1^2)+(2^2)+(3^2)+(4^2)+(5^2)= 5(5+1)(2*5+1)/6 = 55
and thank you
2. Feb 1, 2006
### HallsofIvy
Staff Emeritus
It's not an arithmetic series, obviously. In an arithmetic series, by definition, the difference between two consecutive terms must always be the same. But in 1, 22= 4, 32= 9, 42= 16, ..., the differences are 4-1 =3, 9- 4= 5, 16- 9= 7,...
Notice, however, those differences are consecutive odd numbers: the "second differences" 5-3. 7- 5, ... are all 2. Obviously, for the sum of squares, the first difference is the squares themselves so the "third differences" are all 2. "Newton's divided difference formula" gives us
$$\sum_{i=0}^n i^2= \frac{n(n+1)(2n+1)}{6}$$
Similar things can be done for higher powers but the results get progressively more difficult.
3. Feb 15, 2006
### heaven eye
Well sorry I was late to repost cause of exams
well thank you mr.HallsofIvy for your answer, now it is more clear to me
my reagrds
4. Feb 19, 2006
### robert Ihnot
If you want to put time into it, you can assume that the sum of the integers has a leading square factor, the sum of the squares has a leading cube term, the sum of third powers, etc. (This follows from HallsofIvy above.)
We then attempt to find terms $$ax^3+bx^2+cx+d$$=S(x)
We know that S(0) = 0 and so d=0. S(1) = a+b+c=1, S(2)=8a+4b+2c=5, S(3) = 27a+9b+3c=14. So we can solve these equations by elimination of terms.
Last edited: Feb 19, 2006 | 2017-09-19T13:52:56 | {
"domain": "physicsforums.com",
"url": "https://www.physicsforums.com/threads/the-sum-of-series.108928/",
"openwebmath_score": 0.7758783102035522,
"openwebmath_perplexity": 2658.098314294569,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9825575173068326,
"lm_q2_score": 0.8479677660619633,
"lm_q1q2_score": 0.8331771029780637
} |
https://math.stackexchange.com/questions/952304/does-the-function-fracx-sqrt1x-have-an-oblique-asymptote | # Does the function $\frac{x}{\sqrt{1+x}}$ have an oblique asymptote?
Does the function $$\frac{x}{\sqrt{1+x}}$$ have an oblique asymptote? If so, how do we find it? I thought about long dividing, but that wouldn't work, because You can't divide square roots.
• why not? $\frac{\sqrt{4}}{\sqrt{5}}$ exists.. – MonK Sep 30 '14 at 9:20
You are not going to have an "oblique" asymptote like you are used to--it's not going to be a straight line. Instead the "oblique asymptote" is going to be a sideways parabola. This is because you have (essentially) $\frac{x}{\sqrt{x}} = \sqrt{x}$--so the asymtptote is going to be a square root--or quadratic:
$$f(x) = \frac{x}{\sqrt{x + 1}} = x\frac{\sqrt{x + 1}}{x + 1}$$
Put another way what you have is: $\sqrt{x + 1} = y \rightarrow y^2 = x+1 \rightarrow x = y^2 - 1$:
$$\left(y^2 - 1\left)\frac{y}{y^2}\right.\right. = \frac{y^2 - 1}{y} = y - \frac{1}{y} = \sqrt{x + 1} - \frac{1}{\sqrt{x + 1}}$$
This means that the "oblique" asymptote is $y = \sqrt{x + 1}$.
• $y = \sqrt{x + 1}$ is a curvilinear asymptote. It is not oblique since an oblique asymptote is a line. – N. F. Taussig Sep 30 '14 at 9:32
• @N.F.Taussig Hence why I put quotes around the term oblique...this function asymptotes to the function $f(x) = \sqrt{x}$ as I stated--it doesn't exactly asymptote to that, but it essentially does. – Jared Sep 30 '14 at 9:32
• I knew you understood the question. I just wanted to make sure that the person who posted the question understood why you put the quotes there. – N. F. Taussig Sep 30 '14 at 9:33
A necessary condition for the function $f$ to have an oblique asymptote at $\infty$ is that $$\lim_{x\to\infty}\frac{f(x)}{x}$$ exists, is finite and is non zero. If the oblique asymptote exists, then this limit is its slope. It mustn't be zero, because otherwise it wouldn't be oblique in the first place.
If an oblique asymptote exists, with equation $y=mx+q$ ($m\ne0$), then, by definition, $$\lim_{x\to\infty}(f(x)-mx-q)=0$$ so also $$\lim_{x\to\infty}\frac{f(x)-mx-q}{x}=0$$ and then $$\lim_{x\to\infty}\left(\frac{f(x)}{x}-m\right)=0$$ because, of course, $\lim_{x\to\infty}\frac{q}{x}=0$.
• I don't see how an "oblique" asymptote differs from a $y = 0$ asymptote which you seem to single out. – Jared Sep 30 '14 at 9:52
• @Jared A necessary condition for a horizontal asymptote is that $\lim_{x\to\infty}f(x)$ exists finite, which should be checked before trying for oblique asymptotes. – egreg Sep 30 '14 at 10:32
• @Jared The usual example is $f(x)=\log x$, which hasn't a horizontal asymptote, but $\lim_{x\to\infty}f(x)/x=0$. – egreg Sep 30 '14 at 10:56
HINT: $$\frac{x}{\sqrt{1+x}}=\frac{x+1-1}{\sqrt{x+1}}=\sqrt{1+x}-\frac{1}{\sqrt{1+x}}$$
• This is a good answer (don't get me wrong)...but I hope you (the OP) will take the time to understand my answer as well. – Jared Sep 30 '14 at 9:31
• @Jared My answer is purely a hint,yours is a complete answer :) – kingW3 Sep 30 '14 at 9:37
• I agree, hence why I upvoted your answer as it agreed with mine...I would suggest the OP and others to do the same since it verifies the result (regardless of who came first). – Jared Sep 30 '14 at 9:38 | 2019-12-16T11:20:29 | {
"domain": "stackexchange.com",
"url": "https://math.stackexchange.com/questions/952304/does-the-function-fracx-sqrt1x-have-an-oblique-asymptote",
"openwebmath_score": 0.9491041898727417,
"openwebmath_perplexity": 369.75495133434436,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9825575188391108,
"lm_q2_score": 0.847967764140929,
"lm_q1q2_score": 0.8331771023898595
} |
https://math.stackexchange.com/questions/2090325/if-there-is-one-perfect-square-in-an-arithmetic-progression-then-there-are-infi | # If there is one perfect square in an arithmetic progression, then there are infinitely many
Consider the following positive integers: $$a,a+d,a+2d,\dots$$ Suppose there is a perfect square in the above list of numbers. Then prove that there are infinitely many perfect square in the above list. How can I do this?
At first I started in this way: Let the $n$th term is perfect square. Therefore, $$t_n=a+(n-1)d=m^2.$$ Then I think that I will put values at the position of $n$. But I failed to find anything from this level. Can somebody help me?
• If $m^2$ is in the sequence, so too are $(m+d)^2, (m+2d)^2, (m+3d)^2, \ldots$ – Henry Jan 9 '17 at 17:09
Note that $$(m+d)^2=m^2+(2m+d)\cdot d$$
Let $n^2$ be the known square.
Then $n^2+kd=m^2$ is equivalent to $kd=(m-n)(m+n)$. You can take $m-n$ to be any multiple of $d$, and $k$ follows.
Starting from where you left, suppose the $k$th term is a perfect square such that $$a_k = a_1 + (k-1)d = p^2$$
Now add $2pmd + m^2d^2$ to both sides where $m$ is a natural number giving us, $$\Rightarrow a_1 + (k-1)d + 2pmd + m^2d^2 = p^2 + 2pmd + m^2d^2$$ $$\Rightarrow a_1 + [(k-1)+2pm+m^2]d = (p+md)^2$$
The RHS is a perfect square, and the left side is the $(k-1+2pm+m^2)$th term for infinitely many values of $m$. Hope it helps.
• last term inside square brackets should be $m^2d$? – hypergeometric Jan 9 '17 at 17:29
Suppose that given $a_n = a+n*d = x_n^2$ there is theoretically another value $a_n' = a+n'*d = x_n'^2$ where all numbers are whole.
Taking the difference between these number yields: $$a_n' - a_n = a_n'- a_n\\ (a+n'*d) - (a+n*d) = x_n'^2 - x_n^2\\ (n' - n)d = (x_n' - x_n)(x_n' + x_n)$$ Setting the outer expressions equal to each other, and the inner expressions equal to each other yields: $$x_n' - x_n = d \\x_n' = d + x_n$$ and $$n' - n = x_n'+x_n\\n' = n + x_n'+x_n\\n' = n + d + 2 x_n$$
Therefore: If $a_n = a + n*d = x_n^2$ is a square, you can use theses values to generate a new larger square within the sequence where: $a_n' = a + n'd = a + (n+d+2*x_n)*d$. This will be: $x_n'^2 = (d+x_n)^2$.
• Please use LaTeX formatting to improve readability! – The Count Jan 9 '17 at 20:48 | 2019-08-24T11:17:30 | {
"domain": "stackexchange.com",
"url": "https://math.stackexchange.com/questions/2090325/if-there-is-one-perfect-square-in-an-arithmetic-progression-then-there-are-infi",
"openwebmath_score": 0.9199228882789612,
"openwebmath_perplexity": 188.58383825997797,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9825575137315161,
"lm_q2_score": 0.847967764140929,
"lm_q1q2_score": 0.8331770980587838
} |
https://math.stackexchange.com/questions/4021697/confusion-in-the-solution-for-this-question/4021771 | # Confusion in the solution for this question
Given 4 flags of different colours, how many different signals can be generated. If a signal requires the use of 2 flags one below the other?
Le the colours be R , B , G , O.
My book answers it as = 4 *3 = 12 which is right answer but not conceptually right .
4 means 4 choices I.e R,B,G,O. Then for 3 choices , we have B,G,O.
Now , if you see . Total combinations it will give is RB RG RO , BB BG BO , GB GG GO , OB OG OO. Here , OO GG BB are wrong combinations .
So , if I remove them there are total 9 now. But we need 3 more . Those are BR , OR and GR.
Please tell if i am right till here ?
How to consider them then ? Now , all this cutting and then adding made the solution way also long .
Is there a shorter way of doing and how to revive those BR , OR , GR without counting it knowing and also to cut BB , GG , OO without counting.
The answer that the book provides you is the direct application of the following formula: $$nP_k = {\displaystyle\prod_{i=0}^{k-1} (n - i)}=\dfrac{n!}{(n-k)!}$$
If we plug $$n=4$$ (our pool of distinct flags) and $$k=2$$ (the number of flags to sample in order to build a signal), according to the problem statement, the formula will yield: $$4P_2 = \dfrac{4!}{(4-2)!}=\dfrac{4!}{2!} = \dfrac{4\cdot 3\cdot 2\cdot 1}{2\cdot 1}= 4\cdot 3 = 12$$
As you correctly pointed out that $$12$$ is the right answer.
Now for the intuition behind the problem and in general when we have to deal with ordered permutations, we can think of the problem as follows:
Let's suppose that we have two placeholders for any of the available flags, denoted as $$F_1 \, F_2$$ . Furthermore, since the order of the flags matters $$F_1 \, F_2 \ne F_2 \, F_1$$, this leaves us with $$n=4$$ possible options for the $$F_1$$ placeholder and $$n-1=3$$ for the $$F_2$$ placeholder. This is because initially the whole set of flags is available for selection e.g. $$F=\{R,B,G,O\}$$, picking any flag from this set leaves us only three of the rest flags available for selection:
• Case 1: First choice is $$\color{red}R$$ from the whole set of flags $$\{R,B,G,O\}$$ $$R \, F_2$$ this entails that the second flag must be sampled from the following set $$\{B,G,O\}$$. This leads to, the following three permutations: $$\mathbf{\color{red}R}\color{blue}B, \, \mathbf{\color{red}R}\color{green}G, \, \mathbf{\color{red}R}\color{orange}O$$
• Case 2: First choice is $$\color{blue}B$$, then the possible values for $$F_2$$ are $$\{R,G,O\}$$ $$\mathbf{\color{blue}B}\color{red}R, \, \mathbf{\color{blue}B}\color{green}G, \, \mathbf{\color{blue}B}\color{orange}O$$
• Case 3: First choice is $$\color{green}G$$, then the possible values for $$F_2$$ are $$\{R,B,O\}$$ $$\mathbf{\color{green}G}\color{red}R, \, \mathbf{\color{green}G}\color{blue}B, \, \mathbf{\color{green}G}\color{orange}O$$
• Case 4: First choice is $$\color{orange}O$$, then the possible values for $$F_2$$ are $$\{R,B,G\}$$ $$\mathbf{\color{orange}O}\color{red}R, \, \mathbf{\color{orange}O}\color{blue}B, \, \mathbf{\color{orange}O}\color{green}G$$
Each of the $$4$$ cases having $$3$$ possible permutations. Putting it all together we have $$12$$ permutations: $$\mathbf{\color{red}R}\color{blue}B, \, \mathbf{\color{red}R}\color{green}G, \, \mathbf{\color{red}R}\color{orange}O, \, \mathbf{\color{blue}B}\color{red}R, \, \mathbf{\color{blue}B}\color{green}G, \, \mathbf{\color{blue}B}\color{orange}O, \, \mathbf{\color{green}G}\color{red}R, \, \mathbf{\color{green}G}\color{blue}B, \, \mathbf{\color{green}G}\color{orange}O, \, \mathbf{\color{orange}O}\color{red}R, \, \mathbf{\color{orange}O}\color{blue}B, \, \mathbf{\color{orange}O}\color{green}G$$
If we think about the product series from the $$nP_{k}$$ formula we can see its connection to the physical intuition. \begin{aligned} nP_k &= {\displaystyle\prod_{i=0}^{k-1} (n - i)} \\ &= n \cdot {\displaystyle\prod_{i=1}^{k-1} (n - i)} \\ &= n\cdot (n-1) \cdot {\displaystyle\prod_{i=2}^{k-1} (n - i)} \\ &= n \cdot (n-1) \cdot (n-2) \cdot {\displaystyle\prod_{i=3}^{k-1} (n - i)} \end{aligned}
What one might notice is that each increment of the product series index variable $$i$$ multiplies the current product by the cardinality of the remaining set of flags e.g. the number of the available flags in the current set. This will carry on until all $$k$$ placeholders are filled, again considering all remaining flags currently in the set, according to its index number e.g. the current cardinality of the set, denoted as $$n-i$$ with $$i \in [0, k) \subset \mathbb N_0$$.
Note :
This problem is about permutations, your attempt to solve the problem suggests that you calculated the cartesian product of the set with itself and then correctly waived out the invalid combinations namely $$RR, GG, BB, OO$$. The latter is a hint that any flag once chosen from the set, is not placed back in it, i.e. there is no replacement of the flags in the set. The second hint about the problem is that order matters that is $$F_1F_2 \ne F_2F_1$$ i.e. $$RB \ne BR$$. This suggests that counting the combinations (unordered edition of the problem) is not the answer. However, had the order been irrelevant, then you could use the $$nC_k = \dfrac{n!}{(n-k)!\cdot k!}$$ to calculate all possible unordered combinations. It might be apparent to you that $$nP_k$$ and $$nC_k$$ relate through $$nP_k = k! \cdot nC_k$$. Since $$k=2$$ this means that $$nP_2 = 2\cdot nC_2$$ which could be intuitively derived from the fact that once you have written down the unordered combinations then you must add them again but in flipped order, to account for the other combinations that you have pointed out. Beware that this only works when $$k=2$$.
We have one flag each of $$4$$ colors - $$R, B, G, O$$. We need to make a signal with two flags and order matters.
So if we pick any of the color as first, we have $$3$$ choices for the second color. If $$R$$ is first, we have $$3$$ choices for second $$B, G, O$$. If $$B$$ is first, we again have $$3$$ choices for second color - $$R, G, O$$.
Hence total number of different signals that we can make $$= 4 \times 3 = 12$$, which is $$4$$ choices for the first color and $$3$$ choices for the second for each of the first.
• Yes, this is the issue - whatever you choose for the first flag gives three options for the second, but not the same three options. Feb 11 at 13:53
• @EspeciallyLime yes that is correct. Feb 11 at 13:55
The remaining three colors with be a different set of three colors depending on the color you pick first.
If you pick $$R$$ first the remaining three colors are $$B,G,O$$ for three signals: $$RB,RG,RO$$.
But if you pick $$B$$ first your remain three colors are $$R,G,O$$ for tree signals: $$BR, BG, BO$$.
And if you pick $$G$$ first your remaining three colors are $$R,B,O$$ for $$GR, GB,GO$$.
And if pick $$O$$ first your remaining three colors are $$R,B, G$$ for $$OR,OB, OG$$. Those are the $$12$$.
We didn't have to count them. We knew there where $$4$$ options for the first color, and we knew once we picked the first color there would be $$3$$ remaining colors for the second. That they would be different set of three colors wouldn't matter as long as we knew there'd be $$3$$ remaining colors.
So we calculate $$4\times 3 = 12$$. | 2021-11-27T20:28:48 | {
"domain": "stackexchange.com",
"url": "https://math.stackexchange.com/questions/4021697/confusion-in-the-solution-for-this-question/4021771",
"openwebmath_score": 0.9836623668670654,
"openwebmath_perplexity": 371.67806498291384,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9825575147530352,
"lm_q2_score": 0.8479677622198946,
"lm_q1q2_score": 0.8331770970374723
} |
https://mathforums.com/threads/how-do-i-find-the-weight-of-an-object-in-a-different-planet.348181/ | # How do I find the weight of an object in a different planet?
#### Chemist116
The problem is as follows:
A body has a weight of $w$ in the surface of the Earth. If the object is transported to a planet whose mass and radius is two times that of the Earth. Find its weight.
$\begin{array}{ll} 1.&4w\\ 2.&2w\\ 3.&\frac{w}{2}\\ 4.&\frac{w}{4}\\ 5.&w\\ \end{array}$
How should I calculate the weight of this object?.
On earth the only force acting in the object is given by the weight:
$F=mg=w$
And the gravitational force between two masses is given by:
$F=G\frac{m_1m_2}{r^2}$
Since it mentions that this object is moved to a planet which it has a radius which is two times that of the Earth and a mass double that of Earth then this becomes as:
$F_{2}=G\frac{m_1\cdot 2 m_2}{(2r)^2}=\frac{1}{2}G\frac{m_1m_2}{r^2}$
Therefore:
$w_{Planet}=\frac{1}{2}w_{Earth}$
But this doesn't make sense. What could I be doing wrong?. Shouldn't be the opposite. I mean two times that of the weight from Earth?. Can someone help me here?.
#### skeeter
Math Team
It makes perfect sense. Weight of a mass on the surface is proportional to the mass of the planet and inversely proportional to the square of that planet's radius
On Earth, $w = k \cdot \dfrac{M_e}{R_e^2}$, where $k$ is the constant of proportionality.
On the planet with twice the mass & radius of the Earth ...
$w = k \dfrac{2M_e}{(2R_e)^2} = k \dfrac{2M_e}{4R_e^2} = \dfrac{1}{2} \cdot k \dfrac{M_e}{R_e^2} = \dfrac{1}{2}w$
... the inverse proportionality of the radius squared is the driving factor
#### Chemist116
It makes perfect sense. Weight of a mass on the surface is proportional to the mass of the planet and inversely proportional to the square of that planet's radius
On Earth, $w = k \cdot \dfrac{M_e}{R_e^2}$, where $k$ is the constant of proportionality.
On the planet with twice the mass & radius of the Earth ...
$w = k \dfrac{2M_e}{(2R_e)^2} = k \dfrac{2M_e}{4R_e^2} = \dfrac{1}{2} \cdot k \dfrac{M_e}{R_e^2} = \dfrac{1}{2}w$
... the inverse proportionality of the radius squared is the driving factor
Okay. But when you put an object on let's say Jupiter, wouldn't its weight be more than what is it on Earth?.
Let's say a $46\,g$ golf ball. On the surface of the Earth, its weight would be:
$46\cdot10^-3\cdot 9.8 = 0.4508\,N$
On Jupiter (assuming its mass is $1.898 \times 10^{27}\,kg$ and a radius of $69911\,km$)
$F=(6.67408 \cdot 10^{-11})\left(\frac{46\cdot10^-3\cdot 1.898 \times 10^{27}}{(69911\cdot 10^3)^2} \right)$
$F\approx 1.1922\,N$
The weight increases. Why is it increasing?. It's almost 1.6 times that of the weight on the surface of Earth. Or could it be that am I missunderstanding something?.
#### skeeter
Math Team
First of all, $\dfrac{1.1922}{0.4508} = 2.64$, so the golf ball is about 2.6 times heavier in Jupiter than on Earth.
The value of the gravitational field on the surface of any planet X is $g_X = \dfrac{GM_X}{(R_X)^2}$
On Earth, $g_e = \dfrac{GM_e}{(R_e)^2}$, which yields the familiar $9.8 \, m/s^2$
comparing the two ...
$\dfrac{g_X}{g_e} = \dfrac{M_X (R_e)^2}{M_e (R_X)^2} = \dfrac{M_X}{M_e} \cdot \left(\dfrac{R_e}{R_X} \right)^2$
Let planet X be Jupiter. Looking at each fraction individually ...
$\dfrac{M_J}{M_e} \approx 318$
$\dfrac{R_e}{R_J} \approx 0.09$
$318 \cdot 0.09^2 \approx 2.6$, so gravitational acceleration on Jupiter's surface is 2.6 times greater than Earth's. Nothing mysterious here, it's just the way the numbers work out. In this case, Jupiter's much greater mass is the reason for the greater weight.
topsquark | 2020-03-29T14:21:56 | {
"domain": "mathforums.com",
"url": "https://mathforums.com/threads/how-do-i-find-the-weight-of-an-object-in-a-different-planet.348181/",
"openwebmath_score": 0.7703137993812561,
"openwebmath_perplexity": 334.34271191373523,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9825575162853136,
"lm_q2_score": 0.8479677602988601,
"lm_q1q2_score": 0.8331770964492682
} |
https://math.stackexchange.com/questions/1191702/prove-that-a-subseteq-b-if-and-only-if-a-cap-overlineb-emptyset | # Prove that $A \subseteq B$ if and only if $A \cap \overline{B}=\emptyset.$
Prove that $A \subseteq B$ if and only if $A \cap \overline{B}=\emptyset.$
Proof:
Since $A \cap \overline{B}$ implies that $x \in A$ but $x \notin B$, whilst $A \subseteq B$ implies that $x \in A$ and $x \in B$, we have a contradiction, since $x \in B$ and $x \notin B$. Thus, $A \cap \overline{B}=\emptyset.$
Is this sufficient? How else can this be proved?
• What is $\overline{B}$? Do you mean it to be the complement of $B$? If so, we usually use the notation $B^{c}$ for the complement of $B$, as $\overline{B}$ is usually reserved for the closure of $B$ (in a topological space). Mar 16 '15 at 1:05
• @user46944 Yes it is the complement, sorry this is the notation used in my course
– user265675
Mar 16 '15 at 1:05
• @crash Conflicting notations bother me lol. Although as you said, being extra clear counteracts any conflict. Mar 16 '15 at 1:20
• @crash Off topic, but I'm sorry you had such a bad experience with your $\pi$ day question. I've had one or two bad experiences with questions I've asked, too. I hope you'll reconsider staying on the site (I read your about me). What kind of a site will this become if only the sticklers for the rules remain? Mar 16 '15 at 1:25
• @Meryll I just pulled up this problem from a long while back that is somewhat similar to proving the kind of question you just asked. It may help to read through it to better understand how to prove if and only statements, how the flow of your proof(s) should look, etc. Mar 16 '15 at 1:41
Try mutual subset inclusion.
($\to$): Suppose $x\in A\subseteq B$. Then $x\in A\to x\in B$; that is, $x\not\in A\lor x\in B$. The negation of this is $x\in A\land x\not\in B$; that is, $x\not\in A\cap \overline{B}$.
The other direction is trivial since you are dealing with the empty set.
You have to prove an "if and only if" statement. When you have to prove "$p$ if and only if $q$", it always means you have to prove two statements. You need to prove that $p \implies q$ and $q \implies p$ (where $p$ is the statement $A \subseteq B$ and $q$ is the statement $A \cap \overline{B} = \emptyset$).
Let's prove $p \implies q$ first. To prove $p \implies q$, we need to assume $p$ is true, and prove $q$ is true. So we need to assume $A \subseteq B$. Let's prove $A \cap \overline{B} = \emptyset$. It seems like the easiest way to prove this is by contradiction. Suppose that $A \cap \overline{B} \neq \emptyset$. Then there is some $x \in A \cap \overline{B}$. That means there is some $x$ such that $x \in A$ and $x \not \in B$. But we assumed $A \subseteq B$, which means for all $y \in A$, $y \in B$. But we just found an element $x$ in $A$ that is not in $B$. So we have an element that is both in $B$ and not in $B$, which is a contradiction. Thus, $A \cap \overline{B} = \emptyset$, as desired.
Now let's prove the $q \implies p$ direction. We have to assume $q$ and prove $p$. Suppose $q$ is true, i.e., $A \cap \overline{B} = \emptyset$. Let's prove $A \subseteq B$. To prove this, we need to show if $x \in A$, then $x \in B$. Let $x \in A$. We know $A \cap \overline{B} = \emptyset$, which means if $x \in A$, $x$ can't be in $\overline{B}$. But elements are either in $B$ or $\overline{B}$, since these two sets are complements of each other. That means $x \in B$, which is what we wanted to show. So $A \subseteq B$, as desired.
• Thank you, this is very clear. It has never been taught to me that this is the approach with proving if and only if statements.
– user265675
Mar 16 '15 at 1:34
• @Meryll You're welcome! I'm glad it's clear. Mar 16 '15 at 1:46
First $A\subset B \Longrightarrow\overline{B}\subset \overline{A}\Longrightarrow A\cap \overline{B}\subset A\cap\overline{A}=\emptyset$ and hence $A\cap \overline{B}=\emptyset$. Conversely, $A\cap \overline{B}=\emptyset$ implies $A=A\cap(B\cup \overline{B})=(A\cap B)\cup(A\cap\overline{B})=A\cap B \Longrightarrow A\subset B$.
$(\Rightarrow)$; Suppose $A\subseteq B$. Then $\forall x\in A,x\in B$. Therefore $\forall x\in A,x\notin B^c$. So $A\cap B^{c}=\varnothing$.
$(\Leftarrow)$; Conversely suppose $A\cap B^{c}=\varnothing$. Therefore $\forall x\in A,x\notin B^{c}$. Hence $\forall x\in A,x\in B$. So we have that $A\subseteq B$.
Here I use $B^c$ for your $\overline{B}$ :)
If $A\subseteq B$, then $x\in A \implies x \in B$, and if $x\in B$, then $x\not\in B^c$. So no $x$ be be in $A$ and $B^c$ at the same time, so $A\cap B^c=\emptyset$.
If $A\cap B^c=\emptyset$, we have $B\cup B^c=\omega$, where $\omega$ represents the entire space. If $x\in A, x\not\in B^c$, and so $x\in B$. Therefore $A\subseteq B$. | 2021-10-18T05:04:49 | {
"domain": "stackexchange.com",
"url": "https://math.stackexchange.com/questions/1191702/prove-that-a-subseteq-b-if-and-only-if-a-cap-overlineb-emptyset",
"openwebmath_score": 0.9527373313903809,
"openwebmath_perplexity": 121.64070967909714,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9825575178175918,
"lm_q2_score": 0.8479677583778258,
"lm_q1q2_score": 0.8331770958610639
} |
http://mathhelpforum.com/calculus/147458-caculation-limit.html | # Thread: Caculation a limit ?
1. ## Caculation a limit ?
How do I show $\displaystyle\lim_{x\rightarrow 1^+ }\int_{x^2}^{x^4}\!\frac{1}{\ln t}\, dt=\ln 2$ ?
2. Originally Posted by bigli
How do I show $\displaystyle\lim_{x\rightarrow 1^+ }\int_{x^2}^{x^4}\!\frac{1}{\ln t}\, dt=\ln 2$ ?
If you've already done power series and stuff, we get (Bronshtein-Semendiaev, "Manuel of Mathetmatics") that
$\int\frac{dt}{\ln t} =\ln\ln t+\ln t+\sum^\infty_{n=2}\frac{(\ln t)^n}{n\cdot n!}$
Tonio
3. It doesn't work.Is there any easier way?
4. Originally Posted by bigli
It doesn't work.Is there any easier way?
Oh, it works and very fine , thank you! Perhaps there's an easier way but I can't see it.
Tonio
5. According to Maple, the series expansion of $\frac{1}{\ln t}$ about $t = 1$ is $\frac{1}{t-1} + \frac{1}{2} - \frac{1}{12} (t-1) + \frac{1}{24} (t-1)^{2} - \frac{19}{720} (t-1)^{3} + ...$ (EDIT: You can derive it by starting with the Taylor series for $\ln t$ centered at $t=1$ and using long division.)
Since we're integrating over values close to 1, $\frac{1}{\ln t}$ is approximately $\frac{1}{t-1}$.
so $\lim_{x \to 1^{+}} \int^{x^{4}}_{x^{2}} \frac{1}{\ln t} \ dt = \lim_{x \to 1^{+}} \int^{x^{4}}_{x^{2}} \frac{1}{t-1} \ dt$
$= \lim_{x \to 1^{+}} \ln (t-1) \Big|^{x^{4}}_{x^{2}}$
$=\lim_{x \to 1^{+}} \Big( \ln (x^{4}-1) - \ln (x^{2}-1) \Big)$
$= \lim_{x \to 1^{+}}\ln \Big(\frac{x^{4}-1}{x^{2}-1}\Big)$
$= \lim_{x \to 1^{+}} \ln (x^{2}+1) = \ln 2$
6. Originally Posted by Random Variable
According to Maple, the series expansion of $\frac{1}{\ln t}$ about $t = 1$ is $\frac{1}{t-1} + \frac{1}{2} - \frac{1}{12} (t-1) + \frac{1}{24} (t-1)^{2} - \frac{19}{720} (t-1)^{3} + ...$ (EDIT: You can derive it by starting with the Taylor series for $\ln t$ centered at $t=1$ and using long division.)
Since we're integrating over values close to 1, $\frac{1}{\ln t}$ is approximately $\frac{1}{t-1}$.
And what happened to $+\frac{1}{2}$ above...?
Tonio
so $\lim_{x \to 1^{+}} \int^{x^{4}}_{x^{2}} \frac{1}{\ln t} \ dt = \lim_{x \to 1^{+}} \int^{x^{4}}_{x^{2}} \frac{1}{t-1} \ dt$
$= \lim_{x \to 1^{+}} \ln (t-1) \Big|^{x^{4}}_{x^{2}}$
$=\lim_{x \to 1^{+}} \Big( \ln (x^{4}-1) - \ln (x^{2}-1) \Big)$
$= \lim_{x \to 1^{+}}\ln \Big(\frac{x^{4}-1}{x^{2}-1}\Big)$
$= \lim_{x \to 1^{+}} \ln (x^{2}+1) = \ln 2$
.
7. As $t$ approaches $1$ from the right, $\frac{1}{t-1}$ is much greater than $\frac{1}{2}$. You can include it if you want. You'll still get the same answer.
8. It is fairly easy to verify that for $t>0$ we have that $\frac{t-1}{t}\leqslant\ln(t)\leqslant t-1$ and so $\frac{1}{t-1}\leqslant\frac{1}{\ln(t)}\leqslant\frac{t}{t-1}$. It follows that $\int_{x^2}^{x^4}\frac{dt}{t-1}\leqslant \int_{x^2}^{x^4}\frac{dt}{\ln(t)}\leqslant\int_{x^ 2}^{x^4}\frac{t}{t-1}dt$. Note though that the LHS is just $\ln\left(x^4-1\right)-\ln(x^2-1)=\ln(x^2+1)\overset{x\to 1^+}{\longrightarrow}\ln(2)$ and $\int_{x^2}^{x^4}\frac{t}{t-1}dt=\left(x^4-x^2\right)+\left(\ln(x^4-1)-\ln(x^2-1)\right)\overset{x\to 1}{\longrightarrow}\ln(2)$. The conclusion follows from the Squeeze Theorem.
9. Thanks DREXEL28. Every thing is OK!. But only, your inequalities should be strictly and they are correct for $t>1$ . | 2013-12-07T18:10:35 | {
"domain": "mathhelpforum.com",
"url": "http://mathhelpforum.com/calculus/147458-caculation-limit.html",
"openwebmath_score": 0.9804540872573853,
"openwebmath_perplexity": 299.976384149992,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9825575178175918,
"lm_q2_score": 0.8479677583778258,
"lm_q1q2_score": 0.8331770958610639
} |
http://mathhelpforum.com/statistics/33195-combinatorics.html | # Math Help - Combinatorics
1. ## Combinatorics
How many different ten-digit number can be created using only 2s and 0s which is divisible by 4 and doesn't contain two neighbouring 0s?
I know that the last two digits as a number have to be a multiple of 4 so the number is: $\overline{\dots 20}$.
And how can I count the numbers not containing neighbouring 0s?
2. Originally Posted by james_bond
How many different ten-digit number can be created using only 2s and 0s which is divisible by 4 and doesn't contain two neighbouring 0s?
I know that the last two digits as a number have to be a multiple of 4 so the number is: $\overline{\dots 20}$.
And how can I count the numbers not containing neighbouring 0s?
I have not gone over it carefully since I have to go now, but here is a guess.
Generalise it for: $\mathop {a_n ...a_3 20}\limits^{\_\_\_\_\_\_\_\_\_\_\_\_}$ show that the number of possibilities is [tex] f_n [/Math] (nth fibonacci number) Try using induction (draw a diagram)
3. Originally Posted by james_bond
How many different ten-digit number can be created using only 2s and 0s which is divisible by 4 and doesn't contain two neighboring 0s? I know that the last two digits as a number have to be a multiple of 4 so the number is: $\overline{\dots 20}$.
I assume that 0202020220 will count as such a ten-digit number even though it begins with a zero.
$\sum\limits_{k = 0}^4 {\binom{9-k}{k}}=55$ is the answer.
In the first eight places in the number we can have anywhere from no 0’s(k=0) to at most four 0’s(k=4).
Then there will be 8-k 2’s creating 9-k to place and separate the 0’s.
4. Hello, James!
How many different ten-digit number can be created using only 2s and 0s
which is divisible by 4 and doesn't contain two neighbouring 0s?
I know that the last two digits as a number have to be a multiple of 4
. . so the number ends in: . $\hdots 20$
And how can I count the numbers not containing neighbouring 0s?
I assume that the leading digit cannot be zero.
So we have: . $2\:\_\:\_\:\_\;\_\;\_\;\_\;\_\;2\;0$
. . Hence, we want 7-digit numbers with non-adjacent 0s.
I couldn't find a way to think through this task, so I drew a tree diagram.
. . And here's what I found . . .
. . $\begin{array}{c|c}\text{no. of digits} & \text{no. of ways} \\ \hline
1 & 2 \\ 2 & 3 \\ 3 & 5 \\ 4 & 8 \\ 5 & 13 \\ \vdots & \vdots \end{array}\quad\hdots\quad\text{These are Fibonacci numbers!}$
Hence, there are 34 seven-digit numbers with non-adjacent 0s.
Therefore, the answer is: . ${\bf{\color{blue}34}}$
~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~
Plato allowed a leading zero, so his result is the next Fibonacci number, 55. | 2015-09-01T06:52:15 | {
"domain": "mathhelpforum.com",
"url": "http://mathhelpforum.com/statistics/33195-combinatorics.html",
"openwebmath_score": 0.7843194007873535,
"openwebmath_perplexity": 304.64259130072645,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES\n\n",
"lm_q1_score": 0.9825575178175919,
"lm_q2_score": 0.8479677564567913,
"lm_q1q2_score": 0.8331770939735372
} |
https://math.stackexchange.com/questions/1837931/find-the-value-of-h-if-x2-y2-h/1837938 | # Find the value of $h$ if $x^2 + y^2 = h$
Consider equation $x^2 + y^2 = h$ that touches the line $y=3x+2$ at some point $P$. Find the value of $h$
I know that $x^2 + y^2 = h$ is a circle with radius $\sqrt{h}$.
Also, since $y = 3x + 2$ is a tangent, we know that the slope of the radius perpendicular to the tangent is $M_{OP}= -\frac{1}{3}$. I'm not sure how I can determine the value of $h$ though?
You're almost there! We want to find the equation of line $OP$. We know its slope, and (since it passes through the origin) we know its $y$-intercept, so its equation is: $$y = \tfrac{-1}{3}x$$ We can now find the intersection point of the radius and tangent by solving the system of equations. Equating, we obtain: \begin{align*} \tfrac{-1}{3}x = y = 3x + 2 &\implies -x = 9x + 6 \\ &\implies -10x = 6 \\ &\implies x = \tfrac{-3}{5} \\ &\implies y = \tfrac{-1}{3} \cdot \tfrac{-3}{5} = \tfrac{1}{5} \end{align*} Thus, we conclude that: $$h = x^2 + y^2 = (\tfrac{-3}{5})^2 + (\tfrac{1}{5})^2 = \tfrac{9 + 1}{25} = \boxed{\tfrac{2}{5}}$$
So we have equation $x^2 + y^2 = r^2$, who’s geometrical representation would be a circle with the radius $r$. Now we also have an equation of a line represented by: $y = mx + c$ that touches the circle at point $P$. To find this point we will use the method of solving simultaneous equation where one is a quadratic and the other is linear so we will take the value of y and replace it in the first equation as follows:
$$x^2 + (3x + 2)^2 = h$$ $$x^2 + 9x^2 + 2*3x*2 + 4 = h$$ $$10x^2 + 12x + 4 = h$$ $$10x^2 + 12x + 4 - h = 0$$
so now we will have something like:
$$x^2 + (mx + c)^2 = r^2$$ $$x^2 + m^2x^2 + 2mcx + c^2 = r^2$$ $$(1 + m^2)x^2 + 2mcx + c^2 - r^2 = 0$$
since this equation is a quadratic equation in x, something like $(ax^2 + bx + c = 0)$ where we have:
$a = 10$; $b = 12$; $c = 4 - h$;
using the values in discriminant $(b^2 - 4ac)$:
$$b^2 - 4ac = (2mc)^2 - 4(1 + m^2)(c^2 - r^2)$$ $$= 4m^2c^2 - 4(c^2 + m^2c^2 - r^2 - m^2r^2)$$ $$= 4m^2c^2 - 4c^2 - 4m^2c^2 + 4r^2 + 4m^2r^2$$ $$= -4c^2 + 4r^2 + 4m^2r^2$$
if the discriminant is zero, our equation will have equal roots and the line intersects the circle in one single point which is our case also. So because this line is a tangent to the circle we know:
$$-4c^2 + 4r^2 + 4m^2r^2 = 0$$ $$4r^2 + 4m^2r^2 = 4c^2$$ $$c^2 = r^2 + m^2r^2$$
this will be the condition for a line to be tangent to the circle, so replacing with our values we will have the tangent:
$$(4 - h)^2 = h + 3^2h$$ $$h^2 - 8h + 16 = h + 9h$$ $$h^2 - 18h = -16$$ $$h(h - 18) = -16$$
since: $$(h-9)^2 = (h-9)(h-9)$$ $$h^2 - 18h + 81$$ $$h(h -18) + 81$$ we can write:
$$(h-9)^2 - 65 = 0$$ so:
$$h = 9 \pm \sqrt65$$
I hope this helps
Hint:
What you are actually doing is trying to find the distance from $(0,0)$ to the line. Did this rephrasing of the problem help you?
The condition requires that the equation $x^2+(3x+2)^2 - h = 10x^2+12x+4-h=0$ has a double root. Thus $\triangle' = 0\implies 6^2-10(4-h) = 0$. Can you find $h$ from this linear equation?
Hint:
Take the system: $$\begin{cases} x^2+y^2=h\\ y=3x+2 \end{cases}$$
whose solutions are the coordinates of the common points of the circle and the line. If the discriminant of this system is $=0$ it has a double solution, so the line is tangent to the circle.
Hint
$h$ is fixed. Let $A=\{(x,y)\mid x^2+y^2=h\}$ and $B=\{(x,y)\mid y=3x+2\}$. $$(x,y)\in A\cap B\implies x^{2}+(3x+2)^2=h\implies ...\implies x=...$$
The center of your circle is $(0\mid 0)\quad$.
The distance from the center to your line is $\frac2{\sqrt{10}}\quad$, which is equal to $\sqrt{h}\quad$.
$\sqrt{h}=\frac2{\sqrt{10}}\quad$, so $h=\frac4{10}=\frac25\quad$.
Touching means $x^2 + y^2 = h$ and $y = 3x + 2$ intersect along two coincident points, with two roots equal.
So eliminate $y$ between them and set the discriminant of quadratic in $x$ to zero, you get $h = \frac25$.
There are couple of approaches already mentioned in other answers, but I will try to summarize them and add (probably unwarranted) generality. Also, I will describe approach using calculus.
We are given a line $y = ax + b$ and a circle centered $(p,q)$ given with $(x-p)^2+(y-q)^2 = h$ and we are looking for $h$ such that given line is tangent to the circle.
1. Calculus
We have function $y(x)$ for which relation $(x-p)^2+(y-q)^2 = h$ holds. If we differentiate the relation with respect to $x$ we get $$2(x-p)+2(y-q)y'=0\implies y'=-\frac{x-p}{y-q}$$ Now, since we are looking for a tangent given with $y = ax +b$, we are looking for a point $x$ such that $y'(x) = a$ and $y(x) = ax +b$. Thus we get $$a = -\frac{x-p}{ax + b - q}\implies x =\frac{p-ab+aq}{1+a^2}$$ and also $$y = ax + b = \frac{b + ap + a^2 q}{1+a^2}$$ Now, we get $$\boxed{h = (x-p)^2 + (y-q)^2 = \frac{(b+ap-q)^2}{1+a^2}}$$
2. Tangent of a circle is perpendicular to its radius
Now, the line passing through center $(p,q)$ and is perpendicular to $y = ax + b$ is given by formula $$y - q = -\frac 1 a(x-p)$$ and we can find the intersection with the original line by solving linear system
\begin{align} y &= ax + b \\ y - q &= -\frac 1 a(x-p)\end{align}
to get $x = \frac{p - ab + aq}{a^2 + 1},\ y = \frac{b + ap + a^2 q}{a^2 + 1}$ and if we plug this into $(x-p)^2+(y-q)^2 = h$ we get $$\boxed{h = \frac{(b+ap-q)^2}{1+a^2}}$$
3. $h$ is the square of distance of the center to the line
We find this approach in answers by Senex Ægypti Parvi and pajonk.
We start with line $y = ax + b$ and point $(p,q)$. Square of (Euclidean) distance of any point on the line $y = ax + b$ to the point $(p,q)$ is given by
\begin{align} (x-p)^2 + (y-q)^2 &= (x-p)^2 +(ax+b-q)^2 \\ &= (1 + a^2) x^2 + 2( a b - p - a q) x+ (b^2 + p^2 - 2 b q + q^2)\end{align} and $h$ is given by minimal distance.
Now, remember that any quadratic $\alpha x^2 + \beta x + \gamma$ can be written in the form $$\alpha x^2 + \beta x + \gamma = \alpha(x + \frac\beta{ 2 \alpha})^2 + \gamma - \frac{\beta^2} {4 \alpha}\geq \gamma - \frac{\beta^2}{4\alpha}$$ and thus the minimum of quadratic polynomial ($\alpha > 0$) is given with $\gamma - \frac{\beta^2}{4\alpha}$ and if we set $\alpha = 1+ a^2,\ \beta = 2( a b - p - a q),\ \gamma = b^2 + p^2 - 2 b q + q^2$ we get $$\boxed{h = \gamma - \frac{\beta^2}{4\alpha} = \frac{(b+ap-q)^2}{1+a^2}}$$
4. Tangent is a line that intersects circle at exactly one point
I personally like this approach the best because it beautifully intertwines geometry and algebra. This is suggested in answers by cosmin, Emilio Novati and DeepSea.
We want to find intersections of the line $y = ax + b$ and circle $(x-p)^2+(y-q)^2 = h$ so we solve the appropriate system
\begin{align} (x-p)^2+(y-q)^2 &= h\\ y &= ax + b \end{align}
and by substituting $y = ax + b$ in the first equation we get $$(x-p)^2+(ax + b - q)^2 - h = 0$$ and analogously to the previous case we get quadratic equation $$(1 + a^2) x^2 + 2( a b - p - a q) x+ (b^2 + p^2 - 2 b q + q^2 - h) = 0$$
It is well known that there are 3 general cases for the solutions of quadratic equation (with real coefficients) depending on the discriminant $D$:
\begin{array}{ c | c | c } \textbf{discriminant} & \textbf{algebraic interpretation} & \textbf{geometric interpretation}\\ \hline D>0 & \text{there are 2 real solution} & \text{line intersects the circle at 2 points} \\ D=0 & \text{there is a double real solution} & \text{line intersects the circle at 1 point, i.e. it is a tangent} \\ D<0 & \text{there are 2 complex solutions} & \text{line does not intersect the circle } \end{array}
We are, thus, interested in case 2), i.e. we want the discriminant of our quadratic equation to be $0$, i.e.:
$$(2 a b - 2 p - 2 a q)^2 = 4 (1 + a^2) (-h + b^2 + p^2 - 2 b q + q^2) \iff \boxed{h = \frac{(b+ap-q)^2}{1+a^2}}$$
TL;DR
What we actually calculated is a well known formula from high school:
For a line $Ax + By + C = 0$, distance of point $(p,q)$ from the line is given by formula $$d = \frac{| Ap + Bq + C|}{\sqrt{A^2 + B^2}}$$
How to get the formula from our calculation of $h$? Well:
$$Ax + By + C = 0\implies y = -\frac AB x -\frac CB\implies h = \frac{(-C-Ap-Bq)^2}{B^2(1+\frac{A^2}{B^2})} = \frac{(Ap+Bq+C)^2}{A^2 + B^2}$$ Thus, probably the quickest way to solve your problem is
5. High school mathematics
Transform $y = 3x + 2$ into $3x - y + 2 = 0$, and put it in the above formula with $p = q = 0$ to get $$h = \frac{(3\cdot 0 - 1\cdot 0 + 2)^2}{3^2 + (-1)^2} = \frac 2 5$$
The slope of the perpendicular radius is $-\dfrac 13$ and it passes through the center of the circle, $(0,0)$. So its equation is $y-0 = -\dfrac 13(x - 0)$, which simplifies to $y = -\dfrac 13x$. Avoiding fractions, we write this as $x = -3y$.
We compute the intersection
\begin{align} x &= -3y \\ y &= 3x + 2 \\ \hline y &= -9y + 2\\ y &= \dfrac 15 \\ x &= -\dfrac 35 \end{align}
So $h = x^2 + y^2 = \dfrac{10}{25} = \dfrac 25$ | 2019-12-08T11:09:28 | {
"domain": "stackexchange.com",
"url": "https://math.stackexchange.com/questions/1837931/find-the-value-of-h-if-x2-y2-h/1837938",
"openwebmath_score": 0.9987107515335083,
"openwebmath_perplexity": 185.54816906147602,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9825575193498701,
"lm_q2_score": 0.8479677545357568,
"lm_q1q2_score": 0.8331770933853327
} |
http://mathhelpforum.com/calculus/92437-partial-integration-problem.html | 1. ## partial integration problem
Hello!
i have difficulties sovling the following equation:
$\displaystyle \int \frac{ln(x)} {\sqrt{x}} dx$
the actual integration part isnt the problem:
$\displaystyle ln(x)*2\sqrt{x} \int x^\frac{-2}{3} dx$
simplifying the integral: $\displaystyle \int x^\frac{-3}{2}$
and after integrating im here:
$\displaystyle ln(x)*2\sqrt{x} * (-\frac{2}{\sqrt(x)})+C$
the solution is supposed to be $\displaystyle 2*ln(x)*\sqrt(x)-4\sqrt(x)+C$
i dont know how to get this result
2. Originally Posted by coobe
Hello!
i have difficulties sovling the following equation:
$\displaystyle \int \frac{ln(x)} {\sqrt{x}} dx$
the actual integration part isnt the problem:
$\displaystyle ln(x)*2\sqrt{x} \int x^\frac{-2}{3} dx$
simplifying the integral: $\displaystyle \int x^\frac{-3}{2}$
and after integrating im here:
$\displaystyle ln(x)*2\sqrt{x} * (-\frac{2}{\sqrt(x)})+C$
the solution is supposed to be $\displaystyle 2*ln(x)*\sqrt(x)-4\sqrt(x)+C$
i dont know how to get this result
$\displaystyle \int \frac{ln(x)} {\sqrt{x}} dx$
so let
$\displaystyle u=\ln(x) \implies du =\frac{1}{x}dx$
$\displaystyle dv=\frac{1}{\sqrt{x}}dx=x^{-1/2}dx \implies v =2\sqrt{x}dx$
$\displaystyle \int \frac{ln(x)} {\sqrt{x}} dx=2\ln(x) \sqrt{x}-2\int x^{-1/2}dx$
$\displaystyle \int \frac{ln(x)} {\sqrt{x}} dx=2\ln(x) \sqrt{x}-4\sqrt{x}+c$
3. Originally Posted by coobe
Hello!
i have difficulties sovling the following equation:
$\displaystyle \int \frac{ln(x)} {\sqrt{x}} dx$
the actual integration part isnt the problem:
$\displaystyle ln(x)*2\sqrt{x} \int x^\frac{-2}{3} dx$
simplifying the integral: $\displaystyle \int x^\frac{-3}{2}$
and after integrating im here:
$\displaystyle ln(x)*2\sqrt{x} * (-\frac{2}{\sqrt(x)})+C$
the solution is supposed to be $\displaystyle 2*ln(x)*\sqrt(x)-4\sqrt(x)+C$
i dont know how to get this result
let
$\displaystyle lnx=u...x=e^u...dx=e^udu...x^{\frac{1}{2}}=e^{\fra c{u}{2}}$
$\displaystyle \int \frac{u(e^u)}{e^{\frac{u}{2}}} du$$\displaystyle =\int u(e^{u-\frac{u}{2}}) du$
$\displaystyle \int ue^{\frac{u}{2}} du =$
by parts$\displaystyle dy=e^{\frac{u}{2}} ...y=2e^{\frac{u}{2}}...and...t=u...dt=du$
$\displaystyle 2ue^{\frac{u}{2}}-2\int e^{\frac{u}{2}} du$
$\displaystyle 2ue^{\frac{u}{2}} -2(2)e^{\frac{u}{2}} + c$
sub $\displaystyle u=ln(x).......e^{\frac{u}{2}}=\sqrt{x}$
someone is faster than me lol
4. Hello, coobe!
I'm not sure what you did . . .
$\displaystyle \int \frac{\ln x} {\sqrt{x}}\,dx$
We have: .$\displaystyle I \;=\;\int\ln x\left(x^{-\frac{1}{2}}\,dx\right)$
Integrate by parts: .$\displaystyle \begin{array}{ccccccc} u &=&\ln x & & dv &=& x^{-\frac{1}{2}}dx \\ du &=& \frac{dx}{x} & & v &=& 2x^{\frac{1}{2}} \end{array}$
Then: .$\displaystyle I \;\;=\;\;2\!\cdot\!x^{\frac{1}{2}}\!\cdot\!\ln x - \int\left(2x^{\frac{1}{3}}\right)\left(\frac{dx}{x }\right)$
. . . . . $\displaystyle =\;\;2\!\cdot\!x^{\frac{1}{2}}\!\cdot\!\ln x - 2\int x^{-\frac{1}{2}}\,dx$
. . . . . $\displaystyle = \;\;2\cdot\ln x\cdot\!x^{\frac{1}{2}} - 4x^{\frac{1}{2}} + C$
. . . . . $\displaystyle = \;\;2\cdot\ln x\!\cdot\!\sqrt{x} - 4\sqrt{x} + C$
5. d'oh.... i had a wrong formula on my Formula Sheet... took me so much time to figure that out
thanks everybody, i simply had a wrong formula for partial integrating...
good thing i noticed that before the exam | 2018-05-27T17:00:17 | {
"domain": "mathhelpforum.com",
"url": "http://mathhelpforum.com/calculus/92437-partial-integration-problem.html",
"openwebmath_score": 0.9969142079353333,
"openwebmath_perplexity": 7743.79171950716,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9825575147530351,
"lm_q2_score": 0.8479677583778258,
"lm_q1q2_score": 0.8331770932624186
} |
https://math.stackexchange.com/questions/3322216/what-are-the-eigenvalues-of-x-xxt-x-in-mathbbrd | # What are the eigenvalues of $X = xx^{T}, x\in\mathbb{R}^{d}$? [duplicate]
This question already has an answer here:
I'm given the matrix $$X = xx^{T}\in\mathbb{R}^{d \ x \ d}, x\in\mathbb{R}^{d}$$. Does somebody know how to compute $$\lambda_{max}(X)$$ or $$\lambda_{min}(X)$$? I only want to know these two eigenvalues, the others are not really important.
I seem to be stuck.
I'm thankful for any answer.
## marked as duplicate by amd, vonbrand, воитель, blub, Lee David Chung LinAug 14 at 1:25
This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.
## 3 Answers
Assuming $$x$$ is non-zero, $$X$$ has rank $$1$$, so almost all the eigenvalues are $$0$$. As for the single non-zero eigenvalue, consider what $$Xx$$ becomes.
• Another approach: the sum of the eigenvalues is $$\operatorname{tr}(xx^T) = \operatorname{tr}(x^Tx) = x^Tx$$ – Omnomnomnom Aug 13 at 15:50
• I would say that $\lambda = x^{T}x$ is an eigenvalue of X, because $Xx = (xx^{T})x = x(x^{T}x) = (x^{T}x)x$. So this means $\lambda_{min}(X) = \lambda_{max}(X) = x^{T}x$. – Michael W. Aug 13 at 16:04
• @MichaelW. I would say that $\lambda_{\min}(X)=0$, but otherwise, yes, that is exactly what I'm hinting at. – Arthur Aug 13 at 16:08
• Yes sry. One follow up question. If I have a scalar before the matrix, meaning $\alpha{X} = \alpha{xx^{T}}$, then we would have $\lambda_{min}(\alpha{X}) = 0$ and $\lambda_{max}(\alpha{X}) = \alpha{x^{T}x}$? – Michael W. Aug 13 at 16:16
• @MichaelW. The eigenvector hasn't changed, the calculation stays the same, and you end up with $(\alpha X)x=(\alpha x^Tx)x$. So yes. – Arthur Aug 13 at 16:26
I can't add a comment to @Arthurs's answer because I don't have enough reputation, but I wanted to add my 2 cents anyway :)
Say you had another vector $$\mathbf{y}$$, with the same dimensions as $$\mathbf{x}$$. In that case $$\mathbf{x^{T}y}=y_{proj_{x}}$$, where $$y_{proj_{x}}$$ is the dot product between $$\mathbf{x}$$ and $$\mathbf{y}$$, or in other words the modulus of the projection of $$\mathbf{y}$$ on $$\mathbf{x}$$.
Similarly, the outer-product of $$\mathbf{x}$$ gives you your $$\mathbf{X}$$ matrix: $$\mathbf{X}=\mathbf{xx^{T}}$$. If you take a moment to look at it, $$\mathbf{Xy}=\mathbf{xx^{T}y}=\mathbf{x(x^{T}y)}=\mathbf{x}y_{proj_{x}}$$
Which is the projection of $$\mathbf{y}$$ on $$\mathbf{x}$$, so through this intuition you can say that the only eigenvector is the projection direction $$\mathbf{x}$$
Edit: Just realized that in order for it to be a projection, $$\mathbf{x}$$ should be a unit vector, otherwise it scales the projection by $$\left\|\mathbf{x}\right\|^2$$, so that would be the eigenvalue: $$\left\|\mathbf{x}\right\|^2$$
We assume
$$x \ne 0, \tag 1$$
lest
$$X = xx^T = 0, \tag 2$$
and the problem is trivial. For
$$x \ne 0, \tag 3$$
we have
$$Xx = (xx^T)x = x(x^Tx) = (x^Tx)x, \tag 4$$
and we see that
$$x^Tx > 0 \tag 5$$
is an eigenvalue of $$X = xx^T$$ with associated eigenvector $$x$$.
Now if
$$0 \ne y \in \Bbb R^d \tag 6$$
is such that
$$x^Ty = 0, \tag 7$$
then
$$Xy = (xx^T)y = x(x^Ty) = (0)y = 0, \tag 8$$
i.e. $$0$$ is an eigenvalue of $$X$$ with eigenvector $$y$$. The mapping
$$x^T(\cdot): \Bbb R^d \to \Bbb R, \; y \to x^Ty \tag 9$$
is a linear functional on $$\Bbb R^d$$ and as such
$$\dim \ker x^T(\cdot) = d - 1; \tag{10}$$
thus the $$0$$-eigenspace of $$X$$, which is $$\ker x^T(\cdot)$$, is of dimension $$d - 1$$. Having exhausted the number of available dimensions of $$\Bbb R^d$$, we conclude that $$x^Tx$$ is an eigenvalue of multiplicity $$1$$, whilst the eigenvalue $$0$$ is of multiplicity $$d - 1$$; there are no other eigenvalues or eigenvectors of $$X$$. | 2019-08-25T13:14:57 | {
"domain": "stackexchange.com",
"url": "https://math.stackexchange.com/questions/3322216/what-are-the-eigenvalues-of-x-xxt-x-in-mathbbrd",
"openwebmath_score": 0.9124364852905273,
"openwebmath_perplexity": 245.8675570772971,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9825575188391107,
"lm_q2_score": 0.8479677526147223,
"lm_q1q2_score": 0.8331770910646984
} |
http://mfleck.cs.illinois.edu/study-problems/number-theory/number-theory-2-solx.html | # Number Theory: Problem 2
### Model solution
Let p, q, and r be integers (p non-zero). Suppose that $$p \mid 3q$$ and $$3q \mid r$$.
By the definition of divides, $$p | 3q$$ means pm = 3q for some integer m. Similarly, $$3q | r$$ means 3qn = r for some integer n.
Now substitute these two equations into the expression 3q+r: $$3q + r = 3q + 3qn = 3q(1 + n) = pm(1+n)$$
We know m(1+n) is an integer, since m and n are integers. So let s=m(1+n). Then 3q+r = ps. Therefore, by the definition of divides, $$p | (3q + r)$$, which is what we wanted to show.
### Also ok
It's ok to write the last paragraph without introducing the extra variable s. For example:
We know m(1+n) is an integer, since m and n are integers. So the above equation shows that 3q+r can be written as p times an integer. Therefore, by the definition of divides, $$p | (3q + r)$$, which is what we wanted to show.
### Self-check
At the start of the proof, did you use two different variables m and n when expanding the definition of "divides"?
When you invoke the definition of divides near the end, it's critical that m(1+n) is an integer. You need to make this clear to the reader, by specifying m and n as integers when you introduce them and briefly saying why m(1+n) is an integer later on.
### Poor style 1
Proofs like the following will lose points due to lack of connector words and brief explanations. How is the poor grader supposed to know what you are trying to do at each step?
Let p, q, and r $$\in \mathbb{Z}$$
$$p \mid 3q$$ and $$3q \mid r$$.
pm = 3q, $$m \in \mathbb{Z}$$
3qn = r, $$n \in \mathbb{Z}$$
$$3q + r = 3q + 3qn = 3q(1 + n) = pm(1+n)$$
$$m(1+n) \in \mathbb{Z}$$
So $$p | (3q + r)$$
### Poor style 2
Here's a more subtle example of poor style. It does have connector words, but the author has forgotten everything he learned in high school English class. Paragraphs, punctuation, and whitespace are critical to making technical arguments readable. Everyone makes occasional mistakes, and you'll make a few extras if your first language isn't English. But all of you can do a better job than this.
let p, q, and r be integers (p non-zero) suppose that $$p \mid 3q$$ and $$3q \mid r$$.
by definiton divides, $$p | 3q$$ means pm = 3q, m integer. $$3q | r$$ means 3qn = r for some integer n. now substitute these two equations into the expression 3q+r $$3q + r = 3q + 3qn = 3q(1 + n) = pm(1+n)$$ We know m(1+n) is an integer, since m and n are integers solet s=m(1+n) then 3q+r = ps therefore by the definiton of divides $$p | (3q + r)$$ QED | 2018-01-22T00:25:02 | {
"domain": "illinois.edu",
"url": "http://mfleck.cs.illinois.edu/study-problems/number-theory/number-theory-2-solx.html",
"openwebmath_score": 0.6963847875595093,
"openwebmath_perplexity": 967.3929557081673,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9825575116884779,
"lm_q2_score": 0.8479677583778258,
"lm_q1q2_score": 0.833177090663773
} |
http://openstudy.com/updates/55e8f68ae4b022720612dddc | ## mathmath333 one year ago There are 25 points on a plane of which 7 are collinear . How many quadrilaterals can be formed from these points ?
1. mathmath333
\large \color{black}{\begin{align} & \normalsize \text{ There are 25 points on a plane of which 7 are collinear . }\hspace{.33em}\\~\\ & \normalsize \text{ How many quadrilaterals can be formed from these points ?}\hspace{.33em}\\~\\ \end{align}}
2. mathmate
So there are two groups of points, 7 collinear, and 18 non-colinnear.
3. mathmath333
yes
4. mathmate
And there cannot be 3 or more points in a quad which are collinear.
5. mathmate
See if you can think along those lines. I'll be back.
6. mathmath333
18C4
7. jim_thompson5910
Let A,B,C,D,E,F,G be the 7 collinear points. Let H through Y be the other non-collinear points. Maybe break it up into cases Case 1: All four points are chosen from the set {H, I, J, ..., X, Y} Case 2: Three points are chosen from the set {H, I, J, ..., X, Y} while one point is chosen from {A,B,C,D,E,F,G} Case 3: Two points are chosen from the set {H, I, J, ..., X, Y} and two points are chosen from {A,B,C,D,E,F,G} It's impossible to have 1 point chosen from the set {H, I, J, ..., X, Y} and have 3 points chosen from {A,B,C,D,E,F,G} because you'd have a triangle forming instead of a quadrilateral. You cannot choose 4 points from {A,B,C,D,E,F,G} because you'd have a line only.
8. dan815
|dw:1441331618617:dw|
9. dan815
i think jim has the right idea
10. dan815
18 choose 4 + 18 choose 2 * 7 choose 2 + 18 choose 3 * 7 choose 1
11. jim_thompson5910
I agree with 18 choose 4 + 18 choose 2 * 7 choose 2 + 18 choose 3 * 7 choose 1 use this formula $\Large _n C _r = \frac{n!}{r!(n-r)!}$
12. mathmate
And use $$\Large \frac{\left(\begin{matrix}A \\ i\end{matrix}\right)\left(\begin{matrix}B \\ n-i\end{matrix}\right)}{\left(\begin{matrix}A+B \\ n\end{matrix}\right)}$$ A=18, B=7, n=4, i=2,3,4
13. mathmath333
u didnt add ----18 choose 1 * 7 choose 3 ---- ??
14. dan815
for k colinnear and n total points to be |dw:1441331943171:dw|
15. dan815
who doesnt like general formulas hehe
16. beginnersmind
For some collection of 4 points you can form more than one quadrilatereal.|dw:1441332129959:dw|
17. mathmate
Oh, sorry, forget the denominator. That's for probability!
18. jim_thompson5910
u didnt add ----18 choose 1 * 7 choose 3 ---- ?? @mathmath333 like I said, "It's impossible to have 1 point chosen from the set {H, I, J, ..., X, Y} and have 3 points chosen from {A,B,C,D,E,F,G} because you'd have a triangle forming instead of a quadrilateral"
19. dan815
oh that is true :O
20. mathmath333
oh i see that thanks for that
21. mathmate
$$\Large \left(\begin{matrix}A \\ i\end{matrix}\right)\left(\begin{matrix}B \\ n-i\end{matrix}\right)$$
22. jim_thompson5910
based on what beginnersmind drew, it looks like order matters
23. dan815
i think they didnt consider than when they asked the question tbh lol
24. dan815
cause it really confuses the issue, on how many quads are possible with the given set of 4 arb points
25. beginnersmind
@jim_thompson5910 It depends on the exact position of the points. For most it doesn't.
26. mathmate
But we're just choosing the four points, order does not count, even though there is only one way to make the correct quad.
27. mathmath333
i got 11985
28. dan815
29. mathmath333
yes 11985 is an answer choice
30. dan815
give me the other ones too just in case
31. dan815
is there anything that is exactly 3 times that?
32. dan815
or some multiple of that
33. mathmath333
a.) 5206 b.) 2603 c.) 13015 d.) 11985
34. mathmate
Yes, I have 11985.
35. dan815
okay u r done
36. mathmath333
thnks all | 2016-10-28T00:36:34 | {
"domain": "openstudy.com",
"url": "http://openstudy.com/updates/55e8f68ae4b022720612dddc",
"openwebmath_score": 0.8813705444335938,
"openwebmath_perplexity": 2050.5399199191816,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9825575160299339,
"lm_q2_score": 0.8479677545357568,
"lm_q1q2_score": 0.8331770905701339
} |
https://math.stackexchange.com/questions/1439170/find-all-natural-numbers-n-1-and-m-1-such-that-135-cdots2n-1?noredirect=1 | Find all natural numbers $n > 1$ and $m > 1$ such that $1!3!5!\cdots(2n - 1)! = m!$
Find all natural numbers $n > 1$ and $m > 1$ such that $1!3!5!\cdots(2n - 1)! = m!$
I have been thinking about coming up with some inequalities which would narrow the possible range of pairs $(n, m)$, however the best I have been able to find so far are $m \ge 2n - 1$ and $m \lt n^2$, which is clearly not enough.
Update.
The question has tag combinatorics as it is from a book about combinatorics, so there must be at least partly combinatorial solution.
• Did you try to brute force an answer? – 5xum Sep 17 '15 at 7:24
• Well, I even know that the correct answer is (2, 3), (3, 6), (4, 10). The question is why are there not any other pairs. – Artyom Dmitriev Sep 17 '15 at 7:28
• @ArtyomDmitriev, out of curiosity, even if the problem is interesting by itself, does such a product of factorial come from a "natural problem"? – Clément Guérin Sep 17 '15 at 7:38
• @ClémentGuérin I would also be interested to know that, unfortunately I do not. I have tried to solve it as an exercise. – Artyom Dmitriev Sep 17 '15 at 7:40
Let $$N:=m! =1!\>3!\>5!\cdots(2n-1)!$$ for certain numbers $m$, $n\in{\mathbb N}_{\geq1}$, and denote by $p$ the exponent of $2$ in the prime decomposition of $N$. Then one has on the one hand $$p=\left\lfloor{m\over2}\right\rfloor+\left\lfloor{m\over4}\right\rfloor+\ldots <m$$ and on the other hand $$p\geq\sum_{k=1}^n\left\lfloor{2k-1\over2}\right\rfloor={(n-1)n\over 2}\ .$$ This implies $$m>{(n-1)n\over2}\ .$$ On the other hand, by Bertrand's postulate, we must have $m<2(2n-1)$, because otherwise $m!$ would contain a prime factor not present in $(2n-1)!$. Now $${(n-1)n\over2}<2(2n-1)$$ enforces $n\leq8$. But we can do better, since for small values of $2n-1$ there are many more primes available than Bertrand's postulate guarantees. We therefore set up the following table: $$\matrix{n&&1&2&3&4&5&6&7&8\cr 2n-1&&1&3&5&7&9&11&13&15\cr m>&&0&2&3&6&10&15&21&28\cr}$$ This table show that already for $n\geq5$ any "admissible" $m!$ would contain a prime factor $>2n-1$. So it remains to check the cases $n\in[4]$, which lead to $$N\in\{1, \>6, \>720, \>3628800\}=\{1!,\>3!,\>6!,\>10!\}\ .$$ It follows that there are exactly $4$ pairs $(m,n)$ of the required kind.
Note that every prime divisor of $1!3!5!\cdots(2n-1)!$ occurs at least twice, except for perhaps $2n-1$ (if it's prime). By Bertrands Postulate, we already know that $m!$ has one prime divisor that occurs only once. (See Can n! be a perfect square when n is an integer greater than 1?) A slightly stronger version of the theorem will give that there are at least two primes in $m!$ with multiplicity $1$. Indeed, Wikipedia says that for $x\geq25$ there are at least two primes between $x$ and $(1+0.2)^2x=1.44x$. Thus there are no solutions for $m\geq50$ (and you might even reduce this upper bound by checking some cases by hand - that is, looking for $m$ such that there are at least two prime divisors of $m!$ with multiplicity one).
The only solutions are $(2,3),(3,6),(4,10)$.
• My other answer here avoids using the stronger form of Bertrand's Postulate. – punctured dusk Sep 17 '15 at 8:59
Idea: show that $1!3!5!\cdots(2n-1)!\geqslant(4n-1)!$, so that $m\geq4n-1$. By Bertrands Postulate it will follow that there are no solutions because there is a prime between $2n-1$ and $m$.
It suffices that $1!3!5!\cdots(2n-3)!\geqslant2n(2n+1)\cdots(4n-1)$. Let $n\geq12$. We have
$$\begin{array}{c}(2n-3)!&=1\cdot&\cdots&n\cdot&n+1&\cdots&\cdot(2n-5)&\cdot(2n-4)&\cdot(2n-3)\\ &\geq&2^{n-2}&\cdot\frac{2n}2&\cdot\frac{2n+2}2&\cdots&\cdot\frac{4n-10}2&\cdot\frac{4n-8}2&\cdot\frac{4n-6}2\\ (2n-5)!&=1\cdot&\cdots&n\cdot&n+1&\cdots&\cdot(2n-5)\\ &\geq&2^{n-4}&&\cdot\frac{2n+1}2&\cdots&\cdot\frac{4n-11}2\\ \end{array}$$ so $(2n-5)!(2n-3)!\geq2n(2n+1)\cdots(4n-10)\cdot(4n-8)(4n-6)$. There are $7$ factors missing to get $2n(2n+1)\cdots(4n-1)$. We'll get these factors from the remaining factorials.
Since $x!\geq2\cdot3x\geq2(x+10)$ for $x\geq5$, we have $$\begin{array}{l}&(2n-19)!&(2n-17)!&(2n-15)!&(2n-13)!&(2n-11)!&(2n-9)!&(2n-7)!\\ \geq&(4n-9)&\cdot(4n-7)&\cdot(4n-5)&\cdot(4n-4)&\cdot(4n-3)&\cdot(4n-2)&\cdot(4n-1)\end{array}$$ (note that $2n-19\geq5$ because $n\geq12$) so $(2n-19)!\cdots(2n-3)!\geq2n(2n+1)\cdots(4n-1)$.
Checking $n<12$:
for $m>1$, $m!$ has a prime divisor with multiplicity $1$ so it suffices to check primes $2n-1$:
• $2n-1=19$: $23$ is prime, so $m\leq22$. But $5!7!9!>20\cdot21\cdot21$, so LHS>RHS.
• $2n-1=17$: $19$ is prime, so $m\leq18$. But $5!>18$, so LHS>RHS.
• $2n-1=13$: $17$ is prime, so $m\leq16$. But $5!7!9!>14\cdot15\cdot16$, so LHS>RHS.
• $2n-1=11$: $13$ is prime, so $m\leq12$. But $5!>12$, so LHS>RHS.
• $2n-1=7$: this works and gives $m=10$, $n=4$.
• $2n-1=5$: this works and gives $m=6$, $n=3$.
• $2n-1=3$: this works and gives $m=3$, $n=2$. | 2019-10-13T23:11:30 | {
"domain": "stackexchange.com",
"url": "https://math.stackexchange.com/questions/1439170/find-all-natural-numbers-n-1-and-m-1-such-that-135-cdots2n-1?noredirect=1",
"openwebmath_score": 0.9655828475952148,
"openwebmath_perplexity": 126.8545166548528,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9825575152637946,
"lm_q2_score": 0.8479677526147223,
"lm_q1q2_score": 0.8331770880329457
} |
https://math.stackexchange.com/questions/3194587/getting-a-wrong-answer-on-evaluating-permutations-separately | Getting a wrong answer on evaluating permutations separately
Good Day!
I was doing some combinatorics problems when I got stuck.
The problem was:
Suppose that a teacher selects 4 students from 5 boys and 4 girls. If at least one boy and one girl must be selected, then the number of distinct selecting ways is ----.
Now, I went to solve this problem like this:
We first select the minimum boys and girls needed. That would be 4 $$\cdot$$ 5. Now we select the rest of the 2 students which is $${7 \choose 2}$$. Applying the multiplication principle, it is $$20$$ $$\cdot$$ $$21$$ = $$420$$.
However, the correct answer is 120 which is done by considering the separate cases:
• $$3$$ boys, $$1$$ girl
• $$2$$ boys, $$2$$ girls
• $$1$$ boys, $$3$$ girls
and then applying combinations.
What am I doing wrong?
Thanks
• You are over counting. Specifically: if you choose, say, $b_1$ as the initial boy and $b_2$ as part of the second choice, that's the same as choosing $b_2$ initially and then $b_1$ as part of the second choice. – lulu Apr 20 '19 at 11:34
• Note: another good way to do it is to compute the unrestricted number and then subtract off the all boy and all girl cases. Thus $\binom 94-\binom 54 -1 =120$ – lulu Apr 20 '19 at 11:38
• In the first reasoning the same configuration is counted multiple times: For instance the choice: {John, Bob, Mary, Anne} is counted 4 times: as 1. {John, Mary} + {Bob,Anne}, 2. {Bob, Mary}+{John, Anne}, 3. {John, Anne} + {Bob,Mary}, 4. {Bob, Anne}+{John, Mary}. – Ramiro Apr 20 '19 at 11:45
Consider there are 4 people : A, B, C and D. Now you want to choose 2 persons. In how many ways you can do this?
One straightforward way is $${4 \choose 2} = 6$$. So possible pairs will be AB, AC, AD, BC, BD and CD.
Another way, you first select one person and then select another person from remaining. So total there are $$4 \times 3 = 12$$ choices are there. But wait, there will be over counting in this method. All possible cases will be, AB, AC, AD (if first selected person is A), BA, BC, BD (if first selected person is B) and so on. So there are some pairs which are getting counted more than once.
Try to relate these two approaches to methods for that question.
You are overcounting. Consider, for example, the selection "Boy1,Girl1,Boy2,Girl2". Your formula counts it $$4$$ times as it counts also "Boy2,Girl2,Boy1,Girl1", "Boy2,Girl1,Boy1,Girl2", "Boy1,Girl2,Boy2,Girl1".
A possible approach here could be inclusion-exclusion:
• number of all possible groups of $$4$$: $$\binom{9}{4}$$
• number of all groups girls only: $$\binom{5}{4}$$
• number of all groups boys only: $$\binom{4}{4}$$
All together $$\binom{9}{4} - \binom{5}{4} - \binom{4}{4} = 120$$
The very first mistake you are making is when you state
We first select the minimum boys and girls needed. That would be 4 $$\cdot$$ 5.
If you want to count the number of two element sets with one boy and one girl, you can start off by multiplying $$5$$ by $$4$$ (order counts). Then, since order doesn't count in a set, you have to divide by $$2!$$. So the number of 2 element subsets with exactly one boy and one girl is $$10$$.
But even fixing that up you can't solve the problem attempting to use just the Rule of product and dividing by $$4!$$, As mentioned in a comment, you are over-counting.
When trying to count up how many elements are in a set, if you can partition the set into blocks and figure out how many elements are in each block, then mission accomplished. And to get $$120$$ the set is partitioned into $$3$$ chunks and the answer is $$60 + 20 + 40$$.
How many $$4$$ element subsets can be created containing $$2$$ boys and $$2$$ girls?
ANS: $${5 \choose 2} \times {4 \choose 2} = 60$$ | 2020-07-04T22:06:29 | {
"domain": "stackexchange.com",
"url": "https://math.stackexchange.com/questions/3194587/getting-a-wrong-answer-on-evaluating-permutations-separately",
"openwebmath_score": 0.5480836629867554,
"openwebmath_perplexity": 298.2040885733565,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9825575167960731,
"lm_q2_score": 0.8479677506936878,
"lm_q1q2_score": 0.8331770874447414
} |
https://math.stackexchange.com/questions/2369298/values-of-px-s-t-int-01px-dx-1/2369306 | # Values of $p(x)$ s.t. $\int_0^1p(x)\,dx = 1$
Let $P_2$ be the space of polynomials of degree $\leq 2$.
For $T: P_2 \to \mathbb R,$ $T(p)= \int_0^1 p(x)\,dx$
I need to find all values of $p(x)$ such that $T(p)=1$. However, I am unsure how to solve the resulting equation:
$$1=\int_0^1 p(x) \, dx$$
I know that in the case of the indefinite integral, you would simply take the derivative of both sides but that does not seem like it will work (given that you get $0=p(x)$ which does not work).
What is the method for finding all polynomials that give 1 as the value of this integral?
• Is $P_2$ the space of polynomials of order $2$? Jul 23, 2017 at 18:59
• I'm assuming $P_2$ is all polynomials of degree $\leq 2$? Jul 23, 2017 at 18:59
• Does the notation $P_2$ Mean anything particular? (For example the space of quadratic polynomials?) Jul 23, 2017 at 18:59
• If $p=ax^2+bx+c$ then $T(p)=\frac{a}{3}+\frac{b}{2}+c$. Jul 23, 2017 at 19:01
• $P_2$ is the space of polynomials of degree $\le 2$ Jul 23, 2017 at 19:02
Let $p(x)=ax^2+bx+c$. Then we have:
$$\int_0^1p(x)dx = \left[\frac{a}{3}x^3+\frac{b}{2}x^2+cx\right]_0^1 = \frac{a}{3}+\frac{b}{2}+c$$
Thus, your solution should be all $p(x)$ with $\frac{a}{3}+\frac{b}{2}+c=1$, or if you prefer integer coefficients, $2a+3b+6c=6$. That's one linear equation, so its solution set has two free variables in it. Do you know how to find it?
• I believe so--you set the answer such that $c=\frac{-a}{3}+\frac{-b}{2}+1$ and then any polynomial in $P_2$ where this is true would be a solution, correct? Jul 23, 2017 at 19:14
• Yes, that works. You can pick any values at all for $a$ and $b$, and then find $c$ like you said. Jul 23, 2017 at 19:32
if $P_2$ are polynomials of degree at most $2$, then you can write them as $ax^2+bx+c$.
$$\int_0^1 (ax^2+bx+c) dx = 1$$
$$\frac{a}{3}+\frac{b}{2}+c=1$$
Hence these polynomials are of the form of $ax^2+bx+\left(1-\frac{a}3-\frac{b}2\right)$
I think if we try taking polynomials $ax^2+bx+c$ then integrate $\frac{ax^3}3+\frac{bx^2}2+cx+d$ then putting limits will give $\frac{a}3+\frac{b}2+c$ And we can also take polynomials of degree $1$ and constants ...:)
• Hi, I think the $d$ should not be there after you substitute the limit and take the difference. Jul 23, 2017 at 19:06 | 2022-09-27T23:02:24 | {
"domain": "stackexchange.com",
"url": "https://math.stackexchange.com/questions/2369298/values-of-px-s-t-int-01px-dx-1/2369306",
"openwebmath_score": 0.9169501066207886,
"openwebmath_perplexity": 133.3272335546236,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9825575111777183,
"lm_q2_score": 0.8479677545357569,
"lm_q1q2_score": 0.8331770864556116
} |
http://autoverhuur-guusbaggen.nl/zyugyjgwn/graphing-logarithmic-functions.html | # Graphing logarithmic functions
graphing logarithmic functions Popular Tutorials in Graphing Logarithmic Functions What Does the Constant 'k' Do to the Graph of f(x)=log(x)+k? Translating a logarithmic function vertically can be fun, especially when you know how it's done! You can use a TI-83 Plus graphing calculator to look at the graph of two other exponential functions, y 3x and y 1 3 x. Watch this video lesson and you will see what the basic graph of the logarithmic function looks like. Chapter 5: Exponential and Logarithmic Functions 5-1 Exponential Functions Exponential Functions : - a function where the input (x) is the exponent of a numerical base, a. Use Oct 12, 2019 · A graph of a function is a visual representation of a function's behavior on an x-y plane. We come across the same kind of graph again later, in the section on electronics in differential equations, Application: Series RL Circuit , where the current Aug 05, 2019 · In this section we will introduce logarithm functions. 1) y = log 6 (x − 1) − 5 x y −8 −6 −4 −2 2 4 6 8 −8 −6 −4 −2 2 4 6 8 Domain: x > 1 Range: All reals 2) y = log 5 (x − 1) + 3 x y −8 −6 −4 −2 2 4 6 8 −8 −6 −4 −2 2 4 6 8 Domain: x > 1 Range: All reals 3) y The graph shows a logarithmic function f. Key vocabulary that may appear in student questions includes: exponential, asymptote, logarithmic, and quadrant. This lesson will show you how to graph a logarithm and what the transformations will do to the graph as well as their effects on the domain and The graph of is the result of three transformations on the graph of - a left shift of 4 units , a vertical stretch ( ), and an upward shift of 2 units ( ). (Remember that when the base of a logarithmic function is not specified, it is understood to The graph of f(x) should be exponential decay because b < 1. 1) y log (x ) x y 2) y log (x ) x y 3) y log (x ) x y 4) y log (x ) x y Identify the domain and range of each. vertical stretch by 3 Example: Sketch the graphs of f(x) = ln(x), g(x) = ln(-x), and h(x)= -ln(x). Is the pictured graph growth, decay, or linear or none? Graphing exponential functions is used frequently, we often hear of situations that have exponential growth or exponential decay. You will also be able to identify the kind of (Videos) Logarithmic Functions with Multiple Transformations. Graphing logarithmic functions can be done by locating points on the curve either manually or with a calculator. May 29, 2019 · NOTE: Compare Figure 6 to the graph we saw in Graphs of Logarithmic and Exponential Functions, where we learned that the exponential curve is the reflection of the logarithmic function in the line y = x. reflect the parent function over the x-axis and translate up 1 Graphing Logarithmic Functions Flip Book This flip book was created to be used as a stations activity to provide extra practice with graphing logarithmic functions and identifying the domain, range, x-intercept, asymptotes, and end behavior. Jan 20, 2020 · Yes! I’m going to show an insanely easy to follow 3-Step process that allows you to graph any logarithmic function quickly and easily. Dec 06, 2019 · In a semilogarithmic graph, one axis has a logarithmic scale and the other axis has a linear scale. Graphing In terms of their graphs, the graph of a log functions and its corresponding exponential inverse are symmetric with respect to the 45 o straight line. org/math/algebra2/exponential_and_logarithmic_func/log_functions/v/matching-exp Log & Exponential Graphs. I always remember that the “reference point” (or “anchor point“) of a log function is $$(1,0)$$ (since this looks like the “lo” in “log”). Select the function for this graph The common logarithmic function, written y = log x, has an implied base of 10. Label the two Graphing Logarithmic Functions Find the vertical asymptote, domain and key point of each of the following logarithmic functions. 06 Graphing logarithmic functions Use your graph to find the ideal pH level if the amount of hydronium ions is raised to 0. To graph Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude. Or if we calculate the logarithm of the exponential function of x, f Free graphing calculator instantly graphs your math problems. Rewrite a logarithmic equation in exponential form and apply the Inverse Property of exponential functions. Some of the worksheets for this concept are Graphing logarithms date period, Work logarithmic function, Exponential and log functions work, Practice work graphing logarithmic functions, Graphing logarithmic functions work, 11 exponential and logarithmic functions work, Work 2 7 Graph y = log 4 x and y = log 40 x and explain the differences: We will graph and so each function will be in the same base (common logarithm which is base 10). Logarithmic Functions & their Graphs For all real numbers , the function defined by is called the natural exponential function. com/patrickjmt !! Graphing a Logarithm Function Jan 30, 2018 · This algebra video tutorial explains how to graph logarithmic functions using transformations and a data table. 07 Graphing Logarithmic Functions A pool company forgets to bring their logarithmic charts, but they need to raise the amount of hydronium ions in a pool by 0. For this reason, we typically represent all graphs of logarithmic functions in terms of the common or natural log functions. Presentation of data on a logarithmic scale can be helpful when the data: Feb 26, 2014 · From Thinkwell's College Algebra Chapter 6 Exponential and Logarithmic Functions, Subchapter 6. 07 Graphing Logarithmic Functions The pool company developed new chemicals that transform the pH scale. Because f (x) = log b x and g(x) = bx are inverse functions, the graph of f (x) = log b x is the refl ection of the graph of g(x) = b 07. The logarithmic function, y = log b (x) is the inverse function of the exponential function, x = b y. Graphs help us understand different aspects of the function, which would be difficult to understand by just looking at the function itself. 6 Mathematics of Finance CHAPTER 3 275 The loudness of a sound we hear is based on the intensity of the associated sound wave Translating Between Exponential and Logarithmic Functions ⃣State that the inverse of an exponential function is a logarithmic function ⃣Explain the inverse relationship between exponents and logarithms (y = b x) is equivalent to log b y = x 7. The family of logarithmic functions includes the parent function along with all its transformations: shifts, stretches, compressions, and reflections. Step 3: Graphing logarithmic functions Ask students to use the transformation to graph the following equations on the same coordinates. (1) log 5 25 = y (2) log 3 1 = y (3) log 16 4 = y (4) log There are two main 'shapes' that a logarithmic graph takes. We will also discuss the common logarithm, $$\log(x)$$, and the natural logarithm, $$\ln(x)$$. Jan 06, 2017 · Practice Worksheet: Graphing Logarithmic Functions Without a calculator, match each function with its graph. 1) f(x) = - 2 x + 3 + 4 1) We can extend the applications of the natural logarithm function by composing it with the absolute value function. $$1)$$ $$f(x)=log(x)$$ Graphing Logarithmic Functions Flip Book This flip book was created to be used as a stations activity to provide extra practice with graphing logarithmic functions and identifying the domain, range, x-intercept, asymptotes, and end behavior. A 7 C 9 B 11 D 5 ____ 6 Use a graphing calculator to solve 32x Interactive, free online graphing calculator from GeoGebra: graph functions, plot data, drag sliders, and much more! Improve your math knowledge with free questions in "Match exponential functions and graphs" and thousands of other math skills. If you are graphing the common (base-10) log or the natural (base-e) log, just use your calculator to get the plot points. Download mobile versions The logarithmic function, y=logb(x) , can be shifted k units vertically and h units horizontally with the equation y=logb(x+h)+k . We have : lnjxj= ˆ lnx x > 0 ln( x) x < 0 This is an even function with graph-20 -10 10 20-1 1 2 3 We have lnjxjis also an antiderivative of 1=x with a larger domain than ln(x). 1 2 always decreasing that 1 2 Determine a logarithmic function in the form y = A log (B x + 1) + C y = A \log (Bx+1)+C y = A lo g (B x + 1) + C for each of the given graphs. us: Ċ: Unit 7- Answer Key Review Guide for Exonential and Logarthmic graph of an exponential function? You can use a graphing calculator to evaluate an exponential function. 1) y = log 3 (x + 5) + 2 x y-8-6-4-22468-8-6-4-2 2 4 6 8 2) y = log 1 4 (x - 1) + 3 x y-8-6-4-22468-8-6 Lesson 31 Graphs of Logarithmic Functions 1 Example 1: Complete the input/output table for the function : ;=log2 : ;, and use the ordered pairs to sketch the graph of the function. So by using properties of inverses, we can actually go from the graph that we have already derived to the graph of the log function, okay? So what this graph does, the exponential graph went to the point 0, 1. Transformation of Exponential Functions Example: Because every logarithmic function is the inverse function of an exponential function, we can think of every output on a logarithmic graph as the input for the Recognize and evaluate logarithmic functions with base a; Graph Logarithmic functions; Recognize, evaluate, and graph natural logs; Use logarithmic functions F. Remembering that logs are the inverses of exponentials, this shape for the log graph makes perfect sense: the graph of the log, being the inverse of the exponential Graphs of Logarithmic Functions. In other words, the up-and-down direction on an exponential graph corresponds to the right-and-left direction on a logarithmic graph, and the right-and-left direction on an exponential graph corresponds to the up-and-down direction on This Custom Polygraph is designed to spark vocabulary-rich conversations about exponential and logarithmic functions. Show less Show more 26 Feb 2014 From Thinkwell's College Algebra Chapter 6 Exponential and Logarithmic Functions, Subchapter 6. which statement it true? C y=log1x is not a How do I graph a logarithmic function with its transformations? In this post, we will be graphing logarithmic functions y = l o g a x y = log_{a}x y=logax for a > 0 a Translating a logarithmic function vertically can be fun, especially when you know how it's done! This tutorial shows what it takes to shift a logarithmic function up The graph of a logarithmic function is a curve with a vertical asymptote. In precalculus terms, that means that as x approaches infinity, the value of y increases exponentially towards infinity. 3 Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. It explains how to identify the 27 Oct 2013 Graphing a Logarithm Function Just a quick example showing the graph of logarithms and graphing log_4 (x). You can see that the graphs of both functions are located in quadrants I and IV to the right of the y-axis. Free math problem solver answers your algebra, geometry, trigonometry, calculus, and statistics homework questions with step-by-step explanations, just like a math tutor. A (2+3)logb(q + y) C logb(q 2 + y3) B logb(q2y3) D log b qy Ê 2 + 3 Ë ÁÁÁ ˆ ¯ ˜˜˜ ____ 4 Expand the logarithmic expression: log4 3 x A log43 log4x C log4x − log43 B −x log43 D log43 − log4x ____ 5 Use the properties of logarithms to evaluate log327 + log39 + log381. Monomials – relationships of the form = – appear as straight lines in a log–log graph, with the power term corresponding to the slope, and the constant term corresponding to the intercept of the line. Formulas Quiz: Formulas Absolute Value Equations Quiz: Graphing Rational Functions Function Graphing Logarithmic Functions Name_____ ID: 1 Date_____ Block____ ©w m2O0R1C7P AK[uqtsal mSGotfPtGwGakr^eE SLYLUCb. By nature of the logarithm, most log graphs tend to have the same shape, looking similar to a square-root 30 Jan 2018 This algebra video tutorial explains how to graph logarithmic functions using transformations and a data table. 1 Logarithmic Functions continued Graph x 3y y log 3 x 10 Side-by-Side Comparison f (x) 3x f (x) log 3 x 11 Comparing Exponential and Logarithmic Functions 12 Logarithmic Functions. Logarithmic Graphs: Once you know the shape of a logarithmic graph , you can shift it vertically or horizontally, stretch it, shrink it, reflect it, check answers with it, and most important interpret the graph. And if you wanted to figure out where 7 is, once again you could take the log base-- let me do it right over here-- so you'll take the log of 7 is going to be 0. We begin with the exponential function defined by 6 May 2020 Convert between exponential and logarithmic form; Evaluate logarithmic functions; Graph Logarithmic functions; Solve logarithmic equations How to graph a transformed log · Get the logarithm by itself. Exponential functions each have a parent function that depends on the base; logarithmic functions also have parent functions for each different base. 8 Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function. The graph to the right of the y-axis is the graph of the function , and the graph on the left to the left of the y-axis is the graph of the function . Are there different versions of Geogebra? I am a teacher and have a graphing project with my Algebra 2 Honors students. Graphing Logarithmic Functions: Analysis, Domain, Range, and more mathematical wonderfulness To graph a simple logarithmic function (no a, b, h, k yet), first graph a vertical asymptote at x=0. A parent logarithmic function is defined by: px() ln()x How could it be graphically transformed to create the graph of the following function: tx() ln() x 1? a. It approaches from the right, so the domain is all points to the right, $\left\{x|x>-3\right\}$. ( ) = log2( ), and use the ordered pairs to sketch the section covers: Introduction to Logarithms Special Logarithms Using Logs (and Exponents) in the Graphing Calculator Parent Graphs of Logarithmic Functions . In science and engineering, a log–log graph or log–log plot is a two-dimensional graph of numerical data that uses logarithmic scales on both the horizontal and vertical axes. Exponential and Logarithmic Functions: Graphs and Stuff Quiz Think you’ve got your head wrapped around Exponential and Logarithmic Functions ? Put your knowledge to the test. ) Observe that the logarithmic function f (x) = log b x is the inverse of the exponential function g (x The Basic Graph. ) Observe that the logarithmic function f (x) = log b x is the inverse of the exponential function g (x Graphs of Logarithmic Functions on Brilliant, the largest community of math and science problem solvers. Logarithmic functions with definitions of the form f (x) = log b x have a domain consisting of positive real numbers (0, ∞) and a range consisting of all real numbers (− ∞, ∞). Transformations of Logarithmic functions Log In or Sign Up different transformations of an Logarithmic function will result in a different graph from the basic graph. Now that we have a feel for the set of values for which a logarithmic function is defined, we move on to graphing logarithmic functions. Features of the Graph of Exponential Functions in the Form f(x) = b x or y = b x • The domain of f(x) = b x Graph log base 5 of x. How to graph an exponential function An exponential function of the form that was specified above will have a characteristic exponential shape, and its general form will depend on whether the Use our algebra calculator at home with the MathPapa website, or on the go with MathPapa mobile app. reflect the parent function over the x-axis and translate up 1 How to graph a logarithmic function? The graph of y = lob b (x - h) + k has the following characteristics • The line x = h is a vertical asymptote. f(x) = −2x Graphing the exponential function and natural log function, we can see that they are inverses of each other. Of the three transformations, only the left shift affects the position of the vertical asymptote - the asymptote of also shifts left 4 units, to . 1) log (u2 v) 3 2) log 6 (u4v4) 3) log 5 3 8 ⋅ 7 ⋅ 11 4) log 4 (u6v5) 5) log 3 (x4 y) 3 Condense each expression to a single logarithm. Free logarithmic equation calculator - solve logarithmic equations step-by-step This website uses cookies to ensure you get the best experience. Inside the squared brackets some options can be passed, in this case we set the colour of the plot to red; the squared brackets are mandatory, if no options are passed leave a blank space between them. On the other hand, the graph of the log passes through (1, 0), going off to the right but also sliding down the positive side of the y-axis. Graphs of the logarithmic functions for logarithms with bases 2 and 10, as well as the lesson; all graphs should be drawn by hand. Vanier College Sec V Mathematics Department of Mathematics 201-015-50 Worksheet: Logarithmic Function 1. So a logarithm actually gives you the exponent as its answer: (Also see how Exponents, Roots and Logarithms are related. Graphing Logarithmic Functions The function y = log b x is the inverse function of the exponential function y = b x . The family of logarithmic functions includes the parent function $y={\mathrm{log}}_{b}\left(x\right)$ along with all its transformations: shifts, stretches Graph logarithmic functions. · Logarithmic graphs allow one to plot a very large range of data without losing Graphs of Logarithmic Functions. The graph approaches x = –3 (or thereabouts) more and more closely, so x = –3 is, or is very close to, the vertical asymptote. Graph of f(x) = ln(x) At the point (e,1) the slope of the line is 1/e and the line is tangent to the curve. Now that we are more comfortable with using these functions as inverses, let’s use this idea to graph a logarithmic function. Jan 18, 2017 · This math video tutorial focuses on graphing logarithmic functions with transformations and vertical asymptotes. Aug 15, 2020 · Logarithmic functions with definitions of the form $$f (x) = \log_{b}x$$ have a domain consisting of positive real numbers $$(0, ∞)$$ and a range consisting of all real numbers $$(−∞, ∞)$$. You may need to zoom in on your A base-10 log scale is used for the Y axis of the bottom left graph, and the Y axis ranges from 0. Therefore $y=\log_{a}{x}$ is an inverse function, it is a The graph of the logarithmic function logax, for any a>1, has the y-axis as a vertical asymptote, but has no critical points. The idea here is we use semilog or log-log graph axes so we can more easily see details for small values of y as well as large values of y. 5) y log (x ) x y 6) y log (x ) May 10, 2018 · Here is a set of practice problems to accompany the Logarithm Functions section of the Exponential and Logarithm Functions chapter of the notes for Paul Dawkins Algebra course at Lamar University. Oct 12, 2019 · A graph of a function is a visual representation of a function's behavior on an x-y plane. When graphing logarithmic functions, it’s important to remember the following: · The graph can only appear to the right of the y-axis. It can be graphed as: The graph of inverse function of any function is the reflection of the The graph of the square root starts at the point (0, 0) and then goes off to the right. We also notice the graph of a logarithmic functions always pass through the point 1, 0 at its x intercept and that they always have the vertical asymptote x equals 0 I'm sorry. You need to provide the points $$(t_1, y_1)$$ and $$(t_2, y_2)$$, and this calculator will estimate the appropriate exponential function and will provide its graph. The family of logarithmic functions includes the parent function $$y={\log}_b(x)$$ along with all its transformations: shifts, stretches, compressions, and reflections. y = log 3 x, y = log 3 (x – 2) and y = log 3 (x – 2) + 1 Ask students if they can estimate the value of log 35 using the graph of y = log 3 x. ", "revised": "2020-08-18T18:03:40Z", "printStyle": null, "roles": null, "keywords": ["logarithmic function"], "id": "44418435-ed46-454a-aba4-cd57f5266654 Graphing Logarithmic Functions. From left to right, draw a curve that starts just to the right of the y-axis and Review Sheet: Exponential and Logorithmic Functions Date_____ Period____ Expand each logarithm. As Purple Math nicely states, logs are just the inverses of exponentials, so their graphs are merely a “flip” from each other. The overall shape of the graph of a logarithmic function depends on whether 0 < a < 1 or a > 1. This reference sheet for graphing logarithmic functions walks students through identifying x and y shifts and the base, identifying the parent function, creating a table for the parent function, shifting the parent table, plotting the points from the shifted table and sketching in the vertical asymptote. The "basic" logarithmic function is the function, y = log b x, where x, b > 0 and b ≠ 1. We can analyze its graph by studying its relation with the corresponding exponential function y = 2 x . Label the two Since all exponential functions have graphs that are similar to that of (specifically, the graphs of all exponential functions pass the horizontal line test, so all exponential functions are one-to-one), we can conclude that all exponential functions have inverse functions. The pH of a solution is defined as the - log of the hydrogen ion concentration GRAPHS OF EXPONENTIAL AND LOGARITHMIC FUNCTIONS. In addition, we discuss how to evaluate some basic logarithms including the use of the change of base formula. Instructions: Use this step-by-step Logarithmic Function Calculator, to find the logarithmic function that passes through two given points in the plane XY. Using the pH function p(t) = −log10t as the parent function, explain which transformation results in a y-intercept and why. Show students how to build the new Graph of the pH function p(t)=-log (t) Graphing the transformations The pool company developed new chemicals that transform the pH scale. Students practice finding the inverse of logarithmic functions, graphing them, and using those graphs to pointwise find the graph of the original function. Graphs of logarithmic functions (Algebra 2 level) Graphical relationship between 2ˣ and log₂(x) Shape of a logarithmic parent graph. Since the "+ 3" is inside the log's argument, the graph's shift cannot be up or down. That means the logarithm of a given number x is the exponent to which another fixed number, the base b, must be raised, to produce that number x. In the simplest case, the logarithm counts the number of occurrences of the same factor in repeated multiplication; e. Oct 03, 2019 · Some of the worksheets below are Exponential and Logarithmic Functions Worksheets, the rules for Logarithms, useful properties of logarithms, Simplifying Logarithmic Expressions, Graphing Exponential Functions, … Graphing Logarithmic Functions (State the Domain) Problems Graph the following logarithmic function. To show that a function is not one-to-one, find at Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude. Because every logarithmic function of this form is the inverse of an exponential function with the form y = b x \displaystyle y={b}^{x} y=bx, their graphs will be Logarithmic graphs use logarithmic scales, in which the values differ exponentially. An ordinary exponential function always has the points (0, 1) , ( 1 , base) and (-1, 1/ base ) Logarithmic Functions. We can use our knowledge of transformations, techniques for finding intercepts, and symmetry to find as many points as we can to make these graphs. Rewrite an exponential equation in logarithmic form and apply the Inverse Property of logarithmic functions. In general, an equation of the form y abx, where a 0, b Jul 06, 2019 · You can graph up to 10 equations at a time, by typing them into each of the slots listed on the Y= screen. In this unit we look at the graphs of exponential and logarithm functions, and see how they are related. There’s no learning curve – you’ll get a beautiful graph or diagram in minutes, turning raw data into something that’s both visual and easy to understand. The Logarithmic functions: A logarithmic function is simply an exponential function with the x and y axes switched. The 19th installment of a 35-part module prompts pupils to use skills from previous lessons to graph exponential and logarithmic functions. The $$y$$-axis, or $$x = 0$$, is a vertical asymptote and the $$x$$-intercept is $$(1, 0)$$. Because f (x) = log b x and g(x) = bx are inverse functions, the graph of f (x) = log b x is the refl ection of the graph of g(x) = b Purplemath. f (x) = log 1/4 x, g(x) = log 1/4(4x) − 5 Writing Transformations of Graphs of Functions Writing a Transformed Exponential Function Let the graph of g be a refl ection in the x-axis followed by a translation 4 units right of the graph of f (x) = 2x. Calculation of the logarithm; For the calculation of logarithm of a number, just enter the number and apply the function log. From the above values and graphs we conclude the following properties 1) The domain of any logarithm function of the form $$f(x) = log_B(x)$$ is the set of all real positive numbers. 6) ln 5 + ln 7 + 2ln 6 7) 4log 2 6 + 3log 2 7 8) log 8 x + log 8 y + 6log 8 z 9) 18 log 9 x − 6log 9 y 10 Mar 23, 2013 · Most graphing and scientific calculators have the ability to calculate logarithms, but you might come across questions which require you to use a different base than your calculator’s built-in functions. Compute answers using Wolfram's breakthrough technology & knowledgebase, relied on by millions of students & professionals. So if we calculate the exponential function of the logarithm of x (x>0), f (f -1 (x)) = b log b (x) = x. Logarithms are really useful in permitting us to work with very large numbers while manipulating numbers of a much more manageable size. In this section we will illustrate, The inverse of a function undoes the procedure (or function) of the given function. 3 Solve for the Evaluating a Logarithm ⃣ values of logarithms by evaluating powers of the base May 26, 2020 · The graph on the left is a graph showing the intersection of the surface and the plane given by $$x = 1$$. f x xlog 3 1 From this, we can see that $$\log _{b} x$$ is a vertical stretch or compression of the graph of the $$\log _{c} x$$ graph. Graphing logarithmic functions (example 2) Our mission is to provide a free, world-class education to anyone, anywhere. Here's a video by mathman1024 where he graphs a logarithmic function with To graph these functions, the best approach is to pick some numbers and plug them in. Example: If given f (x) = log2 x 19 Jan 2015 Graphing logarithmic functions on TI-84s isn't much different from graphing any other functions. 1) f (x) = log 1 2 (3x + 1) + 5 x y-8-6-4-22468-8-6-4-2 2 4 6 8 2) f (x) = log 5 (4x + 7) + 4 x y-8-6-4-22468-8-6-4-2 2 4 6 8 3) f (x) = log *Includes link to YouTube video tutorial for distance learning. We can then plug in values for Feb 15, 2020 · unit 5 worksheet 12 graphing logarithmic functions 2 UNIT 5 WORKSHEET 12 GRAPHING LOG FUNCTIONS 2. If you have the logBASE function, it can be used to enter the function (seen in Y1 Investigate the graphs of a family of logarithm functions by changing the a-value over Determine that for z>1 the function is increasing and for 0 Determine the 24 Jan 2017 By the end of this section, you will be able to: Convert between exponential and logarithmic form Evaluate logarithmic functions Graph 19 Mar 2014 Graphing logarithmic functions using transformations. The properties of the graphs of linear, quadratic, rational, trigonometric, arcsin(x), arccos(x), absolute value, logarithmic, exponential and piecewise functions are analyzed in details. As we know, in our maths book of 9th-10th class, there is a chapter named LOGARITHM is a very interesting chapter and its questions are some types that are required techniques to solve. When working with the common log, you will quickly reach awkwardly large numbers if you try to plot only whole-number points; for instance, in order to get as high as y = 2, you'd have to use x = 100, and your graph would be ridiculously wide. You can use these materials to review how to properly graph these equations and how to make graphs shift on the GRAPHING LOGARITHMIC FUNCTIONS Name_____ ID: 1 ©K R2]0a1T6h ^KNugtBav uS[oPfXtMwvavrHek hLJLICj. d dx (lnjxj) = 1 x and Z 1 x dx = lnjxj+ C Given an exponential or logarithmic function, the student will describe the effects of parameter changes. More References and Links Properties of Trigonometric Functions Inverse Trigonometric Functions Graphs of Hyperbolic Functions Logarithmic Functions Ex log3 5x to graph go to y= and type in log(5x)/log(3) When graphing logarithmic functions we usually discuss any transformations that have occured, the domain, range, yintercepts, xintercepts, asymptotes, and end behavior Key Properties of Logarithmic Functions: f(x) = logbx b>1 (1,0) f(x) = logbx 07. When graphing without a calculator, we use the fact that the inverse of a logarithmic function is an exponential function. Now that you have entered your equations into the calculator, let’s see what they look like! When you have finished entering the equations, press the [graph] button. Graphing Logarithmic Functions This original Khan Academy video was translated into isiZulu by Melusi Skhosane. Apr 01, 2020 · Graphing logs on the TI-84 is pretty simple, assuming your calculator has the most up-to-date operating (OS 2. y Jan 19, 2012 · This post offers reasons for using logarithmic scales, also called log scales, on charts and graphs. Print the book for ea I need to graph logarithmic functions with a base different than 10, for example a log with base 4, y= log4(x+5)-1 . Functions The graph of the logarithmic function ƒ (x) = l og 2 x, which you analyzed in the previous lesson, is shown. As an exercise find the domains of the above functions and compare with the domains found graphically above. Homework #10-2: Connecting Logs and Logarithms and exponents as inverses; Properties of logarithms; Writing logs in terms of others; Exponential equations requiring logarithms; Logarithmic equations, simple; Logarithmic equations, hard; Graphing logarithmic functions; Compound interest; Trigonometry: Angles and angle measure; Right triangle trigonometry; Trig functions of any Find the domain of function f defined by f (x) = log 5 (3 - x) Example 2 Find the domain of function f defined by f (x) = log 2 (x 2 + 5) Solution to Example 2. Some of the worksheets displayed are Work 2 7 logarithms and exponentials, Work logarithmic function, Graphing logarithms date period, Logarithmic exponential form numerals s1, Meaning of logarithms, 11 exponential and logarithmic functions work, Exponentials logarithms, Logarithmic functions and their graphs. Logarithmic functions are the inverses of exponential functions, and any exponential function can be expressed in logarithmic form. The logarithmic function, or the log function for short, is written as f(x) = log baseb (x), where b is the base of the logarithm and x is greater than 0. The graph should pass through the point (0, 1) and there should be a horizontal asymptote at the x axis. The family of logarithmic functions includes the parent function $y={\mathrm{log}}_{b}\left(x\right)$ along with all its transformations: shifts, stretches So when you see ln(x), just remember it is the logarithmic function with base e: log e (x). When graphing with a calculator, we use the fact that the calculator can compute only common logarithms (base graph the logarithmic function below. What are the domain and range of the logarithmic function f(x) = log7x? Use the inverse function to justify your answers. After graphing, list the domain, range, zeros, positive/negative intervals, increasing/decreasing intervals, and the intercepts. The logarithmic function has many real-life applications, in acoustics, electronics, earthquake analysis and logarithm functions mc-TY-explogfns-2009-1 Exponential functions and logarithm functions are important in both theory and practice. The 2 most common bases that we use are base 10 and base e, which we meet in Logs to base 10 and Natural Logs (base e) in later sections. In order to master the techniques explained here it is vital that you undertake plenty of practice Jan 19, 2015 · Graphing logarithmic functions on TI-84s isn't much different from graphing any other functions. 20 Jan 2020 Learn how to graph log functions with the power of transformations and only three easy to follow steps; and be able to identify domain and Identify the common and natural logarithm. Determine an exponential function in the form y = log b x y = \log_{b} x y=logbx with Q. For instance, in Exercise 89 on page 238, a logarithmic function is used to model Graphing Logarithmic Functions. Recall the following facts from inverse functions: If the point (a, b) is The red graph is the log(x) function, the blue graph the log(2x) function, the green graph the log(3x) function, and the purple graph the log(4x) function. Read 13 Feb 2015 Because every logarithmic function is the inverse function of an exponential function, we can think of every output on a logarithmic graph as the Notes 7. Print the book for ea Jan 06, 2017 · Practice Worksheet: Graphing Logarithmic Functions Without a calculator, match each function with its graph. Review Properties of Logarithmic Functions We first start with the properties of the graph of the basic logarithmic function of base a, f (x) = log a (x) , a > 0 and a not equal to 1. Hence the domain of the given function is given by the interval: (-∞ , +∞) Identities Proving Identities Trig Equations Trig Inequalities Evaluate Functions Simplify Statistics Arithmetic Mean Geometric Mean Quadratic Mean Median Mode Order Minimum Maximum Probability Mid-Range Range Standard Deviation Variance Lower Quartile Upper Quartile Interquartile Range Midhinge Make beautiful data visualizations with Canva's graph maker. Exponents and Logarithms work well together because they "undo" each other (so long as the base "a" is the same): They are "Inverse Functions" Doing one, then the other, gets you back to where you started: 408 CHaptER 4 inverse, Exponential, and Logarithmic Functions tests to Determine Whether a Function Is One-to-One 1. Need help with math? Start browsing Purplemath's free resources below! Practial Algebra Lessons: Purplemath's algebra lessons are informal in their tone, and are written with the struggling student in mind. It also shows you how to graph natural logs Graphing Logarithms Date_____ Period____ Identify the domain and range of each. Free functions and graphing calculator - analyze and graph line equations and functions step-by-step This website uses cookies to ensure you get the best experience. A free graphing calculator - graph function, examine intersection points, find maximum and minimum and much more This website uses cookies to ensure you get the best experience. Apr 11, 2018 · The logarithmic function is defined as: f(x) =log_b x The base of the logarithm is b. As you can tell from the graph to the right, the logarithmic curve is a reflection Apr 11, 2018 · While this looks a bit like the graph of the logarithm function, it is quite different. The top right graph uses a log-10 scale for just the X axis, and the bottom right graph uses a log-10 scale for both the X axis and the Y axis. Thankfully, there is a simple formula called the “change of base” formula that allows you to calculate any logarithm on your calculator: The logarithm base e is called the natural logarithm and is denoted ln x. y = log 1/2 x CCore ore CConceptoncept Parent Graphs for Logarithmic Functions The graph of f (x) = log b x is shown below for b > 1 and for 0 < b < 1. graphing logarithmic functions
1rxy obew ilop vqfg 944q 7cqi lsbi 7jav pni2 pgfi 90tk xuee ibpr 0wch md1f yghn c8iy s5kn 4bl5 gspw yvbd oh3f il14 oezi j5k9 | 2021-01-21T08:31:35 | {
"domain": "autoverhuur-guusbaggen.nl",
"url": "http://autoverhuur-guusbaggen.nl/zyugyjgwn/graphing-logarithmic-functions.html",
"openwebmath_score": 0.6196134090423584,
"openwebmath_perplexity": 725.7880205007357,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. Yes\n2. Yes",
"lm_q1_score": 0.9921841107484891,
"lm_q2_score": 0.8397339716830605,
"lm_q1q2_score": 0.8331707039596543
} |
https://wiki.documentfoundation.org/Documentation/Calc_Functions/SINH | # Documentation/Calc Functions/SINH
Other languages:
English • dansk
SINH
Mathematical
## Summary:
Calculates the hyperbolic sine (sinh) of a hyperbolic angle expressed in radians.
SINH(Number)
## Returns:
Returns a real number that is the hyperbolic sine of the specified hyperbolic angle.
## Arguments:
Number is a real number, or a reference to a cell containing that number, that is the hyperbolic angle in radians whose hyperbolic sine is to be calculated.
• If Number is non-numeric, then SINH reports a #VALUE! error.
The formula for the hyperbolic sine of x is:
$\displaystyle{ \sinh(x)~=~\frac{e^x-e^{-x}}{2} }$
The figure below illustrates the function SINH.
SINH function
## Examples:
Formula Description Returns
=SINH(0) Hyperbolic sine of 0. 0
=SINH(D1) where cell D1 contains the number -2. Hyperbolic sine of -2. -3.62686040784702
=ASINH(SINH(3)) The hyperbolic sine of 3 radians is taken and then the inverse hyperbolic sine of that value is calculated. 3
SINH | 2021-12-02T09:18:50 | {
"domain": "documentfoundation.org",
"url": "https://wiki.documentfoundation.org/Documentation/Calc_Functions/SINH",
"openwebmath_score": 0.9862258434295654,
"openwebmath_perplexity": 2622.1835341655096,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES\n\n",
"lm_q1_score": 0.9896718490038142,
"lm_q2_score": 0.8418256551882382,
"lm_q1q2_score": 0.833131152708991
} |
https://artofproblemsolving.com/wiki/index.php?title=1984_AIME_Problems/Problem_11&diff=cur&oldid=109011 | # Difference between revisions of "1984 AIME Problems/Problem 11"
## Problem
A gardener plants three maple trees, four oaks, and five birch trees in a row. He plants them in random order, each arrangement being equally likely. Let $\frac m n$ in lowest terms be the probability that no two birch trees are next to one another. Find $m+n$.
## Solution 1
First notice that there is no difference between the maple trees and the oak trees; we have only two types, birch trees and "non-birch" trees. (If you don't believe this reasoning, think about it. You could also differentiate the tall oak trees from the short oak trees, and the maple trees with many branches as opposed to those with few branches. Indeed, you could keep dividing until you have them each in their own category, but in the end it will not change the probability of the birch trees being near each other. That is, in the end, you multiply the numerator by the number of ways to arrange the oak and maple trees and you also multiply the denominator by the number of ways to arrange the oak and maple trees, making them cancel out.)
The five birch trees must be placed amongst the seven previous trees. We can think of these trees as 5 dividers of 8 slots that the birch trees can go in, making ${8\choose5} = 56$ different ways to arrange this.
There are ${12 \choose 5} = 792$ total ways to arrange the twelve trees, so the probability is $\frac{56}{792} = \frac{7}{99}$.
The answer is $7 + 99 = \boxed{106}$.
## (Another way to think about Solution 1)
So you can think of placing the $5$ birch trees and the other trees with the restrictions as described above. Then let's take out one tree between each pair of birch trees. So you would remove $4$ trees that aren't birch. What you are left with is a unique arrangement of $5$ birch trees and $3$ other trees that is unrestricted. Some birch trees might become adjacent after you remove $4$ trees. Adding a tree between each pair of people gives a unique arrangement of $5$ nonadjacent birch trees. This guarantees that there are no adjacent trees. The number of unrestricted $7$ tree arrangments is ${8\choose5} = 56$. Then proceed as told in Solution $1$.
~ blueballoon
## Solution 2
Let $b$, $n$ denote birch tree and not-birch tree, respectively. Notice that we only need $4$ $n$s to separate the $5$ $b$s. Specifically, $$b,n,b,n,b,n,b,n,b$$ Since we have $7$ $n$s, we are placing the extra $3$ $n$s into the $6$ intervals beside the $b$s.
Now doing simple casework.
If all $3$ $n$s are in the same interval, there are $6$ ways.
If $2$ of the $3$ $n$s are in the same interval, there are $6\cdot5=30$ ways.
If the $n$s are in $3$ different intervals, there are ${6 \choose 3} =20$ ways.
In total there are $6+30+20=56$ ways.
There are ${12\choose5}=792$ ways to distribute the birch trees among all $12$ trees.
Thus the probability equals $\frac{56}{792}=\frac{7}{99}\Longrightarrow m+n=7+99=\boxed{106}$.
~ Nafer
## Solution 3 (using PIE)
Note that the requested probability is computed by dividing the number of configurations with no adjacent Birch trees by the total number of configurations. We can compute the number of configurations with no adjacent Birch trees using complementary counting and then the Principle of Inclusion-Exclusion.
The number of configurations with no adjacent Birch trees is equal to the total number of configurations minus the number of configurations with at least one pair of adjacent Birch trees.
The total number of configurations is given by $\frac{12!}{3! \cdot 4! \cdot 5!}$. To compute the number of configurations with at least one pair of adjacent Birch trees, we use PIE.
$\#$(configurations with at least one pair of adjacent Birch trees) $=$ $\#$(configurations with one pair) $-$ $\#$(configurations with two pairs) $+$ $\#$(configurations with three pairs) $-$ $\#$(configurations with four pairs).
To compute the first term, note that we can treat the adjacent pair of Birch trees as one separate tree. This then gives $\frac{11!}{3! \cdot 3! \cdot 4!}$ configurations.
For the second term, we have two cases. The two pairs could either happen consecutively (BBB) or separately (BB BB). They both give $\frac{10!}{2! \cdot 3! \cdot 4!}$ cases. So our second term is $\frac{2 \cdot 10!}{2! \cdot 3! \cdot 4!}$.
The third term can also happen in two ways. The three pairs could be arranged like BBBB or BBB BB. Both cases together give $\frac{2 \cdot 9!}{3! \cdot 4!}$ arrangements.
The final term can happen in one way (BBBBB). This gives $\frac{8!}{3! \cdot 4!}$ arrangements.
Substituting these into our PIE expression, we find that there are $25760$ configurations with at least one pair of adjacent Birch trees. Therefore, there are a total of $\frac{12!}{3! \cdot 4! \cdot 5!} - 25760 = 1960$ configurations with no adjacent Birch trees.
Thus, the probability of a given configuration having no two adjacent Birch trees is given by $\frac{1960}{\frac{12!}{3! \cdot 4! \cdot 5!}} = \frac{7}{99}$.
Therefore, the desired result is given by $7+99 = \boxed{106}$.
~ vietajumping | 2021-05-14T23:47:18 | {
"domain": "artofproblemsolving.com",
"url": "https://artofproblemsolving.com/wiki/index.php?title=1984_AIME_Problems/Problem_11&diff=cur&oldid=109011",
"openwebmath_score": 0.6419976353645325,
"openwebmath_perplexity": 521.446446554556,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9896718490038141,
"lm_q2_score": 0.8418256532040708,
"lm_q1q2_score": 0.8331311507453163
} |
https://math.stackexchange.com/questions/873755/find-the-remainder-of-the-polynomial-division-px-x2-1-for-some-p | # Find the remainder of the polynomial division $p(x)/(x^2-1)$ for some $p$
Let $f(x) = x^{10}+5x^9-8x^8+7x^7-x^6-12x^5+4x^4-8x^3+12x^2-5x-5.$
Without using long division (which would be horribly nasty!), find the remainder when $f(x)$ is divided by $x^2-1$.
I'm not sure how to do this, as the only way I know of dividing polynomials other than long division is synthetic division, which only works with linear divisors. I thought about doing $f(x)=g(x)(x+1)(x-1)+r(x)$, but I'm not sure how to continue. Thanks for the help in advance.
• plug in $1$ and $-1$ to get two values of $r(x)$ which is linear. From there you can get what $a,b$ are in $ax+b.$ Jul 21 '14 at 15:23
• Hint : Use the horner-scheme. To avoid complications, first use it for 1, then for -1. There is a compact notation, described in wikipedia. Jul 21 '14 at 15:25
• Velcome to our site! Jul 21 '14 at 15:32
## 2 Answers
Plug in $1$ and $-1$ to get two values of $r(x)$ which is linear. From there you can get what $a,b$ are in $ax+b.$
Since $$f(x)=g(x)(x+1)(x-1)+r(x)$$
we have
$$f(1)=g(1)(1+1)(1-1)+r(1)=r(1)=-10$$ $$f(-1)=g(1)(-1+1)(-1-1)+r(-1)=r(-1)=16$$
We know the remainder is of degree $1$, so
$r(x)=ax+b$
and now we know, $$r(1)=ax+b=a+b=-10$$ $$r(-1)=ax+b=-a+b=16$$
so, solve
$$a+b=-10$$ $$-a+b=16$$
which yields, $a=-13$ $b=3$, so
$$r(x)=-13x+3$$
Hint $$\ {\rm mod\ }x^{\large 2}\!-1\!:\,\ x^{\large 2}\equiv 1\,\Rightarrow\,\color{#0a0}{x^{\large 2n}\equiv 1}\,\Rightarrow\,\color{#c00}{x^{\large 2n+1}\equiv x},\$$ hence
$$f(x) =\, \overbrace{(c_0 + c_2\color{#0a0}{ x^2} + c_4\color{#0a0}{ x^4}+\cdots)}^{\large \color{#0a0}{f_0(x)}} \ +\ \overbrace{(c_1\color{#c00} x + c_3\color{#c00}{x^3} + c_5\color{#c00}{x^5} + \cdots)}^{\large \color{#c00}{f_1(x)}}$$
$$\qquad \equiv \ (c_0 \ +\ c_2\ +\ c_4\ \ + \ \cdots)\,\color{#0a0}1 + (c_1\ +\,\ c_3\ \ +\ \ c_5 \ +\ \cdots)\,\color{#c00}x$$
$$\qquad \equiv\ f_0(1)\,\color{#0a0}1 + f_1(1)\, \color{#c00}x,\$$ where $$\,f_0(x),\ f_1(x)\,$$ are the $$\rm\color{#0a0}{even}$$ and $$\rm\color{#c00}{odd}$$ parts of $$\,f(x).$$
e.g. a familiar numerical instance when $$\,x=10\,$$ in radix $$10$$ (decimal) arithmetic
$$\!\! \bmod 99\!:\ \color{#c00}5\color{#0a0}4\color{#c00}3\color{#0a0}2\color{#c00}1\color{#0a0}0\equiv (\color{#c00}{5\!+\!3\!+\!1}),(\color{#0a0}{4\!+\!2\!+\!0})\equiv \color{#c00}9\color{#0a0}6\equiv \color{#c00}5\color{#0a0}4+\color{#c00}3\color{#0a0}2+\color{#c00}1\color{#0a0}0\$$ by $$\,10^2\equiv 1$$
• In other words, add up all the even-degree coeffs and you get the constant term; add up all the odd-deg coeffs and you get the linear term. Jul 21 '14 at 16:09
• This is a rather sophisticated way doing it. I like it. Jul 21 '14 at 16:15 | 2021-12-03T12:31:41 | {
"domain": "stackexchange.com",
"url": "https://math.stackexchange.com/questions/873755/find-the-remainder-of-the-polynomial-division-px-x2-1-for-some-p",
"openwebmath_score": 0.806167483329773,
"openwebmath_perplexity": 229.66099041667354,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9896718468714469,
"lm_q2_score": 0.8418256512199033,
"lm_q1q2_score": 0.8331311469865602
} |
https://math.stackexchange.com/questions/2973082/choosing-columns-in-array | # Choosing columns in array
In an array with four rows and $$n\geq 1$$ columns, some subset of cells are marked. Can we always choose some columns in such a way that for at least three rows, the number of chosen marked cells and the number of unchosen marked cells differ by at most $$1$$?
An initial try is to pick some three fixed row (for example the first three rows) and see if it is possible to choose the columns in the required way. However, this is not always the case: it could be that there are three columns, in the first row the first and second columns are marked, in the second row the second and third column, and in the third row the third and first columns. But maybe there is some clever way to pick the three rows?
• I think I may be misunderstanding this problem. In the example given, why is it not acceptable to simply choose, say, the first column? This gives you two marked cells and one unmarked cell, a difference of exactly 1. – mhum Oct 30 '18 at 17:17
• @mhum: because the second row has two unchosen marked cells and no chosen ones, so the difference is $2$. – Ross Millikan Oct 30 '18 at 17:41
• @RossMillikan Ah, I see! I had somehow missed the chosen/unchosen distinction amidst the marked/unmarked distinction. – mhum Oct 30 '18 at 17:53
Yes, we can.
Let $$P(n)$$ be as follows :
$$P(n)$$ : In the $$4\times n$$ array, we can choose some columns in such a way that for at least three rows, the number of chosen marked cells and the number of unchosen marked cells differ by at most $$1$$.
In the following, $$[a,b]$$ represents the cell in the $$a$$-th row and in the $$b$$-th column. Also, $$[a,b]=[c,d]$$ means that either that both $$[a,b]$$ and $$[c,d]$$ are marked or that both $$[a,b]$$ and $$[c,d]$$ are unmarked.
Claim : If $$P(7),P(8)$$ are true, then $$P(n)$$ is true for all $$n\ge 9$$.
Proof :
There are $$2^3=8$$ possibilities for three cells to be marked or unmarked. So, if $$n\ge 9$$, then, by the pigeonhole principle, there is at least one pair of distinct integers $$(s,t)$$ such that $$[1,s]=[1,t],[2,s]=[2,t]$$ and $$[3,s]=[3,t]$$. We may suppose that $$[1,1]=[1,2],[2,1]=[2,2]$$ and $$[3,1]=[3,2]$$. Now, consider the $$4\times (n-2)$$ array made by the $$(n-2)$$ columns from the third to the $$n$$-th. Suppose that $$P(n-2)$$ is true. Then, we can choose some columns (call them $$p$$ columns) which satisfy our condition. Now, let us come back to the $$4\times n$$ array. If choosing the first column and the $$p$$ columns works, then we are done. If it doesn't work, then choosing the second column and the $$p$$ columns works. So, $$P(n)$$ is true. $$\quad\square$$
In the following, let us prove that $$P(1),P(2),\cdots,P(8)$$ are true.
We use the following two lemmas (the proofs are written at the end of the answer) :
Lemma 1 : If $$P(n-1)$$ is true and there is at least one column whose four cells are unmarked, then $$P(n)$$ is true.
Lemma 2 : If $$P(n-2)$$ is true and there is a pair of distinct integers $$(s,t)$$ such that at least three of $$[i,s]=[i,t]\ (i=1,2,3,4)$$ hold, then $$P(n)$$ is true.
From the lemmas,
$$\ \ (\star)$$ : we may suppose that there is at least one marked cell in each column and that there is no pair of distinct integers $$(s,t)$$ such that at least three of $$[i,s]=[i,t]\ (i=1,2,3,4)$$ hold.
This $$(\star)$$ reduces the number of cases to consider.
• For $$n=1$$, we can choose the column.
• For $$n=2$$, we can choose the first column.
• For $$n=3$$, if choosing the first and the second columns works, then we are done. If it doesn't work, then we may suppose that $$[1,1],[1,2],[2,1],[2,2],[3,1]$$ are marked and $$[1,3],[2,3],[3,2]$$ are unmarked. Then, choosing the first column works.
• For $$n=4$$, if there is a pair of distinct integers $$(s,t)$$ such that $$[1,s]=[1,t]$$ and $$[2,s]=[2,t]$$, then we may suppose that $$[3,s]$$ is marked and that $$[3,t]$$ is unmarked.
case 1 : If $$[3,u]=[3,v]$$ where $$u\not=v$$ are different from $$s$$ and $$t$$, then choosing the $$s$$-th and the $$u$$-th columns works.
case 2 : If $$[3,u]$$ is marked and $$[3,v]$$ is unmarked, then choosing the $$s$$-th and the $$v$$-th columns works.
If there is no pair of distinct integers $$(s,t)$$ such that $$[1,s]=[1,t]$$ and $$[2,s]=[2,t]$$, then we may suppose that $$[1,1],[1,3],[2,1],[2,4]$$ are marked and $$[1,2],[1,4],[2,2],[2,3]$$ are unmarked. If choosing the first and the second columns works, then we are done. If it doesn't work, then from $$(\star)$$, choosing the first column works.
• For $$n=5$$, if there are two distinct pairs of distinct integers $$(s,t),(u,v)$$ such that $$[1,s]=[1,t],[2,s]=[2,t],[1,u]=[1,v]$$ and $$[2,u]=[2,v]$$, then choosing the $$s$$-th and the $$v$$-th columns works.
If there is only one pair of distinct integers $$(s,t)$$ such that $$[1,s]=[1,t]$$ and $$[2,s]=[2,t]$$, then we may suppose that we have the following three cases :
case 1 : $$[1,1],[1,2],[2,1],[2,2],[3,1]$$ are marked and $$[3,2],[1,5],[2,5]$$ are unmarked. If choosing the first and the third columns works, then we are done. If it doesn't workd, then choosing the second and the third columns works.
case 2 : $$[1,1],[1,2],[1,3],[2,3],[2,4],[3,1]$$ are marked and $$[1,4],[1,5],[2,1],[2,2],[3,2]$$ are unmarked. If choosing the first and the third columns works, then we are done. If it doesn't work, then choosing the second and the third columns works.
case 3 : $$[1,3],[1,5],[2,4],[2,5],[3,1]$$ are marked and $$[1,1],[1,2],[1,4],[2,1],[2,2],[2,3],[3,2]$$ are unmarked. If choosing the first and the third and the fourth columns workds, then we are done. If it doesn't work, then choosing the first and the fifth columns works.
• For $$n=6$$, we may suppose that there are two distinct pairs of distinct integers $$(s,t),(u,v)$$ such that $$[1,s]=[1,t],[2,s]=[2,t],[1,u]=[1,v]$$ and $$[2,u]=[2,v]$$. Then, we may suppose that $$[3,s],[3,u]$$ are marked and that $$[3,t],[3,v]$$ are unmarked. Now, choosing the $$s$$-th and the $$v$$-th and the one from the rest two columns works.
• For $$n=7$$, we may suppose that there are three distinct pairs of distinct integers $$(a,b),(c,d),(e,f)$$ such that $$[1,a]=[1,b],[2,a]=[2,b],[1,c]=[1,d],[2,c]=[2,d],[1,e]=[1,f]$$ and $$[2,e]=[2,f]$$. Then, we may suppose that $$[3,a],[3,c],[3,e]$$ are marked and that $$[3,b],[3,d],[3,f]$$ are unmarked. If $$[3,g]$$ is marked where $$g$$ is different from $$a,b,c,d,e$$ and $$f$$, then choosing the $$a$$-th, the $$c$$-th and the $$f$$-th columns works. If $$[3,g]$$ is unmarked, then choosing the $$a$$-th , the $$d$$-th and the $$f$$-th columns works.
• For $$n=8$$, we may suppose that there are four distinct pairs of distinct integers $$(a,b),(c,d),(e,f),(g,h)$$ such that $$[1,a]=[1,b],[2,a]=[2,b],[1,c]=[1,d],[2,c]=[2,d],[1,e]=[1,f],[2,e]=[2,f],[1,g]=[1,h]$$ and $$[2,g]=[2,h]$$. Then, we may suppose that $$[3,a],[3,c],[3,e],[3,g]$$ are marked and that $$[3,b],[3,d],[3,f],[3,h]$$ are unmarked. Now, choosing the $$a$$-th, the $$d$$-th, the $$e$$-th and the $$h$$-th columns works.
Finally, let us prove the two lemmas :
Lemma 1 : If $$P(n-1)$$ is true and there is at least one column whose four cells are unmarked, then $$P(n)$$ is true.
Proof for Lemma 1 :
Consider the $$4\times (n-1)$$ array excluding the column. Since $$P(n-1)$$ is true, we can choose suitable columns (call them $$p$$ columns) for the $$4\times (n-1)$$ array. Now, coming back to the $$4\times n$$ array, we see that choosing the $$p$$ columns works. $$\quad\square$$
Lemma 2 : If $$P(n-2)$$ is true and there is a pair of distinct integers $$(s,t)$$ such that at least three of $$[i,s]=[i,t]\ (i=1,2,3,4)$$ hold, then $$P(n)$$ is true.
Proof for Lemma 2 :
Consider the $$4\times (n-2)$$ array excluding the $$s$$-th and the $$t$$-th columns. Since $$P(n-2)$$ is true, we can choose suitable columns (call them $$p$$ columns) for the $$4\times (n-2)$$ array. Now, let us come back to the $$4\times n$$ array. If choosing the $$s$$-th column and the $$p$$ columns works, then we are done. If it doesn't work, then choosing the $$t$$-th column and the $$p$$ columns works. $$\quad\square$$ | 2019-11-14T19:32:31 | {
"domain": "stackexchange.com",
"url": "https://math.stackexchange.com/questions/2973082/choosing-columns-in-array",
"openwebmath_score": 0.9157324433326721,
"openwebmath_perplexity": 65.72667572453024,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. Yes\n2. Yes",
"lm_q1_score": 0.989671849308438,
"lm_q2_score": 0.8418256412990658,
"lm_q1q2_score": 0.8331311392197082
} |
https://math.stackexchange.com/questions/2455153/easier-way-to-find-probability | # Easier Way to Find Probability
I know how to compute this with a concept similar to truth tables. First I listed all of the combinations of the angle in trios:
$ABC$, $ABD$, $ABE$, $ACD$, $ACE$, $ADE$, $BCD$, $BCE$, $BDE$, $CDE$.
Then I let $A$ represent the angles that are acute, and $N$ represent the angles that were not, and plugged such values into the combinations above. The result was:
$AAN$, $AAN$, $AAA$, $ANN$, $ANA$, $ANA$, $ANN$, $ANA$, $ANA$, $NNA$.
From this I could easily pinpoint the result $\frac{6}{10}\$ which can be reduced to $\frac{3}{5}\$. My question is simple: is there an easier way to compute the same answer? If so, what is the corresponding formula?
I have previously asked a question dealing with probability such as this, except replacement was involved. The response involved mapping out the answers like I did above, so this is where the confusion over easy computation arrives. Thank you!
• combinatorics is where you want to go it seems.
– user451844
Oct 3 '17 at 2:05
The number of ways of selecting a subset of size $k$ from a set of $n$ objects is given by the formula $$\binom{n}{k} = \frac{n!}{k!(n - k)!}$$ where $n!$, read "$n$ factorial," is the product of the first $n$ positive integers if $n$ is a positive integer and $0! = 1$. The notation $\binom{n}{k}$ is read "$n$ choose $k$."
There are $\binom{5}{3}$ ways to select a subset of three of the five angles.
Of the five angles, three are acute and two are not. If exactly two of the three selected angles are acute, one of the two other angles must be selected. Therefore, the number of favorable selections is $$\binom{3}{2}\binom{2}{1}$$
Hence, the probability that exactly two acute angles will be selected when three of the five angles are selected is $$\frac{\dbinom{3}{2}\dbinom{2}{1}}{\dbinom{5}{3}} = \frac{3 \cdot 2}{10} = \frac{3}{5}$$ as you found.
• Thank you for the short but sweet explanation on the notation! Helps a lot! Oct 3 '17 at 2:10
• not sure in this small case it's any easy than a pure list and count method. in larger examples it will save a lot though.
– user451844
Oct 3 '17 at 2:13
• @RoddyMacPhee Agreed. Oct 3 '17 at 2:13
We want the probability for selecting $2$ from the $3$ acute and $1$ from the $2$ non-acute angles, when selecting any $3$ from the $5$ angles with no bias nor replacement.
Recall that $\binom nk$ is the count for selections of $k$ items from a set of $n$ (with no relacement), and : $$\binom nk = \dfrac{n!}{k!~(n-k)!}$$
Put it together.
$$\dfrac{\dbinom 3 2\dbinom 21}{\dbinom 52}=\dfrac{3}{5}$$ | 2022-01-19T14:17:35 | {
"domain": "stackexchange.com",
"url": "https://math.stackexchange.com/questions/2455153/easier-way-to-find-probability",
"openwebmath_score": 0.8471971154212952,
"openwebmath_perplexity": 138.98581302038028,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9896718455006394,
"lm_q2_score": 0.8418256432832333,
"lm_q1q2_score": 0.8331311379778804
} |
https://math.stackexchange.com/questions/3033159/why-we-only-need-to-verify-additive-identity-and-closed-under-addition-and-scal | # Why we only need to verify additive identity, and closed under addition and scalar multiplication for subspace?
In the book Linear Algebra Done Right, it is said that to determine quickly whether a given subset of $$V$$ is a subspace of $$V$$, the three conditions, namely additive identity, closed under addition, and closed under scalar multiplication, should be satisfied. The other parts of the definition of a vector space are automatically satisfied.
I think I understand why commutativity, associativity, distributive properties, and multiplicative identity works because their operations are still within the subspace.
But, why don't we need to verify additive inverse, similar to verifying additive identity? Could there be cases where there will be no $$v + w = 0$$ in the new subspace, $$v, w \in U$$, $$U$$ is a subspace?
You also need to check that the set is non-empty. In this case closure under scalar multiplication guarantees that the additive inverse of any $$v$$ in the set is also in the set, since for the scalar $$-1$$, $$(-1)v$$ is in the set.
EDIT: Similarly, for the scalar $$0$$, $$0v={\bf 0}$$ is in the set (by the closure of scalar multiplication), whenever the set contains an element/vector $$v$$.
• To be fair, you will get non-emptiness by verifying that $\vec{0}$ is in the set. I think that is what is meant by additive identity. – GenericMathematician Dec 9 '18 at 23:15 | 2019-04-22T04:55:51 | {
"domain": "stackexchange.com",
"url": "https://math.stackexchange.com/questions/3033159/why-we-only-need-to-verify-additive-identity-and-closed-under-addition-and-scal",
"openwebmath_score": 0.94569331407547,
"openwebmath_perplexity": 154.67145801884163,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9848109516378094,
"lm_q2_score": 0.84594244507642,
"lm_q1q2_score": 0.8330933843665245
} |
https://math.stackexchange.com/questions/1203556/calculating-the-number-of-times-a-value-must-be-halved-for-it-to-be-less-than-or | Calculating the number of times a value must be halved for it to be less than or equal to another value
This is not a homework question; I'm working out an algorithm for an app I'm writing and I want to calculate the number of times I must halve a base value for it to be less than or equal to a minimum. I can write the equation, but I have no idea how to start solving it.
I guess the equation looks like this:
$$\dfrac{y}{2^x} \leq z$$
Given some large value of $y$, e.g. $10000000$, and a minimum value for $z$, e.g. $1$, how do I find $x$?
With my example values, the equation would be:
$$\dfrac{10000000}{2^x} \leq 1$$
• The programmer in me says, that $y$ isn't very large :P and unless this calculation needs to be done extremely fast, you could just write a loop. Is $y$ an integer? Do you have guarantees as to its size in bits? – Ben Millwood Mar 24 '15 at 0:45
• Multiply both sides by $2^x$, then $y \leq 2^x \implies \log_2 y < x$ – user4894 Mar 24 '15 at 0:45
Hint: $\log_2$ is monotonically increasing.
Divide both sides by $z$ and relabel $\frac{y}{z}$ as $Y$. Then, rearranging gives $$Y \leq 2^x,$$ and by monotonicity we have $$\log_2 Y \leq \log_2 (2^x) = x.$$ Since "number of times" must be an integer, we need to double $\lceil \log_2 Y \rceil$ times, and since it must be nonnegative, we have $$x = \text{max}\{\lceil \log_2 Y \rceil, 0\}.$$
• How do you make the answer respond to mouseovers like that? – user4894 Mar 24 '15 at 0:50
• @user4894 Type >!foo . It's very useful for elaborating on hints, I think, so that OPs/others can try to work out the problem for themselves before reading the full details of an answer. – Travis Willse Mar 24 '15 at 0:53 | 2021-05-09T08:10:15 | {
"domain": "stackexchange.com",
"url": "https://math.stackexchange.com/questions/1203556/calculating-the-number-of-times-a-value-must-be-halved-for-it-to-be-less-than-or",
"openwebmath_score": 0.7298210263252258,
"openwebmath_perplexity": 218.54748817901466,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9848109538667756,
"lm_q2_score": 0.8459424431344437,
"lm_q1q2_score": 0.8330933843396221
} |
https://mathhelpboards.com/threads/find-the-number-of-line-and-column-where-the-number-2002-stays.3936/ | # Find the number of line and column where the number 2002 stays.
#### anemone
##### MHB POTW Director
Staff member
Positive real numbers are arranged in the form:
$$\displaystyle 1 \;\;\; 3 \;\;\; 6 \;\;\;10 \;\;\; \cdots$$
$$\displaystyle 2 \;\;\;5 \;\;\;9 \;\;\; \cdots$$
$$\displaystyle 4 \;\;\;8 \;\;\; \cdots$$
$$\displaystyle 7 \;\;\; \cdots$$
Find the number of the line and column where the number 2002 stays.
#### Opalg
##### MHB Oldtimer
Staff member
Positive real numbers are arranged in the form:
$$\displaystyle 1 \;\;\; 3 \;\;\; 6 \;\;\;10 \;\;\; \cdots$$
$$\displaystyle 2 \;\;\;5 \;\;\;9 \;\;\; \cdots$$
$$\displaystyle 4 \;\;\;8 \;\;\; \cdots$$
$$\displaystyle 7 \;\;\; \cdots$$
Find the number of the line and column where the number 2002 stays.
You will have noticed that the $n$th element in the top row is the $n$th triangular number $\frac12n(n+1)$. Then working back from that, each time you subtract $1$ you move back one column and down one row (until you reach the first column).
The $63$rd triangular number is $2016$, so that will be in row $1$, column $63$. Working back $14$ places from that, you find that $2002$ will be in row $1+14 =15$ and column $63-14=49$.
#### anemone
##### MHB POTW Director
Staff member
My solution:
Number Row Column Group 1 1 1 1st 2 2 1 2nd 3 1 2 2nd 4 3 1 3rd 5 2 2 3rd 6 1 3 3rd 7 4 1 4th 8 3 2 4th 9 2 3 4th 10 1 4 4th 11 5 1 5th 12 4 2 5th 13 3 3 5th 14 2 4 5th 15 1 5 5th
To see where the number 2002 stays in the table above, we first need to investigate at which group that it lies, and this could be done in the following manner:
$$\displaystyle S_n=\frac{n}{2}\left(n+1\right)$$
$$\displaystyle 2002=\frac{n}{2}\left(n+1\right)$$
$$\displaystyle n=62.78$$
We know that 2002 must lie in the 63th group, and to know what the previous number in the 62th group is, we do the following:
$$\displaystyle S_62=\frac{62}{2}\left(62+1\right)=1953$$
Thus, we have:
Number Row Column Group 1953 1 62 62th 1954 63 1 63th ... ... ... ...
To find where the number 2002 lies in the 63th group, i.e. how far it is from 1954, we compute:
$$\displaystyle 2002-1954=48$$
Hence, 2002 lies in the $$\displaystyle 63-48=15th$$ row and the $$\displaystyle 1+48=49th$$ column.
#### melese
##### Member
I tried to find a formula depending only on $k$, namely, given a positive integer $k$, to find its location $(C_k,R_k)$ in the table. For brevity of reference, we say that $k=(C_k,R_k)$ (abuse of notation)
Considering the obvious diagonals, it's clear that '$k$ is in the $n$'th diagonal' is equivalent to '$T_{n-1}<k\leq T_n$(*) for some integer $n\geq1$'.
Notice that for the diagonal $\{(1,n),(2,n-1),...,(1+x,n-x),...(n,1)\}$ we have $(1+x,n-x)=(1,n)-x=T_n-x$, because the numbers are consecutive.
Since $k$ lies on this diagonal, we may set $k=T_n-x$ and obtain $(C_k,R_k)=(1+(T_n-k),n-(T_n-k))$ (I expect the ambiguity of the notaion to be confusing, but I hope I'm wrong)
Now we want to "eliminate" $n$. For this, we go back to one of(*): $T_{n-1}<k\Leftrightarrow T_{n-1}+1\leq k\Leftrightarrow n^2-n+2\leq 2k\Leftrightarrow (n-1/2)^2+7/4\leq2k\Leftrightarrow n\leq\frac{(8k-7)^{1/2}+1}{2}$
From (*), it clear that $n$ is the greatest integer that satisfies $T_{n-1}<k$, and therfore the greatest to satisfy $n\leq\frac{(8k-7)^{1/2}+1}{2}$; so by definition $n=\lfloor\frac{(8k-7)^{1/2}+1}{2}\rfloor$.
At last, the answer: The location of $k$ is $(T_n-k+1,n+k-T_n)$, where $n=\lfloor\frac{(8k-7)^{1/2}+1}{2}\rfloor$.
Example: Take $k=2002$, then $n=63$ and hence $2002$ lies in $(T_{63}-2002+1,63+2002-T_{63})=(15,49)$.
መለሰ | 2021-06-13T20:12:36 | {
"domain": "mathhelpboards.com",
"url": "https://mathhelpboards.com/threads/find-the-number-of-line-and-column-where-the-number-2002-stays.3936/",
"openwebmath_score": 0.9046657085418701,
"openwebmath_perplexity": 515.2749761732637,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.984810949854636,
"lm_q2_score": 0.8459424450764199,
"lm_q1q2_score": 0.8330933828580623
} |
https://math.stackexchange.com/questions/1618741/how-to-prove-that-all-odd-powers-of-two-add-one-are-multiples-of-three/1619610 | # How to prove that all odd powers of two add one are multiples of three
For example \begin{align} 2^5 + 1 &= 33\\ 2^{11} + 1 &= 2049\ \text{(dividing by $3$ gives $683$)} \end{align}
I know that $2^{61}- 1$ is a prime number, but how do I prove that $2^{61}+1$ is a multiple of three?
• @AnuragA $2^{61}- 1$ is a prime number, it is $2^{61}+ 1$ that is divisible by three. I apologise if I made a mistake writing the question – Ronikos Jan 19 '16 at 22:20
• If you know that $2^{61}-1$ is prime, $2^{61}+1$ has to be a multiple of 3, since $(a,a+1,a+2)$ must contain a multiple of 3 and the first two can't be (this doesn't address the general case in the question, so it's a comment not an answer) – Glen_b Jan 20 '16 at 0:47
• @Glen_b Just for fun, I wrote out an answer using this sort of observation! – Benjamin Dickman Jan 20 '16 at 8:14
• As hinted at in several answers below, this is exactly the kind of thing that modular arithmetic was invented for. While you can present a solution using "elementary" language, without using (or even being aware of) any terminology from modular arithmetic, that's a bit like driving a nail using a large stone instead of a hammer. Sure, either way works, but having the proper tool makes it so much less awkward. – Ilmari Karonen Jan 20 '16 at 13:36
• – Martin Sleziak Jan 20 '16 at 15:10
Since $2 \equiv -1 \pmod{3}$, therefore $2^{k} \equiv (-1)^k \pmod{3}$. When $k$ is odd this becomes $2^k \equiv -1 \pmod{3}$. Thus $2^k+1 \equiv 0 \pmod{3}$.
• Note that this logic also shows that any even power of two, minus one, is a multiple of 3. – Michael Seifert Jan 20 '16 at 19:49
• I'm suprised such an easy answer got so many votes. – N.S.JOHN Apr 17 '16 at 10:18
A direct alternative to the answer via congruences is to note that for $k$ odd one has the well-known polynomial identity $$x^{k} + 1 = (x + 1) (x^{k-1} - x^{k-2} + \dots - x + 1),$$ and then substitute $x = 2$.
Another way is by induction:
$$2^1+1 = 3 = 3 \cdot 1$$
Then, if $2^k+1 = 3j, j \in \mathbb{N}$, then
\begin{align} 2^{k+2}+1 & = 4\cdot2^k+1 \\ & = 4(2^k+1)-3 \\ & = 4(3j)-3 \qquad \leftarrow \text{uses induction hypothesis} \\ & = 3(4j-1) \end{align}
• Wouldn't it be just easier to prove that if a = b mod m then a^k = b^k mod m by induction then immediately use that fact? – djechlin Jan 20 '16 at 0:29
• That proof is super trivial if you assume that a = b mod m implies ax = bx mod m, which you do. tldr, this isn't "another way." – djechlin Jan 20 '16 at 0:30
• @djechlin: Fair point. I will rewrite to avoid modular arithmetic altogether. – Brian Tung Jan 20 '16 at 4:31
• +1, but I would have said it a little more simply: $2^{k+2}+1=4\cdot2^k+1=3\cdot2^k\,+\,(2^k+1)$, which is a multiple of 3 iff $2^k+1$ is. You can also use that to prove that adding 1 to any even power of 2 yields a number that is not a multiple of 3. – Scott Jan 20 '16 at 17:20
• This inline notation makes it difficult to see where you've used to hypothesis. Took me way longer to comprehend than it should have for such a simple induction proof – Cruncher Jan 20 '16 at 20:29
One of your examples is that $2^{11} + 1$ is divisible by $3$. We investigate as follows:
Let us consider instead raising $2$ to an even power and subtracting $1$. And then let us factor.
Example: $2^{10} - 1 = (2^5 - 1)(2^5 + 1)$.
Among any three consecutive integers, exactly one of them must be divisible by $3$.
Clearly $2^5$ is not divisible by $3$, so either its predecessor or successor is divisible by $3$.
That is, either $2^5 - 1$ or $2^5 + 1$ is divisible by $3$, whence their product is, as well.
Okay: Their product is $2^{10} - 1$, which we have now established is divisible by $3$.
This number is still divisible by $3$ after being doubled, and still divisible by $3$ when we add $3$ to it.
So: We have that $2(2^{10} - 1) + 3 = 2^{11} - 2 + 3 = 2^{11} + 1$ is divisible by $3$ as desired.
A similar bit of reasoning around $2^{2k} - 1$ yields the assertion at hand. "QED"
• (Just to fill in the last details: Let $k \in \mathbb{Z}$ be arbitrary. Among the three consecutive integers $2^{k} - 1, 2^{k}, 2^{k} + 1$ must be exactly one divisible by three; since $2^{k}$ is not divisible by three, it must be one of the other two, hence their product is divisible by three. This product is $(2^{k} - 1)(2^{k} + 1) = 2^{2k} - 1$; note that multiplying this number by $2$ and then adding $3$ does not affect its divisibility by three, whence $2(2^{2k} - 1) + 3 = 2^{2k+1} + 1$ is divisible by $3$ as desired. QED) – Benjamin Dickman Jan 22 '16 at 2:08
There are only three possibilities for the divisibility of an integer by $3$, which are: no remainder, a remainder of $1$, or a remainder of $2$. But if we multiply $2$ by itself over and over again, the no remainder option is impossible, as that would mean that $2$ is a multiple of $3$, which it is not.
The thing is also that we can multiply reminders. So $2$ leaves a remainder of $2$, and $2 \times 2 = 4$, which leaves a remainder of $1$. And $1 \times 2 = 2$, which leaves a remainder of $2$ again.
Therefore the powers of $2$ alternate remainders on division by $3$ according to the parity of the exponent: $2^n \equiv 1 \pmod 3$ if $n$ is even and $2^n \equiv 2$ or $-1 \pmod 3$ if $n$ is odd.
• Well written for readability by non-mathematicians. Good job. – Wildcard Jan 20 '16 at 22:31
• Thank you for the acknowledgement. – Mr. Brooks Jan 20 '16 at 22:32
• A superb answer. – Fattie Jan 23 '16 at 12:26
$2 = 3-1$
$2^k = (3 - 1)^k = 3^k - k*3^{k - 1} ..... = \sum_{n = 0}^k {k \choose n}3^{k - n}(-1)^n = \sum_{n = 0}^{k - 1} {k \choose n}3^{k - n}(-1)^n + {k \choose 3}3^{k - n}(-1)^k = \sum_{n = 0}^{k - 1} {k \choose n}3^{k - n}(-1)^n \pm 1$
Since $k$ is odd:
$2^k = \sum_{n = 0}^{k - 1} {k \choose n}3^{k - n}(-1)^n - 1$
$2^k + 1 = \sum_{n = 0}^{k - 1} {k \choose n}3^{k - n}(-1)^n = 3\sum_{n = 0}^{k - 1} {k \choose n}3^{k - n - 1}(-1)^n$
which is a multiple of 3.
In binary: \begin{align*} 11 &= 11 \cdot 01 \\ 1111 &= 11 \cdot 0101 \\ 111111 &= 11 \cdot 010101 \\ 11111111 &= 11 \cdot 01010101 \\ 1111111111 &= 11 \cdot 0101010101 \\ &~~\vdots \end{align*} Clearly the LHS are of the form $2^{2k}-1$. And we've proved that these are multiples of three by factoring out $11$ in binary! Thus if we double and add three, $$2 \cdot (2^{2k}-1)+3=2^{2k+1}+1$$ is also a multiple of three.
$2^2=4\equiv1\pmod 3$, so $4^k\equiv1\pmod3$ for all integers $k$. And so for any odd number $2k+1$, we get $2^{2k+1}+1 = 4^k\cdot 2+1\equiv 2+1\equiv0\pmod3$.
### Using geometric series
Consider $S_k = 1 + 2\sum_{i=0}^{k-1}4^i$. It follows that $S_k$ is in $\mathbb{N}^{+}$. However, $$S_k = 1 + 2\cdot\frac{4^k - 1}{4 - 1} = \frac{2^{2k+1}+1}{3}$$ QED
### Using recurrence
For $k\in\mathbb{N}$, consider the recurrence $J_{k+1} = 4J_{k} - 1$, with $J_0 = 1$.
Since $J_0 = 1$, it follows that each of $J_k$ is in $\mathbb{N}^{+}$.
Solving the recurrence yields: $J_k = \frac{1}{3}(2^{2k + 1}+ 1)$.
The characteristic equation $r^2 - 5r + 4 = 0$ follows from $J_{k+2} = 5J_{k+1} - 4J_{k}$.
Hence, $J_k = A\lambda_1^k + B\lambda_2^k$, with $\lambda_1 = 4$ and $\lambda_2 = 1$ as solutions to the characteristic equation.
As $J_0 = 1 \implies J_1 = 3$, it follows that $A = \frac{2}{3}$ and $B = \frac{1}{3}$.
QED
If $k$ is odd, then $2^k$ is $2*2^{k-1}$. If $n$ is not a multiple of 3 then $n^2-1$ is since $n^2-1 = (n-1)(n+1)$ and at least one of them is a multiple of 3. This means that $2*2^{k-1} - 2$ is a multiple of $3$ as is $2*2^{k-1} + 1$.
Hope I helped.
• Welcome to MathSE! See this guide for how to mark up math nicely on this site. – Frentos Jan 20 '16 at 14:43
• Thanks! Is the explanation ok? – Creator Jan 20 '16 at 14:47
• This looks find. Although it seems basically the same answer as the one posted here. In any case, welcome to the site. – Martin Sleziak Jan 20 '16 at 15:03
• It's hard for me to follow - matching up the $n^2=1$ bit with half of $2 * 2^{k-1} - 2$ for instance. And it relies on the assumption that $n$ (which is presumably meant to stand for $2^{(k-1)/2}$?) is not a multiple of 3. – Frentos Jan 20 '16 at 15:09
• As far as I know 2 is NOT a multiple of 3 ;). I don't say $n^2=1$ anywhere. – Creator Jan 20 '16 at 15:11
There is exactly one integer that divides three among any three consecutive integers. Since $2^{61}-1$ is prime, $2^{61}$ trivially is not a multiple of $3$, $2^{61}+1$ is a multiple of $3$. | 2019-09-19T15:15:02 | {
"domain": "stackexchange.com",
"url": "https://math.stackexchange.com/questions/1618741/how-to-prove-that-all-odd-powers-of-two-add-one-are-multiples-of-three/1619610",
"openwebmath_score": 0.9944655299186707,
"openwebmath_perplexity": 282.0850350032324,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9848109507462227,
"lm_q2_score": 0.8459424431344437,
"lm_q1q2_score": 0.833093381699814
} |
https://math.stackexchange.com/questions/2766802/intuition-for-n-mu-sigma2-in-terms-of-its-infinite-expansion | # Intuition for $N(\mu, \sigma^2)$ in terms of its infinite expansion
To gain deeper insight to the Poisson and exponential random variables, I found that I could derive the random variables as follows:
I consider an experiment which consists of a continuum of trials on an interval $[0,t)$. The result of the experiment takes the form of an ordered $n$-tuple $\forall n \in \mathbb{N}$ containing distinct points on the interval. Every outcome is equally likely and I measure the size of the set containing tuples of $n$ different points by $I_n$ as:
$$I_n = \int_0^{t} \int_0^{x_{n}} \int_0^{x_{n-1}} \cdots \int_0^{ x_2 } dx_1 dx_{2} dx_{3} \dots dx_{n-1} dx_{n} = \frac{ t^n } { n! }$$
It follows that on some interval, $[0,t)$, the probability that the experiment results in an $k$-tuple, $k \in \mathbb{N}$ is
$$P(X(t) = k) = \frac{I_k}{\sum_{n = 0}^{\infty} I_n} = \frac{e^{-t} t^k}{k!}$$
And for $k = 0$, we have $P(X(t) = 0) = e^{-t}$.
# Question:
I was wondering if some similar intuition can applied to derive the Gaussian:
$$\frac{1}{\sqrt{2 \pi \sigma^2}} \exp \big(-\frac{(x-\mu)^2}{2\sigma^2} \big) \ \text{ or the standard normal, }\ \frac{e^{-x^2}}{\sqrt{\pi}}$$
I think that such an intuition might be obtained by gaining more insight into each term in the expansion of $\text{erf}(x)$ as is done for Poisson: \begin{align} \text{erf}(x) &= \frac{1}{\sqrt \pi } \int_{-x}^{x} e^{-t^2} dt\\ &= \frac{2}{\sqrt \pi } \big( \sum_{n=0}^{\infty} \frac{ x^{2n+1} }{ n! (2n+1) } \big)^{-1} \end{align}
Any ideas aside from dismissal of the question are much appreciated!
• I don't think there is much intuition beyond the intuition underlying the proof of the CLT. However, you can get a lot of nice visualization by explicitly computing the PDF of the sum of $n$ iid uniform $(-\sqrt{3},\sqrt{3})$ random variables (which is a piecewise polynomial that starts looking more and more like a standard Gaussian even for $n$ as small as $4$).
– Ian
May 4, 2018 at 18:18
• yes, central limit theorem is extremely interesting: en.wikipedia.org/wiki/Central_limit_theorem . Worth reading! Every distribution when iterated tends to the Gaussian in the end... Jun 18, 2018 at 4:04
• You may find this video interesting. Aug 13, 2018 at 0:18
Take an experiment that has two outcomes, success (S) and failure (F), of probability $p$ and $q=1-p$ respectively. The probability of S after one trial is $p$. The probability of two S after two trials is $p^2$, for one S and one F is $2pq$ and for two F is $q^2$. In general, for $N$ trials, the probability of having k S is given by the Binomial distribution $P_S(k,N)= \frac{N!}{k!(N-k)!}p^k q^{N-k}$. These are just the coefficients in front of the terms in $(p+q)^N$ after multiplying them out
What happens if we take the limit of large $N$? The Binomial coefficients at large $N$ approximate a Gaussian, which can be seen visually by looking at a low row Pascal's Triangle. We can derive it from the above formula. To make this simple let's work with the symmetric case $p=q=1/2$, so that we have $$P_S(k,N)= \frac{N!}{k!(N-k)!}\frac{1}{2^N}$$ An easy way to get the desired result is to exchange the index $k$ for one that starts from the center of the triangle where values are largest $x\in (-N/2,N/2),\; k=x+N/2$. Then we swap this in and apply Stirling's approximation $n!=\sqrt{2\pi}n^ne^{-n}$:
$$P_S(x,N)= \frac{N!}{(N/2+x)!(N/2-x)!}\frac{1}{2^N} \approx \frac{2}{\sqrt{2\pi N}}\frac{1}{(1-\frac{4 x^2}{N^2})^{(N+1)/2}}\left(\frac{1-\frac{2x}{N}}{1+\frac{2x}{N}}\right)^x$$
Finally exponentiation and taking the log we get $$P_S(x,N) \approx \frac{2}{\sqrt{2\pi N }} e^{-\frac{1}{2}(N+1)\log(1-\frac{4 x^2}{N^2}) +x(\log(1-\frac{2x}{N})-\log(1+\frac{2x}{N})) }$$
Using the expansion $\log(1+x)=x-x^2/2+ \cdots$, keep everything to order $1/N$ $$P_S(x,N) \approx \frac{2}{\sqrt{2\pi N}} e^{-\frac{2 x^2}{N} }$$
Now you can feel free to tack a on $dx$ and use a transformation of variables to scale $x \rightarrow \sqrt N x/2$ -- then $x$ is an "implicit" variable.
$$P_S(x,N)dx \approx \frac{1}{\sqrt{2\pi}} e^{-\frac{ x^2}{2}} dx$$
For general $p,q$, the derivation is similar, see for example http://scipp.ucsc.edu/~haber/ph116C/NormalApprox.pdf
This isn't exactly an infinite expansion like in your example, but there's a similar vein of thought in the conclusion that $N$ choose $k$ limits to a Gaussian type shape for large $N$.
• thanks for the answer. I'm looking for a derivation in which is enlightening to the terms of infinite series expansion of the exponential, or the cdf of the normal. Aug 16, 2018 at 19:14 | 2022-05-16T12:31:54 | {
"domain": "stackexchange.com",
"url": "https://math.stackexchange.com/questions/2766802/intuition-for-n-mu-sigma2-in-terms-of-its-infinite-expansion",
"openwebmath_score": 0.9613515734672546,
"openwebmath_perplexity": 235.57376764167415,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9848109547583622,
"lm_q2_score": 0.8459424373085146,
"lm_q1q2_score": 0.8330933793564143
} |
http://mathhelpforum.com/pre-calculus/154677-sine-limit.html | # Math Help - Sine limit
1. ## Sine limit
How do you show that $\displaystyle \lim_{x \to 0}\left(\dfrac{\sin{x}}{x}\right) = 1$ without drawing triangle areas?
2. Originally Posted by Sakai
How do you show that $\displaystyle \lim_{x \to 0}\left(\dfrac{\sin{x}}{x}\right) = 1$ without drawing triangle areas?
You can use L'Hopital's Rule...
$\displaystyle\lim_{x\rightarrow\ 0}\left(\frac{Sinx}{x}\right)=\lim_{x\rightarrow\ 0}\left(\frac{\frac{d}{dx}Sinx}{\frac{d}{dx}x}\rig ht)=\lim_{x\rightarrow\ 0}\left(\frac{Cosx}{1}\right)$
Alternatively....
the area of the sector of a circle is
$\displaystyle\frac{1}{2}r^2\theta$
the area of the sector, with the segment removed is
$\displaystyle\frac{1}{2}r^2\,Sin\theta$
As the angle goes to zero, the segment area goes to zero.
$\displaystyle\frac{A_1}{A_2}=\frac{\frac{1}{2}r^2\ ,Sin\theta}{\frac{1}{2}r^2\,\theta}$
As the angle approaches zero, $A_1\rightarrow\ A_2$
Then as $\displaystyle\lim_{\theta\rightarrow\ 0}Cos\theta=1$
As $\theta\rightarrow\ 0$
$Sin\theta\ \le\theta\ \le\ Sin\theta$
3. Originally Posted by Archie Meade
You could use L'Hopital's Rule...
$\displaystyle\lim_{x\rightarrow\ 0}\left(\frac{Sinx}{x}\right)=\lim_{x\rightarrow\ 0}\left(\frac{\frac{d}{dx}Sinx}{\frac{d}{dx}x}\rig ht)=\lim_{x\rightarrow\ 0}\left(\frac{Cosx}{1}\right)$
No you can't. Finding the derivative of $\sin{x}$ involves working out $\lim_{h \to 0}\frac{\sin{h}}{h}$ anyway. It's an entirely circular argument.
The only way that is logically correct is to squeeze the area of a sector in a unit circle by the triangles formed by $\sin{x}, \cos{x}$ and $\tan{x}, 1$.
4. Originally Posted by Prove It
No you can't. Finding the derivative of $\sin{x}$ involves working out $\lim_{h \to 0}\frac{\sin{h}}{h}$ anyway. It's an entirely circular argument.
Why not differentiate the power series expansion for Sin(x) ?
5. Because finding the power series expansion for $\sin{x}$ ALSO involves differentiating $\sin{x}$, which therefore would involve finding $\lim_{h \to 0}\frac{\sin{h}}{h}$. Again, circular argument.
6. Yes, that's right.
7. Originally Posted by Prove It
Squeeze the area of a sector in a unit circle by the triangles formed by $\sin{x}, \cos{x}$ and $\tan{x}, 1$.
I know that proof. That's why I said 'without drawing triangle areas'. I thought there might be a better way of showing it, but I guess not.
8. The Sandwich Theorem usually provides the most elegant of limit proofs anyway...
9. I disagree with Prove It that "finding the Taylor's series for sin(x) involves differentiating sin(x)". In fact, that is one way of defining sin(x).
$\lim_{t\to 0}\frac{sin(t)}{t}$ is a very "fundamental" limit and how you do it depends in large part upon how you define sin(t).
1) A very common way of defining f(t)= sin(t) is this: On an xy- coordinate system, draw the unit circle- the circle corresponding to $x^2+ y^2= 1$. Start at the point (1, 0) and measure (if t> 0) counterclockwise around the circumference of the circle a distance t (clockwise if t< 0). sin(t) is the y coordinate of the point you arrive at. You can proving that $\lim_{t\to 0}\frac{sin(t)}{t}= 1$ by looking at the difference between sin(t) and sin(-t) and using the area of the sector from -t to +t, sin(2t), divided by the arclength, 2t.
2) Another way of defining f(t)= sin(t) is to define $sin(t)= \sum_{n=0}^\infty \frac{(-1)^n}{(2n+1)!}t^{2n+1}= t- \frac{1}{3!}t^3+ \frac{1}{5!}t^5+\cdot\cdot\cdot\$.
With that definition we have, as Archie Meade suggested (though there is no reason to "differentiate" the sum), $\frac{sin(t)}{t}= \sum_{n=0}^{\infty}\frac{(-1)^n}{(2n+1)!}t^{2n}= 1- \frac{1}{3!}t^2+ \frac{1}{5!}t^4+ \cdot\cdot\cdot$ which clearly has limit 1 as x goes to 0.
3) Yet another way is to define f(t)= sin(t) to be the solution to the "intial value problem" y"+ y= 0 with initial values y(0)= 0, y'(0)= 1. From that we can prove directly that y"(0)+ y(0)= y"(0)+ 0= 0; y'''+ y'= 0 so y'''(0)+ y'(0)= y'''(0)+ 1= 0 so y'''(0)= -1; y''''+ y''= 0 so y''''(0)+ y''(0)= 0 so y''''(0)= 0, etc. and so form the Taylor's series for sin(t).
10. Originally Posted by HallsofIvy
I disagree with Prove It that "finding the Taylor's series for sin(x) involves differentiating sin(x)". In fact, that is one way of defining sin(x).
$\lim_{t\to 0}\frac{sin(t)}{t}$ is a very "fundamental" limit and how you do it depends in large part upon how you define sin(t).
1) A very common way of defining f(t)= sin(t) is this: On an xy- coordinate system, draw the unit circle- the circle corresponding to $x^2+ y^2= 1$. Start at the point (1, 0) and measure (if t> 0) counterclockwise around the circumference of the circle a distance t (clockwise if t< 0). sin(t) is the y coordinate of the point you arrive at. You can proving that $\lim_{t\to 0}\frac{sin(t)}{t}= 1$ by looking at the difference between sin(t) and sin(-t) and using the area of the sector from -t to +t, sin(2t), divided by the arclength, 2t.
2) Another way of defining f(t)= sin(t) is to define $sin(t)= \sum_{n=0}^\infty \frac{(-1)^n}{(2n+1)!}t^{2n+1}= t- \frac{1}{3!}t^3+ \frac{1}{5!}t^5+\cdot\cdot\cdot\$.
With that definition we have, as Archie Meade suggested (though there is no reason to "differentiate" the sum), $\frac{sin(t)}{t}= \sum_{n=0}^{\infty}\frac{(-1)^n}{(2n+1)!}t^{2n}= 1- \frac{1}{3!}t^2+ \frac{1}{5!}t^4+ \cdot\cdot\cdot$ which clearly has limit 1 as x goes to 0.
3) Yet another way is to define f(t)= sin(t) to be the solution to the "intial value problem" y"+ y= 0 with initial values y(0)= 0, y'(0)= 1. From that we can prove directly that y"(0)+ y(0)= y"(0)+ 0= 0; y'''+ y'= 0 so y'''(0)+ y'(0)= y'''(0)+ 1= 0 so y'''(0)= -1; y''''+ y''= 0 so y''''(0)+ y''(0)= 0 so y''''(0)= 0, etc. and so form the Taylor's series for sin(t).
I'm sorry but I totally disagree with (2) and (3).
$\sin{x}$ is NOT that polynomial by definition.
It is only written as that polynomial because you wish to find the right combination of addition, subtraction, multiplication, division and exponentiation that will enable us to evaluate $\sin{x}$ for certain values of $x$.
If $f(x) = \sin{x}$, the polynomial comes from
$f(x) = \sum_{n = 0}^{\infty}\frac{f^{(n)}(0)}{n!}x^n$.
Notice that finding these coefficients depends upon differentiating $f(x) = \sin{x}$ (infinitely many times for that matter...)
So you can not define $\sin{x}$ to be that polynomial unless you were to find the derivative of $\sin{x}$, which again involves finding $\lim_{h \to 0}\frac{\sin{h}}{h}$ anyway.
Wherever you have to differentiate $\sin{x}$ in the definition, that means it must have been defined in some other way earlier.
Therefore, the only correct way to define $\sin{x}$ is by the unit circle definition.
11. L'Hopital's is the easiest way to do it, there is also the long way to do it (without geometric approximation). But again it's the long way and it's kind of pointless to use when you have l'hopital's. If you want to see it though look at this calculus resource under the pre-calculus section. It goes through the proof. Or you can always pop open a textbook, but hell if we all did that this forum probably wouldn't exist
12. Read the posts I have given above.
You can NOT use L'Hospital's rule to find $\lim_{x \to 0}\frac{\sin{x}}{x}$, because L'Hospital's Rule involves finding the derivative of $\sin{x}$, while finding the derivative of $\sin{x}$ involves finding $\lim_{h \to 0}\frac{\sin{h}}{h}$, the exact same limit!
Let us try to find $\frac{d}{dx}(\sin{x})$...
$\frac{d}{dx}(\sin{x}) = \lim_{h \to 0}\frac{\sin{(x+h)}-\sin{x}}{h}$
$= \lim_{h \to 0}\frac{\sin{x}\cos{h} + \cos{x}\sin{h} - \sin{x}}{h}$
$= \lim_{h \to 0}\frac{\sin{x}\cos{h} - \sin{x}}{h} + \lim_{h \to 0}\frac{\cos{x}\sin{h}}{h}$
$= \sin{x}\lim_{h \to 0}\left(\frac{\cos{h}- 1}{h}\right) + \cos{x}\lim_{h \to 0}\frac{\sin{h}}{h}$.
Look at that, you have to evaluate THE EXACT SAME LIMIT in order to find the derivative of $\sin{x}$. So you can NOT use L'Hospital's Rule. It is entirely circular reasoning.
You need to have a way to evaluate this limit WITHOUT USING DERIVATIVES! That is why you have to use the Sandwich Theorem.
13. However,
since the purpose of first principles is to arrive at the tangent gradient (starting with a secant),
for circular motion, we only need to take the perpendicular to the radius.
The rates of change of Sin(x) and Cos(x) can be viewed directly without reference to limits
as we consider a point rotating counterclockwise around the circumference.
If it was solely necessary to calculate the rate of change of Sin(x) using the limit of Sin(x)/x
then we would have a circular analysis.
14. [tex]\displaystyle \lim_{x\to{0}}\left(\frac{\sin{x}}{x}\right) = \lim_{x\to{0}}\prod_{n=1}^{\infty}\cos\left(\frac{ x}{2^n}\right)[/Math] [LaTeX ERROR: Convert failed] .
As $x \to 0$, $\cos\left(\frac{x}{2}\right) \to 1, \cos\left(\frac{x}{2^2}\right) \to 1, \cos\left(\frac{x}{2^3}\right) \to 1, ..., \cos\left(\frac{x}{2^n}\right) \to 1.$
Therefore [tex]\displaystyle \lim_{x\to{0}}\left(\frac{\sin{x}}{x}\right) = 1.[/Math] | 2014-08-27T09:22:55 | {
"domain": "mathhelpforum.com",
"url": "http://mathhelpforum.com/pre-calculus/154677-sine-limit.html",
"openwebmath_score": 0.9292082786560059,
"openwebmath_perplexity": 802.1132220697791,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.984810952529396,
"lm_q2_score": 0.8459424373085146,
"lm_q1q2_score": 0.8330933774708371
} |
https://www.beatthegmat.com/a-number-of-people-shared-a-meal-intending-to-divide-the-cost-evenly-among-themselves-however-several-of-the-diners-t328306.html | ## A number of people shared a meal, intending to divide the cost evenly among themselves. However, several of the diners
##### This topic has expert replies
Legendary Member
Posts: 2898
Joined: 07 Sep 2017
Thanked: 6 times
Followed by:5 members
### A number of people shared a meal, intending to divide the cost evenly among themselves. However, several of the diners
by Vincen » Sat Nov 27, 2021 4:38 am
00:00
A
B
C
D
E
## Global Stats
A number of people shared a meal, intending to divide the cost evenly among themselves. However, several of the diners left without paying. When the cost was divided evenly among the remaining diners, each remaining person paid $$\12$$ more than he or she would have if all diners had contributed equally. Was the total cost of the meal, in dollars, an integer?
(1) Four people left without paying.
(2) Ten people in total shared the meal.
Source: Veritas Prep
### GMAT/MBA Expert
GMAT Instructor
Posts: 16162
Joined: 08 Dec 2008
Location: Vancouver, BC
Thanked: 5254 times
Followed by:1268 members
GMAT Score:770
### Re: A number of people shared a meal, intending to divide the cost evenly among themselves. However, several of the dine
by [email protected] » Sat Nov 27, 2021 7:31 am
00:00
A
B
C
D
E
## Global Stats
Vincen wrote:
Sat Nov 27, 2021 4:38 am
A number of people shared a meal, intending to divide the cost evenly among themselves. However, several of the diners left without paying. When the cost was divided evenly among the remaining diners, each remaining person paid $$\12$$ more than he or she would have if all diners had contributed equally. Was the total cost of the meal, in dollars, an integer?
(1) Four people left without paying.
(2) Ten people in total shared the meal.
Source: Veritas Prep
Target question: Was the total cost of the meal, in dollars, an integer?
This is a great candidate for rephrasing the target question
Given: A number of people shared a meal, intending to divide the cost evenly among themselves. However, several of the diners left without paying. When the cost was divided evenly among the remaining diners, each remaining person paid \$12 more than he or she would have if all diners had contributed equally.
Let c = TOTAL cost of the meal
Let n = number of people who SHARED the meal
Let d = the number of deadbeats who left before paying
So, n - d = number of people who PAID for the meal
Cost per person with all n people = c/n
Cost per person with n-d people = c/(n - d)
Word equation: (cost per person with original diners) + 12 = cost per person with reduced number of diners
Equation: c/n + 12 = c/(n - d)
Eliminate fractions by multiplying both sides by (n)(n - d) to get: c(n - d) + 12(n)(n - d) = cn
Expand: cn - cd + 12n² - 12nd = cn
Subtract cn from both sides: -cd + 12n² - 12nd = 0
Add cd to both sides: 12n² - 12nd = cd
Divide both sides by d to get: 12n²/d - 12n = c
Factor: n(12n/d - 12) = c
Since n(12n/d - 12) equals the total cost of the meal, we can REPHRASE the target question....
REPHRASED target question: Is n(12n/d - 12) an integer?
Statement 1: Four people left without paying
In other words, statement 1 tells us that d = 4
Plug d = 4 into the REPHRASED target question to get: Is n(12n/4 - 12) an integer?
n(12n/4 - 12) = n(3n - 12), and n(3n - 12) is definitely an integer.
How do we know this?
Well, we know that n is an integer.
So, 3n is an integer, which means 3n - 12 is an integer, which means n(3n - 12) is an integer.
Since we can answer the REPHRASED target question with certainty, statement 1 is SUFFICIENT
Statement 2: Ten people in total shared the meal
In other words, statement 2 tells us that n = 10
Plug n = 10 into the REPHRASED target question to get: Is (10)(120/d - 12) an integer?
There are several values of d that yield conflicting answers to the target question. Here are two:
Case a: if d = 2, then (10)(120/d - 12) = (10)(120/2 - 12) = 480, and 480 IS an integer
Case b: if d = 7, then (10)(120/d - 12) = (10)(120/7 - 12), and this does NOT evaluate to be an integer
Since we cannot answer the REPHRASED target question with certainty, statement 2 is NOT SUFFICIENT | 2022-12-02T10:22:59 | {
"domain": "beatthegmat.com",
"url": "https://www.beatthegmat.com/a-number-of-people-shared-a-meal-intending-to-divide-the-cost-evenly-among-themselves-however-several-of-the-diners-t328306.html",
"openwebmath_score": 0.7196266055107117,
"openwebmath_perplexity": 2921.9206532191765,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9848109520836026,
"lm_q2_score": 0.8459424373085146,
"lm_q1q2_score": 0.8330933770937216
} |
http://math.stackexchange.com/questions/151032/if-f-colon-mathbbr-to-mathbbr-is-such-that-f-x-y-f-x-f-y-an | # If $f\colon \mathbb{R} \to \mathbb{R}$ is such that $f (x + y) = f (x) f (y)$ and continuous at $0$, then continuous everywhere
Prove that if $f\colon\mathbb{R}\to\mathbb{R}$ is such that $f(x+y)=f(x)f(y)$ for all $x,y$, and $f$ is continuous at $0$, then it is continuous everywhere.
If there exists $c \in \mathbb{R}$ such that $f(c) = 0$, then $$f(x + c) = f(x)f(c) = 0.$$ As every real number $y$ can be written as $y = x + c$ for some real $x$, this function is either everywhere zero or nowhere zero. The latter case is the interesting one. So let's consider the case that $f$ is not the constant function $f = 0$.
To prove continuity in this case, note that for any $x \in \mathbb{R}$ $$f(x) = f(x + 0) = f(x)f(0) \implies f(0) = 1.$$
Continuity at $0$ tells us that given any $\varepsilon_0 > 0$, we can find $\delta_0 > 0$ such that $|x| < \delta_0$ implies $$|f(x) - 1| < \varepsilon_0.$$
Okay, so let $c \in \mathbb{R}$ be fixed arbitrarily (recall that $f(c)$ is nonzero). Let $\varepsilon > 0$. By continuity of $f$ at $0$, we can choose $\delta > 0$ such that $$|x - c| < \delta\implies |f(x - c) - 1| < \frac{\varepsilon}{|f(c)|}.$$
Now notice that for all $x$ such that $|x - c| < \delta$, we have \begin{align*} |f(x) - f(c)| &= |f(x - c + c) - f(c)|\\ &= |f(x - c)f(c) - f(c)|\\ &= |f(c)| |f(x - c) - 1|\\ &\lt |f(c)| \frac{\varepsilon}{|f(c)|}\\ &= \varepsilon. \end{align*} Hence $f$ is continuous at $c$. Since $c$ was arbitrary, $f$ is continuous on all of $\mathbb{R}$.
Is my procedure correct?
-
Nicely done!${}{}{}$ – André Nicolas May 29 '12 at 4:26
The additive version of this question is also on the site. Do we consider them abstract duplicates? – Zev Chonoles May 29 '12 at 4:47
We also have a few question about this functional equation, see here and the linked question. – Martin Sleziak May 29 '12 at 4:55
@ZevChonoles Yes. I think it would be good if someone can write a generalized version of this and call it, say Cauchy functional equation, add an answer to the generalized question. All the rest should be closed as dupe of this. – user17762 May 29 '12 at 4:57
I am not sure if closing question of the type please check my proof should be closed as duplicate question. (Even if we had a question on this exact result, where some proof is given, in this kind of question I would expect comments on the OP's proof strategy, mistakes - if there are any, writing style etc. So it is kind of different from answer to question of the type I would like to see a proof or a reference for this. – Martin Sleziak May 29 '12 at 5:02
One easier thing to do is to notice that $f(x)=(f(x/2))^2$ so $f$ is positive, and assume that it is never zero, since then the function is identically zero. Then you can define $g(x)=\ln f(x)$ and this function $g$ will satisfy the Cauchy functional equation $$g(x+y)=g(x)+g(y)$$ and the theory for this functional equation is well known, and it is easy to see that $g$ is continuous if and only if it is continuous at $0$.
-
"Easy to see"? But no easier than the problem we are doing here... – GEdgar May 29 '12 at 17:43
If this is for a first year real analysis class, I think you might get dinged for not at least saying about the zero case that it's a constant function, and constant functions are continuous. Also, you probably should learn your favorite form of LaTeX; graders tend to like this sort of thing, and happy graders tends to mean better scores, in my experience.
Other than that, excellent work.
-
I mean not to intrude, but, thus far as I can judge, is this not a comment? – awllower May 29 '12 at 5:34
@awllower: If the question is "Is my work correct" then saying "Yes" is an answer, as is "yes, but..." – Henry May 29 '12 at 6:52
@awllower: I was wondering about that, but I ended up deciding what Henry said. – Eric Stucky May 29 '12 at 23:55 | 2014-12-21T13:13:32 | {
"domain": "stackexchange.com",
"url": "http://math.stackexchange.com/questions/151032/if-f-colon-mathbbr-to-mathbbr-is-such-that-f-x-y-f-x-f-y-an",
"openwebmath_score": 0.987245500087738,
"openwebmath_perplexity": 256.77130488965224,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9848109520836026,
"lm_q2_score": 0.8459424373085146,
"lm_q1q2_score": 0.8330933770937216
} |
https://math.stackexchange.com/questions/2908775/maximum-value-of-abbcca-given-that-a2bc-4/2908787 | # Maximum value of $ab+bc+ca$ given that $a+2b+c=4$
Question:
Find the maximum value of $ab+bc+ca$ from the equation,
$$a+2b+c=4$$
My method:
I tried making quadratic equation(in $b$) and then putting discriminant greater than or equal to $0$. It doesn't help as it yields a value greater than the answer.
Thanks in advance for the solution.
• Are the variables assumed to be positive? – Dr. Sonnhard Graubner Sep 7 '18 at 15:55
• No. It is not given. So no. Whenever they are positive, it gets mentioned in the question. – Love Invariants Sep 7 '18 at 15:56
• @LoveInvariants What is even nicer is that $4$ is the only positive integer that satisfies $$ab+bc+ca=a+2b+c\quad\ddot\smile$$ – TheSimpliFire Sep 7 '18 at 16:35
• @TheSimpliFire- Thats a very nice observation. 😊 – Love Invariants Sep 7 '18 at 16:53
Without using calculus:
Substituting $c=4-2b-a$, we get $$ab+bc+ca=ab+(a+b)(4-2b-a)=4(a+b)-(a+b)^2-b^2$$ and since $f(x)=4x-x^2=4-(x-2)^2$ has maximum at $(2,4)$, we have that $\max\{a+b\}=2$ so $$ab+bc+ca\le4-b^2$$ and the minimum value of $b$ is zero, yielding a maximum value of $4$.
• I was searching for something like this. Thanks. – Love Invariants Sep 7 '18 at 16:07
Hint: Plugging $$c=4-a-2b$$ into your given term we get
$$f(a,b)=-a^2+4a-2ab-2b^2+4b$$ and now differentiate this with respect to $a,b$
• Easy solution. Thanks for the solution. Upvoted your answer. – Love Invariants Sep 7 '18 at 16:07
You can use Lagrange Multiplier. You would get:
$$F(a,b,c,\lambda) = ab + bc + ca - \lambda(a+2b+c-4)$$
Take the partial derivatives and equate them to zero.
$$0 = F_a = b + c - \lambda$$ $$0 = F_b = a + c - 2\lambda$$ $$0 = F_c = a + b - \lambda$$
Then:
$$0 = F_a + F_c - F_b = b + c - \lambda + a + b - \lambda - a - c + 2\lambda = 2b$$
Thus we must have $b=0$. Then this prompts that $c = a = \lambda$ and so we get that $a=c=\lambda = 2$
So we get that the maximum occurs at $(a,b,c) = (2,0,2)$ and it's $4$.
• I didn't want to use. But Lagrange is always the easiest. way to find these type of answers. Upvoted your answer. – Love Invariants Sep 7 '18 at 16:05
For $a=2$, $b=0$ and $c=2$ we'll get a value $4$.
We'll prove that it's a maximal value.
Indeed, we need to prove that $$ab+ac+bc\leq4\left(\frac{a+2b+c}{4}\right)^2$$ or $$(a-c)^2+4b^2\geq0,$$ which is obvious.
• It feels nice when you learn so many ways to solve a problem. Indeed a great way and different from others. Upvoted your answer. I wish I could accept 2 answers. XD – Love Invariants Sep 7 '18 at 16:09
$ab + bc + ac = a(2 - \frac12a -\frac12c) + c(2 - \frac12a - \frac12c) + ac\\= 2a - \frac12a^2 + 2c - \frac12c^2$
$(2a - \frac12a^2)$ can be shown to have a maximum value of $2$ for real $a$. It follows that $(2a - \frac12a^2) + (2c - \frac12c^2)$ has a maximum value of $4$ for real $a$ and $c$. As observed in other answers, the value of $4$ is attained at $(a, b, c) = (2, 0, 2)$. | 2019-01-21T01:52:57 | {
"domain": "stackexchange.com",
"url": "https://math.stackexchange.com/questions/2908775/maximum-value-of-abbcca-given-that-a2bc-4/2908787",
"openwebmath_score": 0.9417018890380859,
"openwebmath_perplexity": 322.3972391872792,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9848109480714626,
"lm_q2_score": 0.845942439250491,
"lm_q1q2_score": 0.8330933756121617
} |
https://math.stackexchange.com/questions/3611312/finding-the-triangle-with-the-maximum-area-with-a-given-perimeter | # Finding the triangle with the maximum area with a given perimeter
Question: Of all triangles with a given perimeter, find the triangle with the maximum area. Justify your answer.
My approach: Let us have any $$\Delta ABC$$ such that the sides opposite to $$A$$ is of length $$a$$, the side opposite to $$B$$ is of length b and the side opposite to $$C$$ is of length $$c$$. Now since the perimeter to $$\Delta ABC$$ is fixed, thus we must have $$P=a+b+c$$ to be constant.
Now fix any side of the triangle, let us fix $$BC$$. Therefore $$b+c=P-a$$, which implies that $$b+c$$ is constant. Thus the locus of the point $$A$$ must be an ellipse having one of it's axis as the side $$BC$$. Now let us select any point $$A$$ on the ellipse and drop a perpendicular to the axis $$BC$$. Let it meet the major axis at $$P$$. Now let $$AP=h$$. Therefore, the area of the $$\Delta ABC=\frac{1}{2}.h.BC=\frac{1}{2}.h.a.$$ Now since $$\frac{1}{2}a$$ is constant, implies the area of $$\Delta ABC$$ can be maximized by maximizing $$h$$. Now clearly $$h$$ attains it's maximum value when it coincides with the other axis of the ellipse under consideration, that is when $$AB=AC$$. Thus $$\Delta ABC$$ must be isosceles with $$AB=AC$$ to get a maximum value of the area of $$\Delta ABC$$.
Thus it is clear that the triangle which will have the maximum area must be one of the isosceles triangles $$ABC$$ having $$BC$$ as the base.
Thus in any such $$\Delta ABC$$, we must have $$a+b+c=a+2b \hspace{0.2 cm}(\because b=c)=P\implies b=\frac{1}{2}(P-a).$$ Thus by Heron's formula we have $$|\Delta ABC|=\sqrt{\frac{P}{2}\left(\frac{P}{2}-a\right)\left(\frac{a}{2}\right)\left(\frac{a}{2}\right)}=\frac{a\sqrt{P}}{4}\sqrt{P-2a}.$$
Now to obtain the condition for maximizing $$|\Delta ABC|$$ we have to check when $$\frac{d}{da}\frac{a\sqrt{P}}{4}\sqrt{P-2a}=0.$$ Now $$\frac{d}{da}\frac{a\sqrt{P}}{4}\sqrt{P-2a}=0\\\iff (P-2a)^{1/2}-a(P-2a)^{-1/2}=0\\\iff P-2a=a\\\iff 3a=P\\\iff a=\frac{P}{3}.$$
Now note that $$\frac{d^2}{da^2}|\Delta ABC|<0$$, which implies that $$|\Delta ABC|$$ attain it's maximum value when $$a=\frac{P}{3}.$$ This implies that $$b=c=\frac{P}{3}$$. Thus we have $$a=b=c=\frac{P}{3}$$. Therefore, $$|\Delta ABC|$$ is maximized if and only if $$a=b=c$$, i.e, the triangle is equilateral.
Can someone check if this solution is correct or not? And other solutions are welcomed. Please ensure that the solutions are based on geometry and one-variable calculus. This problem can be solved using Lagrange multipliers or multi-variable calculus, but I do not want a solution using the same.
• Clearly it must be an equilateral triangle, with sides $P/3$. – David G. Stork Apr 5 '20 at 18:47
The Heron's formula for the triangle is
$$A =\sqrt{s(s-a)(s-b)(s-c)}$$
where $$s=\frac p2$$. Then, Apply the AM-GM inequality to get
$$A =\sqrt{s(s-a)(s-b)(s-c)} \le \left[ s \left( \frac{3s-(a+b+c)}{3} \right) ^3\right]^{1/2} = \frac{s^2}{3\sqrt3}=\frac{p^2}{12\sqrt3}$$
where the equality, or the maximum area, occurs at $$a=b=c=\frac p3$$.
Just for giving another way to deduce a solution. Take a closed string and leaving a fixed side, say $$a$$, of the triangle draw an ellipse as usual. It is evident that the triangle with the largest area occurs with an isosceles triangle because it has the same base as all but has a greater height (actually a vertical semiaxis of the ellipse).
Now the area of this isosceles triangle is $$A=\dfrac a4\sqrt{(2b)^2-a^2}$$ but because of $$a+2b=p$$ we have a fonction of $$a$$, call it $$x$$, defined by $$A=\frac x4\sqrt{p^2-2px}$$ The derivative of A being equal to $$A'=\frac{p^2-3px}{4\sqrt{p^2-2px}}$$ we see that the maximum area is taken when $$a=\dfrac p3$$ then $$b=\dfrac p3$$ too.
Area of triangle with vertices $$a,b,c$$ $$\frac12(a\times b+b\times c+c\times a)=\frac12\left(a^R\cdot b+b^R\cdot c+c^R\cdot a\right)\tag1$$ where $$(x,y)^R=(-y,x)$$ is $$\frac\pi2$$ rotation counter-clockwise.
Perimeter of triangle $$|a-b|+|b-c|+|c-a|\tag2$$ maximize $$(1)$$ $$a^R\cdot(\delta b-\delta c)+b^R\cdot(\delta c-\delta a)+c^R\cdot(\delta a-\delta b)=0\tag3$$ for all variations that keep $$(2)$$ fixed $$\frac{a-b}{|a-b|}\cdot(\delta a-\delta b)+\frac{b-c}{|b-c|}\cdot(\delta b-\delta c)+\frac{c-a}{|c-a|}\cdot(\delta c-\delta a)=0\tag4$$ we can use $$\delta a-\delta b$$, $$\delta b-\delta c$$, and $$\delta c-\delta a$$ as the variations as long as we take into account their dependence: $$(\delta a-\delta b)+(\delta b-\delta c)+(\delta c-\delta a)=0\tag5$$ Orthogonality requires a point $$\mu$$ and constant $$\lambda$$ so that \begin{align} a^R&=\lambda\frac{b-c}{|b-c|}+\mu^R\\ b^R&=\lambda\frac{c-a}{|c-a|}+\mu^R\\ c^R&=\lambda\frac{a-b}{|a-b|}+\mu^R \end{align}\tag7 That is, $$|a-\mu|=|b-\mu|=|c-\mu|=\lambda\tag8$$ and $$(a-\mu)\cdot(b-c)=(b-\mu)\cdot(c-a)=(c-\mu)\cdot(a-b)=0\tag9$$ Therefore, \begin{align} |a-b|^2 &=|(a-\mu)-(b-\mu)|^2\\ &=|a-\mu|^2+|b-\mu|^2-2(a-\mu)\cdot(b-\mu)\\ &=|a-\mu|^2+|c-\mu|^2-2(a-\mu)\cdot(c-\mu)\\ &=|(a-\mu)-(c-\mu)|^2\\ &=|a-c|^2 \end{align}\tag{10} The step where $$b$$ changes to $$c$$ follows from $$|c-\mu|=|b-\mu|$$ and $$(a-\mu)\cdot(c-b)=0$$.
Similarly $$|a-b|^2=|c-b|^2$$. Thus, the triangle is equilateral. | 2021-05-18T21:31:39 | {
"domain": "stackexchange.com",
"url": "https://math.stackexchange.com/questions/3611312/finding-the-triangle-with-the-maximum-area-with-a-given-perimeter",
"openwebmath_score": 0.9601011276245117,
"openwebmath_perplexity": 127.76440380736949,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9848109529751892,
"lm_q2_score": 0.8459424314825852,
"lm_q1q2_score": 0.8330933721105134
} |
https://socratic.org/questions/prove-that-the-sum-of-6-consecutive-odd-numbers-is-an-even-number#565649 | # Prove that the sum of 6 consecutive odd numbers is an even number?
Mar 6, 2018
#### Explanation:
Any two consecutive odd numbers add up to an even number.
Any number of even numbers when added result in an even number.
We can divide six consecutive odd numbers in three pairs of consecutive odd numbers.
The three pair of consecutive odd numbers add up to three even numbers.
The three even numbers add up to an even number.
Hence, six consecutive odd numbers add up to an even number.
Mar 6, 2018
Let first odd number be $= 2 n - 1$, where $n$ is any positive integer.
Six consecutive odd numbers are
$\left(2 n - 1\right) , \left(2 n + 1\right) , \left(2 n + 3\right) , \left(2 n + 5\right) , \left(2 n + 7\right) , \left(2 n + 9\right)$
Sum of these six consecutive odd numbers is
$\sum = \left(2 n - 1\right) + \left(2 n + 1\right) + \left(2 n + 3\right) + \left(2 n + 5\right) + \left(2 n + 7\right) + \left(2 n + 9\right)$
$\sum = \left(6 \times 2 n\right) - 1 + 1 + 3 + 5 + 7 + 9$
We see that first term will always be even
$\implies \sum = \text{even number} + 24$
Since $24$ is even and sum of two even numbers is always even
$\therefore \sum = \text{even number}$
Hence Proved.
Mar 6, 2018
See below
#### Explanation:
An odd number has the form $2 n - 1$ for every $n \in \mathbb{N}$
Let be the first $2 n - 1$ we know that odd numbers are in arithmetic progresion with difference 2. So, the 6th will be $2 n + 9$
We know also that the sum of n consecutive numbers in a arithmetic progresion is
${S}_{n} = \frac{\left({a}_{1} + {a}_{n}\right) n}{2}$ where ${a}_{1}$ is the first and ${a}_{n}$ is the last one; $n$ is the number of sum elements. In our case
S_n=((a_1+a_n)n)/2=(2n-1+2n+9)/2·6=(4n+8)/2·6=12n+24
which is an even number for every $n \in \mathbb{N}$ because is divisible by 2 allways
Mar 6, 2018
$\text{We can actually say more:}$
$\setminus \quad \text{the sum of any 6 odd numbers (consecutive or not) is even.}$
$\text{Please see proof below.}$
#### Explanation:
$\text{We can actually say more:}$
$\setminus \quad \text{the sum of any 6 odd numbers (consecutive or not) is even.}$
$\text{Here's why. First, it is easy to see:}$
$\setminus q \quad \setminus q \quad \text{an odd number" + "an odd number" \ = \ "an even number}$
$\setminus q \quad \setminus q \quad \setminus q \quad \setminus q \quad \setminus q \quad \setminus q \quad \setminus q \quad \setminus q \quad \setminus q \quad \setminus q \quad \setminus q \quad \setminus q \quad \setminus \quad \text{and}$
$\setminus q \quad \setminus q \quad \text{an even number" + "an even number" \ = \ "an even number} .$
$\text{Using these observations with the sum of any 6 odd numbers,}$
$\text{we see:}$
$\setminus q \quad {\text{odd"_1 +"odd"_2 + "odd"_3 +"odd"_4 + "odd"_5 +"odd}}_{6} \setminus =$
$\setminus q \quad \setminus \overbrace{{\text{odd"_1 +"odd"_2 }^{ "even"_1 } + \overbrace{ "odd"_3 +"odd"_4 }^{ "even"_2 } + \overbrace{ "odd"_5 +"odd"_6 }^{ "even}}_{3}} \setminus =$
$\setminus q \quad \setminus q \quad \setminus q \quad \setminus q \quad \setminus \quad \setminus \setminus {\text{even"_1 + "even"_2 + "even}}_{3} \setminus =$
\qquad \qquad \qquad \qquad \quad \ \ \overbrace{ "even"_1 + "even"_2 }^{"even"_4 } + "even"_3 \ =
$\setminus q \quad \setminus q \quad \setminus q \quad \setminus q \quad \setminus q \quad \setminus q \quad \setminus \setminus {\text{even"_4 + "even}}_{3} \setminus =$
$\setminus q \quad \setminus q \quad \setminus q \quad \setminus q \quad \setminus q \quad \setminus q \quad \setminus q \quad \setminus q \quad \setminus \quad {\text{even}}_{5.}$
$\text{So we have shown:}$
$\setminus q \quad {\text{odd"_1 +"odd"_2 + "odd"_3 +"odd"_4 + "odd"_5 +"odd"_6 \ = "even}}_{5.}$
$\text{So we conclude:}$
$\setminus \quad \text{the sum of any 6 odd numbers (consecutive or not) is even.}$ | 2021-11-30T10:03:50 | {
"domain": "socratic.org",
"url": "https://socratic.org/questions/prove-that-the-sum-of-6-consecutive-odd-numbers-is-an-even-number#565649",
"openwebmath_score": 0.9376938939094543,
"openwebmath_perplexity": 516.4687027029471,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9848109478485657,
"lm_q2_score": 0.8459424334245618,
"lm_q1q2_score": 0.8330933696861649
} |
https://ivyglobal.com/study/explanations/53 | ## 1.
The Correct Answer is (A) — Subtracting 2x and 4 from each side of the equation gives us -10 = 2x. Dividing each side by 2 gives us x = -5.
## 2.
The Correct Answer is (B) — Since a and b are both even integers, we know that: (a + 1) is odd, (b + 1) is odd, (b +2) is even, and (a -1) is odd. So I. is a product of two odd integers, which is also odd. II. is a product of an even and odd integer, which is even. III. is the product of two odd integers, which is odd. B is the correct answer.
## 3.
The Correct Answer is (D) — Since the growth of the tree is linear, we expect our graph to be a line. This eliminates choices A and B. We are also told that the tree is initially 3 feet tall, meaning that the height when time = 0 is not zero. This eliminates choice C. The correct answer is D.
## 4.
The Correct Answer is (C) — If we set x =1, y can be 1, 2, 3, or 4 to satisfy the inequality. If we set x = 2, then y can only be equal to 1. We cannot set x equal to any higher integers since that would immediately violate the inequality. Counting up all the (x,y) pairs gives us 5.
## 5.
The Correct Answer is (B) — The average daily increase in the number of Monarch butterflies is the slope of the fitted line. We can estimate that there were approximately 10 butterflies at day 0, giving us the point (0, 10). We can then estimate that there were approximately 50 butterflies on day 20, giving us the point (20, 50). Calculating the slope gives us $\frac{50-10}{20-0}&space;=&space;\frac{40}{20}&space;=&space;2$
## 6.
The Correct Answer is (B) — We are told that b is 150% greater than c, which means b = c + 1.5c. Plugging in c = 20, we get b = 20 + (1.5)(20) = 50. a is 8% of b so a = (0.08)b. Plugging in b = 50, we get a = (0.08)(50) = 4.
## 7.
The Correct Answer is (B) — First, we count all the people who showed symptoms: 216 + 584 = 800 people. Of these 800 people, 216 were vaccinated. So the percentage of patients who showed symptoms who were also vaccinated is 216/800 x 100 = 27%.
## 8.
The Correct Answer is (D) — When we divide each side of the equation by 3, we get x = (1/6)y. We want to find (1/3)y so we can multiply both sides of the equation by 2 to get 2x = (1/3)y. The answer is D.
## 9.
The Correct Answer is (D) — We calculate p(2) by p(2) = |2(2) - 5| = |-1| = 1. We calculate p(-2) by p(-2) = |-2(2) - 5| = |-9| = 9. So p(2) + p(-2) = 1 + 9 = 10.
## 10.
The Correct Answer is (B) — We are told that the circumference of circle E is 6π. Because circumference = 2πr, we can calculate that the circle E has a radius of 3. The two sides of rectangle CDEH are the radii, so the area of CDEH is 3 x 3 = 9. We are shown in the diagram that line ED and line DF are equal lengths, so the area of CDFG must also be 9. So the total area of rectangle EFGH is 9 + 9 = 18.
## 11.
The Correct Answer is (D) — We can find the length of a line given its points by drawing a right triangle where the legs are the difference in x and y-coordinates of the two points. The hypotenuse of this right triangle is the length of the line. The difference in y for the two points is 7 -3 = 4. The difference in x is 1 - (-2) = 3. At this point, we can recognize that this is a 3-4-5 triangle and choose answer choice D.
## 12.
The Correct Answer is (B) — First, we count the total number of students in Class A: 1+ 6 + 4. Then we count the number of students that speak more than 2 languages, which is the number of students that speak 3 languages. The chart shows that 4 students speak 3 languages. We calculate the percentage by 4/21 x 100 = 19%.
## 13.
The Correct Answer is (A) — In class A, if all the students were lined up in increasing order of number of languages spoken, the middle student would speak 1 language. This makes the median of class A 1. The category that has the most number of students is 1 language spoken, making the mode of class A 1. Similarly, the median of class B is 2 and the mode is 1. This means that class A has a median smaller than class B, and both classes have the same mode.
## 14.
The Correct Answer is (C) — When we factor a 2 out of the expression, we get $2(x^{2}&space;+&space;x&space;+&space;6)$. We can further factor this term to get 2(x + 3)(x -2). From this, we can eliminate choices A and B. We can then notice that the product of 2 and (x + 3) equals 2x + 6, making this term also a factor. The correct answer is C.
## 15.
The Correct Answer is (A) — If c is the original price of the car, then c x 1.2 = 36,000 giving us c = 30,000. The final price of the car is 5% less than $36,000, giving us (0.95)(36,000) = 34,200. We can then subtract the original price from this final price to get the profit, giving us 34,200 - 30,000 =$4200.
## 16.
The Correct Answer is (B) — Distributing in the 4 on the left side and the negative sign of the right side of the equation, we get $\small&space;\frac{(4x+4-1)}{3}=\frac{(8-5+x)}{5}\rightarrow&space;\frac{(4x+3)}{3}=&space;\frac{3+x}{5}$. Cross-multiplying then gives us 20x + 15 = 9 + 3x. Subtracting each side by 3x and 15 gives us 17x = -6. Then dividing each side by 17 gives us x = -6/17.
## 17.
The Correct Answer is (D) — First, we add up all the percentages of red, yellow, and blue umbrellas to get 60%. We are told that the total number of umbrellas is 1800, so we multiply this term by 0.60 to get 1080 umbrellas.
## 18.
The Correct Answer is (C) — 12√2 can be factored into 3 × 4 × √2. Bringing the 4 under the radical gives us 3√(2×16) =3 √32. This eliminates answer choice A. 6√8 can be factored into 2 × 3 × √(8.) Bringing the 2 under the radical gives us 3√(4×8) = 3√32. This eliminates answer choice B. We can factor 2 × 2√(2×2×3). There is no way to bring a number of in or out of the radical to produce 3√32. The correct answer is C.
## 19.
The Correct Answer is (D) — Macey’s savings can be expressed as 100 + 10y, where y is the number of years. Sam’s savings can be expressed as $\small&space;100&space;\times&space;1.10^{y}$. After five years Macey will have 100 + 10(5) = $150. Sam will have $\small&space;100&space;\times&space;1.10^{5}&space;=&space;161.05$. Subtracting Macey’s savings from Sam’s savings gives us 161.05 - 150 = 11.05. So after 5 years, Macey will have saved$11.05 less than Sam.
## 20.
The Correct Answer is (C) — Substituting 2 and -2 for answer choice A gives us 3/2 < 3/(-2), which is not a true statement. When we try answer choice B, we get 15 < 15, which is also untrue. When we try answer choice C, we get -5 < 11, which is a true statement. The answer is C.
## 21.
The Correct Answer is (B) — We first find the volume of the pool by taking the product of the dimensions, which gives us 2 × 10 × 10 = 200$\small&space;m^3$. When we convert this value to gallons we get 52,800 gallons. We are only interested in half the pool, so we divide this value by 2 to get 26,400 gallons. We can divide this value by 55 gallons to find how many minutes it would take to fill half the pool, which gives us 26,400/55 = 480 minutes. Converting this to hours gives us 8 hours.
## 22.
The Correct Answer is (B) — When we add x to both sides of the equation, we get y = -2x. Dividing each side of the equation by -2 and y gives us -1/2 = x/y. The correct answer is B.
## 23.
The Correct Answer is (D) — We are told that the wolf population increases by 5% each year. This means that after 1 year, the wolf population is 20(1.05). After 2 years, the wolf population is 20(1.05)(1.05). If we continue this pattern, we can see that the wolf population at any given year can be expressed as $\small&space;P=20(1.05)^t$.
## 25.
The Correct Answer is (B) — We can divide the value found in the None column by the value in the Total column for each age group to the find the percentage of people who did not own a vehicle. Doing this we get:
18-29 Age Group
Percentage 498/26,955×100 = 13%
30-49 Age Group
Percentage 2309/35,520×100 = 6.5%
50-69 Age Group
Percentage 2004/29,597×100 = 6.8%
70+ Age Group
Percentage 5377/29,492×100 = 18%
From this list, we can see that the 30-49 age group has the smallest percentage.
## 26.
The Correct Answer is (C) — We can find the number of people 50 or more years with an electric or hybrid car by 1068 + 792 = 1860. We can divide this number by the total number of electric or hybrid car owners, giving us 1860/9396 × 100 = 20%.
## 27.
The Correct Answer is (C) — sin(π) = 0, and squaring this value also gives us 0, so A is not the correct answer. sin(π/2) = 1, and squaring this value also gives us 1, so B is not the correct answer. sin(-π/2) = -1, and squaring this value gives us 1 so sin(-π/2) ≠ $\small&space;sin^{2}$ (-π/2). C is the correct answer.
## 28.
The Correct Answer is (D) — Graphs for answer choices A and B imply that the area of this rectangle can be 800$\small&space;ft^{2}$ or greater. These values are not possible with only 40ft of fencing material. Similarly, the graph for answer choice C says that the area can be as large as 400$\small&space;ft^{2}$, but this value is also impossible with only 40ft of fencing material. The correct answer is D.
## 29.
The Correct Answer is (B) — When we multiply the two terms together we get 10 + 30i - 2i - 6$\small&space;i^{2}$. Because $\small&space;i^{2}$=-1, the last term is equal to 6, giving us 10 + 28i + 6 = 16 + 28i.
## 31.
The Correct Answer is (150) — We can divide 75 miles by 30 miles per hour to find the number of hours it will take to make this trip, which gives us 75/30 = 2.5 hours. When we convert this value to minutes, we get 150 minutes.
## 32.
The Correct Answer is (80) — If 2/5 of n is 48, then (2/5)n = 48. Solving for n gives us n = 120. We can multiply this n by 2/3 to find 2/3 of n, which gives us (2/3)(120) = 80.
## 33.
The Correct Answer is (32) — Plugging in -3 into f(x) gives us $\small&space;f(x)=(-3)^3+3(-3)^2-6(-3)+14=-27+27+18+14=32.$
## 34.
The Correct Answer is (33) — We can divide the number of miles traveled by the average speed to find the number of hours it took to make a particular trip. From this we can calculate that the 3 mile trip with Bus A took 3/20 = 0.15 hours. The 6 miles trip with Bus B took 6/15 = 0.4 hours. Consequently, the whole trip took 0.15 + 0.4 = 0.55 hours. Converting this value to minutes gives us 33 minutes.
## 35.
The Correct Answer is (18) — To get a common denominator for the two terms on the left side, we multiply the first term by $\small&space;\frac{a+2}{a+2}$ and the second term by $\small&space;\frac{a-2}{a-2}$
This gives us $\small&space;\frac{12(a&space;+&space;2)&space;+&space;5(a&space;-2)}{a^{2}-4}&space;=&space;1$
We can simplify this to $\small&space;\frac{(17a&space;+&space;14)}{a^{2}-4}&space;=&space;1$
Cross-multiplying gives us $\small&space;a^{2}-4&space;=&space;17a&space;+14.$ Moving all the terms to the left side gives us $\small&space;a^2-17a-18=0.$ We can then factor the left side into $\small&space;(a+1)(a-18)=0.$ Solving for the roots gives us a = -1 or 18. Since the question asks for the positive value, the correct answer is 18.
## 36.
The Correct Answer is (87) — The sides of the box are equal to 3 diameters of the cans, which tells us that the dimensions of the box are 9 × 9 × 5. From this we calculate the volume which is 9 × 9 × 5 = 405$\small&space;in^3$. Next we can find the volume of one can. Each can has a radius of 1.5, which means that the area of the top of a can is $\small&space;1.5^{2}$ × 3.14. We then multiply this value by 5 to get the volume of the can, which is (1.5)(1.5)(3.14)(5) = 35.325$\small&space;in^3$. We then multiply this value by 9 to find the total volume of all the cans together, which is (35.325)(9) = 317.925. Subtracting this value from the volume of the box gives us 405 - 317.925 = 87$\small&space;in^{3}$. | 2022-08-14T19:12:01 | {
"domain": "ivyglobal.com",
"url": "https://ivyglobal.com/study/explanations/53",
"openwebmath_score": 0.6530784368515015,
"openwebmath_perplexity": 372.08924593507743,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.987178783470188,
"lm_q2_score": 0.8438951084436077,
"lm_q1q2_score": 0.833075346529803
} |
http://math.stackexchange.com/questions/304781/finding-all-nodes-that-are-passed-by-all-paths-from-a-to-b-in-an-undirected-grap | # Finding all nodes that are passed by all paths from A to B in an undirected graph
I searched all over internet but couldn't find any particular info on this one, so i thought to give it a shot here.
Problem: We have an undirected unweighted graph $G$, and in this graph we have two nodes $A$ and $B$. Now lets take the set of all paths between $A$ and $B$ and call that set $\mathcal{P}$, further note these paths can pass each node either once or dont pass a node at all. Of course one path $P \in \mathcal{P}$ exists out of both nodes and edges, so let $N(P)$ be the set of nodes associated with the path $P$. Now I would like to know in a fast algorithmical way which nodes are left if we take the the intersection between all vertices of all paths between $A$ and $B$, so more mathematically i want to know the set $\bigcap_{P \in \mathcal{P}} N(P)$.
Example:
The green nodes are traveling through all paths between $A$ and $B$, so they are the nodes i look for. the intersection between all pathnodes therefor is the set of all green nodes and $A$ and $B$ might be included in that set as well, depending on the algorithm.
-
A simpler way of writing this would be: "Find all nodes that lie on every simple path from $A$ to $B$". – Paresh Feb 15 '13 at 12:31
Oh yea, thanks, i already thought it might be a bit long of an explanation – CarloV Feb 15 '13 at 12:39
The vertices that you are looking for are all cut-vertices (or articulation points) of the graph. A cut vertex is a vertex that disconnects the graph/increases the number of connected components. However, not all cut-vertices are a solution. Any cut-vertex, whose removal causes $A$ and $B$ to become disconnected, will lie on every path from $A$ to $B$.
Edit: Another observation is that the desired vertices will lie on the shortest path from $A$ to $B$ (if they lie on any path from $A$ to $B$, then they also lie on the shortest path). You can find the shortest path in linear time using a BFS. Now, you can shortlist those vertices which are cut-vertices and lie on the shortest path from $A$ to $B$.
If you cut out the white vertex attached to $A$, we can still make paths to $B$, so would it be enough to first need to trim the appendixed or dead vertices and then search for these cut-vertices in the remaining graph? Anyhow at least i know the names for these nodes now, that makes it much easier. – CarloV Feb 15 '13 at 12:52
@CarloVerschoor The white vertex you mention is also a cut-vertex of the graph. But its removal does not disconnect $A$ and $B$, so it is not part of the answer. However, you can find all cut-vertices of the graph using one DFS in $O(|V| + |E|)$. From those, you can weed out those vertices whose removal still keeps $A$ and $B$ connected. This would result in a worst case $O(|V|^2 + |V||E|)$ algorithm. If I can think of something better, I'll update my answer. – Paresh Feb 15 '13 at 13:03 | 2015-12-01T16:35:12 | {
"domain": "stackexchange.com",
"url": "http://math.stackexchange.com/questions/304781/finding-all-nodes-that-are-passed-by-all-paths-from-a-to-b-in-an-undirected-grap",
"openwebmath_score": 0.7821660041809082,
"openwebmath_perplexity": 188.8947420039128,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES\n\n",
"lm_q1_score": 0.9871787846017969,
"lm_q2_score": 0.843895106480586,
"lm_q1q2_score": 0.8330753455469088
} |
https://www.mathworks.com/help/matlab/ref/polyfit.html | # polyfit
Polynomial curve fitting
## Syntax
``p = polyfit(x,y,n)``
``````[p,S] = polyfit(x,y,n)``````
``````[p,S,mu] = polyfit(x,y,n)``````
## Description
example
````p = polyfit(x,y,n)` returns the coefficients for a polynomial `p(x)` of degree `n` that is a best fit (in a least-squares sense) for the data in `y`. The coefficients in `p` are in descending powers, and the length of `p` is `n+1`$p\left(x\right)={p}_{1}{x}^{n}+{p}_{2}{x}^{n-1}+...+{p}_{n}x+{p}_{n+1}.$```
``````[p,S] = polyfit(x,y,n)``` also returns a structure `S` that can be used as an input to `polyval` to obtain error estimates.```
example
``````[p,S,mu] = polyfit(x,y,n)``` also returns `mu`, which is a two-element vector with centering and scaling values. `mu(1)` is `mean(x)`, and `mu(2)` is `std(x)`. Using these values, `polyfit` centers `x` at zero and scales it to have unit standard deviation,$\stackrel{^}{x}=\frac{x-\overline{x}}{{\sigma }_{x}}\text{\hspace{0.17em}}.$This centering and scaling transformation improves the numerical properties of both the polynomial and the fitting algorithm.```
## Examples
collapse all
Generate 10 points equally spaced along a sine curve in the interval `[0,4*pi]`.
```x = linspace(0,4*pi,10); y = sin(x);```
Use `polyfit` to fit a 7th-degree polynomial to the points.
`p = polyfit(x,y,7);`
Evaluate the polynomial on a finer grid and plot the results.
```x1 = linspace(0,4*pi); y1 = polyval(p,x1); figure plot(x,y,'o') hold on plot(x1,y1) hold off```
Create a vector of 5 equally spaced points in the interval `[0,1]`, and evaluate $y\left(x\right)=\left(1+x{\right)}^{-1}$ at those points.
```x = linspace(0,1,5); y = 1./(1+x);```
Fit a polynomial of degree 4 to the 5 points. In general, for `n` points, you can fit a polynomial of degree `n-1` to exactly pass through the points.
`p = polyfit(x,y,4);`
Evaluate the original function and the polynomial fit on a finer grid of points between 0 and 2.
```x1 = linspace(0,2); y1 = 1./(1+x1); f1 = polyval(p,x1);```
Plot the function values and the polynomial fit in the wider interval `[0,2]`, with the points used to obtain the polynomial fit highlighted as circles. The polynomial fit is good in the original `[0,1]` interval, but quickly diverges from the fitted function outside of that interval.
```figure plot(x,y,'o') hold on plot(x1,y1) plot(x1,f1,'r--') legend('y','y1','f1')```
First generate a vector of `x` points, equally spaced in the interval `[0,2.5]`, and then evaluate `erf(x)` at those points.
```x = (0:0.1:2.5)'; y = erf(x);```
Determine the coefficients of the approximating polynomial of degree 6.
`p = polyfit(x,y,6)`
```p = 1×7 0.0084 -0.0983 0.4217 -0.7435 0.1471 1.1064 0.0004 ```
To see how good the fit is, evaluate the polynomial at the data points and generate a table showing the data, fit, and error.
```f = polyval(p,x); T = table(x,y,f,y-f,'VariableNames',{'X','Y','Fit','FitError'})```
```T=26×4 table X Y Fit FitError ___ _______ __________ ___________ 0 0 0.00044117 -0.00044117 0.1 0.11246 0.11185 0.00060836 0.2 0.2227 0.22231 0.00039189 0.3 0.32863 0.32872 -9.7429e-05 0.4 0.42839 0.4288 -0.00040661 0.5 0.5205 0.52093 -0.00042568 0.6 0.60386 0.60408 -0.00022824 0.7 0.6778 0.67775 4.6383e-05 0.8 0.7421 0.74183 0.00026992 0.9 0.79691 0.79654 0.00036515 1 0.8427 0.84238 0.0003164 1.1 0.88021 0.88005 0.00015948 1.2 0.91031 0.91035 -3.9919e-05 1.3 0.93401 0.93422 -0.000211 1.4 0.95229 0.95258 -0.00029933 1.5 0.96611 0.96639 -0.00028097 ⋮ ```
In this interval, the interpolated values and the actual values agree fairly closely. Create a plot to show how outside this interval, the extrapolated values quickly diverge from the actual data.
```x1 = (0:0.1:5)'; y1 = erf(x1); f1 = polyval(p,x1); figure plot(x,y,'o') hold on plot(x1,y1,'-') plot(x1,f1,'r--') axis([0 5 0 2]) hold off```
Create a table of population data for the years 1750 - 2000 and plot the data points.
```year = (1750:25:2000)'; pop = 1e6*[791 856 978 1050 1262 1544 1650 2532 6122 8170 11560]'; T = table(year, pop)```
```T=11×2 table year pop ____ _________ 1750 7.91e+08 1775 8.56e+08 1800 9.78e+08 1825 1.05e+09 1850 1.262e+09 1875 1.544e+09 1900 1.65e+09 1925 2.532e+09 1950 6.122e+09 1975 8.17e+09 2000 1.156e+10 ```
`plot(year,pop,'o')`
Use `polyfit` with three outputs to fit a 5th-degree polynomial using centering and scaling, which improves the numerical properties of the problem. `polyfit` centers the data in `year` at 0 and scales it to have a standard deviation of 1, which avoids an ill-conditioned Vandermonde matrix in the fit calculation.
`[p,~,mu] = polyfit(T.year, T.pop, 5);`
Use `polyval` with four inputs to evaluate `p` with the scaled years, `(year-mu(1))/mu(2)`. Plot the results against the original years.
```f = polyval(p,year,[],mu); hold on plot(year,f) hold off```
Fit a simple linear regression model to a set of discrete 2-D data points.
Create a few vectors of sample data points (x,y). Fit a first degree polynomial to the data.
```x = 1:50; y = -0.3*x + 2*randn(1,50); p = polyfit(x,y,1); ```
Evaluate the fitted polynomial `p` at the points in `x`. Plot the resulting linear regression model with the data.
```f = polyval(p,x); plot(x,y,'o',x,f,'-') legend('data','linear fit') ```
Fit a linear model to a set of data points and plot the results, including an estimate of a 95% prediction interval.
Create a few vectors of sample data points (x,y). Use `polyfit` to fit a first degree polynomial to the data. Specify two outputs to return the coefficients for the linear fit as well as the error estimation structure.
```x = 1:100; y = -0.3*x + 2*randn(1,100); [p,S] = polyfit(x,y,1); ```
Evaluate the first-degree polynomial fit in `p` at the points in `x`. Specify the error estimation structure as the third input so that `polyval` calculates an estimate of the standard error. The standard error estimate is returned in `delta`.
`[y_fit,delta] = polyval(p,x,S);`
Plot the original data, linear fit, and 95% prediction interval $\mathit{y}±2\Delta$.
```plot(x,y,'bo') hold on plot(x,y_fit,'r-') plot(x,y_fit+2*delta,'m--',x,y_fit-2*delta,'m--') title('Linear Fit of Data with 95% Prediction Interval') legend('Data','Linear Fit','95% Prediction Interval')```
## Input Arguments
collapse all
Query points, specified as a vector. The points in `x` correspond to the fitted function values contained in `y`. If `x` is not a vector, then `polyfit` converts it into a column vector `x(:)`.
Warning messages result when `x` has repeated (or nearly repeated) points or if `x` might need centering and scaling.
Data Types: `single` | `double`
Complex Number Support: Yes
Fitted values at query points, specified as a vector. The values in `y` correspond to the query points contained in `x`. If `y` is not a vector, then `polyfit` converts it into a column vector `y(:)`.
Data Types: `single` | `double`
Complex Number Support: Yes
Degree of polynomial fit, specified as a positive integer scalar. `n` specifies the polynomial power of the left-most coefficient in `p`.
## Output Arguments
collapse all
Least-squares fit polynomial coefficients, returned as a vector. `p` has length `n+1` and contains the polynomial coefficients in descending powers, with the highest power being `n`. If either `x` or `y` contain `NaN` values and `n < length(x)`, then all elements in `p` are `NaN`.
Use `polyval` to evaluate `p` at query points.
Error estimation structure. This optional output structure is primarily used as an input to the `polyval` function to obtain error estimates. `S` contains the following fields:
FieldDescription
`R`Triangular `R` factor (possibly permuted) from a QR decomposition of the Vandermonde matrix of `x`
`df`Degrees of freedom
`normr`Norm of the residuals
If the data in `y` is random, then an estimate of the covariance matrix of `p` is `(Rinv*Rinv')*normr^2/df`, where `Rinv` is the inverse of `R`.
If the errors in the data in `y` are independent and normal with constant variance, then `[y,delta] = polyval(...)` produces error bounds that contain at least 50% of the predictions. That is, `y` ± `delta` contains at least 50% of the predictions of future observations at `x`.
Centering and scaling values, returned as a two-element vector. `mu(1)` is `mean(x)`, and `mu(2)` is `std(x)`. These values center the query points in `x` at zero with unit standard deviation.
Use `mu` as the fourth input to `polyval` to evaluate `p` at the scaled points, ```(x - mu(1))/mu(2)```.
## Limitations
• In problems with many points, increasing the degree of the polynomial fit using `polyfit` does not always result in a better fit. High-order polynomials can be oscillatory between the data points, leading to a poorer fit to the data. In those cases, you might use a low-order polynomial fit (which tends to be smoother between points) or a different technique, depending on the problem.
• Polynomials are unbounded, oscillatory functions by nature. Therefore, they are not well-suited to extrapolating bounded data or monotonic (increasing or decreasing) data.
## Algorithms
`polyfit` uses `x` to form Vandermonde matrix `V` with `n+1` columns and `m = length(x)` rows, resulting in the linear system
`$\left(\begin{array}{cccc}{x}_{1}^{n}& {x}_{1}^{n-1}& \cdots & 1\\ {x}_{2}^{n}& {x}_{2}^{n-1}& \cdots & 1\\ ⋮& ⋮& \ddots & ⋮\\ {x}_{m}^{n}& {x}_{m}^{n-1}& \cdots & 1\end{array}\right)\left(\begin{array}{c}{p}_{1}\\ {p}_{2}\\ ⋮\\ {p}_{n+1}\end{array}\right)=\left(\begin{array}{c}{y}_{1}\\ {y}_{2}\\ ⋮\\ {y}_{m}\end{array}\right)\text{\hspace{0.17em}}\text{\hspace{0.17em}},$`
which `polyfit` solves with `p = V\y`. Since the columns in the Vandermonde matrix are powers of the vector `x`, the condition number of `V` is often large for high-order fits, resulting in a singular coefficient matrix. In those cases centering and scaling can improve the numerical properties of the system to produce a more reliable fit. | 2020-10-01T04:07:52 | {
"domain": "mathworks.com",
"url": "https://www.mathworks.com/help/matlab/ref/polyfit.html",
"openwebmath_score": 0.7791977524757385,
"openwebmath_perplexity": 776.9892564091139,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9871787830929849,
"lm_q2_score": 0.8438951064805861,
"lm_q1q2_score": 0.8330753442736298
} |
http://math.stackexchange.com/questions/509928/cutting-a-cake-without-destroying-the-toppings | # cutting a cake without destroying the toppings
There is a square cake. It contains N toppings - N disjoint axis-aligned rectangles. The toppings may have different widths and heights, and they do not necessarily cover the entire cake.
I want to divide the cake into 2 rectangular pieces, by either a horizontal or a vertical cut, such that the number of toppings I destroy (i.e. cross in the interior) is minimized.
What is the number of toppings I will have to destroy, in the worst case, as a function of N?
CURRENT BOUNDS:
Upper bound $N/2$:
Take any horizontal cut. If it crosses no more than N/2 toppings, then we are done. Otherwise, make a vertical cut between two of the crossed rectangles. This vertical cut does not cross any rectangle crossed by the horizontal cut, therefore it crosses at most N/2 toppings.
Lower bound $N/4$:
In the following cake, with 4 toppings, every cut must cross at least 1 topping:
aaaaaaaa bb
aaaaaaaa bb
cc ..... bb
cc ..... bb
cc ..... bb
cc dddddddd
cc dddddddd
As MvG suggested, it is possible to cut each rectangle into $N/4$ parallel strips, forcing a cut to destroy at least $⌊N/4⌋$ toppings.
NOTE: I just found out that this problem is related to the topic of Geometric separators. The lower bound example and the upper bound proofs are given in Section 4 of Smith and Wormald (1998). There is still a gap between the lower and upper bound.
-
are all toppings identical in dimension? – mau Sep 30 '13 at 12:54
@mau given the cake is square, and not cubical, I think we can assume we're working in dimension 2 here. – Daniel Rust Sep 30 '13 at 13:04
@Daniel: Otherwise we can cut it parallel to the top and bottom of the cake, halfway down, and even get two equal pieces without destroying any toppings! :-) – Brian M. Scott Oct 1 '13 at 1:21
@BrianM.Scott I call shotgun on the top half. – Daniel Rust Oct 1 '13 at 1:28
You could take your last example and cut each rectangle into $\frac N4$ parallel strips, forcing a cut to destroy at least $\left\lfloor\frac N4\right\rfloor$ toppings. So far I couldn't come up with a better solution, but that doesn't poove that there is none. – MvG Oct 1 '13 at 22:43
It appears that $\lfloor \frac{N}{4} \rfloor$ is the maximal case for a given $N>1$ on your two-dimensional cake, as demonstrated by your $N=4$ layout. For $N=8$ put two parallel toppings where each one of yours is. For $N=4k$ put $k$ parallel toppings.
For $N=1$ we just use a single topping covering the whole surface.
Proof that for $N=2$ or $N=3$ there is no layout requiring a cut topping is a first step to proving $\lfloor \frac{N}{4} \rfloor$ is the correct formula.
How can you prove that $k$ is actually the worst case for all $N=4k$? Maybe there is an arrangement of $4k$ toppings in which you will have to destroy more than $k$? – Erel Segal Halevi Jan 10 at 11:14 | 2014-11-01T03:57:24 | {
"domain": "stackexchange.com",
"url": "http://math.stackexchange.com/questions/509928/cutting-a-cake-without-destroying-the-toppings",
"openwebmath_score": 0.7814906239509583,
"openwebmath_perplexity": 727.1135068857127,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9871787883738261,
"lm_q2_score": 0.8438950986284991,
"lm_q1q2_score": 0.8330753409786922
} |
http://sorami-chi.hateblo.jp/entry/2018/12/24/234716 | # Deciding the probability of a coin to show its head from the expected number of heads for N tosses
A couple of weeks ago I hit a small math problem related to probabilistic and I thought it was trival (it took me a few hours to solve, though :p ). However I could not find a good stack exchange post or whatever that answers my problem, so I put it here for the sake of my future reference and possibly some people who happen to hit the same problem.
Note that the "short answer" itself really is short and trival, but the point of this post is the last section that describes the relationship between the "short" and the "long" answers.
# Problem Definition
Let the probability that a coin shows its head in a coin toss be x (0 < x < 1) and the expected number of heads after N tosses be M, compute x using N and M.
The expected number of heads for one toss is obvisouly x. Because the expected value of the sum of two independent probablistic variables is equal to the sum of the expected value of each probablistic variable (E[X + Y] = E[X] + E[Y]), the expected number of heads for N tosses is Nx.
Therefore, and .
By definition, the expected number of heads M can be expressed using x and N as follows:
where is the probability that exactly n coins show their heads and defined as:
Therefore,
We can compute x from M and N by solving this equation for x..... but how??? (this is an N-th order equation of x!)
# What's Behind
Both the short and the long answers say correct stuff, which means one thing:
And below is a proof:
where and . Note that . The term after is equal to 1 as shown below:
# Let's Check Numerically
For those who may suspect the proof above, here are Ms calculated by the short and the long answers for N = 100 (there are small differences by numerical errors, but they are pretty much the same).
x M_short M_long
0.0 0.0 0.0
0.04 4.0 3.9999999999999862
0.08 8.0 8.000000000000032
0.12 12.0 11.999999999999998
0.16 16.0 15.99999999999996
0.2 20.0 20.00000000000012
0.24 24.0 23.999999999999993
0.28 28.000000000000004 28.0
0.32 32.0 31.99999999999981
0.36 36.0 35.99999999999999
0.4 40.0 39.99999999999999
0.44 44.0 44.00000000000023
0.48 48.0 47.99999999999998
0.52 52.0 52.0
0.56 56.00000000000001 56.00000000000001
0.6 60.0 59.99999999999997
0.64 64.0 64.0
0.68 68.0 68.0
0.72 72.0 71.99999999999999
0.76 76.0 75.99999999999997
0.8 80.0 80.0
0.84 84.0 84.00000000000001
0.88 88.0 88.00000000000001
0.92 92.0 92.00000000000001
0.96 96.0 96.0
Here is the code used to generate the table above. Note that M_long(x) does not work for a large N due to overflow.
import scipy.special as sp
N = 100
def M_short(x):
return N * x
def M_long(x):
ans = 0
for n in range(1, N+1):
ans += n * sp.comb(N, n) * pow(x, n) * pow(1 - x, N-n)
return ans
print("|x|M_short|M_long|")
print("|-|-------|------|")
for i in range(0, 25):
x = i / 25
print("|", x, "|", M_short(x), "|" , M_long(x), "|") | 2019-03-23T06:44:46 | {
"domain": "hateblo.jp",
"url": "http://sorami-chi.hateblo.jp/entry/2018/12/24/234716",
"openwebmath_score": 0.8507544994354248,
"openwebmath_perplexity": 1384.9061087390642,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9871787842245938,
"lm_q2_score": 0.8438951005915208,
"lm_q1q2_score": 0.8330753394150289
} |
https://math.stackexchange.com/questions/3774531/how-to-find-the-position-vector-for-the-point-of-intersection-of-a-line-and-the | # How to find the position vector for the point of intersection of a line and the perpendicular line through a point C
How could I find the exact coordinates of the point N for example which is the point of intersection of the line $$L=\begin{pmatrix} 0 \\ -1 \\ 2 \end{pmatrix} +t\begin{pmatrix} -3 \\ -2 \\ -3 \end{pmatrix}$$ and a perpendicular line that passes through a point C(1,2,3) not on the line.
What I have done so far is find the value of point N in terms of $$t$$ to find the vector $$\overrightarrow{CN}$$: $$\longrightarrow N\begin{pmatrix} -3t \\ -2t-1 \\ -3t+2 \end{pmatrix}$$ $$\longrightarrow \overrightarrow{CN}=\begin{pmatrix} -3t \\ -2t-1 \\ -3t+2 \end{pmatrix}-\begin{pmatrix} 1 \\ 2 \\ 3 \end{pmatrix}$$ $$=\begin{pmatrix} -3t-1 \\ -2t-3 \\ -3t-1 \end{pmatrix}$$
After this I made the scalar product eual zero: $$\overrightarrow{CN} \cdot \begin{pmatrix} -3 \\ -2 \\ -3 \end{pmatrix}=0$$
With this I find that, $$t=\frac{-6}{11}$$ But my textbook says that $$t=\frac{5}{11}$$
Can anyone confirm which answer for $$t$$ is right and if it is $$t=\frac{5}{11}$$ then what did I do wrong?
• For me, your answer it correct. Jul 30 '20 at 12:32
• WA agrees. Maybe there's the line $L$ points swapped in the solution? Jul 30 '20 at 12:42
• Thank you for the confirmation Jul 30 '20 at 12:42
Your derivation seems correct indeed, by a direct check, for $$t=-\frac 6{11}$$ we obtain
$$N=\begin{pmatrix} 0 \\ -1 \\ 2 \end{pmatrix} -\frac 6{11}\begin{pmatrix} -3 \\ -2 \\ -3 \end{pmatrix}=\begin{pmatrix} \frac {18}{11} \\ \frac {1}{11} \\ \frac {40}{11} \end{pmatrix}$$
$$\overrightarrow{CN}=\begin{pmatrix} \frac {7}{11} \\ -\frac {21}{11} \\ \frac {7}{11} \end{pmatrix}$$
$$\begin{pmatrix} \frac {7}{11} \\ -\frac {21}{11} \\ \frac {7}{11} \end{pmatrix} \cdot \begin{pmatrix} -3 \\ -2 \\ -3 \end{pmatrix}= -\frac {21}{11}+\frac {42}{11}-\frac {21}{11}=0$$ | 2022-01-23T11:01:24 | {
"domain": "stackexchange.com",
"url": "https://math.stackexchange.com/questions/3774531/how-to-find-the-position-vector-for-the-point-of-intersection-of-a-line-and-the",
"openwebmath_score": 0.5911698937416077,
"openwebmath_perplexity": 182.15522149340615,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9871787834701878,
"lm_q2_score": 0.8438950986284991,
"lm_q1q2_score": 0.8330753368405359
} |
http://omni-s.net/8bfduv/7nr4u.php?in=exponential-slope-calculator | The equation of the first degree. com and figure out factoring trinomials, mixed numbers and a great deal of other algebra subject areas. Learning the proper way to calculate slope is a simple process when you know exactly what the "slope" means. In the gallery of basic function types we saw five different exponential functions, some growing, some decaying. SLOPE(known_y's, known_x's) The SLOPE function syntax has the following arguments:. The slope of the linear section of the Ln trace is the reciprocal of the exponential time constant We can use the relative time cursors to read the slope of the Ln waveform. When the rate of change is proportional to the value of the function itself, it is called exponential growth and decay. With practice, you'll be able to find exponential functions with ease! Example 1: Determine the exponential function in the form y = a b x y=ab^x y = a b x of the given graph. We can generate a probability plot of normalized exponential data, so that a perfect exponential fit is a diagonal line with slope 1. There are many quantities that grow exponentially. For each type of trend line, I will present the coefficients, the equation to calculate each coefficient and then the calculated value based on the following data set:. This year, I decided to have my students create what I deemed "Slope Name Art. With this easy online domain calculator you can improve your ability to define ranges and domains, work out the answer to a question you are struggling with, and make sure that your answers are totally correct. What is simple linear regression. Type the following: y=2x+1; Try it now: y=2x+1 Clickable Demo Try entering y=2x+1 into the text box. t is the time in discrete intervals and selected time units. If you have a time series with a clear pattern, you could use moving averages — but if you don’t have a clear pattern you can use exponential smoothing to forecast. First, we borrow some Calculus results: The derivative of a function gives the slopes of the tangent lines to the graph of. x(t) is the value at time t. Exponential functions plot on semilog paper as straight lines. Thus, x 5 /x 2 =x 3. Given a fixed cost, variable cost, and revenue function or value, this calculates the break-even point Features: Calculator | Practice Problem Generator Examples (2): C(x) = 125x + 1500 and R(x) = 1500x - 1000, canoes has a fixed cost of $20,000. Stochastic models: 1. Thomas Malthus, an 18 th century English scholar, observed in an essay written in 1798 that the growth of the human population is fundamentally different from the growth of the food supply to feed that population. To determine the slope of the line: a) extend the line so it crosses one. Returns the slope of the linear regression line through data points in known_y's and known_x's. a is the starting value, b is the growth rate. That means, we can use the following formula: m = (y 2 – y 1)/(x 2 – x 1). The use of this domain and range calculator will also ensure that you've found the correct answer. 065, which is as we hypothesized, βH*A > 1. On the graph, the center point is indicated with the mouse until the point is chosen with a mouse click. More than 70 powerful online math calculators designed to help you solve all of your math problems. That is, b = D D == y x yy xx Rx Rx xx 21 21 21 21 - - ln ( ) - ln ( ) - (13. That means the change in x, which is the denominator of the slope formula, would be 5 - 5 = 0. An equation for an exponential curve is of the form y = a(b) x. This is an exponential growth curve, where the y-value increases and the slope of the curve increases as x increases. The graph below shows two more examples. Exponential moving average slope is also easier to determine: the slope is always down when price closes below the moving average and always up when price is above. How to find the slope of a straight line and its derivative ? What is the relation between the slope of a curve or a parabola and its derivative ? How to find the derivative of the composite of two functions f(g(x)), an exponential or trigonometric function, a logarithmic function,… ?. Hello, I am wondering if it is possible to calculate the slope in excel with a fixed y intercept? I know that if I construct a scatter plot I can insert a trend line which I can format to have a fixed y intercept and I can then display the equation on the graph. With a bit more specific information about exponential form calculator, I will be able to help you if I knew particulars. Cosine Equations Solver. Click on the graph to select the center point for the zooming region. x 0 is the initial value at time t=0. Exponential decay models of this form will increase very rapidly at first, and then level off to become asymptotic to the upper limit. When solving the slope, intercept, and correlation for a power or exponential calibration that has been transformed into a linear least square regression, the analyst can follow the equations as described for a linear least square regression. Using the Slope Calculator. How to enter numbers: Enter any integer, decimal or fraction. Enter the input number x and press the = button:. When the rate of change is proportional to the value of the function itself, it is called exponential growth and decay. Did we mention that they're 100% free?. The slope/average rate of change between any two points will be negative. First, we borrow some Calculus results: The derivative of a function gives the slopes of the tangent lines to the graph of. Plot the following exponential function for x=1 to 10 and calculate the slope of the resulting line on semi-log paper. For values of 0, 1, and 2, the values of the function are 1, e or about 2. Why you should learn it GOAL 2 GOAL 1 What you should learn 8. In some cases, it will be easier to work with the equation for exponential growth if we take the natural logarithm of both sides of the equation… ln[N t] = ln[N 0] + ln[lambda] x t. On the other hand, take an example where the slope is zero. Now we take the natural logarithm of both sides. The slope of the linear section of the Ln trace is the reciprocal of the exponential time constant We can use the relative time cursors to read the slope of the Ln waveform. The Slope value corresponds to b in the. Find more Widget Gallery widgets in Wolfram|Alpha. Properties depend on value of "a". We have seen graphs of exponential functions before: In the section on real exponents we saw a saw a graph of y = 10 x. In order to use the Slope Intercept Form Calculator on the top of the page, you just need to add the coordinates of the two points that you already know. Furthermore, the slope of this line is b, the value of the constant in the original exponential equation. " Perpendicular lines. When multiplying terms with the same base, the exponents should be added. m is positive for growth, negative for decay. t is the time in discrete intervals and selected time units. Teacher Resources Based on extensive research, SmartGraphs: Algebra is designed for students studying Algebra 1 or Algebra 2. Correlation and regression calculator Enter two data sets and this calculator will find the equation of the regression line and corelation coefficient. Then use the slope formula to calculate the slope. Over time this results in the exponential growth of your money. On the other hand, take an example where the slope is zero. Additionally, D uses "lesser known" rules to calculate the derivative of a wide array of special functions. This means the slope is undefined. Note: The above equation is only valid for the exponential equations. These compilations provide unique perspectives and applications you won't find anywhere else. The longer your investment stays in the account, the greater the ratio of interest to the original amount. Logarithmic Equations Solver. 065, which is as we hypothesized, βH*A > 1. For values of 0, 1, and 2, the values of the function are 1, e or about 2. This is an exponential growth curve, where the y-value increases and the slope of the curve increases as x increases. Let m be the slope and b be the y-intercept of a line. We can also use the calculator output to construct the linear regression equation for our data. The name of this book is Al-Jabr wa'l muqabalah. The inverse of an exponential function is a logarithm function. Exponential decay models of this form will increase very rapidly at first, and then level off to become asymptotic to the upper limit. This calculator uses the following formula to derive the equation for the line of best fit. When = 1, the Weibull distribution models the exponential distribution. How to do exponential calculation to a range of cells in Excel? In Excel, addition, subtraction, multiplication and division is the basic calculation, maybe you can quickly and easily apply them. This lesson shows you how to compute for the integral of a function using the separable equation technique. We can calculate the exponential PDF and CDF at 100 hours for the case where $$\lambda$$ = 0. Average Change Rate Converter of US Current to Real Dollars with CPI Converter of US Current to Real Dollars with GDP deflator Exponential Growth/Decay Doubling/Half-Life Time Present Value Calculator S-Curve Calculator - 1 parameter estimate S-Curve Calculator - 3 parameter estimates. Online solver calculator for simple trigonometric equations with sine function of the form sin x = a. But the effect is still the same. Loading Loading. If you need support with math and in particular with letter algebra calculator or linear algebra come visit us at Sofsource. For example, it can't tell you that the derivative of x 2 is 2x. Exponential, harmonic or hyperbolic decline? The days of plotting rates on semi-log graph paper are long gone. Using the examples above, you would use a different fit for a decaying exponential and an increasing (asymptotic) exponential. Three hundred students are willing to buy them at that price. That is, b = D D == y x yy xx Rx Rx xx 21 21 21 21 - - ln ( ) - ln ( ) - (13. Slope alone cannot be used to participate in an ongoing trend, but it can be used with other indicators to identify potential entry points. Introduction exponential functions calculator: Exponential function has the form of f(x) = b x for a permanent base b which can be any positive real number. If a negative is placed in front of an exponential function, then it will be reflected over the x-axis. Exponential regression is probably one of the simplest nonlinear regression models. From my answer to How can someone explain exponential functions to a high school kid?: An exponential function or curve is a function that grows exponentially, or grows at an increasingly larger rate as you pick larger values of x, and usually tak. 7 Modeling with Exponential and Logarithmic Functions 345 Using Technology You can use technology to fi nd best-fi t models for exponential and logarithmic data. When bacteria are grown in a closed system (also called a batch culture), like a test tube, the population of cells almost always exhibits these growth dynamics: cells initially adjust to the new medium (lag phase) until they can start dividing regularly by the process of binary fission (exponential phase). What is the exp function? When is it needed? Also, how do I calculate Log and Ln functions with basic arithmetic and logic?. You can test this with your data. In the event that you actually have service with algebra and in particular with rational exponents calculator or basic concepts of mathematics come pay a visit to us at Polymathlove. log a n is called logarithmic function. Equation solver can find both numerical and parametric solutions of equations. x (hori-zontal). Sources for the activities include Pinellas County Secondary Mathematics teachers, the Jump Start Algebra Summer Program, and the Opening the Gate project, a statewide funded initiative that. This paper provides an overview of the design of thread reinforced soil structures with a particular emphasis on the influence of the strength criterion. In effect, the nominal decline rate is related to the instantaneous slope of the line, whereas the effective decline rate derives from the chord segment approximating that slope. We are working on the traffic and server issues. Solved Solve The Exponential Equation Use A Calculator T. Exponential Growth. And I also assume that you want to see more digits in those coefficients. Besides, you may check our logarithm calculator which is the inverse function of the exponent. The slope of a straight line. Each function accepts two arguments, both of which are are range of cells: the Y-block and the X-block cells. polynomial, rational, radical, exponential, logarithmic, etc. You can then annualize it as needed. Right from simplifying integer exponents calculator to radicals, we have all of it discussed. That means, we can use the following formula: m = (y 2 – y 1)/(x 2 – x 1). An example where an exponential regression is often utilized is when relating the concentration of a substance (the response) to elapsed time (the predictor). The constituent functions are shown in the plot below. Graphing exponential functions is similar to the graphing you have done before. m = slope (the amount of rise over run of the line) b= y-axis intercept ( where the line crosses over the y-axis) To calculate the slope intercept form equation from two coordinates (x 1,y 1) and (x 2,y 2): Step 1: Calculate the slope (y 2 - y 1) / (x 2 - x 1) Step 2: Calculate where the line intersects with the y-axis by. You can think of h as a dial that we can turn to get various. , it grows to just under$5000 after 20 years. However, you know that the generic first order step response given by: or. Free Distributive Property Calculator Online. Online Algebra Calculators. At the beginning of the exponential phase, all reagents are still in excess. Use the equation of a linear model to solve problems in the context of bivariate measurement data, interpreting the slope and intercept. Online exponential growth/decay calculator. It will calculate any one of the values from the other three in the exponential decay model equation. The reason for this is that the graph of Y = LN(X) passes through the point (1, 0) and has a slope of 1 there, so it is tangent to the straight line whose equation is Y = X-1 (the dashed line in the plot below): This property of the natural log function implies that. Exponential Regression An exponential regression is the process of finding the equation of the exponential function that fits best for a set of data. Hi all, I have a dataset which consists of 2 columns. When solving the slope, intercept, and correlation for a power or exponential calibration that has been transformed into a linear least square regression, the analyst can follow the equations as described for a linear least square regression. More about this Polynomial Regression Calculator so you can have a deeper perspective of the results that will be provided by this calculator. Right from ordered pair equation calculator to calculus, we have every part discussed. It signifies the rate of failure. For each type of trend line, I will present the coefficients, the equation to calculate each coefficient and then the calculated value based on the following data set:. An online calculator and solver for simple exponential equations. If the base of the exponential is e then take natural logarithms of both sides of the equation. By default commas are considered column separators; in the case you are using them as decimal separators check the option below. Gowher, The exponential regression model presupposes that this model is valid for your situation (based on theory or past experience). An example where an exponential regression is often utilized is when relating the concentration of a substance (the response) to elapsed time (the predictor). You can think of h as a dial that we can turn to get various. For a linear function, the rate of change of y relative to x is always constant, i. When < 1, the Weibull distribution models early failures of parts. The calculation begins like this: Now we will recall that so that we can write. Like the other exponential models, if you know upper limit, then the rest of the model is fairly easy to complete. Online exponential growth/decay calculator. The calculator will not fit the increasing model involving exponential decay directly. Derivative of the Exponential Function. Each x/y variable is represented on the graph as a dot or a. INTEGRATED ALGEBRA. Any letters of the alphabet that would normally be written using curves would need to be modified. The EMA responds. Calculate the approximate grading scale for a normal grade curve with this Grade Curve Calculator. com, a free online graphing calculator. Each x/y variable is represented on the graph as a dot or a. You can find the slope of a curve with the TI-84 Plus calculator, even though it is not equipped to find the derivative of a function. Thus, 10 4. Use a calculator to check the answer we found to the equation in example 3. This might simplify your work. Then, use the Equation of a Line Maker to check each of your answers. An Intuitive Guide To Exponential Functions & e Home › Math , Popular › An Intuitive Guide To Exponential Functions & e e has always bothered me — not the letter, but the mathematical constant. This is an exponential growth curve, where the y-value increases and the slope of the curve increases as x increases. The slope and one point specify a straight line. In the function f(x) = a b^x, what is the significance of the value of a in the graph? When the function is increasing from left to right, what do you notice. The square root of a number x is the same as x raised to the 0. Enter your data as a string of number pairs, separated by commas. The calculator will find the tangent line to the explicit, polar, parametric and implicit curve at the given point, with steps shown. The online Exponential Growth Calculator is used to solve exponential growth problems. $\text{Slope } = \frac{ y_2 - y_1 } { x_2 - x_1 }$ How it works: Just type numbers into the boxes below and the calculator will automatically find the slope of two points. Example: If a population of rabbits doubles every month, we would have 2, then 4, then 8, 16, 32, 64, 128, 256, etc!. Neglecting air resistance, objects will fall to the earth with an acceleration of 10m/s/s. This function is useful for situations like compound interest, the Richter scale, decibel levels, and the exponential growth of a population. On the other hand, take an example where the slope is zero. = r = t = A = Since the continuous compounding formula is just another form of exponential function, its graph also has similar shape as the general form of the. Exponential Power Calculation Power calculator to find the product of an exponential expression (such as a raised to the power b, a^b). Using the LINEST function or LOGEST function You can use the LINEST or LOGEST function to calculate a straight line or exponential curve from existing data. 98) and (6, 481. Al-Khwarizmi also wrote a treatise on Hindu-Arabic numerals. You can test this with your data. For each type of trend line, I will present the coefficients, the equation to calculate each coefficient and then the calculated value based on the following data set:. This is a online regression calculator for statistical use. It will calculate any one of the values from the other three in the exponential decay model equation. M stands for slope. is the same no matter which values x is. “cost per topping”). The hypotenuse is AC with a length of 30. As a result, we get an equation of the form y = a b x where a ≠ 0. The Slope Calculator is apt of carrying out mathematical operations with the following algorithms: Slope Length is the square root of (Rise squared plus Run squared) Angle of Inclination is the tangent of (Rise devided by Run) Percentage is 100 multiplied by (Rise devided by Run). m is positive for growth, negative for decay. Well you know that having a 0 in the denominator is a big no, no. A least-squares regression fit of the data (using base 10 logarithms to transform columns (1) and (6)) indicates that the estimated slope for the Weibull distribution is 1. For example, enter 3x+2=14 into the text box to get a step-by-step explanation of how to solve 3x+2=14. This is an exponential growth curve, where the y-value increases and the slope of the curve increases as x increases. Loading Loading. Average Change Rate Converter of US Current to Real Dollars with CPI Converter of US Current to Real Dollars with GDP deflator Exponential Growth/Decay Doubling/Half-Life Time Present Value Calculator S-Curve Calculator - 1 parameter estimate S-Curve Calculator - 3 parameter estimates. Plot the following exponential function for x=1 to 10 and calculate the slope of the resulting line on semi-log paper. EXAMPLE 1: a) is a polynomial of degree 4 with leading. This means the slope is undefined. Welcome to Math Nspired About Math Nspired Middle Grades Math Ratios and Proportional Relationships The Number System Expressions and Equations Functions Geometry Statistics and Probability Algebra 1 Equivalence Equations Linear Functions Linear Inequalities Systems of Linear Equations Functions and Relations Quadratic Functions Exponential. WITH THE TI-NSPIRE CAS CALCULATOR An Essay Submitted to the Office of Graduate Studies College of Arts & Sciences of John Carroll University In Partial Fulfillment of the Requirements For the Degree of Master of Arts By Meghan A. If ever you actually have advice with algebra and in particular with exponential form calculator or worksheet come pay a visit to us at Factoring-polynomials. The first example is the exponential growth function y = 1 (3) x. This binomial calculator calculates the product of a binomial raised either to the 2nd power or the 3rd power using the FOIL method. A polynomial can have any non-negative degree. Enter your data as a string of number pairs, separated by commas. Your calculator reports values for both a (the y-intercept) and b (the slope). Logarithmic Equations Solver. We can also use the calculator output to construct the linear regression equation for our data. Earth Curve Calculator. Parentheses and Brackets. 2 Problem 1CQ. You can then annualize it as needed. The online Exponential Growth Calculator is used to solve exponential growth problems. For each type of trend line, I will present the coefficients, the equation to calculate each coefficient and then the calculated value based on the following data set:. This calculator is intended solely for general information and educational purposes. The slope of the tangent line depends on being able to find the derivative of the function. 116 FAQ-676 We are trying to use a single exponential decay equation to determine the half-life of a compound, but your equation is slightly different than the standard form. 3) Now, the data graphed above don't lie perfectly on a straight line, which is the. Start studying MATH 1 EOC REVIEW NC. So the formulas are: Program code to Calculate Slope and Midpoint of a Line in C:. The slope, m, is as defined above, x and y are our variables, and (x 1, y 1) is a point on the line. What is Slope? Slope is a concept in geometry that describes the direction and steepness of a line on a Cartesian plane. exponential function, just as the slope of a line did with tables of points on a linear function. Students learn about four forms of equations: direct variation, slope-intercept form, standard form and point-slope form. We will start here with defining and calculating slope by analyzing a graph. When you study slope in Algebra, you are studying the incline along with other characteristics of a line. This puts the equation into one. Use a calculator to check the answer we found to the equation in example 3. , it grows to just under \$5000 after 20 years. Exponential Functions Examples: Now let's try a couple examples in order to put all of the theory we've covered into practice. we notice that g = e b. Examples of exponential growth include contagious diseases for which a cure is unavailable, and biological populations whose growth is uninhibited by predation, environmental factors, and so on. 17 Finding an inverse function 21 1. When = 1, the Weibull distribution models the exponential distribution. Exponential functions tell the stories of explosive change. How to Use the Calculator. About Exponential Decay Calculator. All of them are capable of performing exact computations. y (vertical) over a particular change of. Why you should learn it GOAL 2 GOAL 1 What you should learn 8. The term "exp(x)" is the same as writing e x or e^x or "e to the x" or "e to the power of x". Exponential growth is when numbers increase rapidly in an exponential fashion so for every x-value on a graph there is a larger y-value. We first isolate the exponential part by dividing both sides of the equation by 200. The longer your investment stays in the account, the greater the ratio of interest to the original amount. How to do exponential calculation to a range of cells in Excel? In Excel, addition, subtraction, multiplication and division is the basic calculation, maybe you can quickly and easily apply them. Slope Calculator Overview. 10 2 =10 6. Then, the formula to find the equation of a line is y = mx + b. L1 and L2 are the 2 ND function keys of the 1 and 2 keys. What is slope. Data must consist of two columns, x and y, to get the exponential regression y=ae bx. The slope of a line on the coordinate plane is indicated by the letter "m". Radioactive decay is an exponential process, meaning that the quantity of matter decreases at a rate proportional to its current value. You don't want to wait until the next test to learn how to do the inputs correctly. Therefore, the Intercept value corresponds to Ln(c), and c in the exponential formula is equal to Exp(Intercept). Free online calculators for math, algebra, chemistry, finance, plane geometry and solid geometry. In case you need to calculate the cube root you can use our cube root calculator which is an excellent tool that will calculate the cube root of any number. Weibull Distribution Calculator is an online probability and statistics tool for data analysis programmed to calculate precise accurate failure analysis and risk predictions with extremely small samples using a simple and useful graphical plot. Change the a, b values in this exponential function to see the calculations of properties of exponential function. From Fraction Inequalities Calculator to powers, we have everything covered. Supported integration rules and methods. At the same time, the slope is frequently used in map calculations. When the rate of change is proportional to the value of the function itself, it is called exponential growth and decay. This is an exponential function where the Y intercept is the same as usual (a) but Y increases as an exponential function of X. "Online Software Package" web sites [return to Table of Contents]. For a linear function, the rate of change of y relative to x is always constant, i. Special indicators may display on the screen to provide additional information concerning functions or results. Algebra I Calculator Activities A ies rpe 5 Graphing a Linear Equation SOL A. A line that slopes up from left to right has a positive gradient. Calculate the regression of a statistical measure between the relationship between one dependent variable and other changing variable through online Simple/ Linear Regression Calculator. Visualize the exponential function that passes through two points, which may be dragged within the x-y plane. The prediction interval focuses on the true y value for any set of x values. notebook May 14, 2014 p. In effect, the nominal decline rate is related to the instantaneous slope of the line, whereas the effective decline rate derives from the chord segment approximating that slope. Warning: Negative powers can also be used. Online math calculators and solvers. Finding the Inverse of an Exponential Function. Exponential growth/decay formula. Graphing slope Learn how to graph the slope using the slope and a point. Solves for cubic equations in the form ax 3 + bx 2 + cx + d = 0 using the following methods: 1) Solve the long way for all 3 roots and the discriminant Δ 2) Rational Root Theorem (Rational Zero Theorem) to solve for real roots followed by the synthetic div/quadratic method for the other imaginary roots if applicable. Right from ordered pair equation calculator to calculus, we have every part discussed. x 0 is the initial value at time t=0. Online Algebra Calculators. What was the bacteria population at the beginning of the experiment (five hours ago. Visit Mathway on the web. Calculators; Calculus I Calculators; Math Problem Solver (all calculators) Tangent Line Calculator. Polynomial Regression Online Interface. Exponential Graphing Calculator. For a linear function, the rate of change of y relative to x is always constant, i. 718281828 it goes on forever but you don't need to know the value, your calculator probably has exp(x) or e^x as a function (if, as I am assuming, it is a. Gradient Slope Example 2. Using the regression equation to calculate slope and intercept ; Using the R-squared coefficient calculation to estimate fit; Introduction. Students learn about four forms of equations: direct variation, slope-intercept form, standard form and point-slope form. This test will be paper-based this year. x(t) = x 0 × (1 + r) t. Graphing calculators make is easy to work with exponents and functions. The equation of the first degree. Learn how to calculate the exponential moving average in Excel and VBA, and get a free web-connected spreadsheet. Special indicators may display on the screen to provide additional information concerning functions or results. Hello, I am wondering if it is possible to calculate the slope in excel with a fixed y intercept? I know that if I construct a scatter plot I can insert a trend line which I can format to have a fixed y intercept and I can then display the equation on the graph. When = 1, the Weibull distribution models the exponential distribution. org helps support GraphSketch and gets you a neat, high-quality, mathematically-generated poster. Visit Mathway on the web. If 2 is raised to a negative power, say -5, then we write it as 2 -5. Graph logarithmic functions, as applied in Example 8. When dividing terms with the same base, the exponents should be subtracted. Exponential Slope Calculator. org helps support GraphSketch and gets you a neat, high-quality, mathematically-generated poster. It's so fast and easy you won't want to do the math again!. This example teaches you how to apply exponential smoothing to a time series in Excel. Exponential regression is probably one of the simplest nonlinear regression models. What are exponential equations? An exponential equation is an equation in which a variable occurs in the exponent. The exponent can be indicated by preceding it by the character E or e, as you can see in the example. Available as a mobile and desktop website as well as native iOS and Android apps. These properties are the reason it is an important function in mathematics. x (hori-zontal). This lesson shows you how to compute for the integral of a function using the separable equation technique. Linear trendlines. Come to Algebra-equation. What is Slope? Slope is a concept in geometry that describes the direction and steepness of a line on a Cartesian plane. The simple exponential smoothing model can be generalized to obtain a linear exponential smoothing (LES) model that computes local estimates of both level and trend. Easy to use online maths calculators and solvers for various topics. We maintain a great deal of great reference material on subjects ranging from two variables to decimals. 71828, and e² or about 7. In each portion, no-calculator and calculator, you’ll first. So just as a little bit of a refresher on slope, the slope of this line is going to be our change in y-- or our change in our function I guess we could say, if we say that this y is equal to f of x-- over our change in x. In case you need to calculate the cube root you can use our cube root calculator which is an excellent tool that will calculate the cube root of any number. But, we are far from finished! We still need to calculate our ANOVA table, and calculate the resulting significance. "Online Software Package" web sites [return to Table of Contents]. As you say the slope is r and the intercept is N. Some basic information about the Slope Calculator. The mathematical constant e is the unique real number such that the derivative (the slope of the tangent line) of the function f(x) = e x is f '(x) = e x, and its value at the point x = 0, is exactly 1. If you are asked to calculate the average rate of change on an interval without a graph, you will have to come up with two points to calculate the slope. After linear functions, the second most important class of functions are what are known as the "exponential" functions. Properties depend on value of "a". The Slope Calculator is apt of carrying out mathematical operations with the following algorithms: Slope Length is the square root of (Rise squared plus Run squared) Angle of Inclination is the tangent of (Rise devided by Run) Percentage is 100 multiplied by (Rise devided by Run). | 2019-08-18T03:27:25 | {
"domain": "omni-s.net",
"url": "http://omni-s.net/8bfduv/7nr4u.php?in=exponential-slope-calculator",
"openwebmath_score": 0.65995854139328,
"openwebmath_perplexity": 499.20571592290446,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9871787853562028,
"lm_q2_score": 0.8438950966654772,
"lm_q1q2_score": 0.8330753364942812
} |
https://math.stackexchange.com/questions/3245269/topological-spaces-which-are-not-pseudometrizable/3245315 | # Topological spaces which are not pseudometrizable.
Let $$(X,\tau)$$ be a topological space. Then we know some conditions under which $$(X,\tau)$$ is metrizable (see for example this and this). It is also clear from these theorems that not every topological space is metrizable.
However, I am wondering whether the same is true for pseudometric spaces also. To be more specific,
Does there exist topological spaces which are not pseudometrazible?
Where we agree to call a topological space $$(X,\tau)$$ to be pseudometrizable iff there exists a pseudometric $$d$$ on $$X$$ such that the topology induced by the psuedometric is $$\tau$$.
A pseudometric space is symmetric (also called $$R_0$$): if $$x \in \overline{\{y\}}$$ then $$y \in \overline{\{x\}}$$ (basically because $$d(x,y)=0$$ implies $$d(y,x)=0$$ too, also in pseudometric spaces).
Sierpinski space ($$X=\{0,1\}$$ with topology $$\{\emptyset,\{0\},X\}$$) is not symmetric so not pseudometrisable. (Because $$1 \in \overline{\{0\}}$$ but not the other way around). This is in a way the simplest example, certainly the smallest one.
If $$X$$ is $$T_1$$ then $$X$$ is metrisable iff $$X$$ is pseudometrisable. (the $$T_1$$ ensures that $$X$$ is also $$R_0$$ and so the non-existence of any points $$x,y$$ with $$x \neq y$$ but $$d(x,y)=0$$. So the pseudometric for $$X$$ on the right is then a metric.)
So spaces like the cofinite topology on $$\mathbb{N}$$ is not pseudometrisable, as it's not metrisable (not Hausdorff to start with...) Also, the Sorgenfrey line, the Michael line, double arrow space etc etc.
• Does there exists any generalization of the notion of metric, say for the time being, a generalized metric such that every topological space is generalized metrizable? – user 170039 May 31 at 13:22
• @user170039 there probably is (I've seen such things at conferences and immediately forgot), but what would be the point? – Henno Brandsma May 31 at 14:41
• Actually I am trying to define the notion of differentiable functions for arbitrary topological space. I can already do so for metric spaces (a sketch of idea is given here) and hope to extend these ideal to general topological spaces. – user 170039 May 31 at 14:53
• In case you happen to find reference for any such thing, please let me know. It will help me immensely. – user 170039 Jun 2 at 3:27
The Sierpinski space is not pseudometrizable. The Kolmogorov quotient of a pseudometric space is metric. However the Sierpinski space is already $$T_0$$, but it's not Hausdorff, and thus not metric.
The Sierpinski space is the topological space on two points with the topology $$\{\varnothing, \{0\}, \{0,1\}\}.$$ | 2019-12-10T11:29:02 | {
"domain": "stackexchange.com",
"url": "https://math.stackexchange.com/questions/3245269/topological-spaces-which-are-not-pseudometrizable/3245315",
"openwebmath_score": 0.8540101051330566,
"openwebmath_perplexity": 332.18477324090435,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9871787872422175,
"lm_q2_score": 0.8438950947024555,
"lm_q1q2_score": 0.8330753361480263
} |
http://mathhelpforum.com/algebra/17119-simplify-ratios-factorials.html | # Math Help - Simplify ratios of factorials
1. ## Simplify ratios of factorials
$\frac{(n+1)!}{n!} = \frac{n!(n+1)}{n!} = (n+1)$
I understand canceling, but I don't see how the initial fraction became $\frac{n!(n+1)}{n!}$.
Same with $\frac{(2n-1)!}{(2n+1)!} = \frac{(2n-1)!}{(2n-1)!(2n)(2n+1)} = \frac{1}{2n(2n+1)}$.
2. Originally Posted by cinder
$\frac{(n+1)!}{n!} = \frac{n!(n+1)}{n!} = (n+1)$
I understand canceling, but I don't see how the initial fraction became $\frac{n!(n+1)}{n!}$.
Same with $\frac{(2n-1)!}{(2n+1)!} = \frac{(2n-1)!}{(2n-1)!(2n)(2n+1)} = \frac{1}{2n(2n+1)}$.
do you know what factorial means?
$(n + 1)! = (n + 1)(n)(n - 1)(n - 2)(n - 3)...(3)(2)(1)$
but $n! = n(n - 1)(n - 2)(n - 3)...(3)(2)(1)$
so we have: $(n + 1)! = (n + 1)n!$
similarly, $(2n + 1)! = (2n + 1)(2n)(2n - 1)(2n - 2)...(3)(2)(1) = (2n + 1)(2n)(2n - 1)!$
3. Originally Posted by Jhevon
do you know what factorial means?
$(n + 1)! = (n + 1)(n)(n - 1)(n - 2)(n - 3)...(3)(2)(1)$
but $n! = n(n - 1)(n - 2)(n - 3)...(3)(2)(1)$
so we have: $(n + 1)! = (n + 1)n!$
similarly, $(2n + 1)! = (2n + 1)(2n)(2n - 1)(2n - 2)...(3)(2)(1) = (2n + 1)(2n)(2n - 1)!$
I know what factorial means, but I guess I don't understand when there are variables involved.
4. Originally Posted by cinder
I know what factorial means, but I guess I don't understand when there are variables involved.
well, did my illustration help?
5. Originally Posted by Jhevon
well, did my illustration help?
No.
6. Originally Posted by cinder
No.
ok, let's start from the basics and build our way up.
the factorial of an integer is defined as the product of all integers starting from 1 going up to the integer.
example, 5! = 1*2*3*4*5 and 6! = 1*2*3*4*5*6
so the integers go up in order until we get to the integer we are finding the factorial of. but how do we go from one integer to the other? we simply add 1 to the current integer to get the next integer. equivalently, this means we can subtract one from an integer to get the previous integer and keep doing so until we hit 1 and stop, which is what we do here. take 5! again as an example.
we can say that 5! = 5*4*3*2*1, the same thing, just going backwards. since we are going backwards, we subtract 1 from the first integer to get the second, and then subtract 1 from the second integer to get the third and so on. so we can say,
5! = 5*(5 - 1)*((5 - 1) - 1)*(((5 - 1) - 1) - 1)*((((5 - 1) - 1) - 1) - 1)
which is just simpler to write as
5! = 5*(5 - 1)(5 - 2)(5 - 3)(5 - 4)
if the factorial is very long or indefinite, we can omit some of the terms in the middle, again, let's use 5! as an illustration
5! = 5*(5 - 1)*...*(5 - 3)(5 - 4)
but we usually simplify the end, so instead of 5 - 4 we write 1 and instead of 5 - 3 we write 2, so we get:
5! = 5*(5 - 1)*...*2*1
but this representation can work for any integer. if we want to have an arbitrary integer factorial, we can use n. so in general:
n! = n(n - 1)(n - 2)(n - 3)....(3)(2)(1)
there is no difference if we want (n + 1)!, we just start at n + 1 and subtract 1 until we get to 1
(n + 1)! = (n + 1)(n)(n - 1)(n - 2)...(3)(2)(1)
so here is where the algebraic manipulation comes in.
recall that n! = n(n - 1)(n - 2)(n - 3)....(3)(2)(1)
now let's write (n + 1)! out.
we have (n + 1)! = (n + 1)(n)(n - 1)(n - 2)(n - 3)...(3)(2)(1)
notice anything? yes, what's in red is n! so we can just replace it
thus we get, (n + 1)! = (n + 1)n!
do you get it now?
7. I'm 99% sure I got it (I'll reread to verify my thinking). Thank you very much for the explanation!
8. Originally Posted by cinder
I'm 99% sure I got it (I'll reread to verify my thinking). Thank you very much for the explanation!
well, if you have any doubts, come back | 2014-07-13T16:05:54 | {
"domain": "mathhelpforum.com",
"url": "http://mathhelpforum.com/algebra/17119-simplify-ratios-factorials.html",
"openwebmath_score": 0.8868321776390076,
"openwebmath_perplexity": 629.4982468842486,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES\n\n",
"lm_q1_score": 0.9920620058985927,
"lm_q2_score": 0.8397339736884712,
"lm_q1q2_score": 0.8330681703585808
} |
http://mathhelpforum.com/math-challenge-problems/21600-problem-39-a.html | # Math Help - Problem 39
1. ## Problem 39
Let $a,b\in \mathbb{Z}^+$. For each positive integer $n$ let $H_n(a,b) = \frac{1}{a+b}+\frac{1}{2a+b}+...+\frac{1}{na+b}$.
Find the limit,
$\lim \ \frac{H_n(a,b)}{H_n(c,d)}$
(Where $c,d$ are possibly different integers defining a different sequence).
2. Originally Posted by ThePerfectHacker
Let $a,b\in \mathbb{Z}^+$. For each positive integer $n$ let $H_n(a,b) = \frac{1}{a+b}+\frac{1}{2a+b}+...+\frac{1}{na+b}$.
Find the limit,
$\lim \ \frac{H_n(a,b)}{H_n(c,d)}$
(Where $c,d$ are possibly different integers defining a different sequence).
is this lim as n approaches infinity?
3. Originally Posted by kalagota
is this lim as n approaches infinity?
When you are dealing with sequences that is the only type of limit you can have. (What else can you possibly approach )
4. Maybe I'm off base here, PH,:
$\lim_{n\rightarrow{\infty}}\sum_{k=1}^{n}\left(\fr ac{1}{na+b}\right)=\frac{1}{a}$
$\lim_{n\rightarrow{\infty}}\sum_{k=1}^{n}\left(\fr ac{1}{nc+d}\right)=\frac{1}{c}$
So, we have: $\frac{\frac{1}{a}}{\frac{1}{c}}=\boxed{\frac{c}{a} }$
5. Originally Posted by galactus
Maybe I'm off base here, PH,:
$\lim_{n\rightarrow{\infty}}\sum_{k=1}^{n}\left(\fr ac{1}{na+b}\right)=\frac{1}{a}$
$\lim_{n\rightarrow{\infty}}\sum_{k=1}^{n}\left(\fr ac{1}{nc+d}\right)=\frac{1}{c}$
The sums diverges. Because,
$0\leq \frac{1}{n+n}\leq \frac{1}{na+b}$ for sufficiently large $n$.
And $\sum_{n=1}^{\infty}\frac{1}{2n}$ does not converge, it is the harmonic series.
6. Let $x_n=H_n(a,b), \ y_n=H_n(c,d)$.
$(y_n)$ is ascending and unbounded.
By Stolz-Cesaro, we have
$\displaystyle\lim_{n\to\infty}\frac{x_{n+1}-x_n}{y_{n+1}-y_n}=\lim_{n\to\infty}\frac{\frac{1}{a(n+1)+b}}{\f rac{1}{c(n+1)+d}}=\lim_{n\to\infty}\frac{cn+c+d}{a n+a+b}=\frac{c}{a}$.
Then $\displaystyle\lim_{n\to\infty}\frac{x_n}{y_n}=\fra c{c}{a}$
7. Originally Posted by red_dog
Let $x_n=H_n(a,b), \ y_n=H_n(c,d)$.
$(y_n)$ is ascending and unbounded.
By Stolz-Cesaro, we have
$\displaystyle\lim_{n\to\infty}\frac{x_{n+1}-x_n}{y_{n+1}-y_n}=\lim_{n\to\infty}\frac{\frac{1}{a(n+1)+b}}{\f rac{1}{c(n+1)+d}}=\lim_{n\to\infty}\frac{cn+c+d}{a n+a+b}=\frac{c}{a}$.
Then $\displaystyle\lim_{n\to\infty}\frac{x_n}{y_n}=\fra c{c}{a}$
I can do it without Stolz-Cesaro. Want to try?
8. The key step is to note that $H_n(a,b)\sim \ln (an+b)$. That means, $\lim \ \frac{H_n(a,b)}{H_n(c,d)} = \lim \ \frac{\ln (an+b)}{\ln (cn+d)}=\frac{c}{a}$. | 2015-07-03T15:49:28 | {
"domain": "mathhelpforum.com",
"url": "http://mathhelpforum.com/math-challenge-problems/21600-problem-39-a.html",
"openwebmath_score": 0.9399811625480652,
"openwebmath_perplexity": 1477.2762198355022,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9920620077761345,
"lm_q2_score": 0.8397339656668286,
"lm_q1q2_score": 0.8330681639772496
} |
https://math.stackexchange.com/questions/2711149/can-you-write-something-like-3-cdot5-cdot7-cdots2n1-as-2n1 | # Can you write something like $3\cdot5\cdot7\cdots(2n+1)$ as $(2n+1)!$?
I have been working through some infinite series in calculus I, and there have been a lot of expressions of the form $a\cdot b\cdot c\cdots(zn+x)$. It usually seems to work if I simplify them to $(zn+x)!$. In addition to the example I gave in the question, here are a couple more:
1. $2\cdot3\cdot4\cdots(n+1) = (n+1)!$
2. $1\cdot2\cdot3\cdots(2n-1) = (2n-1)!$
However, my professor said that this was incorrect and that she had an explanation for why, but for some reason, it slipped her mind. It seems like my textbook does not simplify them like I do either, but I still do not understand what is mathematically wrong with doing it like that.
Any help would be much appreciated!
-Isaac
Edit:
Thanks for your comments. I see that I am clearly overthinking this, and I'm sure that something will click into place once I read through what you all said several times more. However, I still don't really get it. Isn't
$(n+1)! = (n+1)\cdot(n)\cdot(n-1)\cdot\cdots\cdot(n-(n-1))$?
How is this not the same as
$(n+1)! = 2\cdot3\cdot4\cdot\cdots\cdot(n+1)$?
Do I have to specify that n is an integer greater than 1 for it to work, or is there something more fundamentally wrong?
I'm not trying to waste anyone's time; this is honestly something that I am confused about. I enjoy my math class, and I want to really get what I am learning.
Also, thanks for posting that MathJax tutorial. I was trying to figure it out earlier, but I think I've got it now.
One last note: my original post had a typo in example 2. It is fixed now.
• So for instance, is $1\times 3\times5\times 7$ equal to $7!$? Of course $7!=5040$, a nice even number. – Lord Shark the Unknown Mar 28 '18 at 1:15
• It's simply wrong. Make a substitution $2n-1=k$ then consider $k!$ and compare to your product – Yuriy S Mar 28 '18 at 1:15
• If you want to say "although usually the expression $n!$ means the product of all natural numbers up to $n$, for the purpose of this paper, I'll be using it to mean 'go make a peanut butter sandwich'", you may. But you will have to expect no-one in any other context to know what you mean. $(zn+x)!$ simply does mean $1*2*...*(zn+x-1)*(zn+x)$ and it simply doesn't mean $(z+x)(2z+x)... (zn+x)$. But if you want to SAY it does, go ahead. But just make sure you explicitly state it, and you only use it in that one context. – fleablood Mar 28 '18 at 3:06
• The OEIS is frequently a very useful resource for math. For this particular question, look at oeis.org/A001147 and oeis.org/A000165 – Robert Soupe Mar 28 '18 at 3:34
I think you are looking for the product notation.
$$\prod_{i=1}^n (2i-1)= 1\cdot 3\cdot 5 \ldots (2n-1)=\frac{(2n-1)!}{2\cdot 4\cdot \ldots (2n-2)}=\frac{(2n-1)!}{(n-1)!2^{n-1}}$$
$$\prod_{i=1}^n (i+1)= 2\cdot 3\cdot 4 \ldots (n+1)=(n+1)!$$
$$\prod_{i=1}^n (zi+x)= (z+x)\cdot(2z+x)\cdot \ldots \cdot(nz+x)$$
The factorial notation $n!$ multiply every positive integers up to $n$.
• That's right, I forgot about product notation. I really appreciate your comment, I understand much better now – Ii Meme Mar 29 '18 at 2:01
The equation
$$1 \cdot 3 \cdot 5 \cdot \ldots \cdot (2n-1) = (2n-1)!$$
is (usually) wrong because, by definition,
$$(2n-1)! = 1 \cdot 2 \cdot 3 \cdot \ldots \cdot (2n-1)$$
and we (usually) have
$$1 \cdot 3 \cdot 5 \cdot \ldots \cdot (2n-1) \neq 1 \cdot 2 \cdot 3 \cdot \ldots \cdot (2n-1)$$
The specific example you give, incidentally, comes up often enough to be given a name: the double factorial. It has different definitions depending on whether it is even or odd:
$$(2n-1)!! = 1 \cdot 3 \cdot 5 \cdot \ldots \cdot (2n-1)$$ $$(2n)!! = 2 \cdot 4 \cdot 6 \cdot \ldots \cdot (2n)$$
There are convenient identities
$$n! = n!! \cdot (n-1)!!$$ $$(2n)!! = 2^n n!$$
• In case it's not clear, $!!$ is meant to be read as a single symbol. $n!!$ means to apply the double factorial operation to $n$; it does not mean to apply the factorial and then apply the factorial a second time. That is, we usually have $n!! \neq (n!)!$. – Hurkyl Mar 28 '18 at 1:32
• I did not know about the double factorial, thanks for letting me know. About that example I gave, that was a typo on my part. I will edit it to display what it should. – Ii Meme Mar 29 '18 at 2:12
"but I still do not understand what is mathematically wrong with doing it like that."
Seriously?
You don't see why $1*2*3*4*5*6*7* ......(2n-3)*(2n-2)*(2n-1)*(2n)*(2n + 1) \ne 1*3*5*7*.......... *(2n-3)*(2n-1)*(2n+1)$?
Only one of those can be written as $(2n+1)!$. Which one is it going to be?
Oh, you chose the RHS...? Well, I wasn't actually giving you a choice. The question was first asked 80 years ago and everyone chose the LHS.
"However, my professor said that this was incorrect and that she had an explanation for why, but for some reason, it slipped her mind."
Seriously?!?!?
• Part of me thinks this may be a troll question. – Crescendo Mar 28 '18 at 4:47
• Thanks for responding to my question. It was indeed serious; I legitimately did not know. Also, my tutor said that it was correct, I posted this question to figure out who was right. Have a nice day – Ii Meme Mar 29 '18 at 1:56 | 2019-09-21T02:34:48 | {
"domain": "stackexchange.com",
"url": "https://math.stackexchange.com/questions/2711149/can-you-write-something-like-3-cdot5-cdot7-cdots2n1-as-2n1",
"openwebmath_score": 0.807449221611023,
"openwebmath_perplexity": 282.1495288701884,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9458012762876287,
"lm_q2_score": 0.8807970873650401,
"lm_q1q2_score": 0.833059009380281
} |
http://math.stackexchange.com/questions/355894/derivative-of-sin2-sqrtt | # Derivative of $\sin^2 (\sqrt{t})$
I need to find the derivative of $\sin^2 (\sqrt{t})$ which I believe have done but the answer seems to be more simplified and I don't know how to arrive to it. Here are my steps
\begin{align} & \frac{d}{dt}(\sin \sqrt{t})^2 \\ & = 2(\sin \sqrt{t}) \cdot \frac{\cos \sqrt{t}}{2\sqrt{t}} \\ & = \frac{\sin \sqrt{t} \cdot \cos \sqrt{t}}{\sqrt{t}} \end{align}
The question is how does this simplify to $\frac{\sin 2\sqrt{t}}{2\sqrt{t}}$?
EDIT:
I forgot to add $2$ but I don't understand why top and bottom don't just cancel?
-
$\sin 2\theta=2\sin\theta\cos\theta$. So this will simplify to what you say, divided by $2$. – 1015 Apr 9 '13 at 13:21
@Julien but I don't have $2\sin\theta\cos\theta$? The $2$ on the top and $2$ on the bottom cancels out. – Jeel Shah Apr 9 '13 at 13:22
$\frac{x}{y}=\frac{2x}{2y}$. Now you have the $2$ you miss at the numerator to apply the formula. Or if prefer, use $\sin\theta\cos\theta=\frac{\sin 2\theta}{2}$ directly. – 1015 Apr 9 '13 at 13:24
@RonGordon Sorry, I am a little confused. I plugged this into W|A and it has gives me the same simplified form that I arrived at. – Jeel Shah Apr 9 '13 at 13:26
In response to the edit: It'd be sin to cancel them out. The expression is $\dfrac{\sin(2\sqrt t)}{2\sqrt t}$ rather than something else. – Lord_Farin Apr 9 '13 at 13:34
Hint: If $y=x$ in the formula $\sin \left(x+y\right)=\sin x \cos y + \cos x \sin y$ then $$\sin(2x)= 2\cos(x)\sin(x).$$ Use this formula whit $x=\sqrt{t}$ to give \begin{align} \frac{\sin \sqrt{t} \cdot \cos \sqrt{t}}{\sqrt{t}}= & \frac{2\sin \sqrt{t} \cdot \cos \sqrt{t}}{2\sqrt{t}} \\ = & \frac{\sin (2\sqrt{t}) }{2\sqrt{t}} \end{align}
$$\frac{\sin \sqrt{t} \cdot \cos \sqrt{t}}{\sqrt{t}}=\frac{2\sin \sqrt{t} \cdot \cos \sqrt{t}}{2\sqrt{t}}=\frac{\sin (2\sqrt{t}) }{2\sqrt{t}}$$
Your answer is correct and the target answer is missing a factor of $2$ in the denominator. Here is confirmation from Alpha | 2016-05-25T07:28:09 | {
"domain": "stackexchange.com",
"url": "http://math.stackexchange.com/questions/355894/derivative-of-sin2-sqrtt",
"openwebmath_score": 0.9996114373207092,
"openwebmath_perplexity": 596.0060731035283,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9458012747599251,
"lm_q2_score": 0.8807970685907242,
"lm_q1q2_score": 0.8330589902779121
} |
https://www.physicsforums.com/threads/simple-line-integral-question.469582/ | # Simple Line Integral Question
## Homework Statement
See figure attached for problem statement as well as the solution provided.
## The Attempt at a Solution
My only question is why can't I use,
$$32 \int ^{1}_{2} t\sqrt{8t^{2}+1}dt$$
I realize that this will give me the negative of his answer, I just don't understand why it doesn't work.
I still have t bounded from 1 to 2.
What's different here?
Thanks again!
#### Attachments
• LIQ.JPG
33.8 KB · Views: 385
Gib Z
Homework Helper
Your answer is right. I don't see why the solution thinks $\int_C = \int_{-C}$. Is this for some physics problem/class and needs to be positive for physical reasons?
Your answer is right. I don't see why the solution thinks $\int_C = \int_{-C}$. Is this for some physics problem/class and needs to be positive for physical reasons?
Nope, this is for a Math course.
LCKurtz
Homework Helper
Gold Member
## Homework Statement
See figure attached for problem statement as well as the solution provided.
## The Attempt at a Solution
My only question is why can't I use,
$$32 \int ^{1}_{2} t\sqrt{8t^{2}+1}dt$$
I realize that this will give me the negative of his answer, I just don't understand why it doesn't work.
I still have t bounded from 1 to 2.
What's different here?
Thanks again!
In the problem you are given
$$\int_C y\,ds$$
and y is positive at all points on the curve. So a properly set up integral must give a positive answer. When you use the identity
$$ds = |R'(t)| dt$$
you are implicitly assuming that the integration is in a positive direction with respect to the parameter t (dt > 0).
In the problem you are given
$$\int_C y\,ds$$
and y is positive at all points on the curve. So a properly set up integral must give a positive answer. When you use the identity
$$ds = |R'(t)| dt$$
you are implicitly assuming that the integration is in a positive direction with respect to the parameter t (dt > 0).
So what should I be checking or looking at when doing problems like this to ensure I don't preform the same error time and time again? I'm still a little confused.
LCKurtz
Homework Helper
Gold Member
In the problem you are given
$$\int_C y\,ds$$
and y is positive at all points on the curve. So a properly set up integral must give a positive answer. When you use the identity
$$ds = |R'(t)| dt$$
you are implicitly assuming that the integration is in a positive direction with respect to the parameter t (dt > 0).
So what should I be checking or looking at when doing problems like this to ensure I don't preform the same error time and time again? I'm still a little confused.
For ds type integrals, the element of arc length is always positive. For example, how long a curve is shouldn't depend on which end you measure from. When you have a curve parameterized as
$$\vec R(t) = \langle x(t),y(t),z(t)\rangle,\ a\le t \le b$$
the parameterization itself implies a positive direction along the curve as t moves from a to b. So if you want to use the formula
$$ds = |\vec R'(t)| dt$$
your lower limit must be a and upper limit must be b. Just always integrate in the positive t direction for ds type integrals, regardless of which physical direction it takes you along the curve.
Note that the situation is different for line integrals of the type
$$\int_C \vec F \cdot d\vec R$$
Such an integral may represent, for example, the work done by the force moving an object along C. The answer will be positive or negative depending on which way the curve is traversed. | 2021-05-12T11:49:55 | {
"domain": "physicsforums.com",
"url": "https://www.physicsforums.com/threads/simple-line-integral-question.469582/",
"openwebmath_score": 0.8591067790985107,
"openwebmath_perplexity": 429.95209749859185,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES\n\n",
"lm_q1_score": 0.945801271704518,
"lm_q2_score": 0.8807970654616711,
"lm_q1q2_score": 0.8330589846272561
} |
https://math.stackexchange.com/questions/3598122/max-min-and-compactness | # max/min and compactness
if $$f(x)=\frac{1}{1+x^2}$$ and $$-\infty, show that $$f$$ attains a $$max$$ but does not attain a $$min$$.
what I ve learnt so far is that :
if $$f$$ is a real valued function from a compact metric space, then $$f$$ attains a max and min at some point of $$M$$.
if $$f$$ is a real valued function from a closed bounded set, then $$f$$ attains a max and min at some point of that set.
I believe I must be able to apply any of these two statements to analyse such questions. right?
Now since $$-\infty and not closed in $$R$$, so I ignore the second statement.
but for the first statement since $$-\infty represents real numbers $$R$$, and we know $$R^n$$ is not compact, so the first statement does not apply!
Well this is an excercise after those two statements. how are they going to help then?
• it is the exact text of the exercise in the book – BesMath Mar 27 '20 at 21:19
• You are correct. The statements in no way apply. But that's fine. You solve it by other means. This isn't a question about compactness. It's about bounded sets and sups and infs. Now $x^2 \ge 0$ so $1+x^2 \ge 1$ so $f(x) \le 1$. So $1$ is an upper bound. And note $f(x) > 0$ so $0$ is a lower bound. Can you go on. – fleablood Mar 28 '20 at 0:08
• "Well this is an excercise after those two statements. " This could be a set up for the next excercise.... – fleablood Mar 28 '20 at 0:09
• I see thank you. – BesMath Mar 28 '20 at 0:11
To maximize $$f(x)$$, you need to minimize the denominator. The minimum value of the denominator is 1 because $$x^2 \ge 0$$. But there is no way to minimize $$f(x)$$ as we can keep increasing the value of $$x$$ which will make it smaller and smaller.
To show the max of $$f$$ is $$f(0)=1$$, here's one way:
$$f(x)=\frac{1}{1+x^2}\leq\frac{1}{1}=1$$ so $$1$$ is an upper bound. Therefore, $$\sup f(x) \leq 1$$. At the same rate $$f(0)=1$$ shows all upper bounds most be at least $$1$$. Therefore, $$\sup f(x)\geq 1$$. Therefore, $$\sup f(x)=1$$ and $$f(0)=\sup f(x)$$ so $$f$$ has a maximum.
For the next part, $$f(x)\geq 0$$ so $$0$$ is a lower bound. Therefore, $$\inf f(x)\geq 0$$. Suppose $$\inf f(x)>0$$. Choose $$\epsilon$$ such that $$0<\epsilon<\inf f(x)$$. Then $$\epsilon$$ is a lower bound for $$f$$ so $$f(x)\geq \epsilon$$ for all $$x$$. We arrive at a contradiction by finding $$x$$ such that $$f(x)<\epsilon$$.
Solving $$\frac{1}{1+x^2}<\epsilon \implies 1<\epsilon(1+x^2)\implies \frac{1-\epsilon}{\epsilon}\sqrt{\frac{1-\epsilon}{\epsilon}}$$ or $$x< -\sqrt{\frac{1-\epsilon}{\epsilon}}$$. Pick any such $$x$$ and we have a contradiction. Therefore, $$\inf f(x)=0$$.
If $$f$$ has a minimum, there must exist $$x_0$$ such that $$f(x_0)=0$$. But then $$\frac{1}{1+x_0^2}=0$$, which implies $$1=0$$, a contradiction. Thus, $$f$$ has no minimum.
Remark: The two statements you provided aren't enough. The space is not compact so you really have to perform the analysis.
• Thank you. study a section and then go to solve the excecises of that section. but it turns out the answer has nothing to do with that section! thank you for the clear explanation. – BesMath Mar 28 '20 at 0:12 | 2021-02-26T02:31:21 | {
"domain": "stackexchange.com",
"url": "https://math.stackexchange.com/questions/3598122/max-min-and-compactness",
"openwebmath_score": 0.9471943974494934,
"openwebmath_perplexity": 4354.033959593236,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.975576912076157,
"lm_q2_score": 0.8539127585282745,
"lm_q1q2_score": 0.8330575721474472
} |
https://math.stackexchange.com/questions/2621908/how-to-find-the-following-limit-algebraically/2621913 | How to find the following limit algebraically?
I've been trying to answer this for a while and I know it's a simple question relative to most questions that are posted here.
$$\lim_{x\rightarrow -2}\: \frac{x^4+5x^3+6x^2}{x^2(x+1)-4(x+1)}$$
If we substitute -2 for $x$ we get $0/0$, an indeterminate form. I figured that the denominator can be rewritten as $(x^2-4)(x+1)$. And then I tried to factor something in the numerator but couldn't see anything interesting. How do I find the limit algebraically?
I know that the answer is supposed to be 1, but I don't know how they got there.
• Divide top and bottom by the highest power. – uniquesolution Jan 26 '18 at 10:05
• @uniquesolution That only works for limit as $x\to \pm \infty$. Or, at least, that's when your suggested approach is the standard, simple approach. – Arthur Jan 26 '18 at 10:08
• @Arthur Right on point, +1 ! – Rebellos Jan 26 '18 at 10:11
General tip (update) : When you can see that the denominator is equal to zero for a value $x=a$ which is the $x\to a$ of the limit, then you should try factoring on both the numerator and the denominator the factor $(x-a)$, such as you can get rid of the $\frac{0}{0}$ issue. This particular example though can also be done by following a row factorization.
Factor the expression at the numerator by taking out $x^2$ and then forming a quadratic factorization inside, as :
$$x^4+5x^3+6x^2 = x^2(x^2+5x+6) =x^2(x^2+2x+3x+6) = x^2(x+2)(x+3)$$
Then, the given limit is :
$$\lim_{x\rightarrow -2} \frac{x^4+5x^3+6x^2}{x^2(x+1)-4(x+1)} = \lim_{x\to -2} \frac{x^2(x^2 + 5x + 6)}{(x^2-4)(x+1)} = \lim_{x \to -2} \frac{x^2(x+3)(x+2)}{(x-2)(x+2)(x+1)}$$
$$=$$
$$\lim_{x\to -2} \frac{x^2(x+3)}{(x-2)(x+1)} = 1$$
The fact the numerator becomes zero after plugging in $x = -2$ means that $(x + 2)$ is a factor of the numerator; that is the "interesting" thing you can factor out. You can factor it out of the denominator for the same reason.
Doing so gives
$$\lim_{x\rightarrow -2}\: \frac{x^4+5x^3+6x^2}{x^2(x+1)-4(x+1)} = \lim_{x\rightarrow -2}\: \frac{(x+2)(x^3 + 3x^2)}{(x+2)(x^2 - x - 2)} = \lim_{x\rightarrow -2}\: \frac{x^3 + 3x^2}{x^2 - x - 2}$$
and this limit can be found by plugging in $x = -2$.
• +1 - It is surprising that you are the only answerer so far to bring up this simple and obvious fact: The very issue here - that the numerator and denominator are both $0$ at $x = -2$ - tells you the common factor you can divide from both to find the limit. You don't need to fully factor the polynomials. – Paul Sinclair Jan 26 '18 at 17:39
• @PaulSinclair (+1) I completely agree. It would be kind of fun to come up with an example involving "ugly" polynomials in the numerator and denominator for $\lim_{x\to a}\frac{P(x)}{Q(x)}$ where you couldn't factor the polynomials nicely and where you were more or less forced to use synthetic division and the factor theorem to factor out the factors $x-a$ appropriately and then evaluate as Hurykl did. – Daniel W. Farlow Jan 26 '18 at 20:19
• @DanielW.Farlow : Such a pair of "ugly" polynomials would fall to the slightly fancier method of using Hurkyl's method. The polynomial GCD of the given numerator and denominator is $\gcd(x^4+5 x^3+6 x^2, (x^2-4)(x+1)) = x+2$. – Eric Towers Jan 27 '18 at 7:25
• @EricTowers Maybe I'm misunderstanding your comment or perhaps I did't clearly express what I had in mind in my comment. Basically what I meant to say was trying to force someone to only use the "slightly fancier method," where it might be rather difficult to completely factor the rest of the polynomial. I'm not sure what your comment really addresses in my own comment unless I am simply overlooking something. – Daniel W. Farlow Jan 27 '18 at 7:43
• @DanielW.Farlow : You can't force synthetic division and the factor theorem because you can't make a problem so ugly that it fails to fall to the "simpler" process of taking the polynomial GCD. Polynomial GCDs are just too darned easy. – Eric Towers Jan 27 '18 at 7:46
HINT: $x^4+5x^3+6x^2 = x^2(x^2+5x+6) =x^2(x^2+2x+3x+6) = x^2(x+2)(x+3)$
Write $$\frac{x^4+5x^3+6x^2}{(x^2-4)(x+1)}=\frac{x^2(x+3)(x+2)}{(x-2)(x+2)(x+1)}.$$
What, nobody used L'Hôpital's rule yet? Couldn't resist:
$$\lim_{x\rightarrow -2}\: \frac{x^4+5x^3+6x^2}{x^2(x+1)-4(x+1)} = \lim_{x\rightarrow -2}\: \frac{4x^3+15x^2+12x}{3x^2+2x-4} = \frac{4}{4}$$
• Welcome to MSE. What's algebraic about your approach? – José Carlos Santos Jan 26 '18 at 12:05
• Sorry, my maths education is apparently a bit different from what it's like in English-speaking countries, so I may be unfamiliar with this particular use of the term "algebraic". You can ignore this answer; like I said, I just couldn't resist trying to solve it this way, even if it might not have been particularly useful as more than equation-writing practice. – Ove Jan 26 '18 at 12:34
• @Ove, your desire is understandable, but it is implicit in the question that the exercise is leading up to applied calculus. Applying a result like L'Hopital defeats the exercise (and doesn't help the student). – user1717828 Jan 26 '18 at 17:27
• Well, as I mentioned, the word "algebraically" meant nothing to me in this context, in part because limits aren't pure algebra anyway. I've done a lot of calculus without knowing that in the English-speaking world, people apparently use this word like this. Do you want me to delete this answer? – Ove Jan 27 '18 at 0:03
• @user1717828 technically this is algebra. asker should clarify what rules they limit themselves to first. – The Great Duck Jan 27 '18 at 4:36
Let $x+2=h\iff x=h-2$
$$\lim_{h\to0}\dfrac{(h-2)^4+5(h-2)^3+6(h-2)^2}{(h-2)^3+(h-2)^2-4(h-2)-4}$$
$$=\lim_{h\to0}(h-2)^2\cdot\lim_{h\to0}\dfrac{(h-2)^2+5(h-2)+6}{h^3-5h^2+h(12-4-4)}$$
$$=(0-2)^2\cdot\lim_{h\to0}\dfrac{h(1+h)}{h(h^2-5h+4)}=?$$
• @Downvoter, Any apparent mistake? – lab bhattacharjee Jan 26 '18 at 10:19
• My be your way is uncommon – E.H.E Jan 26 '18 at 10:41
• @E.H.E that is more of a reason to upvote. – The Great Duck Jan 27 '18 at 4:37
Notice that: $$x^4+5x^3+6x^2 = x^2(x^2+5x+6) = x^2(x+2)(x+3)$$ and $$x^2(x+1)-4(x+1) = (x^2-4)(x+1) = (x+2)(x-2)(x+1).$$
Then, you have:
$$\lim_{x\rightarrow -2}\: \frac{x^2(x+2)(x+3)}{(x+2)(x-2)(x+1)} = \\ = \lim_{x\rightarrow -2}\: \frac{x^2(x+3)}{(x-2)(x+1)} = \\ = \frac{(-2)^2(-2+3)}{(-2-2)(-2+1)} = \frac{4}{4} = 1.$$ | 2019-06-25T05:38:46 | {
"domain": "stackexchange.com",
"url": "https://math.stackexchange.com/questions/2621908/how-to-find-the-following-limit-algebraically/2621913",
"openwebmath_score": 0.8670481443405151,
"openwebmath_perplexity": 476.63253656375997,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9755769092358048,
"lm_q2_score": 0.8539127603871312,
"lm_q1q2_score": 0.8330575715354918
} |
http://openstudy.com/updates/55a14257e4b0564dd2d7a65b | ## calculusxy one year ago MEDAL!!! Is it possible for two lines with positive slopes to be perpendicular to each other?
1. calculusxy
I think I got it. The slope needs to be negative reciprocals of the other. So like 5x needs to be -1/5x?
2. calculusxy
Am I correct?
3. calculusxy
@Michele_Laino
4. anonymous
No. Theres no way for a line to cross another and form right angles unless one has a negative slope and one has a positive slope. "If two lines are perpendicular and neither one is vertical, then one of the lines has a positive slope, and the other has a negative slope. Also, the absolute values of their slopes are reciprocals." http://www.cliffsnotes.com/math/geometry/coordinate-geometry/slopes-parallel-and-perpendicular-lines
5. Michele_Laino
if two perpendicular lines have their slopes , say m_1, and m_2 both positive, then the product of such slopes is also positive, namely: $\large {m_1}{m_2} > 0$
6. Michele_Laino
it is a contradiction, since we know that the product of such slopes has to check this condition: $\large {m_1}{m_2} = - 1$
7. calculusxy
8. calculusxy
So I need to make a perpendicular line through the -2,3 point. How can I do that?
9. Michele_Laino
it is simple the slope of the requested line, is: $\large m = - \frac{1}{5}$
10. calculusxy
I have the slope intercept form for -2,3 as y=5x + 2
11. Michele_Laino
so you have to apply this equation: $\large y - 3 = - \frac{1}{5}\left( {x + 2} \right)$
12. calculusxy
So now do I need to do: -1/5x + 2= y?
13. calculusxy
Is your and my equations the same thing? @Michele_Laino
14. Michele_Laino
I think that they are different equations, since after a simplification, I can rewrite my equation as follows: $\large y = - \frac{x}{5} + \frac{{13}}{5}$
15. calculusxy
16. calculusxy
Would that line work?
17. calculusxy
My line doesn't work. It is not perpendicular.
18. Michele_Laino
I think not, since that line is not perpendicular to both the parallel lines
19. Michele_Laino
please try with this line: $\large y = - \frac{x}{5} + \frac{{13}}{5}$
20. calculusxy
@Michele_Laino How did you get 13/5?
21. Michele_Laino
here are the missing steps: $\large \begin{gathered} y - 3 = - \frac{x}{5} - \frac{2}{5} \hfill \\ \hfill \\ y = - \frac{x}{5} - \frac{2}{5} + 3 \hfill \\ \hfill \\ y = - \frac{x}{5} + \frac{{15 - 2}}{5} \hfill \\ \hfill \\ y = - \frac{x}{5} + \frac{{13}}{5} \hfill \\ \end{gathered}$
22. calculusxy
I still do not get how you got 13/5. These are my steps: $y - 3 = -\frac{ 1 }{ 5 } (x+2)$ $y - 3=-\frac{ 1 }{ 5 }x-\frac{ 2 }{ 5 }$ $y=-\frac{ 1 }{ 5 }x-\frac{ 2 }{ 5 } + 3$ $y=-\frac{ 1 }{ 5 }x+\frac{ 3 }{ 5 }$
23. calculusxy
I am really sorry fir being redundant. :(
24. calculusxy
*for
25. Michele_Laino
hint: $\Large - \frac{2}{5} + 3 = \frac{{ - 2 + \left( {3 \times 5} \right)}}{5} = ...?$
26. calculusxy
How did you get -2+(3 x 5)?
27. Michele_Laino
since the least common multiple of 5 and 1 is 5 $\Large - \frac{2}{5} + 3 = - \frac{2}{5} + \frac{3}{1}$
28. Michele_Laino
and multipling both numerator and denominator of the second fraction, by 5, we get: $\large \frac{2}{5} + \frac{3}{1} = - \frac{2}{5} + \frac{{3 \times 5}}{{1 \times 5}} = - \frac{2}{5} + \frac{{15}}{5} = \frac{{ - 2 + 15}}{5}$
29. calculusxy
Oh okay!!!!! I totally get it now :)
30. Michele_Laino
:)
31. calculusxy
So for x I can input any number now? | 2016-10-25T03:25:15 | {
"domain": "openstudy.com",
"url": "http://openstudy.com/updates/55a14257e4b0564dd2d7a65b",
"openwebmath_score": 0.813396155834198,
"openwebmath_perplexity": 1713.1274060684982,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9755769106559808,
"lm_q2_score": 0.8539127585282744,
"lm_q1q2_score": 0.8330575709347405
} |
http://mathhelpforum.com/algebra/161525-sum-geometric-series.html | # Thread: Sum of Geometric series........
1. ## Sum of Geometric series........
Sum to n groups the following series
$(x+y) + (x^2+xy+y^2) + {x^3+(x^2)*y+x(y^2)+y^3}$
Need help fast..............Thanks a lot in advance.
2. Sum to n groups the following series
$(x+y) + (x^2+xy+y^2) + (x^3+(x^2)*y+x(y^2)+y^3)$
Need help fast..............Thanks a lot in advance.[/QUOTE]
Sorry..... I forgot to give brackets in the 3rd group........
3. And what have you done or tried so far?
4. Originally Posted by Arka
Sum to n groups the following series
$(x+y) + (x^2+xy+y^2) + (x^3+(x^2)*y+x(y^2)+y^3)$
Need help fast..............Thanks a lot in advance.
sure you didn't mean ...
$(x+y) + (x^2 + 2xy + y^2) + (x^3 + 3x^2y + 3xy^2 + y^3) + ...$
???
5. No I did not mean that.........
I tried till
$(x+y) + (x(x+y) + y^2) + (x(x^2+xy+y^2) + y^3) + .......$
Now, from here if I get the sum of the nth group , I can easily find the sum to n groups.............but that's what I'm having trouble finding.......
6. Originally Posted by skeeter
sure you didn't mean ...
$(x+y) + (x^2 + 2xy + y^2) + (x^3 + 3x^2y + 3xy^2 + y^3) + ...$???
Originally Posted by Arka
No I did not mean that.........
Then that is not a geometric series.
7. NO.....It's not......but see....
$(x+y) + (x(x+y) + y^2) + (x^2(x+y) + y^2(x+y)) +......$
I definitely see a pattern up there... and I belive it is possible to find the sum of the nth group and then the sum to n groups.........but I can't figure out how to do it........
And the chapter in which this sum is given is called "Geometrical Progression".......so I believe that even if the series as a whole is not in G.P. some part of it must be in G.P.
8. $x^{4}-y^{4} = (x-y)(x+y)(x^{2}+y^{2})$
therefore, Series reduces to,
$(x^{2}-y^{2})/(x-y) + (x^{3}-y^{3})/(x-y) + (x^{4}-y^{4})/(x-y)+....$
Simplify further, you'll obtain something resembling a geometric series
Obviously this is under the assumption that x and y are not equal.
9. How do I simplify further.......?
10. See the common denominator,
u have, $(1/(x-y))((x^{2}+x^{3}+......)-(y^{2}+y^{3}+......))$ upto n terms
so there are two geometric progressions in x and y. Now sum each upto n terms
11. Hello, Arka!
Sum to $\,n$ groups the following series:
. . $S_n \;=\;(x+y) + (x^2+xy+y^2) + (x^3 + x^2y+xy^2+y^3) + \hdots$
Multiply each group by $\frac{x-y}{x-y}$
. . $\begin{array}{ccccccc}
\dfrac{x+y}{1}\cdot\dfrac{x-y}{x-y} &=& \dfrac{x^2-y^2}{x-y} \\ \\[-3mm]
\dfrac{x^2+xy+y^2}{1}\cdot\dfrac{x-y}{x-y} &=& \dfrac{x^3-y^3}{x-y} \\ \\[-3mm]
\dfrac{x^3+x^2y+xy^2+y^3}{1}\cdot\dfrac{x-y}{x-y} &=& \dfrac{x^4 - y^4}{x-y} \\ \\[-3mm]
\vdots && \vdots \\
\dfrac{x^n + x^{n-1}y + \hdots + xu^{n-1} + y^n}{1}\cdot\dfrac{x-y}{x-y} &=& \dfrac{x^{n+1}-y^{n+1}}{x-y} \end{array}$
$\text{Hence: }\;S_n \;=\;\dfrac{(x^2+x^3+x^4 + \hdots + x^{n+1}) - (y^2 + y^3 + y^4 + \hdots + y^{n+1})}{x-y}$
$\text{The numerator has two geometric series:}$
. . $S_x \:=\:x^2\cdot\dfrac{1-x^n}{1-x}\;\text{ and }\;S_y \;=\;y^2\cdot\dfrac{1-y^n}{1-y}$
$\text{Therefore: }\;S_n \;=\;\dfrac{x^2\cdot\frac{1-x^n}{1-x} - y^2\cdot\frac{1-y^n}{1-y}}{x-y}$ | 2016-12-06T11:24:07 | {
"domain": "mathhelpforum.com",
"url": "http://mathhelpforum.com/algebra/161525-sum-geometric-series.html",
"openwebmath_score": 0.859839916229248,
"openwebmath_perplexity": 1555.7051033894008,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9755769092358048,
"lm_q2_score": 0.8539127585282744,
"lm_q1q2_score": 0.833057569722034
} |
https://math.stackexchange.com/questions/3678889/proving-riemann-integrability-for-piecewise-function | # Proving Riemann integrability for piecewise function
Question: Prove the function $$f:[0,1] \to \mathbb R$$ given by $$f(x) = \begin{cases} 1, & \text{if x=\frac{1}{n} for any positive integer n} \\ 0, & \text{otherwise} \end{cases}$$ is Riemann integrable.
My attempt:
To prove this, I need to show that there exists a partition $$P$$ such that the difference between the upper and lower Darboux sums (denoted $$\mathcal U(P,f)$$ and $$\mathcal L(P,f)$$ respectively) is less than any $$\epsilon>0$$.
I have the following formulae:
• $$\mathcal U(P,f)=\sum^n_{i=1} M_i\Delta x_i$$
• $$\mathcal L(P,f)=\sum^n_{i=1} m_i\Delta x_i$$
• $$m_i= \inf\{f(x):x_{i-1} \le x \le x_i\}$$
• $$M_i=\sup\{f(x):x_{i-1} \le x \le x_i\}$$
• $$\Delta x_i= x_i - x_{i-1}$$
However, to begin this computations I need a partition $$P$$. How should one go about determining this?
Any help would be greatly appreciated.
• Try to see how many of the points $1/n$ lie outside the interval $[0,\epsilon /2]$. Say at most there are $k$ such points $x_1,x_2,\dots, x_k=1$. Now your partition $P$ should include points $0,\epsilon/2, 1$ and points around $x_i$ which are $\epsilon/2k$ apart. – Paramanand Singh May 17 '20 at 10:45
• @ParamanandSingh so are you saying $P = \{0, \frac{\epsilon}{2}, 1\}$? – Viv4660 May 17 '20 at 13:41
• Read again. The partition include these three points and more points around $x_i$. Something like $0,\epsilon /2,y_1,x_1,y_2,y_3,x_2,y_4,y_5,x_3,y_6,\dots,y_{2k-3},x_{k-1},y_{2k-2},y_{2k-1},1$. The thing to note is that $y_{2i}-y_{2i-1}<\epsilon/2k$. – Paramanand Singh May 17 '20 at 14:18
• Thus each $x_i$ lies between two $y_j$ and these two $y$'s differ by at most $\epsilon/2k$. I don't want to write a full answer because that takes away the joy from you. Try to understand the reason why this kind of partition would work and write an answer yourself. Do let me know if you need more clarity. The question is not difficult. – Paramanand Singh May 17 '20 at 14:21
• So this could be $P=\{ 0, \frac{\epsilon}{2}, \frac{\epsilon}{2}+\frac{\epsilon}{2k}, 1\}$ where $\epsilon > 0$ and $k \in \mathbb Z^{>0}$? – Viv4660 May 17 '20 at 14:23
pick arbitrary $$x>0$$ we know that there is $$n \in \mathbb{N}$$ s.t $$\frac{1}{n}< x$$ because $$\lim_{n \rightarrow \infty} \frac{1}{n} = 0$$ . Now we know that there are atmost finite number of discontinuities on $$[\frac{1}{n}, 1]$$ hence your function is integrable on $$[\frac{1}{n},1]$$ Since this holds for arbitrary $$x$$ we can say that function is integrable on interval $$[x,1]$$ for any $$x>0$$.Hence it is Riemann Integrable on $$[0,1]$$ using following theorem.
$$f : [a, b] \rightarrow \mathbb{R}$$ is bounded, and $$f$$ is integrable on $$[c, b]$$ for all $$c \in$$ $$(a, b)$$, then f is integrable on $$[a, b]$$.
infact you can calculate value integral of function by proving following
$$\int_{0}^1 f = \lim_{n \rightarrow \infty} \sum_{k=1}^{n} \int_{\frac{1}{k+1}}^{\frac{1}{k}} f = 0$$
can you prove the threorem and above statement?
• I'm not sure I can prove the theorem. Intuitively, it makes sense though since the "for all $c \in (a,b)$" part pretty much includes $c=a$. Do you know where I can find a proof of this theorem? – Viv4660 May 17 '20 at 14:08
I guess you want a elementary detailed proof.
Fix $$\epsilon>0$$, there exists $$n_0\in\Bbb N$$ such that $$\frac{1}{n_0}<\epsilon/2$$. Then we can split $$[0,1]$$ into $$[0,\frac{1}{n_0}]$$ and $$[\frac{1}{n_0},1]$$. Notice that $$[\frac{1}{n_0},1]$$ contains only finitely many $$\frac{1}{n}$$, therefore $$f$$ has only finitely many discontinuous points on $$[\frac{1}{n_0},1]$$. Thus $$f$$ is integrable on $$[\frac{1}{n_0},1]$$(why?). Then we can choose a partition $$P:x_0,x_1,...x_N$$ of $$[\frac{1}{n_0},1]$$ such $$U(P,f)-L(P,f)<\epsilon/2$$.
We consider $$Q=P\cup\{0\}$$. Let $$Q:y_0,...y_{N+1}$$.
Notice that $$y_1=x_0,y_2=x_1,...y_{N+1}=x_N$$.
Thus
$$U(Q,f)-L(Q,f)$$
$$=(M_1-m_1)\Delta y_1+\sum_{i=2}^{N+1}(M_i-m_i)\Delta y_i$$
$$=(M_1-m_1)\Delta y_1+\sum_{i=1}^{N}(M_i-m_i)\Delta x_i$$
$$<(1-0)(\frac{1}{n_0}-0)+U(P,f)-L(P,f)$$
$$=\frac{1}{n_0}+\epsilon/2$$
$$<\epsilon/2+\epsilon/2$$
• I have a few questions: (1) Why does $[\frac{1}{n_0}, 1]$ contain only finitely many $\frac{1}{n}$? (2) For the partition $P$, what is $x_N$? Is this $1$? (3) How did you get the difference of the Darboux sums for $P$ being $< \frac{\epsilon}{2}$? – Viv4660 May 17 '20 at 14:05
• (1): if $\frac{1}{n}\in[\frac{1}{n_0},q]$, then $n<=n_0$ – user743633 May 17 '20 at 15:55
• (2): $x_N$ is 1, since $P$ is defined to be a partition of $[0,1]$ – user743633 May 17 '20 at 15:57
• (3) since $f$ contains only finitely many discontinuous points on $[\frac{1}{n_0},1]$, we have $f$ is integrable on $[\frac{1}{n_0},1]$, therefore, by definition, we can choose a partition $P$ such that $U(P,f)-L(P,f)<\epsilon/2$ (does this answer your question? Do you know how to prove "discontinuous on only finitely many points implies integrability"?) – user743633 May 17 '20 at 15:59 | 2021-05-07T05:43:37 | {
"domain": "stackexchange.com",
"url": "https://math.stackexchange.com/questions/3678889/proving-riemann-integrability-for-piecewise-function",
"openwebmath_score": 0.9145681262016296,
"openwebmath_perplexity": 152.58396415810975,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.975576912786245,
"lm_q2_score": 0.8539127548105611,
"lm_q1q2_score": 0.833057569126885
} |
http://math.stackexchange.com/questions/61981/calculating-the-limit-lim-limits-x-rightarrow-0-frac-cos5x-ax-bx | # Calculating the Limit $\lim\limits_{x\rightarrow 0} \frac{\cos^5{x} + ax + b}{x^2}$
I am studying for an entrance exam and would like somebody to confirm my answer or point out mistakes I made. Answers are greatly appreciated!
Find a and b so that the following Limit exists. $$L = \lim_{x\rightarrow 0} \frac{\cos^5{x} + ax + b}{x^2}$$
My solution approach was using l'Hôpital's rule so I set a to 0 and b to -1. -1 cancels the 1 from cos 0 so I get 0/0 then I can use l'Hôpital's rule. Having a = 0 I can use it again.
Is this approach right?
-
Sure. First set $b=-1$. Use L'Hospital's Rule. Then set $a=0$. End up with $\lim_{x\to 0}\frac{-5\cos^4 x\sin x}{2x}$. Limit of this could be done with L'Hospital's Rule, but shouldn't be, since $\lim_{x\to 0}\frac{\sin x}{x}=1$. – André Nicolas Sep 5 '11 at 9:34
@André: You're missing a minus sign from the cosine derivative there. – joriki Sep 5 '11 at 9:37
@joriki: Thanks. Middle of the night. – André Nicolas Sep 5 '11 at 9:38
@joriki.. But the problem asks to find all pairs $(a,b)$ such that the limit exists and if you start saying: "ok let's put $b=-1$..", you are already considering a particular case. How would you proceed in order to eliminate all possible pairs except for $(a,b)=(0,-1)$? – uforoboa Sep 5 '11 at 10:09
The OP perhaps described the process non-optimally, in terms of a strategy based on L'H. Rule. It really should have been something like this. If $b \ne -1$, the thing blows up near $0$. So $b=-1$ is the only possibility worth chasing. Use L'H. Rule. In the expression we obtain, if $a \ne 0$, we get blow-up. So $a=0$ is the only thing worth chasing. – André Nicolas Sep 5 '11 at 10:24
Your approach is correct. Since $\lim_{x\rightarrow 0}\left( \cos ^{5}x+ax+b\right) =1+b$, for the limit $$L=\lim_{x\rightarrow 0}\frac{\cos ^{5}x+ax+b}{x^{2}}$$ to exist, $1+b$ must be $0$, which means $b=-1$. And since $$\lim_{x\rightarrow 0}\left( \frac{d}{dx}\left( \cos ^{5}x+ax-1\right) \right) =a,$$ for L to exist, $a$ must be $0$. The limit is
$$L=\lim_{x\rightarrow 0}\frac{\frac{d}{dx}\left( \cos ^{5}x-1\right) }{\frac{d }{dx}\left( x^{2}\right) }=\lim_{x\rightarrow 0}-\frac{5}{2}\left( \cos ^{4}x\right) \frac{\sin x}{x}=-\frac{5}{2}.$$
-
If you can use the Maclaurin series for $\cos(x)=1-\frac{1}{2}x^2+O(x^4)$, then try using the binomial theorem to get the first two non-zero terms for $\cos^5(x)$.
-
@Thijs: It would seem that for an entrance exam, one should be able to use whatever they know. So I guess I should have said, "If you know the Maclaurin series..." – robjohn Sep 5 '11 at 13:48
Your approach is right, however in an exam i would reason as follows. The problem in your limit is the $x^2$ at the denominator. As long as you can factor it out from the fraction everything is allright. So try to write $$\cos(x)=1-\frac{x^2}{2}+o(x^2)$$ and you can easily see that $$\cos^5(x)=1+p(x)$$ where $p(x)$ is an infinite converging sum of monomials of degree at least $2$. Having noticed this, there is no way for your limit to exists unless $a=0$ and $b=-1$. This part of the reasinonig shows that, if you want your limit to exists, then necessarily $a=0,\: b=-1$. On the other hand, if $a=0,\: b=-1$, then a simple evaluation shows that $$\lim_{x\to 0}\frac{\cos^5(x)-1}{x^2}=\lim_{x\to 0}-\frac{5\cos^4(x)\sin(x)}{2x}=\lim_{x\to 0}-\frac{5\cos^4(x)}{2}\cdot\frac{\sin(x)}{x}=-\frac{5}{2}.$$ Hope everything is clear.
-
Robjohn was quicker... – uforoboa Sep 5 '11 at 9:54
Generally, "speed" should not play a role in accepting answers. Rather, one should strive to accept the answer that one thinks will help the most readers. For example, if one answer serves a fish on a platter, but another answer teaches one how to fish, then generally one should accept the latter (this remark is not specific to this thread - it's a general remark). – Bill Dubuque Sep 5 '11 at 16:44
HINT $\$ If as $\rm\ x\to 0:\:$ $\rm\ f(x)\to f_0,\ \ f{\:\:'}(x)\to f_1,\ \ f(x)/x^2\to f_2\$ for $\rm\:f_i \in\mathbb R\$ then $\rm\ f_0 = 0 = f_1\:.$
Proof $\$ If $\rm\ f_0\ne 0\$ then $\rm\:f(x)/x^2\to\: f_0/0^+ = \infty\not\in\mathbb R\:.\:$ Thus $\rm\:f_0 = 0\:.\:$ Similarly $\rm\:f_1 = 0\:,\:\:$ else
$$\rm f_1 \ne 0\:\ \ \Rightarrow\ \ f_2\: =\ \lim_{x\:\to\: 0}\ \frac{f(x)}{x^2}\ =\ \lim_{x\: \to\: 0}\ \frac{\frac{f(x)-f(0)}{x}}{x}\ \to\ \frac{f_1}{0}\ =\ \pm \infty\ \not\in\: \mathbb R$$ $\$
So for $\rm\ f(x)\: =\: cos^5(x)+a\:x+b\:,\ \ f(0) = 0\:\Rightarrow\: b=-1\:,\:$ and $\rm\ f{\:\:'}(x)\: =\: -5\ cos^4(x)\ sin(x)+a\$ hence $\rm\: f{\:\:'}(0) = 0\:\Rightarrow\:a = 0\:.$
REMARK $\$ If you know about Taylor series then it should be clear that the above amounts to computing a Taylor series approximant.
-
Something without l'Hospital. It is clear that $b=-1$ as mentioned above.
$$L = \lim_{x\rightarrow 0} \frac{\cos^5{x} + ax -1}{x^2}=\lim_{x \to 0}\left( \frac{\cos^5 x-1}{x^2}+\frac{a}{x} \right)$$
But
$$\lim_{x \to 0} \frac{\cos^5 x-1}{x^2}=\lim_{x \to 0} \frac{\cos x-1}{x^2}(1+\cos x+...+\cos^4 x)=\frac{5}{4} \lim_{x \to 0} \frac{-2\sin^2 \frac{x}{2}}{\frac{x}{2}^2}=-\frac{5}{2}$$
Therefore, if $a \neq 0$ the limit does not exist.
- | 2016-07-27T07:59:47 | {
"domain": "stackexchange.com",
"url": "http://math.stackexchange.com/questions/61981/calculating-the-limit-lim-limits-x-rightarrow-0-frac-cos5x-ax-bx",
"openwebmath_score": 0.956944465637207,
"openwebmath_perplexity": 288.8882262881785,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.975576912431201,
"lm_q2_score": 0.8539127548105611,
"lm_q1q2_score": 0.8330575688237084
} |
https://math.stackexchange.com/questions/2255297/coin-toss-probability-of-getting-4-heads | # (Coin Toss) Probability of Getting 4 Heads
I am stuck on a problem for my stats class. I have not done any work but I will explain why. Here is the problem;
• A coin is flipped until you get a tails. What is the probability of getting at least 4 heads?
I have done probability with coins before, but this question stumped me. How? Because we only have ONE coin, and we don't know how many times the coin is tossed. I know that with one coin, the probability of getting a head is 1. And the number of outcomes is 2. However, I don't know the next step after this, especially when I'm not given information on how many times the coin should be tossed.
Any help would be great. Or maybe just a tip on looking at this problem from a different perspective? Thank you!
• Can you work out the probability that you’ll get tails in four tosses or fewer? – amd Apr 27 '17 at 20:23
• But there is information: the coin is flipped until you get tails – drhab Apr 27 '17 at 20:25
• The coin will be tossed until you get a tails, then the number of times you got a heads will be counted. What is the chance that you got 4 heads? you can get one heads then another then another, you will only stop when you get tails or get 4 heads (when you get 4 heads you don't care about what happens if you keep flipping the coin). That means the probability of getting at least 4 heads is the probability of getting 4 heads consecutively from the beginning, that is 1/16. – Donat Pants Apr 27 '17 at 20:26
I've also started statistics as well.
How I look into this question is the other way around:
Rather than looking for 4 in a row, I look at the probability of not having 4 heads in a row (having the compliment of the probability).
Let say P(A) = Having at least 4 heads before first tail
$P(A') = P(T)+P(H\cap T)+P(H\cap H \cap T)+P(H \cap H \cap H \cap T)$
$P(A')+P(A) = 1 \rightarrow P(A) = 1-P(A')$
$1-P(T)+P(H\cap T)+P(H\cap H \cap T)+P(H \cap H \cap H \cap T)$
$1-[1/2 + 1/2^2 + 1/2^3 +1/2^4] = 1-15/16 = 1/16$
You want the probability of flipping at least four heads before obtaining the first tail.
Thus you want the probability that at least the first four tosses are heads.
This is $1/2^4$.
The probability of getting a heads first is 1/2. The probability of getting 2 heads in a row is 1/2 of that, or 1/4. The probability of getting 3 heads in a row is 1/2 of that, or 1/8. The probability of getting 4 heads in a row is 1/2 of that, or 1/16.
After that... it doesn't matter... you have at least 4 heads.
• This works for $a=1$, but not for other values. – Trurl Apr 27 '17 at 21:06
• Good point... I took the phrase "a tails" to mean one tails. Pretty sure that was the intention of the OP, but your version is more interesting. – Jed Apr 27 '17 at 21:23
There are several possible approaches.
Getting 4 heads to start has probability $(1/2)^4 = 1/16$ as in the comment by @DonatPants.
More formally, outcomes that satisfy your condition are HHHHT, HHHHHT, HHHHHHT, etc. So the total probability is the geometric series with probability $A = (1/2)^5 + (1/2)^6 + (1/2)^7 + \dots .$
There is a formula for summing this series. If you don't know it, you can note that $(1/2)A = (1/2)^6 + 1/2)^7 + \dots,$ so that $A - (1/2)A = = (1/2)A = (1/2)^5$ and $A = (1/2)^4,$ which is the same as the previous answer.
I do not know why @MarkusStuhr has withdrawn his Answer. The explanation by @Joel (+1) as does the answer by @manmood (+1) that appeared while was typing this. I hope one of these explanations is clear to you. The key points throughout is that we're assuming the coin is fair [$P(H) = 1/2$] and that the tosses are independent.
Also, if your book includes the geometric distribution, you should look at that because it is related to this problem. I don't want to discuss the geometric distribution in this Answer because there are at least two versions of it, and discussing the wrong one might be confusing.
Well, h=heads, t=tails Sample space={(hhhh),(hhht),(hhth),(hhtt),(hthh),(htht),(htth),(httt), (thhh),(thht),(thth),(thtt),(tthh),(ttht),(ttth),(tttt)} There are 16 total outcomes, and only 1 of these outcomes results in 4 heads. This means the probability of landing all 4 heads in 4 tosses is 1 out of the 16. So the answer is 1/16. As for the "before flipping a tail", it doesn't seem to matter because no matter the outcome there is still only 1 in 16 chances to get 4 heads flipped. | 2019-10-18T18:29:47 | {
"domain": "stackexchange.com",
"url": "https://math.stackexchange.com/questions/2255297/coin-toss-probability-of-getting-4-heads",
"openwebmath_score": 0.8152938485145569,
"openwebmath_perplexity": 201.7704946153579,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.975576913496333,
"lm_q2_score": 0.8539127529517043,
"lm_q1q2_score": 0.8330575679197805
} |
https://math.stackexchange.com/questions/73368/a-continuously-differentiable-map-is-locally-lipschitz | # A continuously differentiable map is locally Lipschitz
Let $$f:\mathbb R^d \to \mathbb R^m$$ be a map of class $$C^1$$. That is, $$f$$ is continuous and its derivative exists and is also continuous. Why is $$f$$ locally Lipschitz?
### Remark
Such $$f$$ will not be globally Lipschitz in general, as the one-dimensional example $$f(x)=x^2$$ shows: for this example, $$|f(x+1)-f(x)| = |2x+1|$$ is unbounded.
• Well, $x \mapsto x^2$ is not particularly Lipschitz, is it? You seem to miss either a locally or that $df$ must be bounded.
– t.b.
Oct 17 '11 at 15:37
• @t.b. In some contexts $C^1$ implies that $\sup |f| + \sup |df| < \infty$. (At least, when I write $C^1(M,\mathbb{R}^k)$ with $M$ being non-compact, that's what I would mean.) Of course, this is not what the OP wrote in the parenthetical. Oct 17 '11 at 15:41
• yes you're right. I meant locally Lipschitz.
– bass
Oct 17 '11 at 15:41
• @bass: use that continuous functions are locally bounded. Oct 17 '11 at 15:43
• First reduce to the scalar-valued case (this is easy). Then note that $\left|f\left(x\right)-f\left(y\right)\right|=\left|\int_{0}^{1}\frac{d}{dt}f\left(tx+\left(1-t\right)y\right)dt\right|$. Then use the chain rule and...
– Mark
Oct 17 '11 at 17:20
If $f:\Omega\to{\mathbb R}^m$ is continuously differentiable on the open set $\Omega\subset{\mathbb R}^d$, then for each point $p\in\Omega$ there is a convex neighborhood $U$ of $p$ such that all partial derivatives $f_{i.k}:={\partial f_i\over \partial x_k}$ are bounded by some constant $M>0$ in $U$. Using Schwarz' inequality one then easily proves that $$\|df(x)\|\ \leq\sqrt{dm}\>M=:L$$ for all $x\in U$. Now let $a$, $b$ be two arbitrary points in $U$ and consider the auxiliary function $$\phi(t):=f\bigl(a+t(b-a)\bigr)\qquad(0\leq t\leq1)$$ which computes the values of $f$ along the segment connecting $a$ and $b$. By means of the chain rule we obtain $$f(b)-f(a)=\phi(1)-\phi(0)=\int_0^1\phi'(t)\>dt=\int_0^1df\bigl(a+t(b-a)\bigr).(b-a)\>dt\ .$$ Since all points $a+t(b-a)$ lie in $U$ one has $$\bigl|df\bigl(a+t(b-a)\bigr).(b-a)\bigr|\leq L\>|b-a|\qquad(0\leq t\leq1)\>;$$ therefore we get $$|f(b)-f(a)|\leq L\>|b-a|\ .$$ This proves that $f$ is Lipschitz-continuous in $U$ with Lipschitz constant $L$.
• I have a little doubt, isn't $df$ with respect to $t$ , then how can we justify the inequality ? Nov 18 '13 at 10:05
• @Theorem: $df\bigl(a+t(b-a)\bigr)$ is the derivative ("Jacobian") of $f$, evaluated at the point $a+t(b-a)\in U$. Nov 18 '13 at 10:14
• @kam:Consider the function $f(x):=x^2\sin(1/x^3)$. With $f(0):=0$ it is differentiable on all of ${\mathbb R}$, but it is not Lipschitz continuous near $x=0$. Mar 8 '20 at 9:55
Maybe this can help. The Lipschitz condition comes many times from the Mean Value Theorem. Search the link for the multivariable case. The fact that $f$ is $C^1$ helps you to see that when restricted to a compact set the differential is bounded. That's why you only have local Lipschitz condition.
A function is called locally Lipschitz continuous if for every x in X there exists a neighborhood U of x such that f restricted to U is Lipschitz continuous. Equivalently, if X is a locally compact metric space, then f is locally Lipschitz if and only if it is Lipschitz continuous on every compact subset of X. In spaces that are not locally compact, this is a necessary but not a sufficient condition.The function $f(x) = x^2$ with domain all real numbers is not Lipschitz continuous. This function becomes arbitrarily steep as x approaches infinity. It is however locally Lipschitz continuous. | 2022-01-27T20:05:14 | {
"domain": "stackexchange.com",
"url": "https://math.stackexchange.com/questions/73368/a-continuously-differentiable-map-is-locally-lipschitz",
"openwebmath_score": 0.9598161578178406,
"openwebmath_perplexity": 113.99504942963681,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9755769099458927,
"lm_q2_score": 0.8539127548105611,
"lm_q1q2_score": 0.8330575667014719
} |
https://nbviewer.jupyter.org/github/gokererdogan/Notebooks/blob/master/Reparameterization%20Trick.ipynb | # Reparameterization Trick¶
Here we will understand the reparameterization trick used by Kingma and Welling (2014) to train their variational autoencoder.
Assume we have a normal distribution $q$ that is parameterized by $\theta$, specifically $q_{\theta}(x) = N(\theta,1)$. We want to solve the below problem $$\text{min}_{\theta} \quad E_q[x^2]$$ This is of course a rather silly problem and the optimal $\theta$ is obvious. We want to understand how the reparameterization trick helps in calculating the gradient of this objective $E_q[x^2]$.
One way to calculate $\nabla_{\theta} E_q[x^2]$ is as follows $$\nabla_{\theta} E_q[x^2] = \nabla_{\theta} \int q_{\theta}(x) x^2 dx = \int x^2 \nabla_{\theta} q_{\theta}(x) \frac{q_{\theta}(x)}{q_{\theta}(x)} dx = \int q_{\theta}(x) \nabla_{\theta} \log q_{\theta}(x) x^2 dx = E_q[x^2 \nabla_{\theta} \log q_{\theta}(x)]$$
For our example where $q_{\theta}(x) = N(\theta,1)$, this method gives $$\nabla_{\theta} E_q[x^2] = E_q[x^2 (x-\theta)]$$
Reparameterization trick is a way to rewrite the expectation so that the distribution with respect to which we take the expectation is independent of parameter $\theta$. To achieve this, we need to make the stochastic element in $q$ independent of $\theta$. Hence, we write $x$ as $$x = \theta + \epsilon, \quad \epsilon \sim N(0,1)$$ Then, we can write $$E_q[x^2] = E_p[(\theta+\epsilon)^2]$$ where $p$ is the distribution of $\epsilon$, i.e., $N(0,1)$. Now we can write the derivative of $E_q[x^2]$ as follows $$\nabla_{\theta} E_q[x^2] = \nabla_{\theta} E_p[(\theta+\epsilon)^2] = E_p[2(\theta+\epsilon)]$$
Now let us compare the variances of the two methods; we are hoping to see that the first method has high variance while reparameterization trick decreases the variance substantially.
In [64]:
import numpy as np
N = 1000
theta = 2.0
eps = np.random.randn(N)
x = theta + eps
grad1 = lambda x: np.sum(np.square(x)*(x-theta)) / x.size
grad2 = lambda eps: np.sum(2*(theta + eps)) / x.size
3.86872102149
4.03506045463
Let us plot the variance for different sample sizes.
In [66]:
Ns = [10, 100, 1000, 10000, 100000]
reps = 100
means1 = np.zeros(len(Ns))
vars1 = np.zeros(len(Ns))
means2 = np.zeros(len(Ns))
vars2 = np.zeros(len(Ns))
est1 = np.zeros(reps)
est2 = np.zeros(reps)
for i, N in enumerate(Ns):
for r in range(reps):
x = np.random.randn(N) + theta
eps = np.random.randn(N)
means1[i] = np.mean(est1)
means2[i] = np.mean(est2)
vars1[i] = np.var(est1)
vars2[i] = np.var(est2)
print means1
print means2
print
print vars1
print vars2
[ 4.10377908 4.07894165 3.97133622 4.00847457 3.99620013]
[ 3.95374031 4.0025519 3.99285189 4.00065614 4.00154934]
[ 8.63411090e+00 8.90650401e-01 8.94014392e-02 8.95798809e-03
1.09726802e-03]
[ 3.70336929e-01 4.60841910e-02 3.59508788e-03 3.94404543e-04
3.97245142e-05]
In [67]:
%matplotlib inline
import matplotlib.pyplot as plt
plt.plot(vars1)
plt.plot(vars2)
plt.legend(['no rt', 'rt'])
/usr/local/lib/python2.7/dist-packages/matplotlib/__init__.py:872: UserWarning: axes.color_cycle is deprecated and replaced with axes.prop_cycle; please use the latter.
warnings.warn(self.msg_depr % (key, alt_key))
Out[67]:
<matplotlib.legend.Legend at 0x7facb844ae50>
Variance of the estimates using reparameterization trick is one order of magnitude smaller than the estimates from the first method! | 2020-09-19T00:40:36 | {
"domain": "jupyter.org",
"url": "https://nbviewer.jupyter.org/github/gokererdogan/Notebooks/blob/master/Reparameterization%20Trick.ipynb",
"openwebmath_score": 0.8423262238502502,
"openwebmath_perplexity": 959.889406610184,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9755769099458927,
"lm_q2_score": 0.8539127529517044,
"lm_q1q2_score": 0.8330575648880143
} |
http://math.stackexchange.com/questions/205769/simplify-a-factorial | # Simplify a factorial
I have the problem to evaluate the following:
$$(2n)!\over 2^n(n!)$$
Does this reduce to anything in particular?
I stuck it into a computer and it's
1: 1
2: 3
3: 15
4: 105
5: 945
6: 10395
No pattern immediately apparent.
-
Have you noticed that each term divides the next term? – Qiaochu Yuan Oct 2 '12 at 2:32
Were you computing the possible arrangements of a commutative, non-associative, operator over n terms, by any chance? – Philippe Oct 2 '12 at 9:34
@Philippe No, it's just an exercise from a book. – Luigi Plinge Oct 2 '12 at 22:09
@LuigiPlinge OK.. My colleague's whiteboard has the same sequence written all over it, complete with lots of little hand-drawn trees, that's why I was asking :) Also, for such things the On-Line Encyclopedia of Integer Sequences is a great resource. – Philippe Oct 3 '12 at 8:02
Since $(2n)! = (2n) \times (2n-1) \times \cdots \times 2 \times 1$. Split the product into products of even factors and odd factors: $$(2n)! = \prod_{m=1}^{n} (2m) \cdot \prod_{m=1}^{n} (2m-1) = 2^n \prod_{m=1}^n m \cdot \prod_{m=1}^{n} (2m-1) = 2^n n! \prod_{m=1}^{n} (2m-1)$$ Therefore: $$\frac{(2n)!}{2^n n!} = \prod_{m=1}^n (2m-1) = (2n-1)!!$$ where $m!!$ denotes double factorial.
-
Great, the book doesn't mention double factorials but this must be what they were alluding to. Thanks! – Luigi Plinge Oct 2 '12 at 3:02
You won’t get a nice closed form, but there is another way to write it that is sometimes useful. Notice that
\begin{align*}2^nn!&=\underbrace{2\cdot2\cdot2\cdot\ldots\cdot2}_n\cdot1\cdot2\cdot3\cdot\ldots\cdot n\\&=(2\cdot1)(2\cdot2)(2\cdot3)\dots(2\cdot n)\\&=2\cdot4\cdot6\cdot\ldots\cdot 2n\;,\end{align*}
do some cancelling, and look at Qiaochu’s comment.
-
You can use the identity
$$(2n)! = \Gamma(2n+1) = {\frac {{2}^{2n} \Gamma \left( n + 1\right) \Gamma \left( n + \frac{1}{2} \right) }{\sqrt {\pi }}}\,.$$
$$\frac{(2n)!}{2^n n!} = \frac{ 2^n \Gamma(n+\frac{1}{2})}{\sqrt{\pi}} = \frac{ 2^n (n-\frac{1}{2})!}{\sqrt{\pi}}\,.$$ | 2014-07-25T14:17:57 | {
"domain": "stackexchange.com",
"url": "http://math.stackexchange.com/questions/205769/simplify-a-factorial",
"openwebmath_score": 0.85395348072052,
"openwebmath_perplexity": 1211.7499047069002,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9755769078156283,
"lm_q2_score": 0.8539127529517043,
"lm_q1q2_score": 0.8330575630689543
} |
http://laartagency.com/kate-and-jqzxi/0720ad-radius-of-circle-inscribed-in-a-right-angled-triangle | Many geometry problems involve a triangle inscribed in a circle, where the key to solving the problem is relying on the fact that each one of the inscribed triangle's angles is an inscribed angle in the circle. F, Area of a triangle - "side angle side" (SAS) method, Area of a triangle - "side and two angles" (AAS or ASA) method, Surface area of a regular truncated pyramid, All formulas for perimeter of geometric figures, All formulas for volume of geometric solids. In the figure at right, given circle k with centre O and the point P outside k, bisect OP at H and draw the circle of radius OH with centre H. OP is a diameter of this circle, so the triangles connecting OP to the points T and T′ where the circles intersect are both right triangles. Before proving this, we need to review some elementary geometry. It is given that ABC is a right angle triangle with AB = 6 cm and AC = 8 cm and a circle with centre O has been inscribed. Yes; If two vertices (of a triangle inscribed within a circle) are opposite each other, they lie on the diameter. First, form three smaller triangles within the triangle… The radius Of the inscribed circle represents the length of any line segment from its center to its perimeter, of the inscribed circle and is represented as r=sqrt((s-a)*(s-b)*(s-c)/s) or Radius Of Inscribed Circle=sqrt((Semiperimeter Of Triangle -Side A)*(Semiperimeter Of Triangle -Side B)*(Semiperimeter Of Triangle -Side C)/Semiperimeter Of Triangle ). Given the side lengths of the triangle, it is possible to determine the radius of the circle. cm. Triangle PQR is right angled at Q. QR=12cm, PQ=5cm A circle with centre O is inscribed in it. Can you please help me, I need to find the radius (r) of a circle which is inscribed inside an obtuse triangle ABC. All formulas for radius of a circumscribed circle. An equilateral triangle is inscribed in a circle. A circle is inscribed in it. The incircle or inscribed circle of a triangle is the largest circle contained in the triangle; it touches (is tangent to) the three sides. 2 twice the radius) of the unique circle in which $$\triangle\,ABC$$ can be inscribed, called the circumscribed circle of the triangle. In a right angle Δ ABC, BC = 12 cm and AB = 5 cm, Find the radius of the circle inscribed in this triangle. Fundamental Facts i7 circle inscribed in the triangle ABC lies on the given circle. Approach: Formula for calculating the inradius of a right angled triangle can be given as r = ( P + B – H ) / 2. Given: SOLUTION: Prove: An inscribed angle of a triangle intercepts a diameter or semicircle if and only if the angle is a right angle. It is given that ABC is a right angle triangle with AB = 6 cm and AC = 8 cm and a circle with centre O has been inscribed. With this, we have one side of a smaller triangle. Then Write an expression for the inscribed radius r in . ABC is a right triangle and r is the radius of the inscribed circle. askedOct 1, 2018in Mathematicsby Tannu(53.0kpoints) Calculate the Value of X, the Radius of the Inscribed Circle - Mathematics If AB=5 cm, BC=12 cm and < B=90*, then find the value of r. Triangle ΔABC is inscribed in a circle O, and side AB passes through the circle's center. ABC is a right angle triangle, right angled at A. Pythagorean Theorem: The circle is the curve for which the curvature is a constant: dφ/ds = 1. Problem. and is represented as r=b*sqrt (((2*a)-b)/ ((2*a)+b))/2 or Radius Of Inscribed Circle=Side B*sqrt (((2*Side A) … Hence the area of the incircle will be PI * ( (P + B – H) / 2)2. This problem involves two circles that are inscribed in a right triangle. Question from akshaya, a student: A circle with centre O and radius r is inscribed in a right angled triangle ABC. This problem looks at two circles that are inscribed in a right triangle and looks to find the radius of both circles. Find its radius. A circle is inscribed in a right angled triangle with the given dimensions. 1 8 isosceles triangle definition I. A website dedicated to the puzzling world of mathematics. Thus, in the diagram above, \lvert \overline {OD}\rvert=\lvert\overline {OE}\rvert=\lvert\overline {OF}\rvert=r, ∣OD∣ = ∣OE ∣ = ∣OF ∣ = r, (the circle touches all three sides of the triangle) I need to find r - the radius - which is starts on BC and goes up - up course the the radius creates two right angles on both sides of r. = = = = 3 cm. radius of a circle inscribed in a right triangle : = Digit 6 The radius … Remember that each side of the triangle is tangent to the circle, so if you draw a radius from the center of the circle to the point where the circle touches the edge of the triangle, the radius will form a right angle with the edge of the triangle. The radius of the circle is 21 in. Hence, the radius is half of that, i.e. Radius of the Incircle of a Triangle Brian Rogers August 4, 2003 The center of the incircle of a triangle is located at the intersection of the angle bisectors of the triangle. Let W and Z 5. All formulas for radius of a circle inscribed, All basic formulas of trigonometric identities, Height, Bisector and Median of an isosceles triangle, Height, Bisector and Median of an equilateral triangle, Angles between diagonals of a parallelogram, Height of a parallelogram and the angle of intersection of heights, The sum of the squared diagonals of a parallelogram, The length and the properties of a bisector of a parallelogram, Lateral sides and height of a right trapezoid, Calculate the radius of a inscribed circle of a right triangle if given legs and hypotenuse (. A circle of radius 3 cm is drawn inscribed in a right angle triangle ABC, right angled at C. If AC is 10 Find the value of CB * - 29943281 10 Over 600 Algebra Word Problems at edhelper.com, Tangent segments to a circle from a point outside the circle, A tangent line to a circle is perpendicular to the radius drawn to the tangent point, A circle, its chords, tangent and secant lines - the major definitions, The longer is the chord the larger its central angle is, The chords of a circle and the radii perpendicular to the chords, Two parallel secants to a circle cut off congruent arcs, The angle between two chords intersecting inside a circle, The angle between two secants intersecting outside a circle, The angle between a chord and a tangent line to a circle, The parts of chords that intersect inside a circle, Metric relations for secants intersecting outside a circle, Metric relations for a tangent and a secant lines released from a point outside a circle, HOW TO bisect an arc in a circle using a compass and a ruler, HOW TO find the center of a circle given by two chords, Solved problems on a radius and a tangent line to a circle, A property of the angles of a quadrilateral inscribed in a circle, An isosceles trapezoid can be inscribed in a circle, HOW TO construct a tangent line to a circle at a given point on the circle, HOW TO construct a tangent line to a circle through a given point outside the circle, HOW TO construct a common exterior tangent line to two circles, HOW TO construct a common interior tangent line to two circles, Solved problems on chords that intersect within a circle, Solved problems on secants that intersect outside a circle, Solved problems on a tangent and a secant lines released from a point outside a circle, Solved problems on tangent lines released from a point outside a circle, PROPERTIES OF CIRCLES, THEIR CHORDS, SECANTS AND TANGENTS. an isosceles right triangle is inscribed in a circle. Find the radius of the circle if one leg of the triangle is 8 cm.----- Any right-angled triangle inscribed into the circle has the diameter as the hypotenuse. A circle with centre O has been inscribed inside the triangle. Abc is a Right Angles Triangle with Ab = 12 Cm and Ac = 13 Cm. A triangle has 180˚, and therefore each angle must equal 60˚. Since ΔPQR is a right-angled angle, PR = sqrt(7^2 + 24^2) = sqrt(49 + 576) = sqrt625 = 25 cm Let the given inscribed circle touches the sides of the given triangle at points A, B and C respectively. The center of the incircle is a triangle center called the triangle's incenter.. An excircle or escribed circle of the triangle is a circle lying outside the triangle, tangent to one of its sides and tangent to the extensions of the other two. is a right angled triangle, right angled at such that and .A circle with centre is inscribed in .The radius of the circle is (a) 1cm (b) 2cm (c) 3cm (d) 4cm math. Problem. Let P be a point on AD such that angle … Angle Bisector: Circumscribed Circle Radius: Inscribed Circle Radius: Right Triangle: One angle is equal to 90 degrees. Calculate the value of r, the radius of the inscribed circle. Find the radius of the inscribed circle of this triangle, in the cases w = 5.00, w = 6.00, and w = 8.00. The center of the incircle is called the triangle’s incenter. Figure 2.5.1 Types of angles in a circle Answer. Calculate the radius of a inscribed circle of a right triangle if given legs and hypotenuse ( r ) : radius of a circle inscribed in a right triangle : = Digit 2 1 2 4 6 10 F The center point of the inscribed circle is … This formula was derived in the solution of the Problem 1 above. Pythagorean Theorem: a) Express r in terms of angle x and the length of the hypotenuse h. b) Assume that h is constant and x varies; find x for which r is maximum. Right Triangle Equations. Right triangle or right-angled triangle is a triangle in which one angle is a right angle (that is, a 90-degree angle). The radius of the inscribed circle is 3 cm. Now, use the formula for the radius of the circle inscribed into the right-angled triangle. Right Triangle Equations. The inscribed circle has a radius of 2, extending to the base of the triangle. 4 Find the circle's radius. Radius of the inscribed circle of an isosceles triangle is the length of the radius of the circle of a triangle is the largest circle contained in the triangle; it touches (is tangent to) the three sides. Solution to Problem: a) Let M, N and P be the points of tangency of the circle and the sides of the triangle. Therefore, in our case the diameter of the circle is = = cm. 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. Determine the side length of the triangle … This common ratio has a geometric meaning: it is the diameter (i.e. By the inscribed angle theorem, the angle opposite the arc determined by the diameter (whose measure is 180) has a measure of 90, making it a right triangle. Angle Bisector: Circumscribed Circle Radius: Inscribed Circle Radius: Right Triangle: One angle is equal to 90 degrees. a Circle, with Centre O, Has Been Inscribed Inside the Triangle. 2 And we know that the area of a circle is PI * r2 where PI = 22 / 7 and r is the radius of the circle. Since the triangle's three sides are all tangents to the inscribed circle, the distances from the circle's center to the three sides are all equal to the circle's radius. The length of two sides containing angle A is 12 cm and 5 cm find the radius. Problem 3 In rectangle ABCD, AB=8 and BC=20. Using Pythagoras theorem, we get BC 2 = AC 2 + AB 2 = (8) 2 + (6) 2 = 64 + 36 = 100 ⇒ BC = 10 cm Tangents at any point of a circle is perpendicular to the radius … Centre O, has been inscribed inside the triangle, right angled at a possible determine! Triangle ’ s incenter to 90 degrees the side lengths of the incircle will be PI * ( P. 12 cm and 5 cm find the radius of the incircle will PI... Radius r in possible to determine the radius of the problem 1 above, right angled triangle the... R in One side of a smaller triangle case the diameter ( i.e = 1 through the circle 's.., has been inscribed inside the triangle diameter ( i.e angle must equal 60˚ must equal.... Is possible to determine the radius of the incircle radius of circle inscribed in a right angled triangle be PI * ( ( +. And r is the diameter of the triangle therefore, in our case the diameter of problem. Right angled triangle with the given circle is possible to determine the of! Radius: inscribed circle triangle with the given dimensions triangle and looks find. Of two sides containing angle a is 12 cm and 5 cm find the is... And 5 cm find the radius is half of that, i.e a geometric meaning: it the. Review some elementary geometry called the triangle ’ s incenter extending to the puzzling world of mathematics a... To the puzzling world of mathematics, use the formula for the radius of the triangle 3! One side of a smaller triangle, right angled triangle with the given dimensions Bisector... The diameter of the inscribed circle the circle is the diameter of the.! The diameter ( i.e a circle with centre O has been inscribed the., right angled triangle with the given circle is half of that,.... Half of that, i.e and BC=20 curve for which the curvature is a right angle triangle right. Need to review some elementary geometry the solution of the problem 1 above AB=8 and BC=20 to determine radius! ( P + B – H ) / 2 ) 2 is 12 cm and 5 cm the... Containing angle a is 12 cm and 5 cm find the radius the! Right triangle: One angle is equal to 90 degrees P + B – )! With this, we have One side of a smaller triangle circle,... Ab passes through the circle inscribed into the right-angled triangle was derived in the triangle s. This problem looks at two circles that are inscribed in a circle with centre O, has been inside! O has been inscribed inside the triangle is inscribed in a right triangle and r the!: right triangle and r is the diameter of the circle is 3 cm involves two that! + B – H ) / 2 ) 2 this, we need to review some elementary geometry meaning... Of that, i.e on the given circle ΔABC is inscribed in right... Possible to determine the radius in our case the diameter of the circle is = =.. Pi * ( ( P + B – H ) / 2 ) 2 ( +. A geometric meaning: it is possible to determine the radius of the abc... We have One side of a smaller triangle fundamental Facts i7 circle inscribed into the right-angled.! The inscribed circle is = = cm hence, the radius … Bisector! Circle has a geometric meaning: it is possible to determine the radius of circle... Called the triangle abc lies on the given dimensions called the triangle two circles that are inscribed in the of. ) 2 … a triangle has 180˚, and therefore each angle must equal 60˚ = cm world of.! Two circles that are inscribed in the solution of the incircle will PI! This common ratio has a radius of the triangle circle with centre O has been inscribed inside the.. Need to review some elementary geometry and 5 cm find the radius of the circle 's center the of! * ( ( P + B – H ) / 2 ) 2 2 ) 2 ) 2 before this... That are inscribed in a circle, with centre O, and side AB passes through the circle the... Diameter of the circle is = = cm be PI * ( ( P + B H! Involves two circles that are inscribed in a right triangle: One angle is to., right angled triangle with the given circle this, we have One side of a smaller.! This common ratio has a radius of the triangle, it is the radius a constant: dφ/ds 1... ( ( P + B – H ) / 2 ) 2 ) / )! The base of the triangle is 12 cm and 5 cm find the radius at two circles that inscribed. 12 cm and 5 cm find the radius … angle Bisector: Circumscribed circle radius right. Triangle abc lies on the given circle the formula for the inscribed circle is half of,. H ) / 2 ) 2 of both circles, the radius of the incircle will PI... And side AB passes through the circle inscribed into the right-angled triangle curve. Cm find the radius … angle Bisector: Circumscribed circle radius: right triangle the side of. The given circle a geometric meaning: it is the curve for which the curvature is a right triangle... Radius is half of that, i.e … angle Bisector: Circumscribed circle:. Through the circle the triangle has 180˚, and therefore each angle must 60˚... Side AB passes through the circle is = = cm be PI * ( ( +. Expression for the radius is equal to 90 degrees length of the triangle at two that! Diameter ( i.e curvature is a right angle triangle, right angled triangle the... Smaller triangle the length of the inscribed circle radius: inscribed circle radius: right and. Has a geometric meaning: it is possible to determine the radius of the is... A geometric meaning: it is the curve for which the curvature a!: Circumscribed circle radius: right triangle: One angle is equal to 90 degrees the side lengths the! Angled at a website dedicated to the puzzling world of mathematics P + B – H ) / 2 2! Extending to the puzzling world of mathematics review some elementary geometry is = =.... 2, extending to the base of the triangle, right angled triangle with the given.... Circle, with centre O, and side AB passes through the circle is diameter..., AB=8 and BC=20 that are inscribed in the triangle … a triangle has,. Triangle abc lies on the given circle possible to determine the radius of the circle is inscribed a! This problem involves two circles that are inscribed in a right triangle and looks to find the of... This, we need to radius of circle inscribed in a right angled triangle some elementary geometry 180˚, and therefore each angle must equal.... = = cm PI * ( ( P + B – H ) / 2 2... Circle O, has been inscribed inside the triangle ’ s incenter right triangle! It is the curve for which the curvature is a constant: dφ/ds 1! Extending to the puzzling world of mathematics the circle extending to the puzzling world of mathematics +... A right triangle and radius of circle inscribed in a right angled triangle to find the radius circle has a radius of circle... Derived in the triangle abc lies on the given dimensions inscribed radius r in it is the (... Passes through the circle inscribed into the right-angled triangle i7 circle inscribed into the right-angled triangle inscribed into the triangle. Inside the triangle this common ratio has a geometric meaning: it is possible to determine the radius 2... Circle radius: inscribed circle is inscribed in a right triangle is inscribed in right... The base of the problem 1 above the area of the problem 1 above = 1 dedicated the... Was derived in the solution of the circle will be PI * (. A is 12 cm and 5 cm find the radius is half of that i.e! The given dimensions for the radius of the problem 1 above inscribed circle the length of the 's... 5 cm find the radius of the incircle will be PI * ( ( P + B – )... 90 degrees value of r, the radius of the problem 1.... Circle O, and therefore each angle must equal 60˚ a website dedicated to the base of the circle. Triangle, right angled triangle with the given circle Theorem: this ratio! Is equal to 90 degrees center of the incircle will be PI * ( P! 5 cm find the radius meaning: it is possible to determine the of! Possible to determine the radius of both circles AB passes through the circle 's.... Base of the incircle will be PI * ( ( P + B – H ) 2... Inscribed radius r in the formula for the radius of the circle inscribed in right!, we have One side of a smaller triangle equal to 90 degrees incircle is called the abc! Of two sides containing angle a is 12 cm and 5 cm find the radius of the circle degrees... O, and therefore each angle must equal 60˚ dedicated to the puzzling world of mathematics, is... This formula was derived in the solution of the inscribed circle radius inscribed... Triangle ’ s incenter of that, i.e One angle is equal to 90 degrees triangle: One angle equal. Circumscribed circle radius: inscribed circle radius: right triangle and looks to find the radius the! | 2022-12-02T13:55:42 | {
"domain": "laartagency.com",
"url": "http://laartagency.com/kate-and-jqzxi/0720ad-radius-of-circle-inscribed-in-a-right-angled-triangle",
"openwebmath_score": 0.733421266078949,
"openwebmath_perplexity": 308.6810973504068,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. Yes\n2. Yes",
"lm_q1_score": 0.9755769042651874,
"lm_q2_score": 0.8539127548105611,
"lm_q1q2_score": 0.8330575618506452
} |
https://math.stackexchange.com/questions/1874562/compare-two-coin-tossing-games | # Compare two coin tossing games
Compare the following two games:
1. You have a fair coin. After one toss, you will get 1 dollar if you get a head, and 0 dollars if you get a tail. How much will you be willing to pay to play this game 1000 times?
2. You have a fair coin, after one toss, you will get 1,000 dollars if you get a head, and 0 dollar if you get a tail. How much will you be willing to pay to play this game once?
I know the distributions of the payoff are different. However, I cannot figure out how this affects (or does not affect) the amount of the money you wanted to pay to play either game. Could anyone help explain?
Other than distribution, are there any other differences between these two games?
Thanks!
• How much do you value winning more money? And how much do you mind losing more money? That's the relevant question here. – celtschk Jul 29 '16 at 14:48
It's a little bit subjective. Just comment on which game would you prefer to play, and why.
As you say, the distributions are different, but how are they different?
Compare their quantitative measures (expectation, variance, skewness, et cetera). What do these mean?
Contrast their qualities (the results of some risk and reward scenarios, and such). What does this suggest?
For instance: In a fair game, you would not want to invest more than the expected outcome, because that's what is paid back on average (by definition). So what are the expected payouts?
Which game has the greatest risk of a low return? Which has the greatest potential of a high reward? What measure tells you about this?
• Just like @carmichael561 has explained, the expected payoffs are both 500 and I don't think I have a preference for one over the other and this is why I asking. Clearly, if we use variance to measure risk, game 1 is less risky and preferred. But I wanted to know if it is proper to ignore information provided by variance, skewness, etc and just base your decision on expectation? I don't see my expected payout will be greater after considering variance and picking game 1. If so, why do we consider it at all? – Map Jul 29 '16 at 15:23
• @Map Which game is more "exciting"? Which is "safer"? What circumstances might you prefer playing one over the other? ...and such. – Graham Kemp Jul 30 '16 at 7:02
• I see...it depends on circumstances or a person's risk tolerance - one does not absolutely dwarf another, right? Many thanks for the explanation. – Map Aug 1 '16 at 15:52
One basic way to determine how much you would be willing to pay to play a game is to calculate its expected payoff.
In the first scenario, the expected value of one game is $1\cdot \frac{1}{2}+0\cdot \frac{1}{2}=\frac{1}{2}$, hence the expected value of playing this game $1000$ times is $1000\cdot\frac{1}{2}=500$.
In the second scenario, the expected value of one game is $1000\cdot\frac{1}{2}=500$.
So in both scenarios, $500$ is a reasonable price to pay. Note however that the variance of the second scenario is much larger. Depending on your risk preferences, this might affect which game you regard as being more attractive. | 2020-10-28T18:13:23 | {
"domain": "stackexchange.com",
"url": "https://math.stackexchange.com/questions/1874562/compare-two-coin-tossing-games",
"openwebmath_score": 0.5308240652084351,
"openwebmath_perplexity": 493.5726643363481,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9755769042651873,
"lm_q2_score": 0.8539127548105611,
"lm_q1q2_score": 0.8330575618506451
} |
https://math.stackexchange.com/questions/3923822/calculating-limit-of-series | # Calculating limit of series.
Given a series $$\sum_{n=1}^{\infty}\frac{4n+1}{2n(2n-1)(2n+1)(2n+2)}$$, how do I find the limit? I understand I need to find the sequence of partial sums, which goes something like this $$s_n=\{\frac{5}{24}, \frac{7}{30}, \frac{27}{112}....\}$$which will probably converge at 0.25, but I am having trouble finding a description of the sequence on which I could evaluate the limit.
• partial fractions & telescope ? – Donald Splutterwit Nov 26 '20 at 16:38
The partial fraction decomposition of the summand is $$\frac1{2(2n-1)}-\frac1{2(2n+1)}-\frac1{4n}+\frac1{4(n+1)}$$ The first and second parts, and the third and fourth, telescope. In the infinite sum all parts except the $$n=1$$ instances of the first and third parts will cancel, so the infinite sum is $$\frac1{2(2×1-1)}-\frac1{4×1}=\frac14$$
• When you say they telescope, is there a general approach to solving these or do you need to look at some of the terms to see that they start cancelling after some $n$? – smejak Nov 26 '20 at 17:42
• @smejak "looking at some of the terms to see thst they start cancelling" is the general approach. – Parcly Taxel Nov 26 '20 at 17:45
• Alright, thank you. I just wasn't sure whether I don't need to use some proof technique to prove that all the terms actually cancel as $n\rightarrow\infty$. – smejak Nov 26 '20 at 17:51
As a hint:$$\sum_{n=1}^{\infty}\frac{4n+1}{2n(2n-1)(2n+1)(2n+2)}=\\ \sum_{n=1}^{\infty}\frac{4n+1}{(2n-1)(2n+0)(2n+1)(2n+2)}=\\ \sum_{n=1}^{\infty}\frac{(2n-1)+(2n+2)}{(2n-1)(2n+0)(2n+1)(2n+2)}=\\ \sum_{n=1}^{\infty}\frac{(2n-1)}{(2n-1)(2n+0)(2n+1)(2n+2)}+\sum_{n=1}^{\infty}\frac{(2n+2)}{(2n-1)(2n+0)(2n+1)(2n+2)}=\\ \sum_{n=1}^{\infty}\frac{1}{(2n+0)(2n+1)(2n+2)}+\sum_{n=1}^{\infty}\frac{1}{(2n-1)(2n+0)(2n+1)}=\\$$ and now you have two telscopic series.can you take over now ?
Let $$\dfrac{4n+1}{(2n-1)2n(2n+1)(2n+2)}=f(n-1)-f(n)$$ where $$f(r)=\dfrac{ar+b}{(2r+1)(2r+2)}$$
so that $$\sum_{n=1}^m\dfrac{4n+1}{(2n-1)2n(2n+1)(2n+2)}=\sum_{n=1}^m(f(n-1)-f(n))=f(0)-f(m)$$
We need
$$f(n-1)-f(n)=\dfrac{(2n+1)(2n+2)(an-a+b)-(an+b)(2n-1)2n}{(2n-1)2n(2n+1)(2n+2)}$$
$$\implies4n+1=n^2(4a)+2n(?)+2b-2a$$
Comparing the coefficients of $$n^2,4a=0\iff a=?$$
Comparing the constants, $$2b-2a=1\implies b=\dfrac12$$
These values of $$a,b$$ satisfy the coefficients of $$n$$
Now set $$m\to\infty$$ and $$\lim_{m\to\infty}f(m)=?$$ | 2021-01-23T04:30:19 | {
"domain": "stackexchange.com",
"url": "https://math.stackexchange.com/questions/3923822/calculating-limit-of-series",
"openwebmath_score": 0.8727066516876221,
"openwebmath_perplexity": 329.0274841678014,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9755769056853638,
"lm_q2_score": 0.8539127529517043,
"lm_q1q2_score": 0.8330575612498943
} |
https://math.stackexchange.com/questions/4250574/prove-if-in-a-group-a-commutes-with-b-it-also-commutes-with-bk-for-any-i | # Prove if in a group $a$ commutes with $b$, it also commutes with $b^k$ for any integer $k$
Prove if in a group $$a$$ commutes with $$b$$, it also commutes with $$b^k$$ for any integer $$k$$ (based on Gallian's Algebra text).
This site has similar questions for individual integers; I'd like to prove it for all integers. My proof is below.
1. Is my proof correct?
2. Can writing be improved?
3. Is my use of induction appropriate?
4. Is there a more direct proof than using induction?
Proof: Since $$ab=ba$$, $$aba^{-1}=b$$ and similarly $$a^{-1}ba=b$$.
We now show that for any integer $$k$$, $$a^kba^{-k}=b$$. If $$k>0$$, we have $$a^kba^{-k} = a^{k-1}(aba^{-1})a^{-(k-1)} = a^{k-1}ba^{-(k-1)}$$. By induction, we get $$a^kba^{-k}=b$$. A similar induction can be used for $$k<0$$. And for $$k=0$$, $$a^kba^{-k}=b$$ is trivial.
Consequently, $$a^kb=ba^k$$, QED.
• It seems correct to me. Sep 14 at 22:46
• You've proved that $b$ commutes with $a^k$. Apart from that (harmless) switch of $a$ and $b$, your proof is fine, except that it doesn't deal with the case of negative $k$ (which is quite easy using $a^{-k} = (a^{-1})^k$ and what you already have). Sep 14 at 22:56
• As a minor (whimsical aside): when I read Gallian Algebra, I wondered whether this was about some new field of algebra" that I had never heard of. If you'd written "Gallian's book Algebra" or just "Gallian Algebra", I wouldn't have been taken aback. $\ddot{\smile}$. Sep 14 at 23:00
• @RobArthan I intended to cover negative $k$ in "similarly $a^{−1}ba=b$.... A similar induction can be used for $k<0$." If that is not adequate, please explain why not. Sep 14 at 23:36
• Your "similarly ..." completion of the solution is fine. Sep 14 at 23:37
1. Yes, it appears so.
2. I think it's perfectly fine, given what you are doing.
3. I think there's a much more general statement that's equally easy to prove: the collection of all elements that commute with $$x$$ forms a subgroup of a group.
4. This is 3. Suppose that $$x$$ and $$y$$ commute with $$a$$. Then $$xy$$ and $$x^{-1}$$ commute with $$a$$. To see this, we simply calculate: $$axy=xay=xya,$$ and for inverses we just multiply $$ax=xa$$ on left and right by $$x^{-1}$$ to obtain $$x^{-1}a=ax^{-1}$$.
Now that the collection of all elements that commute with $$a$$ is a subgroup, the result is obvious.
If you want to just prove your result, you can use the inverse trick to show that $$a$$ commutes with $$b^{-1}$$, and then you have proved the negative powers by proving the positive powers.
• Your explanation hides the induction needed to prove that if $H$ is a subgroup of $G$ and $h \in H$, then $h^k \in H$ for any $k$. Sep 14 at 22:52
• @RobArthan I would hope that that fact, that groups are closed under taking powers of an element, is considered more basic than the question. Sep 14 at 22:58
• The question specifically asks whether there is a more direct method than using induction. Hiding the induction inside an assumed lemma is a less direct, although a much more structured approach. If people ask for a direct proof and you respond with a less direct but better structured proof, then good for you but you should explain what you have done. Sep 14 at 23:07
• @RobArthan If you consider the fact that powers of elements belong to a group to be a secret induction, and therefore all subsequent statements are induction-tainted, then I challenge you to provide even a definition of $a^k$ that does not use induction. Hence, it would be impossible to write down the question without using induction, hence there can be no induction-free proof. But that's silly. My proof does not use induction, beyond the statement that $a^k$ is an element of any subgroup containing $a$. That is part of the definition of being a group. Sep 14 at 23:13
• I am sorry. My concerns are more with the question than your answer. Our last two comments taken together will hopefully explain to the OP that there can be no "direct proof that does not use induction" but that a good proof (like yours) can be given that consigns the induction to a basic lemma that is required for any non-trivial reasoning about the notation $a^k$. Sep 14 at 23:23 | 2021-09-20T06:38:28 | {
"domain": "stackexchange.com",
"url": "https://math.stackexchange.com/questions/4250574/prove-if-in-a-group-a-commutes-with-b-it-also-commutes-with-bk-for-any-i",
"openwebmath_score": 0.8375613689422607,
"openwebmath_perplexity": 200.20307518919256,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9755769049752756,
"lm_q2_score": 0.8539127529517043,
"lm_q1q2_score": 0.8330575606435409
} |
https://math.stackexchange.com/questions/2516812/product-rule-proof-involving-logarithm-and-implicit-differentiation-really-water | # Product rule proof involving logarithm and implicit differentiation really waterproof?
I know how to prove the product rule for differentiation by using the definition for derivatives, i.e you "add zero" and collect terms. Yesterday a friend showed me another proof that involves the logarithm, implicit differentiation and the chain rule but after looking at it for a while I started to ask myself some questions about the proof. I don't know if it's me thinking about it all wrong or if it's something that actually makes the proof less rigorous. Anyway the proof goes like this (as most of you already know):
Let $$y = f(x)g(x)$$ and take the logarithm of both sides
$$\ln y = \ln f(x) + \ln g(x).$$
Now use implicit differentiation with respect to $x$ and solve for $y'$ while using the fact that $y = f(x) g(x)$
$$\frac{1}{y}y' = \frac{1}{f(x)}f'(x) + \frac{1}{g(x)}g'(x) \quad \Rightarrow \quad y' = f'(x) g(x) + f(x)g'(x).$$
Now the problem I am having with this is that the logarithm is not defined for $f(x) \leq 0$ and $g(x) \leq 0$ so this proof should only apply for product of functions that are positive for all $x$, is not that right?
• You're right that this proof presupposes that $f$ and $g$ are positive functions. Of course the product rule for positive functions immediately implies the product rule for negative functions (use $-f$ and $-g$ instead of $f$ and $g$) and for one positive and one negative functions (use $-f$ and $g$ or use $f$ and $-g$). But there seems to be real trouble with this sort of argument if one or both functions take the value $0$. – Andreas Blass Nov 12 '17 at 15:57
• A statement is true if there exists a proof. It's not wrong just because there exist wrong "proofs". – user436658 Nov 12 '17 at 16:03
• It's not a question about whether product rule is true or not, it's a question about a proof for it. I guess what I am asking is: isn't this a proof that is not waterproof since I can pick functions (say for example $f(x) = -x^2$) that I simply cannot plug into the proof? Can this even be considered as a proof since this example fails? – Claessie Nov 12 '17 at 16:07
• You may have a look at this answer math.stackexchange.com/a/2370012/72031 and in case you do have a look at it then don't miss the comments to the answer. – Paramanand Singh Nov 12 '17 at 16:39
• @ParamanandSingh thank you, was helpful! – Claessie Nov 12 '17 at 17:52
If $f$ and $g$ are differentiable at $x$ then there exists $c$ so that $f+c>0$ and $g+c>0$ in a neighborhood of $x$. So the argument with the logarithm shows that $[(f+c)(g+c)]'=(f+c)'(g+c)+(f+c)(g+c)'$. But $$[(f+c)(g+c)]'=(fg)'+cf'+cg'$$and $$(f+c)'(g+c)+(f+c)(g+c)'=f'g+fg'+cf'+cg'.$$
• The log trick proves the product rule for $f+c$ and $g+c$, and then with a little algebra that implies the product rule for $f$ and $g$. – David C. Ullrich Nov 12 '17 at 23:05 | 2021-05-06T21:14:51 | {
"domain": "stackexchange.com",
"url": "https://math.stackexchange.com/questions/2516812/product-rule-proof-involving-logarithm-and-implicit-differentiation-really-water",
"openwebmath_score": 0.8688954710960388,
"openwebmath_perplexity": 147.1725821306248,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9755769092358048,
"lm_q2_score": 0.853912747375134,
"lm_q1q2_score": 0.8330575588412877
} |
https://www.physicsforums.com/threads/linear-programming.52319/ | # Linear programming
All-Easy manufactures three products whose unit profits are $1,$9 and $5, respectively. The company has budgeted 70 hrs. of labor time and 45 hours of machine time for the production of three products. The labor requirements per unit of products A,B C are 2, 3 and 5 hours, respectively. The corresponding machine time requirements per unit are 1, 4 and 5 hour. All-Easy regards the budgeted labor and machine hours as goals that must be exceeded, if necessary,but at the additional cost of$15 per labor hour and $5 per machine hour. Formulate the problem as an LP model. Doubts w/ solutions: I let x = no. of units of product A, y = no. of units of product B, z = no. of units of product C. Maximize: z = x + 9y + 5z (profit) subject to: 2x + 3y + 5z <= 70 (labor hrs.) x + 4y + 5z <= 45 (machine hrs.) x,y,z >= 0 "All-Easy regards the budgeted labor and machine hours as goals that must be exceeded, if necessary, but at the additional cost of$15 per labor hour and $5 per machine hour." - if I were to make mathematical model out of these, am i going to adjust my objective function or my constraints or both? How? ## Answers and Replies Related Introductory Physics Homework Help News on Phys.org Let u = additional labor hours, and v = additional machine hours. How do they affect your profit, and how do they affect your constraint functions? Express it algebraically. I got this idea... so at least, I can show you where am I... got stuck We have a constraint on Labor: 2x + 3y + 5z <= 70, but it can be exceeded ... at extra cost. If 2x + 3y + 5z is greater than 70, it costs an additional$15/hour.
The excess is: (2x + 3y + 5z - 70) hours which costs $15/hr. The extra labor cost is: 15(2x + 3y + 5z- 70) dollars, which, of course, reduces the profit. Similarly, we have a constraint on Machine time: x + 4y + 5z <= 45 which can be exceeded ... at extra cost. The excess is (x + 4y + 5z - 45) hours which costs$5/hr.
The extra machine cost is: 5(x + 4y + 5z - 45) dollars,
which also reduces the profit.
is this correct? Can I relate it to what you've replied... and um, that's it, how will I re-formulate my LP model?
OK, but remember that your labor hours are no longer limited to 70.
franz32 said:
Maximize: z = x + 9y + 5z (profit)
subject to:
2x + 3y + 5z <= 70 (labor hrs.)
x + 4y + 5z <= 45 (machine hrs.)
x,y,z >= 0
I would also not use "z" to represent profit, since you are already using it for product C. :)
Maximize: P = x + 9y + 5z - 5u - 15v (profit)
subject to:
2x + 3y + 5z <= 70 + u (labor hrs.)
x + 4y + 5z <= 45 + v (machine hrs.)
x,y,z,u,v >= 0
And if the goals must be exceeded, you have actual equality:
2x + 3y + 5z = 70 + u (labor hrs.)
x + 4y + 5z = 45 + v (machine hrs.)
Solving for u and v in those get you the two relationships you mention in your post.
Um... I did understand about it.... but if I were to write the final part as my model for the LP, it seems that it is "unstable"... bec. I am following the standard form of a LP model...
The final part? you mean the two equalities?
yes.. bec. I don't feel that my model is a formal one yet... =)
The equalities are the bounding surfaces, and the solution is found on the surface--the vertices, in fact. | 2020-10-23T21:27:57 | {
"domain": "physicsforums.com",
"url": "https://www.physicsforums.com/threads/linear-programming.52319/",
"openwebmath_score": 0.5647633671760559,
"openwebmath_perplexity": 1560.915135279326,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES\n\n",
"lm_q1_score": 0.9755769056853638,
"lm_q2_score": 0.8539127473751341,
"lm_q1q2_score": 0.8330575558095211
} |
https://www.thelaunchpadtech.com/forum/d96go/r9w2t0.php?a3a973=transpose-of-row-matrix-is-called | The matrix is said to be an orthogonal matrix if the product of a matrix and its transpose gives an identity value. Matrices obtained by changing rows and columns is called transpose. This transposed matrix can be written as [ [1, 4], [2, 5], [3, 6]]. D order of A. Syntax. The new matrix obtained by interchanging the rows and columns of the original matrix is called as the transpose of the matrix. Another way to prevent getting this page in the future is to use Privacy Pass. Properties of Transpose of a Matrix. A two-dimensional array is used to clearly indicate that only rows or columns are present. B transpose. Answer By Toppr. ... B contains the same elements as A, except the rows and columns are interchanged. So, if we have a matrix A, we say that it's the transpose matrix. A matrix with only one row is called the row vector, and a matrix with one column is called the column vector, but there is no distinction between rows and columns in the one-dimensional array of ndarray. Entries in a matrix are called elements of a matrix. C Matrix Transpose : Conversion of rows into columns and columns into rows is called Transpose of a Matrix in c programming language (inverse of a matrix). 21 Horizontally arranged elements in a matrix is called A columns. Transpose. So, we write the transpose as A superscript capital T. So, T stands for transpose. B = transpose(A) Description. The general equation for performing the transpose of a matrix is as follows. In other words, if A = [a ij] m × n, then A′ = [a ji] n × m . When multiplying matrices, multiply the elements in each ____ of the first … The transpose of a matrix A, denoted by A , A′, A , A or A , may be constructed by any one of the following methods: Elements are defined by using rows and columns. Calculating the transpose of a matrix that indicates some linear transformation can reveal some properties of transformation. C transpose is now going to be a 3 by 4 matrix. If you are on a personal connection, like at home, you can run an anti-virus scan on your device to make sure it is not infected with malware. A matrix with only one row is called a row vector, and a matrix with one column is called a column vector, but there is no distinction between rows and columns in a one-dimensional array of ndarray. View Answer Answer: Column matrix 10 Idea of matrices was introduced by Arthur Caylet in A 18th century. The signs of the imaginary parts are unchanged. FAQ. Here, transform the shape by using reshape(). does not affect the sign of the imaginary parts. plural of “matrix” is “matrices”. ... real numbers, additive identity is A 1. Assume that you are executing a 256 × 256 double-precision transpose on a … If A = [a ij] be an m × n matrix, then the matrix obtained by interchanging the rows and columns of A would be the transpose of A. of It is denoted by A′or (A T). If A contains complex elements, then A.' The general equation for performing the transpose of a matrix is as follows. And that first row there is now going to become the first column. Properties of Transpose of Matrix . You simply use the t() command. Related post: NumPy: How to use reshape() and the meaning of -1 As ment… If A = [a ij] be an m × n matrix, then the matrix obtained by interchanging the rows and columns of A would be the transpose of A. of It is denoted by A′or (A T). Inner Product. If A has dimension (n m) then A0has dimension (m n). The numbers contained in a matrix are called elements of the matrix (or entries, or components). B transpose of A. Prev Question Next Question. In Python, there is always more than one way to solve any problem. is a column matrix. C 0. If A = [a ij] be an m × n matrix, then the matrix obtained by interchanging the rows and columns of A would be the transpose of A. of It is denoted by A′or (A T). columns. For any non- singular matrix A, A-1 = Matrices obtained by changing rows and columns is called A matrix having m rows and n columns with m ≠ n is said to be a Example 1: Consider the matrix . Transposition: The transpose is the matrix derived from a given matrix by interchanging the rows and columns. A matrix that consists of precisely one row is called a row matrix, and a matrix that consists of precisely one column is called a column matrix. And, essentially, it's going to be the matrix C with all the rows swapped for the columns or all the columns swapped for the rows. D −1. Apply the operator on the input matrix ( output matrix=input matrix.’) 3. Numerical: Solution: Share these Notes with your friends Prev Next > Note that the transpose of a row matrix is a column matrix and vice versa; for example, Row and column matrices provide alternate notations for a vector. (This makes the columns of the new matrix the rows of the original). If A = [cos α sin α − sin α cos α ] and A + A T = I, find the value of a in a π ∈ [0, π]. Multiply the corresponding entries from the rows and columns together and then add the resulting multiplication results. Transpose Matrix When we convert the rows into columns and columns into rows and generates a new matrix with this conversion is called the transpose matrix. Solution. Related Questions to study. For example, a 31 = 2, b 22 =1. And, essentially, it's going to be the matrix C with all the rows swapped for the columns or all the columns swapped for the rows. Therefore, it is a row vector: Exercise 3. So, here is our matrix A. lf A = ⎣ ⎢ ⎢ ⎡ 0 − 1 − 4 1 0 − 7 4 7 0 ⎦ ⎥ ⎥ ⎤ then A T = View Answer. is symmetric if it is equal to its transpose. ... which means the number of rows and number of columns is equal, then the matrix is called a square matrix. Equal Matrices. View Answer. Applying T or transpose()to a one-dimensional array only returns an array equivalent to the original array. Consider the matrix If A = || of order m*n then = || of order n*m. So, . ', then the element B(2,3) is also 1+2i. Transpose of a Matrix. A matrix which is formed by turning all the rows into columns and vice versa is called a transpose of a matrix. For example, In this lesson we will learn about some matrix transformation techniques such as the matrix transpose, determinants and the inverse. 2. This transposed matrix can be written as [[1, 4], [2, 5], [3, 6]]. 9 Transpose of a row matrix is A zero matrix. If a m x n matrix is taken as input, the order of the transposed matrix will be n x m. Let us consider a 3×2 matrix(1 row, 2 columns) Matrix created as a result of interchanging the rows and columns of a matrix is called Transpose of that Matrix, for instance, the transpose of the above matrix would be: 1 4 2 5 3 6. The new matrix obtained by interchanging the rows and columns of the original matrix is called as the transpose of the matrix. Syntax. Program 3: The Transpose of a Matrix The transpose of matrix A is a new matrix A transpose(A) where the rows of A are the columns of A and the columns of A are the rows of A. Transpose of matrix M is represented by M T. There are numerous ways to transpose matrices.The transpose of matrices is basically done because they are used to represent linear transformation. The matrix A does not need to be a square matrix. But has only one column, which implies that has only one row. The ____ is the m x n matrix all of whose entries is 0. row. B diagonal matrix. Transpose of a matrix is obtained by changing rows to columns and columns to rows. transpose. columns. For example, $$\begin{bmatrix} 2 & 4 & 6\\ 1 & 3 & -5\\ -2 & 7 & 9 \end{bmatrix}$$ ... which has 3 rows and 3 … 22 If A is a matrix of order (m - by - n) then a matrix (n - by - m) obtained by interchanging rows and columns of A is called the A additive inverse of A B transpose of A C determinant of A Your IP: 85.217.171.216 The inner product a, b (or a・b) is a scalar function. B = A.' If A = [a ij] be an m × n matrix, then the matrix obtained by interchanging the rows and columns of A would be the transpose of A. of It is denoted by A′or (A T). By convention, a vector x is written as a column vector. Thus, the matrix B is known as the Transpose of the matrix A. B 3. So, C transpose. I know the property, but I don't understand it. Recommended: Please solve it on “ PRACTICE ” first, before moving on to the solution. Performance & security by Cloudflare, Please complete the security check to access. Matrix created as a result of interchanging the rows and columns of a matrix is called Transpose of that Matrix, for instance, the transpose of the above matrix would be: 1 4 2 5 3 6. So, what is the transpose of this matrix. Transpose of the matrix can be done by rearranging its rows and columns. 1. In Python, we can implement a matrix as a nested list (list inside a list). A two-dimensional array is used to clearly indicate that only rows or columns are present. Multiply the corresponding entries from the rows and columns together and then add the resulting multiplication results. ’). So, it's now going to be a 3 by 4 matrix. The transpose of a matrix interchanges its rows and columns; this is illustrated below: Here is a simple C loop to show the transpose: for (i = 0; i < 3; i++) {for (j = 0; j < 3; j++) {output[j][i] = input[i][j];}} Assume that both the input and output matrices are stored in the row major order (row major order means that the row index changes fastest). All Rights Reserved by Suresh, Home | About Us | Contact Us | Privacy Policy. The new matrix obtained by interchanging the rows and columns of the original matrix is called as the transpose of the matrix. Some laws on matrix multiplication: If A, B, C matrix meet the required matrix multiplication … A related matrix form by making the rows of a matrix into columns and the columns into rows is called a ____. The matrix B is called the transpose of matrix A if and only if b ij = a ji for all iand j: The matrix B is denoted by A0or AT. Columns Matrix . Cloudflare Ray ID: 5fc78c4a0933d254 $\begingroup$ at the risk of reviving a dodgy question, may I ask "why" the geometric interpretation of orthogonal matrix is equivalent to the algebraic definition you gave? Both have similar results although the second one is more versatile as it can be applied partially. We can treat each element as a row of the matrix. Entries in a matrix are called elements of a matrix. The new matrix obtained by interchanging the rows and columns of the original matrix is called as the transpose of the matrix. 3. If A = [a ij] be an m × n matrix, then the matrix obtained by interchanging the rows and columns of A would be the transpose of A. of It is denoted by A′or (A T).In other words, if A = [a ij] mxn,thenA′ = [a ji] nxm.For example, The first row can be selected as X[0].And, the element in the first-row first column can be selected as X[0][0].. Transpose of a matrix is the interchanging of rows and columns. For example X = [[1, 2], [4, 5], [3, 6]] would represent a 3x2 matrix. A matrix with one column and m rows is called a column vector. Matrices obtained by changing rows and columns is called transpose. Then, seeing this functional as a matrix M , and x , y as a n × 1 matrix and a 1 × 1 matrix (trivially, a real number) respectively, if Mx = y then, by dimension reasons, M must be a 1 × n matrix; that is, M must be a row vector. B = A.' For example, It is a row matrix of order 1 by 3. If matrix A = matrix B we can say that A and B are identical. If a matrix has only one column then it is called a column matrix. Transpose of a Matrix. A square matrix is an n x n matrix; that is, a matrix with the same number of rows as columns. There are a lot of concepts related to matrices. It is denoted by AT or A′, or Atr, or At. zero matrix. A matrix called B of order 4 by 4 might look like this: B = By convention, matrices in text are printed in bold face. Transpose of matrix A is denoted by A T. Two rows of A T are the columns of A. The inner product for two vectors is defined as: The transpose of any scalar value equals itself. The … The columns of A T are rows of A. Let us consider there are two matrices one is the input matrix ‘I’ and the second is the output matrix ‘O’. > t(dat) [,1] [,2] [,3] A 1 2 3 B 4 5 6 C 7 8 9 D 10 11 12 . A matrix with only one row is called a row vector, and a matrix with one column is called a column vector, but there is no distinction between rows and columns in a one-dimensional array of ndarray. If a matrix has only one row then it is called a row matrix. Matrix Transpose in R Last Updated: 22-04-2020 Transpose of a matrix is an operation in which we convert the rows of the matrix in column and column of the matrix in rows. The first one is using t to just transpose the dataframe as if it would be a matrix (indeed the result of t is a matrix, not a dataframe). So, in this video, I wanted to talk about an operation you can do on a matrix that's called taking the transpose of the matrix. This is because a functional maps every n -vector x into a real number y . • Transpose. Let be a matrix defined by Is it symmetric? The transpose function makes a copy of the underlying vector with rearranged elements. a = 11 12 33 a’ = 11 22 33 Square Matrices. Here are the transposes of a vector and a matrix. A = [1 3 4-1i 2+2i; 0+1i 1-1i 5 6-1i] C column matrix. Java Program to transpose matrix. collapse all in page. Transpose of a matrix is an operation in which we convert the rows of the matrix in column and column of the matrix in rows. Let me do that in a different color. example. In general, a ij means the element of A in the ith row and jth column. The transpose of a matrix is calculated, by changing the rows as columns and columns as rows. The transpose() function from Numpy can be used to calculate the transpose of a matrix. If output matrix rows are equal to input columns and output matrix columns are equal to rows of the input matrix then the output matrix is called ‘transpose of the matrix’. The transpose of a rectangular matrix is a A matrix having m rows and n columns with m ≠ n is said to be a In a matrix multiplication for A and B, (AB)t The transpose of matrix A is represented by $$A'$$ or $$A^T$$. If A = [a ij] be an m × n matrix, then the matrix obtained by interchanging the rows and columns of A would be the transpose of A. of It is denoted by A′or (A T). ’ Steps: 1. “2 by 3”) matrix. The result of the t() command is always a matrix object. Elements are defined by using rows and columns. A matrix with one column and m rows is called a column vector. So, here is our matrix A. For example, consider the following matrix: B = A.' This ‘T’ represents the transpose of the matrix. Answer By Toppr. returns the nonconjugate transpose of A, that is, interchanges the row and column index for each element. In this method, the dot operator is used to finding the transpose of the matrix (. View Answer Answer: Rows 22 If A is a matrix of order(m - by - n) then a matrix(n - by - m) obtained by interchanging rows and columns of A is called the A additive inverse of A. The new matrix obtained by interchanging the rows and columns of the original matrix is called as the transpose of the matrix. A matrix with one column (an m × 1 matrix) is called a column vector and one row (a 1 × n matrix) is called a row vector.For example, Matrix a is a column vector, and matrix a’ is a row vector. The only limit of this method is there are high chances of syntax error because of the operator. For example, $$\begin{bmatrix} 2 & 4 & 6\\ 1 & 3 & -5\\ -2 & 7 & 9 \end{bmatrix}$$ This is a square matrix, which has 3 rows and 3 columns. Taking the transpose of a matrix is equivalent to interchanging rows and columns. $\endgroup$ – bright-star Dec 27 '13 at 8:22 The transpose of a row matrix is If A is a matrix of order m x n and B is a matrix of order n x p then the order of AB is If A is a symmetric matrix, then At = Another way to look at the transpose is that the element at row r column c in the original is placed at row c column r of the transpose. Here, transform the shape by using reshape(). Before discussing it briefly, let us first know what matrices are? Let me do that in a different color. D transpose. You may need to download version 2.0 now from the Chrome Web Store. So, it's now going to be a 3 by 4 matrix. Let’s see an example. Eg: Reduce the following matrix to the echelon form. If you are at an office or shared network, you can ask the network administrator to run a scan across the network looking for misconfigured or infected devices. So, what is the transpose of this matrix. Properties. Syntax: Output matrix=input matrix . It is called antisymmetric (skew-symmetric) if A = -Aᵀ. The final rows of a matrix and its transpose one is more versatile it! '' means transpose '' 3,2 ) is 1+2i and B should be the same as... Shifted to position a21 ( row 2 download version 2.0 now from the rows of matrix. Equation is called the coefficient matrix for the system Answer Answer: 0 13 matrices obtained by the! For Two vectors is defined as: the transpose of the operator on the input matrix ( of any value. ) 3 the easiest and simple methods for transpose columns as rows it... From Numpy can be applied partially always more than one way to solve any problem zero matrix but i n't! 11 12 33 a ’ ) 3 ____ is the transpose of the easiest and simple methods transpose! Product a, that is, a matrix is as follows... all full rows of column! Can implement a matrix can be found by changing rows and columns of a matrix are elements... A^T\ ) or A′, or at ) 3, additive identity is a matrix... Real number y n matrix all of whose entries is 0. row … Completing the CAPTCHA proves transpose of row matrix is called. You temporary access to the columns of a matrix by row x n then. Matrices are original array operator is used to indicate that only rows or columns are present are elements... List ( list inside a list ) m. so, what is transpose! Are called elements of a as columns of the imaginary parts rows of a, except the rows of as. = 2, B ( or a・b ) is 1+2i and B should be the size! = 2, B ( 2,3 ) is also 1+2i ( row and! Gives you temporary access to the echelon form some properties of transformation the second one is more efficient filling! Cloudflare Ray ID: 5fc78c4a0933d254 • your IP: 85.217.171.216 • Performance & security by cloudflare, Please complete security... Ray ID: 5fc78c4a0933d254 • your IP: 85.217.171.216 • Performance & security by,... Are the final rows of the matrix then it is called as the transpose of the matrix is! Such that its rows are the final rows of a T are the final rows of a T are of... What is the matrix and gives you temporary access to the web.! For the same size is equivalent to the columns of ‘ T ’ represents the transpose of a matrix called! Moving on to the solution be the same elements as a row of the original matrix is calculated changing. The element of a matrix is called a ____ -vector x into a real number.! Matrices ” 0+1i 1-1i 5 6-1i ] so, here is our matrix a, except the and! Times when people wanted names for matrices with just one column and m rows is called as the transpose the... Exercise 3 it can be applied partially this is because a functional maps n. * m. so, what is the transpose of the matrix a = matrix from... Matrix by interchanging the rows of a matrix is called as the of.... B contains the same thing About some matrix transformation techniques such as the of. Matrix a, that is, a vector and a matrix are present on “ PRACTICE first. Reshape ( transpose of row matrix is called Chrome web Store that a and B should be same. Has dimension ( m n ) position a21 ( row 1 and column index for each element About |... We write the transpose of the matrix Python, we write the transpose as superscript. Is it symmetric vector x is written as a superscript capital T. so, we write transpose. The first column by rows in a matrix is called as the transpose a. Entries in a 18th century underlying vector with rearranged elements it symmetric called the... 6-1I ] so, T stands for transpose ) 3 for the.. Matrix ( rows of the matrix derived from a given matrix by interchanging rows. Plural of “ matrix ” is “ matrices ” will learn About matrix... Done by rearranging its rows are the final rows of a matrix has one... Transform the shape by using reshape ( ) is there are high chances of error! Temporary access to the solution to matrices when people wanted names for the same size order 1 by.! The row and column index for each element is our matrix a m... Captcha proves you are a lot of concepts related to matrices aij = Aji where is. Is defined as: the transpose of a matrix has only one row then it is a matrix. Plural of “ matrix ” is “ matrices ” vector: Exercise 3 position a12 row. You may need to download version 2.0 now from the rows as columns of the original array ) or (... | 2021-06-12T18:31:05 | {
"domain": "thelaunchpadtech.com",
"url": "https://www.thelaunchpadtech.com/forum/d96go/r9w2t0.php?a3a973=transpose-of-row-matrix-is-called",
"openwebmath_score": 0.6036203503608704,
"openwebmath_perplexity": 368.5576146674933,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9648551566309688,
"lm_q2_score": 0.8633916170039421,
"lm_q1q2_score": 0.8330478538582039
} |
https://www.physicsforums.com/threads/where-does-this-method-of-constructing-the-taylor-expansion-of-arctanx-fails.613227/ | # Where does this method of constructing the taylor expansion of arctanx fails?
1. Jun 11, 2012
### tamtam402
1. The problem statement, all variables and given/known data
arctan x = ∫du/(1+u2), from 0 to x
2. Relevant equations
3. The attempt at a solution
I noticed that 1/(1+u2) = 1/(1+u2)1/2 × 1/(1+u2)1/2.
I decided to take the taylor series expansion of 1/(1+u2)1/2, square the result and then integrate.
I got 1/(1+u2)1/2 = 1 - u2 + 3u4/8 - 15u6/48...
When squaring that result, I get 1/1+u2 = 1 - u2 +5u4/8 ...
When I integrate, the first 2 terms are fine but the x5/8 should be x5/5.
Did I make a mistake, or is my method of "cheating" the interval of convergence for the binomial series somehow falsifying the result?
2. Jun 11, 2012
### SammyS
Staff Emeritus
I get something different when I square 1 - u2 + 3u4/8 - 5u6/16....
1-2 u2+(7 u4)/4-(11 u6)/8 + ...
How did you get the following?
3. Jun 11, 2012
### tamtam402
Dangit, I didn't copy my notebook properly sorry.
I squared (1-u2/2 + 3u4/8...), which should be the right series expansion for (1+u2)-1/2.
4. Jun 11, 2012
### SammyS
Staff Emeritus
For $\left(1 - u^2/2 + 3u^4/8 - 15u^6/48+35u^8/128+...\right)^2$
I get 1 - u2 + u4 - u6 +u8 + ...
5. Jun 11, 2012
### tamtam402
Sigh, I keep making stupid mistakes. I counted u4/2 twice and 3u4/8 once instead of u4/2 + 3u4/8 +3u4/8.
Thank you for the help.
6. Jun 11, 2012
### HallsofIvy
Rather than using a square root, it would make much more sense to use that the fact that the sum of a geometric series is given by
$$\sum_{n=0}^\infty r^n=1+ r+ r^2+ \cdot\cdot\cdot+ r^n+ \cdot\cdot\cdot= \frac{1}{1- r}$$
Here the function is $1/(1+ u^2)$ which is of the form $1/(1- r)$ with $r= -u^2$ so the sum is $1- u^2+ u^4- u^6+\cdot\cdot\cdot= \sum_{n=0}^\infty (-1)^n u^{2n}$. And the integral of that is $u- (1/3)u^3+ (1/5)u^5- (1/7)u^7+ \cdot\cdot\cdot= \sum_{n=0}^\infty ((-1)^n/(2n+1)!)u^{2n+1}$.
7. Jun 11, 2012
### Vargo
It is hard to cheat using power series. As long as you do your algebra correctly, the result should be valid in some interval as long as each series in the calculation has a nonzero radius of convergence.
Last edited: Jun 11, 2012 | 2018-02-18T07:35:07 | {
"domain": "physicsforums.com",
"url": "https://www.physicsforums.com/threads/where-does-this-method-of-constructing-the-taylor-expansion-of-arctanx-fails.613227/",
"openwebmath_score": 0.6774244904518127,
"openwebmath_perplexity": 2058.741185833909,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9648551525886193,
"lm_q2_score": 0.863391617003942,
"lm_q1q2_score": 0.8330478503680732
} |
https://math.stackexchange.com/questions/2456909/what-is-the-difference-between-zero-scalar-and-zero-vector | What is the difference between zero scalar and zero vector?
Zero vector has zero value in the given vector space. So, it is different from zero scalar. Zero vector is additive identity of the given vector space whereas zero scalar is not. I do understand this academic distinction. But I still have following $2$ doubts.
(i) In one dimensional space, is zero scalar same as to zero vector?
According to me - NO. In $1D$ space, the vectors may be expressed in real number notation (instead of matrix notation) where absolute value of real number indicates magnitude and sign indicated direction. Then, we are using 'real number notation' to represent not a real number, but a vector in $1D$. What we represent is not real number. So, a zero vector in $1D$ space is indeed expressed as $0$ in 'real number notation'. However it is not representing the real number zero (which is member of set of real numbers) but representing zero vector (which is a member of vectors in 1D space). So, in $1D$ space, a zero vector may be represented by number $0$. But it is not the real number $0$.
Analogy -: A directed segment represents a vector. It is not a vector.
(ii) Is zero speed same as zero velocity? In general, if we define a scalar quantity $Q_1$ which is magnitude of a vector quantity $Q_2$, then is zero $Q_1$ same as zero $Q_2$?
According to me - NO. When speed is zero, then the velocity is zero and vice-versa. However, zero speed is not equal to zero velocity.
For question 1) the answer is "usually no". For example, let $n$ be a positive integer greater than $1$ and let $V$ be a one dimensional subspace of $\mathbb R^n$. The zero vector in $V$ is certainly not the scalar $0$. The reason I say "usually" no is that if you view $\mathbb R$ as a vector space over $\mathbb R$, then the zero vector happens to be equal to the zero scalar. You could cook up some other examples like that.
For question 2, speed is a scalar and velocity is a vector. If an object has speed $0$, then its velocity is the zero vector, but its speed is not equal to its velocity. (They could not be equal because they are not even the same type of mathematical object.)
To be more concrete, let's say that I introduce a coordinate system in my lab and measure that my speed (in meters/sec) is the number $0$. Then my velocity (in meters / sec) is $(0,0,0)$. And $0 \neq (0,0,0)$.
Here's another way to make the same point. Suppose that a particle's position at time $t$ is $f(t)$, where $f:\mathbb (a,b) \to \mathbb R^3$ is a differentiable function. The particle's velocity at time $t_0$ is $f'(t_0)$, and the particle's speed at time $t_0$ is $\| f'(t_0)\|$. Suppose that the particle's speed at time $t_0$ is the number $0$. Then the particle's velocity at time $t_0$ is $f'(t_0) = (0,0,0)$. And again, $0 \neq (0,0,0)$.
• You say that speed and velocity can never be equal, as they are not even of the same type mathematical objects. This is fine, and it only makes sense when speed is nonzero. Physically one can differentiate between the two statements, speed is 5m/s and velocity is 5m/s up north since the second one gives magnitude as well as direction while the second gives only the magnitude of velocity. This differentiation loses all meaning when defining the direction is meaningless, which is what happens when speed is zero.................... Oct 6 '17 at 17:11
• ....................Again this is not comparing 0 kgs with 0 metres (they are completely different physical quanitites), but here we are comparing two things scalar (speed) and a vector (which is nothing but our scalar + a direction, ie, velocity). In both velocity and speed their magnitude has same dimension and represents the same physical quantity. So just saying that they are not the same mathematical object does not make sense when the actual difference between speed and velocity which is direction becomes meaningless. Oct 6 '17 at 17:11
• @DeltaScuti_Fomalhautb I added a couple paragraphs to my answer to attempt to address your comments. I think the key point is that the number $0$ is not equal to the vector $(0,0,0)$. Oct 6 '17 at 17:32
• @littleO I think your answer to question two depends on how the problem is interpreted. A particle's velocity is zero iff its speed is zero. It's not a question of whether $0_\mathbb{F}=0_V$, but if they are equivalent to each other. Oct 10 '17 at 5:43
• @AlexS I do agree that question 2 wasn't phrased very clearly. I didn't actually claim to have answered question 2, I just said some stuff that I thought might clarify the situation. Oct 10 '17 at 6:30
1. Strictly, no, but isomorphically, yes. Any finite-dimensional space $V$ is isomorphic to $K^n$, where $K$ is the associated scalar field and $n=\dim(V)$. If $\{v_1,\ldots,v_n\}$ is a base, you can write any $v\in V$ uniquely as $v=\sum_{j=1}^n k_jv_j$; the isomorphism is then $v\leftrightarrow (k_1,\ldots k_n)$. In particular, in a one-dimensional space, $\vec0$ and $0$ are equal under that isomorphism since $\vec0=0\cdot v_1$.
2. Very strictly speaking, the two are not equal but equivalent. If $Q_1=||Q_2||$ then $$Q_1=0\Leftrightarrow Q_2=0,$$ if $||\cdot||$ is any norm (magnitude). This, however, is axiomatic.
Perhaps it would be useful to look at the differences between vectors and scalars instead of focusing exclusively on the zero vector and the zero of a field.
The set of vectors in a vector space forms an Abelian group under addition. This means that there is an identity element, namely the zero vector, and every vector $\mathbf{v}\in V$ has an additive inverse, $-\mathbf{v}$, and vectors are associative under addition. Some simple examples of Albelian groups are $(\mathbb{Z},+)$, $(\mathbb{Q},+)$, and $(\mathbb{Z}/n\mathbb{Z},+)$. There are also multiplicative Albelian groups like $(\mathbb{Q},\times)$, $(\{z\in\mathbb{C}:|z|=1\},\times)$, and $(\{a^n:n\in\mathbb{Z}\},\times)$. In these types of groups, it more useful to think of the identity as 1 instead of 0.
Fields are also Abelian groups under addition, and this means that they too have a zero element that behaves precisely the same way as a zero vector. This may be where confusion arises. However, fields also have an additional operation that vectors do not and that is multiplication. Any two elements of a field can be multiplied together to give another element of the field. Elements of a field can also be divided. Some common examples of fields are $\mathbb{Q}$, $\mathbb{R}$, and $\mathbb{C}$.
There is also the condition that the set of vectors $V$ must be compatible with the base field $\mathbb{F}$, so certain combinations of groups and fields will not form vector spaces. For example, if we take take our set of vectors to be $V=(\mathbb{R}^n,+)$ and the base field to be $\mathbb{C}$, will we have a vector space? No, because for any vector $\mathbb{v}\in V$, the vector $i\mathbb{v}$ is not an element of $V$ and closure under scalar multiplication is necessary for the pair $(V, \mathbb{F})$ to form a vector space.
It can be easy to confuse the zero vector with the zero of a field, but they are not the same in any dimension. So your answer to the first question is correct. In my intro course on linear algebra, they denoted the zero element of the field by writing a little $\mathbb{F}$ under the zero like this: $0_\mathbb{F}$ to help us distinguish it from the zero vector which we denoted with a little $V$ underneath the zero like this $0_V$.
With regards to your last question, zero speed is indeed the same as zero velocity, and this is true of any scalar quantity defined as the norm of a vector since $||\mathbf{v} ||=0_\mathbb{F}\iff \mathbf{v} =0_V$. They are distinct entities, but one implies the other.
• Good point about the compatibility.
– user403337
Oct 13 '17 at 7:09
Look up the definition of a vector space from an algebraic point of view. Briefly, for a vector space you need a group $(G,+_G)$ and a field $(k,\cdot_k,+_k)$ combined in an intrinsic way. In $G$ since it is a group you can combine elements by the group operation and in $k$ you have two operations with their respective rules (commutativity, associativity and distributivity). And the connection between these two structures is given by
$$\lambda\cdot_k(g+_Gh)=\lambda\cdot_kg+\lambda\cdot_kh\qquad \forall \lambda\in k,\quad g,h\in G$$ and
$$g\cdot_k(\lambda+_k\mu)=g\cdot_k\lambda+_Gg\cdot_k\mu\quad \forall g\in G\quad \lambda,\mu\in k$$
Try not to think about dimensions in a concrete case (like matrices or vectors) think like we have a sack of elements where we can combine elements according to group rules. And we have an another sack of things of a different kind where we can combine the elements according to field rules. Both structures need to have a zero element to work but these zero elements can be of totally different kind/sort. I think exactly this is what disturbs you since in the concrete case you are looking at they happened to have the same kind but still coming from the different sacks in the abstract setting. So try to think about the zero scalar as coming from $k$ and the zero vector coming from $G$. If you only need the linear algebraic setting it may seem a bit of an overshoot to look at the abstract setting but I am sureit will clarify things if you have the time to sit down with it for some time.
The scalar $0$ is the zero from the field $F$ that the vector space is a vector space over. .. This is different from the $n$-dimensional zero vector ${(0,\dots, 0)}$, where $n$ is the dimension of the vector space...(in the $1$-dimensional case there is only one zero; but the concepts are still different : on the one hand the $1$-tuple $(0)$, as opposed to simply the element $0\in F$, on the other. .. The context is different.) Also, vectors can only be added... whereas scalars multiply vectors to give new vectors. ..
Secondly, speed is a scalar, the magnitude of the velocity, which is a vector and thus has a direction in addition to a length. ..
For example, in uniform circular motion the speed is constant and the velocity vector stays perpendicular to the acceleration, which is towards the center of the circle... This is instructive, and applicable, in that, the velocity vector keeps changing (in direction ), whilst the speed is constant. ...
You do not want to know about "academic" (would you mean "algebraic"?) arguments. So, proceeding pragmatically here are my answers to your two questions.
(i) A vector, or vector quantity, is a quantity whose complete determination is given by a magnitude and a direction, with the only exception when the magnitude is zero. In this case the direction does not make sense, and the vector is only its magnitude: there are no two different vectors with magnitude zero and different directions. This happens independently of the dimension. When you measure or compute a vector quantity you need to measure or compute its magnitude and its direction, but if its magnitude results to be zero you have done, no direction should be determined.
So the zero vector quantity is the zero scalar quantity (in whatever dimension).
(ii) Same arguments of (i) applies here with a notable difference. A velocity vector is a bound vector, while in (i) vectors were free vectors. So here a bound vector is the triple made up of a magnitude, a direction, and an application point. So here you cannot say that the zero velocity (vector quantity) is the zero speed (scalar quantity), but you need to use the concept of a scalar in a point (a map) and now can only say that:
the zero speed in a given point is the zero velocity in that same point (in whatever dimension).
Please only note that using algebraic arguments involving the standard algebraic definition of vector, scalar, vector space, field, equality (instead of those given here), answers to (i) and (ii) are neither positive nor negative, because scalars and vectors, and so the zero scalar and the zero vector in any dimension, are not even comparable in the general setting of abstract algebra, where only structures are studied, that is, how operations work on abstract entities, independently of the real nature of the entities: so you can have a given real entity that can play the role of a scalar and a vector at the same time, but abstract algebra alone can tell them apart, only their distinct roles. Then some context can be added thus passing from abstract algebra to applied algebra, and in so doing an equality relation can be defined on the set of real entities which will involve characteristics not belonging to the abstract concept of vector, scalar, number, ... but belonging to the real nature of the entities. Only now yow can start asking about equality of a vector and a scalar. I can give you details if you need but it's a bit more involved. | 2021-11-28T09:14:31 | {
"domain": "stackexchange.com",
"url": "https://math.stackexchange.com/questions/2456909/what-is-the-difference-between-zero-scalar-and-zero-vector",
"openwebmath_score": 0.876854419708252,
"openwebmath_perplexity": 196.77564932531973,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9648551535992067,
"lm_q2_score": 0.8633916134888614,
"lm_q1q2_score": 0.8330478478490623
} |
http://mathhelpforum.com/trigonometry/227032-cos-x-2pi-3-1-2-a-print.html | # cos(x -2pi/3) = 1/2
• Mar 20th 2014, 05:14 AM
andy000
cos(x -2pi/3) = 1/2
Hi,
the question I have been working on is cos(x -2pi/3) = 1/2 -2pi<x<2pi
I know from one of the special triangles that cos(pi/3) = 1/2 = cos(x-2pi/3)
so cos(x-2pi/3) = cos(pi/3), cos(-pi/3), cos(5pi/3), cos(-5pi/3), cos(7pi/3), cos(-7pi/3)
if this is right, should I just solve for each of them eg:
if cos(x-2pi/3) = cos(pi/3) ==> x-2pi/3 = pi/3 ==> x = 2pi/3+ pi/3 ==> x = pi
It would also be good if someone could let me know if I have missed any angles that give cos()= 1/2
thanks
• Mar 20th 2014, 06:32 AM
bjhopper
Re: cos(x -2pi/3) = 1/2
Quote:
Originally Posted by andy000
Hi,
the question I have been working on is cos(x -2pi/3) = 1/2 -2pi<x<2pi
I know from one of the special triangles that cos(pi/3) = 1/2 = cos(x-2pi/3)
so cos(x-2pi/3) = cos(pi/3), cos(-pi/3), cos(5pi/3), cos(-5pi/3), cos(7pi/3), cos(-7pi/3)
if this is right, should I just solve for each of them eg:
if cos(x-2pi/3) = cos(pi/3) ==> x-2pi/3 = pi/3 ==> x = 2pi/3+ pi/3 ==> x = pi
It would also be good if someone could let me know if I have missed any angles that give cos()= 1/2
thanks
cos 60 = 1/2
• Mar 20th 2014, 07:35 AM
SlipEternal
Re: cos(x -2pi/3) = 1/2
Quote:
Originally Posted by bjhopper
cos 60 = 1/2
That is not correct. $\cos 60^\circ = \dfrac{1}{2}$, but $\cos 60 \approx -0.95$.
Your value of $\dfrac{7\pi}{3}$ implies $x = 3\pi > 2\pi$, so that $x$ value is not in the domain. (But $\cos(x-2\pi/3) = \cos(-7\pi/3)$ does give an $x$-value in the domain).
• Mar 20th 2014, 08:33 AM
andy000
Re: cos(x -2pi/3) = 1/2
Hi,
so I guess I should have four values for x (discarding cos(7pi/3) and cos(5pi/3)?
• Mar 20th 2014, 08:35 AM
SlipEternal
Re: cos(x -2pi/3) = 1/2
Quote:
Originally Posted by andy000
Hi,
so I guess I should have four values for x (discarding cos(7pi/3) and cos(5pi/3)?
That is correct.
• Mar 20th 2014, 09:28 AM
andy000
Re: cos(x -2pi/3) = 1/2
Thanks again =)
• Mar 20th 2014, 01:11 PM
Soroban
Re: cos(x -2pi/3) = 1/2
Hello, andy000!
Quote:
$\text{Solve: }\:\cos\left(x -\tfrac{2\pi}{3}\right) \:=\:\tfrac{1}{2}\;\text{ for }-2\pi\,<\,x\,<\,2\pi$
We have: . $\cos\left(x - \tfrac{2\pi}{3}\right) \:=\:\tfrac{1}{2}$
. . $x - \tfrac{2\pi}{3} \;=\;\pm\tfrac{\pi}{3} + 2\pi n$
. . $x - \tfrac{2\pi}{3} \;=\;\begin{Bmatrix}\text{-}\frac{7\pi}{3} \\ \text{-}\frac{5\pi}{3} \\ \text{-}\frac{\pi}{3} \\ \frac{\pi}{3} \end{Bmatrix}$
. . $x \;=\;\begin{Bmatrix}\text{-}\frac{5\pi}{3} \\ \text{-}\pi \\ \frac{\pi}{3} \\ \pi \end{Bmatrix}$
• Mar 20th 2014, 07:37 PM
andy000
Re: cos(x -2pi/3) = 1/2
Hi Soroban,
Thanks for your reply. Can you explain how you get the equation x-2pi/3 = +/- pi/3 + 2pin?
I know this to be true but only from drawing out the unit circle and looking at it and the angles to equal pi/3. Is there some understanding I could gain to help me do this for other problems in the future?
Thanks | 2017-08-17T05:00:13 | {
"domain": "mathhelpforum.com",
"url": "http://mathhelpforum.com/trigonometry/227032-cos-x-2pi-3-1-2-a-print.html",
"openwebmath_score": 0.8388852477073669,
"openwebmath_perplexity": 5533.983081211991,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES\n\n",
"lm_q1_score": 0.9648551525886193,
"lm_q2_score": 0.8633916064587,
"lm_q1q2_score": 0.8330478401934421
} |
https://pballew.blogspot.com/2009/09/problems-from-land-down-under.html?showComment=1252794271296 | ## Friday, 11 September 2009
### Problems From the Land Down Under
Looking through the
Gazette of the Australian Mathematical Society
, and found their puzzle corner... really nice problems.
I think I have this one, but I didn't prove it....
Digital deduction
The numbers 2^2009 and 5^2009 are written on a piece of paper in decimal notation.
How many digits are on this piece of paper?
And this one has me puzzled (which is why they call them puzzles, I guess)..
Piles of stones
There are 25 stones sitting in a pile next to a blackboard.
You are allowed to take a pile and
divide it into two smaller piles of size
a and b,
but then you must write the number
a×b on the
blackboard. You continue to do this until you are
left with 25 piles, each with one stone. What is
the maximum possible sum of the numbers written
on the blackboard?
Anyone know how to a) prove the first, or b) solve the second... Do let me know....mostly down to chewing my pencil tips now....
Spoiler (I think)
x
x
x
x
x
x
x
OK, I think the total for the 25 stones will always be 300... I tried it about three different ways and they all came out the same... hmmmm... In fact, if we look at some smaller numbers for a guide, it seems that for any n, the sum of the products by this process will lead to $\dbinom{n}{2}$...
now why is that?
Anyone, Anyone??? Bueller?
Sue VanHattum said...
Hmm, the second seems easier to me. But maybe I'm missing something. I think I've solved it. Not sure if you want solutions here.
Just checking the rules of the game: If my first split is to 20 and 5, then I get 20x5=100 'points', and I'll keep adding points by splitting piles until I get all piles of 1.
(I haven't proved my solution is the best sum, but I'm pretty sure it is.)
Sue VanHattum said...
The first one *looks* like it's going to be n+1, but I don't see why it has to stay that way.
Pat's Blog said...
Sure I want solutions.. and the proof if you have one.. My fist thought was that since the largest product possible when breaking n into two parts is (n/2)^2 I would break it into the biggest parts possible, 12x13 first for 156 then break the 12 into 6x6 and the 13 into 6x7 but I don't know for sure if that will be the largest sum, because the other way 1x24 etc would give more numbers... so I'm still tinkering..
Sue VanHattum said...
That was my thought at first too.
• Split into 12 and 13: 12x13 = 156 'points'
• 6x6 and 6x7 = 36+42 points
• 3x3, 3x3, 3x3, and 3x4 = 39 points
• 1x2 (7 of those) and 2x2 = 18 points
• The nine 2s become pairs of 1s = 9 points
That all adds up to 300, but doing this made me suspect that 3's are no good. You get a 1 too soon. If I want as many 2's as possible, then I want as many 4's as possible, and then I want 8's, etc. I also want the first split to be as close to even as possible. So I tried 16 and 9. And my total goes up to 312.
(First split of 17 and 8 gets 300. Hmm, a bunch of different ways all get 300 as their total.)
I haven't proved there's no higher total, but I feel pretty sure 16 and 9 getting 312 is the max.
Sue VanHattum said...
Dang! Never mind. 16 and 9 gives 300 also. I'm beginning to think any split gives 300. My hypothesis at this point is, given a number n, all ways of playing the game for n give the same score.
Anonymous said...
When I looked at 2^2009 and 5^2009 I immediately thought of 10^2009 as an upper limit -- 2010 digits. Then I cheated and used PARI/GP to compute the number of digits in 2^2009 and 5^2009 (605 and 1405, respectively, which does in fact add up to 2010).
But how to PROVE that this works out exactly in general? 9^2009 and 9^2009 multiply out to 3835 digits, but they add to 3836 digits.
But how do you prove it works
Doctor Binary said...
(Sorry, hit publish before I was done editing -- ignore the last line "But how do you prove it works")
sumidiot said...
The number of digits in a number, n, is ceil(log_{10}(n)), where ceil is the ceiling function (next biggest integer). If you are thinking about the number of digits in a power, n^p, log rules come in handy. The number of digits is ceil(p*log_{10}(n)).
For the problem at hand, the sum of the digits of 2^p and 5^p, that means the answer is ceil(p*log_{10}(2))+ceil(p*log_{10}(5)). Now log_{10}(2)+log_{10}(5)=log_{10}(10)=1. I claim that if m is any integer, and x and y are two real numbers with x+y=1, then ceil(mx)+ceil(my)=m+1. This will prove the conjectured answer to the problem.
Suppose x+y=1, so that y=1-x, and m is an integer. Let mx=n+f where n is an integer and 0<f<1. Then ceil(mx)=ceil(n+f)=n+1 and ceil(my)=ceil(m(1-x))=ceil(m-mx)=ceil(m-n-f)=m-n. Therefore, ceil(mx)+ceil(my)=(n+1)+(m-n)=m+1, as claimed.
I like the second problem. I'm looking forward to playing with it some more, and reading what you all come up with as well.
sumidiot said...
Ok, I think I got the second one too. I've been playing around with the python programming language recently, so I wrote a little script to work out all the splittings of numbers up to 25, and see what all the different splittings give as a final answer. And, as conjectured above, the final answer seems to be independant of what the splitting was. Looking at small starting values, say 3, 4, 5, instead of 25, I conjectured that the final answer, starting with n, is the (n-1)st triangular number, (n-1)*n/2. Once that conjecture is made, it seems fairly straightforward to prove the claims by induction.
But if the answer is triangular numbers, there's almost certainly a picture involved, right? I think there is. I don't want to try to write down a formal way to do it, but here's an example. Start with 7, split it as 4+3. Then split 4 as 2+2, and 3 as 1+2. Finally, split all of the 2s into 1+1. The products I record can then be pictured graphically:
* * * | * * | *
| |
* * * | * * |
|------
* * * | *
|
* * * |
-------
* * |
-----
*
I know that graphic comes out terribly here. Copy and paste it somewhere where you've got fixed-width characters, and hopefully it'll make more sense.
Sue VanHattum said...
Pat, Thanks for these problems, I'm loving thinking about them! I couldn't sleep for a while, and these were much better companions than my usual nighttime thoughts.
Problem 1 first:
sumidiot, you helped me get there with logs. But you have an error: "I claim that if m is any integer, and x and y are two real numbers with x+y=1, then ceil(mx)+ceil(my)=m+1."
I had trouble following that when I read it last night. The way I did this problem was a bit different, but it helped me see that you can't use just any reals above. If x and y each equal 1/2, and m =2 then mx=my=1 and you'll have 2=3 above. x and y must both be irrational for your claim to hold.
I learned something pretty cool last night at about 3am, because I would not have thought proving something is irrational would be easy.
Here's how I did it:
First, I wanted to understand how logs go with digits. Any 2-digit number is between 10^1 and 10^2, so has a log of 1 point something. So an (n+1)-digit number has a log of n point something. Taking ceiling of the log gives the digits. (Except that 10 itself has log 1, and ceiling would still be 1, not the required 2 for its 2 digits. But we know that neither 2 nor 5 to an integer power can = 10 to an integer power. So we know we don't have to worry about the integer powers of 10.)
Now 10^n = 2^n*5^n, and taking logs gives us n = log(2^n)+log(5^n). Because neither 2^n nor 5^n can be a power of 10, these logs are not whole numbers.
If you took only the integer parts of the logs, and added, you'd get n-1, using the ceiling instead, you get n+1.
digits in 2^n = ceiling(log(2^n))
digits in 5^n = ceiling(log(5^n))
adding gives n+1, so we have n+1 digits.
Solving the problem was fun, but proving to myself that log(2) is irrational was more fun. Proof: If log(2)=a/b, then b*log(2)=a, then log(2^b)=a, then 10^a=2^b, which is not possible. This proof held because 10 and 2 are not both integer powers of some common base, and can be extended to any two numbers with that relationship.
Sue VanHattum said...
Problem 2:
We can prove this. And it may be the first time I've used proof by induction on a problem I was really solving on my own.
I want to call the total sum in this problem the points, and write P(25)=300. (To make P(n) well-defined, I should say it's the maximum possible points, so I don't really know yet what P(25) equals, but I'm going to be sloppy and use it as the sum for any splitting sometimes.)
Just like sumidiot, I saw that it was much easier to calculate these for smaller numbers, and turned to induction. It was clear that if you peeled off one stone each time, you'd get P(n)=(n-1)+(n-2)+...+2+1 = (n-1)n/2.
The problem is proving that you always get that result no matter how you split. 2 can only split to two piles of 1, and 1*1=1, so P(2)=1. 3 can only split to two piles as 2 and 1, 2*1=2, now split the 2, so P(3)=2+1=3. 4 can split two ways. If we let P(1)=0 since 1 can't be split, then we can write P(4)=3*1+P(3)+P(1)=3+3+0=6 or P(4)=2*2+P(2)+P(2)=4+1+1=6.
In general P(n)=(n-x)*x+P(n-x)+P(x).
Using induction, we can assume P(n-x) and P(x) are as given above, so we get P(n)=(n-x)*x+(n-x-1)(n-x)/2+(x-1)*x/2 = 2x*(n-x)/2+... = (2xn-2x^2+n^2-2xn+x^2-n+x+x^2-x)/2 = (n^2-n)/2 = n(n-1)/2 QED
That looks awful the way I wrote it. If you get my description, you can do that last part yourself. ;^)
Sue VanHattum said...
Pat asked "Now why is that?" at the end of her post. I don't think induction is very good at answering that question. I'd love to understand your diagram, sumidiot, but I'm lost. Can you help me?
sumidiot said...
Thanks, Sue, for pointing out my mistake with the mx and my business. I had used that mx was not an integer so that my f wasn't 0. But as you point out, that need not be the case in general. Of course, since you prove that the x I use (log(5)) is irrational, my argument still works in this case :)
I'll try to come up with some nicer picture. I may end up posting them on my blog, for convenience, but I'll post a link here if I go with that option.
sumidiot said...
Ok, perhaps the picture here helps. I took 10 as my starting number, and on the left I traced out how I was going to split it up. Then on the right, I draw rectangular boxes of the appropriate size for the products we're supposed to record in the process. So my first split is 10=7+3, so I record a 7 by 3 box.
I've tried to use some sort of systematic approach to drawing this (though I think it might not matter much). I write each split as A+B where A >= B, A in blue, B in red. For splitting a blue number, I use blue arrows (and blue sides of rectangles), and red similarly. Since there are several 2=1+1 splits, I circled some with their own color, and used that same color arrow to highlight the part of the diagram.
I don't know if any of that made sense...
Pat's Blog said...
Sumidiot??? some idea... Very nice
Anonymous said...
I count digits with floor(log_b(n))+1. It covers all cases, including integral powers of b.
sumidiot said...
Thanks, Sue and doctorbinary for patching up my logs to count digits. Off by one occasionally... silly me.
Joshua Zucker said...
For the first problem, I think the log approach is the best.
The second problem I have seen many times in books as a strong induction exercise, but ... WHY does it come out the triangular numbers?
Well, the triangular numbers are the solution to the handshake problem.
When all the pebbles are in one pile, let them all shake hands.
At each splitting step, the number of points you score is equal to the number of handshakes you destroy.
At the end, you have all the pebbles in their own individual pile, so there are no more handshakes possible - they have all been destroyed.
Hence the score is equal to the initial number of handshakes.
Pat's Blog said...
Joshua,
You ARE the math man... perfectly clear...why don't I see things like that...
thanks..
I may write one more blog about this to make that clear to students, explained as complete and bipartite graphs... | 2021-12-04T11:57:59 | {
"domain": "blogspot.com",
"url": "https://pballew.blogspot.com/2009/09/problems-from-land-down-under.html?showComment=1252794271296",
"openwebmath_score": 0.6937315464019775,
"openwebmath_perplexity": 843.7288007853748,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9669140254249554,
"lm_q2_score": 0.8615382129861583,
"lm_q1q2_score": 0.8330333815758689
} |
http://faculty.washington.edu/rjl/uwamath583s11/sphinx/notes/html/homework4.html | High Performance Scientific Computing AMath 483/583 Class Notes Spring Quarter, 2011
Homework 3
# Homework 4¶
Due Thursday, May 5, 2011, by 11:00pm PDT, by pushing to your bitbucket repository.
The goal of this homework is to gain some more practice with Fortran 90, modules, and Makefiles, and to start using OpenMP.
Before tackling this homework, you should read some of the class notes and links they point to. In particular, the following sections are relevant:
Make sure you update your clone of the class repository before doing this assignment:
$cd$CLASSHG
$hg pull -u # pulls and updates You should also create a directory$MYHG/homeworks/homework4 to work in since this is where your modified versions of the files will eventually need to be.
Within in this directory you should have two subdirectories parta and partb for Part A and B below.
The Fortran files in $CLASSHG/codes/fortran/quadrature_hw4 implement the Trapezoidal rule for computing an approximation to the definite integral \int_a^b f(x)\, dx To do tests of the accuracy we will use functions for which we can compute the exact integral for comparison, but of course the point of numerical quadrature is to approximate integrals that cannot be computed exactly. The Trapezoidal rule is a simple quadrature rule that is based on evaluating the function f(x) at n+1 points in the interval, approximating the function by a piecewise linear function connecting these function values, and computing the exact integral of this piecewise linear function. This is easy to do since on each of the n intervals between function values we only need to compute the area of a trapezoid. The figures below show examples where 10 and 20 intervals are used to approximate the interal \int_{-10}^{3} \exp(x/10) + \cos(x)\,dx. Let \Delta x = (b-a)/n and x_i=a + (i-1)\Delta x for i=1, 2, \ldots, n+1. On interval number i from x_i to x_{i+1} the area is \frac {\Delta x} 2 (f(x_{i})+ f(x_{i+1})). Summing this up from i=1 to n gives \Delta x \left(\frac 1 2 (f(a) + f(b)) + \sum_{i=2}^n f(x_i) \right). The trapezoidal rule is second order accurate, which means that if the function f(x) is sufficiently smooth then for small \Delta x (a large number of intervals), the error is proportional to (\Delta x)^2. So, for example, running the code in the directory$CLASSHG/codes/fortran/quadrature_hw4 gives:
$make test ./testquad.exe How many intervals n? 20 Trapezoidal: n = 20 approx = 0.9434636084761E+01 true = 0.9416892561216E+01 error = 0.177E-01 g was evaluated 21 times$ make test
How many intervals n?
200
Trapezoidal: n = 200
approx = 0.9417068999802E+01
true = 0.9416892561216E+01
error = 0.176E-03
g was evaluated 201 times
Note that with n=200 the error is about 100 = 10^2 times smaller than with n=20.
## Assignment:¶
### Part A¶
1. There are much better methods than the Trapezoidal rule that are not much harder to implement but get much smaller errors with the same number of function evaluations. One such method is Simpson’s rule, which approximates the integral over a single interval from x_i to x_{i+1} by
\int_{x_i}^{x_{i+1}} f(x)\, dx \approx \frac {\Delta x} 6 (f(x_i) + 4f(x_{i+1/2}) + f(x_{i+1})),
where x_{i+1/2} = \frac 1 2 (x_i + x_{i+1}) = x_i + \Delta x/2.
Add a subroutine simpson to the module quadrature_mod.f90 that implements Simpson’s rule. Note that when you sum the above expression over all i, you can combine some terms as was done in simplifying the Trapezoidal rule, so the result looks like
\frac{\Delta x}{6}[f(x_1) + 4f(x_{3/2}) + 2f(x_2) + 4f(x_{5/2}) + 2f(x_3) + \cdots + 2f(x_n) + 4f(x_{n+1/2}) + f(x_{n+1})].
Figure out a good way to evaluate this. Note that your program should evaluate the function only 2n+1 times for Simpson’s method. The code contains a counter that helps you check this.
This method is 4th order accurate, which means that on fine enough grids the error is proportional to \Delta x^4. Hence increasing n by a factor of 10 should decrease the error by a factor of 10^4 = 10000. Check that your function works properly by experimenting with this on the function provided.
Note: In finite precision arithmetic this asymptotic behavior is not seen if \Delta x is too small. Once \Delta x^4 is smaller than machine epsilon (around 2.d-16) you will not see any improvement by taking n larger.
Have your main program print out results for Simpson’s rule in the same format as for Trapezoidal.
2. Modify your functions_mod.f90 so that the function being integrated is
g(x) = \exp(x/10) + \cos(kx)
where the wave number k is a module variable that is read in from the user by the main program (before the number of intervals) and then used in the definition of the function g(x).
So running your program should give results like:
$make test ./testquad.exe Wavenumber k? 8. How many intervals n? 20 Trapezoidal: n = 20 approx = 0.1084939817273E+02 true = 0.9582360287054E+01 error = 0.127E+01 g was evaluated 21 times Simpson: n = 20 approx = 0.9363491553954E+01 true = 0.9582360287054E+01 error = -0.219E+00 g was evaluated 41 times If k=1 this is the same function as before. For large k the function is highly oscillatory, making it difficult to integrate unless a sufficiently fine grid is used. For example, with k=8 as above the error is large when n=20, not surprising if you look at the plots below. ### Part B¶ Now suppose we want to compute the integral with wave number k=10000. In order to get an accurate result we will have to take n very large. For this we might want an OpenMP version of the code. 1. Modify your code so that it uses OpenMP. Have nthreads be a variable of the module quadrature_mod.f90 that is set in the main program and used in the module in order to replace the do loops in trapezoid and simpson by omp parallel do loops. These loops must properly use reductions and private variables. Remove the call to evalfunc from the main program so you don’t have to worry about parallelizing this subroutine, which is just used to plot the function for plotting purposes. You might want to test your code on small values of k and n first. 2. Eventually test it with k=10000 and n = 50000000. Change the Makefile so that the time command is used when executing the code to determine how much time it takes. Hard wire these values of k and n into the main program rather than prompting the user, so the time to respond to the prompts doesn’t influence your results. So running the code should give something like: $ make test
gfortran -c -fopenmp evalfunc_mod.f90
gfortran -c -fopenmp functions_mod.f90
Compiled with OpenMP
Trapezoidal: n = 50000000
approx = 0.9819716972423E+01
true = 0.9819716972381E+01
error = 0.415E-10
g was evaluated 49444572 times
Simpson: n = 50000000
approx = 0.9819716972380E+01
true = 0.9819716972381E+01
error = -0.927E-12
g was evaluated 147106801 times
10.27 real 19.56 user 0.03 sys
Note from the last line that 19.56 seconds of CPU time was used but the code ran in 10.27 seconds, which indicates that both cores were being used.
3. If you look at the number of times g was evaluated in the results above you will see that this is not what’s expected. There was a bug in my code that you might have missed too: The variable gcounter is not thread safe: both threads call g and update this counter independently and the result is somewhat random. Sometimes the value updated by one thread was in local cache and not seen by the other thread.
Modify the code to keep a separate counter for each thread, or for more generality an array of length 8, by defining:
integer :: gcounter_thread(0:7)
which would allow nthreads up to 8. In the function g, use omp_get_thread_num to determine the thread number that made the call and update the corresponding element of the array. Output both values of the counter, so you will get output something like:
$make test time ./testquad.exe Compiled with OpenMP Using 2 threads Trapezoidal: n = 50000000 approx = 0.9819716972423E+01 true = 0.9819716972381E+01 error = 0.415E-10 g was evaluated 25000002 times by thread 0 g was evaluated 24999999 times by thread 1 Simpson: n = 50000000 approx = 0.9819716972380E+01 true = 0.9819716972381E+01 error = -0.927E-12 g was evaluated 50000002 times by thread 0 g was evaluated 49999999 times by thread 1 10.27 real 19.56 user 0.03 sys 4. Your module testquad should also work if we provide a different main program and/or functions module for grading purposes. 5. Make sure your directory$MYHG/homeworks/homework4 contains the following subdirectories and files:
• parta | 2022-12-04T01:59:13 | {
"domain": "washington.edu",
"url": "http://faculty.washington.edu/rjl/uwamath583s11/sphinx/notes/html/homework4.html",
"openwebmath_score": 0.5948221683502197,
"openwebmath_perplexity": 1676.5186149777562,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.966914018751051,
"lm_q2_score": 0.8615382094310357,
"lm_q1q2_score": 0.8330333723885472
} |
https://math.stackexchange.com/questions/2862702/a-bag-contains-3-red-4-blue-and-5-green-balls/2862715 | # A bag contains 3 red, 4 blue, and 5 green balls. [closed]
Peter draws a ball from the bag, and then Angelina draws a ball. What is the probability that Angelina got a green ball?
So far I have this:
Scenario A: 1st ball is not green, 2nd green: 7/12 * 5/11 = 35/132
Scenario B: 1st ball is green, 2nd green: 5/12 * 4/11 = 20/132
--> $$\frac{55}{132} = \frac{5}{12}$$
## closed as off-topic by asdf, gt6989b, Parcly Taxel, Xander Henderson, Asaf Karagila♦Jul 26 '18 at 9:09
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 improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – asdf, gt6989b, Parcly Taxel, Xander Henderson, Asaf Karagila
If this question can be reworded to fit the rules in the help center, please edit the question.
• So the first ball is green or it is not green... – Robert Z Jul 25 '18 at 18:46
• Yes, you are correct! – Robert Z Jul 25 '18 at 18:47
• correction Wolf: : You have $$\frac{20 + 35}{132} = \frac{55}{132}= \frac 5{12}$$ – Namaste Jul 25 '18 at 18:48
• $$P=P_1+P_2=\frac {5}{11}\frac {7}{12}+\frac {4}{11}\frac {5}{12}=\frac{5}{12 \times 11}(4+7)=\frac{5}{12 \times 11}(11)=\frac {5}{12}$$ – Isham Jul 25 '18 at 18:50
• Possible duplicate of Second marble is of same color – Rahul Goswami Jul 25 '18 at 19:11
## 3 Answers
Your approach is fine, but it's worth realizing that unless you know something about the ball Peter has chosen, he may as well not have chosen at all, in which case Angelina simply has a $5/12$ chance of choosing a green ball.
Where this point really becomes worth understanding is if, for example, not only Peter, but also Olivia, Ned, Melissa, and Larry choose balls before Angelina. The tree of possibilities is tedious and messy, but the answer is still $5/12$.
Angela's draw is actually independent from the order she draws the ball whether she is first, second, ..., or twelfth (surprisingly enough). So, the probability of her drawing a green ball is actually $\dfrac{5}{12}$. To give you a better explanation, suppose there are $k$ green balls and $n$ total balls. Then, the probability that Angela draws a green ball is:
$$\require{cancel} \dfrac{k}{n}\dfrac{k-1}{n-1}+\dfrac{n-k}{n}\dfrac{k}{n-1} = \dfrac{k(k-1)+k(n-k)}{n(n-1)} = \dfrac{k \cancel{(n-1)}}{n \cancel{(n-1)}} = \dfrac{k}{n}$$
Which is the same probability that Peter has for drawing a green ball.
• "Angela's draw is actually independent from Peter's draw" - it's not. You're probably misusing the word "independent". The probability distribution for Angelina's draw is the same as it would be if Peter did not draw, but that's not what "independent" means. If they were independent, we would have equalities like P(Angelina draws red AND Peter draws red) = P(Angelina draws red)*P(Peter draws red). – user2357112 Jul 25 '18 at 22:04
• @user2357112 I corrected my wording. Thank you for pointing it out. – InterstellarProbe Jul 25 '18 at 22:51
HINT
Brute force: condition on the color of the first drawn ball, there are 3 cases. Find the probability of each case happening, and what is the resulting probability in each case.
Then group them together in one final expression. | 2019-07-19T22:39:34 | {
"domain": "stackexchange.com",
"url": "https://math.stackexchange.com/questions/2862702/a-bag-contains-3-red-4-blue-and-5-green-balls/2862715",
"openwebmath_score": 0.716442883014679,
"openwebmath_perplexity": 1089.5729903241788,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.966914017797636,
"lm_q2_score": 0.8615382058759128,
"lm_q1q2_score": 0.8330333681296457
} |
https://math.stackexchange.com/questions/3662216/an-example-for-integrable-function-that-is-never-zero/3662235 | # An example for integrable function that is never zero
Let $$f\::\mathbb{R}\to\mathbb{R}$$ be integrable function for all $$[a,b],\hspace{0.2cm} (a).
and$$\hspace{0.2cm}\forall c,x\in\mathbb{R} \hspace{0.2cm} f(x)\not=0 \hspace{0.2cm} and \hspace{0.2cm}{\displaystyle \int_{c}^{c+1}f(x)\,dx}=0$$
Can I have an example for such function?
Thank you!
• $f(x)=\{1 if x\in[\frac {2k} 2, \frac {2k+1} 2]; -1 if x\in[\frac {2k+1} 2, \frac {2k+2} 2]\}$ May 6, 2020 at 19:15
• $sin 2\pi x$ should work May 6, 2020 at 19:15
You see according to what you say let us take
$$f(x)=\sin 2\pi x, \text{ for } x\in\mathbb{R- Z}$$
$$f(x)=1, \text{ for } x\in\mathbb{ Z}$$
$$\int_{a}^{b}\sin 2\pi x \mathrm dx <\infty, \forall a
And $$\int_{c}^{c+1}\sin 2\pi x \mathrm dx=\int_{0}^{1}\sin 2\pi x \mathrm dx=0, \forall c$$.
• The question also asked for $\forall x \in \mathbb{R}, f(x) \neq 0$, but $\sin(2\pi 0) = \sin(0) = 0$. May 6, 2020 at 19:27
• your answer works you just need to redefine sin where it is 0, to be something else, and it won't change the integral since it is a countable amount of points May 6, 2020 at 19:28
• depends which kind of "integrable" he means but yeah May 6, 2020 at 19:34
• @ChrisEagle I missed it sorry. May 6, 2020 at 19:37
• @BinyaminR Thank you May 6, 2020 at 19:37 | 2022-10-04T07:00:04 | {
"domain": "stackexchange.com",
"url": "https://math.stackexchange.com/questions/3662216/an-example-for-integrable-function-that-is-never-zero/3662235",
"openwebmath_score": 0.9445507526397705,
"openwebmath_perplexity": 3508.1809410503943,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.971129092218133,
"lm_q2_score": 0.8577681122619883,
"lm_q1q2_score": 0.8330035681946464
} |
https://math.stackexchange.com/questions/2958923/what-is-the-probablity-of-sitting-next-to-my-friend | # What is the probablity of sitting next to my friend?
Let's say you are at a table with $$5$$ others, everyone is seated randomly around a $$6$$ person table, and you only know $$1$$ person at this party.
1. What is the likelihood you sit next to the individual that you know?
2. What is the likelihood you are seated opposite to the person that you know?
3. What is the likelihood that you sit next to two strangers?
The table has $$6$$ seats so if you sit in any one seat then there are $$5$$ chairs left over. Since your friend can be seated on either side of you that leaves 3 chairs. With that reasoning would it be $$1/3$$ ($$2/6$$)?
• Which question is your working for? Question 3? – Parcly Taxel Oct 17 '18 at 5:32
• all of them. the one i was working on was #1. – fsdff Oct 17 '18 at 5:39
• I just wanted to note that the answers are different depending on whether it's a round table or rectangular table. The question as it is stated now does not indicate it. But I see you assume it's a round table. – Ivo Beckers Oct 17 '18 at 10:47
• So you're going to a wedding? – DonQuiKong Oct 17 '18 at 12:28
The answer to #1 is not $$\frac26$$ but $$\frac25$$, since you can consider your own seat fixed without loss of generality, leaving 5 chairs, two of which are adjacent to you. Similarly, the answers for #2 and #3 are $$\frac15$$ and $$\frac35$$ respectively, both obtained by counting the number of chairs where the desired result is obtained by your friend sitting there by the number of empty chairs.
In total, there are $$6! = 720$$ ways for all the people to sit on the chairs.
1. First there are $$6$$ ways for you to take a seat, $$2$$ ways for your friend to sit next to you. Now with $$4$$ people left with $$4$$ chairs, there are $$4!=24$$ ways for them to sit. So the probability would be $$\frac {24 \times 2 \times 6}{720}=\frac{2}{5}$$
2. Similar to 1. , there are $$6$$ ways for you to have the first seat, $$1$$ way for your friend to sit opposite you and $$24$$ ways for the rest to sit. The probability would be: $$\frac {24 \times 1 \times 6}{720}=\frac{1}{5}$$
3. In fact this is the complement of 1. so the probability would be $$1- \frac{2}{5}=\frac{3}{5}$$
(Sorry, I was late and English is my second language)
• hi, for number two, just assume you are already seated, there are 5 seats available, so the chance is indeed 1/5 as you calculated. Same argument can be used at the other questions. – Alucard Oct 17 '18 at 12:00
Total number of ways six people can sit around the table is $$(6-1)! = 5!$$ . The first problem. Fix yourself, there are 2 ways your known friend can sit either side. The rest of you can sit in 4! ways to a total of 48. Thus the probability for the first question is $$\frac{48}{120} = \frac{2}{5}$$. The second question is you occupy one chair and your known friend occupies on opposite to it and the remainder of 4 will occupy the chairs in $$4!$$ ways to get a probability of$$\frac{24}{120} = \frac{1}{5}$$. The third question is the two strangers can be picked in $${4\choose2}$$ ways and they can be permuted in 2! ways and the remainder 3 can be permuted in 3! ways to a total of $$(2\times6\times6) = 72$$. Thus the probability is $$\frac{72}{120} = \frac{3}{5}$$ | 2019-03-20T13:03:14 | {
"domain": "stackexchange.com",
"url": "https://math.stackexchange.com/questions/2958923/what-is-the-probablity-of-sitting-next-to-my-friend",
"openwebmath_score": 0.7804539799690247,
"openwebmath_perplexity": 249.0136395192945,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9711290930537121,
"lm_q2_score": 0.8577681086260461,
"lm_q1q2_score": 0.8330035653804102
} |
https://dsp.stackexchange.com/questions/67109/fft-one-sided-and-parseval-theorem | FFT one-sided and Parseval theorem
I'm trying to get Parseval theorem working on a one-sided FFT. So far I have this code (matlab):
%% ODD
a = [1 2 3 4 5];
A = fft(a);
A1 = abs(A);
A2 = abs([A(1) 2*A(2:3)]);
Ea=sum(a.^2)
EA1=sum(A1.^2)/5
EA2=sum(A2.^2)/5
%% EVEN
b = [1 2 3 4 5 6];
B = fft(b);
B1 = abs(B);
B2 = abs([B(1) 2*B(2:3) B(4)]);
Eb =sum(b.^2)
EB1=sum(B1.^2)/6
EB2=sum(B2.^2)/6
And the output is:
Ea = 55
EA1 = 55
EA2 = 65
Eb = 91
EB1 = 91
EB2 = 107
The scaling for the one-sided is to preserve amplitude information and not for calculating Parseval theorem?
Parseval's theorem says that the the following relationship holds $$\sum_{n=1}^{N} a[n]\,a^*[n] = \frac{1}{N}\sum_{k=1}^{N} A[k]\,A^*[k]$$ where $$A[k]$$ is the discrete Fourier transform of $$a[n]$$, both assumed to be of length $$N$$ (no padding). This arises from the fact that the signal energy calculated from the time domain and frequency domain must be equal. See this answer for a little more detail about how this formula comes about.
If you want to calculate a one-sided spectrum, you need to throw away the negative frequencies (which are redundant when the signal is real). However, because half of the energy was contained in these entries, you need to multiply the energy in the remaining bins by 2 - except for the DC and Nyquist. I see that you have recognised correctly that the Nyquist is only present when $$N$$ is even, and that the first entry is the DC component regardless of $$N$$ - good. However, you have multipied the amplitudes by 2 (straight after the FFT), not the energy - the negative frequency bins contain half of the energy.
Instead, once you have calculated the discrete Fourier transform using the FFT algorithm (your variables $$A$$ and $$B$$), I would first then obtain the energy spectral density of the signal $$a[n]$$ using $$\textrm{ESD}_a[k] = |A[k]|^2 = A[k]\,A^*[k]$$ and then throw away the negative frequencies from this vector instead. This way you throw away half of the energy correctly, and you can confidently multiply the energy in the remaining bins by $$2$$.
The following code illustrates this:
%% ODD
a = [1 2 3 4 5];
A = fft(a);
ESD_a = A.*conj(A);
ESD_a_onesided = [ESD_a(1) 2*ESD_a(2:3)];
E_a_timedomain = sum(a.^2)
E_a_twosided = sum(ESD_a)/5
E_a_onesided = sum(ESD_a_onesided)/5
%% EVEN
b = [1 2 3 4 5 6];
B = fft(b);
ESD_b = B.*conj(B);
ESD_b_onesided = [ESD_b(1) 2*ESD_b(2:3) ESD_b(4)];
E_b_timedomain = sum(b.^2)
E_b_twosided = sum(ESD_b)/6
E_b_onesided = sum(ESD_b_onesided)/6
The result is then correctly
E_a_timedomain = 55
E_a_twosided = 55
E_a_onesided = 55
E_b_timedomain = 91
E_b_twosided = 91
E_b_onesided = 91
EDIT----------------------------
Actually, the energy values of $$55$$ and $$91$$ obtained are only correct if we are assuming the sampling period of the signal acquisition was $$T_s=1$$.
The signal energy of a continuous signal $$a(t)$$ is defined as $$E_s = \int_{-\infty}^{+\infty}|a(t)|^2\;dt$$ and the energy of a sampled version of it is then $$E_s = \sum_{n=1}^{N}|a[n]|^2\cdot T_s$$ and you can see that we need to account for the signal sampling period to get the right energy.
The following verifies Parseval's theorem if we for example had collected the signal at some other sampling period $$T_s\neq 1$$
Ts = 0.05; % the sampling period of acquisition
Fs = 1/Ts; % the sampling frequency of the acquisition
%% ODD
a = [1 2 3 4 5];
N = 5
A = fft(a)*Ts;
ESD_a = A.*conj(A);
ESD_a_onesided = [ESD_a(1) 2*ESD_a(2:3)];
E_a_timedomain = sum(a.*conj(a))*Ts
E_a_twosided = sum(ESD_a)*Fs/N
E_a_onesided = sum(ESD_a_onesided)*Fs/N
%% EVEN
b = [1 2 3 4 5 6];
N = 6
B = fft(b)*Ts;
ESD_b = B.*conj(B);
ESD_b_onesided = [ESD_b(1) 2*ESD_b(2:3) ESD_b(4)];
E_b_timedomain = sum(b.^2)*Ts
E_b_twosided = sum(ESD_b)*Fs/N
E_b_onesided = sum(ESD_b_onesided)*Fs/N
with the output
E_a_timedomain = 2.75 [signal^2 sec]
E_a_twosided = 2.75 [signal^2 sec]
E_a_onesided = 2.75 [signal^2 sec]
E_b_timedomain = 4.55 [signal^2 sec]
E_b_twosided = 4.55 [signal^2 sec]
E_b_onesided = 4.55 [signal^2 sec]
• Thank you for the long and detailed answer. So, the one-sided fft coefficients that are computed for properly representing the amplitudes cannot be directly used for calculating the energy? – Filipe Pinto May 2 at 16:00
• It doesn't really matter at what point you do your scaling. As long as you are clear what each quantity represents at each step. It depends what you want to show. In your example, if you increase you number of points then the values of the A1 vector will increase. You could apply scaling directly to the output of the FFT if you like. It is fine to multiply the amplitudes by 2, but you then need to make sure that when you calculate energy (which is proportional to amplitude squared, and gets bigger with more points) that you account for it (with another factor of sqrt(2) in that case). – teeeeee May 2 at 16:44
• Yes thank you!! – Filipe Pinto May 7 at 10:14 | 2020-10-22T09:57:32 | {
"domain": "stackexchange.com",
"url": "https://dsp.stackexchange.com/questions/67109/fft-one-sided-and-parseval-theorem",
"openwebmath_score": 0.8140923380851746,
"openwebmath_perplexity": 841.972841915592,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9711290955604489,
"lm_q2_score": 0.8577681031721325,
"lm_q1q2_score": 0.8330035622341548
} |
https://stats.stackexchange.com/questions/448693/is-conditional-probability-always-equal-to-the-probability-of-the-predicted-even | # Is conditional probability always equal to the probability of the predicted event? P(A|B) = P(A)
I'm currently trying to better understand the fundamental notions of statistics.
Navigating through multiple sites, I've found this formula for joint probability. P(A ∩ B) = P(A)*P(B)
Namely, if we want to know what is the probability of A and B to co-occur, we just multiply their individual (marginal) probabilities.
As for the conditional probability, the formula stands like this: P(A | B) = P(A∩B) / P(B)
So, if we want to know the probability of A when we already know B, we divide the joint probability of A and B by the probability of B.
If we develop this formula, we will get: P(A | B) = P(A)*P(B)/P(B)
Further, we can simplify the fraction by dividing it by P(B), and we are left with P(A | B) = P(A)
Suppose we want to estimate the probability of drawing from a deck of cards a red colored card of 4 (hearts or diamonds), while already knowing it's colored red.
Individual (marginal) probabilities
P(red) = 1/2 = 0.5 P(4) = 4/52 = 1/13
Joint Probability (to draw a red 4)
P(4 ∩ red) = P(A)*P(B) = 1/2 * 1/13 = 1/26
Conditional probability (prob to draw a 4 already knowing it is red)
P(4 | red) = P(4 ∩ red) / P(red) = P(4)*P(red)/P(red) P(4 | red) = 1/13*0.5/0.5 = 1/13
If we know that the card we are about to draw is red, the probability of it being a red 4 is 1/13 (or 7.6%), which is just the probability of drawing a 4 from the deck. This makes sense because we have eliminated half of the possibilities (black cards), and we are actually drawing from a set of 26 cards that has 2 fours in it. So, 2 out of 26 (or 1 of 13) of the drawn cards will be a red 4.
The results look fine, but it really seems that I am missing something, because in the relationship P(A | B) = P(A) doesn't take into account the probability of B.
I will be highly grateful is someone could look into my question and give a response.
• Check out this question. Your probability for the intersection isn't always right. Feb 9 '20 at 22:37
• All clear, thank you all for the explanations. Feb 18 '20 at 17:13
$$P(A\cap B)=P(A)P(B)$$ is true if and only if $$A$$ and $$B$$ are independent. So, $$P(A|B)\neq P(A)$$ in general. In your example, it holds because the events are fortunately independent. You already informally found it in your reasoning paragraph, by thinking about the sample space.
For a simple counter example, consider the case where $$A=B$$. $$P(A|B)=1$$ clearly, because if you know that $$B=A$$ happened, the probability of $$A$$ happening is $$1$$.
• Typo in the last sentence: because if you know that A happened. It's B, obviously. Feb 10 '20 at 16:46
• @RuiBarradas It wasn't meant to be a typo, but maybe this is more clear. In the reason part, I was just using the fact that $B=A$ to make sense. Feb 10 '20 at 16:50
• Yes, it is, thanks. Feb 10 '20 at 16:52
P(A ∩ B) = P(A)*P(B) this expression is true only when events A and B are independent. It is not true in general.
It might help to think of P(A|B) as meaning "What percentage of B is A", and to think of all probabilities as conditional; if a probability is not given explicitly as conditional, there's an implicit universal condition; P(A) can be taken as P(A|S) where S is the entire set you're considering. In this case, S is just "any card". So if A is "card is red" and B is "card is a 4", then P(A|S) is "probability that what we draw is red, given that it's from a standard deck" or "percentage of all the cards that are red". P(A|B) is "percentage of all the 4s that are red". P(A ∩ B) = P(A ∩ B|S) = percentage of cards that are red and 4s.
So when you ask "Is P(A) = P(A|B)", you're asking "Is there percentage of all cards that are red equal to the percentage of 4s that are red?" In this case, the answer is "yes". However, this is not in general the case. The property of these percentages being equal is known as "independence".
It's easy to come up with examples of variables that are dependent. For instance, P(Diamond|Red) != P(Diamond). Any time a subpopulation has a property at a rate different from the general population, the conditional probability is different. E.g. P(child goes to a private school|parents make more than \$100k) != P(child goes to private school). | 2021-12-01T19:28:41 | {
"domain": "stackexchange.com",
"url": "https://stats.stackexchange.com/questions/448693/is-conditional-probability-always-equal-to-the-probability-of-the-predicted-even",
"openwebmath_score": 0.7300189137458801,
"openwebmath_perplexity": 447.8158561799293,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.971129090546975,
"lm_q2_score": 0.8577681049901037,
"lm_q1q2_score": 0.8330035596992416
} |
https://mathematica.stackexchange.com/questions/238254/how-can-i-find-all-solutions-of-this-equation | # How can I find all solutions of this equation?
I am trying to solve this equation with integer solution $$x^3 + (x + 1)^3 + (x + 2)^3 + (x + 3)^3 + (x + 4)^3 + (x + 5)^3 + (x + 6)^3 + (x + 7)^3 = y^3.$$ I tried
Solve[{x^3 + (x + 1)^3 + (x + 2)^3 + (x + 3)^3 + (x + 4)^3 + (x +
5)^3 + (x + 6)^3 + (x + 7)^3 == y^3}, {x, y}, Integers]
and got massage
{{y -> ConditionalExpression[
Root[-784 - 420 x - 84 x^2 - 8 x^3 + #1^3 &,
1], (x |
Root[-784 - 420 x - 84 x^2 - 8 x^3 + #1^3 &, 1]) \[Element]
Integers]}}
If I tried
Solve[{x^3 + (x + 1)^3 + (x + 2)^3 + (x + 3)^3 + (x + 4)^3 + (x +
5)^3 + (x + 6)^3 + (x + 7)^3 == y^3, -1000 <= x <= 2000}, {x, y}, Integers]
I got
{{x -> -5, y -> -6}, {x -> -4, y -> -4}, {x -> -3, y -> 4}, {x -> -2,
y -> 6}}
It seems all solutions of the given equation. How can I find all solutions of the given equation?
• On version 12.2.0 the command Solve[{x^3 + (x+1)^3 + (x+2)^3 + (x+3)^3 + (x+4)^3 + (x+5)^3 + (x+6)^3 + (x+7)^3 == y^3}, {x, y}, Integers] gives all solutions: {{x->-5, y->-6}, {x->-4, y->-4}, {x->-3, y->4}, {x->-2, y->6}}. – Roman Jan 15 at 12:56
• Or Reduce[{x^3 + (x + 1)^3 + (x + 2)^3 + (x + 3)^3 + (x + 4)^3 + (x + 5)^3 + (x + 6)^3 + (x + 7)^3 == y^3}, {x, y}, Integers] – cvgmt Jan 15 at 12:58
• Or FindInstance[{x^3 + (x + 1)^3 + (x + 2)^3 + (x + 3)^3 + (x + 4)^3 + (x + 5)^3 + (x + 6)^3 + (x + 7)^3 == y^3}, {x, y}, Integers, 10] which results in {{x -> -5, y -> -6}, {x -> -4, y -> -4}, {x -> -3, y -> 4}, {x -> -2, y -> 6}} in 12.2. – user64494 Jan 15 at 13:51
• One way would be to show that there can be no solutions outside of certain bounds, and then check within those bounds exhaustively. – Daniel Lichtblau Jan 15 at 15:05
Your equation for the sum of 8 consecutive cubes becomes $$y^3=u^3+63u$$ on substitution of $$x=(u-7)/2$$.
y^3 == Simplify[784 + 420 x + 84 x^2 + 8 x^3 /. x -> (u - 7)/2]
y^3 == u (63 + u^2)
The substitution is equivalent to $$u=2x+7$$, implying $$u$$ must be odd.
$$y<$$0 requires $$u<0$$ because $$63+u^2$$ is always positive. $$y>0$$ requires odd $$u>0$$.
Rewrite the equation as the difference of two cubes $$y^3-u^3=63u$$, and expand on the comment by @DanielLichtblau.
For $$y>0$$, the minimum difference between two positive cubes occurs when $$y=u+1$$. This minimum difference is $$1+3u+3u^2$$. As positive odd $$u$$ increases, the minimum difference will eventually exceed $$63u$$ for some $$u$$.
Reduce[{1 + 3 u + 3 u^2 > 63 u, u > 0}, u, Integers]
u [Element] Integers && u >= 20
Therefore, the only possible odd positive $$u$$ are $$1\le u\le 19$$. Similarly for negative odd $$u$$ with $$-19\le u \le -1$$.
A simple search of these 20 candidates gives all 4 solutions.
{x, y} -> Select[
Table[{(u - 7)/2,
CubeRoot[784 + 420 x + 84 x^2 + 8 x^3 /. x -> (u - 7)/2]},
{u, -19, 19, 2}],
IntegerQ[#[[2]]] &]
{x, y} -> {{-5, -6}, {-4, -4}, {-3, 4}, {-2, 6}} | 2021-04-19T03:24:00 | {
"domain": "stackexchange.com",
"url": "https://mathematica.stackexchange.com/questions/238254/how-can-i-find-all-solutions-of-this-equation",
"openwebmath_score": 0.18030065298080444,
"openwebmath_perplexity": 599.4042632390767,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9711290930537121,
"lm_q2_score": 0.8577681013541613,
"lm_q1q2_score": 0.8330035583184712
} |
https://math.stackexchange.com/questions/1766351/inequality-with-a-square-root | # Inequality with a square root
If the inequality $(x+2)^{\frac{1}{2}} > x$ is satisfied. what is the range of x ?
My approach - I squared both the sides and proceeded on to solve the quadratic obtained in order to solve the inequality. However, this method only gives -1 and 2 as the roots of the quadratic. But obviously this is not the correct answer as this inequality ( with equality ) even if x is put equal to -2.
What am I doing wrong , why am I obtaining the wrong interval ?
What is the correct method to solve inequalities of this kind ?
• You can't always square inequalities. You should make sure that both sides are positive. – S.C.B. May 1 '16 at 5:50
• Ahh yes , I had not considered that . Can you give me a hint on how to proceed while keeping this in mind ? – Noob101 May 1 '16 at 5:52
HINT
You can't always square inequalities.
You should make sure that both sides are positive. And you also have to make sure the radicand is positive.
However, in this case, the inequality is not to hard to prove that it is true for $-2 \le x<0$, so it does not really matter. Now you merely have to check $x \ge 0$.
Now, you can square both sides and get $$x+2 \ge x^2 \Leftrightarrow (x+1)(x-2) \le 0$$
When faced with problems such as $$f(x) \ge g(x)$$Where both sides involve square roots, you should split it into two cases (Well, maybe not just two.) in order to avoid the above mentioned problem.
Split the range into $3$ parts:
• $\color\red{x<-2}\implies(x+2)^{\frac12}\not\in\mathbb{R}$
• $\color\red{-2\leq x<0}\implies(x+2)^{\frac12}\geq0>x$
So $\color\green{-2\leq x<0}$ is part of the solution.
• $\color\red{x\geq0}\implies(x+2)^{\frac12},x\geq0$, therefore:
You can square each side of the inequality without affecting the sign.
As you've noted, solving the quadratic gives you $-1<x<2$.
Intersecting this range with $x\geq0$ gives you $0\leq x<2$.
So $\color\green{0\leq x<2}$ is another part of the solution.
Hence the combined solution is the union of:
• $-2\leq x<0$
• $0\leq x<2$
Or simply $-2\leq x<2$.
You could also approach this by considering the properties of the curves for the functions $\ y \ = \ x \$ and $\ y \ = \ \sqrt{x+2} \$ . The square-root function only gives non-negative values and the curve $\ y \ = \ \sqrt{x+2} \$ is the curve for $\ y \ = \ \sqrt{x} \$ shifted horizontally "to the left" by 2 units; so the curve is always above the $\ x-$ axis except at the point $\ (-2, \ 0 ) \$ . Thus, the curve plainly lies above the line $\ y \ = \ x \$ in the interval $\ [ \ -2 \ , \ 0 \ ] \$ .
The intersection point(s) of the two curves is found by setting the functions equal and solving for values of $\ x \$ (so it is all right to square both sides here):
$$\ \sqrt{x + 2} \ = \ x \ \ \Rightarrow \ \ x^2 \ - \ x \ - \ 2 \ = \ 0 \ \ \Rightarrow \ \ x \ = \ -1 \ \ , \ \ x \ = \ 2 \ \ ,$$
as you've already found. The solution $\ x \ = \ -1 \$ is "spurious" since the square-root curve would have to meet $\ y \ = \ x \$ at $\ ( -1, \ -1 ) \$ , so there is only an intersection at $\ (2, \ 2 ) \$ . For $\ x \ > \ 2 \$ , the inequality is not satisfied: the line continues to rise above the square-root curve. So the inequality $\ \sqrt{x + 2} \ > \ x \$ is satisfied only on the interval $\ [ \ -2 \ , 2 \ ) \$ . | 2021-03-02T18:54:24 | {
"domain": "stackexchange.com",
"url": "https://math.stackexchange.com/questions/1766351/inequality-with-a-square-root",
"openwebmath_score": 0.8775291442871094,
"openwebmath_perplexity": 144.55372637134187,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9711290955604489,
"lm_q2_score": 0.8577680977182187,
"lm_q1q2_score": 0.8330035569377005
} |
https://math.stackexchange.com/questions/3108814/minimum-point-of-x2y2-given-that-xy-10 | # Minimum point of $x^2+y^2$ given that $x+y=10$
How do you I approach the following question:
Find the smallest possible value of $$x^2 + y^2$$ given that $$x + y = 10$$.
I can use my common sense and deduce that the minimum value is $$5^2 + 5^2 = 50$$. But how do you approach this mathematically?
Since the condition $$x+y=10$$ is simple and allows easy elemination of one variable, one possible approach is to put the resulting $$y=10-x$$ into the term to mimimize:
$$x^2+y^2=x^2+(10-x)^2:=f(x)$$
You now have a function in one variable (named $$f(x)$$) where you are looking for the minimal value over all real $$x$$.
Since this is a quadratice function, it can be minimized with a bit of algebra and without calculus:
$$f(x)=x^2+(10-x)^2=2x^2-20x+100=2(x^2-10x)+100 = 2(x^2-10x+25) + 50 = 2(x-5)^2+50 \ge 50.$$
So we have $$f(x)\ge 50$$ and equality happens at $$x=5$$ (which implies $$y=5$$).
Since the tag "derivatives" is used, I'll also use the usual calculus approach:
We have $$f(x) = 2x^2-20x+100$$, which implies $$f'(x)=4x-20$$ and $$f''(x)=4$$.
A local mimimum $$x_m$$ has $$f'(x_m)=0$$ as necessay condition, and $$4x_m-20=0$$ easily leads to the only solution $$x_m=5$$ and $$f''(x_m)=4 > 0$$ shows this is a local mimimum, with $$f(x_m)=f(5)=50$$.
Also, one needs to check the behaviour of $$f(x)$$ when $$x$$ tends to $$+\infty$$ and $$-\infty$$, as $$f'(x_m)=0$$ only finds local extrema. Since $$f(x)$$ is a quadratic with positive constant before $$x^2$$, the function tends to $$+\infty$$ in either case, so no interference with the looked for minimum.
Take the vectors $$u=(1,1)$$ and $$v=(x,y)$$ then apply Cauchy-Schwarz inequality. It comes : $$2(x^2+y^2)\geq (x+y)^2=100$$ and for $$x=y=5$$ the equality holds. The minimum value is therefore $$50$$.
Showing the problem graphically
The line is defined by the equation $$f: x + y = 10$$. Minimizing $$x^2 + y^2$$ amounts to finding the smallest circle around the origin touching the line. Since all radii are perpendicular, we search for the intersection of $$f$$ and $$y - x = 0$$ which is solved by $$B: x = y = 5$$.
The general, "If all you have is a hammer, everything looks like a nail" method requiring very little creative thinking is to use Lagrange multipliers.
(Note that there are some nontrivial conditions on when the method of Lagrange multipliers can be used; for example things get a bit messier if $$\nabla g(x,y) = 0$$ is possible when $$g(x,y)=0$$.)
You want to minimize $$f(x,y) = x^2 + y^2$$ subject to the condition $$g(x,y)=x+y-10 = 0$$. The point of using Lagrange multipliers is that you get simple conditions for the critical point of the constrained problem with the cost of having to add another unknown, $$\lambda$$, to the problem:
\begin{align*} \frac{\partial}{\partial x} f(x,y) &= \lambda \frac{\partial}{\partial x} g(x,y), \\ \frac{\partial}{\partial y} f(x,y) &= \lambda \frac{\partial}{\partial y} g(x,y), \\ g(x,y) &= 0. \end{align*}
Plugging in $$f$$ and $$g$$ there gives \begin{align*} 2x &= \lambda, \\ 2y &= \lambda, \\ x+y - 10 &= 0, \end{align*} which is a system of linear equations for $$3$$ variables, giving you $$x=y=5$$.
This method might seem like an overkill for such a simple problem, but once you're familiar with it, it's quite straightforward and effortless to write down the equations.
You can approach it geometrically. Condition $$x+y=10$$ means that the solution lies on the line $$y = -x + 10$$. $$x^2+y^2$$ is the square of distance between $$(0,0)$$ and $$(x, y)$$. That means that you want the closest point on the line $$y = -x + 10$$ to the origin. To get this, project the origin onto the line to get $$(5,5)$$.
$$\large \text{How about this approach:}$$
$$x + y = 10, \text{ so } y = 10 - x \\ x^2 + y^2 = k, y = \sqrt{k - x^2}\\ \rightarrow 10 - x = \sqrt{k - x^2}\\ \rightarrow 100 - 20x + x^2 = k - x^2\\ \rightarrow 2x^2 - 20x + 100 = k\\ f\prime = 4x - 20\\ \text{When x is 5, } f\prime = 0\\~\\ \text{Testing values on the right and left: }\\ f(4) = 62, f(6) = 52\\~\\ \therefore \textbf{There is a minimum point at } x = 5, \textbf{ hence } y = 5.\\ 5^2 + 5^2 = 50$$
$$x^2+y^2\:=\: (x+y)^2-2xy \:=\: 100 -2xy$$ is minimal if $$\,xy\,$$ is maximal (hence $$x,y\geqslant 0$$). Which is maximal if $$\,\sqrt{xy}\:$$ is maximal.
The inequality of arithmetic and geometric means $$\sqrt{xy}\:\leqslant\:\frac{x+y}2 \:=\: 5$$ gets an equality only for $$y=x$$, which means the LHS is maximal.
Thus your common sense is mathematically, especially algebraically, confirmed $$\:\ddot\smile$$
To minimize $$f(x,y) = x^2+y^2$$; given that $$y=10-x$$.
Then: $$f(x,10-x)=x^2+(x-10)^2$$
$$f$$ is minimized for $$f'=0$$, so:
$$f'=2x+2(x-10)=0$$
Gives:
$$x=5\land y=5$$
Thus we have: $$\min (x^2+y^2)=5^2+5^2=50$$
• Thanks, I fixed it. – Max Feb 11 at 15:01
Most answers use functions or derivatives... I'll use another approach: Inequalities!
It's easy to prove that the minimum will be achieved for positive values of $$x$$ and $$y$$. Thus, in virtue of the QM-AM inequality:
$$\sqrt{\frac{x^2+y^2}{2}}\geq \frac{x+y}{2}=5\iff x^2+y^2\geq 50$$
Proof of the Quadratic Mean - Arithmetic Mean inequality ($$x,y\geq0)$$: $$(x-y)^2\geq 0\iff x^2+y^2\geq 2xy\iff 2(x^2+y^2)\ge (x+y)^2\iff\color{red}{\sqrt{\frac{x^2+y^2}{2}}\geq \frac{x+y}{2}}$$ | 2019-10-17T15:03:11 | {
"domain": "stackexchange.com",
"url": "https://math.stackexchange.com/questions/3108814/minimum-point-of-x2y2-given-that-xy-10",
"openwebmath_score": 0.9988733530044556,
"openwebmath_perplexity": 206.74433831532826,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9895109065748189,
"lm_q2_score": 0.8418256432832333,
"lm_q1q2_score": 0.8329956554631223
} |
http://iquk.acpm.it/nonlinear-spring-mass-system.html | Nonlinear Spring Mass System
org are unblocked. The mass of a vertically aligned spring system will exert a net force on the spring, thereby compressing or extending the spring. Simulates are conducted under different conditions, and it is found that when the spring mass is large, the phase plane of particle's motion trajectories change significantly to the condition when spring is. In some cases, the mass, spring and damper do not appear as separate components; they are inherent and integral to the system. Currently the code uses constant values for system input but instead I would like to vectors as input. iii) Write down mathematical formula for each of the arrows (vectors). A linear spring k 1 and a linear damper c 11 are attached to the mass m 1, whereas a linear spring k 2 and a nonlinear damper connects the two masses m 1 and m 2. equilibrium, the spring balance and mass attached to the hook causes the spring to extend from an initial position until the resultant force is zero. Abstract In this paper, we study the nonlinear response of the nonlinear mass-spring model with nonsmooth stiffness. LECTURE 14: DEVELOPING THE EQUATIONS OF MOTION FOR TWO-MASS VIBRATION EXAMPLES Figure 3. A novel nonlinear seat suspension structure for off-road vehicles is designed, whose static characteristics and seat-human system dynamic response are modeled and analyzed, and experiments are conducted. Tools needed: ode45 , plot Description: For certain (nonlinear) spring-mass systems, the spring force is not given by Hooke's Law but instead satisfies F spring = ku + u 3 , where k > 0 is the spring constant and is small but may be positive or negative and represents the. The spring-mass system has also a cubic nonlinearity. 1 Mass-Spring-Damper System The most basic system that is used as a model for vibrational analysis is a block of mass m connected to a linear spring (with spring constant K and unstretched length ℓ0) and a viscous damper (with damping coefficient c). Questions: Suppose a nonlinear spring-mass system satis es the initial value problem (u00+ u+ u3 = 0. Thus, v0= y00= k m y. To start with, a linear two degrees of freedom, spring-mass system was taken and its response was generated using the fourth-order Runge-Kutta method. Structural Dynamics prototype single degree of freedom system is a spring-mass-damper system in which the spring has no damping or mass, the mass has no stiffness or damp-ing, the damper has no stiffness or mass. Free Vibration of Nonlinear Conservative system; Free vibration of nonlinear single degree of freedom conservative systems with quadratic and cubic nonlinearities; Free vibration of nonlinear single degree of freedom nonconservative systems; Free vibration of systems with negative damping. This means that the force needed to travel one inch, millimeter, or degree might not double when it travels two inches, millimeters, or degrees like a linear spring would. 5 m beyond its undisturbed length. Understand solutions to nonlinear differential equations using qualitative methods. A 2 m/s initial velocity pushes the concentrated masses against each other. Dancing Manhole Cover: A Nonlinear Spring-Mass System. The Spring-Mass System Simple Harmonic Motion of Class 11. Dynamic characteristics of nonlinear systems. solve the initial vaule problem by hand first for w not equal to w. Example of a General Nonlinear System. Departmentof PhysicalSciencesandEngineering Prince George’s Community College December31,2010 1 The Simple Plane Pendulum A simple plane pendulumconsists, ideally, of a point mass connected by a light rod of lengthL to a frictionless pivot. In the actual crash test, the spring is highly nonlinear to represent the operation of seatbelt and airbag. Define state variables: x1=x and x2=dx/dt The model in state-space format:. We can introduce. We shall now generalize the simple model problem from the section Finite difference discretization to include a possibly nonlinear damping term $$f(u^{\prime})$$, a possibly nonlinear spring (or restoring) force $$s(u)$$, and some external excitation $$F(t)$$:. This equation is called. -Compatible with pre-2010 vers. When the mass is in motion and reaches the equilibrium position of the spring, the mechanical energy of the system has been completely converted to kinetic energy. Instead of Hooke's linear law for the spring, assume that the force of the spring on the mass is given by a function such as. Only 3 of the 36 DOF have mass (m = 3) and 33 are massless (n = 33). Note as well that while we example mechanical vibrations in this section a simple change of notation (and corresponding change in what the. The system is attached to a dashpot that imparts a damping force equal to 14 times the instantaneous velocity of the mass. The upper wave indicates the prescribed motion which is a given vehicle crash pulse. For the SHM part of the experiment, a single mass of 4kg was hung from the spring and the time required for the system of mass plus spring to execute an integer N number of oscillations was measured with a digital stopwatch. In the actual crash test, the spring is highly nonlinear to represent the operation of seatbelt and airbag. If a force is applied to a translational mechanical system, then it is. (For most of our springs, starting with 50 gm and proceeding in 50 gm increments will be fine, but use some judgment and keep your eye on the graph. 3 Recommended. If the full 36 by 36 matrix were available (and properly partitioned), the 3 by 3 matrix. 1 – Mechanical model of the CMSD system The second mass m 2 only feels the nonlinear restoring force from the elongation, or compression, of the second spring. A horizontal spring-mass system The system in Example 1 is particularly easy to model. A Hajati, SG Kim, Nonlinear Resonator for Ultra Wide Bandwidth Energy Harvesting, MRS Spring Meeting, 2011 (invited). At this instant the spring is in tension and so is providing a restoring force to the left. 10; Lissajous figures - scope 3A80. Keywords: Time integration, implicit Euler method, mass-spring systems. Note that the combination of the two tanks is a closed system with constant mass and fixed volume. The spring force magnitude is a general function of displacement. Between the mass and plane there is a 1 mm layer of a viscous fluid and the block has an area of. The outer product abT of two vectors a and b is a matrix a xb x a xb y a yb x a yb y. It will be shown that CCC for a regular wave is equivalent to a power factor of one in electrical power networks, equivalent to mechanical resonance in a mass-spring-damper. Design Linear and non-linear mendelian randomisation analyses. •Holm, Darryl D. Not only are they ubiquitous in science and engineering, but their mathematics is also v astly harder, man y standard time-series. Solution using analog electronic circuits. The first uses one air track glider and the second uses two similar gliders, so the mass is doubled. It provides a fast way to model geometrically nonlinear parts in system model. Fs takes whatever value, between its limits, to keep the mass at rest. Graphical, iteration, perturbation, and asymptotic methods. Links: DL PDF VIDEO WEB 1 Introduction Mass-spring systems provide a simple yet practical method for mod-eling a wide variety of objects, including cloth, hair, and deformable solids. The average stiffness of the nonlinear spring over each cycle of the motion increases with amplitude, so the period reduces. Free, out of plane vibration of a rotating beam with nonlinear spring-mass system has been investigated. The purpose of this lab experiment is to study the behavior of springs in static and dynamic situations. It will be shown that CCC for a regular wave is equivalent to a power factor of one in electrical power networks, equivalent to mechanical resonance in a mass-spring-damper. Ben-Gal ∗ K. Suppose that the mass of a system is 4 kg and the stiffness is 100 N/m. In particular we will look at mixing problems in which we have two interconnected tanks of water, a predator-prey problem in which populations of both are taken into account and a mechanical vibration problem with two masses, connected. Wu, Submitted Fall 2019. displacement for a linear spring will always be a straight line, with a constant slope. sm = spring-mass system. An analytical approach is developed for nonlinear free vibration of a conservative, two-degree-of-freedom mass–spring system having linear and nonlinear stiffnesses. A 2 kg (20 N) mass is attached to a spring, thereby stretching it 0. The upper wave indicates the prescribed motion which is a given vehicle crash pulse. You can add a Point Mass to body 1 to make up the difference between the current mass and the desired mass. Track design for the new dual track railway line. Applications Nonlinear vibration of mechanical systems. Now, the state may be defined as and the input as u= f. Between the mass and plane there is a 1 mm layer of a viscous fluid and the block has an area of. This was repeated for. Period of vibration is determined. of the chain of mass points. For example, consider a spring with a mass hanging from it suspended from the ceiling. Consider a mechanical system consisting of a mass � sliding on a horizontal bar and connected to a spring with constant � as shown in Figure 2. This model also rises in circuit theory (RLC circuits) and in physics of particles. The second spring is stretched, or compressed, based upon the relative locations of the two masses. This is an example of a nonlinear second-order ode. An equilibriumpoint in a nonlinear system is asymptotically Lyapunov stable if all the eigen-values of the linear variational equations have negative real parts. It is based on the Runge-Kutta series expansion method and zero-order hold assumption. Applying the 1 Hz square wave from earlier allows the calculation of the predicted vibration of the mass. The simplest model for mechanical vibration analysis is a MASS-SPRING system: Mass m Mass m k k with m = mass, and k = spring constant k is defined as the amount of force required to deflect a certain amount of the spring = F/δ =. [email protected] Spring, 2015 This document describes free and forced dynamic responses of single degree of freedom (SDOF) systems. 71 Elton Avenue Watertown, MA 02472 USA tel. When the mass is in motion and reaches the equilibrium position of the spring, the mechanical energy of the system has been completely converted to kinetic energy. Ben-Gal ∗ K. Thus a point particle of mass $$m$$ connected to a harmonic spring with natural. Here the vector L points from the part where the spring is attached to the platform to the mass and L 0 is the unstretched length of the spring. 10; Coupled pendula 3A70. physical model, so called Maxwell model, which is composed of a serial spring-mass-damper model to simulate a vehicle crash event. is the vector of external inputs to the system at time , and is a (possibly nonlinear) function producing the time derivative (rate of change) of the state vector, , for a particular instant of time. Our analysis will be divided into two parts:. To thus mass–spring combination is attached an identical oscillator, the spring of the latter being connected to the mass of the former. Essentially, a linear system is one where doubling the perturbation doubles the response. Not only are they ubiquitous in science and engineering, but their mathematics is also v astly harder, man y standard time-series. The spring is stretched 2 cm from its equilibrium position and the mass is released from rest. , Hou, Qingzhi, and Bozkuş, Zafer. 71 Elton Avenue Watertown, MA 02472 USA tel. Periodic Solutions to Two Nonlinear Spring-Mass Systems N. cantilever system modeled as a nonlinear single mass-spring-damper system with electrostatic actuation (m = 3. I'm having trouble starting the problem off. A mass m m is attached to a nonlinear linear spring that exerts a force F =−kx|x| F = − k x | x |. The spring is stretched 2 cm from its equilibrium position and the mass is. These systems may range from the suspension in a car to the most complex robotics. 1007/s11071-014-1402-5 ORIGINAL PAPER Nonlinear vibration of an axially loaded beam carrying multiple mass–spring–damper systems. Additionally, the one dimensional mass spring simulator is validated for a micro-electro-mechanical system band structure. Free Vibration of Nonlinear Conservative system; Free vibration of nonlinear single degree of freedom conservative systems with quadratic and cubic nonlinearities; Free vibration of nonlinear single degree of freedom nonconservative systems; Free vibration of systems with negative damping. " Proceedings of the. • Spring – Stiffness Element – Idealization • Massless • No Damping • Linear – Stores Energy Basic (Idealized) Modeling Elements – Reality • 1/3 of the spring mass may be considered into the lumped model. Damping might be provided by a dashpot that exerts a continuous force that is proportional to the velocity (F(t)=-cv(t), where c is a constant). Express the total potential energy of the spring, and use this potential energy to. Chaotic Pendulum CP 5 Linear Dynamics Despite being nonlinear, the system can still behave like a harmonic oscillator as long as the total energy remains small compared to the height of the potential barrier at = 0. Nonlinear Dyn (2012) 70:25-41 DOI 10. 78 × 10−6 kg s−1 and the initial gap g =3 µm). The block represents a translational spring with nonlinear force-displacement curve. Not only are they ubiquitous in science and engineering, but their mathematics is also v astly harder, man y standard time-series. The arbitrary constant C that appears in the equation can be expressed in terms of the initial conditions. Nonlinear Continuum Mechanics for Finite Element Analysis. You can use the System Identification app or commands to estimate linear and nonlinear models of various structures. cations, but almost all of the systems we study here rely on the above three examples. Dynamics of rotational motion Read more Nonlinear Pendulum. of the chain of mass points. No damping in the system. jump property to establish UGpAS for a nonlinear mass-spring system with impacts having a (non-necessarily periodic) time varying restitution coefficient. 3) Don’t worry, we’ll show you in Section 6. and Cani, M. 0-cm ruler, a mass scale, a C-clamp and rod attachment, a skew clamp, a regular weight hanger and a few sheets of Cartesian graph paper. Session 2: Mass-Spring-Damper with Force Input, Mass-Spring-Damper with Displacement Input, Pattern for Correct Models for Forces Exerted by Springs and Dampers (8-14). The values of the spring and mass give a natural frequency of 7 Hz for this specific system. In this section we’ll take a quick look at some extensions of some of the modeling we did in previous chapters that lead to systems of differential equations. Client: Swiss Federal Railways, Bahn 2000. When modeling various types of structural systems, one of the goals of the analysis could be to come up with an effective value of stiffness and interpret its scope based on how we compute it from the structural problem at hand. one order, using some lookup table to solve the non-linearities:. Nonlinear Springs Goal: Investigate the behavior of nonlinear springs. Nonlinear Dyn (2012) 70:25-41 DOI 10. If you do not know the equation of. Only horizontal motion and forces are considered. We have used this book extensively in our LS DYNA training programmes which has an equal focus on nonlinear continuum mechanics and a hands on training on the software involving problems ranging from the most simple spring mass dasahpot system to full car crash simulations. Links: DL PDF VIDEO WEB 1 Introduction Mass-spring systems provide a simple yet practical method for mod-eling a wide variety of objects, including cloth, hair, and deformable solids. Thus, researchers have focused on finding methods to effectively isolate or control low-frequency vibrations. SKU: 700009135 High-Pressure Seal, Dual-Spring, 2/pk. Webb and Y. We model an unstructured triangular mesh as a M-S system by treating each triangle edge as a spring (Fig. The response is found by using two different perturbation approaches. M is the vehicle mass, m is the occupant mass, k is the spring stiffness, and δ is the initial slack between the occupant and restraint system. The system of interest for the stability analysis is therefore m 1 q 00 c 1 q 0+ k 1q 1 + c 2 (q 0 q0 2. Determining the displacement of q1 and q2 of two spring attached to one and other and hang from a ceiling, in-terms of W1, W2, K1 and K2. Reducing complexity of models for vibrations in mechanical systems. Extension of the theory to general non-linear multiple body dynamic systems is then made. However, as with other methods for modeling elasticity, ob-. Nonlinear Oscillation Up until now, we've been considering the di erential equation for the (damped) harmonic oscillator, y + 2 y_ + !2y= L y= f(t): (1) Due to the linearity of the di erential operator on the left side of our equation, we were able to make use of a large number of theorems in nding the solution to this equation. When > and > the spring is called a hardening spring. Define state variables: x1=x and x2=dx/dt The model in state-space format:. Non-linear springs •Material Elastic/Plastic •Non-linear soil behavior •Non-linear behavior between soil and structure (i. 3 PSD Analysis of a Wind Load. with systems of DEs. At this instant the spring is in tension and so is providing a restoring force to the left. non-linear time domain solution is used to represent the variation of the shear modulus (G) and the damping ratio ( ) during shaking. [6] Jerison, D. Lumping half the mass from two consecutive layers at their common boundary forms the mass matrix. , device which exhibits both linear springiness and linear damping) • Pure and ideal spring element: • K s = spring stiffness (N/m or N-m/rad) • 1/K s = C s = compliance (softness parameter) ( ) ( ) s12s s12s fKxxKx TKK =−= =θ−θ=θ s s xCf. Consider the mass-spring nonlinear model with the equation x00(t) + kx0(t) + g(x) = 0; where gis continuous and it satis es xg(x) >0 for x6= 0 and k>0 is the friction constant. Example 18 from Introductory Manual for LS-DYNA Users by James M. Systems of Nonlinear Differential Equations 2 / 36 We often work with systems in the general form: x˙ = f(t,x) where solutions x(t) take values in Rn and f(t,x) is a function defined on a subset of R×Rn. $E_{spring}=\int_{0}^{x} q*x^3 dx$ which evaluates to $E_{spring}=q*x^4/4$ when the mass is at it's maximum displacement and the mass is not moving all the energy in the system is stored in the spring. Solution using analog electronic circuits. Tools needed: ode45 , plot Description: For certain (nonlinear) spring-mass systems, the spring force is not given by Hooke's Law but instead satisfies F spring = ku + u 3 , where k > 0 is the spring constant and is small but may be positive or negative and represents the. Indeed, decomposing a deformable system to point mass particles and springs is an intuitive, versatile and powerful model for simulating a great amount of physical phenomena. This equivalency can be exploited to. The simplest solution to this is to linearize the equation of motion around a desired operating point, then apply traditional linear controls methods. Zhang and Whiten noted that Tsuji's non-linear contact model is more realistic and closer to the experimental. We thus obtain an approximate analytical solution of the steady-state response of an SMA mass-spring system. Mass, in kg, is plotted against elongation, in cm, in the graph in Figure 2. Duffing oscillator is an example of a periodically forced oscillator with a nonlinear elasticity, written as $\tag{1} \ddot x + \delta \dot x + \beta x + \alpha x^3 = \gamma \cos \omega t \ ,$ where the damping constant obeys $$\delta\geq 0\ ,$$ and it is also known as a simple model which yields chaos, as well as van der Pol oscillator. Mass Spectrometry Systems. This technique also offers the periodic solutions to the nonlinear free vibration of a conservative, coupled mass–spring system having linear and nonlinear stiffnesses with cubic nonlinearity. Thus, researchers have focused on finding methods to effectively isolate or control low-frequency vibrations. Adams Free oscillations only occur when systems contain both mass and stiffness. Nonlinear Continuum Mechanics for Finite Element Analysis. We thus obtain an approximate analytical solution of the steady-state response of an SMA mass-spring system. Not only are they ubiquitous in science and engineering, but their mathematics is also v astly harder, man y standard time-series. Linear Spring-Mass-System Nonlinear Spring-Mass-System Thin Walled Cylinder Buckling Membrane with Hot Spot 1D Heat Transfer (Radiation) 1D Heat Transfer (Bar) 2D Heat Transfer (Convection) 3D Thermal Load Cooling via Radiation Pipe Whip. For example, a nonlinear system might be described by a set of n first-order nonlinear differential equations. Figure 4 shows a spring dashpot mass system. The values of the spring and mass give a natural frequency of 7 Hz for this specific system. [6] Jerison, D. (The default calculation is for an undamped spring-mass system, initially at rest but stretched 1 cm from its neutral position. First, we will explain what is meant by the title of this section. ADINA Theory and Modeling Guide Volume I: ADINA Solids & Structures December 2012 ADINA R & D, Inc. In terms of energy, all systems have two types of energy: potential energy and kinetic energy. Chapter 21 Explaining the difference between linear and non linear analysis - Duration: 8:32. (For most of our springs, starting with 50 gm and proceeding in 50 gm increments will be fine, but use some judgment and keep your eye on the graph. (When you see this kind of spring-mass system, each Mass is the building block of the system). The Nonlinear Pendulum D. Questions: Suppose a nonlinear spring-mass system satis es the initial value problem (u00+ u+ u3 = 0. Thus, v0= y00= k m y. The system dynamics must be described by a state-space model. 78 × 10−6 kg s−1 and the initial gap g =3 µm). 22 To design a linear, spring-mass system it is often a matter of choosing a spring constant such that the resulting natural frequency has a specified value. Here ‘ ’ is the extension of the spring after suspension of the mass on the spring. •Craig, Kevin: Spring Pendulum Dynamic System Investigation. Advancing to second-order differential equations (those involving both the first and second derivatives), examine a mass-spring system, also known as a harmonic oscillator. Free-body diagrams. Masses and springs are energy storage elements. An external force is also shown. In this paper we study the nature of periodic solutions to two nonlinear spring-mass equations; our nonlinear terms are similar to. Generalization: damping, nonlinear spring, and external excitation¶. Thermo-mechanical nonlinear vibration analysis of a spring-mass-beam system MH Ghayesh, S Kazemirad, MA Darabi, P Woo Archive of Applied Mechanics 82 (3), 317-331 , 2012. If the full 36 by 36 matrix were available (and properly partitioned), the 3 by 3 matrix. u g (2) where M is the mass matrix,. Consider the mass-spring-damper system, described in About Dynamic Systems and Models. Hi All, I need to do modal analysis for a model consists of 10 meter of soil rested on a masonry wall. Assume that the end-mass is much greater than the mass of the beam. Eint D (heat-like terms) Internal energy The non-kinetic non-potential part of a system’s total energy. The second figure denotes a two rotor system whose motion can be specified in terms of θ1 and θ2. Applying the 1 Hz square wave from earlier allows the calculation of the predicted vibration of the mass. To verify the formula for the period, T, of an oscillating mass-spring system. An analytical approach is developed for areas of nonlinear science such as the nonlinear free vibration of a conservative, two-degree-of-freedom mass-spring system having linear and nonlinear stiffnesses. A nonlinear relationship is a type of relationship between two entities in which change in one entity does not correspond with constant change in the other entity. Visit Stack Exchange. As before, the spring mass system corresponds to the DE y00 +4y = 0. This is one of the most famous example of differential equation. its velocity), we turn the above ODE into the following system of two 1st order ODEs involving two unknowns y(t) and v(t): ˆ y0(t) = v(t); v0(t) = g: (8) Systems of linear ODEs will be covered later in. We learned how to solve linear equations here in the Systems of Linear Equations and Word Problems Section. (a) Derive an expression for the equilibrium position of the mass. Most mechanical resonators operate in a linear damping regime, but the behaviour of nanotube and graphene resonators is best described by a model with nonlinear damping. It need not satisfy Hooke's law. 2 1 Nonlinear Dynamics of Nanomechanical and Micromechanical Resonators combination of practical needs as well as fundamental questions. Featured on Meta What posts should be escalated to staff using [status-review], and how do I…. The mass of asymptotically hyperbolic manifolds with non-compact boundary. A prototypical system, namely a thin plate carrying a concentrated hardening cubic spring-mass, is explored. Spring constant kspecifies the intensity of load (force or torque) which causes unit deformation (shift or turning) of the spring. Vibration of Mechanical Systems Figure 7. A block of mass M is attached to a spring of mass m and force constant K. 5 p 7 i 8: @ xi A yi B T is the position of the i-th mo vable spring-mass. 5 m d 2 p C i D dt2 is the inertial force for the i-th mo vable spring-mass. ME8230 Nonlinear Dynamics Prof. A 1-kg mass stretches a spring 20 cm. For example, if I have spring and I pull on it slightly (a small distance x on the figure below), it will undergo oscillations that are nice and regular. *cos (w*t), y (0) = 0. The system is attached to a dashpot that imparts a damping force equal to 14 times the instantaneous velocity of the mass. The analytical solution can describe both stable and unstable behaviors of the vibration system and therefore offer a comprehensive understanding of the nonlinear responses. Chakraverty 1, * Abstract: The dynamic analysis of damped structural system by using finite element method leads to nonlinear eigenvalue problem (NEP) (particularly, quadratic eigenvalue problem). Extension of the theory to general non-linear multiple body dynamic systems is then made. uence the dynamics. (a) Derive an expression for the equilibrium position of the mass. Since the mass is displaced to the right of equilibrium by 0. CMES-Computer Modeling in Engineering & Sciences, 121(3), 947–980. Now pull the mass down an additional distance x', The spring is now exerting a force of. Control Systems”, grant agreement 257462. You can change the system parameters and initial conditions that determine this system; the Demonstration then solves the equations numerically. The mass of asymptotically hyperbolic manifolds with non-compact boundary. spring = ku+ u3; where k > 0 is the spring constant and is small but may be positive or negative and represents the \strength" of the spring ( = 0 gives Hooke’s Law). The following plot shows the system response for a mass-spring-damper system with Response for damping ratio=0. Consider the mass-spring-damper system, described in About Dynamic Systems and Models. The centroid e ectively de nes the geometric center of an object, x = R xdA R dA y = R ydA R dA: 6. The spring is called a hard spring if ± > 0 and a soft spring if ± < 0. In this section we’ll take a quick look at some extensions of some of the modeling we did in previous chapters that lead to systems of differential equations. Consider the mass-spring system shown in Figure 1. Solution: The system is given by (x0 = y y0 = ky g(x): (b) Show that the function V(x;y) := 1. Abstract In this paper, we study the nonlinear response of the nonlinear mass-spring model with nonsmooth stiffness. We learned how to solve linear equations here in the Systems of Linear Equations and Word Problems Section. Mass, in kg, is plotted against elongation, in cm, in the graph in Figure 2. A simple pendulum consists of a mass m hanging from a string of length L and fixed at a pivot point P. Additionally, the one dimensional mass spring simulator is validated for a micro-electro-mechanical system band structure. To define the "strategy" of qualitative methods one has to note that the solutions of equations of non linear dynamic systems are in general non classical transcendental functions of the calculus, which are very complex. Session 2: Mass-Spring-Damper with Force Input, Mass-Spring-Damper with Displacement Input, Pattern for Correct Models for Forces Exerted by Springs and Dampers (8-14). The system is attached to a dashpot that imparts a damping force equal to 14 times the instantaneous velocity of the mass. When the object is displaced horizontally by u (to the right, let’s say), then the spring exerts a force ku to the left, by Hooke’s law. •(M1) 1/4 bus body mass - 2500 kg •(M2) suspension mass - 320 kg •(K1) spring constant of suspension system-80,000 N/m •(K2) spring constant of wheel and tire - 500,000 N/m •(B1) damping constant of suspension system -350 N. Seyedalizadeh Ganji 1, A. This is an example of a nonlinear second-order ode. Apply the appropriate constraints and load. 1 Mass-Spring-Damper System The most basic system that is used as a model for vibrational analysis is a block of mass m connected to a linear spring (with spring constant K and unstretched length ℓ0) and a viscous damper (with damping coefficient c). Jos van Kreij 29,780 views. " Proceedings of the. The Duffing equation may exhibit complex patterns of periodic, subharmonic and chaotic oscillations. Find the equation of motion if the mass is released from equilibrium with an upward velocity of 3 m/sec. Driven mass (dragon) on spring 3A60. The vibration characteristic of a Timoshenko beam resting on non-linear viscoelastic foundation subjected to any number of springs – mass systems (sprung masses) is governed by system of non – linear partial differential equations. A graph showing force vs. Time graph. When displaced to an initial angle and released, the pendulum will swing back and forth with periodic motion. , another nonlinear system x_1 = 1 x3 1 x_2 = x1 x22 equilibrium points are described by x1 = 1 and x2 = 1 note: the equilibrium points of a nonlinear system can be nite (2 in the previous examples, but any other number is possible, including zero) or in nite, and they can be isolated points in state space Oriolo: Stability Theory for. For the combined system, there is no boundary work and no changes in kinetic or potential energy. I am asked to solve analytically. I am also given the nonlinear force deflection plot of the nonlinear spring. The Nonlinear Pendulum D. Vibration suppression for mass-spring-damper systems with a tuned mass damper using interconnection and damping assignment passivity-based control. Explore the dynamics of a double spring mass. The finite element method (FEM), or finite element analysis (FEA), is a computational technique used to obtain approximate solutions of boundary value problems in engineering. Objectives of Analysis of Nonlinear Systems Similar to the objectives pursued when investigating complex linear systems Not interested in detailed solutions, rather one seeks to characterize the system behavior---equilibrium points and their stability properties A device needed for nonlinear system analysis summarizing the system. Oscillations of a mass-spring system; Q-factor; Driven oscillations; First order linear differential equations; Second order linear differential equations; Mass-spring system; Nonlinear spring. The purpose of this lab experiment is to study the behavior of springs in static and dynamic situations. Whitaker) Math 534, Introduction to Partial Differential Equations, Spring 06. However, as with other methods for modeling elasticity, ob-. In this case, the system dynamics become mx¨+ cx˙ + kx3 = 0. 1m^2 in contact the plane. 1) (a) Spring Mass (b) Static Condition (c) Dynamic Condition Figure 7. The block represents a translational spring with nonlinear force-displacement curve. Keywords: Time integration, implicit Euler method, mass-spring systems. You can change parameters in the simulation such as mass, gravity, and damping. The damper is assumed to provide 5% of critical damping. The differential equation is m + + = ∂ ∂2 t2 y(t) b. Provided the structure of the spring is unaltered by these forces, the tension in the spring is proportional to the extension of the spring from the natural length of the spring. Only horizontal motion and forces are considered. A prototypical system, namely a thin plate carrying a concentrated hardening cubic spring-mass, is explored. You need to combine them with dampers to have a realistic simulation. Such type of nonlinear energy harvester with magnetic-spring offers many advantages of high output performance, high tunability, less prone to failure, ease of construction, and low cost. When a spring is stretched or compressed, it stores elastic. Alternative free-body diagram. If the spring has some nonlinear component to it (and many do), then this will result in a different spring constant at the given “zero position”. To linearise the system, we first have to find the operating point. Tools needed: ode45, plot Description: For certain (nonlinear) spring-mass systems, the spring force is not given by Hooke’s Law but instead satisfies Fspring = ku + u3 , where k > 0 is the spring constant and is small but may be positive or negative and represents the “strength” of the spring ( = 0 gives Hooke’s Law). I is the area moment of inertia. Nonlinear Dynamics of a Mass-Spring-Damper System Background: Mass-spring-damper systems are well-known in studies of mechanical vibrations. 4) about either minimum 0 = e gives V = kr2 2 e +mglcos e (15) + 1 2 (2kr. 2: System of two masses and two springs. And the nonlinear equations that arise in engineering and physics might be more complex still, with 10 or 15 variables. Our simple example system is a mass on a spring. Lorenz convection equations for flow produced by temperature gradient and non-linear forced spring-mass system described by the so called Buffing equation. Determining the displacement of q1 and q2 of two spring attached to one and other and hang from a ceiling, in-terms of W1, W2, K1 and K2. A diagram of this system is shown below. M is the vehicle mass, m is the occupant mass, k is the spring stiffness, and δ is the initial slack between the occupant and restraint system. Consider the mass-spring system shown in Figure 1. An ideal mass m=10kg is sitting on a plane, attached to a rigid surface via a spring. When the block is displaced through a distance x towards right, it experiences a net restoring force F. These requirements are even more stringent for nonlinear systems. One approach for describing linear systems, Asymptotic Modal Analysis (AMA), has been extended to nonlinear systems in this paper. Unlike a mass, spring, dashpot system or an LRC circuit, the equation of motion of this levitator is nonlinear in both the input variable (i) and the state variable (x). Nonlinear Spring-Mass-System A mass is attached to a nonlinear spring. it is just kidding. Mass, in kg, is plotted against elongation, in cm, in the graph in Figure 2. This model is for an active suspension system where an actuator is included that is able to generate the control force U to control the motion of the bus body. To verify the formula for the period, T, of an oscillating mass-spring system. In most cases, you choose a model structure and estimate the model parameters using a single command. Manoj Srinivasan Mechanical and Aerospace Engineering srinivasan. 40; Coupled oscillations 3A70; Wilberforce pendulum 3A70. Questions: Suppose a nonlinear spring-mass system satisfies the initial value problem {u + u + u^3 = 0 u(0) = 0, u'(0) = 1 Use ode45 and plot to answer the following: 1. A nonlinear elastic string is considered here as a main structure to be passively controlled using a Nonlinear Energy Sink (NES). This was repeated for. DESCRIPTION. That is, springs in series combine like resistors in parallel (capacitors in series). Tools needed: ode45, plot Description: For certain (nonlinear) spring-mass systems, the spring force is not given by Hooke’s Law but instead satisfies Fspring = ku + u3 , where k > 0 is the spring constant and is small but may be positive or negative and represents the “strength” of the spring ( = 0 gives Hooke’s Law). ⇒ linear model a + Imposing acceleration at x = 0 (control) ⇒ nonlinear model b a M. An important kind of second-order non-linear autonomous equation has the form (6) x′′ +u(x)x′ +v(x) = 0 (Li´enard equation). The response is found by using two different perturbation approaches. Analytical Mechanics (7th ed. • UCY: ECE 424: Introduction to Fault Tolerant Systems (Spring 19, Spring 18, Spring 17, Spring 16, Spring 15) • UCY: ECE 621: Random Processes (Fall 14, Spring 13, Fall 13) • UCY: ECE 690: Fault Tolerant Systems (Fall 2018, Spring 2012) • UCY: ECE 800: Fault Diagnosis and State Classification (Spring 2013, Fall 10). A Model for a General Spring-Mass System with Damping. For the SHM part of the experiment, a single mass of 4kg was hung from the spring and the time required for the system of mass plus spring to execute an integer N number of oscillations was measured with a digital stopwatch. Polynomial and table lookup parameterizations provide two ways to specify the force-displacement relationship. The experimental modal analysis is carried out by three typical types of s. Since Andronov (1932), traditionally three different approaches are used for the study of dynamical systems: qualitative methods, analytical methods, and numerical methods. 090604 Systems []. 1007/s11071-014-1402-5 ORIGINAL PAPER Nonlinear vibration of an axially loaded beam carrying multiple mass–spring–damper systems. ^ (2)* (y) = A. spring–mass model for walking is fully defined by four state variables, which are the center of mass position and velocity [x,y,x,˙ y˙] and three parameters (mass m, leg length l 0, and spring stiffness k). iii) Write down mathematical formula for each of the arrows (vectors). Period of vibration is determined. What is the time period of oscillation of the block spring system? If a mass is attached to a spring, what happens to the frequency of the mass when another identical spring is connected to the mass parallel t. Mass-spring systems, mostly cloth simulators, have recently become popular effects in demos, thanks to the ever increasing processing capabilities. When the suspension system is designed, a 1/4 model (one of the four wheels) is used to simplify the problem to a 1-D multiple spring-damper system. Let’s use the following expression for the force the spring exerts on the mass. Considering the uncertainties and disturbances, the nonlinear mass-spring system (1) in the state-space form can be represented by. , another nonlinear system x_1 = 1 x3 1 x_2 = x1 x22 equilibrium points are described by x1 = 1 and x2 = 1 note: the equilibrium points of a nonlinear system can be nite (2 in the previous examples, but any other number is possible, including zero) or in nite, and they can be isolated points in state space Oriolo: Stability Theory for. Stack Exchange network consists of 176 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. Spring constant kspecifies the intensity of load (force or torque) which causes unit deformation (shift or turning) of the spring. Structural Dynamics prototype single degree of freedom system is a spring-mass-damper system in which the spring has no damping or mass, the mass has no stiffness or damp-ing, the damper has no stiffness or mass. Step 1: Euler Integration We start by specifying constants such as the spring mass m and spring constant k as shown in the following video. Because it is impossible to show animation on paper, the animation figures shows 4 consecutive frames. No damping in the system. · Uniform Boundedness for Reaction-Diffusion Systems with Mass Dissipation, with B. 1 – Mechanical model of the CMSD system The second mass m 2 only feels the nonlinear restoring force from the elongation, or compression, of the second spring. Element types SPRING1 and SPRING2 can be associated with displacement or rotational degrees of freedom (in the latter case, as torsional springs). For examples, I would like to replace my force amplitude F0 with a vector value. Mohammed, EEL5285 Lecture Notes, Spring 2013 Energy Systems Research Laboratory, FIU Power in the Wind • Power in the wind is also proportional to A • For a conventional HAWT, A = (π/4)D2, so wind power is proportional to the blade diameter squared. No system in the macroscopic world is a simple oscillator. The one dimensional mass spring model is developed and the simulator operation is validated through comparison with the published simulation data in the orig-inal paper by J. Free-body diagrams. In the first approach, the method of multiple scales is applied directly to the nonlinear partial differential equations and boundary conditions. To do this, the mass-spring-damper system shown above will be used as an example. Translational mechanical systems move along a straight line. Figure 1 : Nonlinear Mass-Spring System. In this paper, we show that the mathematical structure of the new discretization scheme is explored and characterized in order to represent. The mass of each triangular element, computed as the product of its area, thickness and density, is distributed equally among its three vertices. Sometimes we need solve systems of non-linear equations, such as those we see in conics. Damping might be provided by a dashpot that exerts a continuous force that is proportional to the velocity (F(t)=-cv(t), where c is a constant). Generalization: damping, nonlinear spring, and external excitation¶. But there are examples which are modeled by linear systems (the spring-mass model is one of them). Lumping half the mass from two consecutive layers at their common boundary forms the mass matrix. If you do not know the equation of. 10; Lissajous figures - scope 3A80. In some cases, the mass, spring and damper do not appear as separate components; they are inherent and integral to the system. 1 Phase portrait of a mass-spring system_____ k =1 m =1 0 (a) (b) x x& Most nonlinear systems cannot be easily solved by either of the above two techniques. Dragan Marinković, Zoran Marinković, Goran Petrović: Mass-Spring Systems for Geometrically Nonlinear Dynamic Analysis; Machine Design, Vol. This first of three volumes from the inaugural NODYCON, held at the University of Rome, in February of 2019, presents papers devoted to Nonlinear Dynamics of Structures, Systems and Devices. Consider the mass-spring-damper system, described in About Dynamic Systems and Models. The nonlinear constraint is connected to the beam between two points on the beam through a rigid rod. The main contribution of this research is twofold. previous paper 7\ linear system theory had been employed to estimate the dynamic characteristics of hydraulic mounts[ Recently\ a non!linear mathematical model has been developed for a generic hydraulic mount with only an inertia track "Figure 0"b##\ by formulating ~uid system equations and measuring non!linear system parameters 8[ It has. The spring force acting on the mass is given as the product of the spring constant k (N/m) and displacement of mass x (m) according to Hook's law. The vibration characteristic of a Timoshenko beam resting on non-linear viscoelastic foundation subjected to any number of springs – mass systems (sprung masses) is governed by system of non – linear partial differential equations. which position are equilibrium position. Most mechanical resonators operate in a linear damping regime, but the behaviour of nanotube and graphene resonators is best described by a model with nonlinear damping. CMES-Computer Modeling in Engineering & Sciences, 121(3), 947–980. Those parameters can be further utilized to characterize a physical model, so called Maxwell model, which is composed of a serial spring-mass-damper model to simulate a vehicle crash event. Tools for Analysis of Dynamic Systems: Lyapunov Modeling the Mass-Spring System For nonlinear systems the state may initially tend away from the equilibrium state of interest but subsequently may return to it. It could also be a cork floating in water, coffee sloshing back and forth in a cup of coffee, or any number of other simple systems. The Overflow Blog Podcast 244: Dropping some knowledge on Drupal with Dries. Rout 1 and S. 1 is the familiar linear second-order differential equation. y(0) = 1. Of course, you may not heard anything about 'Differential Equation' in the high school physics. F spring = - k x. When a spring is stretched or compressed, it stores elastic. 0 and plot the solutions of the above initial value. A harmonic spring has potential energy of the form $$\frac{k}{2}x^2\ ,$$ where $$k$$ is the spring's force coefficient (the force per unit length of extension) or the spring constant, and $$x$$ is the length of the spring relative to its unstressed, natural length. Lumping half the mass from two consecutive layers at their common boundary forms the mass matrix. A block of mass M is attached to a spring of mass m and force constant K. In other words all three springs are currently at their natural lengths and are not exerting any forces on either of the two masses and that there are no external forces acting on either mass. · Simple mass-spring system with damping (Linear) · Coupled oscillators: Two mass/spring hanging system (Linear) 1. Modeling and numerical simulation of the nonlinear dynamics of the parametrically forced string pendulum Veronica Ciocanel Advisor: Thomas Witelskiy June 12, 2012 Abstract The string pendulum consists of a mass attached to the end of an inextensible string which is fastened to a support. Element types SPRING1 and SPRING2 can be associated with displacement or rotational degrees of freedom (in the latter case, as torsional springs). In some cases, the mass, spring and damper do not appear as separate components; they are inherent and integral to the system. Math 597-697Y, Nonlinear Dynamical Lattices, Spring 09: Math 132, Calculus II, Course Chair and Section E Fall 08: Math 534, Introduction to Partial Differential Equations, Spring 07: Math 691, Applied Math M. Int J Robust Nonlinear Control 2016; 26: 235 - 251. 5m, we have y(0) = 1 2. Questions: Suppose a nonlinear spring-mass system satis es the initial value problem (u00+ u+ u3 = 0. The Nonlinear Pendulum D. and passenger traffic of up to 230 km/h. Geometry of the structure and the bar properties are given. All vibrating systems consist of this interplay between an energy storing component and an energy carrying (massy'') component. Free-body diagrams. The transient response of undamped non-linear spring mass systems subjected to a constant force excitation is investigated. The low-energy dynamics of a two-dof system composed of a grounded linear oscillator coupled to a lightweight mass by means of a spring with both cubic nonlinear and negative linear components is investigated. The free-body diagram of the system is Figure A-2. Since the mass is displaced to the right of equilibrium by 0. Fs takes whatever value, between its limits, to keep the mass at rest. In this case, the linear function fitting the straight part of the data gives a spring constant of 17. Viscoelastic spring with a rigid moving mass and a viscous dashpot at the end x = 1. We express the widely used implicit Euler method as an energy minimization problem and introduce spring directions as auxiliary unknown variables. The model is formulated with. where y n is a state vector, A(t) n×m is a bounded matrix, which elements are time dependent, B n×m is a constant matrix, u m is a control vector, and g(y) n is a vector, which elements are continuous nonlinear functions, g(0) = 0. Generalization: damping, nonlinear spring, and external excitation¶. The spring is called a hardening spring if >0and a softening spring if <0. Applications Nonlinear vibration of mechanical systems. Linear and nonlinear system. Thermo-mechanical nonlinear vibration analysis of a spring-mass-beam system MH Ghayesh, S Kazemirad, MA Darabi, P Woo Archive of Applied Mechanics 82 (3), 317-331 , 2012. 4 N/mm, you will need to edit the system to set that up. physical model, so called Maxwell model, which is composed of a serial spring-mass-damper model to simulate a vehicle crash event. Example: Simple Mass-Spring-Dashpot system. The Duffing equation is used to model different Mass-Spring-Damper systems. To linearise the system, we first have to find the operating point. This is one of the most famous example of differential equation. A Model for a General Spring-Mass System with Damping. Linear and nonlinear. For the nonlinear mass-spring-damper system the equations of motion can be written mx&&+ bx&+ fK (x)=f (t) where fK(x) represents the nonlinear spring force at displacement x. 1 Lecture 2 Read textbook CHAPTER 1. It is a 2nd order ODE. These are called Lissajous curves, and describe complex harmonic motion. Eint D (heat-like terms) Internal energy The non-kinetic non-potential part of a system’s total energy. flux in the upper coil, &(kg) is the combined mass of the armature and valve, (N) is the magnetic force generated by the lower coil, : (N) is the magnetic force generated by the upper coil, (N/mm) is the spring constant, # (mm) is half the total armature travel, ((kg/s) is the damping coefficient,- (V) is the voltage applied to the lower coil,-. and Peter Lynch, 2002: Stepwise Precession of the Resonant Swinging Spring, SIAM Journal on Applied Dynamical Systems, 1, 44-64. Here the vector L points from the part where the spring is attached to the platform to the mass and L 0 is the unstretched length of the spring. Since the mass an initial velocity of 1 m/s toward equilibrium (to the left) y0(0) = −1. Phase plane dynamics on an X-Y Recorder. BASIC RESULTS FOR LINEAR SYSTEMS The virtual spring-mass-damper approach to passive controller design is first described for linear dynamic systems as in Ref. 1) (a) Spring Mass (b) Static Condition (c) Dynamic Condition Figure 7. Client: Swiss Federal Railways, Bahn 2000. Here ‘ ’ is the extension of the spring after suspension of the mass on the spring. To start with, a linear two degrees of freedom, spring-mass system was taken and its response was generated using the fourth-order Runge-Kutta method. 1: Spring connected to a sliding mass 2. The following Matlab project contains the source code and Matlab examples used for neural network simulation of non linear mass spring damper. That is, springs in series combine like resistors in parallel (capacitors in series). The first spring with spring constant k1 provides a force on m1 of k1x1. Stack Exchange network consists of 176 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. One part is the system, the other part is the rest of the universe called the surroundings. For vibration suppression, we adopt the interconnection and damping assignment passivity‐based control, whereby the system is transformed to a system with a skyhook damper with an artificial structure matrix. use the trigonometric identity cos(n) - cos(m) = 2sin((m-n)/2)*sin((m+n)/2. 3 for the spring-mass-damper system of Figure 2. In this case, the system dynamics become mx¨+ cx˙ + kx3 = 0. Large Mass-Spring-Damper Networks. equilibrium, the spring balance and mass attached to the hook causes the spring to extend from an initial position until the resultant force is zero. The main contribution of this research is twofold. The first attempt to study a nonlinear mass-spring system dates back to Fermi-Pasta-Ulam. For example, a dynamic system is a system which changes: its trajectory → changes in acceleration, orientation, velocity, position. Determining the displacement of q1 and q2 of two spring attached to one and other and hang from a ceiling, in-terms of W1, W2, K1 and K2. is the vector of external inputs to the system at time , and is a (possibly nonlinear) function producing the time derivative (rate of change) of the state vector, , for a particular instant of time. x2 K x1 fS fS =−Kx x() 21 (x2 −x1) fS. You can use the System Identification app or commands to estimate linear and nonlinear models of various structures. Stanford CS248, Spring 2018 INTERACTIVE COMPUTER GRAPHICS This course provides a comprehensive introduction to computer graphics, focusing on fundamental concepts and techniques, as well as their cross-cutting relationship to multiple problem domains in interactive graphics (such as rendering, animation, geometry, image processing). spring are weightless. A 2 m/s initial velocity pushes the concentrated masses against each other. Generalization: damping, nonlinear spring, and external excitation¶. In this paper, we show that the mathematical structure of the new discretization scheme is explored and characterized in order to represent. AP1 Oscillations Page 3 3. system once, then we know all about any other situation where we encounter such a system. i) ceduees to the well-known Kepler problem when the potential is of the form V(iql) = -k/lq! for some real number k>0. Keywords Nonlinear mass–spring system, fast terminal sliding mode, Lyapunov theory, finite time convergence, uncertainties Introduction The position control of the nonlinear second-order systems is one of the principal issues in the fields of control engineering, mechanics, and robotics. To avoid this reduction in the stable time increment, dashpots should be used in parallel with spring or truss elements, where the stiffness of the spring or truss elements is chosen so that the stable time increment of the dashpot and spring or truss is larger than the stable critical time. Linear Design for Nonlinear System. For example, if I have spring and I pull on it slightly (a small distance x on the figure below), it will undergo oscillations that are nice and regular. Hello, I plan to write a bunch of posts about simulating dynamic systems using Python. and passenger traffic of up to 230 km/h. If a force is applied to a translational mechanical system, then it is. The spring with k =500N/m is exerting zero force when the mass is centered at x=0. The system is globally linear in the node. The systems on the boundaries between different phase portrait types are structurally unstable. Then the nonlinear state equation may be written as. m — show oscillations of linear mass & spring system mspr. Molecular & Cellular Biosciences The fundamental unit of all living organisms is invariably the cell, a self-contained system of numerous biomolecules that responds to and interacts with its neighbors and its environment. Definition: Non-linear springs are helical coil springs that exert an inconsistent amount of force as it is under a working load or torque. The damping is linear viscous (ξ = 0. The Duffing equation may exhibit complex patterns of periodic, subharmonic and chaotic oscillations. The objective has been to develop and test different numerical integration techniques that are proposed. Adams Free oscillations only occur when systems contain both mass and stiffness. Dual pitch springs are non-linear compression springs with different amounts of pitch between coils in different sections of the spring. When you compress the spring 10. Admissible Systems In order to fit into the framework of vibrations, linear or nonlinear, systems must have certain properties whose physical description is :. Then the nonlinear state equation may be written as. Aoki, T, Yamashita, Y, Tsubakino, D. This parameter is determined by the system: the particular mass and spring used. 50; Pump a swing 3A95. Two different curves for describing force versus displacement during loading and unloading are given. A 2 m/s initial velocity pushes the concentrated masses against each other. The Spring-Mass System Simple Harmonic Motion of Class 11. Applying this to the example of a mass–spring system in Fig. Chakraverty 1, * Abstract: The dynamic analysis of damped structural system by using finite element method leads to nonlinear eigenvalue problem (NEP) (particularly, quadratic eigenvalue problem). Two Spring-Coupled Masses Consider a mechanical system consisting of two identical masses that are free to slide over a frictionless horizontal surface. 1), and the equivalent spring is nonlinear "hardening" spring of the form k = k1 + k2*x^2, where k1 = 400 kN/m, and k2 = 40 kN/m3. By applying Newton's secont law for rotational systems, the equation of motion for the pendulum may be obtained , and rearranged as. 1 Look for keywords: Hooke's Law, oscillation 9. Since Andronov (1932), traditionally three different approaches are used for the study of dynamical systems: qualitative methods, analytical methods, and numerical methods. The nonlinear systems are very hard to solve explicitly, but qualitative and numerical techniques may help shed some information on the behavior of the solutions. 20; Non-linear systems 3A95; Chaos systems 3A95. Conse-quently, there exists a practical need to understand this behavior in order to avoid it. * * We will compute : force displacement curve. Finally, suppose that there is damping in the spring-mass system. A novel nonlinear seat suspension structure for off-road vehicles is designed, whose static characteristics and seat-human system dynamic response are modeled and analyzed, and experiments are conducted. For example, if I have spring and I pull on it slightly (a small distance x on the figure below), it will undergo oscillations that are nice and regular. For each trial, record the total mass, the starting position of the spring (before hanging the mass) and the ending position of the spring (while it is being stretched). Thus a point particle of mass $$m$$ connected to a harmonic spring with natural. • Example: simple spring. Terms offered: Spring 2016, Spring 2014, Spring 2012 Oscillations in nonlinear systems having one or two degrees of freedom. Nonlinear Spring-Mass-System A mass is attached to a nonlinear spring. A particle of mass m is attached to a rigid support by a spring with force constant κ. Google Scholar. Chapter 21 Explaining the difference between linear and non linear analysis - Duration: 8:32. The system is globally linear in the node. First, we will explain what is meant by the title of this section. These damping estimates, expressed as both the quality factor and the viscous damping equivalent c , are shown in Fig. The spring supporting the mass is assumed to be of negligible mass. Nonsmooth modal analysis: investigation of a 2-dof spring-mass system subject to an elastic impact law Anders Thorin1*, Mathias Legrand1, Stephane Junca´ 2 Abstract The well-known concept of normal mode for linear systems has been extended to the framework of nonlinear dynamics over the course of the. The Mass-Spring System Warren Weckesser Department of Mathematics, Colgate University This Maple session uses the mass-spring system to demonstrate the phase plane, direction fields, solution curves (‘‘trajectories’’), and the extended phase space. In this paper we study the nature of periodic solutions to two nonlinear spring-mass equations; our nonlinear terms are similar to. 3 Recommended. You can also note that when you let the spring go with a mass on the end of it, the mechanical energy (the sum of potential and kinetic energy) is conserved: PE 1 + KE 1 = PE 2 + KE 2. introduced a non-linear damping term, which is a function of displacement δn and velocityδ n. A spring of spring constant k is hung vertically from a fixed surface, and a block of mass M is attached to the bottom of the spring. Example: Simple Mass-Spring-Dashpot system. harne,r,l; wang,k,w, 2014, "mass detection via bifurcation sensing with multistable microelectromechanical system. His research interests include the design and control of intelligent high performance coordinated control of electro-mechanical/hydraulic systems, optimal adaptive and robust control, nonlinear observer design and neural networks for virtual sensing, modeling, fault detection, diagnostics, and adaptive fault-tolerant control, and data fusion. We shall now generalize the simple model problem from the section Finite difference discretization to include a possibly nonlinear damping term $$f(u^{\prime})$$, a possibly nonlinear spring (or restoring) force $$s(u)$$, and some external excitation $$F(t)$$:. The mass of m (kg) is suspended by the spring force. Solution: The system is given by (x0 = y y0 = ky g(x): (b) Show that the function V(x;y) := 1. The nonlinear response of a simply supported beam with an attached spring-mass system to a primary resonance is investigated, taking into account the effects of beam midplane stretching and damping. Spring, 2015 This document describes free and forced dynamic responses of single degree of freedom (SDOF) systems. Reducing complexity of models for vibrations in mechanical systems. Finally, L-hat is a unit vector in the direction of the spring (without this the spring force would just be a scalar). Seyedalizadeh Ganji 1, A. In this last chapter of the course, we handle two physical phenomena which involve a linear second order constant of coefficients differential equations, say the spring mass system and the motion of the pendulum. * system-level energy versus time. Sample: M Q2 B. Виктор Лемпицкий. 17 N m−1, b =1. With more than two variables, nonlinear equations can get immensely complicated. Energy variation in the spring-damping system. One might think of this as a model for a spring-mass system where the damping force u(x) depends on position (for example, the mass might be moving through a viscous medium. Our method converges to the same final result as Fast Simulation of Mass-Spring Systems 209:3). EK 8 <: 1 2 P miv2 i discrete 1 2 R v2dm continuous Kinetic energy A scalar measure of net system motion. 10; Lissajous figures - scope 3A80. Tijsseling, Arris S. The first attempt to study a nonlinear mass-spring system dates back to Fermi-Pasta-Ulam. When a spring is stretched or compressed, it stores elastic. In terms of energy, all systems have two types of energy: potential energy and kinetic energy. Whitaker) Math 534, Introduction to Partial Differential Equations, Spring 06. Then we sampled initial. Extension of the theory to general non-linear multiple body dynamic systems is then made. What is the time period of oscillation of the block spring system? If a mass is attached to a spring, what happens to the frequency of the mass when another identical spring is connected to the mass parallel t. A Hajati, SG Kim, Nonlinear Resonator for Ultra Wide Bandwidth Energy Harvesting, MRS Spring Meeting, 2011 (invited). Step 1: Euler Integration We start by specifying constants such as the spring mass m and spring constant k as shown in the following video. Determining the displacement of q1 and q2 of two spring attached to one and other and hang from a ceiling, in-terms of W1, W2, K1 and K2. To avoid this reduction in the stable time increment, dashpots should be used in parallel with spring or truss elements, where the stiffness of the spring or truss elements is chosen so that the stable time increment of the dashpot and spring or truss is larger than the stable critical time.
o8v57pdr67v fgrius39td1w139 4q67uh7xlbav1 dkhcbhtamp lgynnews68fm435 ca43m48prtceea 679f3u4efqzx j2v42ctkgrapz8m plrv101eu0or0 kuhrqnfifmn7sjv zks67gubvuwemy qtsna78p9ev qi4eb1thv8 mhfke2x13n00 2tcf3s5jryi e5dr52u0lj51kg jklua6hf8k2a7q 8idwnvw0mx33 k5qfn9nh7bmwwm8 0r3rlvmfy4z qunq839n2m l9ve4bs6hk 5q9krqfpv3nx bk7c4lmf5j zh9rsdus7cob6 pr5cy6wjxt9 | 2020-10-26T12:23:32 | {
"domain": "acpm.it",
"url": "http://iquk.acpm.it/nonlinear-spring-mass-system.html",
"openwebmath_score": 0.5827639102935791,
"openwebmath_perplexity": 937.6084491184243,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9895109081214212,
"lm_q2_score": 0.8418256393148982,
"lm_q1q2_score": 0.8329956528383808
} |
https://math.stackexchange.com/questions/274633/if-you-toss-1000-fair-coins-10-times-each-what-is-the-probability-the-some | # If you toss $1000$ fair coins $10$ times each, what is the probability the *some* coin will get $10$ heads?
The answer to this is supposedly close to $0.63$. However, I get approximately $0.9765625$ for the following reason:
The probability of a fair coin flipped $N$ times resulting in all heads is $1/2^N$. In this case, $1/2^{10}=1/1024$. If I flip $M$ coins, $N$ times each, there are $M$ ways for some coin to result in all heads (from the binomial coefficient "$M$ choose $1$"), so the probability of some coin resulting in all heads is $M(1/2^N)$. In this case, $1000\times 1/1024 \approx 0.9765625$.
Can someone please explain the flaw in my reasoning?
• So if there were instead $2000$ coins, the probability would be $1.953125$? – Chris Eagle Jan 9 '13 at 17:58
• Ack, of course that makes no sense. – user1332148 Jan 9 '13 at 18:03
• About $1-\exp(-1)$. Tossing $2^{10}$ coins would be even closer. – Henry Nov 16 '14 at 16:13
• thanks. tried also to find the answer from youtu.be/MEG35RDD7RA?t=2813 – Claudiu Creanga Sep 4 '16 at 11:03
The probability that you get no tails when you flip a fair coin $10$ times is $\left(\frac12\right)^{10}$. The probability that you get at least one tail is therefore $1-\left(\frac12\right)^{10}$. The probability that each of the $1000$ coins comes up tails at least once is then $$\left(1-\left(\frac12\right)^{10}\right)^{1000}\approx0.37642\;.$$ We want the probability of the complementary event, which is therefore about $0.62357$.
As a quick crude estimate note that $$\left(1-\left(\frac12\right)^{10}\right)^{1000}\approx\left(1-\frac1{1000}\right)^{1000}\approx\frac1e\approx0.37\;.$$
• pure genius Brian. – ILoveMath Mar 20 '14 at 22:48
The flaw in your reasoning is that it is possible for the first coin and the tenth coin to come up all heads, in which case, your $1000/1024$ counts them both.
Essentially, $1000/1024$ is the average number (or "expected" number) of coins that will have come up all heads, but that includes the cases where more than one coin comes up heads all the time, so it doesn't work as a probability.
Consider the case where you flip two coins once each. What is the odds that one coin ended up heads? Your calculation would say, "The probability of any one coin coming up heads is $1/2$ so the probability that one of the two coins comes up heads is $2/2=1$." That is nonsensical. In this case, the probability is $3/4$.
The actual probability are $$1-\left(\frac{1023}{1024}\right)^{1000}$$
Basically, $\left(\frac{1023}{1024}\right)^{1000}$ is the odds that none of the coins are heads for all tosses.
In general, if you toss $N$ coins each $M$ times, the odds that at least one coin will have come up all heads is:
$$1-\left(1-\frac{1}{2^M}\right)^N$$
To calculate this, it is easier to deal with the complement, i.e. what it the probability that no coin will have 10 heads. Probability that a coin won't have 10 heads is $1-2^{-10}$, so the result to your question is $1-(1-2^{-10})^{1000} \approx 0.6235762$, the more accurate data you can find here.
You can also analyze this as an example of a binomial distribution, which the probability of success $p = 1/2^{10}$. The probability of none of the coins being all heads is
$$P' = \binom{1000}{0} p^0 (1-p)^{1000}$$
and the probability in question is just $P = 1 - P'$, which gives the results stated above.
• I think this is the understanding I was trying to resurrect after all these years: $$P = \binom{n}{k} p^k q^{n-k}$$ This, and the utility of the complement. – user1332148 Jan 9 '13 at 19:46
There are two binomial distributions here: (1) the number of heads when a single fair coin is tossed 10 times-- $X \sim \mbox{bin}(n=10,p=\frac{1}{2})$; (2) the number of coins (out of 1000 fair coins) that show up heads on all 10 tosses-- $Y \sim \mbox{bin}(n=1000,p=\frac{1}{2^{10}}$) (here, the Bernoulli trial is "all 10 heads" vs. "at least 1 tail").
If the probability that some coin will get 10 heads'' is the probability that exactly 1 of the 1000 coins shows up heads on all 10 tosses, then we're looking at $$\mbox{P}\{Y= 1\}=\binom{1000}{1} \left(\frac{1}{2^{10}}\right)^1 \left(1-\frac{1}{2^{10}}\right)^{999} \enspace.$$ $\big($OP missed a term in the expression, $M \frac{1}{2^N}\; \left(1-\frac{1}{2^N}\right)^{M-1}$, which makes the correct probability approximately $0.368$ not $0.9766$.$\big)$
If the probability that some coin will get 10 heads'' is the probability that at least 1 of the 1000 coins shows up heads on all 10 tosses, then we're looking at $$\mbox{P}\{Y\geq 1\}=1- \mbox{P}\{Y= 0\} = 1 - \binom{1000}{0} \left(\frac{1}{2^{10}}\right)^0 \left(1-\frac{1}{2^{10}}\right)^{1000} = 1 - \left(1-\frac{1}{2^{10}}\right)^{1000}\enspace,$$ which is approximately $0.624$. | 2020-02-23T12:11:47 | {
"domain": "stackexchange.com",
"url": "https://math.stackexchange.com/questions/274633/if-you-toss-1000-fair-coins-10-times-each-what-is-the-probability-the-some",
"openwebmath_score": 0.9265347719192505,
"openwebmath_perplexity": 255.53809236688198,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9777138164089085,
"lm_q2_score": 0.851952809486198,
"lm_q1q2_score": 0.8329660327630424
} |
https://csharp-book.softuni.org/Content/Chapter-8-2-exam-preparation-part-2/grades/grades.html | # Problem: Grades
Write a program that calculates statistics for grades in an exam. At the beginning, the program reads the number of students who attended the exam and for each student – their grade. At the end, the program must print the percentage of students that have grades between 2.00 and 2.99, between 3.00 and 3.99, between 4.00 and 4.99, 5.00 or more, as well as the average grade of the exam.
Note: we use the Bulgarian grading system, where the grade scale starts from 2.00 (Fail) and ends at 6.00 (Excellent): https://en.wikipedia.org/wiki/Grading_systems_by_country#Bulgaria.
## Input Data
Read from the console a sequence of numbers, each on a separate line:
• On the first line – the number of students who attended the exam – an integer within the range [1 … 1000].
• For each individual student on a separate line – the grade on the exam – a real number within the range [2.00 … 6.00].
## Output Data
Print on the console 5 lines that hold the following information:
• "Top students: {percentage of students with grade of 5.00 or more}%".
• "Between 4.00 and 4.99: {between 4.00 and 4.99 included}%".
• "Between 3.00 and 3.99: {between 3.00 and 3.99 included}%".
• "Fail: {less than 3.00}%".
• "Average: {average grade}".
The results must be formatted up to the second symbol after the decimal point.
## Sample Input and Output
Input Output Comments
6
2
3
4
5
6
2.2
Top students: 33.33%
Between 4.00 and 4.99: 16.67%
Between 3.00 and 3.99: 16.67%
Fail: 33.33%
Average: 3.70
5 or more: 2 students = 33.33% of 6
Between 4.00 and 4.99: 1 student = 30% of 6
Between 3.00 and 3.99: 1 student = 20% of 6
Below 3: 2 students = 20% of 6
The average grade is: 2 + 3 + 4 + 5 + 6 + 2.2 = 22. 2 / 6 = 3.70
Input Output Comments
10
3.00
2.99
5.68
3.01
4
4
6.00
4.50
2.44
5
Top students: 30.00%
Between 4.00 and 4.99: 30.00%
Between 3.00 and 3.99: 20.00%
Fail: 20.00%
Average: 4.06
5 or more: 3 students = 30% of 10
Between 4.00 and 4.99: 3 students = 30% of 10
Between 3.00 and 3.99: 2 students = 20% of 10
Below 3: 2 students = 20% of 10
The average grade is: 3 + 2.99 + 5.68 + 3.01 + 4 + 4 + 6 + 4.50 + 2.44 + 5 = 40.62 / 10 = 4.062
## Hints and Guidelines
We will divide the problem into smaller sub-problems, as described below.
### Reading the Input Data and Creating Helper Variables
By the requirements we see that first we will read the number of students, and then, their grades. For that reason, firstly in a variable of int type we will read the number of students. In order to read and process the grades themselves, we will use a for loop. The value of the int variable will be the end value of the i variable from the loop. This way, all iterations of the loop will read each one of the grades.
Before executing the code of the for loop, we will create variables where we will store the number of students for each group: poor results (up to 2.99), results from 3 to 3.99, from 4 to 4.99 and grades above 5. We will also need one more variable, where we will store the sum of all grades, via which we will calculate the average grade of all students.
### Allocating Students into Groups
We run the loop and inside it we declare one more variable, in which we will store the currently entered grade. The variable will be double type and upon each iteration we will check what is its value. According to this value, we increase the number of students in the relevant group by 1, as we should not forget to also increase the total amount of the grades, which we also track.
We can calculate what percentage is taken by a particular group of students from the total number by multiplying the number of students in the relevant group by 100 and then dividing this by the total number of students.
Pay attention to the numeric data type that you work with upon doing these calculations.
The end result is formed in the well know fashion – up to the second symbol after the decimal point.
## Testing in the Judge System
Test your solution here: https://judge.softuni.org/Contests/Practice/Index/517#3. | 2019-09-15T18:40:58 | {
"domain": "softuni.org",
"url": "https://csharp-book.softuni.org/Content/Chapter-8-2-exam-preparation-part-2/grades/grades.html",
"openwebmath_score": 0.4448505938053131,
"openwebmath_perplexity": 936.4552916512112,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES\n\n",
"lm_q1_score": 0.9777138183570424,
"lm_q2_score": 0.8519528076067262,
"lm_q1q2_score": 0.832966032585175
} |
https://math.stackexchange.com/questions/3453558/getting-different-answers-when-integrating-using-different-techniques | # Getting different answers when integrating using different techniques
Question: Is it possible to get multiple correct results when evaluating an indefinite integral? If I use two different techniques to evaluate an integral, and I get two different answers, have I necessarily done something wrong?
Often, an indefinite integral can be evaluated using different techniques. For example, an integrand might be simplified via partial fractions or other algebraic techniques before integration, or it might be amenable to a clever substitution. These techniques give different results. For example, looking over a few other questions on MSE:
1. From this question: evaluate $$\int x(x^2+2)^4\,\mathrm{d}x.$$
• Via the substitution $$u = x^2+2$$, this becomes $$\int x(x^2+2)^4\,\mathrm{d}x = \frac{1}{10}x^{10} + x^8 + 4x^6 + 8x^4 + 8x^2 + \frac{32}{5} + C.$$
• However, multiplying out the polynomial and integrating using the power rule gives $$\int x(x^2+2)^4\,\mathrm{d}x = \frac{1}{10}x^{10} + x^8 + 4x^6 + 8x^4 + 8x^2 + C$$
2. From this question: evaluate $$\int \frac{1-x}{(x+1)^2} \,\mathrm{d}x.$$
• Simplifying the integrand using partial fractions then integrating gives $$\int \frac{1-x}{(x+1)^2} \,\mathrm{d}x = \frac{-2}{(x+1)} - \ln|x+1| + C.$$
• Via integration by parts, we get $$\int \frac{1-x}{(x+1)^2} \,\mathrm{d}x = \frac{x-1}{(x+1)} - \ln|x+1| + C.$$
3. From this question: evaluate $$\int \frac {\tan(\pi x)\sec^2(\pi x)}2\,\mathrm{d}x.$$
• Using the substitution $$u = \sec(\pi x)$$, this becomes $$\int \frac {\tan(\pi x)\sec^2(\pi x)}2\,\mathrm{d}x = \frac {\sec^2(\pi x)}{4\pi} + C.$$
• Using the substitution $$u = \tan(\pi x)$$, this becomes $$\int \frac {\tan(\pi x)\sec^2(\pi x)}2\,\mathrm{d}x = \frac {\tan^2(\pi x)}{4\pi} + C.$$
• Interesting SAQ. What motivated you to post this. – Don Thousand Nov 27 '19 at 18:52
• This question has been posted in order to provide an abstract duplicate target for questions involving problems where students have found multiple antiderivatives using multiple techniques. More often than not, the problem comes down to recognizing that there is a constant of integration running around. – Xander Henderson Nov 27 '19 at 18:56
• Ah, I see. Makes sense. – Don Thousand Nov 27 '19 at 18:56
• There are some relevant chat conversations: chat.stackexchange.com/transcript/message/52713109#52713109 and chat.stackexchange.com/transcript/message/52713523#52713523. – Xander Henderson Nov 27 '19 at 18:57
• ok, good, I still not familiar with that procedure! – dfnu Nov 27 '19 at 23:51
It is entirely possible to correctly evaluate an indefinite integral using different methods and arrive at functions which look different. However, once the constant of integration is taken into account, the functions will agree. The moral of the story is to be mindful of the constants of integration.
### Antiderivatives
When working with indefinite integrals, it is important to remember that the symbol $$\int f(x) \,\mathrm{d}x$$ does not represent a single function, but rather an entire family of antiderivatives of $$f$$. Phrasing things a little bit more formally:
Definition: Suppose that $$f$$ is integrable on the on interval $$(a,b)$$. An antiderivative of $$f$$ is a function $$F$$ such that $$F'(x) = f(x)$$ for all $$x \in (a,b)$$.
With respect to this definition, antiderivatives are not unique. For example, both $$F(x) = \frac{1}{2} x^2 \qquad\text{and}\qquad G(x) = \frac{1}{2} x^2 + 1$$ are antiderivatives of the function $$f(x) = x$$, since $$F'(x) = x = G'(x).$$ Thus any particular integrable function may have many, many distinct antiderivatives. However, it can be shown that if $$F$$ and $$G$$ are both antiderivatives of a function $$f$$ (in the sense defined above), then $$F$$ and $$G$$ differ by at most a constant.
Since antiderivatives differ from each other by at most a constant, it is common to adopt a notation for "the" antiderivative of a function which captures this typically inessential distinction. Thus $$\int f(x)\,\mathrm{d}x$$ represents the entire collection antiderivatives. Moreover, as antiderivatives differ by at most a constant, we also often write $$\int f(x) \,\mathrm{d}x = F(x) + C,$$ where $$F$$ is any particular antiderivative, and $$C$$ is a "constant of integration."
### Error Detection
As noted above, it is entirely possible to tackle a single indefinite integral in multiple ways and get entirely different looking results. However, the primary motivation for doing a problem in more than one way is to detect mistakes. Hence the fact that antiderivates are not unique may be troublesome. Therefore, from a pedagogical or learning point of view, detecting antiderivatives which differ by a constant might be helpful.
In general, one wants to show that if $$F$$ and $$G$$ are both purported antiderivatives of a given function, then $$F - G$$ is a constant function. Showing that $$F-G$$ is constant might not be trivial, but there are a couple of strategies which come to mind:
1. Inspection: Sometimes, it is obvious that two functions differ by only a constant. For example, in example (1), above, the two polynomials differ by $$\frac{32}{5}$$. This can be seen without doing too much work. Thus if either function is an antiderivative of the given function, then both are.
2. A Little Algebra: Other times, it is not immediately obvious that two functions differ by a constant. In example (2), above, it might require a little work: \begin{align} &\left[\frac{-2}{(x+1)} - \ln|x+1|\right] - \left[\frac{x-1}{(x+1)} - \ln|x+1| \right] \\ &\qquad\qquad= \frac{-2}{x+1} - \frac{x-1}{x+1} \\ &\qquad\qquad= \frac{-2 - x + 1}{x+1} \\ &\qquad\qquad= \frac{-(x+1)}{x+1} \\ &\qquad\qquad= -1, \end{align} which is a constant.
3. Differentiate: One possibility is to simply differentiate the two results and see if they are the same. However, one might be able to save a little work: if two functions differ by a constant, then their difference will have derivative $$0$$. In example (3), above: \begin{align} &\frac{\mathrm{d}}{\mathrm{d}x} \left[ \frac {\sec^2(\pi x)}{4\pi} - \frac {\tan^2(\pi x)}{4\pi} \right] \\ &\qquad\qquad= \frac{1}{4\pi} \left( 2\sec(\pi x)\cdot \sec(\pi x)\tan(\pi x)\cdot \pi - 2\tan(\pi x) \cdot \sec(\pi x)^2 \cdot \pi \right) \\ &\qquad\qquad= 0. \end{align} As the derivative of the difference is zero, the original functions differ by (at most) a constant.
Of course, in this example, one could also use trig identities (as suggested in one of the linked answers), but then I wouldn't get to discuss this alternative. ;)
More generally, it is helpful to keep various identities in mind. In particular, if two different procedures give seemingly different results, think about the kinds of functions which are involved, and search out relevant identities. For example, $$\cos(x)^2 + \sin(x)^2 = 1 \qquad\text{and}\qquad \tan(x)^2 = 1 - \sec(x)^2$$ for real $$x$$ such that the functions involved are defined, and $$\log(xy) = \log(x) + \log(y),$$ for all positive $$x$$ and $$y$$. Different substitutions or integration by parts steps may lead
• I quibble a bit with #3 as a check to see that the answers are essentially the same. You're just checking each antidifferation by differentiating. That's something you could or should have done with either one separately. – Ethan Bolker Nov 27 '19 at 19:03
• @EthanBolker I did note at the beginning of that entry that one can simply differentiate both antiderivatives and see that they give the same thing. However, it is often easier to check that the difference has zero derivative---in this case, there are no extra terms which cancel out, so differentiating the difference is no easier or harder than differentiating each function independently. However, there is a principle that I wanted to demonstrate---if you have a better example, please add it. – Xander Henderson Nov 27 '19 at 19:07 | 2021-06-18T12:11:08 | {
"domain": "stackexchange.com",
"url": "https://math.stackexchange.com/questions/3453558/getting-different-answers-when-integrating-using-different-techniques",
"openwebmath_score": 0.9906453490257263,
"openwebmath_perplexity": 345.22351263090866,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9777138144607745,
"lm_q2_score": 0.8519528057272543,
"lm_q1q2_score": 0.832966027428153
} |
https://math.stackexchange.com/questions/3645232/proof-of-explicit-formula-for-recursive-sequence-by-induction | # Proof of explicit formula for recursive sequence by induction
I am new to proof-writing and have just started working on Math for Computer Science on MIT OCW. I encountered the following problem on one of the assignments and would like feedback on my proof.
Let the sequence $$G_{0},G_{1},G_{2},...$$ be defined recursively as follows: $$G_{0}=0,G_{1}=1,$$ and $$G_{n}=5G_{n-1}-6G_{n-2},$$ for every $$n\in \mathbb{N}, n\geq2.$$ Prove that for all $$n \in \mathbb{N}, G_{n}=3^{n}-2^{n}$$.
Proof by Induction:
Base Case: We first check that the hypothesis is true for $$n=0$$ and $$n=1$$.
$$3^{0}-2^{0}=1-1=0=G_{0} \\ 3^{1}-2^{1}=3-2=1=G_{1}$$ Inductive Step: Assume that $$G_{n}=3^{n}-2^{n}$$ for all $$n \in \mathbb{N}$$, for purposes of induction. \begin{align*} \displaystyle G_{n+1}&=5G_{n}-6G_{n-1} \\ &=5(3^{n}-2^{n})-6(3^{n-1}-2^{n-1}) \\ &=5\cdot3^{n}-5\cdot2^n-2\cdot 3 \,\cdot3^{n-1}+3\,\cdot2 \, \cdot2^{n-1} \\&=5\cdot3^{n}-5\cdot2^n-2\cdot3^n+3\,\cdot2^n \\ &=3\cdot3^n-2\cdot2^n \\ &=3^{n+1}-2^{n+1} \tag*{\blacksquare} \end{align*} So I have a few specific questions regarding my proof.
1. Did I do the base case right and is it connected to the inductive step? I was confused by how the recursive definition applies for $$n\geq2$$ and the explicit formula applies for all $$n$$ so I wasn't really sure what my base case should be.
2. In a formal proof, do you have to state the hypothesis in predicate logic? Also, how would you do that here?
3. How did they derive the explicit formula for this problem? Or more generally, how do you derive the explicit formula for a recursively-defined sequence? I understand that this probably requires a long answer, so even resources on the topic would be appreciated.
4. Any other feedback on my proof would be greatly appreciated.
• Your proof looks good. To derive the formula, find the roots of the characteristic equation, which in this case is $x^2=5x-6$; cf. this Wikipedia resource about that Apr 26, 2020 at 18:37
Your base case is correct. There is a fundamental error at the beginning of your induction step, however: when you assume that $$G_n=3^n-2^n$$ for all $$n\in\Bbb N$$, you are assuming precisely the result that you’re supposed to be proving, which makes your argument circular. What you should be assuming as your induction hypothesis is that $$G_k=3^k-2^k$$ for all $$k\le n$$; then you’ll use that hypothesis to show that $$G_{n+1}=3^{n+1}-2^{n+1}$$, using exactly the calculation that you in fact made. In that calculation you used the induction hypothesis when you replaced $$G_n$$ by $$3^n-2^n$$ and $$G_{n-1}$$ by $$3^{n-1}-2^{n-1}$$. (In fact you didn’t need the whole induction hypothesis that the result is true up through $$n$$: you needed only the last two cases, $$n-1$$ and $$n$$.) It wouldn’t be a bad idea to conclude the argument by saying something like The result now follows by induction. | 2022-05-27T20:01:31 | {
"domain": "stackexchange.com",
"url": "https://math.stackexchange.com/questions/3645232/proof-of-explicit-formula-for-recursive-sequence-by-induction",
"openwebmath_score": 0.9947578310966492,
"openwebmath_perplexity": 118.19055403178588,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9777138144607745,
"lm_q2_score": 0.8519528038477825,
"lm_q1q2_score": 0.8329660255905674
} |
http://gasmar.com.br/transworld-cfs-sibfp/5ede8b-volume-of-parallelepiped-proof | The triple scalar product can be found using: 12 12 12. Proof of the theorem Theorem The volume 푉 of the parallelepiped with? The volume of the parallelepiped is the area of the base times the height. Consider three vectors , , and .The scalar triple product is defined .Now, is the vector area of the parallelogram defined by and .So, is the scalar area of this parallelogram times the component of in the direction of its normal. Click here to edit contents of this page. \begin{align} So the first thing that we need to do is we need to remember that computing volumes of parallelepipeds is the same thing as computing 3 by 3 determinants. General Wikidot.com documentation and help section. Something does not work as expected? The Volume of a Parallelepiped in 3-Space, \begin{align} h = \| \mathrm{proj}_{\vec{u} \times \vec{v}} \vec{w} \| = \frac{ \mid \vec{w} \cdot (\vec{u} \times \vec{v}) \mid}{\| \vec{u} \times \vec{v} \|} \end{align}, \begin{align} V = \| \vec{u} \times \vec{v} \| \frac{ \mid \vec{w} \cdot (\vec{u} \times \vec{v}) \mid}{\| \vec{u} \times \vec{v} \|} \\ V = \mid \vec{w} \cdot (\vec{u} \times \vec{v}) \mid \end{align}, \begin{align} V = \mathrm{abs} \begin{vmatrix} w_1 & w_2 & w_3 \\ v_1 & v_2 & v_3\\ u_1 & u_2 & u_3 \end{vmatrix} \end{align}, \begin{align} \begin{vmatrix} 1 & 0 & 1\\ 1 & 1 & 0\\ w_1 & 0 & 1 \end{vmatrix} = 0 \end{align}, Unless otherwise stated, the content of this page is licensed under. (b × c) ? Checking if an array of dates are within a date range, I found stock certificates for Disney and Sony that were given to me in 2011. The three-dimensional perspective … Vector Triple Product Up: Vector Algebra and Vector Previous: Rotation Scalar Triple Product Consider three vectors , , and .The scalar triple product is defined .Now, is the vector area of the parallelogram defined by and .So, is the scalar area of this parallelogram multiplied by the component of in the direction of its normal. We can now define the volume of P by induction on k. The volume is the product of a certain “base” and “altitude” of P. The base of P is the area of the (k−1)-dimensional parallelepiped with edges x 2,...,x k. The Lemma gives x 1 = B + C so that B is orthogonal to all of the x i, i ≥ 2 and C is in the span of the x i,i ≥ 2. Track 11. Watch headings for an "edit" link when available. Calculate the volume and the diagonal of the rectangular parallelepiped that has … Hence the volume $${\displaystyle V}$$ of a parallelepiped is the product of the base area $${\displaystyle B}$$ and the height $${\displaystyle h}$$ (see diagram). \end{align} Parallelepiped is a 3-D shape whose faces are all parallelograms. \text{volume of parallelopiped} &= \text{area of base} \times \text{height}\\ Recall uv⋅×(w)= the volume of a parallelepiped have u, v& was adjacent edges. Substituting this back into our formula for the volume of a parallelepiped we get that: We note that this formula gives up the absolute value of the scalar triple product between the vectors. Notice that we Notify administrators if there is objectionable content in this page. rev 2021.1.20.38359, The best answers are voted up and rise to the top, Mathematics Stack Exchange works best with JavaScript enabled, Start here for a quick overview of the site, Detailed answers to any questions you might have, Discuss the workings and policies of this site, Learn more about Stack Overflow the company, Learn more about hiring developers or posting ads with us. Theorem 1: If $\vec{u}, \vec{v}, \vec{w} \in \mathbb{R}^3$, then the volume of the parallelepiped formed between these three vectors can be calculated with the following formula: $\mathrm{Volume} = \mathrm{abs} ( \vec{u} \cdot (\vec{v} \times \vec{w}) ) = \mathrm{abs} \begin{vmatrix}u_1 & u_2 & u_3\\ v_1 & v_2 & v_3\\ w_1 & w_2 & w_3 \end{vmatrix}$. One such shape that we can calculate the volume of with vectors are parallelepipeds. How to get the least number of flips to a plastic chips to get a certain figure? c 1 c2 c3 In each case, choose the sign which makes the left side non-negative. How can I cut 4x4 posts that are already mounted? Suppose three vectors and in three dimensional space are given so that they do not lie in the same plane. Click here to toggle editing of individual sections of the page (if possible). It follows that is the volume of the parallelepiped defined by vectors , , and (see Fig. Tetrahedron in Parallelepiped. Change the name (also URL address, possibly the category) of the page. By the theorem of scalar product, , where the quantity equals the area of the parallelogram, and the product equals the height of the parallelepiped. View and manage file attachments for this page. See pages that link to and include this page. Wikidot.com Terms of Service - what you can, what you should not etc. My previous university email account got hacked and spam messages were sent to many people. Then the area of the base is. Finally we have the volume of the parallelepiped given by Volume of parallelepiped = (Base)(height) = (jB Cj)(jAjjcos()j) = jAjjB Cjjcos()j = jA(B C)j aIt is also possible for B C to make an angle = 180 ˚which does not a ect the result since jcos(180 ˚)j= jcos(˚)j 9 The height of the parallelogram is orthogonal to the base, so it is the component of $\vec c$ onto $\vec a \times \vec b$ which is perpendicular to the base, \text{comp}_{\vec a \times \vec b}\vec c=\frac{|c. These three vectors form three edges of a parallelepiped. Male or Female ? View wiki source for this page without editing. ; Scalar or pseudoscalar. We can build a tetrahedron using modular origami and a cardboard cubic box. What should I do? Corollary: If three vectors are complanar then the scalar triple product is equal to zero. As a special case, the square of a triple product is a Gram determinant. First, let's consult the following image: We note that the height of the parallelepiped is simply the norm of projection of the cross product. The cross product a × b is shown by the red vector; its magnitude is the area of the highlighted parallelogram, which is one face of the parallelepiped. Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. &= \mathbf a\cdot(\mathbf b \times \mathbf c) \end{align} It is obviously true for m=1. What difference does it make changing the order of arguments to 'append'. + x n e n of R n lies in one and only one set T z. A parallelepiped can be considered as an oblique prism with a parallelogram as base. Therefore if w_1 = 1, then all three vectors lie on the same plane. Page 57 of 80 Geometric Interpretation of triple scalar product Geometrically, one can use triple scalar product to obtain the volume of a parallelepiped. Check out how this page has evolved in the past. Proof of (1). With How were four wires replaced with two wires in early telephone? The altitude is the length of B. One nice application of vectors in \mathbb{R}^3 is in calculating the volumes of certain shapes. Depending on how rigorous you want the proof to be, you need to say what you mean by volume first., How to prove volume of parallelepiped? a 1 a2 a3 (2) ± b 1 b2 b3 = volume of parallelepiped with edges row-vectors A,B,C. Multiplying the two together gives the desired result. &= (\mathbf b \times \mathbf c) \times A \cos \theta\\ If it is zero, then such a case could only arise when any one of the three vectors is of zero magnitude. The surface area of a parallelepiped is the sum of the areas of the bounding parallelograms: Let $\vec a$ and $\vec b$ form the base. (Poltergeist in the Breadboard). Male Female Age Under 20 years old 20 years old level 30 years old level 40 years old level 50 years old level 60 years old level or over Occupation &= \mathbf a\cdot(\mathbf b \times \mathbf c) This restates in vector notation that the product of the determinants of two 3×3 matrices equals the determinant of their matrix product. So the volume is just equal to the determinant, which is built out of the vectors, the row vectors determining the edges. What environmental conditions would result in Crude oil being far easier to access than coal? It displays vol(P) in such a way that we no longer need theassumption P ‰ R3.For if the ambient space is RN, we can simply regard x 1, x2, x3 as lying in a 3-dimensional subspace of RN and use the formula we have just derived. The volume of one of these tetrahedra is one third of the parallelepiped that contains it. &= (\mathbf b \times \mathbf c) \times A \cos \theta\\ View/set parent page (used for creating breadcrumbs and structured layout). An alternative method defines the vectors a = (a 1, a 2, a 3), b = (b 1, b 2, b 3) and c = (c 1, c 2, c 3) to represent three edges that meet at one vertex. This is a … Truesight and Darkvision, why does a monster have both? Proof: The volume of a parallelepiped is equal to the product of the area of the base and its height. Online calculator to find the volume of parallelepiped and tetrahedron when the values of all the four vertices are given. How many dimensions does a neural network have? \text{volume of parallelopiped} &= \text{area of base} \times \text{height}\\ How can I hit studs and avoid cables when installing a TV mount? SSH to multiple hosts in file and run command fails - only goes to the first host. The sum of two well-ordered subsets is well-ordered. $$, site design / logo © 2021 Stack Exchange Inc; user contributions licensed under cc by-sa. After 20 years of AES, what are the retrospective changes that should have been made? The height is the perpendicular distance between the base and the opposite face. The volume of a parallelepiped based on another. The height is the perpendicular distance between the base and the opposite face. Hence, the theorem. Prism is a 3D shape with two equal polygonal bases whose corresponding vertices can be (and are) joined by parallel segments.Parallelepiped is a prism with parallelogram bases. The volume of the spanned parallelepiped (outlined) is the magnitude ∥ (a × b) ⋅ c ∥. To improve this 'Volume of a tetrahedron and a parallelepiped Calculator', please fill in questionnaire. Proof: The proof is straightforward by induction over the number of dimensions. The volume of a parallelepiped is the product of the area of its base A and its height h.The base is any of the six faces of the parallelepiped.$$ It only takes a minute to sign up. An alternative method defines the vectors a = (a 1, a 2, a 3), b = (b 1, b 2, b 3) and c = (c 1, c 2, c 3) to represent three edges that meet at one vertex. Of course the interchanging of rows does in this determinant does not affect the determinant when we absolute value the result, and so our proof is complete. Is it possible to generate an exact 15kHz clock pulse using an Arduino? Is cycling on this 35mph road too dangerous? For permissions beyond … Or = a. volume of parallelepiped with undefined angles, Volume of parallelepiped given three parallel planes, tetrahedron volume given rectangular parallelepiped. area of base of parallelepiped (parallelogram) = $\mathbf b \times \mathbf c$, the vector $\mathbf b \times \mathbf c$ will be perpendicular to base, therefore: Theorem: Given an $m$-dimensional parallelepiped, $P$, the square of the $m$-volume of $P$ is the determinant of the matrix obtained from multiplying $A$ by its transpose, where $A$ is the matrix whose rows are defined by the edges of $P$. The volume of this parallelepiped (is the product of area of the base and altitude) is equal to the scalar triple product. If you want to discuss contents of this page - this is the easiest way to do it. The direction of the cross product of a and b is perpendicular to the plane which contains a and b. Append content without editing the whole page source. Code to add this calci to your website . Volume of parallelepiped by Duane Q. Nykamp is licensed under a Creative Commons Attribution-Noncommercial-ShareAlike 4.0 License. u=−3, 5,1 v= 0,2,−2 w= 3,1,1. The volume of any tetrahedron that shares three converging edges of a parallelepiped has a volume equal to one sixth of the volume of that parallelepiped (see proof). [duplicate], determination of the volume of a parallelepiped, Formula for n-dimensional parallelepiped. \begingroup Depending on how rigorous you want the proof to be, you need to say what you mean by volume first. \vec{u}, \vec{v}, \vec{w} \in \mathbb{R}^3, \mathrm{Volume} = \mathrm{abs} ( \vec{u} \cdot (\vec{v} \times \vec{w}) ) = \mathrm{abs} \begin{vmatrix}u_1 & u_2 & u_3\\ v_1 & v_2 & v_3\\ w_1 & w_2 & w_3 \end{vmatrix}, V = (\mathrm{Area \: of \: base})(\mathrm{height}), h = \| \mathrm{proj}_{\vec{u} \times \vec{v}} \vec{w} \|, \begin{vmatrix}u_1 & u_2 & u_3\\ v_1 & v_2 & v_3\\ w_1 & w_2 & w_3 \end{vmatrix} = 0, \mathrm{abs} ( \vec{u} \cdot (\vec{v} \times \vec{w})) = 0, w_1 \begin{vmatrix}0 & 1\\ 1 & 0\end{vmatrix} + \begin{vmatrix} 1 & 0\\ 1 & 1 \end{vmatrix} = 0, Creative Commons Attribution-ShareAlike 3.0 License. Let's say that three consecutive edges of a parallelepiped be a , b , c . Stack Exchange network consists of 176 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. It is obtained from a Greek word which means ‘an object having parallel plane’.Basically, it is formed by six parallelogram sides to result in a three-dimensional figure or a Prism, which has a parallelogram base. For each i write the real number x i in the form x i = k i, + α i, where k i, is a rational integer and α i satisfies the condition 0 ≤ α i < 1. The length and width of a rectangular parallelepiped are 20 m and 30 m. Knowing that the total area is 6200 m² calculates the height of the box and measure the volume. The volume of any tetrahedron that shares three converging edges of a parallelepiped is equal to one sixth of the volume of that parallelepiped (see proof). In particular, all six faces of a parallelepiped are parallelograms, with pairs of opposite ones equal. The point is \begin{align} So we have-- … Can Pluto be seen with the naked eye from Neptune when Pluto and Neptune are closest. Volume of the parallelepiped equals to the scalar triple product of the vectors which it is build on: . How do you calculate the volume of a 3D parallelepiped? Find out what you can do. As we just learned, three vectors lie on the same plane if their scalar triple product is zero, and thus we must evaluate the following determinant to equal zero: Let's evaluate this determinant along the third row to get w_1 \begin{vmatrix}0 & 1\\ 1 & 0\end{vmatrix} + \begin{vmatrix} 1 & 0\\ 1 & 1 \end{vmatrix} = 0, which when simplified is -w_1 + 1 = 0. Area and volume interpretation of the determinant: (1) ± a b1 1 a b2 = area of parallelogram with edges A = (a1,a2), B = (b1,b2). The volume of a parallelepiped is the product of the area of its base A and its height h.The base is any of the six faces of the parallelepiped. How does one defend against supply chain attacks? Why are two 555 timers in separate sub-circuits cross-talking? Volumes of parallelograms 3 This is our desired formula. (\vec a \times \vec b)|}{|\vec a \times \vec b|}. How would a theoretically perfect language work? As soos as, scalar triple product of the vectors can be the negative number, and the volume of geometric body is not, one needs to take the magnitude of the result of the scalar triple product of the vectors when calculating the volume of the parallelepiped: Surface area. Given that $\vec{u}, \vec{v}, \vec{w} \in \mathbb{R}^3$ and $\vec{u} = (1, 0, 1)$, $\vec{v} = (1, 1, 0)$, and $\vec{w} = (w_1, 0, 1)$, find a value of $w_1$ that makes all three vectors lie on the same plane. $\endgroup$ – tomasz Feb 27 '17 at 15:02 add a comment | 2 Answers 2 Then how to show that volume is = [a b c] The triple product indicates the volume of a parallelepiped. ], determination of the volume of parallelepiped with edges row-vectors a, b,.... There is objectionable content in this page has evolved in the past cardboard cubic box notation that the of... Can calculate the volume 푉 of the area of the parallelepiped defined by vectors, the of! It follows that is the easiest way to do it the least number of dimensions are complanar then scalar. Calculate the volume of parallelepiped with undefined angles, volume of parallelepiped with row-vectors! A cardboard cubic box 4x4 posts that are already mounted can Pluto be with! Parallelepiped defined by vectors, the row vectors determining the edges do it 5,1 v= 0,2, −2 w=.! Equals to the plane which contains a and b parallelepiped volume of parallelepiped proof is the perpendicular between. Equal to the determinant, which is built out of the base and height... The category ) of the parallelepiped with edit '' link when available \vec b $form base... Can I cut 4x4 posts that are already mounted + x n e n of R n lies one. Are already mounted of opposite ones equal equals the determinant, which is built of. Question and answer site for people studying math at any level and professionals in fields! Which it is build on: same plane edges of a parallelepiped are parallelograms, with pairs opposite. Messages were sent to many people TV mount of dimensions math at any level and in... Parallelepiped defined by vectors, the square of a parallelepiped have u, v & was edges. That the product of a parallelepiped be a, b, c two 555 timers in sub-circuits. Product indicates the volume of a parallelepiped be a, b,.. Proof: the volume is just equal to the scalar triple product vector notation that the product a! How rigorous you want the proof to be, you need to say what you,. Oblique prism with a parallelogram as base the opposite face when any of. Product of area of the parallelepiped that contains it$ \begingroup $Depending on how rigorous you want the is... The retrospective changes that should have been made I hit studs and avoid cables when installing a TV?. Conditions would result in Crude oil being far easier to access than?... When Pluto and Neptune are closest ± b 1 b2 b3 = volume of parallelepiped by Duane Nykamp... All parallelograms triple scalar product can be found using: 12 12 to improve this of! Three vectors form three edges of a and b which is built out the... \Vec a \times \vec b$ form the base retrospective changes that should have been made file and run fails! To zero fill in questionnaire a question and answer site for people studying at. Math at any level and professionals in related fields in one and only one set T z that already! A b c ] Or = a is just equal to zero our desired formula in this page parallelepiped. Retrospective changes that should have been made rectangular parallelepiped category ) of the theorem... Proof: the volume of the parallelepiped defined by vectors, the row vectors determining edges... Can calculate the volume of parallelepiped given three parallel planes, tetrahedron volume given rectangular parallelepiped than. Tetrahedron using modular origami and a parallelepiped a3 ( 2 ) ± b b2. Editing of individual sections of the parallelepiped with edges row-vectors a, b, c if is! Parallelepiped equals to the first host ssh to multiple hosts in file and run command fails - only to... A Gram determinant parallelepiped ( is the easiest way to do it mean by volume first c c2. = the volume of the volume of parallelepiped by Duane Q. Nykamp is licensed under a Commons! Be a, b, c you volume of parallelepiped proof to say what you can, are... Previous university email account got hacked and spam messages were sent to many people to generate an exact 15kHz pulse... ) ± b 1 b2 b3 = volume of this parallelepiped ( is the area of the theorem the... Matrix product, possibly the category ) of the base and the opposite face to access than coal we... Is equal to the determinant, which is built out of the parallelepiped defined by vectors,... Mathematics Stack Exchange is a Gram determinant all three vectors is of zero magnitude case, the row determining! Opposite face early telephone ] Or = a least number of dimensions of shapes! By induction over the number of dimensions multiple hosts in file and run command fails only. The least number of flips to a plastic chips to get the least number dimensions! W= 3,1,1 have u, v & was adjacent edges math at level! Given rectangular parallelepiped that contains it the volume of a parallelepiped is equal to zero so volume... Complanar then the scalar triple product of area of the rectangular parallelepiped that has volume of parallelepiped proof tetrahedron in parallelepiped were wires. Determination of the determinants of two 3×3 volume of parallelepiped proof equals the determinant, which is out. Messages were sent to many people the plane which contains a and b avoid cables when installing TV! The category ) of the three vectors are complanar then the scalar triple product in parallelepiped a. Has evolved in the past out of the three vectors form three of. Equal to the determinant of their matrix product then such a case only! As base this is the volume is = [ a b c ] =! Volumes of certain shapes an edit '' link when available on: the. Vectors is of zero magnitude cross product of the parallelepiped with undefined angles, of. Then such a case could only arise when any one of these tetrahedra is one third of rectangular..., b, c triple product is a question and answer site for people studying math at any level professionals. Perpendicular distance between the base times the height you should not etc arguments! Of flips to a plastic chips to get a certain figure all three vectors are complanar then the triple! B, c AES, what you can, what are the changes. Of a and b wires in early telephone parallelepiped Calculator ', please fill in.... It follows that is the volume of parallelepiped with undefined angles, volume of by. Ones equal licensed under a Creative Commons Attribution-Noncommercial-ShareAlike 4.0 License headings for an ''. If there is objectionable content in this page using an Arduino 3 this the. Wires replaced with two wires in early telephone in calculating the volumes of certain shapes cut posts. Gram determinant application of vectors in $\mathbb { R } ^3$ is in calculating volumes. N $-dimensional parallelepiped installing a TV mount -dimensional parallelepiped 3 this is our desired formula Gram determinant =.! Triple scalar product can be considered as an oblique prism with a parallelogram as base evolved. The height 3 this is our desired formula 푉 of the vectors which it is zero then. Side non-negative$ w_1 = 1 $, then such a case could arise. Answer site for people studying math at any level and professionals in related fields 3-D shape faces! Then such a case could only arise when any one of these tetrahedra is one third of the and! Chips to get the least number of flips to a plastic chips to get least! By Duane Q. Nykamp is licensed under a Creative Commons Attribution-Noncommercial-ShareAlike 4.0 License -dimensional parallelepiped its height difference... The page volume and the opposite face need to say what you mean by first! '' link when volume of parallelepiped proof wikidot.com Terms of Service - what you can, what you mean volume. Two 555 timers in separate sub-circuits cross-talking ( used for creating breadcrumbs and structured layout ) times height... Out how this page has evolved in the past that the product of the parallelepiped with to editing... Run command fails - only goes to the scalar triple product indicates the volume of this (. The diagonal of the theorem theorem the volume of a parallelepiped 2 ) ± 1. Avoid cables when installing a TV mount hit studs and avoid cables when installing a TV mount product! If possible ) perspective … volume of a and b is perpendicular to the plane contains... Improve this 'Volume of a triple product indicates the volume is just equal to the plane which a. } ^3$ is in calculating the volumes of certain shapes plastic chips to get the least of. 'Volume of a parallelepiped is the perpendicular distance between the base cross product area. B2 b3 = volume of the vectors,, and ( see Fig a cardboard cubic box cardboard... How were four wires replaced with two wires in early telephone, −2 w= 3,1,1 equals the. Which is built out of the parallelepiped defined by vectors,, and ( see Fig layout ) mean. Or = a, determination of the rectangular parallelepiped that has … tetrahedron parallelepiped... … tetrahedron in parallelepiped category ) of the parallelepiped that contains it, determination of the and. What are the retrospective changes that should have been made, which is built out of parallelepiped... The direction of the page matrix product a 3-D shape whose faces are all.! This page how do you calculate the volume of a parallelepiped can be found using 12! Can I hit studs and avoid cables when installing a TV mount link to and this... Parent page ( used for creating breadcrumbs and structured layout ) least number of dimensions this parallelepiped ( is easiest! Then how to show that volume is just equal to the product of the parallelepiped with undefined angles, of... | 2021-04-20T09:21:54 | {
"domain": "com.br",
"url": "http://gasmar.com.br/transworld-cfs-sibfp/5ede8b-volume-of-parallelepiped-proof",
"openwebmath_score": 0.9352316856384277,
"openwebmath_perplexity": 672.3718183009221,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9777138157595305,
"lm_q2_score": 0.8519528000888386,
"lm_q1q2_score": 0.8329660230218748
} |
http://math.stackexchange.com/questions/52380/how-to-define-weak-solution-for-an-elliptic-pde-with-non-zero-dirichlet-boundary | How to define weak solution for an elliptic PDE with non-zero Dirichlet boundary condition?
For homogeneous Dirichlet boundary condition, for example \left\{\!\! \begin{aligned} &-\Delta u+c(x)u=f(x),x\in\Omega\\ &u|_{\partial\Omega}=0 \end{aligned} \right. The weak solution is defined as a function $u\in H_0^1(\Omega)$ satisfying $$\int_\Omega\left(\sum_{i=1}^n\frac{\partial u}{\partial x_i} \frac{\partial v}{\partial x_i}+c(x)uv\right)\,dx=\int_\Omega fv\,dx$$ for every $v\in H_0^1(\Omega)$.
I wonder how to define weak solution for an elliptic PDE with non-zero Dirichlet boundary condition.
For example, \left\{\!\! \begin{aligned} &-\Delta u+c(x)u=f(x),x\in\Omega\\ &u|_{\partial\Omega}=g \end{aligned} \right.
Evans's Partial Differential Equations (1st edition, Section 6.1.2) says:
... is is necessary for $g$ to be the trace of some $H^1$ function, say $w$. But then $\tilde u:=u-w$ belongs to $H_0^1(\Omega)$, and is a weak solution of the boundary-value problem \left\{\!\! \begin{aligned} &-\Delta \tilde u+c(x)\tilde u=\tilde f(x),x\in\Omega\\ &\tilde u|_{\partial\Omega}=0 \end{aligned} \right. where $\tilde f:=f-(-\Delta w+c(x)w)$
The problem is: how to find the function $w$, in a constructive way?
-
More succinctly, the problem boils down to finding or recovering an $H^1$ function given its values on $\partial\Omega$. Remind me again what $H^1$ denotes? – anon Jul 19 '11 at 11:57
@anon: $H^1(\Omega)$ (Sobolev Space) is defined as: $H^1(\Omega)=\{u\in L_2(\Omega)\mid \frac{\partial u}{\partial x_i}\in L_2(\Omega),i=1,2,\ldots,n\}.$ You've reminded me that my question seems not well asked. Need I rewrite it more concisely? – Roun Jul 19 '11 at 12:27
I don't see providing background behind a problem as a negative - just that the PDE stuff isn't necessary to understand the second bolded question. Also, I have a habit of never remembering the labels of various continuity/differentiability classes. In any case, if $\Omega$ happens to be a ball, then you can find a harmonic function $w$ using the Poisson kernel - otherwise I'm not sure what explicit constructions exist. – anon Jul 19 '11 at 13:04
You don't find $w$. $w$ is a "given", in the following sense. For the weak formulation of the problem to make sense, the statement
$$u|_{\partial\Omega} = g$$
is in fact the following statement: $\exists$ a fixed $w \in H^1(\Omega)$ such that the trace of $w$ to $\partial \Omega$ is equal to the trace of $u$.
The relevant section in Evans is trying to explain this. Basically what he is trying to say is that the intuition for the Dirichlet problem with "strong" solutions, where you prescribe boundary value as some continuous function $g$ on the boundary, must be replaced by an appropriate weak version defined relative to the trace operator to hypersurfaces, when you consider the weak formulation of the problem. This is because a solution $u$, as an object in the space $W^{1,2} = H^1$, is only an equivalent class of functions defined up to sets of measure zero. If $\Omega$ is a sufficiently regular open set, $\partial\Omega$ has measure zero, so it is meaningless to state that $u$ coincides with $g$ on $\partial\Omega$, since $u$ can always be modified on just $\partial\Omega$ to give any value you want there.
You should compare this to, for example, Theorem 8.3 in Gilbarg and Trudinger, Elliptical partial differential equations of second order, which states
Let [$L$ be an elliptic operator]. Then for $\psi \in W^{1,2}(\Omega)$ and $g,f^i\in L^2(\Omega)$, $i = 1 , \ldots, n$, the generalized Dirichlet problem, $Lu = g + D_i f^i$ in $\Omega$, $u = \psi$ on $\partial\Omega$ is uniquely solvable.
-
I ran across this old question and I would like to say something about it. While the point of view provided by Willie Wong is satisfactory, there is a theory of "inverse traces" which answers OP's question in bold more or less directly. For an example, have a look at the following theorem taken from Kufner, John, Kufcik, "Function Spaces".
6.9.2 Theorem Let $p>1$ and $\Omega\in C^{0, 1}$ [meaning that its boundary is Lipschitz and bounded]. Then there exists a continuous linear mapping $T$ from $W^{1-1/p, 1}(\partial \Omega)$ into $W^{1,p}(\Omega)$ such that for $v=Tu$ we have $v=u$ on $\Omega$ [in trace sense].
The idea of the proof is simpler than one might expect (at least, it is simpler than I expect). Suppose that $$\Omega=\mathbb{R}^{n+1}_+=\{(x, t)\in \mathbb{R}^n\times \mathbb{R}\ :\ t>0\}.$$ If you have a function $u\in C^{\infty}_0(\mathbb{R}^n)$ you can construct a function $v\in C^\infty(\mathbb{R}^{n+1}_+)$ by convolution against the heat kernel: $$v(x,t)=\frac{1}{(4\pi t)^{n/2}}\int_{\mathbb{R}^n} e^{-\frac{\lvert x-y\rvert^2}{4t}}u(y)\, dy.$$ Then the trace of $v$ on the boundary $\{t=0\}$ is $u$, as we already know.
If $\Omega$ is a domain with a nice enough boundary, we can use charts to perform this construction locally and then patch everything up with a partition of unity. This is the idea. Unfortunately, there are some technical difficulties which require the introduction of the cumbersome fractionary order Sobolev spaces $W^{1-1/p, p}$.
- | 2014-08-20T22:49:44 | {
"domain": "stackexchange.com",
"url": "http://math.stackexchange.com/questions/52380/how-to-define-weak-solution-for-an-elliptic-pde-with-non-zero-dirichlet-boundary",
"openwebmath_score": 0.9662479162216187,
"openwebmath_perplexity": 165.87094676732104,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9777138144607744,
"lm_q2_score": 0.8519528000888386,
"lm_q1q2_score": 0.8329660219153959
} |
https://math.stackexchange.com/questions/576903/integral-int-0-infty-1f-2-left-beginarrayc-tfrac12-1-tfrac32-endarr | # Integral $\int_0^\infty{_1F_2}\left(\begin{array}{c}\tfrac12\\1,\tfrac32\end{array}\middle|-x\right)\frac{dx}{1+4\,x}$
I need to evaluate this integral to a high precision: $$\large I=\int_0^\infty{_1F_2}\left(\begin{array}{c}\tfrac12\\1,\tfrac32\end{array}\middle|-x\right)\frac{dx}{1+4\,x}$$ Symbolic integration in Mathematica cannot handle this integral. When I try to evaluate it numerically, it converges extremely slowly and the result is very unstable, so I am not even sure how many correct digits I got. It looks kinda $I\stackrel?\approx0.6212...$
So, my only hope is to find a closed form for $I$ in terms of functions for which fast and robust numerical algorithms exist. Could you please help me to find it?
• An equivalent form of you integral: $\displaystyle I=\frac12\int_0^\infty J_0(x)\operatorname{arccot} x\,dx$. – Vladimir Reshetnikov Nov 22 '13 at 18:59
• And another closed form (although not as simple and nice as in Ron Gordon's answer): $\displaystyle I=\frac\pi4-\frac18G_{1,3}^{3,0}\left(\tfrac14\middle|\begin{array}c\tfrac12\\ - \tfrac12,0,0\\\end{array}\right)$. – Vladimir Reshetnikov Nov 22 '13 at 19:11
• @CarlMummert: the ocntext (which I have observed myself) is that Mathematica takes an unusually long time to evaluate this integral numerically. At least with v. 8.0.4, one can numerically integrate improper integrals such as those above, but in this case, over two hours of computation on a current processor yielded nothing. Truncating the integration interval yielded results, but the convergence was maddeningly slow. That alone should be sufficient motivation to ask if there is a closed form that may be evaluated much more quickly. (1/2) – Ron Gordon Nov 25 '13 at 14:51
• (2/2) Turns out that, indeed this is the case. Fast algorithms exists for even the modified Struve function, so that the time to evaluate the closed form was quite negligible. (Thus, satisfying my definition of "closed form": math.stackexchange.com/questions/562769/…) I think this should have satisfied the OP's reasonable request. – Ron Gordon Nov 25 '13 at 14:54
• @CarlMummert My personal feeling is that always asking users to explain context would place an unnecessary burden on them. A certain integral could came up in many-pages-long chain of computations that originated in some physical or economical model. I do not see how it would be useful in general to post the full context here. Many integrals are interesting self-contained problems that could be used to demonstrate some ingenious approaches and techniques that could be widely applicable irrespectively of where the integral came from. – Vladimir Reshetnikov Nov 25 '13 at 20:41
OK, I have an analytical result:
$$\frac{\pi}{4} \left [K_0(1) \mathbf{L}_{-1}(1) + K_1(1) \mathbf{L}_{0}(1)\right ] \approx 0.621255$$
where $K_0$ and $K_1$ are modified Bessel functions of the second kind, and $\mathbf{L}_{0}$ and $\mathbf{L}_{-1}$ are modified Struve functions of the first kind.
This may be obtained by recognizing that (+)
$${_1F_2}\left(\begin{array}{c}\tfrac12\\1,\tfrac32\end{array}\middle|-x\right) = \int_0^1 du \, J_0\left ( 2 u \sqrt{x}\right )$$
The integral is then, upon reversing order,
$$\int_0^1 du \, \int_0^{\infty} dx \frac{J_0\left ( 2 u \sqrt{x}\right )}{1+4 x}$$
The following will need a derivation (++):
$$\int_0^{\infty} dx \frac{J_0\left ( 2 u \sqrt{x}\right )}{1+4 x} = \frac12 K_0(u)$$
The stated result is then
$$\frac12 \int_0^1 du \, K_0(u) = \frac{\pi}{4} \left [K_0(1) \mathbf{L}_{-1}(1) + K_1(1) \mathbf{L}_{0}(1)\right ]$$
which may be found in the DLMF.
Derivation of (+)
Note that the coefficient of $x^n$ in ${_1F_2}\left(\begin{array}{c}\tfrac12\\1,\tfrac32\end{array}\middle|-x\right)$, by definition, is
$$a_n = \frac{\frac12 \left (\frac12 + 1 \right ) \left (\frac12 + 2 \right ) \cdots \left (\frac12 + n-1 \right )}{n! \frac{3}{2} \left (\frac{3}{2} + 1 \right ) \left (\frac{3}{2} + 2 \right ) \cdots \left (\frac{3}{2} + n-1 \right )} \frac{(-1)^n}{n!}$$
which, after simplification, is
$$a_n = \frac{(-1)^n}{(2 n+1) (n!)^2}$$
Then, note that
$$J_0(2 u \sqrt{x}) = \sum_{n=0}^{\infty} (-1)^n \frac{u^{2 n}}{(n!)^2} x^n$$
Then
$$\int_0^1 du \, J_0(2 u \sqrt{x}) = \sum_{n=0}^{\infty} \frac{(-1)^n}{(2 n+1)(n!)^2} x^n$$
and one may see that the coefficients of the respective power series are equal.
Derivation of (++)
Sub $x=r^2$ to get
$$\int_0^{\infty} dr \, r \frac{J_0(2 u r)}{1+4 r^2}$$
Now write
$$J_0(2 u r) = \frac{1}{2 \pi} \int_0^{2 \pi} d\theta \, e^{i 2 u r \cos{\theta}}$$
so that we now have as the integral
$$\frac{1}{2 \pi} \int_0^{2 \pi} d\theta \, \int_0^{\infty} dr \, r \frac{e^{i 2 u r \cos{\theta}}}{1+4 r^2}$$
Note that we may simply convert back to rectangular coordinates to get
$$\frac{1}{2 \pi} \int_{-\infty}^{\infty} dx \, e^{i 2 u x} \, \int_{-\infty}^{\infty} \frac{dy}{1+4 x^2+4 y^2}$$
The inner integral is simple, so we are back to a single integral:
$$\frac18 \int_{-\infty}^{\infty} dx \frac{e^{i 2 u x}}{\sqrt{1+4 x^2}}$$
By subbing $x=\frac12 \sinh{t}$ and using the definition
$$K_0(u) = \int_0^{\infty} dt \, \cos{(u \, \sinh{t} )}$$
we obtain the stated result.
• @VladimirReshetnikov: thanks, that's really kind of you. – Ron Gordon Nov 22 '13 at 18:47
• @ Ron Gordon: Take a look at my timing 1.014 s when it is calculated numerically . – user64494 Nov 22 '13 at 19:30
• @ Ron Gordon: This is an old-fashioned math, isn't it? – user64494 Nov 22 '13 at 19:33
• @user64494: not sure what you mean by "old-fashioned math." But i can say that, even at 1 sec evaluation time, if we had to evaluate many thousands of values (say, $-p x$ rather than $x$), then finding the analytical result is even more valuable. – Ron Gordon Nov 22 '13 at 19:36
• @cleo: words fail. Thank you. – Ron Gordon Nov 26 '13 at 18:42 | 2020-02-23T11:54:46 | {
"domain": "stackexchange.com",
"url": "https://math.stackexchange.com/questions/576903/integral-int-0-infty-1f-2-left-beginarrayc-tfrac12-1-tfrac32-endarr",
"openwebmath_score": 0.8220704197883606,
"openwebmath_perplexity": 558.335581733399,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9777138099151278,
"lm_q2_score": 0.8519528038477824,
"lm_q1q2_score": 0.8329660217178909
} |
https://www.physicsforums.com/threads/what-type-of-functions-can-satisfy-f-x-f-a-x.542279/ | # Homework Help: What type of functions can satisfy f(x)=f(a-x)?
1. Oct 20, 2011
### McLaren Rulez
1. The problem statement, all variables and given/known data
Given a function such that $f(x)=f(a-x)$ where a is some constant, what can we say about this function? This isn't actually homework; so please forgive the fact that I'm being a little vague
2. Relevant equations
None.
3. The attempt at a solution
I think that a periodic function (with period a) which is also even satisfies this but is there something else that can also satisfy this condition? Not really sure how to proceed in proving either that this is the only answer or to show an alternative.
2. Oct 20, 2011
### lurflurf
All you can say about the function is it is symmetric about x=a/2.
What are the domain and range?
If domain were real numbers for example we could define f to be an arbitrary function when x<=a/2 then define f when x>a using the given functional equation.
or just take an arbitrary function g and define
f=(1/2)g(x)+(1/2)g(a-x)=(1/2)g(a/2+(x-a/2))+(1/2)g(a/2-(x-a/2))
Last edited: Oct 20, 2011
3. Oct 20, 2011
### McLaren Rulez
The domain and range are the set of real numbers.
Thank you for the reply lurflurf. But a function which is symmetric about a should be of the form f(a+x) = f(a-x). Not quite the same as mine, right?
The second part looks fine but what is that function g(x)+g(a-x) for an arbitrary g?
Last edited: Oct 20, 2011
4. Oct 20, 2011
### lurflurf
That should have been symmetric about x=a/2
If
f(x)=(1/2)g(x)+(1/2)g(a-x)
then
f(a-x)=(1/2)g(a-x)+(1/2)g(x)=f(x)
in fact f=g for functions of the type desired
or write any function f
f(x)=(1/2)[f(x)+f(a-x)]+(1/2)[f(x)-f(a-x)]
if
(1/2)[f(x)-f(a-x)]=0
regardless
(1/2)[f(x)+f(a-x)] is a function of your type and in some sense the function of your type most like f
5. Oct 20, 2011
### Ray Vickson
There are many functions that are NOT symmetric about x = a/2, but satisfy the OPs equation. All you can say is that the function is periodic, with period a.
RGV
6. Oct 20, 2011
### McLaren Rulez
Ray, could you post an example? Thank you both for replying
7. Oct 20, 2011
### Ray Vickson
Sorry: I mis-read the original as f(x-a) instead of your f(a-x). So, the statement regarding symmetry about x = a/2 is correct: the function _is_ symmetric. And: it is not (necessarily) periodic. Mea culpa.
RGV | 2018-08-18T09:02:00 | {
"domain": "physicsforums.com",
"url": "https://www.physicsforums.com/threads/what-type-of-functions-can-satisfy-f-x-f-a-x.542279/",
"openwebmath_score": 0.7468229532241821,
"openwebmath_perplexity": 970.9050389347731,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9777138183570424,
"lm_q2_score": 0.8519527963298946,
"lm_q1q2_score": 0.8329660215596609
} |
https://mathoverflow.net/questions/168641/counting-chains-of-inclusions/168647 | # Counting chains of inclusions
Let $g(n,k)$ be the number of chains
$$A_k \subset A_{k-1} \subset\dots\subset A_1 \subset A_0$$
of $k$ proper subset inclusions, where $A_k\neq\emptyset$ and $A_0$ is a standard $n$-element set. Then
$$\sum_{k\ge 0} (-1)^k g(n,k) = (-1)^{n-1}.$$
I can prove this by a fairly boring and unilluminating induction. (And also by a category-theoretic argument involving traces of geometric realizations in derivators, which is how I first noticed it.) Does it have a nicer combinatorial proof, by (say) bijections, generating functions, Mobius inversion, etc.?
• This not an answer to the question as you posed it, but I suspect that your "category-theoretic argument involving traces of geometric realizations in derivators" reduces to a (probably) simpler homotopy theoretic proof. Namely, $g(n,k)$ is the number of $(k-1)$-simplices in the barycentric subdivision of the boundary of the standard $(n-1)$-simplex. Hence your sum (up to the $k=0$ summand and up to sign) computes the Euler characteristic of the $(n-2)$-sphere. – Karol Szumiło May 30 '14 at 17:43
• Hall's formula for the Möbius function of a finite bounded ranked poset says that any such poset $P$ satisfies $\mu\left(P\right) = \sum_{k\geq 0} \left(-1\right)^k \left(\text{number of chains } 0 = x_0 < x_1 < \cdots < x_k = 1 \text{ in } P\right)$, where $0$ and $1$ denote the lower bound and the upper bound of $P$. In your case, the poset is the Boolean lattice, but notice that your $g\left(n,k\right)$ counts chains $0 = x_0 < x_1 < \cdots < x_{k+1} = 1$ rather than $0 = x_0 < x_1 < \cdots < x_k = 1$ so you are skipping the $k = 0$ addend of Hall's formula. – darij grinberg May 30 '14 at 17:55
• (The $k = 0$ addend, of course, is only relevant for $n = 0$. It is more important that your sign is different because of the $k$ shift.) The Möbius function of the Boolean lattice is easily computed, as it is multiplicative and the Boolean lattice is a product of $2$-chains. – darij grinberg May 30 '14 at 17:59
• @darijgrinberg, that looks like an answer; why don't you post it as one? – Mike Shulman May 31 '14 at 22:18
• @KarolSzumiło, you're probably right. – Mike Shulman May 31 '14 at 23:35
A chain $$A_k \subset A_{k-1} \subset\dots\subset A_1 \subset A_0$$ can be represented by the ordered partition $(B_1, B_2, \dots, B_{k+1})$ of the set $A_0=\{1, 2, \dots, n\}$ where $B_1=A_k$, $B_2=A_{k-1}-A_k$, $\dots,$ $B_{k+1}=A_0-A_1$.
First, a generating function proof. If we wanted to count such chains (or ordered partitions) without signs, the exponential generating function would be $\sum_{j=0}^\infty (e^x-1)^j = 1/(2-e^x)$ (http://oeis.org/A000670). With signs, the generating function is $$\sum_{j=0}^\infty (-1)^{j+1}(e^x-1)^j= -e^{-x} = \sum_{n=0}^\infty (-1)^{n+1} \frac{x^n}{n!}.$$
For a combinatorial proof using a sign reversing involution that changes the parity of the number of blocks, write the entries of each block in increasing order, with a bar between blocks, so $\{2,4,6\}\{1,3\}\{5\}$ would be written as $2\, 4\, 6 \,|\, 1\, 3 \,|\, 5$. Find the first position, if there is one, where a number is followed by a larger number. If there is a bar there, remove it, and if there is no bar there then put one in. So our example would be mapped to $2 \,|\, 4 \,6 \,|\, 1 \,3 \,|\, 5$ . As another example, $4 \,|\, 3 \,|\, 1 \,2\,|\,$ would map to $4 \,|\, 3\,|\, 1 \,|\, 2$ . The only ordered partitions not paired up are of the form $n \,|\, n-1\,|\, \cdots \,|\,2 \,|\, 1\,$ .
• Thank you! (As a helpful note to other readers, those are exponential generating functions.) – Mike Shulman Jun 1 '14 at 4:15
• I added the word "exponential". – Ira Gessel Jun 1 '14 at 4:47
There is quite direct proof from point of view of topological combinatorics.
Let $\Delta$ be the standard $(n-1)$-simplex with vertex set $A_0$. Considering the barycentric subdivision $\Delta'$ of $\Delta$, the vertices of $\Delta'$ are nonempty subsets of $A_0$ and faces of $\Delta'$ are subsets $\{A_k, \dots, A_1\}$ of vertices of $\Delta'$ such that $$\emptyset \neq A_k \subset A_{k-1} \subset \cdots \subset A_1 \subseteq A_0.$$ (Note that the last inclusion needn't be proper.)
In this correspondence, your chains correspond to $(k-1)$-faces of $\Delta'$ which do not contain the barycentre $A_0$. These are the faces of the barycentric subdivision of the boundary of $\Delta$, which is topologically an $(n-2)$-sphere. There, if you were summing in your sum for $k \geq 1$, the value you obtain is the minus Euler characteristic of the $(n-2)$-sphere, which is $-(1 + (-1)^{n-2})$. Since you sum for $k \geq 0$, you get $(-1)^{n-1}$.
• Well, this answer now only explains the comment of Karol S. (I did not refresh the comments before posting an answer.) – Martin Tancer May 30 '14 at 18:06
• That's okay; there is a distressing tendency on MO to post answers as comments, so I'm glad to have it here as an answer. – Mike Shulman May 31 '14 at 23:37
This can indeed be seen as a consequence of the properties of the Möbius function of a poset. We recall two important properties of this function (see §7.2.1 in Vic Reiner's and my Hopf Algebras in Combinatorics (arXiv:1409.8356v5), which is the first reference that comes into my mind because it is currently open in my editor):
Property P1. If $$P$$ is a finite bounded poset, then $$\mu\left(P\right) = \sum_{k\geq 0} \left(-1\right)^k \left(\text{number of chains } 0 = x_0 < x_1 < \cdots < x_k = 1 \text{ in } P\right)$$.
Property P2. If $$P$$ and $$Q$$ are two finite bounded posets, then the Cartesian product $$P \times Q$$ (with componentwise order) satisfies $$\mu\left(P \times Q\right) = \mu\left(P\right) \cdot \mu\left(Q\right)$$.
Now, let's return to the question at hand. We assume that $$n$$ is positive, because otherwise your claim is only valid under a very creative interpretation of the $$g\left(n,k\right)$$ and the sum. Let $$B_n$$ be the poset of all subsets of $$\left\{1,2,\ldots ,n\right\}$$, ordered by inclusion. Then, $$B_n$$ is the $$n$$-fold Cartesian product $$\underbrace{B_1 \times B_1 \times \cdots \times B_1}_{n \text{ times}}$$; thus, by iterated application of Property P2, we obtain $$\mu\left(B_n\right) = \mu\left(B_1\right)^n = \left(-1\right)^n$$ (since $$B_1$$ is a $$2$$-element chain and thus has $$\mu\left(B_1\right) = -1$$). Hence,
$$\left(-1\right)^n = \mu\left(B_n\right)$$
$$= \sum_{k\geq 0} \left(-1\right)^k \left(\text{number of chains } 0 = x_0 < x_1 < \cdots < x_k = 1 \text{ in } B_n\right)$$ (by Property P1)
$$= \sum_{k\geq 0} \left(-1\right)^k \left(\text{number of chains } \emptyset = A_0 \subsetneq A_1 \subsetneq \cdots \subsetneq A_k = \left\{1,2,\ldots,n\right\}\right)$$
$$= \sum_{k\geq 1} \left(-1\right)^k \underbrace{\left(\text{number of chains } \emptyset = A_0 \subsetneq A_1 \subsetneq \cdots \subsetneq A_k = \left\{1,2,\ldots,n\right\}\right)}_{\substack{=g\left(n,k-1\right) \\ \text{(not }g\left(n,k\right)\text{, since your chain does not start at }\emptyset\text{)}}}$$ (we got rid of the $$k = 0$$ addend here, since this addend is $$0$$)
$$= \sum_{k\geq 1} \left(-1\right)^k g\left(n,k-1\right) = \sum_{k\geq 0} \left(-1\right)^{k+1} g\left(n,k\right)$$.
Dividing by $$-1$$, we transform this into
$$\left(-1\right)^{n-1} = \sum_{k\geq 0} \left(-1\right)^k g\left(n,k\right)$$,
qed. | 2020-04-04T21:11:43 | {
"domain": "mathoverflow.net",
"url": "https://mathoverflow.net/questions/168641/counting-chains-of-inclusions/168647",
"openwebmath_score": 0.9530738592147827,
"openwebmath_perplexity": 331.08590639242794,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9777138196557983,
"lm_q2_score": 0.8519527944504227,
"lm_q1q2_score": 0.832966020828554
} |
http://mathhelpforum.com/algebra/138004-help-solving-equation.html | # Math Help - Help solving equation
1. ## Help solving equation
How is $\sqrt{50}y=100$ solved? The answer in my book is $2\sqrt{50}$
2. Originally Posted by Mike9182
How is $\sqrt{50}y=100$ solved? The answer in my book is $2\sqrt{50}$
I get
$\sqrt{50}y=100$
$5\sqrt{2}y=100$
$y=\frac{100}{5\sqrt{2}}$
$y=\frac{20}{\sqrt{2}}=10\sqrt{2}$
3. How did $\frac{20}{\sqrt{2}}$ become $10\sqrt{2}$?
4. Originally Posted by Mike9182
How did $\frac{20}{\sqrt{2}}$ become $10\sqrt{2}$?
$\frac{20}{\sqrt{2}} = \frac{20}{\sqrt{2}} \times \frac{\sqrt{2}}{\sqrt{2}} = \frac{20 \times \sqrt{2}}{2} = 10 \times \sqrt{2}$
Note: The answer you have written in the question is $2\sqrt{50} = 2 \sqrt{5 \times 5 \times 2} = 2 \times 5 \times \sqrt{2} = 10 \times \sqrt{2}$
5. $
\frac{20}{\sqrt{2}}=\frac{20}{\sqrt{2}}\times {\color{red}\frac{\sqrt{2}}{\sqrt{2}}}= \frac{20\sqrt{2}}{\sqrt{2}\times \sqrt{2}}=\frac{20\sqrt{2}}{2}=10\sqrt{2}
$
6. Thanks, I understand now.
7. I have another equation I need help solving, the correct answer is 1/8.
$\frac{2}{3}x^{-1/3}-\frac{1}{3}x^{-2/3}=0$
8. Originally Posted by Mike9182
I have another equation I need help solving, the correct answer is 1/8.
$\frac{2}{3}x^{-1/3}-\frac{1}{3}x^{-2/3}=0$
Let $u = x^{-1/3}$.
then the given equation becomes:
$\frac{2u}{3}-\frac{u^2}{3}=0$
$\frac{2u - u^2}{3} = 0$
$2u - u^{2} = 0$
$u^{2} - 2u = 0$
$u(u-2) = 0$
So, $u = 0$ OR $u = 2$
substitute $u= x^{\frac{-1}{3}}$, you get
$x^{\frac{-1}{3}} = 2$
cube both sides to get:
$x^{-1} = 8$
that is
$\frac{1}{x} = 8$
therefore,
$x = \frac{1}{8}$
9. ## another question
Originally Posted by harish21
Let $u = x^{-1/3}$.
then the given equation becomes:
$\frac{2u}{3}-\frac{u^2}{3}=0$
$\frac{2u - u^2}{3} = 0$
$2u - u^{2} = 0$
$u^{2} - 2u = 0$
$u(u-2) = 0$
So, $u = 0$ OR $u = 2$
substitute $u= x^{\frac{-1}{3}}$, you get
$x^{\frac{-1}{3}} = 2$
cube both sides to get:
$x^{-1} = 8$
that is
$\frac{1}{x} = 8$
therefore,
$x = \frac{1}{8}$
A direct solution if a-b =0 a=b Simplify the equals to find x^1/3 =1/2 so x=1/8
bjh | 2014-12-19T14:50:36 | {
"domain": "mathhelpforum.com",
"url": "http://mathhelpforum.com/algebra/138004-help-solving-equation.html",
"openwebmath_score": 0.892774224281311,
"openwebmath_perplexity": 2903.009132366608,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9777138131620183,
"lm_q2_score": 0.8519528000888386,
"lm_q1q2_score": 0.832966020808917
} |
http://mathhelpforum.com/calculus/130409-improper-integrals.html | 1. ## improper integrals
Suppose that
g(x)=1 for x<=0 and
g(x)=-1 for x>0
a) Find the limit as R-> infinity of integral (from -R to R) of g(x).
b) Repeat a), but from -R to R+n with n a positive number
c) Does the integral from -infinity to infinity of g(x) converge or diverge
2. I think the answers are:
a) 0 because of cancellation, but is there a good mathematical way to show this?
b) n, the -R and R region will essentially cancel leaving you with n, but I'm not sure if this is correct, and it is certainly not formal.
c) I think it diverges, but I'm not sure since the limit is 0...this seems contradictory.
A strange problem . . . I think I've solved it.
. . But my approach isn't very formal either.
Suppose that: . $g(x) \;=\;\begin{Bmatrix} 1 && x \leq 0 \\ -1 && x > 0 \end{Bmatrix}$
(a) Find: . $\lim_{R\to\infty} \int^R_{\text{-}R} g(x)\,dx$
The graph of $y \,=\,g(x)$ looks like this, for $x \in [-R,\,R]$
Code:
|
|
* - - - *1
: | R
- + - - - + - - - + - -
-R | :
-1o - - - *
|
|
Since the integral represents area,
. . $\int^R_{\text{-}R} g(x)\,dx$ is the total area of the shaded regions below.
Code:
|
|
* - - - *1
|:::A:::| R
- + - - - + - - - + - -
-R |:::B:::|
-1o - - - *
|
|
Since the area of $B$ is the negative of the area of $A$,
. . the total area is 0 (zero).
That is: . $\int^R_{\text{-}R}g(x)\,dx \;=\;0$
Therefore: . $\lim_{R\to\infty}\int^R_{\text{-}R} g(x)\,dx \;=\;0$
(b) Repeat (a), but: . $\lim_{R\to\infty} \int^{R+n}_{\text{-}R}\!\! g(x)\,dx$ . with $n > 0.$
The graph is a variation of the one in part (a).
Code:
|
|
* - - - *1
|:::A:::| R R+n
- + - - - + - - - + - - * - -
-R |:::B:::|::C::|
-1o - - - * - - *
|
|
Once again, the area of $B$ is the negative of the area of $A.$
. . And the area of $C$ is: . $-n$
Hence: . $\int^{R+n}_{\text{-}R}g(x)\,dx \;=\;-n$
Therefore: . $\lim_{R\to\infty}\int^{R+n}_{\text{-}R}\!\! g(x)\,dx \;=\;-n$
c) Does: . $\int^{\infty}_{\text{-}\infty} g(x)\,dx$ . converge or diverge?
As shown in part (a), it converges to 0.
4. I'm afraid Soroban is wrong about part (c). (Which is very, very unusual!).
The definition of $\int_{-\infty}^{\infty} f(x) dx$ is $\lim_{\alpha\to -\infty} \int_{-\infty}^a f(x) dx+ \lim_{\beta\to\infty}\int_a^\infty f(x)dx$
where $\alpha$ and $\beta$ go to infinity independently.
The integral in part (a), $lim_{R\to\infty}\int_{-R}^R f(x)dx$ is the "Cauchy principal value". If the integral itself exists, then so does the Cauchy principla value and they are equal. But the Cauchy principal value may exist when the integral itself does not.
The integral in (b) shows that the integral in (c) does NOT converge.
5. Thanks guys. I'm still a little confused on part c). So it diverges, essentially because the value of the limit that we found depended on the limit of integration, i.e., on n? I'm just not entirely sure how b) shows c) diverges...I think it is because of the dependence on n. | 2016-08-28T12:52:43 | {
"domain": "mathhelpforum.com",
"url": "http://mathhelpforum.com/calculus/130409-improper-integrals.html",
"openwebmath_score": 0.9365847706794739,
"openwebmath_perplexity": 687.2324735177972,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9777138151101525,
"lm_q2_score": 0.8519527963298947,
"lm_q1q2_score": 0.8329660187934641
} |
https://math.stackexchange.com/questions/463884/show-that-s-fracp2i-p-in-bbb-z-i-in-bbb-n-is-dense-in-bbb-r/463893 | Show that $S=\{\frac{p}{2^i}: p\in\Bbb Z, i \in \Bbb N \}$ is dense in $\Bbb R$. [duplicate]
Show that $S=\{\frac{p}{2^i}: p\in\Bbb Z, i \in \Bbb N \}$ is dense in $\Bbb R$.
Just found this given as an example of a dense set while reading, and I couldn't convince myself of this claim's truthfulness. It kind of bugs me and I wonder if you guys have any idea why it is true. (I thought of taking two rational numbers that I know exist in any real neighborhood and averaging them in some way, but I didn't get far with that idea..)
Thank you!
marked as duplicate by Adar Hefer, Ayman Hourieh, Cameron Buie, Amzoti, Tom OldfieldAug 9 '13 at 21:13
• $S$ is a proper subgroup of $\mathbb Q$ – mrs Aug 9 '13 at 20:32
• @BabakS. Thanks for the idea, but I'm unfamiliar with group theory at the moment.. I'll read up on the proof for $\Bbb Q$ being dense in $\Bbb R$, I don't remember it at all. Thanks for the ideas! – Adar Hefer Aug 9 '13 at 20:37
• Yes. That is just a small point. I see the tag. See Cameron's idea. – mrs Aug 9 '13 at 20:40
• This has been asked here before. – Pedro Tamaroff Aug 9 '13 at 20:40
• You're right. I didn't find it before, but a "related" question just came up at the right so this is indeed a duplicate. – Adar Hefer Aug 9 '13 at 20:44
Try writing out elements in the set: these are rational numbers whose denominators are powers of $2$. So the elements look like
$$\frac{1}{2}, \frac{1}{4}, \frac{3}{4}, \frac{1}{8}, ...$$
The "gap" between $\frac{n}{2^{m}}$ and $\frac{n + 1}{2^{m}}$ can be made as small as you like by simply letting $m$ be large enough; so if you think of the numbers as points on the line, they can be really close together.
An idea of how to put rigor into this: Let $r \in \mathbb{R}$ and $\epsilon > 0$. Choose $m$ large enough that $\frac{1}{2^m} < \epsilon$ and consider the numbers of the form $\frac{n}{2^m}$. Choosing $n$ correctly will then give
$$|r - \frac{n}{2^m}| < \epsilon$$
which gives density.
• So a good choice of $n$ could be $n= \lfloor {r2^m}\rfloor$, right? – Adar Hefer Aug 9 '13 at 20:41
Suppose not, so that there exist $a,b\in\Bbb R$ with $a<b$ such that for all $p\in\Bbb Z$ and all $n\in\Bbb N,$ we have $\frac{p}{2^n}\le a$ or $\frac{p}{2^n}\ge b$. For each $n\in\Bbb N,$ let $p_n$ the greatest integer $p$ such that $\frac{p}{2^n}\le a.$ (Why must there exist such a $p$?) From this, it follows by hypothesis that $$\frac{p_n}{2^n}\le a<b\le\frac{p_n+1}{2^n}$$ for all $n\in\Bbb N$. But then $$0<b-a\le\frac{p_n+1}{2^n}-\frac{p_n}{2^n}=\frac1{2^n}$$ for all $n\in\Bbb N.$ (Why?) Can you derive a contradiction from this?
• Alright, I think I followed you through. A contradiction would come from taking $n$ large enough so that $a,b$ are arbitrarily close to one another. – Adar Hefer Aug 9 '13 at 20:47
• Loosely speaking, yes, that's the idea. Alternately, rewrite as $$2^n\le\frac1{b-a}$$ for all $n\in\Bbb N,$ whence we've put a real upper bound on the natural numbers (since $n<2^n$ for all $n\in\Bbb N$), which is impossible by the Archimedean property. – Cameron Buie Aug 9 '13 at 20:52
I like to think of the answer intuitively. Represent $p$ in binary (base 2). Then $\frac{p}{2^i}$ is simply a number with finitely many binary digits. Conversely, any number whose binary representation has finitely many digits can be written as $\frac{p}{2^i}$.
To show a set is dense, we have to show that given an element $a$ in the set, we can always find an element $b\neq a$ such that $a$ is arbitrarily close to $b$. As a consequence, the density is infinite: You can find infinite numbers from the set in an unit interval. That's the intuitive meaning of "dense".
If you look at the intuition, it should be clear that the set is dense: You can always find a number that is as close as you want to any other number. | 2019-07-24T08:57:10 | {
"domain": "stackexchange.com",
"url": "https://math.stackexchange.com/questions/463884/show-that-s-fracp2i-p-in-bbb-z-i-in-bbb-n-is-dense-in-bbb-r/463893",
"openwebmath_score": 0.8881745338439941,
"openwebmath_perplexity": 164.223642148513,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9777138138113964,
"lm_q2_score": 0.8519527963298947,
"lm_q1q2_score": 0.8329660176869853
} |
https://math.stackexchange.com/questions/498584/factor-xy7-x7y7 | # Factor $(x+y)^7-(x^7+y^7)$
I encountered the following problem while preparing for upcoming math contests.
Factor $(x+y)^7-(x^7+y^7)$.
I got zero for $(x+y)^7-(x^7+y^7)$, however, the solutions said it's
$$7xy(x+y)(x^2+xy+y^2)(x^2+xy+y^2)$$
Can someone explain how this is possible?
• If you got zero, did you by any means work in characteristic$~7$? If you don't know what that means, you probably misapplied the binomial theorem. Sep 19, 2013 at 13:31
Be careful: if you obtained $0$, did you equate $$(x+y)^7 \text{ with }(x^7+y^7)?$$
That's a careless oversight of distributing the exponent over a sum (and is not valid). As a general rule (barring cases like $n = 1)$, $$(x + y)^n \neq x^n + y^n$$
Use the binomial theorem to easily expand $(x + y)^7$,
$$(x+y)^n = {n \choose 0}x^n y^0 + {n \choose 1}x^{n-1}y^1 + {n \choose 2}x^{n-2}y^2 + \cdots + {n \choose n-1}x^1 y^{n-1} + {n \choose n}x^0 y^n,$$ then factor. No quick way around that.
Recall Pascal's Triangle (image from Wikipedia) for recalling the coefficients of the expansion of a binomial:
The bottom row gives the coefficients in the expansion of $(x + y)^7$.
• Yes that's exactly what I did, I have made the stupidest math mistake ever, thanks! Sep 19, 2013 at 13:16
• Hahaha, we've all likely been there, done that! Sep 19, 2013 at 13:17
Expand $(x+y)^7$, you would get:
\begin{align*} (x+y)^7-(x^7+y^7)&=x^7+7x^6y+21x^5y^2+35x^4y^3+35x^3y^4+21x^2y^5+7xy^6+y^7-x^7-y^7 \\ &=7x^6y+21x^5y^2+35x^4y^3+35x^3y^4+21x^2y^5+7xy^6 \\ &=7xy(x^5+3x^4y+5x^3y^2+5x^2y^3+3xy^4+y^5) \\ &=7xy(x+y)(x^4+2x^3y+3x^2y^2+2x^3+y^4) \\ &=7xy(x+y)(x^2+xy+y^2)^2,\text{which is your given answer.} \end{align*}
• I was being stupid, I made (x+y)^7=x^7+y^7 hahaha ty Sep 19, 2013 at 13:13
Motivation for this (late) answer is to demistify the derivation of the square factor, and show that it reduces to the routine factorization of an associated quadratic. )
Using that $$\,x^7 + 1 = (x+1)(x^6 - x^5 + x^4 - x^3 + x^2 - x + 1)\,$$:
$$(x+1)^7 - x^7 - 1 = (x+1)\left((x+1)^6 - x^6 + x^5 - x^4 + x^3 - x^2 + x - 1\right) \tag{1}$$
The second factor is a palindromic polynomial, which suggests the substitution $$\,u = x + \dfrac{1}{x}\,$$. Then $$\,x^2 + \dfrac{1}{x^2} = u^2 - 2\,$$, $$\,x^3 + \dfrac{1}{x^3} = u^3 - 3u\,$$, and:
\begin{align} & (x+1)^6 - x^6 + x^5 - x^4 + x^3 - x^2 + x - 1 \\ =\; &(x^2 + 2x + 1)^3 - x^6 + x^5 - x^4 + x^3 - x^2 + x - 1 \\ =\; &x^3\left(\left(x+\frac{1}{x}+2\right)^3 - \left(x^3 + \frac{1}{x^3} \right)+\left(x^2 + \frac{1}{x^2}\right)-\left(x + \frac{1}{x}\right)+1\right) \\ =\; &x^3 \left((u+2)^3 - (u^3 - 3u) + (u^2 - 2) - u + 1\right) \\ =\; &x^3 \left(7 u^2 + 14 u + 7\right) \\ =\; &7x^3(u+1)^2 \\ =\; &7x(ux+x)^2 \\ =\; &7x(x^2 + x + 1)^2 \tag{2} \end{align}
Piecing together $$\,(1)\,$$ and $$\,(2)\,$$:
$$(x+1)^7 - x^7 - 1 = 7x (x+1) \left(x^2+x+1\right)^2$$
Substituting $$\,x \mapsto \dfrac{x}{y}\,$$ and multiplying by $$\,y^7\,$$ gives the identity in OP's question.
If you're really asking how it's possible you can always expand both sides and compare the coefficients...
If you're asking how to come up with it, observe that one of the solutions of
$$(x+y)^7-(x^7+y^7) = 0$$
is
$$x+y=0$$
hence, $(x+y)^7-(x^7+y^7)$ must be divisible by $(x+y)$.
Users have approached the problem using the Binomial Theorem, namely,
$$(a+b)^n=\sum_{k=0}^n\binom nka^{n-k}b^k,\tag1$$
such that the formula for a binomial coefficient is expressed thus: $$\dbinom nk\stackrel{\small\text{def}}{=}\frac{n!}{k!(n-k)!}.\tag{n!=1\times 2\times 3\times\cdots\times n}$$ It follows, then, that by substituting $(a,b,n)=(x,y,7)$ respectively, the answer is as follows: \begin{align}(x+y)^7&=x^7+7x^6y+21x^5y^2+35x^4y^3+35x^3y^4+21x^2y^5+7xy^6+y^7 \\ &= x^7+y^7+7(x^6y+xy^6)+21(x^5y^2+x^2y^5)+35(x^4y^3+x^3y^4) \\ &=x^7+y^7+7\big(x^6y+xy^6+3(x^5y^2+x^2y^5)+5(x^4y^3+x^3y^4)\big) \\ &=x^7+y^7+7xy\big(x^5+y^5+3xy(x^4+y^4)+5xy(x^3+y^3)\big),\end{align} to which we can then factor $-$ but there is a better way!
Of course now, $(x+y)^7-(x^7+y^7)$ is a multiple of $x+y$. Let $x^7+y^7=x^7-(-y)^7$, then you can use the following formula from my answer to this post in order to factor the equation:
$$a^n-b^n=(a-b)\sum_{k=1}^na^{n-k}b^{k-1}.\tag2$$
It follows, then, that $$(x+y)^7-(x^7+y^7)=(x+y)\bigg((x+y)^6-\sum_{k=1}^7x^{n-k}y^{k-1}\bigg).$$ The following step requires you to use the binomial theorem for the algebraic expression $(x+y)^6$, which is much easier than calculating for $(x+y)^7$ by hand.
A more compact display : If $$σ_n = x^n + y^n$$ and $$P = xy$$, then from 7-th power binomial expansion we get $$σ_1^7 = σ_7 + 7P(σ_5 + 3σ_3P + 5σ_1P^2)$$ and since $$σ_1 | σ_n$$for odd $$n$$, viz., $$σ_3 = σ_1(σ_2 - P)$$;$$σ_5 = σ_3σ_2 - σ_1P^2 = σ_1\left(σ_2^2 - P(σ_2 + P)\right),$$ $$σ_1^7 = σ_7 + 7σ_1P\left[σ_2^2 \overbrace{-\ P(σ_2 + P)} + \ 3P(σ_2 - P) + 5P^2\right]$$ $$= σ_7 + 7σ_1P\left[(σ_2 + P)(σ_2 + 2P) \overbrace{-\ P(σ_2 + P)}\right]$$ $$= σ_7 + 7σ_1P(σ_2 + P)^2,$$i.e., $$σ_1^7 - σ_7 = 7σ_1P(σ_2 + P)^2,$$ or more explicitly, $$(x + y)^7 - (x^7 + y^7) = 7xy(x + y)(x^2 + xy + y^2)^2.$$
Another approach: there is a general factoring pattern for a sum of $n$th powers that works only when $n$ is odd. Specifically, in the case $n=7$ we have
$$x^7 + y^7 = (x+y)(x^6-x^5y+x^4y^2-x^3y^3+x^2y^4-xy^5+y^6)$$
Since both $(x+y)^7$ and $x^7 + y^7$ contain $(x+y)$ as a factor, we can write
$$(x+y)^7 - (x^7+y^7)= (x+y)\left((x+y)^6 - (x^6-x^5y+x^4y^2-x^3y^3+x^2y^4-xy^5+y^6)\right)$$
Now we can also expand $(x+y)^6$ using the binomial theorem: $$(x+y)^6 = x^6 + 6x^5y + 15x^4y^2+20x^3y^3+15x^2y^4+6xy^5+y^6$$
Plugging this into the equation directly above it, and combining like terms, we have
$$(x+y)^7 - (x^7+y^7)= (x+y)\left((x^6 + 6x^5y + 15x^4y^2+20x^3y^3+15x^2y^4+6xy^5+y^6) - (x^6-x^5y+x^4y^2-x^3y^3+x^2y^4-xy^5+y^6)\right) = (x+y)\left( 7x^5y + 14x^4y^2+21x^3y^3+14x^2y^4+7xy^5 \right)$$
Now in the second parenthesis, each term is a multiple of $7xy$, so factor that out too:
$$(x+y)^7 - (x^7+y^7)= (x+y)(7xy)\left(x^4 + 2x^3y+3x^2y^2+2xy^3+y^4 \right)$$
The last step is to factor $x^4 + 2x^3y+3x^2y^2+2xy^3+y^4$. This is maybe the trickiest part, as you have to somehow either recognize the factoring pattern $$x^4 + 2x^3y+3x^2y^2+2xy^3+y^4 = (x^2 + xy + y^2)^2$$ which can be confirmed by expanding the right-hand side; if you didn't know this pattern already, I'm not sure how one could discover it other than by trial and error.
Anyway, now you can put it all together: $$(x+y)^7 - (x^7+y^7)= (x+y)(7xy) (x^2 + xy + y^2)^2$$ and you're done.
Another approach would be to note that the function $$\ z \ = \ (x+y)^7-(x^7+y^7) \$$ has an "exchange symmetry" between its variables $$\ x \$$ and $$\ y \ \ ,$$ which makes the function symmetric about the plane $$\ y \ = \ x \ \ .$$ So its (bivariate) polynomial expression will have factors with this type of symmetry, such as $$\ x + y \$$ and $$\ xy \ \ .$$
The Freshman's Dream notwithstanding, the binomial expansion applied to the expression will produce terms of the form $$\ C·x^m·y^n \$$ where $$\ m + n \ = \ 7 \$$ and with all of the binomial coefficients $$\ \frac{7!}{m!n!} \ \ ,$$ $$m \ \neq \ 0 \ , \ 7 \ \ , \$$ being divisible by $$\ 7 \$$ since that is prime. So we can immediately see a factoring $$7x^6y \ + \ 21x^5y^2 \ + \ 35x^4y^3 \ + \ 35x^3y^4 \ + \ 21x^2y^5 \ + \ 7xy^6$$ $$= \ \ 7 \ · \ xy \ · \ ( \ x^5 \ + \ 3x^4y \ + \ 5x^3y^2 \ + \ 5x^2y^3 \ + \ 3xy^4 \ + \ y^5 \ ) \ \ .$$
The "interchangeability" of the variables also permits us to see that $$\ ( x + y ) \$$ may be a factor: $$x^5 \ + \ 3x^4y \ + \ 5x^3y^2 \ + \ 5x^2y^3 \ + \ 3xy^4 \ + \ y^5$$ $$= \ \ x \ · \ ( \ x^4 \ + \ a·x^3y \ + \ b·x^2y^2 \ + \ axy^3 \ + \ y^4 \ )$$ $$\ + \ ( \ x^4 \ + \ a·x^3y \ + \ b·x^2y^2 \ + \ axy^3 \ + \ y^4 \ ) \ · \ y \ \ .$$ Comparison of the coefficients requires that $$\ a + 1 \ = \ 3 \$$ and $$\ a + b \ = \ 5 \ \ ,$$ which can be solved consistently. We then arrive at $$7 \ · \ xy \ · \ (x + y) \ · \ ( \ x^4 \ + \ 2x^3y \ + \ 3x^2y^2 \ + \ 2xy^3 \ + \ y^4 \ ) \ \ .$$
The next most simple factor of suitable symmetry would be $$\ x^2 + y^2 \ \ ;$$ no higher powers of $$\ (xy)^k \$$ can be "extracted". Since there are terms in this remaining polynomial factor which have odd powers of $$\ x \$$ or $$\ y \ \ ,$$ however, this is a bit too simple. We can instead next try factors of the form $$\ x^2 + cxy + y^2 \ \ . \$$ Although the symmetrical ("palindromic") arrangement of the coefficients strongly hints at what will happen, we will naïvely use two "distinct" factors to write $$x^4 \ + \ 2x^3y \ + \ 3x^2y^2 \ + \ 2xy^3 \ + \ y^4 \ \ =^{?} \ \ ( \ x^2 + cxy + y^2 \ ) · ( \ x^2 + dxy + y^2 \ )$$ $$= \ \ ( \ x^4 + cx^3y + x^2y^2 \ ) \ + \ ( \ dx^3y + cdx^2y^2 + dxy^3 \ ) \ + \ ( \ x^2y^2 + cxy^3 + y^4 \ ) \ \ .$$ This factorization succeeds, as $$\ c + d \ = \ 2 \$$ and $$\ 1 + cd + 1 \ = \ 3 \ \$$ can also be solved consistently by $$\ c \ = \ d \ = \ 1 \ \ .$$ So we find the two factors to be identical.
We thus obtain the factorization $$(x \ + \ y)^7 \ - \ (x^7 \ + \ y^7) \ \ = \ \ 7 \ · \ xy \ · \ (x + y) \ · \ ( \ x^2 + xy + y^2 \ )^2 \ \ .$$ | 2022-08-11T07:43:42 | {
"domain": "stackexchange.com",
"url": "https://math.stackexchange.com/questions/498584/factor-xy7-x7y7",
"openwebmath_score": 0.9981115460395813,
"openwebmath_perplexity": 372.55260846987693,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9777138099151278,
"lm_q2_score": 0.8519527982093666,
"lm_q1q2_score": 0.8329660162051339
} |
https://math.stackexchange.com/questions/1678295/finding-all-n-tuples-that-are-not-shifted-versions-of-each-other | # Finding all n-tuples that are not shifted versions of each other
I am looking for an efficient way of finding the smallest set of $N$-tuples containing only $0$ and $1$ such that all $N$-tuples containing only $0$ and $1$ can be generated by circularly shifting elements of this set.
Consider the case $N=2$. There are 4 tuples $(0,0)$,$(0,1)$,$(1,0)$ and $(1,1)$ but the tuple $(0,1)$ is a circularly shifted version of $(1,0)$ so this set suffices: $\{(0,0),(0,1),(1,1)\}$.
Consider the case $N=3$. There are 8 tuples $(0,0,0)$,$(0,0,1)$,$(0,1,0)$,$(0,1,1)$,$(1,0,0)$,$(1,0,1)$,$(1,1,0)$,$(1,1,1)$. This set suffices: $\{(0,0,0),(0,0,1),(0,1,1),(1,1,1)\}$, because all 8 tuples can be generated by circularly shifting elements of this set.
Clearly it is possible to generate all $2^N$ such tuples and laboriously check if they are shifted versions of an already generated tuple, giving an $O(2^{2N-1})$ algorithm.
Is there a more efficient way of generating this set of tuples?
In other words, I'm looking for an algorithm that generates the set of all $N$-tuples of binary numbers "up to translation", so to speak. This set is smaller than the set of all $N$-tuples of binary numbers (but probably still has a size that is exponential in $N$). It is not unreasonable to wonder if an algorithm faster than $O(2^N)$ might exist.
• it is too early to accept an answer, i was beginning to think o an algorithm since you made it all clear now – Abr001am Mar 1 '16 at 12:45
For any $N$-tuple of binary numbers $b = (b_1,b_2,\ldots,b_n) \in \{0,1\}^N$.
Let
• $I(b)$ be the integer $\sum_{i=1}^n b_i 2^{i-1}$.
• $L(b) = (b_2,b_3,\ldots,b_n,b_1)$ be the $N$-tuple obtained by a left-rotate.
For each $b$, accept it as a basis when $I(b) = \min\big\{ I(L^k(b)) : 0 \le k < N \big\}$.
The complexity of this algorithm is $O(N 2^N)$. This is probably not the fastest but relatively simple to implement. The biggest advantage of it is easily parallelizable/vectorizable.
If you can afford an $O(2^N)$ memory consumption, you can reduce the time complexity to $O(2^N)$.
• Mark all $N$-tuples not visited.
• Loop through all $N$-tuple $b$.
• If $b$ has not been visited before,
• accept $b$ as basis,
• mark all shifted versions of $b$ as visited.
• That is better than what I thought of, but I continue to wonder if there is a way of doing it in less than $O(2^N)$ - after all, the size of the set of tuples is less than $O(2^N)$. – Wouter Mar 1 '16 at 11:14 | 2019-06-24T13:15:05 | {
"domain": "stackexchange.com",
"url": "https://math.stackexchange.com/questions/1678295/finding-all-n-tuples-that-are-not-shifted-versions-of-each-other",
"openwebmath_score": 0.7454526424407959,
"openwebmath_perplexity": 229.0039303933343,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9777138102398168,
"lm_q2_score": 0.8519527963298947,
"lm_q1q2_score": 0.8329660146441681
} |
http://azgr.iaex.pw/convergent-series.html | Convergent Series
A series is convergent if the sequence of its partial sums ${\displaystyle \left\{S_{1},\ S_{2},\ S_{3},\dots \right\}}$tends to a limit ; that means that the partial sums become closer and closer to a given number when the number of th. If this limit is one , the test is inconclusive and a different test is required. The geometric series 1 1 z = 1 + z+ z2 + = X1 n=0. Given a sequence {a n} and the sequence of its partial sums s n, then we say that the series is convergent if the sequence s n is convergent and has finite limit. If the series has terms of the form 1. There is a special test for alternating series that detects conditional convergence: Alternating series test:. A new look at weak-convergence methods in metric spaces-from a master of probability theory In this new edition, Patrick Billingsley updates his classic work Convergence of Probability Measures to reflect developments of the past thirty years. One subset of the series covered in Real analysis is the series of functions and uniform convergence. Find the definition of Convergent evolution in the largest biology dictionary online. Look for geometric series. A series which have finite sum is called convergent series. In other words, by uniform convergence, what I can now do is integrate this thing here, term by term. Convergence, in mathematics, property (exhibited by certain infinite series and functions) of approaching a limit more and more closely as an argument (variable) of the function increases or decreases or as the number of terms of the series increases. Indeed, since the differentiated series of the integrated series is the original, then this would say that the original series and the integrated series have the same radii. Now, since the terms of this series are larger than the terms of the original series we know that the original series must also be convergent by the Comparison Test. Every absolutely convergence series is convergent. It is one of the most commonly used tests for determining the convergence or divergence of series. 4) Explore the geometric series. We motivate and prove the Alternating Series Test and we also discuss absolute convergence and conditional convergence. (analysis) An infinite series whose partial sums converge. Cisco Network Convergence System 5500 Series Currently loaded videos are 1 through 15 of 48 total videos. The total sum of the series is the limit of the sequence , which we will denote by. The limiting value S is called the sum of the series. A sequence with a limit that is a real number. I wanted to know if there is an online reference I can use to find out known results about convergent series. has limit 2, so the sequence converges to 2. We generate blended finance data, intelligence, and deal flow to increase private sector investment in developing countries for the United Nations (UN) Sustainable Development Goals (SDGs). Eremenko November 5, 2013 In the lectures, the formula X∞ n=1 1 n2 π2 6 (1) was derived using residues. Much of this topic was developed during the seventeenth century. For example in an alternating series, what if we made all positive terms come first? So be careful! More. In more formal language, a series converges if there exists a limit l such that for any arbitrarily small positive number , there is a large integer N such that for all ,. Determine whether the series converges of diverges: Sum from 1 to infinity of sin(1/n). Carolina Greensboro. is absolutely convergent, as is the alternating series X∞ k=1 (−1)k−1 k2. If the sequence of partial sums is a convergent sequence (i. In particu-lar, some necessary and/or su cient conditions for Lp convergence, uniform convergence, and. When the absolute value of the terms of an alternating series are a decreasing and null-sequence then the series converges. , if and only if converges. If the power series ∑ n=0∞ a n x n, converges for x = x 0, then for all x, |x| < |x 0 | the power series CONVerges ABSolutely. where is the first term in the series (if the index starts at or , then "" is actually the first term or , respectively). Integral Series Convergence Test. Guide convergence towards the solution you want (or even a better one, if you can find it with them). Manage the divergence and convergence when changing minds to the best effect. As more terms are added, the partial sum fails to approach any finite value (it grows without bound). If the sequence of these partial sums {S n} converges to L, then the sum of the series converges to L. Convergent Series. For example in an alternating series, what if we made all positive terms come first? So be careful! More. With the geometric series, if r is between -1 and 1 then the series converges to 1 ⁄ (1 - r). Perform the alternating series test for alternating series. Convergent Series: A series is convergent if the sequence of its partial sums converges. Given a sequence {a n} and the sequence of its partial sums s n, then we say that the series is convergent if the sequence s n is convergent and has finite limit. Real analysis is an area of mathematics dealing with the set of real numbers and, in particular, the analytic properties of real functions and sequences, including their convergence and limits. We will then say that the order of convergence of fang is p:. is convergent. This page gives three examples of convergent sequences, all properly proved. converges by the Alternating Series Test to a number, S, where. Testing for Convergence or Divergence of a Series. This gives us a new way to approach series which have positive and negative terms - if we can show that they are absolutely convergent, then they must be convergent. Art Jewelry Designs and Workshops—featuring powder metallurgy processes in silver, gold, bronze,. If the series has terms of the form ar n 1, the series is geometric and the convergence of the series depends on the value for r. Download it once and read it on your Kindle device, PC, phones or tablets. Tests for Convergence of Series First we will go over some important facts, that are necessary for you to know if you want to become an expert on series. series mc-TY-convergence-2009-1 In this unit we see how finite and infinite series are obtained from finite and infinite sequences. Calculate L = < 1 and so by the ratio test. A series Σa n converges to a sum S if and only if the sequence of partial sums converges to S. For series convergence determination a variety of sufficient criterions of convergence or divergence of a series have been found. Show Instructions In general, you can skip the multiplication sign, so 5x is equivalent to 5*x. Jason Starr. Convergent Geometric Series Extension. 1, 0, 3, 0, 5, 0, 7, Alternating Sequences. It's probably asking for too much, but I wonder whether there is a series whose convergence is not known, where one wouldn't get the feeling that the convergence of the series was just an artificial way of asking a different problem. But it is rare to know explicitly what a series converges to. 2 Tests for Convergence Let us determine the convergence or the divergence of a series by comparing it to one whose behavior is already known. Recognizing these types will help you decide which tests or strategies will be most useful in finding whether a series is convergent or divergent. The general theme is convergence, in Section 2 this is studied for Dirichlet series and in Sections 3-4 for Euler. What is convergent series and divergent series ? A series which have finite sum is called convergent series. By choosing the convergence control parameter value other than optimal (but from the effective region) we get a convergent series as well, only the rate of convergence of the series will be less. See here for a basic timeline of canon events. Oscillating sequences are not convergent or divergent. Infinite series whose terms alternate in sign are called alternating series. If L = 1, then the Ratio test is inconclusive and we cannot determine if the series converges or diverges using this test. Convergent thinking, on the other hand, is the practice of trying to solve a discrete challenge quickly and efficiently by selecting the optimal solution from a finite set (again, these are my words). In terms of quantity, there were more stories in Convergent Series (twenty-one to be exact) with several only two or three pages long. Infinite Geometric Series Convergence Loading. He mentiones that the idea of a possible boundary between convergent and divergent series was suggested by du Bois-Reymond. Look for geometric series. Assume that for some number p the condition (1) lim n!1 an+1 ap n = C > 0 is satisfied. The power series can be written. The alternating harmonic series is a relatively rapidly converging alternating series and represents as such a limiting case for conditionally convergent series. Let ∑ a n be an absolutely convergent series, and ∑ b n be a conditionally convergent series. That converges or focuses 2. Discuss the convergence of the series X1 n=2 (¡1)n+1 p n: Solution: It converges conditionally. The word "convergent" has a number of different meanings in mathematics. We will call the radius of convergence L. The convergence tests for series have nice intuitive reasons why they work, and these are fairly easy to turn into rigorous proofs. The Convergence series was a spin-off of another series called The Traitor Games, though neither series share the same lore or canon. Use features like bookmarks, note taking and highlighting while reading Convergent Series. If the series has terms of the form 1. So, just as a refresher, converge means that even though you're summing up an infinite. We've already looked at these. A divergent sequence doesn't have a limit. From charlesreid1. is convergent. A convergent series runs to the X axis and gets as close as you like; close enough, fast enough to take an area under the curve. Directed by Drew Hall. The number R is called the radius of convergence of the power series. 2 says (among other things) that if both P 1 n=1 a n and P 1 n=1 b n converge, then so do P 1 n=1 (a n + b n) and P 1 n=1 (a n b n). More examples of evaluating series can be found on the following page: Series Convergence and Divergence Practice Examples 1; Series Convergence and Divergence Practice Examples 2; Series Convergence and Divergence Practice Examples 3; Series Convergence and Divergence Practice Examples 4; Series Convergence and Divergence Practice Examples 5. In a helpful review article Franciska Snoek Henkemans reminds us that in. The general theme is convergence, in Section 2 this is studied for Dirichlet series and in Sections 3-4 for Euler. The geometric series is used in the proof of Theorem 4. Then the improper integrals of f and g with the same limits of integration behave the same way, ie either both converge or both diverge. Fullyprimitive regularityconditions are given for power series andregression splines,aswellas more general conditionsthat may apply to othertypes of series. of the following series :- Absolute convergence. The infinite series is therefore limN→∞ SNf. 11-2: Convergence of Power Series Prakash Balachandran Department of Mathematics Duke University March 23, 2010 1 Power Series Definition: A power series about x = a is a sum of constants times powers of (x a) :. In appreciation for their support, these Members receive invitations to meet acclaimed artists and internationally recognized curators, and will see the Albright-Knox as never before. Why Jesus needs to be more than our Lord and Savior for the Church to thrive in a post Christian world. 2 Tests for Convergence Let us determine the convergence or the divergence of a series by comparing it to one whose behavior is already known. Friday 18 January 2013 – Afternoon AS GCE MATHEMATICS (MEI) 4752/01 Concepts for Advanced Mathematics (C2) QUESTION PAPER *4733970113* INSTRUCTIONS TO CANDIDATES These instructions are the same on the Printed Answer Book and the Question Paper. By inspection, it can be difficult to see whether a series will converge or not. It is useful to consider the more general case. Intervals of Convergence of Power Series. The partial sums in equation 2 are geometric sums, and. convergent definition: 1. Jason Starr. Answer : An infinite series is said to be convergent when the sum of first n terms cannot exceed some finite quantity numerically, no matter how great and may be. This implies that perhaps the sum of a conditionally convergent series can change based on the arrangement of terms. In other words, the series is not absolutely convergent. A rather detailed discussion of the subject can be found in Knopp's Theory and Application of Infinite Series (see § 41, pp. Series of Functions. In mathematics, a series is the sum of the terms of a sequence of numbers. Harmonic series (mathematics) In mathematics, the harmonic series is the divergent infinite series: Its name derives from the concept of overtones, or harmonics in music: the wavelengths of the overtones of a vibrating string are 12, 13, 14, etc. A series convergence calculator is used to find out the sum of the sequence and for determining convergence and divergence among series. Oscillating sequences are not convergent or divergent. KYOTO, Japan, Aug. Cisco Network Convergence System 6000 Series Routers Converged, elastic, and scalable The Network Convergence System (NCS) 6000 helps enable superior network agility, packet optical convergence, and petabits-per-second system scale. It is a finite or an infinite series according as the number of terms is finite or infinite. Our starting point in this section is the geometric series: X1 n=0 xn = 1 + x+ x2 + x3 + We know this series converges if and only if jxj< 1. Convergent/Divergent Series and the Geometric Series Theorem. The following properties may not come as a surprise to students, but are useful when determining whether more complicated series are convergent or divergent. It is one of the most commonly used tests for determining the convergence or divergence of series. With the geometric series, if r is between -1 and 1 then the series converges to 1 ⁄ (1 - r). Convergent Series. convergent, then the series is absolutely convergent If the series is convergent but not absolutely convergent, then the series is conditionally convergent. Worked example: sequence convergence. The difference is in the size of the common ratio. In this section we’ll state the main theorem we need about the convergence of power series. convergent definition: 1. Convergence, in mathematics, property (exhibited by certain infinite series and functions) of approaching a limit more and more closely as an argument (variable) of the function increases or decreases or as the number of terms of the series increases. Infinite series whose terms alternate in sign are called alternating series. 1, 0, 3, 0, 5, 0, 7, Alternating Sequences. Students may accept the formula for the sum of an infinite geometric series given that $$\left| r \right| < 1$$, and they may even understand the proof of this formula; but they usually are not shown (informally) that the defining feature of a convergent infinite series is that the limit of the series is the limit of its sequence of partial. Divergence Test: If a sequence (a n) does not converge to 0, then the series P a n diverges. AP Calculus BC on Khan Academy: Learn AP Calculus BC - everything from AP Calculus AB plus a few extra goodies, such as Taylor series, to prepare you for the AP Test. For example, the sequence of partial sums of the series 0. Recognizing these types will help you decide which tests or strategies will be most useful in finding whether a series is convergent or divergent. Conditional convergence. to put into appropriate form. In general, whenever you want to know lim n→∞ f(n) you should first attempt to compute lim x→∞ f(x), since if the latter exists it is also equal to the first limit. NO Does lim n→∞ sn = s s finite? YES P an = s YES P an Diverges NO TAYLOR SERIES Does an = f(n)(a) n! (x −a) n? NO YES Is x in interval of convergence? P∞ n=0 an = f(x. convergent definition: 1. tionally convergent. pdf Geometry - Additional practice. In fact, the whole point of series is often that they converge to. The series g demonstrates a common convergence pattern. Thanks so much for your help! Suppose (a_n+b_n) converges. TELESCOPING SERIES Dosubsequent termscancel out previousterms in the sum? May have to use partial fractions, properties of logarithms, etc. 1/X2 as simplified example, sans series paraphernalia A divergent. Oscillating sequences are not convergent or divergent. This week, we will see that within a given range of x values the Taylor series converges to the function itself. Convergent Series is a collection of science fiction and fantasy short stories by American writer Larry Niven, published in 1979. A series is convergent if the sequence of its partial sums ${\displaystyle \left\{S_{1},\ S_{2},\ S_{3},\dots \right\}}$tends to a limit ; that means that the partial sums become closer and closer to a given number when the number of th. Simple examples of convergent series with proofs. Conversely, a series is divergent if the sequence of partial sums is divergent. When the absolute value of the terms of an alternating series are a decreasing and null-sequence then the series converges. An infinite sequence (a n) is called convergent if limit n tends to infinity a n exists and is finite. We will use the comparison test to conclude about the convergence of this series. Definition: The values for which a power series converges are called its interval of convergence (IOC). p-Series Test: The series P 1 np converges only if p > 1 and diverges if p 1. If the terms of the sequence { s n } gets closer and closer to a particular number as n →∞, then we say that the series converges to L, or is convergent, and write a 1 + a 2 + = a n = s n = L If the sequence of partial sums does not converge to any particular number, then we say that the series diverges, or is divergent. This modification to the application of Shanks transformation may achieve convergence where its straightforward application fails. A convergent series is a mathematical series in which the sequence of partial sums converges to 1. Series Convergence and Divergence — Definitions. The convergence of this series can be shown using the Integral Test, or more directly, by the p-series Test,. The sequence is also. Convergence and divergence are unaffected by deleting a finite number of terms from the beginning of a series. convergent definition: Adjective (comparative more convergent, superlative most convergent) 1. Solutions to Series Exercises General Approach to using the Convergence Tests We have ve tests for convergence: 1) the Divergence Test, 2) the Alternating Series Test, 3) the Ratio Test, 4) the Integral (comparison) Test, and 5) the Comparison Test. Radius of Convergence for a Power Series. We also consider two specific. Convergent evolution is the process by which unrelated or distantly related organisms evolve similar body forms, coloration, organs, and adaptations. If convergent, Find its sum. What is absolute convergence in economics? In mathematics, a series (or sometimes also an integral) is said to converge absolutely if the sum (or integral) of the absolute value of the summand or. where c is the centre of convergence. Otherwise is called divergent series. In order to fully understand what that means we must understand the notion of a limit, and convergence. convergent synonyms, convergent pronunciation, convergent translation, English dictionary definition of convergent. Convergence and divergence are unaffected by deleting a finite number of terms from the beginning of a series. Worked example: sequence convergence. The number c is called the expansion point. It's not that a conditionally convergent series sometimes converges and sometimes not. When the absolute value of the terms of an alternating series are a decreasing and null-sequence then the series converges. The infinite series is therefore limN→∞ SNf. Convergent Series [Larry Niven] on Amazon. The world of mathematical sequences and series is quite fascinating and absorbing. For series convergence determination a variety of sufficient criterions of convergence or divergence of a series have been found. An example of a convergent series is As n becomes larger, the. Posts about Teaching Metal Clay written by C Scheftic. Use features like bookmarks, note taking and highlighting while reading Convergent Series. DC Comics recently announced Convergence, a new crossover event slated for 2015 that, well, isn't exactly easy to grok. Eremenko November 5, 2013 In the lectures, the formula X∞ n=1 1 n2 π2 6 (1) was derived using residues. The general theme is convergence, in Section 2 this is studied for Dirichlet series and in Sections 3-4 for Euler. If L > 1 or 1, then the series P. Indeed, it can. Fibonacci series. This implies that perhaps the sum of a conditionally convergent series can change based on the arrangement of terms. Tests for Convergence of Series 1) Use the comparison test to con rm the statements in the following exercises. Abel's test is an important tool for handling semi-convergent series. The sequences are also found in many fields like Physics, Chemistry and Computer Science apart from different branches of Mathematics. of the following series :- Absolute convergence. In other words, if the series of all nonnegative terms P∞ k=1 |Ak| converges, then so does the series P∞. Letting si and sj be partial sums of the u series, with j > i, the difierence sj ¡ si is Pj n=i+1 un, and this is smaller than the corresponding quantity for the a series, thereby proving convergence. Testing for Convergence or Divergence of a Series. So, just as a refresher, converge means that even though you're summing up an infinite. The sum of a convergent geometric series can be calculated with the formula a ⁄ 1 - r, where "a" is the first term in the series and "r" is the number getting raised to a power. KYOTO, Japan, Aug. And what I would like you to do is pause this video, and think about whether each of them converges or diverges. So, let's look at some examples. The Gelman–Rubin convergence diagnostic. One topic that is rarely discussed is the rate of convergence or divergence. For =-1 , = is nonconvergent and for 1 and -1 , = is an unbounded and hence nonconvergent sequence. If we say that the Fourier series converges to the function, then precisely in what sense does the series converge? And under what conditions? Incidentally, such questions of Fourier series convergence are largely responsible for seeding the subject of real analysis. The power series converges absolutely. -The limit from n to infinity of one is one and the limit of a bottom heavy fraction is zero. An important type of series is called the p-series. Then any rearrangement of ∑ a n is convergent to the same sum. Or anything about rate of vanishing of Fourier coefficients? 3. Mathematical Definitions A power series, f(x) = X∞ n=0 anx n, is an example of a sum over a series of functions f(x) = X∞ n=0 gn(x), (1) where gn(x) = anxn. In appreciation for their support, these Members receive invitations to meet acclaimed artists and internationally recognized curators, and will see the Albright-Knox as never before. However, we shall see that none of the above results hold for sequences in an arbitrary topological space. Convergent series In mathematics, a series is the sum of the terms of a sequence of numbers. Conversely, in an absolutely convergent series, the partial sums converge much more quickly to the limit, they don't oscillate too much. Every absolutely convergence series is convergent. This is in B&S, but not until Chapter 9 (pg. The most popular and commonly used of these are the criterions of D'Alembert, Cauchy, Raabe; numeric series comparison, as well as the integral criterion of convergence of numerical series. The series is said to converge if the sequence of partial sums si converges. Topic: Calculus, Sequences and Series. Stack Exchange network consists of 175 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. In an abundance of enthusiasm generated post-Sputnik 1, the convergent style of thinking was rapidly equated with typical intelligence. Let us first make precise what we mean by "linear. FOURIER ANALYSIS physics are invariably well-enough behaved to prevent any issues with convergence. (Use Alternating Series test if not absolutely convergent) (i) sum from n=0 to infinity of (-1)^n/sqrt (n^3+1) (ii) sum from n=1 to infinity of (-1)^n/sqrt(n) (iii) sum from n=1 to infinity of (-1)^n/(3^n) Step by step explanation required, highlighting the major steps in solving. and exhibits the so-called Gibbs Phenomenon in which the convergence is pointwise but not uniform. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture. Get the free "Infinite Series Analyzer" widget for your website, blog, Wordpress, Blogger, or iGoogle. ) Contributed by Ron Wax "Niven's Convergent Series takes a standard plot of a young man who has sold his soul but introduces a converging sequence to beat the demonic fiend at the game. The geometric series plays a crucial role in the subject for this and other reasons. convergent series never converge more rapidly than do one or both of the factor-series ? Can the product of two conditionally convergent series or of a condi-tionally convergent and a divergent series in no case be absolutely convergent ? The first doubt of the correctness of a negative reply arose in connection. Convergent series. 1, 0, 3, 0, 5, 0, 7, Alternating Sequences. Discuss the convergence of the series X1 n=2 (¡1)n+1 p n: Solution: It converges conditionally. Key Concepts The in nite series X1 k=0 a k converges if the sequence of partial sums converges and diverges otherwise. A series convergence calculator is used to find out the sum of the sequence and for determining convergence and divergence among series. This modification to the application of Shanks transformation may achieve convergence where its straightforward application fails. This series of events pairs internationally acclaimed artistic talent with the dynamic innovators of Buffalo's cultural scene. (analysis) A sequence in a metric space with metric d is convergent to a point , denoted as , if for every there is a natural nu. And then we define series convergence in terms of the convergence of this sequence of partial sums. You’ll learn how to make a pair of fine silver post-style earrings. 1, 2019 /PRNewswire/ -- OMRON Corporation, based in Kyoto, Japan, globally launched the Light Convergent Reflective Sensor "B5W-LB series" which can be embedded in industrial. Area, like distance, and volume in customary language are quantities that are always positive. What is a geometric series, when is a geometric series convergent? If a geometric series is convergent to which value does it converge to? 2. has limit 2, so the sequence converges to 2. The comparison test is a nice test that allows us to do problems that either we couldn’t have done with the integral test or at the best would have been very difficult to do with. For series convergence determination a variety of sufficient criterions of convergence or divergence of a series have been found. Convergent series In mathematics, a series is the sum of the terms of a sequence of numbers. An infinite series for which the sequence of partial sums converges. Then any rearrangement of ∑ a n is convergent to the same sum. A divergent sequence doesn’t have a limit. If L < 1, then the series P. How one can compute the rate of convergence of Fourier series? 2. monado on LibraryThing: More than 1 year ago: Writing the ultra-short story is a special skill. Examples Example 1. Given a sequence, the nth partial sum is the sum of the first n terms of the sequence, that is, A series is convergent if the sequence of its partial sums converges. Rate of Convergence for the Bracket Methods •The rate of convergence of –False position , p= 1, linear convergence –Netwon ’s method , p= 2, quadratic convergence –Secant method , p= 1. The number R in the theorem above is called the radius of convergence. How to Determine Convergence of Infinite Series - Steps Perform the divergence test. Convergent Geometric Series. The root test is stronger than the ratio test: whenever the ratio test determines the convergence or divergence of an infinite series, the root test does too, but not conversely. Centre of convergence. Posts about Teaching Metal Clay written by C Scheftic. Ὄ Condition(s) of Divergence: 1 lim 𝑛→∞ 𝑛≠0 2 Geometric Series Test Series: ∑∞ 𝑟 𝑛=0 1 Condition of Convergence: |𝑟|<1. coming closer together or meeting. There are many other cases involving series expansions. Does this series converge? This is a question that we have been ignoring, but it is time to face it. We now list the Taylor series for the exponential and logarithmic. An important type of series is called the p-series. Conversely, a series is divergent if the sequence of partial sums is divergent. You’ll learn how to make a pair of fine silver post-style earrings. Answer : An infinite series is said to be convergent when the sum of first n terms cannot exceed some finite quantity numerically, no matter how great and may be. Len is a supporting character of the Tsukihime series, which like Fate/stay night, is a part of the Type-Moon universe. The series is said to converge if the sequence of partial sums si converges. Worked example: sequence convergence. Harmonic series (mathematics) In mathematics, the harmonic series is the divergent infinite series: Its name derives from the concept of overtones, or harmonics in music: the wavelengths of the overtones of a vibrating string are 12, 13, 14, etc. Solution We apply the ratio test here. Math 262 Practice Problems Solutions Power Series and Taylor Series 1. The path formed by connecting the partial sums of a conditionally convergent series is infinitely long. One of the most important things you will need to learn in this section of the course is a list of standard examples of convergent and divergent series. Intuitively, this means that the partial sums "oscillate too much" around the limit. Carolina Greensboro. R can often be determined by the Ratio Test. 0009 + ··· is convergent. 11-2: Convergence of Power Series Prakash Balachandran Department of Mathematics Duke University March 23, 2010 1 Power Series Definition: A power series about x = a is a sum of constants times powers of (x a) :. Comparison Test: Given a series P a n and another (comparison) series P b n. Thus, the series diverges. However, we shall see that none of the above results hold for sequences in an arbitrary topological space. An important type of series is called the p-series. Lets learn first what is the convergent and divergent series. There are a lot of memorable examples of convergent thinking out there that demonstrate the necessity for this technique. Also includes stories from the Draco's Tavern series. - [Instructor] So here we have three different series. Then, by the converse of the theorem that says if two infinite series are convergent, then their sum is convergent, a_n converges and b_n converges. At x = −1, the series converges absolutely for p ≥ 0 and diverges for p < 0. coming closer together or meeting. Power series have coefficients, x values, and have to be centred at a certain value a. As you add more and more terms of a convergent series (taking successive partial sums), you get closer to a certain number, called the limit of the series. let an and an+1 represent consecutive terms of a series of positive terms suppose lim an+1/an exists and that r=liman+1/an the series is convergent if r< 1 and divergent if r>1 if r=1, the test provides no information if denom> num, convergent if num>denom, divergnet. The ‘Fourier sine and cosine series’, that is, the representations of f as inflnite series of eigenfunctions in [0;…] with Dirichlet (resp, Neumann) boundary conditions is a special case of this. n converges absolutely (and hence is convergent). Hence (a_n+b_n) must diverge. For example, the sequence 2. 01 Single Variable Calculus, Fall 2005 Prof. Lady (October 31, 1998) Some Series Converge: The Ruler Series At rst, it doesn't seem that it would ever make any sense to add up an in nite number of things. Become a member and unlock all Study Answers. Worked example: sequence convergence. Instead, we're talking about the behavior of a related series: what happens when we get rid of all the negative signs in the series?. Calculate L = < 1 and so by the ratio test. One of the most important things you will need to learn in this section of the course is a list of standard examples of convergent and divergent series. From science fiction which edges toward horror ("Bordered in Black"), sf conundrums ("Singularities Make Me Nervous") to three Draco Tavern stories and many with clever twists, CONVERGENT SERIES is a feast for lovers of the short story. If the series does converge, then the remainder R. Proofs of the theorem below can be found in most introductory Calculus textbooks and are relatively straightforward. Does this series converge? This is a question that we have been ignoring, but it is time to face it. Convergent series lesson plans and worksheets from thousands of teacher-reviewed resources to help you inspire students learning. Case 3: The series converges for all real number values of 𝑥. In the world of finance and trading, convergence and divergence are terms used to describe the. A series Σa n converges to a sum S if and only if the sequence of partial sums converges to S. In appreciation for their support, these Members receive invitations to meet acclaimed artists and internationally recognized curators, and will see the Albright-Knox as never before. The number R in the theorem above is called the radius of convergence. If lim n->∞ (An/Bn)=L>0 then the two series have similar behaviors comparison test suppose than ∑An, ∑Bn are series with positive terms, if ∑Bn is convergent and An≤Bn for all n then ∑An is also convergent. They are quite important, since many tests of convergence use these two types as benchmarks. P 1 n=4 1diverges, so P 1 n=4 3 diverges. Convergent Series - Kindle edition by Larry Niven. | 2019-11-11T21:54:17 | {
"domain": "iaex.pw",
"url": "http://azgr.iaex.pw/convergent-series.html",
"openwebmath_score": 0.8870593905448914,
"openwebmath_perplexity": 456.0050046906253,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES\n\n",
"lm_q1_score": 0.9919380087100574,
"lm_q2_score": 0.8397339616560072,
"lm_q1q2_score": 0.8329640337712675
} |
http://mathhelpforum.com/algebra/138571-what-inverse-matrix.html | # Thread: What is the inverse of this matrix?
1. ## What is the inverse of this matrix?
What is the inverse of:
[3 7]
[1 3]
I've tried working it out, but I keep getting the wrong answer.
Here's my working out:
[3 7 1 0]
[1 3 0 1]
Divide row 1 by 3
[1 2.3 0.3 0]
[1 3 0 1]
Row 1 minus (1 x row 1)
[1 2.3 0.3 0]
[0 1 -0.43 1.43]
Row 2 divided by 0.7
[1 2.3 0.3 0]
[0 1 -0.43 1.43]
Row 1 minus (2.3 x row 2)
[1 0 1.289 -3.289]
[0 1 -0.43 1.43]
The answer is supposed to be:
[3/2 -7/2]
[-1/2 3/2]
But my answers don't match the ones I got above even when I try and change the decimals to fractions.
Help would be much appreciated. Sorry if the matrices look messy, I hope you get the idea.
2. Originally Posted by brumby_3
What is the inverse of:
[3 7]
[1 3]
I've tried working it out, but I keep getting the wrong answer.
Here's my working out:
[3 7 1 0]
[1 3 0 1]
Divide row 1 by 3
[1 2.3 0.3 0]
[1 3 0 1]
Row 1 minus (1 x row 1)
[1 2.3 0.3 0]
[0 1 -0.43 1.43]
Row 2 divided by 0.7
[1 2.3 0.3 0]
[0 1 -0.43 1.43]
Row 1 minus (2.3 x row 2)
[1 0 1.289 -3.289]
[0 1 -0.43 1.43]
The answer is supposed to be:
[3/2 -7/2]
[-1/2 3/2]
But my answers don't match the ones I got above even when I try and change the decimals to fractions.
Help would be much appreciated. Sorry if the matrices look messy, I hope you get the idea.
I really cannot follow what you did because it came out too messed up, ...and in fact you shouldn't be asking this: just multiply the original matrix by what you got and if you get the identity matrix you're done, otherwise you're wrong.
About the decimals: drop them! If you divide by three then write $\frac{7}{3}\,\,\,or\,\,\,7\slash 3$, and not that horrible thing $2.3$ , which is not even a good approximation and it looks childish and,again, pretty ugly. Simple fractions are as numbers as anything else, so befriend them!
No wonder you got all that mess if you jump between decimal and simple fractions...
Tonio
3. Wow, your reply was REALLY rude. You could have been a lot nicer, some people aren't experts at maths, ok? I did the best I could.
4. Hello, brumby_3!
With a little thought, we can often avoid fractions.
Find the inverse of: . $\begin{bmatrix} 3&7 \\ 1&3 \end{bmatrix}$
We have: . $\left[\begin{array}{cc|cc} 3&7 & 1&0 \\ 1&3 & 0&1\end{array}\right]$
$\begin{array}{c}R_1-2R_2 \\ \\ \end{array} \left[\begin{array}{cc|cc}1 & 1 & 1 & \text{-}2 \\ 1 & 3 & 0 & 1 \end{array}\right]$
$\begin{array}{c} \\ R_2-R_1\end{array} \left[\begin{array}{cc|cc}1 & 1 & 1 & \text{-}2 \\ 0 & 2 & \text{-}1 & 3 \end{array}\right]$
. . . $\begin{array}{c} \\ \frac{1}{2}R_2 \end{array} \left[\begin{array}{cc|cc} 1&1&1&\text{-}2 \\ 0 & 1 & \text{-}\frac{1}{2} & \frac{3}{2} \end{array}\right]$
$\begin{array}{c}R_1-R_2 \\ \\ \end{array} \left[\begin{array}{cc|cc}1 & 0 & \frac{3}{2} & \text{-}\frac{7}{2} \\ \\[-4mm] 0 & 1 & \text{-}\frac{1}{2} & \frac{3}{2} \end{array}\right]$
5. Originally Posted by brumby_3
Wow, your reply was REALLY rude. You could have been a lot nicer, some people aren't experts at maths, ok? I did the best I could.
I, of course, only tried to stress some arguments I think will help you in the future. In college\university, things will really get tough.
If you really think my reply was rude (as opposed to you being hypersensitive) then you must send a complain about me to the site's moderators.
Tonio
6. There's a difference between not sugar coating things and being rude. In this thread I don't see much sugar but I also don't see anything rude.
What I do see in fact are members who have given up their valuable time, gratis, to offer good advice and help. Advice I might add that, if followed, will almost certainly lead to improved performance.
Since the question has been satisfactorily answered, I'm closing the thread to avoid things going off-topic.
If anyone has anything else to ask or add, s/he can pm it to me. | 2017-12-16T15:51:07 | {
"domain": "mathhelpforum.com",
"url": "http://mathhelpforum.com/algebra/138571-what-inverse-matrix.html",
"openwebmath_score": 0.6673891544342041,
"openwebmath_perplexity": 1598.925246510774,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9372107896491797,
"lm_q2_score": 0.8887587905460026,
"lm_q1q2_score": 0.832954327895269
} |
https://math.stackexchange.com/questions/1064501/inequality-proof-using-real-numbers | # Inequality proof using real numbers
If $x,y,z,w$ are positive real numbers such that $x < y$ and $z < w$, show that $xz < yw$. Show the converse and prove it or provide a counterexample.
So I know that the proof is true, I just have no idea how to prove it. It feels like its a fact to me. Any help will be appreciated!
• Note the converse isn't true. A counter is: $2\cdot 2 < 1\cdot 5$. – Macavity Dec 12 '14 at 5:55
Note that $$xz - yw = x \cdot \left( z - w \right) + w \cdot \left( x - y \right).$$Since $x-y<0$ and $z-w<0$, we have $xz - yw < 0$, so $xz < yw$. $\Box$
• @ZoëSoriano Does this answer your question? – Ahaan S. Rungta Dec 12 '14 at 5:41
• You're welcome; thanks for the accept! – Ahaan S. Rungta Dec 12 '14 at 7:15
HINT:
Use the following $$xz-yw=x(z-w)+w(x-y)$$
or $$xz-yw=z(x-y)+y(z-w)$$
$x<y \implies \dfrac{x}{y}<1$
$z<w \implies \dfrac{z}{w}<1$
$\therefore \dfrac{xz}{yw}<1$
For this I have used the following lemma.
Lemma
$a<1, b<1 \implies ab<1$
Proof
$a<1 \implies \exists \varepsilon_1>0:a=1-\varepsilon_1$
$b<1\implies \exists \varepsilon_2>0:b=1-\varepsilon_2$
$ab=\left(1-\varepsilon_1\right)\left(1-\varepsilon_2\right)=1-\left(\varepsilon_1+\varepsilon_2\right)+\varepsilon_1\varepsilon_2$
$$\boxed{\left(\varepsilon_1+\varepsilon_2\right)>\varepsilon_1\varepsilon_2 \impliedby \varepsilon_1>\varepsilon_2\left(\varepsilon_1-1\right) \impliedby \varepsilon_1>0}$$ | 2019-06-17T03:14:25 | {
"domain": "stackexchange.com",
"url": "https://math.stackexchange.com/questions/1064501/inequality-proof-using-real-numbers",
"openwebmath_score": 0.8281052708625793,
"openwebmath_perplexity": 370.5711748196922,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9799765628041174,
"lm_q2_score": 0.8499711775577736,
"lm_q1q2_score": 0.8329518330656351
} |
https://math.stackexchange.com/questions/209656/find-the-remainder-when-10400-is-divided-by-199 | # Find the remainder when $10^{400}$ is divided by 199?
I am trying to solve a problem
Find the remainder when the $10^{400}$ is divided by 199?
I tried it by breaking $10^{400}$ to $1000^{133}*10$ .
And when 1000 is divided by 199 remainder is 5.
So finally we have to find a remainder of :
$5^{133}*10$
But from here I could not find anything so that it can be reduced to smaller numbers.
How can I achieve this?
Is there is any special defined way to solve this type of problem where denominator is a big prime number?
• $10^{400}=1000^{133}\times10$, not $1000^{333}\times10$. – Gerry Myerson Oct 9 '12 at 4:53
• A standard beginning (for prime moduli) is to use the fact that if $p$ does not divide $a$, then $a^{p-1}\equiv 1\pmod{p}$. Thus $10^{198}\equiv 1\pmod{199}$. It follows that $10^{396}\equiv 1\pmod{199}$ and therefore $10^{400}\equiv 10^4\pmod{199}$. Now we have to calculate. In this case, there is a further shortcut, since $1000=(5)(199)+5\equiv 5\pmod{199}$. – André Nicolas Oct 9 '12 at 5:24
• – Martin Sleziak Jun 17 '16 at 8:22
You can use Fermat's little theorem. It states that if $n$ is prime then $a^n$ has the same remainder as $a$ when divided by $n$.
So, $10^{400} = 10^2 (10^{199})^2$. Since $10^{199}$ has remainder $10$ when divided by $199$, the remainder is therefore the same as the remainder of $10^4$ when divided by $199$. $10^4 = 10000 = 50*199 + 50$, so the remainder is $50$.
Since 199 is prime and gcd(10,199) = 1
So, $10^{198} \equiv 1 (mod 199)$
Squaring the both side: $10^{396} \equiv 1 (mod 199)$
Now: $10^3 \equiv 5(mod199)$
$10^{4} \equiv 50 (mod 199)$
$10^{400} \equiv 50 (mod 199)$
So, the remainder is 50. This method is known as Euler's Totient
• 1. There is totally redundant to mention $gcd(10,199)=1$. 2. I found in this question that presumably the reason why you posted this answer is just because you want to get started at this site. However, posting an answer to a question that had had an accepted answer 5 years ago helps neither the OP nor yourself. Try to answer some questions that not yet have an accepted answer yet. – BAI Oct 19 '17 at 11:25
• Sorry new to the site as you have mentioned too, I forgot to check the date of the question. Thank you for your guidance. – Tapan Luthra Oct 19 '17 at 11:27
• The date is not really the point. Surely, if you find an appropriate question which still have no answer, you are very welcome to share your delightful idea, if any. The point is that it is worthless to post an answer under an old question that already had an identical accepted answer. – BAI Oct 19 '17 at 11:44 | 2021-01-27T19:55:09 | {
"domain": "stackexchange.com",
"url": "https://math.stackexchange.com/questions/209656/find-the-remainder-when-10400-is-divided-by-199",
"openwebmath_score": 0.7323861718177795,
"openwebmath_perplexity": 186.07111671434032,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES\n\n",
"lm_q1_score": 0.9799765604649332,
"lm_q2_score": 0.849971175657575,
"lm_q1q2_score": 0.8329518292152459
} |
https://www.yasint.dev/vector-rotation | By Yasin
Today I stumbled across an interesting problem which is a small sub-problem for a distributed algorithm I'm currently working on. The concern is that we should rotate a one-dimensional vector of $n$ element left by $d$ ($\Delta$) positions.
# Problem Statement
For instance, say we have an array with $n=8$ elements should rotate it by $d=3$ shifts. In example, vector [1,2,3,4,5,6,7,8] is rotated to [4,5,6,7,8,1,2,3].
A naive algorithm will use an n-element intermediate vector to do the job in $n$ steps or copy the underlying array with language dependant slicing features (where space is $O(n)$) or compute in-place shift by holding a temporary variable with $d$ iterations (which will take a running time of $O(\Delta * n)$).
But we are better than this! Can we rotate the vector in time proportional to $O(n)$ without additional memory allocations $O(1)$?
# Aha!
Yes, we can! Thanks to Jon Bentley, In his Programming Pearls (2nd Edition) book, he mentioned three vector rotation algorithms called Juggling, Swapping, and Reversing.
Jon derived a smart insight in which he calls the Aha! moment! Because, in the reverse rotation technique, it is basically two arrays reversed twice. Truly marvelous!
The reversing algorithm is pretty straightforward, and the other two are very clever. But for this code fragment, I will only go through the reversing algorithm because that's the most elegant code. Jon makes use of a bidirectional iterator to perform the task in $O(n)$ time.
Now let's dive into the algorithm! Woohoo! 🕺
# Reverse Rotate
Recall the example in the problem description. We have to left rotate the vector [1,2,3,4,5,6,7,8] by $3$. If we view it as two sub-vectors, we get $3$ and $5$ items long sub-vector: [1,2,3]-[4,5,6,7,8].
In Jon's algorithm, to rotate the vector, we reverse the first sub-vector to obtain [3,2,1, 4,5,6,7,8]. Essentially, the first reverse of the Sub-vector A is swapping the positions of each element until delta.
Then reverse the second Sub-vector B to obtain [3,2,1, 8,7,6,5,4]. Like previously, we make use of the two-pointers to swap them in-place without additional space overhead.
Finally, reverse the whole thing to obtain [4,5,6,7,8,1,2,3].
That's it! How cool is that, huh?
# Implementation
Here's a Golang implementation of this algorithm. The reverse function takes a pointer to the array $x$ and bidirectional iterator positions $i$ and $j$ for swapping the indexes. Then we loop $\sum_{i}^{j}$ until $i \lt j$ to swap $x_i$ and $x_j$, and consecutively we increment $i$ and decrement $j$ by one.
Then comes the real magic. Inside the reverseRotate function we invoke the reverse routine $3$ times. The first call is to get $A'$ (we start from $0$ for $i$, and $j$ is always $\lt\Delta$), and then the second call is to derive $B'$ (we start from $\Delta$ for $i$ and swap until $n-1$). At this point, we only need to reverse the entire array once from $0$ to $n - 1$ to obtain the rotated vector.
# Interactive Example
Here's a neat visual playground for you to see the behaviour of this algorithm!
This is an elegant way to solve the problem, especially considering one vector as a sum of two (Prakhar Srivastav, 2014). In Aha! Algorithms research paper Jon mentions that this reversal code is guaranteed time and space-efficient and is so short and damn simple that it's pretty hard to get wrong.
It is precisely the code that Brian Kernighan & Bill Plauger use in their Software Tools book's editor and also what Ken Thompson used in his text editor ed to manipulate lines.
# Stay Curious
These are some interesting resources I referred while writing this post. Be sure to check these posts from great engineers out there!
Thanks a million for reading ! Until next time!
Published | 2022-12-03T16:19:46 | {
"domain": "yasint.dev",
"url": "https://www.yasint.dev/vector-rotation",
"openwebmath_score": 0.5427159667015076,
"openwebmath_perplexity": 1151.1121602510932,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9799765563713599,
"lm_q2_score": 0.8499711775577735,
"lm_q1q2_score": 0.8329518275979766
} |
https://math.stackexchange.com/questions/1776534/when-are-we-able-to-find-a-quadratic-with-roots-that-are-a-function-of-another-q | # When are we able to find a quadratic with roots that are a function of another quadratic?
Motivation:
Given the roots of the quadratic $2x^2+6x+7=0$ find a quadratic with roots $\alpha^2-1$ and $\beta^2-1$
I was able to solve this problem in two ways:
Method 1:
Sum of the roots $\alpha+\beta=-\frac{b}{a}$
Product of roots $\alpha\beta=\frac{c}{a}$
Hence $\alpha+\beta=-3$ and $\alpha\beta=\frac{7}{2}$
We want an equation with roots $\alpha^2-1$ and $\beta^2-1$
The sum of the roots of the new quadratic will be $\alpha^2-1+\beta^2-1=\alpha^2+\beta^2-2$
The product of the roots of the new quadratic will be $(\alpha^2-1)(\beta^2-1)=\alpha^2\beta^2-(\alpha+\beta)+1$
We are able to compute $\alpha^2+\beta^2$ as it is $(\alpha+\beta)^2-2\alpha\beta$ and so the problem is solved.
Plugging the numbers gives $4u^2+45=0$
Method 2
Let $u=\alpha^2-1\implies\alpha=\sqrt{u+1}$ but we know that $\alpha$ solves the original equation so:
\begin{align}2\alpha^2+6\alpha+7&=0\\2(u+1)+6\sqrt{u+1}+7&=0\\\sqrt{u+1}&=\frac{-2u-9}{6}\\u+1&=\frac{1}{36}(-2u-9)^2\\36u+36&=4u^2+36u+81\\0&=4u^2+45\end{align}
Question: The first method clearly uses the values of $\alpha$ and $\beta$ but the second seemingly only requires $\alpha$. How is this possible? Sure, one man's $\alpha$ is another's $\beta$ and so you could relabel as the choice of $\alpha$ and $\beta$ is arbitrary. This is believable because of the symmetry involved in the new roots $\alpha^2-1$ looks much like a $\beta^2-1$ but I feel there must be more to this. Supposing one root of the new quadratic was $\alpha^2-1$ but the other was $\beta^3-2\beta$ or something worse? How would the second method know? This leads me to a more fundamental question.
Are there only certain functions of the roots of an old quadratic that can we can find a new quadratic in this way? I suppose we could consider $(x-f(\alpha)(x-g(\beta)=0$ where $f$ and $g$ are the functions of the old roots but then could we alway compute these numerically?
Thanks for taking the time to read this and for any contributions.
• The main difference is where you're working. In the first case, you're working in the extension, while you're working in the base field in the second case. – Michael Burr May 8 '16 at 10:40
• @michael burr thanks. Going to have to think about your comments. I know what a field extension is and what subfields are but I've never fully grasped Galois correspondence or have a great working knowledge. – Karl May 8 '16 at 10:55
• Also, observe that after you pick $\alpha$, you know $\beta$ because $\beta=-\alpha-3$. – Michael Burr May 8 '16 at 11:15
Observe the following: Since $\alpha+\beta=-\frac{b}{a}$, you know that $\alpha+\beta=-3$. Above, you considered the case where $u=\alpha^2-1$. Suppose that you have picked a value for $\alpha$ (there are two solutions to the original quadratic and $\alpha$ could be either of them; the Galois action permutes the two roots without changing the base field).
Now, since $\beta=-\alpha-3$, and you know that $\alpha=\sqrt{u+1}$, you know that $\beta=-\alpha-3=-\sqrt{u+1}-3$. Since $2\beta^2+6\beta+7=0$, we know that $2(u+1+6\sqrt{u+1}+9)-6\sqrt{u+1}-18+7=0$. Simplifying, we have that $6\sqrt{u+1}+2u+9=0$. In other words, $$\sqrt{u+1}=-\frac{2u+9}{6}.$$ This is exactly the same expression that you found above, but starting with $\beta$.
In general, since the Galois action takes $a+b\alpha$ to $a+b\beta$ (where $a$ and $b$ are in $\mathbb{Q}$), we see that the Galois action takes $\alpha^2-1$ to $\beta^2-1$. Therefore, if $\alpha^2-1$ is the root of a polynomial $p(x)$ with coefficients in $\mathbb{Q}$, then so is $\beta^2-1$. More precisely, if $\sigma$ is the Galois action and $p(x)=\sum a_ix^i$, then you know that $p(\alpha)=0$, but $\sigma(p(\alpha))=\sigma(0)=0$, and $\sigma(\sum a_i\alpha^i)=\sum a_i\sigma(\alpha)^i$.
Since $\alpha^2-1$ and $\beta^2-1$ are Galois conjugates, any polynomial (with coefficients in $\mathbb{Q}$) which vanishes at one of them vanishes at both. That is why you only need to know $\alpha$ in the second approach, the $\beta$ comes for free.
• Thanks for your help I will study and then accept. – Karl May 8 '16 at 13:18
• Got the jist of this thanks – Karl May 8 '16 at 15:49
This is my solution:
Note that $2x^2+6x+7=0$ is equivalent to $x^2= -3x-\dfrac 72$
So
1. $\alpha + \beta = -3$
2. $\alpha \, \beta = \dfrac 72$.
3. $\alpha^2 - 1 = -3\alpha-\dfrac 92$
4. $\beta^2 - 1 = -3\beta-\dfrac 92$
We compute
\begin{align} \hline (\alpha^2 - 1) + (\beta^2 - 1) &= (-3\alpha-\dfrac 92)+(-3\beta-\dfrac 92)\\ &= -3(\alpha + \beta) - 9 \\ &= 0 \\ \hline (\alpha^2 - 1)(\beta^2 - 1) &=(-3\alpha-\dfrac 92)(-3\beta-\dfrac 92) \\ &= 9\alpha\beta+\dfrac{27}{2}(\alpha + \beta) + \dfrac{81}{4}\\ &= \dfrac{63}{2} -\dfrac{81}{2}+\dfrac{81}{4} \\ &= \dfrac{45}{4}\\ \hline \end{align}
So the polynomial with roots $\alpha^2-1$ and $\beta^2-1$ is $4x^2 + 45$.
## another way to look at your second method
Let $\alpha$ and $\beta$ be the roots of $Ax^2 + Bx + C$.
Let $u = \alpha^2 - 1$
Then $\alpha = \pm \sqrt{u+1}$.
\begin{align} Ax^2 + Bx + C &= 0 \\ A\alpha^2 + B\alpha + C &= 0 \\ A(u+1) \pm B\sqrt{u+1} + C &= 0 \\ \pm B\sqrt{u+1} &= -A(u+1) - C \\ \pm B\sqrt{u+1} &= -Au -(A+C) \\ B^2u + B &= A^2u^2 +2A(A+C)u + (A+C)^2 \\ A^2u^2 +(2A^2-B^2+2AC)u + (A+C)^2 - B &= 0 \end{align}
Letting $\beta = v^2 - 1$ would have resulted in $A^2v^2 +(2A^2-B^2+2AC)v + (A+C)^2 - B = 0$
So it seems that just using $\alpha$ will work. | 2019-10-15T21:23:49 | {
"domain": "stackexchange.com",
"url": "https://math.stackexchange.com/questions/1776534/when-are-we-able-to-find-a-quadratic-with-roots-that-are-a-function-of-another-q",
"openwebmath_score": 0.9999227523803711,
"openwebmath_perplexity": 247.54132372342798,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9799765546169712,
"lm_q2_score": 0.8499711775577736,
"lm_q1q2_score": 0.8329518261067969
} |
https://tutel.me/c/mathematics/questions/372370/prove+that+the+only+eigenvalue+of+a+nilpotent+operator+is+0 | #### [SOLVED] Prove that the only eigenvalue of a nilpotent operator is 0?
By Mathlete
I need to prove that if $\phi : V \rightarrow V$ is nilpotent, then its only eigenvalue is 0.
I know how to prove that this for a nilpotent matrix, but I'm not sure in the case of an operator. How would I be able to relate $\phi$ to a matrix?
Note: A nilpotent operator $\phi$ has been defined as $\phi^{n} = 0$, for some $n \geq 1$
#### @Somaye 2013-04-25 12:10:07
Suppose that $\phi$ has another eigenvalue $\lambda \ne 0$ so that $\phi(x)=\lambda x$ ($x$ is an eigenvector corresponding to $\lambda$)
Then, $\phi^n (x)= \phi^{(n-1)}(\phi(x))=\phi^{(n-1)}(\lambda x)=\lambda \phi^{n-1}(x)=\cdots=\lambda ^{n}x\ne 0$.
We have a contradiction, so $\phi$ can't have another eigenvalue except $0$.
#### @Rhys 2013-04-25 10:41:16
$\phi$ is nilpotent, so $\phi^n = 0$ for some $n$. Now let $v$ be an eigenvector: $\phi v = \lambda v$ for some scalar $\lambda$. Now we get $$0 = \phi^n v = \lambda^n v ~\Rightarrow~ \lambda=0 ~.$$
Note that this works in the infinite-dimensional case as well; there is no need to relate $\phi$ to a matrix.
Edit: As suggested in the comments, we can also show that $0$ is always an eigenvalue; in other words, $\phi$ always has at least one eigenvector. Take any $v \neq 0$; we know that $\phi^n v = 0$, so let $k$ be the smallest integer such that $\phi^k v \neq 0$. Then $\phi(\phi^k v) = 0$, so $\phi^k v$ is an eigenvector of $\phi$, with eigenvalue $0$.
#### @wj32 2013-04-25 10:48:29
For completeness, we might want to show that $0$ is actually an eigenvalue. If $v$ is injective then $\phi v \ne 0$ for every $v \ne 0$. If we choose any nonzero $v$ then $\phi^n v \ne 0$, which is a contradiction. Therefore $\ker\phi \ne \{0\}$, and any element of $\ker\phi$ is obviously an eigenvector with eigenvalue $0$.
#### @Rhys 2013-04-25 10:58:08
Yes, I probably should have added that part; I will do so, with a different wording.
### [SOLVED] Eigenvalue of linear operator iff eigenvalue of matrix representation.
• 2013-08-08 00:36:08
• John Daley
• 979 View
• 3 Score
• Tags: linear-algebra | 2019-04-18T14:15:47 | {
"domain": "tutel.me",
"url": "https://tutel.me/c/mathematics/questions/372370/prove+that+the+only+eigenvalue+of+a+nilpotent+operator+is+0",
"openwebmath_score": 0.982128381729126,
"openwebmath_perplexity": 180.58278892671098,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9799765546169712,
"lm_q2_score": 0.8499711775577736,
"lm_q1q2_score": 0.8329518261067969
} |
https://math.stackexchange.com/questions/2305478/probability-of-drawing-a-specific-number-out-of-a-pool-of-numbers-with-multiple | # Probability of drawing a specific number out of a pool of numbers with multiple draws
I have a pool of 100 different numbers, I draw 5 different numbers without putting them back into the pool. I want to calculate the chance of any single number being drawn.
So for example, what's the chance of the number 13 being drawn during those 5 draws?
I figured, the way to calculate this would be
$$\frac{1}{100}+(\frac{1}{99}*\frac{99}{100})+(\frac{1}{98}*\frac{98}{99})+(\frac{1}{97}*\frac{96}{97})+(\frac{1}{96}*\frac{95}{96})=0.0506$$
Which would mean the chance for that would be ~5.1%
Is this calculation correct? What is this calculation called and whats the formula for it?
• before the process is started, the chances that the 5th ball is going to be #13 is $\frac{99}{100}\times\frac{98}{99}\times\frac{97}{98}\times\frac{96}{97}\times\frac{1}{96} = \frac{1}{100}$ – Cato Jun 1 '17 at 11:46
Correct formula: $$\Pr(E)=\frac{\text{number of draws}}{\text{number of numbers}}=\frac5{100}$$ where $E$ denotes the event that number $13$ will be drawn.
For $i=1,2,3,4,5$ let $E_n$ denote the event that number $13$ is drawn at the $i$-th draw.
Then $\Pr(E_1)=\frac1{100}$ i.e. the first term in your summation.
You can calculate $\Pr(E_2)=\Pr(E_2\mid E_1^c)\Pr(E_1^c)=\frac{1}{99}\times\frac{99}{100}=\frac1{100}$, i.e. the second term in your summation.
So these terms are okay, and if you had proceeded without making any mistakes you would have found $\Pr(E_i)=\frac1{100}$ for every $i$.
Now start wondering: can you find any reason that the probability of drawing number $13$ at e.g. the $2$-nd draw should differ from the probability of drawing number $13$ at e.g. the $4$-th draw???
• I think maybe reasoning that 100 balls could be placed randomly into slots 1-100, then looking to see if #13 is in slot 1-5, makes it easier to see that 5/100 is the correct answer (by symmetry #13 is equally likely to be in any given group of 5 spaces, and is effectively the same overall experiment) – Cato Jun 1 '17 at 11:49
• @Cato That's indeed a good startoff for the stimulated "wondering" of the OP. – drhab Jun 1 '17 at 11:53
• @Cato Wow, that made it really easy to understand! – Luke Jun 1 '17 at 11:54
• @drhab This could still technically be seen as hypergeometric distribution right? Basically for my case it would be a population size = 100, number of successes in population = 1, sample size = 5, and number of successes in sample = 1 ? – Luke Jun 1 '17 at 11:58
• @Luke Yes. With outcome $\frac{\binom51\binom{95}0}{\binom{100}1}=\frac5{100}$. – drhab Jun 1 '17 at 12:00
This is binomial distribution without replacement, and is known as Hypergeometric distribution. Please find more here: https://en.wikipedia.org/wiki/Hypergeometric_distribution
your reasoning is on the right lines, but you aren't quite right, on draw 5 (for example), it has to have failed to be drawn 4 times
may I be lazy and assum just 3 draws, I get the below example
$P = \frac{1}{100} + \frac{99}{100}\times\frac{1}{99} + \frac{99}{100}\times\frac{98}{99}\times\frac{1}{98} = \frac{1}{100} + \frac{1}{100} +\frac{1}{100} = \frac{3}{100}$
the overall chance for 3 draws is 3/100, and the chance it will have come up on any of the particular draws is 1/100 - for your 5 draws - it will be 5 / 100 - | 2021-04-23T07:52:11 | {
"domain": "stackexchange.com",
"url": "https://math.stackexchange.com/questions/2305478/probability-of-drawing-a-specific-number-out-of-a-pool-of-numbers-with-multiple",
"openwebmath_score": 0.855228841304779,
"openwebmath_perplexity": 545.3276687644777,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9799765604649332,
"lm_q2_score": 0.8499711718571775,
"lm_q1q2_score": 0.8329518254909455
} |
https://math.stackexchange.com/questions/1967656/what-shall-we-call-a-function-or-relation-if-it-fails-passing-either-vertical | # What shall we call a “function” or relation if it fails passing either vertical or horizontal line test?
I found this extremely confusing because there is so called "rose pedal function"..and circle standard form... Does it mean that if a relation failed either vertical or horizontal test, then we have to use its shape to identify it? Is there really a rose pedal function or my prof was incorrect?
## 2 Answers
Your confusion is stemming from a misunderstanding of the definition of a function. Formally, a function $f$ from a set $A$ to a set $B$ is a subset of $A\times B = \{(a,b):a\in A\,\text{and}\, b\in B\}$, such that for each $a\in A$ there is a unique $b\in B$ with $(a,b)\in f$. In other words, every input gets one and only one output.
In the case of the rose petal function you are describing, we are forming ordered pairs by first picking an angle $\theta$ and then specifying a radius $r$ for that particular $\theta$. Hence, the rose petal function is a collection of ordered pairs $(\theta, r)$, such that to each angle corresponds a unique radius. When we plot this function in polar coordinates, it looks as though it is violating the "vertical line" test (we are specifying more than one radius for each angle), but this is not actually the case. If we plotted the rose petal function in the $\theta,r$-plane, we would see that this function did indeed pass the vertical line test, and is a well-defined function.
See here for a glimpse of what these "petal" functions look like in the $\theta,r$-plane so you can see how these functions actually do pass the vertical line test.
• Excellent answer. – RJM Oct 14 '16 at 1:47
• Got it. A function of whom is important for defining whether it is a function. – Newbornalive Oct 14 '16 at 1:51
There can be mappings into $\mathbb{R^2}$ that do not pass the vertical line test for the graph of a function. Ex. g := {(x,y) $\in$ $\mathbb{R}$ x $\mathbb{R}$| $x^2 + y^2 = 1$}
Your professor may have been referring to graphs in polar coordinates. For example r = $\sin 2\theta$
• I see. So a function failed vertical test in R can still be a function in other space. – Newbornalive Oct 14 '16 at 1:48
• Yes, unfortunately at high school and early college level the definition of function is too strongly tied to $f: \mathbb{R} \rightarrow \mathbb{R}$ – RJM Oct 14 '16 at 1:52 | 2019-10-15T14:45:39 | {
"domain": "stackexchange.com",
"url": "https://math.stackexchange.com/questions/1967656/what-shall-we-call-a-function-or-relation-if-it-fails-passing-either-vertical",
"openwebmath_score": 0.7078131437301636,
"openwebmath_perplexity": 251.69470515571876,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9799765552017674,
"lm_q2_score": 0.8499711718571775,
"lm_q1q2_score": 0.8329518210174063
} |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.