text
stringlengths 6
976k
| token_count
float64 677
677
| cluster_id
int64 1
1
|
|---|---|---|
College Algebra -Text Only - 7th edition wi
th withP.1 Review of Real Numbers and Their Properties P.2 Exponents and Radicals P.3 Polynomials and Special Products P.4 Factoring Polynomials P.5 Rational Expressions P.6 Errors and the Algebra of Calculus P.7 The Rectangular Coordinate System and Graphs
| 677.169 | 1 |
Similar Products From Other Merchants
Advanced Algebra: Expanded Edition is the second book in the Life of Fred High School Mathematics Series, and is designed for students in 10th grade who have already finished the preceding Beginning Algebra, Expanded Edition. This new edition of Algebra replaces the both the earlier Life of Fred...
Beginning Algebra: Expanded Edition is the first book in the Life of Fred High School Mathematics Series, and is designed for students in 9th grade. This expanded edition of Algebra replaces the both the earlier Life of Fred Beginning Algebra and Fred's Home Companion Algebra books; it also...
Pre-Algebra 2 with Economics is the third book in the Life of Fred Pre-Algebra Series (the fifth and last in the Getting Ready for High School Math series that includes both Arithmetic & Pre-Algebra books), and is designed for students who have completed the elementary, intermediate, and...
Pre-Algebra 1 with Biology is the second book in the Life of Fred Pre-Algebra Series (the fourth in the Getting Ready for High School Math series that includes both Arithmetic & Pre-Algebra books), and is designed for students who have completed the elementary, intermediate, and arithmetic series...
Zillions of Practice Problems: Advanced Algebra accompanies the sold-separately Life of Fred: Advanced Algebra, Expanded Editionand provides students with problems that directly correspond to the chapters in the text. Each chapter's questions are divided into two parts; the first part offers a...
Life of Fred Pre-Algebra O with Physics was formerly-titled Elementary Physics. The content has remained the same and it is still the first book in the Life of Fred Pre-Algebra Getting Ready for High School Math Series which is designed for students who have completed the elementary, intermediate,...
| 677.169 | 1 |
X-tending the Fibonacci Sequence
Glenn T. Moran
Article outlines a lesson on the Fibonacci and Lucas sequences, giving opportunity for computation practice, mental mathematics, and proof; for algebra students, the article discusses an extension for solving simultaneous equations.
This is available to members of NCTM. Please log in now to view this article. If you are interested in a NCTM membership join now.
| 677.169 | 1 |
Importance of maths not fully understood by students
Jun 25, 2014
Too many sixth form students do not have a realistic understanding of either the relevance of Mathematics and Statistics to their discipline or of the demands that will be put upon them in undergraduate study, according to a new report published today by the Higher Education Academy (HEA). The report examines the mathematical and statistical needs of students in undergraduate disciplines including Business and Management, Chemistry, Economics, Geography, Sociology and Psychology.
Professor Jeremy Hodgen, lead author of the report from the Department of Education & Professional Studies, said: 'Too few students in the UK study Mathematics after the age of 16, yet the study demonstrates that Mathematics matters across a range of subjects at university. The report recommends that prospective undergraduates are better informed of this when applying to higher education.'
Lack of confidence and anxiety about Mathematics and Statistics is also a problem for many students, making the transition into higher education particularly challenging. A number of recommendations are made within the report to address this problem, but overall it calls for better dialogue between the sectors so that pre-university students have a better understanding of what is expected of them and the higher education sector has a better understanding of what their undergraduates can do.
The report also draws attention to developments at pre-university level, where new 'Core Maths' courses are being designed to meet the needs of the many students (the report estimates at least 200,000 a year) who need Mathematics but for whom a full A-level would not be appropriate. It calls for higher education to become actively involved in and to influence this work.
Dr Mary McAlinden, Discipline Lead for Mathematics, Statistics and Operational Research at the HEA said: 'Many students are surprised at the amount of mathematical content in their undergraduate programmes and some struggle to cope with this content.
'This project, and the accompanying reports, seeks to promote greater understanding between the higher education and pre-university sectors so that students will arrive at university better prepared and better able to cope with the mathematical and statistical demands of their undergraduate studies.'
Dr Janet De Wilde, Head of STEM at the HEA said: 'This report demonstrates the importance that the HEA places on this topic. The recommendations it contains are valuable to the sector to help further the discussion between the secondary and tertiary sector to inform policy development and teaching practice to address the importance of mathematical and statistical skills.'
The solution to England's poor participation rate in post-16 maths education could lie in a new qualification that provides a clear and attractive alternative for students who don't currently go on to study ...
For many years, studies have shown that American students score significantly lower than students worldwide in mathematics achievement, ranking 25th among 34 countries. Now, researchers from the University of Missouri have ...
A key issue facing the UK is how to inspire the next generation of scientists, technologists, engineers and mathematicians on which our future well-being and economy depends. A new report, published yesterday [8 Nov], examines ...
All young people should continue to study maths at least until they are 18, even if they have already gained a good GCSE in the subject, the Sutton Trust said today, because the GCSE curriculum fails to give them the practical
| 677.169 | 1 |
Courses
MATH 085
Pre Algebra
Developmental in nature, this course is designed to strengthen the arithmetic skills of the students. Topics covered include: operation of whole numbers, fractions, percents, decimals, ratios, integers, exponents, operations with signed numbers, an introduction to geometry, and an introduction to algebra. Information on math test taking strategies and study skills will be presented and practiced along with the principles of On Course. This course is developmental in nature and cannot be used toward any associate degree. No calculators are permitted.
Credit Hrs: 4.00 Contact Hrs: 4.00
MATH 095
Basic Algebra
A study of the fundamental concepts and operations of algebra, polynomials, equations, application problems, factoring, introduction to functions and graphs, systems of linear equations, exponents, radicals, and simple quadratic equations. This course is developmental in nature and cannot be used toward any associate degree. Prerequisite: MATH-085 with "C" grade or better or an appropriate score on the mathematics placement test. No calculators are permitted.
Credit Hrs: 4.00 Contact Hrs: 4.00
MATH 097
Mathematical Literacy
MATH-097 is designed for students in majors that do not require Intermediate or College Algebra. It can serve as a pre-requisite to MATH-107 Liberal Arts Math (which meets the MTA math requirement) or BUSN-177 Business Math. The course integrates numeracy, algebraic reasoning, data literacy, unctions, equations, and modeling along with college success content. It is a Transtional Studies course and does not meet the MTA math requirement. A scientific calculator is required. Prerequisite:MATH-085 Pre-Algebra with a "C" or better OR appropriate score on a math placement test.
Credit Hrs: 0.00 Contact Hrs: 0.00
MATH 102
Introduction to Technical Math
Topics include basic algebra formula usage, signed numbers, practical measurements, metrics and conversions, relative error, basic geometry, and right triangle trigonometry. This course is designed to meet occupational program requirements or as a preparation for MATH-106 for those needing more advanced mathematics. Scientific/graphing calculator. Prerequisite: MATH-085 with a "C" or better or appropriate score on the mathematics placement test.
Credit Hrs: 4.00 Contact Hrs: 4.00
MATH 105
Intermediate Algebra
A study of real numbers, algebraic expressions, exponents, complex numbers, solution of linear inequalities, quadratic equations and absolute value equations, equations of lines, conic sections, functions, exponential functions, logarithmic functions, exponential and logarithmic equations, and systems of equations. This course is designed to prepare students for MATH-110 College Algebra. This course may be used as an elective course; however, it does not fulfill the natural science requirement for the Associate of Science degree or the MACRAO agreement. A scientific calculator is required. Prerequisite: MATH-095 with a "C" grade or better or appropriate score in the mathematics placement test.
Credit Hrs: 4.00 Contact Hrs: 4.00
MATH 107
Liberal Arts Mathematics
Upon successful completion of this course, the student will understand and be able to use mathematics in a variety of practical applications, including topics in graph theory, probability, statistics, theory of numbers, coding theory, symmetry, and financial math. These topics will be presented along with real world applications such as street networks, planning and scheduling, and voting schemes, with an emphasis on problem solving. This course is designed for transfer students in the Associate of Arts program who do not need College Algebra at their transfer institution. A calculator is required. Prerequisite: MATH-095 with a "C" or better or appropriate scoreon the mathematics placement test.
Credit Hrs: 4.00 Contact Hrs: 4.00
MATH 110
College Algebra
A study of equations, systems of equations, inequalities, functions and their graphs, polynomial and rational functions, exponential and logarithmic functions, complex numbers, theory of equations. Prerequisite: MATH-105 with a grade of "C" or better or an appropriate score on the mathematics placement test. Graphing calculator is required.
Credit Hrs: 4.00 Contact Hrs: 4.00
MATH 111
Trigonometry
A study of the trigonometric functions, their properties, solutions of right and oblique triangles, radian measure, graphs, identities, trigonometric equations, applications, with optional topics of vectors in the plane, complex numbers, and polar coordinates. A graphing calculator is required. Prerequisite: MATH-110 with a "C" or better or appropriate score on the mathematics placement test. Qualified students may enroll in MATH-111 and MATH-141 during the same semester.
Credit Hrs: 3.00 Contact Hrs: 3.00
MATH 141
Analytical Geometry and Calculus I
Functions and graphs, limits, differentiation of algebraic and trigonometric functions, exponential, and logarithmic functions, applications, the Mean Value theorem, definite and indefinite integrals, and the Fundamental Theorem of integral calculus. Prerequisites: MATH-110 and MATH-111 (or high school trigonometry) with a "C" or better or appropriate score on mathematics placement test. Qualified students may enroll in MATH-111 and MATH-141 during the same semester. Graphing calculator required.
Credit Hrs: 5.00 Contact Hrs: 5.00
| 677.169 | 1 |
1447111257
9781447111252
Details about MATLAB® for Engineers Explained:
This book is written for students at bachelor and master programs and has four different purposes, which split the book into four parts: 1. To teach first or early year undergraduate engineering students basic knowledge in technical computations and programming using MATLAB. The first part starts from first principles and is therefore well suited both for readers with prior exposure to MATLAB but lacking a solid foundational knowledge of the capabilities of the system and readers not having any previous experience with MATLAB. The foundational knowledge gained from these interactive guided tours of the system will hopefully be sufficient for an effective utilization of MATLAB in the engineering profession, in education and in research. 2. To explain the foundations of more advanced use of MATLAB using the facilities added the last couple of years, such as extended data structures, object orientation and advanced graphics. 3. To give an introduction to the use of MATLAB in typical undergraduate courses in electrical engineering and mathematics, such as calculus, algebra, numerical analysis and statistics. This part also contains introductions and mini-manuals to the most used MATLAB toolboxes. Thus, some chapters require additional MATLAB toolboxes. The idea is to give a brief tutorial on each subject and show the possibilities for applying MATLAB to each application area. We have focused on basic concepts in the applications, without trying to explain all theory behind the examples.
| 677.169 | 1 |
Topics in Algebra
Browse related Subjects
New edition includes extensive revisions of the material on finite groups and Galois Theory. New problems added throughout.New edition includes extensive revisions of the material on finite groups and Galois Theory. New problems added throughout.Read Less
Very good. ? nside: owner's name (RRector), owner's RR initials on bottom edge; then all rather clean and newish; no marks; OUTSIDE spine NOT faded, not sunned, not bleached, Spivak CALCULUS, 1967, p519: Herstein's book is meant to be intermediate in difficulty between (Birkhoff) A Survey...and one of the great classics (content is now grad level? ); handsome black cloth with gilt (not silver]; 342pp.
All Editions of Topics in Algebra
Customer Reviews
Great book for math majors
This lovingly written and exquisitely crafted book provides a detailed introduction to abstract algebra. First published in 1964, it is still an excellent textbook and will probably never go out of date.
The text does not assume any background beyond high school math, but the mathematics it covers is intense and detailed. This is a book for college math majors, not for someone who is looking for a casual introduction to
| 677.169 | 1 |
Contains fully worked-out solutions to all of the odd-numbered exercises in the text and all the Cumulative Review exercises in the student textbook, giving students a way to check their answers and ensure that they took the correct steps to arrive at an answer.
| 677.169 | 1 |
Precise Calculator has arbitrary precision and can calculate with complex numbers, fractions, vectors and matrices. Has more than 150 mathematical functions and statistical functions and is programmable (if, goto, print, return, for).
| 677.169 | 1 |
A Cook-Book Of Mathematics This text provides the students with simple cookbook recipes which covers the most significant issues of mathematical economics.
24375
A Gentle Introduction to the Art of Mathematics This open-source textbook covers several topics in the foundations of mathematics (logic, sets, relations, functions and cardinality) and introduces the reader to many techniques of mathematical proof.
A=B Shows how several recently developed computer algorithms can simplify complex summations, presenting the underlying mathematical theory of these methods, the principle theorems and proofs, and the implementation using Maple packages.
15073
Advanced Calculus, Revised Edition This book divides roughly into a first half which develops the calculus (principally the differential calculus) in the setting of normed vector spaces, and a second half which deals with the calculus of differentiable manifolds.
28482
Algorithmic Mathematics Introduces the basic algorithms for computing and provides a constructive approach to abstract mathematics.
Analytic Combinatorics Provides a unified treatment of analytic methods in combinatorics. Many examples are given that relate to words, integer compositions and partitions, paths and walks, graphs, mappings and allocations, lattice paths, permutations, trees, and planar maps.
Calculus Without Limits Introduces differentiability as a local property without using limits. The course is designed for life science majors who have a precalculus background, and whose primary interest lies in the applications of calculus.
12189
Calculus, Applications and Theory This book gives complete proofs of all theorems in one variable calculus and to at least give plausibility arguments for those in multiple dimensions. Serious students will find complete explanations in this book.
24432
California Free Digital Calculus Textbook This is a freely available calculus book, covering a fairly standard course sequence: single variable calculus, infinite series, and multivariable calculus. There is no chapter on differential equations.
Derivations of Applied Mathematics An applied exposition of proofs of numerous mathematical results useful in the modeling of physical systems. If you have seen a mathematical result, if you want to know why the result is so, you can look for the proof here.
14172
Elementary Mathematics Develops fundamental skills in algebra, trigonometry, indices and logarithms, equations and inequalities, as well as progressions. Includes differential and integral calculus in a reasonable level.
24103
Essential Mathematics Provides training for a compulsory examination, designed to address the lack of fluency in elementary arithmetic and algebra.
16970
Introduction to Methods of Applied Mathematics This open source textbook contains material on calculus, functions of a complex variable, ordinary differential equations, partial differential equations and the calculus of variations. Includes exercises and solutions.
| 677.169 | 1 |
Elementary Algebra
I. Course Prefix/Number: MAT 070
Course Name: Elementary Algebra
Credits: 4 (4 lecture; 0 lab)
II. Prerequisite
MAT 060 or appropriate score on Mathematics Placement Test.
III. Course (Catalog) Description
Course prepares students for an intermediate algebra course by covering the fundamental concepts, operations, and applications of basic algebra. Algebraic topics include linear equations and inequalities, polynomial operations, graphing equations and inequalities in two variables, and systems of equations. Course objectives will be achieved using computer-assisted learning, group discussions, and individual tutoring.
IV. Learning Objectives
Module 6 Objectives: Solve and graph first degree equations in one variable. Solve formulas for specific variables. Solve applied problems involving first degree equations in one variable. Solving and graph first degree inequalities in one variable.
Module 7 Objectives: Simplify expressions using the laws of exponents. Calculate using Scientific Notation. Perform addition and subtraction of polynomials. Perform multiplication of polynomials including some special products. Perform division of a polynomial by a monomial.
Module 8 Objectives: Factor out the greatest common factor from a polynomial. Factor trinomials successfully. Factor polynomials using the difference of squares. Solve quadratic equations by factoring.
Module 9 Objectives: Solve and graph first degree equations in two variables. Calculate slope and intercepts of linear equations in two variables. Solve applied problems involving slope. Solve and graph first degree inequalities in two variables.
Students and employees at Oakton Community College are required to demonstrate academic integrity
and follow Oakton's Code of Academic Conduct. This code prohibits:
• cheating,
• plagiarism (turning in work not written by you, or lacking proper
citation),
• falsification and fabrication (lying or distorting the truth),
• helping others to cheat,
• unauthorized changes on official documents,
• pretending to be someone else or having someone else pretend to
be you,
• making or accepting bribes, special favors, or threats, and
• any other behavior that violates academic integrity.
There are serious consequences to violations of the academic integrity
policy. Oakton's policies and procedures provide students a fair
hearing if a complaint is made against you. If you are found to have violated
the policy, the minimum penalty is failure on the assignment and, a disciplinary
record will be established and kept on file in the office of the Vice
President for Student Affairs for a period of 3 years.
Details of the Code of Academic
Conduct can be found in the Student Handbook.
Methods of instruction include one-on-one and/or small group discussion, and required website ancillaries. Calculators/computers will be used. Course may be taught as face-to-face, media-based, hybrid or online course.
VIII. Course Practices Required
This course will be taught by a classroom instructor with the use of an interactive computer website. Course participants must attend scheduled class hours as well as one computer lab hour per week. Students may be dropped from the course if they miss more than three class sessions or three lab hours.
Each of the first four modules must be completed with the minimal post-test score as prescribed by the department to proceed to the final module for the course. All course work must be completed in a notebook.
Students may complete a course at any time during the semester. Upon completion of a course, the student can start the next sequential course. A new access code must be purchased at that time. If all modules of a course are not successfully completed within a semester, the student can re-enroll in the same course the following semester beginning with their first uncompleted module.
IX. Instructional Materials
Note: Current textbook information for each course and section is available on Oakton's Schedule of Classes.
Within the Schedule of Classes, textbooks can be found by clicking on an individual course section and looking for the words "View Book Information".
If you have a documented learning, psychological, or physical disability you may be entitled to reasonable academic accommodations or services. To request accommodations or services, contact the Access and Disability Resource Center at the Des Plaines or Skokie campus. All students are expected to fulfill essential course requirements. The College will not waive any essential skill or requirement of a course or degree program.
| 677.169 | 1 |
Our online classes bring outstanding students together with highly accomplished instructors to prepare
the students for the rigors of top-tier colleges and internationally competitive careers.
The AoPS Online School is accredited by the Western Association of Schools and Colleges.
The Art of Problem Solving mathematics curriculum is designed
for outstanding math students in grades 6-12.
Our texts offer broader, deeper, and more challenging instruction
than other curricula. Our Beast Academy elementary school curriculum
currently covers grades 3 and 4, and will span grades 2 through 5 upon completion.
The Art of Problem Solving texts have been used by tens of thousands of outstanding students,
including many winners of major national contests such as MATHCOUNTS and the American
Mathematics Competitions.
Contests Collections
View our collections of contest problems from around the world, with tens of thousands of problems
from national and international competitions including American Mathematics Competitions,
International Math Olympiad, Putnam, Harvard-MIT Math Tournament, Math Prize for Girls,
and the USAMTS. We also host problems from national olympiads and IMO Team Selection Tests
from countries around the world, such as Russia, China, Japan, Romania, Vietnam,
France, the United Kingdon, Germany, Korea, and many more.
Who is online?
There are 802 users online, including the 265
registered members listed below, and 22 other hidden registered users.
Alcumus is our free adaptive online learning system.
It offers students a customized learning experience, adjusting to student
performance to deliver appropriate problems and lessons.
Alcumus is aligned to our Introductory online courses
and to our Introduction
series of textbooks. Teacher tools are available for instructors and
parents to monitor student progress.
| 677.169 | 1 |
Detailed Course Information
Feb 28, 2015
Select the desired Level or Schedule Type to find available classes for the course.
MATH 1001 - Quantitative Skill & Reasoning
Prerequisite: Exit or exemption from Learning Support
mathematics.
This course places quantitative skills and reasoning
in the context of experiences that students will
be likely to encounter. It emphasizes processing
information in context from a variety of representations,
understanding of both the information and the
processing, and understanding which conclusions
can be reasonably determined.
NOTE: This course is an alternative in Area A of the
Core Curriculum and is not intended to supply
sufficient algebraic background for students who
intend to take Precalculus or the Calculus sequences
for mathematics and science majors.
| 677.169 | 1 |
Mathematical miniatures by Svetoslav Savchev(
) 10
editions published
between
2003
and
2011
in
English
and held by
1,265 WorldCat member
libraries
worldwide
Mathematical Olympiad challenges by Titu Andreescu(
Book
) 39
editions published
between
2000
and
2009
in
English
and held by
1,035 WorldCat member
libraries
worldwide
"Mathematical Olympiad Challenges is a rich collection of problems put together by two experienced and well-known professors and coaches of the U.S. International Mathematical Olympiad Team. Hundreds of beautiful, challenging, and instructive problems for algebra, geometry, trigonometry, combinatorics, and number theory were selected from numerous mathematical competitions and journals. An important feature of the work is the comprehensive background material provided with each grouping of problems." "Aimed at motivated high school and beginning college students and instructors, this work can be used as a text for advanced problem-solving courses, for self-study, or as a resource for teachers and students training for mathematical competitions and for teacher professional development, seminars, and workshops."--Jacket
Complex numbers from A to--Z by Titu Andreescu(
) 33
editions published
between
2004
and
2014
in
English and German
and held by
837 WorldCat member
libraries
worldwide
"The book reflects the unique experience of the authors. It distills a vast mathematical literature, most of which is unknown to the western public, and captures the essence of an abundant problem culture. The target audience includes undergraduates, high school students and their teachers, mathematical contestants (such as those training for Olympiads or the W.L. Putnam Mathematical Competition) and their coaches, as well as anyone interested in essential mathematics."--Page 4 of cover
Mathematical olympiad treasures by Titu Andreescu(
) 27
editions published
between
2002
and
2012
in
English
and held by
637 WorldCat member
libraries
worldwide
"Collection of problems in geometry and trigonmometry, algebra, number theory, and combinatorics. It encourages readers to think creatively about techniques and strategies for solving real-world problems, with new sections, revisions, and many more Olympiad-like problems at various levels of difficulty. The problems are clustered by topic into three self-contained chapters. The book begins with elementary facts, followed by carefully selected problems and detailed, step-by-step solutions, which then lead to more complicated, challenging problems and their solutions"--Back cover
104 number theory problems from the training of the USA IMO team by Titu Andreescu(
) 23
editions published
between
2006
and
2010
in
English and Japanese
and held by
560 WorldCat member
libraries
worldwide
This challenging problem book by renowned US Olympiad coaches, mathematics teachers, and researchers develops a multitude of problem-solving skills needed to excel in mathematical contests and research in number theory. Offering inspiration and intellectual delight, the problems throughout the book encourage students to express their ideas, conjectures, and conclusions in writing. Applying specific techniques and strategies, readers will acquire a solid understanding of the fundamental concepts and ideas of number theory. Key features: @* Contains problems developed for various mathematical contests, including the International Mathematical Olympiad (IMO) @* Builds a bridge between ordinary high school examples and exercises in number theory and more sophisticated, intricate and abstract concepts and problems @* Begins by familiarizing students with typical examples that illustrate central themes, followed by numerous carefully selected problems and extensive discussions of their solutions @* Combines unconventional and essay-type examples, exercises and problems, many presented in an original fashion @* Engages students in creative thinking and stimulates them to express their comprehension and mastery of the material beyond the classroom 104 Number Theory Problems is a valuable resource for advanced high school students, undergraduates, instructors, and mathematics coaches preparing to participate in mathematical contests and those contemplating future research in number theory and its related areas
103 trigonometry problems from the training of the USA IMO team by Titu Andreescu(
) 22
editions published
between
2005
and
2010
in
English and Japanese
and held by
552 WorldCat member
libraries
worldwide
"103 Trigonometry Problems is a cogent problem solving resource for advanced high school students, undergraduate and mathematics teachers engaged in competition training."--Jacket
Geometric problems on maxima and minima by Titu Andreescu(
) 20
editions published
between
2005
and
2006
in
English
and held by
536 WorldCat member
libraries
worldwide
Questions of maxima and minima have great practical significance, with applications to physics, engineering, and economics; they have also given rise to theoretical advances, notably in calculus and optimization. Indeed, while most texts view the study of extrema within the context of calculus, this carefully constructed problem book takes a uniquely intuitive approach to the subject: it presents hundreds of extreme value problems, examples, and solutions primarily through Euclidean geometry. Key features and topics: * Comprehensive selection of problems, including Greek geometry and optics, Newtonian mechanics, isoperimetric problems, and open questions, such as Malfatti?s problem * Unified approach to the subject, with emphasis on geometric, algebraic, analytic, and combinatorial reasoning * Presentation and application of classical inequalities, including Cauchy--Schwarz and Minkowski?s Inequality; important results in real variable theory, such as the Intermediate Value Theorem; and emphasis on simple but useful geometric concepts, including transformations, convexity, and symmetry * Clear solutions to the problems, often accompanied by figures * Hundreds of exercises of varying difficulty, from straightforward to Olympiad-caliber Written by a team of established mathematicians and teachers, this work draws on the authors? experience in the classroom and as Olympiad coaches. By exposing readers to a wealth of creative problem-solving approaches, the text communicates not only geometry but also algebra, calculus, and topology. Ideal for use at the junior and senior undergraduate level, as well asin enrichment programs and Olympiad training for advanced high school students, this book?s breadth and depth will appeal to a wide audience, from secondary school teachers and pupils to graduate students, professional mathematicians, and puzzle enthusiasts. TOC:Preface * Methods for Solving Geometric Problems on Maxima and Minima * Selected Types of Geometric Problems on Maxima and Minima * Miscellaneous Problems * Hints and Solutions * Glossary
Putnam and beyond by Rǎzvan Gelca(
) 15
editions published
between
2006
and
2007
in
English and Undetermined
and held by
520 WorldCat member
libraries
worldwide
Putnam and Beyond takes the reader on a journey through the world of college mathematics, focusing on some of the most important concepts and results in the theories of polynomials, linear algebra, real analysis in one and several variables, differential equations, coordinate geometry, trigonometry, elementary number theory, combinatorics, and probability. Using the W.L. Putnam Mathematical Competition for undergraduates as an inspiring symbol to build an appropriate math background for graduate studies in pure or applied mathematics, the reader is eased into transitioning from problem-solving at the high school level to the university and beyond, that is, to mathematical research. Key features of Putnam and Beyond @* Preliminary material provides an overview of common methods of proof: argument by contradiction, mathematical induction, pigeonhole principle, ordered sets, and invariants. @* Each chapter systematically presents a single subject within which problems are clustered in every section according to the specific topic. @* The exposition is driven by more than 1100 problems and examples chosen from numerous sources from around the world; many original contributions come from the authors. @* Complete solutions to all problems are given at the end of the book. The source, author, and historical background are cited whenever possible. This work may be used as a study guide for the Putnam exam, as a text for many different problem-solving courses, and as a source of problems for standard courses in undergraduate mathematics. Putnam and Beyond is organized for self-study by undergraduate and graduate students, as well as teachers and researchers in the physical sciences who wish to expand their mathematical horizons
Number theory structures, examples, and problems by Titu Andreescu(
) 20
editions published
in
2009
in
English
and held by
474 WorldCat member
libraries
worldwide
While the forefront of number theory is replete with sophisticated and famous open problems, at its foundation are basic, elementary ideas that can stimulate and challenge beginning students. This introductory text focuses on a problem-solving approach to the subject, situating each concept within the framework of an example for readers to solve
An introduction to diophantine equations a problem-based approach by Titu Andreescu(
) 17
editions published
in
2010
in
English
and held by
471 WorldCat member
libraries
worldwide
"This problem-solving book is an introduction to the study of Diophantine equations, a class of equations in which only integer solutions are allowed. The material is organized in two parts: Part I introduces the reader to elementary methods necessary in solving Diophantine equations, such as the decomposition method, inequalities, the parametric method, modular arithmetic, mathematical induction, Fermat's method of infinite descent, and the method of quadratic fields; Part II contains complete solutions to all exercises in Part I. The presentation features some classical Diophantine equations, including linear, Pythagorean, and some higher degree equations, as well as exponential Diophantine equations. Many of the selected exercises and problems are original or are presented with original solutions. [This book] is intended for undergraduates, advanced high school students and teachers, mathematical contest participants - including Olympiad and Putnam competitors - as well as readers interested in essential mathematics. The work uniquely presents unconventional and non-routine examples, ideas, and techniques."--From back cover
Problems in real analysis advanced calculus on the real axis by Teodora-Liliana T Rădulescu(
) 18
editions published
in
2009
in
English
and held by
468 WorldCat member
libraries
worldwide
"Problems in Real Analysis: Advanced Calculus on the Real Axis features a comprehensive collection of challenging problems in mathematical analysis that aim to promote creative, non-standard techniques for solving problems. This self-contained text offers a host of new mathematical tools and strategies Which develop a connection between analysis and other mathematical disciplines, such as physics and engineering. A broad view of mathematics is presented throughout; the text is excellent for the classroom or self-study. It is intended for undergraduate and graduate students in mathematics, as well as for researchers engaged in the interplay between applied analysis, mathematical physics, and numerical analysis."--Jacket
102 combinatorial problems : from the training of the USA IMO team by Titu Andreescu(
Book
) 11
editions published
between
2002
and
2003
in
English
and held by
369 WorldCat member
libraries
worldwide
"The book is systematically organized, gradually building combinatorial skills and techniques and broadening the student's view of mathematics. Aside from its practical use in training teachers and students engaged in mathematical competitions, it is a source of enrichment that is bound to stimulate interest in a variety of mathematical areas that are tangential to combinatorics."--BOOK JACKET
Essential linear algebra with applications : a problem-solving approach by Titu Andreescu(
) 7
editions published
between
2004
and
2014
in
English and Undetermined
and held by
193 WorldCat member
libraries
worldwide
This textbook provides a rigorous introduction to linear algebra in addition to material suitable for a more advanced course while emphasizing the subject?s interactions with other topics in mathematics such as calculus and geometry. A problem-based approach is used to develop the theoretical foundations of vector spaces, linear equations, matrix algebra, eigenvectors, and orthogonality. Key features include: ? a thorough presentation of the main results in linear algebra along with numerous examples to illustrate the theory; ? over 500 problems (half with complete solutions) carefully selected for their elegance and theoretical significance; ? an interleaved discussion of geometry and linear algebra, giving readers a solid understanding of both topics and the relationship between them. Numerous exercises and well-chosen examples make this text suitable for advanced courses at the junior or senior levels. It can also serve as a source of supplementary problems for a sophomore-level course
| 677.169 | 1 |
Contact information ( * required )
The Vernon Area Public Library in Lincolnshire suggests these titles about calculus:
• "A Tour of Calculus" by David Berlinski
• "Calculus Demystified" by Stephen G. Krantz
• "Calculus: A Complete Introduction" by Paul Abbott
• "Isaac Newton: Organizing the Universe" by William Boerst
• "Great Mathematicians" by Raymond Flood
Calculus was invented more than 300 years ago by two mathematicians -- Sir Isaac Newton and Gottfried Wilhelm von Leibniz.
The building blocks of calculus, which measures rate of change using equations called derivatives and area using equations called integrals, were developed in ancient Egypt, Greece, India, the Middle East and China.
Basically, calculus proves comparisons of quantities and allows you to measure and determine points on curves without seeing the entire curved object. As a result, by using calculus, you can find answers to questions in fields as varied as physics, statistics, engineering, rocketry, chemistry and even economics.
The study of calculus in high school is divided into two sessions, generally offered first semester and second semester, called Calc AB and Calc BC.
"There are two big ideas: tangent lines and areas. In algebra, you can write the equation of a line given two points," said Rick Brenner, a math teacher and math team coach at Libertyville High School. "But how do you do that with one point on a curve? That's Calc I (AB). How do you find the area of shapes that are not polygons? That's Calc II (BC)."
Fan rivalries between the Cubs and White Sox are a friendly walk around the bases compared to the curve ball that placed Newton and von Leibniz in constant competition.
Both brilliant mathematicians worked on breakthrough formulas that led to the development of calculus about the same time, but von Leibniz was first to publish his findings. Not to be outdone, Newton and his supporters claimed von Leibniz copied Newton's concepts.
"Many people credit Newton with the discovery of calculus, but it was sort of coinvented by Sir Isaac Newton and Gottfried Wilhelm von Leibniz," Brenner said.
"Some think von Leibniz stole Newton's work. Some think Newton dropped some hints that von Leibniz ran with. Newton did not publish his work while he worked on it -- it was published decades later. This caused quite a controversy in the early 1700s. Today, we actually use many of von Leibniz's ideas and notations
| 677.169 | 1 |
This book puts Maths into contexts that make sense to pupils, showing them how it relates to other subjects and how useful it is in everyday life. Each concept is presented in a clear, relevant and engaging way, ensuring that pupils are inspired to succeed. Key points and practice questions are all arranged by level to provide explicit differentiation. Stimulating, fun and exciting activities provide a memorable learning experience with high impact images to help put maths in an exciting context. Extended activities give pupils plenty of opportunities for problem solving and peer discussion. SAT-style questions at the end of every Unit ensure that all readers are fully prepared.
{"currencyCode":"GBP","itemData":[{"priceBreaksMAP":null,"buyingPrice":15.94,"ASIN":"043553730X","isPreorder":0},{"priceBreaksMAP":null,"buyingPrice":5.99,"ASIN":"0435537385","isPreorder":0},{"priceBreaksMAP":null,"buyingPrice":5.99,"ASIN":"0435537393","isPreorder":0}],"shippingId":"043553730X::u%2BZf9DGcYq3O%2BoP2SmdNJ9SExqVnDdzjPYGw%2F0zlH%2FthM3jV%2B3HurKuYWazNzZeXNWwHaCgRy5G4rcRSIRXtszjPA09rhwq1,0435537385::RBkhqu4PlMzqyvLPm8MDMDLAPiyLH6h3VfCi%2F3zFRwbj%2BE1%2BoRoJSmVkCJfwaKk7%2BO%2FEMB9OBJB3V6fhhieIdomrox9sMeJU,0435537393::I6qD6BPiaK%2FN4yXWm1JU2fbujuOaDzFNIskiaGz8WGyWdq8055C7v4V8%2BjennwfHxeZHWsTNXuHGWibBUnOFePKZ2JuBF8I Level Up Maths: Pupil Book (Level 3-5) (Level Up Maths) for an Amazon Gift Card of up to £4.00, which you can then spend on millions of items across the site. Trade-in values may vary (terms apply). Learn more
Most Helpful Customer Reviews
This book is impossible to use in schools because it flies through the topics without enough practice for the students. In one exercise there can be up to 5 different sub topics which all require different skills. There is probably about 2/3 questions on each where really you need 15 questions to practice properly. Before you know it the kids are stuck and messing about. If you are thinking of buying this for your school, please reconsider. We've wasted £1000's on them to use in Year 7 and Year 8. I have now vowed never to use them again. Oh Yeah, the good thing is that they have the level by each question but surely a decent teacher would know all this stuff anyway...
I am giving 5 star ratings to this product. Because of this my daughter got 5a in internal test in year 6. And now her teacher recommend her for 6c grade. It is very colorful, well explained and each example in detail so parent can understand easily and help their kid in process of level up.
The level up series are fantastic books with some good ideas. Really useful for working out what level to set you work. Shame there are no answer...but we can't have everything I suppose. A must have for class teachers.
| 677.169 | 1 |
More About
This Textbook
Overview
The Bittinger System for Success–Make it Work forEditorial Reviews
Booknews
Bittinger (mathematics, Indiana U. and Purdue U.) uses a five step problem solving approach with real data applications to make algebra both straightforward and connected to everyday life. Detailed graphs and color drawings and photographs also help students to visualize mathematical concepts. The book is designed to assist in every step of curriculum, from review exercises with answers, to pre and post-tests. There are also a number of supplemental materials available for the instructor to use in conjunction with this
| 677.169 | 1 |
This book provides an introduction to hypergraphs, its aim being to overcome the lack of recent manuscripts on this theory. In the literature hypergraphs have many other names such as set systems and families of sets. This work presents the theory of hypergraphs in its most original aspects, while also introducing and assessing the latest concepts... more...
| 677.169 | 1 |
Commentary on the NYC Mathematics Scope and
Sequence
By Frederick Greenleaf, New York University
With Assistance from Ralph Raimi, University of Rochester
December, 2002
Introduction
What follows is a point-by-point commentary on the New York City Scope and Sequence
for Mathematics. The comments were originally written in
connection with a meeting between NYC mathematicians and Mr. Evan
Rudall, Chair of the Children First numeracy working group, arranged
by Mr. Rudall and Elizabeth Carson of New York City
HOLD. The NYC Scope and Sequence was discussed, with reference to
a hand-written version of the present commentary, in the Talking Points for that meeting.
As is seen from the extensive comments, my colleagues and I feel that
the present NYC Math Scope and Sequence is deeply flawed. While one
could attempt a detailed revision, some have questioned whether the
considerable effort needed would be worth it. California has already
gone through the agony of crafting a similar document (at least for
grades K - 7), and I have repeatedly compared the NYC Scope and
Sequence with the CA Standards in my commentary. Although the CA
Standards are not perfect, I believe they represent a better starting
point for an effort to revise the NYC Scope and Sequence than the
present document.
In any case, revising the NYC Scope and Sequence will not be a small
task, though it is important. A very large proportion of students in
K-8 will ultimately seek entry to college level programs (including
community colleges or technical institutes) where basic math skills
are required to succeed. The current NYC Scope and Sequence reflects
a curriculum that is inadequate to prepare students for the sort of
high school work needed to qualify for such programs.
Preliminary Comments
1. The Scope and Sequence lists (many) goals for math instruction
grade-by-grade. Sometimes these abstract goals are accompanied by
brief illustrative examples indicating what is appropriate for the
grade level being discussed; often there is no elaboration, and in a
distressing number of cases the stated goal is vague and unclear.
The CA Standards present a list of goals in the manner of the NYC
Scope and Sequence, but these are accompanied by a whole manual that
provides multiple (and specific) illustrative examples for each goal
listed in the main document. We could make the NYC Scope and Sequence
much more useful for teachers (and parents) by following California's
lead. This is another incentive for using the CA Standards as a
starting point - they have already created the file of illustrative
examples.
2. It would have been helpful if the bulleted items in the NYC Scope
and Sequence were *numbered*. It would be even more useful if the
items for each grade were ranked in order of importance, or if the
goals posed for each grade and area were somehow organized to
distinguish critical topics from those of lower priority. Teachers
would find that particularly helpful.
3. As one progresses through the grades in the NYC Scope and Sequence
there is an unfortunate trend for the writing to become increasingly
vague and unfocused. By the time I got to grade 8 the writing is so
terse and careless I felt as if the writing team had gotten tired of
their task and were just ``phoning it in'' while reading from the
table of contents of grade 8 CMP. The Scope and Sequence for the
grades 4 - 8 is really quite lame.
4. Many items in the NYC Scope and Sequence fail to distinguish
between the *skills/conceptual goals* a student should achieve by
year's end, and curriculum-specific *processes* by which those goals
are to be achieved. The Scope and Sequence document should be about
*goals*, and should not favor processes specific to a particular set
of curricular materials - e.g. TERC, CMP, ARISE, and IMP. The present
version of the Scope and Sequence is clearly biased toward those
programs.
An inordinate number of items begin with the word ``Explore ...'' .
While a certain amount of time spent in ``exploration'' is useful, the
process of exploration is hardly the whole story. Many items in the
NYC Scope and Sequence would be more forceful if they emphasized
*mastery* of skills or content, as in ``Know ...'' or ``Understand
...''.
5. In its present form the New York City Scope and Sequence promotes
indiscriminate use of calculators as early as grade 1. This is
characteristic of programs such as TERC, and is truly misguided. Use
of calculators when children are first trying to understand numbers
and arithmentic processes seriously undermines the development of
algebraic intuition, which is crucial in higher level courses. USE OF
CALCULATORS SHOULD BE AVOIDED IN GRADES K-3, and a revised Scope and
Sequence should reflect this.
6. For many items the suggested illustrations ``(e.g. ... ''
mentioned in Comment 1 tend to be extremely simplistic, and appear to
be written to conform with NCTM-based curricular materials. Exemplars
accompanying a goal are important, and should be much more diverse.
7. The same item (or nearly the same) is often repeated from one year
to the next. Even when the intent is to reinforce certain topics from
grade to grade, the description in year 2 should represent some
advance beyond the description in year 1.
After reading all this I'm reminded that sometimes ``Less is more.''
Many goals seem relatively unimportant, or are repeated verbatim from
grade to grade; the main goals are not clearly identified (cf. Item 2
above) and get buried under this mountain of micromanagerial details.
There must be a better way.
(A note about page format. In the sequel, the headings and text in
bold constitute the complete Scope and Sequence document [
The text in normal font are the present author's comments.)
GRADES K-3. ARITHMETIC AND NUMBER CONCEPTS
Grade K
* 1. Identify written numerals to ten.
OK
* 2. Use ordinal number names from first to tenth.
OK
* 3. Explore numbers through 20.
OK
* 4. Use a number line to count forward and backward.
OK
* 5. Practice the skill of counting on from a particular number
(e.g., starting from the number three and counting on by ones -
3,4,5,6,7).
OK
* 6. Understand quantities represented by the numerals one to ten
and that the last number counted in a set tells how many things are in
a set.
OK
* 7. Using concrete materials, explore putting two sets of objects
together to produce a new set whose sum is less than ten.
This is constructivist boilerplate; instead of a concrete learning
goal it gives a game to be played (and not a very instructive game at
that). Try the CA Standards version:
* 9. Develop the concept of first, middle and last within a set of
three objects.
OK
* 10. Explore benchmark fractions as they relate to daily life
(e.g., dividing an apple in 1/2, dividing a cake into four equal
parts).
OK
* 11. Explore bills and coins.
OK
Grade 1
* 1. Match words and symbols from zero to twenty.
OK, but why limit to numbers 1-20 unless you want to restrict students
to counting on their fingers and toes? The CA Standards (rightly)
suggest ``numbers 1 - 100''. In view of item #3 below, perhaps what
the writers meant is: students should become thoroughly familiar with
integers 0 - 20, while exploring the larger range of numbers 1 - 100.
The writers were not clear about this.
* 2. Explore ordinal numbers from first to thirty-first.
OK, but why the arbitrary cutoff at 31st?
* 3. Explore numbers to 100.
OK
* 4. Count forward and backward up to 50 by ones and twos using
concrete materials, number lines and number charts.
Although extensive use of concrete objects is important in the early
part of G.1, the goal described here seems needlessly restrictive. By
the end of G.1 shouldn't students begin to be able to do these counts
*without the use of artificial aids*? I suggest that, as with the CA
Standards, the ultimate goal by end of G.1 should be:
``Count forward and backward up to 100 by 1's, 2's, 5's and 10's
*without the use of artificial aids*. (Concrete materials, number
lines, number charts, etc should be used early on in the path toward
fluency).
* 5. Learn about the meaning of each digit in a two-digit number
(place value).
OK
* 6. Explore the concept of even and odd numbers using sets of
concrete objects.
Grade 2
Very weak; has already been done in G.1. A more appropriate G.2 level
goal would be:
Count, read, and write whole numbers 1 - 1000, and identify the place
value of each digit.
* 2. Use ordinal numbers from first to thirty-first and
beyond.
OK
* 3. Use concrete materials such as base-ten blocks to represent
numbers between ten and nine hundred ninety nine.
Weak, and specifically biased toward the use of manipulatives promoted
in TERC. In the Scope and Sequence it is not appropriate to mandate
use of a particular proprietary product such as the ``base 10
blocks''. I suggest a less sectarian goal:
Use words, pictures, and expanded forms (e.g. 45 = 4 tens + 5 ones) to
represent numbers 1 - 1000. Work should become less dependent on
manipulatives toward the end.
OK, except that ``Understand ...'' is more to the point than ``Explore
...''
* 5. Explore the role of zero in two- and three-digit numbers.
OK, but definitely replace ``Explore'' by ``Understand'' here.
* 6. Count forward up to 100 by twos, threes, fours, fives, and
tens and backward by twos, fives, and tens, using concrete materials,
number lines and number charts.
OK if you drop the NCTM-style implication that it should *only* be
done using concrete aids. Students should be able to do this
*unassisted* - i.e. orally, using no concrete aids at all - by the end
of G.2. That would distinguish this G.2 item from G.1 #4.
* 7. Explore the relationship between addition and
subtraction.
THIS TOPIC BELONGS IN G.1 because addition and subtraction are
introduced there. Especially, one should not delay *committing to
memory* number facts about one-digit addition!
* 8. Know single-digit addition and subtraction facts.
THIS TOPIC BELONGS IN G.1 because addition and subtraction are
introduced there. Especially, one should not delay *committing to
memory number facts about one-digit addition!
* 9. Learn about the associative [e.g.,(3+4)+6=13 and 3+(4+6)=13]
and commutative (e.g., 5+3=8 and 3+5=8) properties of addition.
Grade 1
OK, but this is a verbatim repeat of K, #3. It would help to specify
some examples to indicate what is appropriate for G.1.
* 3. Explore more than one object belonging to one set (e.g., five
fingers to one hand, two eyes to one face).
This item is really dumb, as well as vague. Rewrite to have some
point.
NOTE REGARDING G.1: Something important is missing. Throughout G.2
reference is made to (+), (-) operations, but nothing at all is said
about these symbols in G.1! Teaching elementary notions of addition
and subtraction, real-world interpretations, and the symbols used to
indicate these operations, should be a key objective in G.1.
SERIOUS TOPIC OMISSIONS:
A. Understand the meaning of addition, subtraction, and the symbols
(+), (-).
Really idiotic, completely slanted toward TERC, and avoids mentioning
the real skill G.3 students should acquire. Delete this item
entirely, and replace it with the following important goal, which has
not been explicitly mentioned.
* 6. Predict the outcome of an experiment and compare the result
to the prediction.
Vague; the writers give no idea *what kind of experiments* they have
in mind at this grade level, or what their purpose should be.
Furthermore, it seems premature to launch into this at level G.3. I
think #7 below is quite enough at the present stage.
* 7. Understand and use fractional notation to show the
probability of the outcome of an experiment (e.g., one out of three
chances of making a specific selection is the same as the fraction
1/3).
OK
* 8. Explain why a game is fair or unfair.
OK
* 9. Develop orderly ways to determine the number of possible
arrangements and combinations.
* 4. Talk about mathematics and problem solving in everyday life
(e.g., attendance, time, weather).
Misleading. As written, this seems an open invitation to avoid math
content entirely, and spend a lot of time just talking about math. To
give it some content, I would rewrite along lines of CA Standards:
Formulate arithmetic problems in terms of easily understood models
drawn from everyday life (e.g. attendance, time, weather), explaining
what features of the model appear as numbers and operations.
The use of calculators should be avoided throughout grades K-3, where
students are just beginning to understand addition and subtraction on
their own! A more appropriate, and clearer, version of #5 might be:
Use physical measuring devices such as rulers, scales, clocks,
measuring cups, and diagrams to illustrate numerical results of
calculations.
This revised item might well be repeated in G.2 and G.3, with some
indication of the increasingly sophisticated calculations that should
be done.
Grade 2
* 1. Use a variety of strategies to solve problems (e.g., using
estimation, objects or drawings).
Incomplete owing to its narrowly focused list of suggested methods.
In G.2 students should begin to see how word problems can be
represented as mathematical word equations (``open sentences''), as
asserted in G.2 #4 (Function and Algebra Concepts). Therefore I
believe the present item should be expanded to include the additional
sentence:
Solve simple word problems by describing them as open sentences, which
are solved using elementary arithmetic.
* 3. Discuss, justify, organize, and write about solutions to
problems using content specific language to describe, explain, and
compare.
Idiotic ed-speak. What does the second half of this sentence mean??
Furthermore, by G.2 the emphasis should be on *actually solving* some
problems, and explaining how those solutions were arrived at. This
item should be completely rewritten.
Once again: the use of calculators should be avoided throughout grades
K-3. Early use of calculators will interfere with understanding of
algebraic processes later on. It would be better to replace this item
with an appropriately modified version of my rewrite for G.1 #5
Grade 3
* 2. Use a variety of strategies to solve and represent
problems/solution (e.g., logical thinking, estimation, number sense,
pictures, diagrams, and charts).
Would be OK if it did not completely overlook the use of *mathematical
equations* in favor of purely descriptive modes of inquiry. Item
should be rewritten to read:
Use ... diagrams, charts, and mathematical equations).
* 3. Work individually and collaboratively to discuss, justify,
organize, and write about solutions to problems using content specific
mathematical language.
Focused entirely on process and fails to state any objective; very
strongly slanted toward the TERC philosophy. The tone is changed
considerably, and the correct emphasis is provided, if you replace
... and write about solutions to .... mathematical language.
with
... and SOLVE problems using appropriate mathematical tools.
Nowhere in the present description is it suggested that children
should acquire the ability to actually *solve* problems, as opposed to
writing or talking about them. This is the difference between
``math'' and ``math appreciation'', similar to the difference between
``art'' and ``art appreciation''.
* 4. Recognize the use of mathematics in other subject areas such
as science, social studies, and music.
* 12. Identify use of fractions and decimals in daily life (e.g.,
.75 = $ .75 = 3/4 of a dollar).
Delete this verbatim repeat of a G.3 item, or indicate what new
emphasis is intended for G.4.
* 13. Add and subtract decimals with two places (hundredths).
OK
* 14. Learn about percents as part of one hundred (e.g.,
twenty-five out of one hundred is the same as 25%).
Another really lame exemplar. I suggest:
... 35 out of 100 is the same as 35%.
* 15. Compare relationships between fractions, decimals, and
percents as they relate to daily life (e.g., Ten students were asked
to name their favorite sport. 1/2 chose soccer = 0.5 chose soccer =
50% chose soccer).
Yet another really lame exemplar. Try:
... (e.g. 20 students were asked to name their favorite sport; 3/5
chose soccer = 60%, or 12 students).
SERIOUS TOPIC OMISSIONS:
A. Use concept of *negative number*. Be able to interpret negative
numbers in terms of the number line and in practical settings (e.g.
in counting, in ``owing'', in reading temperature, etc).
NOTE: The first mention I can find of negative numbers in this Scope
and Sequence occurs in G.6 #9, and then again in G.7 #3. That seems
late in the game to start talking about negative numbers. However,
some of my colleagues disagree, and would place this important topic
in G.5. The important point is that the writers of the present Scope
and Sequence do not indicate clearly where this crucial topic should
be taken up. G.6 #9 presumes that it has already been covered.
B. Explain *different interpretations of fractions*, including the
use of charts. Explain equivalence of fractions (e.g. 3/7 = 9/21).
A. Solve problems involving *ratios and proportions* (e.g. find N if
4/7 = N/21 using cross-multiplication as a method, as an illustration
of the principle that the same thing is being done to both sides of an
identity).
NOTE: Ratios and proportions are not the same thing as percentages,
although they are related. Furthermore, many types of word problems
get cast into mathematical form using the concepts of ratio and
proportion. The G.6 sequence makes no mention of this extremely
important topic.
Grade 7
* 1. Read and write numbers through trillions.
This item is pretty useless at level G.7, and just repeats similar
items from G.5 and G.6. Delete it.
As presently worded this item is overly oriented toward process
(``Explore ...'' as opposed to ``Understand...''), and does not state
clear goals regarding skills or concepts to be mastered. For
instance, having discussed powers and (small) exponents in G.6 #3 one
should now discuss the general inverse operation of taking (small
order) roots, *not restricting the discussion to square roots only*.
The square root should be portrayed as an example of an *inverse
process*, and should receive most of the stress at this grade level,
but the concept of n th root should also be mentioned at this stage.
It would suffice, for G.7, to provide a few simple illustrations
(e.g. cube root of 8 = 2)
* 6. Understand terminating and repeating decimals.
Seems weaker than necessary. Try:
Differentiate between rational and irrational numbers, and understand
terminating/repeating decimals.
* 7. Find the percent of a number.
Incredibly lame; what is this doing in G.7? Maybe what they had in
mind is the following item lifted from the CA Standards:
Convert fractions to decimals and percents, and use these
representations in estimations, computations, and applications.
With this rewrite item #9 below becomes redundant and should be
deleted.
* 8. Apply the associative, commutative, and distributive
properties; and the inverse and identity element.
NOTE: The treatment of exponents and exponent notation is not well
delineated in this Scope and Sequence. This is not good, as this very
important topic underlies much later work. The goals regarding this
topic should be spelled out clearly and consistently!
B. Understand the meaning of *absolute value* and its number line
interpretation.
Grade 8
NOTE: The following section is so badly written that it is hopeless to
criticize it item by item. It is repetitous and vague, maybe
deliberately evasive, and any serious rewrite will take time and
personnel.
* 1. Use scientific notation to express and compare very large and
very small numbers.
What is this doing here?? This is a G.6 level topic. Failure to cover
it earlier than G.8 will preclude discussion of MANY interesting real
world applications. It should be covered thoroughly by the end of
G.7, and initially introduced no later than G.6.
ADDITIONAL USEFUL TOPICS:
NOTE: This capability is necessary to effectively apply Pythagoras'
Theorem, a key G.8 geometry topic, in real world situations.
B. Understand the concept of *logarithmic scales* and their use in
real world applications (e.g. the sizes of things in the universe).
NOTE: This is a natural companion to the discussion of scientific
notation.
GRADES 4-8. GEOMETRY AND MEASUREMENT CONCEPTS
NOTE: The whole Geometry section for grades 4-8 is incoherent: the
subject matter is not organized to reveal which facts are related to
each other. For example, area of a triangle can only be gotten
*after* one has the formula for the area of a parallelogram, which is
obtained after one derives the area formula for a rectangle.
Furthermore, there is no clear emphasis on the axiomatic properties of
``area'' and ``volume'', for instance: if you cut a figure into
pieces, the area of the figure is the sum of areas of the pieces; if
figure A is contained in figure B then the area of A is less than or
equal to that of B; etc.
If one uses these basic laws (backed up by intuitive explorations with
real objects earlier on), many things follow directly. NCTM-based
curricula, however, eschew clear definitions and logical deduction
(``proofs'') based on simple collections of fundamental laws. The
fact that this all-important topic receives no mention suggests the
failings of this whole section.
Grade 4
Quite vague, and nearly a repeat of an item that first appeared in
G.3. Rewrite for clarity and emphasize what's new. It might help to
split into several items:
Identify lines that are parallel or perpendicular
Identify congruent figures and discuss the meaning of congruence
* 2. Explore the properties of circles, including diameter, radius
and circumference.
OK, but change ``Explore ...'', which is vague and focused on process,
to read ``Identify ...''
* 3. Explore the use of formulas to find the area and volume.
VAGUE. Replace with more specific G.4 level objectives:
Develop formulas for the area and perimeter of squares and rectangles,
and the volumes of rectangular solids.
NOTE: The skills listed here are required in item #5 below.
* 4. Select units of measure (pounds, inches, minutes, and
degrees) for estimating and determining quantities such as weight,
area, time, and temperature.
Too vague to be meaningful, and it seems to repeat a G.3 item. Delete
or rewrite to indicate G.4 level content.
* 5. Estimate, measure, and represent length, width, perimeter,
and area of objects in the real world.
OK
* 6. Read and draw simple maps using coordinates.
OK
Grade 5
* 1. Represent and create models of two- and three-dimensional
shapes including cubes and prisms.
OK
* 2. Use concrete and artistic activities to explore the concepts
of similarity, symmetry, and congruence in plane geometric
figures.
Emphasizes process rather than content. Rewrite to say:
Explore the concepts of similarity, symmetry, and congruence of plane
geometric figures.
* 3. Develop formulas for the area and perimeter of rectangles and
squares.
Delete. This should be done earlier, in G.4, and I have moved it into
G.4 item #3.
* 4. Measure area and perimeter of triangles, regular and
irregular polygons by using graph paper and square tiles.
Weak and pointless. In G.4 students already should have done
measurements to arrive at area formulas for rectangles. It is then
easy to derive formulas for triangles, parallelograms, and polygons by
dissection and rearrangement. (a much more effective approach than
playing around with square tiles). This item should be replaced by:
Derive and apply area formulas for triangles, parallelograms, and
polygons using the area formula for rectangles plus dissection and
rearrangement.
* 5. Explore the relationships among diameter, radius, and
circumference of circles.
OK
* 6. Investigate three-dimensional shapes to begin to develop a
method for finding the volume of rectangular prisms.
Hopelessly vague and convoluted. Would be better as several items,
each with somewhat different objectives:
A. Use units and *dimensional analysis* to check resonableness of
answers.
Grade 8
GENERAL COMMENTS on G.8 Geometry. Basically the text of this section
is impossible. It sounds vaguely like what used to be part of a 10th
grade course, but it got garbled in transcription. There's no use
trying to fix it item-by-tiem, especially with all the repetitions. I
can't take the authors of this section seriously.
``Understand ... area'' of what?? In what way is this G.8 item
supposed to differ from G.6 #5 or G.7 #5? If it is to remain, this
item needs serious clarification of its intent for level G.8.
* 5. Understand and apply the geometry of right triangles
(including the Pythagorean Theorem and trigonometric ratios).
OK
* 6. Use appropriate units of measure to the correct degree of
accuracy.
OK
* 7. Estimate, make, and use measurements in real-world
situations.
Really vague. What kind of measurements do you have in mind for G.8
level? Measure Angles? Areas? Temperatures? And which connections
should be emphasized at this grade level relating the measuring
process to the mathematical concepts of geometry?
* 8. Determine the image of a shape under a transformation in the
coordinate plane.
Replace ``under a transformation'' with ``under various
transformations
The concept of ``variable'' is tricky and sophisticated. In G.4 I
would stick with specific skills, as in #4 immediately below where the
objective is to get students used to letting letters stand for
numbers. That should suffice for G.4.
* 4. Use letters, boxes, or other symbols to stand for any number
or object.
Weak and unfocused. It should have the use of symbols in
*mathematical equations* as its main goal, once the notion of
``variable'' has been introduced as in item #3. Rewrite as:
Use various symbols (e.g. letters, boxes, etc) to stand for any number
*in simple algebraic expressions and mathematical equations*.
* 5. Find missing number in an open sentence (e.g., 7 x ____ =
56).
OK, but the mathematical point (distinct from that in #4) becomes
clearer if this item is rewritten to say:
Solve simple mathematical equations (e.g. find the missing number in
an open sentence such as 7 x __ = 56).
Furthermore, it is not clear how the present item differs from G.3 #5
(Functions and Algebra Concepts). Perhaps the goal at level G.4
should be more advanced than the one stated here.
SERIOUS TOPIC OMISSIONS:
A. Understand use of simple equations (e.g. 3x + 5 = y) as a
prescription for determining one number when the other is given.
B. *Use and interpret simple formulas* in both symbolic and verbal
form (e.g. Area = length x width or A = lw ), to answer questions
about relationships between quantities.
SUGGESTED ADDITIONAL TOPICS:
C. Know *use of parentheses* in algebraic expressions.
Grade 5
NOTE: The whole set of goals in this area is pretty weak for G.5, much
weaker than the G.5 goals set forth in the CA Standards. The reason
is a strong aversion to the use of symbolic processes (so prevalent in
TERC and CMP), evident in the way many of these items are crafted.
This will not do! G.5 is the place where students should start to
become accustomed to abstraction and symbolic processes (read:
elementary algebra).
Grade 7
NOTE: The goals set forth for G.7 are really pretty weak. A number of
items from G.8 should be taken up in G.7, and some important topics
are not mentioned at all. I have suggested changes that would
strengthen the G.7 goals.
* 1. Find the missing term in a sequence and write the rule.
Very weak for G.7, and repeats G.6 item #1. This should be deleted
from G.7 goals.
* 2. Find the missing term in a proportion where terms can be
fractions, decimals, or percents.
OK
* 3. Describe functions and generalize them by the use of rules
and algebraic expressions.
Pretty lame for such an important topic. Try:
Identify word problems in which one must find the missing term in a
proportionality relation. Know how to solve such relations when the
known terms can be fractions, decimals, or percents.
* 4. Use algebra to translate verbal phrases into mathematical
form.
Would be OK if you add at end some mention of the use of equations:
Use algebra ... form as equations or inequalities (e.g. perimeter of a
rectangle is P = 2A + 2B if the side lengths are A,B).
However, item #4 here is pretty much redundant if one rewrites #6
below as suggested, since linear equations and inequalities are the
only ones dealt with at this grade level. So, item #4 could be
deleted.
* 5. Evaluate algebraic expressions.
Awfully vague and uninformative. Try:
Simplify and evaluate algebraic expressions using the laws of algebra.
* 6. Solve an equation and check the solution set by
substitution.
Could be a lot better. Try:
Solve linear equations and inequalities in one variable. Interpret
solutions in the context from which they arose, and verify that the
result is reasonable.
* 7. Explore the concept of rates (distance, time, and unit
pricing).
Weak, and process-oriented. State some explicit goals, as in the CA
Standards:
A. If you begin the discussion of functions in G.7, as in item #3
above, it is educational malpractice not to discuss *graphical
representations of functions at the same time*. Defering it to G.8,
item #2, is absurd. Shift G.8, #2, to here!
B. Represent quantitative relationships graphically.
C. Know how to graph linear functions. Explore the meaning of
``slope of a line''; know how it is determined from the equation of
the line.
Grade 8
NOTE: Items for this G.8 area are so vague or terse one gets the
feeling that the writers got tired of their task and were just
``phoning it in'' while reading from the table of contents of the G.8
materials for CMP. The G.8 goals deserve much more serious attention
than is evident here. Many important G.8 topics seem to be missing.
* 1. Identify, describe, represent, extend, and create
patterns.
Ridiculously vague statement! This could mean anything. Besides,
numerical patterns (sequences) have already been mentioned twice, in
G.7, #1, and G.6, #1. Unless there is something specific and new that
should be taken up in G.8 - for instance, initial discussion of
mathematical induction - this item should be deleted.
Mashes together two distinct goals. What G.8 students are expected to
know about quadratic equations must be specified carefully, and the
statement about linear equations leaves too much to the reader. I
suggest replacing this with several items, as in the CA Standards.
Know how to set up the equation of a straight line from verbal
description of a relationship between two variables.
Solve multistep problems, including word problems, involving linear
equations and linear inequalities in one variable.
Know how to simplify expressions, such as 3(2x - 5) + 4(x - 2) = 12,
before solving linear equations and inequalities in one variable.
Understand the concept of ``slope of a line'', and how to find the
equation of a line given its slope and a point on the line. Know how
slopes are related for parallel and perpendicular lines.
Solve quadratic equations by factoring or completing the square.
* 5. Apply the order of operations including the use of
parentheses.
Vague, and misses the point. Try:
Correctly apply the rules of algebra in handling operations and
parentheses to simplify algebraic expressions.
* 6. Model and solve problems involving rate, average speed,
distance and time, or direct variation.
OK, but ``direct variation'' is 19th century jargon; the word
``proportionality'' would be more appropriate. Even so, this item is
fairly vague. I would recommend the following, adapted from the CA
Standards, which is somewhat more specific and demanding:
A. Understand the use of *laws of exponents* when finding
reciprocals, taking roots, and *raising to fractional power*.
Understand the meaning of fractional powers of positive numbers.
B. Solve a system of two linear equations in two variables
*algebraically*, and interpret the solution in terms of the graphs of
the lines.
C. Know the quadratic formula, its proof by completing the square,
and its use to solve quadratic equations with real roots.
D. Understand how to graph quadratic functions, and the significance
of the x-axis intercepts.
E. Apply quadratic equations to physical problems, such as the motion
of an object under the influence of gravity.
GRADES 4-8. STATISTICS AND PROBABILITY CONCEPTS
NOTE: The whole ``Statistics'' section from grades K-8 is extremely
inflated, and could well be put into two of the grades, maybe 4 and 8.
Its underlying premise that students are in a position to make
meaningful predictions or assess sampling results is mistaken, except
in so primitive a sense as to be meaningless. The best they can do at
this level is understand that the sample resembles the total, and the
future resembles the past. Not much mathematics there.
Students should learn about graphical presentations of data, how to
construct them on their own, and how to interpret such presentations.
Another approach appropriate to this level would be to run them
through the little book ``How to Lie with Statistics'' (which is by
now antiquated in its 1920's language), or it really effective modern
successor (Statistics: Concepts and Controversies, by David S. Moore,
W.H. Freeman, 1991). The math required to understand all the common
fallacies is simple; the practical importance of understanding this
topic is great; nothing at all is said about it in this Scope and
Sequence.
Items #2,3 in this G.5 area are redundant, almost verbatim repeats of
G.4, #2,3. Delete this, or state new concepts or skills to be
mastered at level G.5.
* 3. Read and interpret double bar graphs and circle graphs.
See comments to #2. Delete, or seriously revise.
* 4. Use circle graphs to explore the concept of percent.
Silly and useless, as stated here. Percentages have already been
covered and are not a probability topic. In this G.5 area it WOULD be
relevant to discuss the notions of ``frequency'' and ``relative
frequency'' of events in a data set, and examine how these empirical
notions might be related to mathematical probabilites. Unfortunately,
the authors of the Scope and Sequence do not say that.
* 8. Identify events that are impossible (that have a chance or
probability of happening equal to zero), events that are certain (that
have a chance or probability of happening equal to one), and events
that occur sometimes (expressed as a proper fraction).
OK
* 9. Examine random and unbiased samples such as market
surveys.
OK
SERIOUS TOPIC OMISSIONS:
A.See item #4 above for a really serious omission.
B. Use fractions and percents to *compare data sets of different
sizes*.
* 6. Conduct a variety of simulation techniques to estimate
probability of events.
OK
* 7. Develop and explore combinations and permutations.
OK
* 8. Identify sample spaces by listing all elements.
OK
Grade 8
NOTE:
In this G.8 area many topics are stated in a vague and perfunctory
way. If I were a teacher I'd have little idea how to procede based on
what is said below. I have some suggestions, but this area needs
serious thought and a total rewrite.
* 1. Collect, organize, and display data with tables, charts, and
graphs that are appropriate for the data.
OK
* 2. Construct scatter plots and box and whisker plots.
Weak. Stronger and more to the point if rewritten to say:
Know various ways to display data sets, including histograms (bar
charts), scatter plots, and box-and-whisker plots. Construct plots
based on real-world data sets, and know how to use them to compare two
data sets.
* 3. Consider the effects of missing or incorrect information.
Mushy wording. ``Consider ... '' is aimless; ``Understand ...'' has a
point. Furthermore, it would help to add a related topic in this
area, so the item reads:
Understand ... information. Examine the effects of outliers on the
mean and median.
* 4. Analyze data with respect to frequency and distribution.
Vague and convoluted statement. I think the following goal is more
suitable for students by the end of G.8:
Understand the meaning of a histogram as a plot describing the
frequency distribution of some property of a population (e.g. as a
description of weight distribution, income distribution, etc.).
* 5. Formulate hypotheses to answer a question and use data to
test hypotheses.
This is so vague I'm not at all sure what is intended. One reasonable
and explicit interpretation, which fits well with the proposed rewrite
of #4, is:
Know how to plot histograms from real data sets. Know how to
interpret histograms, and make predictions based on the information
they provide.
* 6. Represent and determine probability.
What is this supposed to mean?? How is it related to goals specified
in this area for lower grade levels? This item is impossibly vague.
Total rethinking and rewrite needed.
* 7. Use estimation to check the reasonableness of results.
Quite vague; moreover, it hardly sounds like a topic in probability
and statistics. Maybe in ``Number Sense'' (where it has already
appeared several times)?
* 8. Estimate the probability of an event.
Not at all clear what the distinction is between this item and #9
below. Both are extremely vague: in what context are students
supposed to do this? Maybe the writer had something like this in mind:
Use histograms describing a population to estimate the probability of
events associated with that population.
* 9. Make predictions based on experimental and theoretical
probabilities.
See comments to item #8, which this closely resembles.
* 10. Understand combinations and permutations.
Incomplete. At G.8 level this really should include the connection
with calculation of probabilities:
GRADES 4-8. MATHEMATICAL PROCESS
Grade 4
* 2. Use a variety of strategies to solve and represent
problems/solutions (e.g., logical thinking, estimation, number sense,
pictures, diagrams, and charts).
OK, except for a crucial omission in the list of suggested strategies.
No mention is made of the use of *mathematical equations as a solution
strategy*; this relentless de-emphasis of algebraic and symbolic
methods is typical of most NCTM-based curricular materials. It is
completely wrong-headed, as these methods are the heart of real
mathematics! This same omission occurs repeatedly throughout all grade
levels in the area of Mathematical Process.
* 3. Work individually and collaboratively to discuss, justify,
organize, and write about solutions to problems using content specific
mathematical language.
This item is process-oriented ed-speak mush, typical of TERC at this
grade level. Here's a more goal-oriented item from the G.4 CA
Standards. In fact those standards address the issues in item #3 as
two separate statements:
Use a variety of methods, such as words, numbers, symbols, charts,
graphs, models, to explain mathematical reasoning.
Grade 7
* 1. Create, analyze, and solve word problems in all of the
concept areas.
OK
* 2. Identify pertinent, extraneous, and missing information.
OK
* 3. Use a variety of strategies to solve and represent
problems/solutions (e.g., logical thinking, estimation, number sense,
pictures, diagrams, and charts).
Delete. This is same as G.5, #3, and G.6, #4; what is it doing in
G.7?? Furthermore, the use of algebra and mathematical equations as
strategies has once again been omitted.
* 4. Work individually and collaboratively to discuss, justify,
organize, and write about solutions to problems using content specific
mathematical language.
Delete. This item is entirely process-oriented, and tailored
specifically to the ideology of NCTM-based math curricula.
* 5. Apply basic math skills to real-world situations.
Extremely vague. It would be more meaningful if split into two
separate parts:
Apply basic math skills to real-world situations, expressing solutions
clearly and logically using appropriate mathematical terms and
notation. Support solutions with evidence in both verbal and symbolic
work.
Decide whether solutions are reasonable in terms of the original
situation.
* 6. Talk about the uses of mathematics and its importance to
their present and future lives.
Vague, and a nearly verbatim repeat of G.6 #6. By G.7 this item
should indicate some new and more advanced objectives. First of all,
write ``Use appropriate ...'' in place of the convoluted ``Explore the
use of appropriate ... ''. Second, this item speaks of using tools -
to do what?? Either add some meaningful G.7 level exemplars or delete
this item.
SUGGESTED ADDITIONAL TOPICS.
See Topic A suggested for G.6. If that is not made part of the G.6
goals, it certainly should become one of the G.7 goals.
Grade 8
* 1. Create, analyze, and solve word problems in all of the
concept areas.
OK, but emphasis slightly askew. Better if rewriten:
Apply basic math skills to create, ... concept areas.
* 2. Identify pertinent, extraneous, and missing information.
OK, but a verbatim repeat of G.7 #2. Nothing new at level G.8?
* 3. Use a variety of strategies to solve and represent
problems/solutions (e.g., logical thinking, estimation, number sense,
pictures, diagrams, and charts).
Once again, the use of algebraic methods and mathematical methods as
solution strategies go unmentioned. Add:
This is a verbatim repeat of G.7 #7. Delete it or rewrite to indicate
that the G.8 goals represent some advance beyond those of G.7. A more
relevant G.8 goal might be the following, which appears in the CA
Standards:
| 677.169 | 1 |
precalculusActually most of the stuff you learn in pre-calc doesn't rear its head to Calc II or Calc III.
| 677.169 | 1 |
The two-line display scientific calculator combines statistics and advanced scientific functions and is a durable and affordable calculator for the classroom. The two-line display helps students explore math and science concepts in the classroom.
Ideal for:
General Math
Algebra 1 & 2
Geometry
Trigonometry
Statistics
Science
Display Two-Line
Shows entries on the top line and results on the bottom line.
Scrolling
Entry line (top) shows up to 11 characters and can scroll left and right up to 88. Result line (bottom) shows up to a 10-digit answer and 2-digit exponent.
Key features for math and science Previous entry
Lets you review previous entries and look for patterns.
Most Recent Reviews
great calculator
9/2/2011
I recieved this product just in time for my algebra 2 test. They will not let you use graphing calculators, and because this one has a two line display you can check your work much easier. I would recomend this product to anyone not needing to do graphs specifically.
Unless you need to graph on your calculator buy th
10/1/2010
This calculator is good for everything accept graphing. Most people think of Scientific calculators and go, oh no I need the one that costs a $100 or more. No all you need is this one. This is the one that was recommended by the college I'm attending. Great name, great price, free, fast shipping! Can't go wrong!
Easiest way to shop!!
9/11/2010
Getting the calculator my son needed for school had me running in all directions with no success.
I checked the internet and had an order placed in minutes, with a great price and quick delivery right to my door.
Thanks, Buy.Com
| 677.169 | 1 |
Analytical Integrated Mathematics (AIM) is a turnkey solution for teaching Engineering Mathematics (§130.367) in Texas. Students use robotics to solve a series of real world design challenges by applying mathematics and engineering concepts. Program staff have four years of experience implementing AIM in over 25 Texas schools.
A Video Introduction to AIM
What's included with AIM?
The AIM program includes everything you need to hit the ground running. We deliver a turnkey solution so you can focus on instruction in the classroom.
AIM Curriculum
The online content includes lessons, reviews, quizzes and tests that guide students toward mastery of course concepts. Information is presented using multiple instructional strategies, helping students with varying knowledge, abilities and interests become active learners. More
The Online Learning System
Intelitek's LearnMate Learning Management System is at the center of the student's interactive learning experience. Students log on to access their lessons, take quizzes and tests, while teachers utilize embedded assessment features to get prompt feedback on students' mastery of the course content. More
AIM Short Videos
Educational Robots
We use educational robots to bring the mathematics learning experience to life by utilizing design challenges based on real world scenarios. Vex robots are kitted specifically for use in AIM high school classrooms. More
Teachers attend an immersive one-week training seminar taught by veteran AIM instructors. Learn the ins and outs of the program from those who know it best! More
Installation & Support
All equipment and software is quickly and professionally installed so that your teachers and classrooms are ready to start the first day of class. Ongoing support is provided for hardware, software and program needs.
| 677.169 | 1 |
Functions
1. Understanding concept of functions, including piecewise ones
2. Ability to graph functions, using appropriate calculus techniques
3.Understanding periodicity, evenness and oddness and using it to solve computational and graphical problems
4. Ability to graph f(ax+b), given the graph of f(x)
5. Ability to evaluate and graph piecewise functions
Differentiation
1. Understanding and application of derivative as a rate of change; understanding its graphical interpretation
2. Ability to differentiate polynomials in standard form and all powers of x, including higher derivatives
3. Ability to use the product, quotient and chain rules
4. Ability to use differentiation to solve optimisation problems
Integration
1. Ability to evaluate an integral by anti-differentiation
2. Understanding an integral as a sum
3. Ability to integrate polynomials in standard form and all powers of x
4. Ability to use simple rearrangements (trigonometric and partial fractions) and simple substitution
5. Ability to construct integrals using the summation definition, with applications
Trigonometric functions
1. Ability to evaluate all six ratios from given information
2. Ability to use addition formulae and multiple angle-formulae, including their reversals
3. Ability to calculate amplitude, period and phase for sinusoidal functions
4. Ability to differentiate and integrate sin, cos, tan
5. Ability to integrate squares and products of sin and cos
Logarithms and Exponentials
1. Understanding the definition of a log as the inverse of exponentiation and ability to solve simple problems using this
2. Ability to manipulate exponential functions
3. Ability to use the log rules
4. Ability to differentiate ln x
5. Ability to integrate 1/(ax+b) and f'/f; ability to differentiate and integrate ekx
6. Ability to use log-linear and log-log graphs, including understanding of exponential processes
| 677.169 | 1 |
Mathematics plays an important part in every person's life, so why isn't everyone good at it? The Routledge International Handbook of Dyscalculia and Mathematical Learning Difficulties brings together commissioned pieces by a range of hand-picked influential, international authors from a variety of...Now in a second edition, the award-winning The Trouble with Maths offers important insights into the often confusing world of numeracy. By looking at learning difficulties in maths from several perspectives, including the language of mathematics, thinking styles and the demands of individual topics...
Published August 16
| 677.169 | 1 |
CalcComplex
By Marcus Staloff
Description
Complex Arithmetic is a simple app for performing the five operations of addition, subtraction, multiplication, division and conversion of complex numbers. It is useful for the student and practitioner in the many technical disciplines which use complex numbers. These numeric operations which require a number of steps and trig functions are instead done with the touch of a button. Direct button operations are provided for the operations with the exception of subtraction.
Two complex numbers in rectangular form are entered into the number 1 and number 2 fields labeled real and imag. Pressing one of the four function buttons then gives the input numbers in polar form. Concurrently the corresponding answer is displayed in both rectangular and polar form. The subtraction operation is done by entering the negative of the number which is the subtrahend. The square root operation is the square root of the product of the two numbers entered.
The conversion button accesses a screen which provides for the conversion of a complex number between the rectangular and polar forms.
| 677.169 | 1 |
Additional product details
Foundations of Geometry, Second Edition is written to help enrich the education of all mathematics majors and facilitate a smooth transition into more advanced mathematics courses. The text also implements the latest national standards and recommendations regarding geometry for the preparation of high school mathematics teachers—and encourages students to make connections between their college courses and classes they will later teach. This text's coverage begins with Euclid's Elements, lays out a system of axioms for geometry, and then moves on to neutral geometry, Euclidian and hyperbolic geometries from an axiomatic point of view, and then non-Euclidean geometry. Good proof-writing skills are emphasized, along with a historical development of geometry. The Second Edition streamlines and reorganizes material in order to reach coverage of neutral geometry as early as possible, adds more exercises throughout, and facilitates use of the open-source software Geogebra.
This text is ideal for an undergraduate course in axiomatic geometry for future high school geometry teachers, or for any student who has not yet encountered upper-level math, such as real analysis or abstract algebra. It assumes calculus and linear algebra as prerequisites.
| 677.169 | 1 |
Learning Math: Patterns, Functions, and Algebra is the first of five video- and Web-based mathematics courses for elementary and middle school teachers. These courses, organized around the content standards of the National Council of Teachers of Mathematics (NCTM), will help you better understand the mathematics concepts underlying the content that you teach.
Patterns, Functions, and Algebra explores the "big ideas" in algebraic thinking, such as finding, describing, and using patterns; using functions to make predictions; understanding linearity and proportional reasoning; understanding non-linear functions; and understanding and exploring algebraic structure. The concluding case studies show you how to apply what you have learned in your own classroom. The course consists of 10 two-and-a-half hour sessions with a half-hour of video programming each, problem-solving activities available in print and on the Web, and class discussions.
Video 1. Algebraic Thinking
Begin to explore what it means to think algebraically and learn to use algebraic thinking skills to make sense of different situations. This session covers describing situations through pictures, charts, graphs, and words; interpreting and drawing conclusions from graphs; and creating graphs to match written descriptions of real-life situations. Go to this unit.
Video 2. Patterns in Context
Explore the processes of finding, describing, explaining, and predicting using patterns. Topics covered include how to determine if patterns in tables are uniquely described and how to distinguish between closed and recursive descriptions. This session also introduces the idea that there are many different conceptions of what algebra is. Go to this unit.
Video 3. Functions and Algorithms
Investigate algorithms and functions. Topics covered include the importance of doing and undoing in mathematics, determining when a process can or cannot be undone, using function machines to picture and undo algorithms, and the unique outputs produced by functions. Go to this unit.
Video 4. Proportional Reasoning
Look at direct variation and proportional reasoning. This investigation will help you to differentiate between relative and absolute meanings of "more" and to compare ratios without using common denominator algorithms. Topics include differentiating between additive and multiplicative processes and their effects on scale and proportionality, and interpreting graphs that represent proportional relationships or direct variation. Go to this unit.
Video 5. Linear Functions and Slope
Explore linear relationships by looking at lines and slopes. Using computer spreadsheets, examine dynamic dependence and linear relationships and learn to recognize linear relationships expressed in tables, equations, and graphs. Also, explore the role of slope and dependent and independent variables in graphs of linear relationships, and the relationship of rates to slopes and equations. Go to this unit.
Video 6. Solving Equations
Look at different strategies for solving equations. Topics include the different meanings attributed to the equal sign and the strengths and limitations of different models for solving equations. Explore the connection between equality and balance, and practice solving equations by balancing, working backwards, and inverting operations. Go to this unit.
Video 7. Non-Linear Functions
Continue exploring functions and relationships with two types of non-linear functions: exponential and quadratic functions. This session reveals that exponential functions are expressed in constant ratios between successive outputs and that quadratic functions have constant second differences. Work with graphs of exponential and quadratic functions and explore exponential and quadratic functions in real-life situations. Go to this unit.
Video 8. More Non-Linear Functions
Investigate more non-linear functions, focusing on cyclic and reciprocal functions. Become familiar with inverse proportions and cyclic functions, develop an understanding of cyclic functions as repeating outputs, work with graphs, and explore contexts where inverse proportions and cyclic functions arise. Explore situations in which more than one function may fit a particular set of data. Go to this unit.
Video 9. Algebraic Structure
Take a closer look at "algebraic structure" by examining the properties and processes of functions. Explore important concepts in the study of algebraic structure, discover new algebraic structures, and solve equations in these new structures. Go to this unit.
Video 10. Classroom Case Studies, Grades K-2 K-2 grade band. Go to this unit.
Video 11. Classroom Case Studies, Grades 3-5 3-5 grade band. Go to this unit.
Video 12. Classroom Case Studies, Grades 6-8 6-8 grade band. Go to this unit.
| 677.169 | 1 |
Table of Contents
MATHEMATICS BEYOND MYTHS
Undergraduate mathematics is the linchpin of mathematics education. . . . No reform of mathematics education is possible unless it begins with revitalization of undergraduate mathematics in both curriculum and teaching style.
—Everybody Counts, 1989
Moving Beyond Myths: Revitalizing Undergraduate Mathematics
MATHEMATICS BEYOND MYTHS
Undergraduate mathematics is the linchpin of mathematics education. . . . No reform of mathematics education is possible unless it begins with revitalization of undergraduate mathematics in both curriculum and teaching style.
—Everybody Counts, 1989OCR for page 43
Moving Beyond Myths: Revitalizing Undergraduate Mathematics
The evidence from exemplary programs is clear: In mathematics, the American dream of equal educational opportunity for all need not be a myth. The dream can be achieved. We know how to do it. We know where it is being done. And we know why it must be done. The nation cannot afford to ignore this opportunity. The resources needed are not negligible, but the cost of ignoring the opportunity is incalculable. The national revitalization of mathematics is within our reach, if only we are prepared to make a serious intellectual and financial commitment to our children's and our nation's future.
OCR for page 43
Moving Beyond Myths: Revitalizing Undergraduate Mathematics
SUMMARY
Elevate the importance of undergraduate teaching.
Engage mathematics faculty in issues of teaching and learning.
Teach in a way that engages students.
Achieve parity for women and minorities and the disabled.
Establish effective career paths for college teaching.
Broaden attitudes and value systems of the mathematics profession.
Increase the number of students who succeed in college mathematics.
Ensure sufficient numbers of school and college teachers.
Elevate mathematics education to the same level as mathematical research.
Link colleges and universities to school mathematics.
Provide adequate resources for undergraduate mathematics.
OCR for page 43
Moving Beyond Myths: Revitalizing Undergraduate Mathematics
| 677.169 | 1 |
Geometry, Revised Edition describes geometry in antiquity. Beginning with a brief description of some of the geometry that preceded the geometry of the Greeks, it takes up the story of geometry during the European Renaissance as well as the significant mathematical progress in other areas of the world. It also discusses the analytic geometry of Ren... more...
Covering the many aspects of geometry, this volume of the History of Mathematics series presents a compelling look at mathematical theories alongside historical occurrences. The engaging and informative text, complemented by photographs and illustrations, introduces students to the fascinating story of how geometry has developed. Biographical information... more...
| 677.169 | 1 |
Description: Topics covered: dimensional analysis and scaling; perturbation methods; the calculus of variations; the theory of partial differential equations; Sturm-Liouville theory, the theory for the corresponding generalized Fourier series and some further methods for solving PDE; transform theory with applications; Hamiltonian theory and isoperimetric problems; the theory of integral equations; the theory of dynamical systems, chaos, stability and bifurcations; discrete mathematics.
| 677.169 | 1 |
Coursework Requirements
Students majoring in mathematics must complete MATH 115 and one of 116/120 (or the equivalent) and at least eight units of 200-level and 300-level courses. These eight units must include 205, 206, 302, 305, and two additional 300-level courses. (Thus a student who places out of 115/116 and starts in 205 requires only eight courses.) At most two of 206, 210 and 215 may be counted towards the major. These courses must be completed for the mathematics major:
Math 115: Calculus I and Math 116: Calculus II, or the equivalent
Math 205: Multivariable Calculus
Math 206: Linear Algebra
Math 302: Elements of Analysis I
Math 305: Abstract Algebra
At least two elective 300-level courses not counting any of 350, 360, 370.
A student may count Math 215/Phys 215 towards her mathematics major. However, she may count at most two of the course 206, 210, and 215 toward the major. Credit for Math 216/Phys 216 satisfies the requirement that a math major take 205, but cannot be counted as one of the 200- or 300-level units required for the major.
Major Presentation Requirement
Majors are also required to present one classroom talk in either their junior or senior year. This requirement can be satisfied with a presentation in the student seminar, but it can also be fulfilled by giving a talk in one of the courses whose catalog description says"Majors can fulfill the major presentation requirement in this course." In addition, a limited number of students may be able to fulfill the presentation requirement in other courses, with permission of the instructor
| 677.169 | 1 |
Math Matters
MATH MATTERS: Why do I need to know this?
Math Matters: Why Do I Need To Know This? originally appeared during the Spring 2006 semester as a half-hour program on WKU's
campus cable system. Dr. Bruce Kessler, at that time Assistant Dean (now the Head
of the Mathematics Department in WKU's Ogden College of Science & Engineering) and
a WKU math professor, wrote and hosted the program, which focused on how the mathematics
being taught in WKU's General Math, College Algebra, and Math for Elementary Teachers
courses was and still is applied in the real world. The shows have since been segmented
into roughly 8 minute vignettes that can be downloaded from this website and from iTunes U.
This program was produced by WKYU-PBS, using their studio and students in the production,
and was supported by a Provost's Initiative for Excellence grant, the Ogden College
of Science and Engineering, the Department of Mathematics, and Housing and Residence
Life. For questions regarding the show, contact Bruce Kessler at [email protected] or at (270) 745-3651.
About the Host
A 1989 graduate of Western Kentucky University, Dr. Bruce Kessler completed his Masters
at Vanderbilt in 1991, and returned to WKU as a math instructor at the Glasgow Regional
Campus. Kessler returned to Vanderbilt in 1994 to work on his Ph.D. under the direction
of Doug Hardin, and finished in 1997. Today, he is a professor and the Head of the
WKU Department of Mathematics.
| 677.169 | 1 |
Enter your mobile number or email address below and we'll send you a link to download the free Kindle Reading App. Then you can start reading Kindle books on your smartphone, tablet, or computer - no Kindle device required.
Comment: This amazing book is in near mint condition and is ready to be read by you! This book has no hand writing or highlighting in it. Ships direct from Amazon and will arrive at your door in a flash! Thanks for reading, enjoy your day history traces the development of mathematical ideas and the careers of the men responsible for them. Volume 1 looks at the discipline's origins in Babylon and Egypt, the creation of geometry and trigonometry by the Greeks, and the role of mathematics in the medieval and early modern periods. Volume 2 focuses on calculus, the rise of analysis in the nineteenth century, and the number theories of Dedekind and Dirichlet. The concluding volume covers the revival of projective geometry, the emergence of abstract algebra, the beginnings of topology, and the influence of Godel on recent mathematical study.
{"currencyCode":"USD","itemData":[{"priceBreaksMAP":null,"buyingPrice":25.31,"ASIN":"0195061365","isPreorder":0},{"priceBreaksMAP":null,"buyingPrice":23.74,"ASIN":"0195061373","isPreorder":0},{"priceBreaksMAP":null,"buyingPrice":23.7,"ASIN":"0195061357","isPreorder":0}],"shippingId":"0195061365::1bOW%2FcKeYrQ%2BB7b31x5Fca3GvsD9z5NWbXMOgw%2Ff9%2F769iYc4JBA9fFEcqxmRpxFEZgspZZ77WiNF1NCzmpybOfdh5JmnNc33TgS9iH%2FYG0%3D,0195061373::Yhlq5LI%2Ftl0cp%2FNCXwPnrI6SPMx0TZuRBMsOp3jyACyDFla5GFA7zK5y6mZOUbBNMvzydKuQSU5l0uAgHCAv64D7hBO6cyetwzGKN6b0%2Bao%3D,0195061357::7Zx1JMMaoU%2FDyEEd1q5iHt3LcPjtSGodW9FITB6uYJJMl7NSNYoBFaTIsyZMtb%2B7foMHEzIlR3uNyEu%2Fc8r5Gg%2BhOTFKalol6zdH%2FyVJy Kindle edition of this work is incomplete . Key examples are missing notably the non- commutativity of quaternian multiplication. I would be unlikely to by Kindle editions of any textbook or technical work.
Morris Kline is one of the best math writers of all time. His books are intended for folks with only the most basic math, and he then uses physical and intuitive analogies to explain what's going on "beneath" the equations. MANY of his books are now available in inexpensive Dover editions, just put Morris Kline in the search bar at the top of any Amazon page!
This book starts with calculus after leaving off there in the first volume. Like his intuitive calculus book, it reads like an adventure novel! This is a MUST buy if you're interested in the history of math. His Mathematics and the Physical World book (Mathematics and the Physical World (Dover books explaining science)) is one of the all time best introductions to advanced math ever written, from High School to beginning undergrad. ALL his books make any other text MUCH more clear, due to his wonderful "real world" examples and palpable enthusiasm.
If you see reviews trashing this or any other math book due to Kindle, don't fault the book! In general ALL e-readers (not just Kindle) still have trouble with LaTex, especially older "scanned" texts! If the book has complex exponential equations, just assume they'll be problematic as e-books, especially if the edition is pre-2005. Of course I agree with reviewers who point this out, but it is sad that it effects the overall rating of the BOOK itself, which deserves many more stars!!!
Library Picks reviews only for the benefit of Amazon shoppers and has nothing to do with Amazon, the authors, manufacturers or publishers of the items we review.Read more ›
| 677.169 | 1 |
Find a Woodhaven TrigonometryThe laws of exponents are extended to the cases of zero, negative and fractional exponents. The idea of a function and its inverse is introduced. Extensive use is made of exponential and logarithmic functions, including graphing and solving equations
| 677.169 | 1 |
The best way to master math is to practice, practice, practice?and 1,001 Math Problems offers ?mathophobes? and others who just need a little math tutoring the practice they need to succeed. Whether students need help calculating a tip or facing a standardized math test that could determine their future, the 1,001 math questions in this useful... more...
This book, in honor of Hari M. Srivastava, discusses essential developments in mathematical research in a variety of problems. It contains thirty-five articles, written by eminent scientists from the international mathematical community, including both research and survey works. Subjects covered include analytic number theory, combinatorics, special... more...
Intended for researchers, numerical analysts, and graduate students in various fields of applied mathematics, physics, mechanics, and engineering sciences, Applications of Lie Groups to Difference Equations is the first book to provide a systematic construction of invariant difference schemes for nonlinear differential equations. A guide to methods... more...
Arnold's Problems' contains mathematical problems which have been brought up by Vladimir Arnold in his famous seminar at Moscow State University over several decades. In addition, there are problems published in his numerous papers and books. Many of these problems are still at the frontier of research today. more...
Imagine that you've finally found a parking space after a long and harrowing search, but are now encountering some difficulty in trying to enter this space. Wouldn't it be great if you knew a formula that allowed you to enter the space without difficulty? Are you annoyed because your soda can doesn't remain upright during a picnic? Would... more...
Boundary Value Problems is a translation from the Russian of lectures given at Kazan and Rostov Universities, dealing with the theory of boundary value problems for analytic functions. The emphasis of the book is on the solution of singular integral equations with Cauchy and Hilbert kernels. Although the book treats the theory of boundary value problems,... more...
The CliffsTestPrep series offers full-length practice exams that simulate the real tests; proven test-taking strategies to increase your chances at doing well; and thorough review exercises to help fill in any knowledge gaps. CliffsTestPrep California High School Exit Exam: Mathematics can help you pass this critical competency exam necessary for... more...
| 677.169 | 1 |
self-contained introduction to ordinary differential equations and dynamical systems suitable for beginning graduate students. It contains almost three hundred exercises. Additionally, the use of mathematical software systems is incorporated throughout, showing how they can help in the study of differential equations
| 677.169 | 1 |
books.google.com - The book conveys a distinctive approach, stimulating readers to develop a broader, deeper understanding of mathematics through active participation—including discovery, discussion, and writing about fundamental ideas. It provides a series of interesting, challenging problems, then encourages readers... Geometry
Experiencing Geometry: In Euclidean, Spherical, and Hyperbolic Spaces
The book conveys a distinctive approach, stimulating readers to develop a broader, deeper understanding of mathematics through active participation—including discovery, discussion, and writing about fundamental ideas. It provides a series of interesting, challenging problems, then encourages readers to gather their reasonings and understandings of each problem.
About the author (2001)
Dr. David W. Henderson is a senior pastor in West Lafayette, Indiana, and a regular contributor to Discipleship Journal. He has served widely as a conference and chapel speaker in the United States and overseas.
| 677.169 | 1 |
Table of Contents
1: The Elements of Analysis in the Eighteenth Century.- 2: The Lagrangian Calculus and Fourier Series.- 3: New Trends in Rigor.- 4: Complex Functions and Integration.- 5: The Convergence of Fourier Series.- 6: Riemann's Theory of Functions.- 7: The Arithmetization of Analysis.- Appendix: On the History of "Dirichlet's Principle
| 677.169 | 1 |
Algebra 1
Quick Links
Webinars
Unit Downloads
Equivalence
Rewriting expressions into equivalent forms is a central part of doing mathematics. In these lessons students investigate the concepts of variable, equivalent expressions and the distributive property. As they evaluate expressions, they can also observe patterns such as rate of change and order of operations in the results.
| 677.169 | 1 |
Algebra Images
algebra - One On One -
Educational/Mathematics ... algebra One on One is an educational game for those wanting a fun way to learn and practice algebra. This program covers 21 functions which includes maximums, minimums, absolute values, averages, x/y, ax + b, axy + b, ax + by + c, squares, cubes, and so on. It has a practice and a game area. It has a great help system that makes it easy for the beginner to do and understand algebra. It also has a "Einstein" level that even algebra experts will find fun and challenging. You can choose from a ten ...
2.
algebra Advanced -
Mobile/Education ... algebra practice is *CAKE* when you play this FUN GAME designed by TOP TEACHERS who bring you exciting math skills practice that will explore all the mathematics you need to master and prepare for college, high school or university pre-calculus and calculus courses. Challenge and upgrade your algebra skills while having mega fun playing a game that takes you through intermediate algebra practice including factoring, polynomials, rational expressions, all the way to sequences and series and even ...
3.
algebra I Review -
Mobile/Education ... YayMath.org and Study By App, LLC have partnered to create a comprehensive algebra 1 app. The Yay Math movement and this project are built on an understanding of ...
4.
EMMentor algebra -
Educational/Mathematics ... Interactive multilingual mathematics software for training problem-solving skills offers 70439 algebraic problems, a variety of appropriate techniques to solve problems and a unique system of performance analysis with methodical feedback. Included are linear, quadratic, biquadratic, reciprocal, cubic and complex fractional algebraic expressions, identities, equations and inequalities. The software allows students at all skill levels to practice at their own pace, learn from both errors and ...
5.
Pre algebra!!! -
Mobile/Education ... Powered by EasyWorksheet, used in over 15,000 schools worldwide, Pre algebra!!! is one of the largest collection of Pre algebra Worksheets on iPad. This App covers Pre algebra and Honors Pre algebra, helping you prepare almost all that you need to practice before you venture into algebra. The worksheets in the App allows students to practice each pre-algebra lesson one by one until the student achieves mastery in each topic. To achieve mastery, student is required to score a 100 points. If an ...
6.
MathAid algebra II -
Educational/Mathematics ... Highly interactive tutorials and self-test system for individual e-learning, home schooling, college and high school computer learning centers, and distance learning. The product emphasizes on building problem-solving skills. Tutorials include the reviews of basic concepts, interactive examples, and standard problems with randomly generated parameters. The self-test system allows selecting topics and length for a test, saving test results, and getting the test review. Topics covered: rectangular ...
algebra Touch -
Mobile/Education ... Have you forgotten most of your algebr* algebra Touch will refresh your skills using touch-based techniques built from the ground up for your iPhone/iPad/iPod Touch.
Say you have x + 3 = 5. You can drag the 3 to the other side of the equation.
Enjoy the wonderful conceptual leaps of algebra, without getting bogged down by the tedium of traditional methods.
"With this app, the author has successfully made algebra beautiful. I really wish this app existed when I was studying algebra 25 ...
9.
algebra Helper 1 -
Mobile/Education ... *** On sale for back to school ***
Learn solving two equations system by example, by plugging the equation Coefficients. You will be able to build your examples, which is the best way to learn algebra.
========================================
This application provide step by step solution to the two equation system in additions to providing the value of x and y. Enter the equation Coefficients and see the full solution which can be sent as an email message. ...
10.
algebra and Sequences -
Mobile/Education ... Fun, colourful and interactive GCSE maths revision apps providing thousands of examples, clues and easy-to-remember revision notes. Passing your maths exam starts here!
Why did we create these apps?
Because we know how it feels to read through big, boring, heavy maths textbooks. And we didn't like it either!
What's so good about them?
Unlike textbooks and some other maths apps out there, most of the examples in our apps are randomly generated and can be shuffled an unlimited number of ...
Algebra Images
From Short Description
1.
DragonBox algebra 12+ -
Mobile/Education ... DragonBox algebra 12+: An inexpensive algebra tutor, you can keep in your pocket.
DragonBox algebra 12+ is a must-have tool for students who wish to earn better grades and gain confidence in algebra and mathematics. It covers the same topics as the award winning game DragonBox algebra 5+, but moves on to cover more advanced topics in mathematics and algebra. It also includes practice tasks that allow students to practice specific types of algebraic operations.
DragonBox algebra 12+ covers ...
2.
TutorVideo: algebra -
Mobile/Education ... TutorVideo: algebra is the premier iPhone and iPod Touch video collection of algebra tutorials.
Want an on-the-go algebra study guide?
algebra is the branch of mathematics concerning the study of the rules of operations and the things which can be constructed from them, including terms, polynomials, equations and algebraic structures. Together with geometry, analysis, topology, combinatorics, and number theory, algebra is one of the main branches of pure mathematics. With TutorVideo, ...
3.
algebra Explained c. 1 Order of Operations LITE -
Mobile/Education ... Try algebra Explained and see why people are saying, "Love this app", "Bravo", "Fun and interesting", "Terrific" and "[Jamie] is a great teacher... 'THE BEST'!".
The algebra Explained series teaches algebra using ENTERTAINING VIDEO LESSONS, study cards and practice problems to make learners successful and interested. Expertly designed by a former math teacher with a Masters in Curriculum and Development, algebra Explained connects abstract concepts to relevant real-world topics using fun ...
4.
MathProf -
Educational/Education ... MathProf can display mathematical correlations in a very clear and simple way. The program covers the areas Analysis, Geometry, algebra, Stochastics, Vector algebra. It helps Junior High School students with problems in Geometry and algebra. High School and College students, seeking to expand their knowledge into further reaching mathematical concepts find this program very useful as well. ...
5.
Video algebra Tutor -
Mobile/Education ... Video algebra Tutor, by MathVids
Having difficulty with some parts of algebra or just need a refresher course,? Then this video series is for you. With 17 MAJOR CATEGORIES and 35 VIDEO LECTURES totaling over 7 HOURS of lessons, Video algebra Tutor covers all the essential areas you need to know about algebra.
* These 35 VIDEOS are in the app, not streamed from the Internet. You can use this product anywhere.
MathVids, in collaboration with iPREPpress has created mobile VIDEO tutors for ...
Khan Academy: Linear algebra 7 -
Mobile/Education ... Ximarc Studios Inc is proud to bring you Khan Academy Linear algebra 7 (videos 121-1389.
Khan Academy: Linear algebra 2 -
Mobile/Education ... Ximarc Studios Inc is proud to bring you Khan Academy Linear algebra 2 (videos 21-40). Khan Academy Linear algebra allows students to learn Linear algebra through various videos which are downloaded directly on your iPhone or iPod touch and in the future to your iPad. Students can watch the video anywhere, anytime, all the time and NEVER be concerned about having access to the internet while you are going through a Khan Academy lesson.
Ximarc Studios will continue to bring you great video ...
10.
Khan Academy: Linear algebra 4 -
Mobile/Education ... Ximarc Studios Inc is proud to bring you Khan Academy Linear algebra 4 (videos 61-80Gaigen -
Utilities/Other Utilities ... Gaigen is a Geometric algebra Implementation Generator. You specify the geometric algebra you want to use in your (C++) project, and then Gaigen generates C++ code that implemenents this algebra. Requires FLTK library for the user interface. ...
3.
Innoexe Visual algebra -
Educational/Mathematics ... Innoexe Visual algebra works in three modes. Work with others over the internet, network, or alone. Chat with others and solve problems at the same time. IVA is perfect for tutors teaching students over the internet or a network connection. Innoexe Visual algebra will solve your problems step by step and explain as it goes. Innoexe Visual algebra will change the way you look at algebra problems. All registered users will receive free up grades. ...
4.
algebra Quiz Game - Learn to simplify, factor, & solve math equations for your test -
Mobile/Education ... "This game saved me hundreds of dollars in tutoring costs!" - A very happy parent
algebra Quiz Game is a unique and exciting game that helps you get ready for your next algebra test or quiz. Answer algebra questions correctly as fast as you can to get points and reach higher levels. Topics include:
- Factoring equations
- Multiplying monomials, binomials, and polynomials (FOIL)
- Distributing
- Simplifying expressions/combining like terms
- Solving equations for x (equality and ...
5.
Wolfram algebra Course Assistant -
Mobile/Education ... Taking algebr* Then you need the Wolfram algebra Course Assistant. This definitive app for algebrafrom the world leader in math softwarewill help you quickly solve your homework problems, ace your tests, and learn algebra concepts so you're prepared for your next courses. Forget canned examples! The Wolfram algebra Course Assistant solves your specific algebra problems on the fly, often showing you how to work through the problem step by step.
This app covers the following topics applicable ...
6.
Learning algebra -
Mobile/Education ... Excellent Application on Learning algebra From Beginner to Master.
algebra is one of the broad parts of mathematics, together with number theory, geometry and analysis.
Video Training and Practice Exam ...
7.
algebra Vision -
Educational/Mathematics ... algebra Vision is a unique educational software tool to help students develop algebraic problem solving strategies. It provides an environment to play and see algebra in a more tangible light. You can literally move expressions around! Draw lines connecting distributive elements! ...
Symbol Manipulations in algebra -
Mobile/Education ... The apps consist of lessons created as multiple-choice questions, and when you answer a question, you receive an immediate response. Unlike multiple-choice tests, the app lessons use multiple-choice questions as a teaching tool. The 'algebra through Function' series contains five (5) apps and has 41 embedded lessons.
Since the primary mode of operation of the brain is pattern generalizing, almost all of the lessons are structured as pattern-building activities. Users are encouraged
| 677.169 | 1 |
Elementary Linear Algebra
9780030973543
ISBN:
0030973546
Edition: 5 Pub Date: 1994 Publisher: Thomson Learning
Summary: Intended for the first course in linear algebra, this widely used text balances mathematical techniques and mathematical proofs. It presents theory in small steps and provides more examples and exercises involving computations than competing texts.
Grossman is the author of Elementary Linear Algebra, published 1994 under ISBN 9780030973543 and 0030973546. Eleven Elementary Linear Algebra textbooks are availa...ble for sale on ValoreBooks.com, three used from the cheapest price of $2.91, or buy new starting at $233
| 677.169 | 1 |
... create an immersive, media-rich 3D environment that engages students in unique science learning. By leveraging existing Flash assets and the rapid development environment of Director, Science Alberta Foundation was able ...
... facilitates the teaching of Java to first year students. Special emphasis has been placed on visualisation and interaction techniques to create a highly interactive environment that encourages experimentation and ...
... purpose graphing program designed for scientists, engineers, and students. It features multiple scaling types, including linear, logarithmic, and probability scales, as well as several special purpose graphs: tripartite grids (shock ...
... of the concept of phase transition for the students of very broad ranges of initial background. Very moderate scientific background is needed to understand a material in this section. Preliminary ...
UltimaCalc Professional is a highly functional graphing math toolbox with a compact and intuitive design offering the ultimate balance of power and convenience. Its algebra module lets you perform manipulations and ...
A handy, fast, reliable, precise tool if you need to perform calculations with comlex functions. Complex Number Calculator Precision 36 is programmed in C#. All calculations are done in proprietary data ...
A handy, fast, reliable, precise tool if you need to perform mathematical calculations. The calculator was designed with purpose to fit Netbooks and Notebooks with small display. Of course, the calculator ...
... make your Microsoft Word documents readable to your students with vision and learning disabilities? The MathDaisy application ... in the DAISY Digital Talking Book format. Your students can use MathPlayer-enabled DAISY player software to read ...
... XML and CSV formats · Very flexible students (pupils) structure, organized into sets: years, groups and ... and non-overlapping subgroups. You can even define individual students (as separate sets) · Each constraint has ...
... the way that communities of users (teachers, moviegoers, students...) will use these self-publishing tools to share their audiovisual "readings", and to envision new editing and viewing interfaces for interactive comment ...
| 677.169 | 1 |
New and Published Books
Research Methodology and Mathematical Models
The development of advanced materials has become extremely important in the last decade, being widely used in academic and industrial research. This book examines the potential of advanced materials as well as nanotechnology to improve fiber science from fibril to fabric mode, to create better...
A Course in Real Analysis provides a rigorous treatment of the foundations of differential and integral calculus at the advanced undergraduate level. The book's material has been extensively classroom tested in the author's two-semester undergraduate course on real analysis at The George Washington...Design and Analysis of Experiments with R presents a unified treatment of experimental designs and design concepts commonly used in practice. It connects the objectives of research to the type of experimental design required, describes the process of creating the design and collecting the data,...
An Introduction
A Contemporary Approach to Teaching Differential Equations
Applied Differential Equations: An Introduction presents a contemporary treatment of ordinary differential equations (ODEs) and an introduction to partial differential equations (PDEs), including their applications in engineering and the...
A Course in Ordinary Differential Equations, Second Edition teaches students how to use analytical and numerical solution methods in typical engineering, physics, and mathematics applications. Lauded for its extensive computer code and student-friendly approach, the first edition of this popular...
A Primer
Fulfilling the need for a practical user's guide, Statistics in MATLAB: A Primer provides an accessible introduction to the latest version of MATLAB® and its extensive functionality for statistics. Assuming a basic knowledge of statistics and probability as well as a fundamental understanding of...
The Essentials of a First Linear Algebra Course and More
Linear Algebra, Geometry and Transformation provides students with a solid geometric grasp of linear transformations. It stresses the linear case of the inverse function and rank theorems and gives a careful geometric treatment of the...
The Monte Carlo method has become the de facto standard in radiation transport. Although powerful, if not understood and used appropriately, the method can give misleading results.
Monte Carlo Methods for Particle Transport teaches appropriate use of the Monte Carlo method, explaining the method's...
Approximate Analytical Methods for Solving Ordinary Differential Equations (ODEs) is the first book to present all of the available approximate methods for solving ODEs, eliminating the need to wade through multiple books and articles. It covers both well-established techniques and recently...
| 677.169 | 1 |
Most helpful customer reviews
To put it quite simply, if you are a physics student, you must own this book. What does this book do for you? Consider this... In my school, we do not have a mathematical methods course for science, so I decided to take on a math minor to take all the classes neccesary to do physics "right." This included a class on ODEs, Fourier Series & PDEs, Linear Algebra, and Complex Variables. These classes, although helpful, cover a lot of stuff that is not quite useful for understanding physics concepts, often undermining or dampening the stuff that is actually applicable. What makes this book so great is that it combines all the essential math concepts into one compact, clearly written reference. If I could do it all over again, I would easily rather take a two semester Math Methods course (like they do in many schools) using a book like Boas than take all these obtuse math courses. With this book, it makes it so handy to review previously learned concepts or actually learn poorly presented topics ( for a physicist anyway) in mathematics classes... (Things like Coordinate Transformations, Tensors, Special Functions & PDEs in spherical & cylindrical coordinates, Diagonilzation, the list goes on.....) Keep this gem handy when doing homework and studying for exams, learning the math tools from this book enables you to concentrate squarely on the physics in your other textbooks... (since mathematical background information, understandably, is often cut short...)
Yes, everyone loves the book - and so do I (see? I have given it 5 stars!). There is one little problem: this excellent book cannot replace the "real" mathematical books. When I first started using this book I was always concerned about the completness of the material. In other words, when she gave a "receipe" for solving a problem I would always think to myself "how do I know, that this solution is complete? are there not any other solutions? WHERE IS THE PROOF? etc." You are always given the receipe, and, yes, this receipes will help you solve most problems and prepare for most examinations, but will you really understand MATHEMATICS behind the problems? The solution in my opinion is to get hold of a few good and rigorous books on calculus, advanced calculus, variational methods, elements of complex analysis and basics of functional analysis. Once you have worked through them you can read M.Boas and really understand and appreciate the book. But the question is: will you need M.Boas then?
This book has a bit of everything from Linear Algebra, Calculus, Analysis, Probability and Statistics, ODE, PDE, Transforms just to name a few. If you get a chance to study everything from this book, you will probably learn more from this book than all your undergraduate math courses combined. Some concepts on this book may be difficult to understand due to the lack of in depth coverage. But I guess the main intention of this book is to focus on the applied side and cover as much material that is relevant to physics and engineering as possible and not go into much detail on the theory side. If you are a graduate student in physics or engineering and want to buy this book for reference, it will be a good start for the first year courses but won't help you much after that. Readibility of this book is excellent. You will understand most of the concepts and examples presented. Bottomline: This is a must have book for engineers and physicists.
Everyone doing a physics degree knows that it is often the case that there is no ideal book for the area that you're studying. For instance, in atomic physics, you need to use Eisberg, Bransden, and Rae (or at least, you do for my course). Here's the good news: when it comes to maths, there is only one book that you will ever need, and it's this one. No matter what stage of your degree you're at, this book will always come in handy. The maths behind quantum mechanics or thermodynamics is made trivial thanks to a few clarifying words from good ol' Mary L. When it comes to revising for those dreaded finals, Boas is the ideal aid - you just do a quick couple of questions to remind yourself how Lagrange Multipliers work and you're home and dry. I really can't say enough good things about this wonderful little book. If you only buy one book for your physics course, do yourself a favour and buy this one.
I'm a physics undergraduate. Out of all my books on math, this is far and away the most comprehensive and useful book! It has supplanted my other, thicker books and is the one thing I turn to whenever I need to refresh myself on a math method. It covers practically every useful math technique for physics, and never assumes that you're a genius (unlike other books). Each step is explained in clear, refreshing language and in a very logical order. From Laplacian transforms to Fourier series to ODEs, each subject is introduced so well that, even when I've missed a lecture, I can understand the topic just from reading it. Highly recommended and worth the price, this is one book physics undergraduates should have. The only thing else needed with it is the solutions manual.
| 677.169 | 1 |
Students can begin with calculus (115 or 116), or an introduction to statistics (101), or a course to explore applications of mathematics without calculus (102). Students with an extensive background in mathematics may begin in upper level courses such as Linear Algebra (206), Combinatorics (225), or Number Theory (223). You can learn about our courses from the Wellesley College Course Catalog.
The logical thinking and quantitative reasoning skills you gain in math classes are valuable in all fields, even if you do not end up using the particular course material that you learned.
Calculus is not a required course at Wellesley College, but it is required for many majors, including economics and most pure and applied sciences. In addition, it is frequently required for admission to medical school. Every entering student is sent a brief placement questionnaire to assess precalculus and calculus skills. We use your placement questionnaire and your SAT scores (as well as your AP score, if applicable) to determine the calculus course that we believe is most appropriate for you. Letters are sent out in July with placement information. During First-Year Orientation we are available at "Advising Day" to answer any questions that you have. Our calculus placement FAQs may also be helpful.
All mathematics courses satisfy the Mathematical Modeling requirement and the courses focusing on statistics also satisfy the Quantitative Reasoning Overlay requirement
| 677.169 | 1 |
17Mathematical Methods for Physicists
Summary
This best-selling title provides in one handy volume the essential mathematical tools and techniques used to solve problems in physics. It is a vital addition to the bookshelf of any serious student of physics or research professional in the field. The authors have put considerable effort into revamping this new edition. * Updates the leading graduate-level text in mathematical physics * Provides comprehensive coverage of the mathematics necessary for advanced study in physics and engineering * Focuses on problem-solving skills and offers a vast array of exercises * Clearly illustrates and proves mathematical relations New in the Sixth Edition: * Updated content throughout, based on users' feedback * More advanced sections, including differential forms and the elegant forms of Maxwell's equations * A new chapter on probability and statistics * More elementary sections have been deleted
| 677.169 | 1 |
FOREWORD:
1. Basically targeted to all engineering entrance exams.
2. Useful to 11th Class and 12th Class (Intermediate) Mathematics students for public examinations.
3. We wish that this Question Bank will win the hearts of the students and teaching faculty also.
4. Useful to Class-I competitive exams like Civil Services, Bank Probationary Officers and Staff Selection Commission etc.
| 677.169 | 1 |
There are four types of buses — Type A, Type B, Type C and Type D — ranging from 6,000 pounds to more than 33,000 pounds. A conventional school bus is a Type C bus, which can weigh from 19,501 to 26,000 pounds, according to School Transportation News.
Precalculus generally uses algebraic concepts taught in college-level algebra, but if there is a strong understanding of algebraic problems, precalculus may not be difficult. Both forms of mathematics courses involve a significant number of new concepts and whether a person finds one course more difficult than the other depends on the person's strengths as a mathematician.
| 677.169 | 1 |
Democracy may be a word familiar to most, but it is a concept still misunderstood and misused at a time when dictators,...
see more
Democracy may be a word familiar to most, but it is a concept still misunderstood and misused at a time when dictators, single-party regimes, and military coup leaders alike assert popular support by claiming the mantle of democracy. Yet the power of the democratic idea has prevailed through a long and turbulent history, and democratic government, despite continuing challenges, continues to evolve and flourish throughout Democracy in Brief to your Bookmark Collection or Course ePortfolio
Select this link to close drop down of your Bookmark Collection or Course ePortfolio for material Democracy in Brief
Select this link to open drop down to add material Democracying Information Systems to your Bookmark Collection or Course ePortfolio
Select this link to close drop down of your Bookmark Collection or Course ePortfolio for material Discovering Information Systems
Select this link to open drop down to add material Discovering Information Systems to your Bookmark Collection or Course ePortfolio
Elementary Algebra is a textbook that covers the traditional topics studied in a modern elementary algebra course. It is...
see more
Elementary Algebra is a textbook that covers the traditional topics studied in a modern elementary algebra course. It is intended for students who (1) have no exposure to elementary algebra, (2) have previously had an unpleasant experience with elementary algebra, or (3) need to review algebraic concepts and
A textbook that covers the traditional topics studied in a modern prealgebra course, as well as topics of estimation,...
see more
A textbook that covers the traditional topics studied in a modern prealgebra course, as well as topics of estimation, elementary analytic geometry, and introductory algebra. It is intended for students who (1) have had a previous course in prealgebra, (2) wish to meet the prerequisite of a higher level course such as elementary algebra, and (3) need to review fundamental mathematical concepts and techniques
Pick a Bookmark Collection or Course ePortfolio to put this material in or scroll to the bottom to create a new Bookmark Collection
Name the Book Mathematics to your Bookmark Collection or Course ePortfolio
Select this link to close drop down of your Bookmark Collection or Course ePortfolio for material Fundamentals of Mathematics
Select this link to open drop down to add material FundamentalsWhen I teach computer science courses, I want to cover important concepts in addition to making the material interesting and...
see more
'When I teach computer science courses, I want to cover important concepts in addition to making the material interesting and engaging to students. Unfortunately, there is a tendency for introductory programming courses to focus far too much attention on mathematical abstraction and for students to become frustrated with annoying problems related to low-level details of syntax, compilation, and the enforcement of seemingly arcane rules. Although such abstraction and formalism is important to professional software engineers and students who plan to continue their study of computer science, taking such an approach in an introductory course mostly succeeds in making computer science boring. When I teach a course, I don't want to have a room of uninspired students. I would much rather see them trying to solve interesting problems by exploring different ideas, taking unconventional approaches, breaking the rules, and learning from their mistakes. In doing so, I don't want to waste half of the semester trying to sort out obscure syntax problems, unintelligible compiler error messages, or the several hundred ways that a program might generate a general protection fault.One of the reasons why I like Python is that it provides a really nice balance between the practical and the conceptual. Since Python is interpreted, beginners can pick up the language and start doing neat things almost immediately without getting lost in the problems of compilation and linking. Furthermore, Python comes with a large library of modules that can be used to do all sorts of tasks ranging from web-programming to graphics. Having such a practical focus is a great way to engage students and it allows them to complete significant projects. However, Python can also serve as an excellent foundation for introducing important computer science concepts. Since Python fully supports procedures and classes, students can be gradually introduced to topics such as procedural abstraction, data structures, and object-oriented programming — all of which are applicable to later courses on Java or C++. Python even borrows a number of features from functional programming languages and can be used to introduce concepts that would be covered in more detail in courses on Scheme and Lisp Think Like a Computer Scientist - Learning with Python to your Bookmark Collection or Course ePortfolio
Select this link to close drop down of your Bookmark Collection or Course ePortfolio for material How to Think Like a Computer Scientist - Learning with Python
Select this link to open drop down to add material How to Think Like a Computer Scientist - Learning with Python to your Bookmark Collection or Course ePortfolio
In the last decade the demand for human rights has become a revolutionary force in the world. In this essay, scholar Jack...
see more
In the last decade the demand for human rights has become a revolutionary force in the world. In this essay, scholar Jack Donnelly traces the development of human rights from their origins as a political theory in 17th-century Europe to their present-day acceptance as an international standard Rights in Brief to your Bookmark Collection or Course ePortfolio
Select this link to close drop down of your Bookmark Collection or Course ePortfolio for material Human Rights in Brief
Select this link to open drop down to add material Human Rights
| 677.169 | 1 |
To log in and use all the features of Khan Academy, please enable JavaScript in your browser.
Pre-algebra
No way, this isn't your run of the mill arithmetic. This is Pre-algebra. You're about to play with the professionals. Think of pre-algebra as a runway. You're the airplane and algebra is your sunny vacation destination. Without the runway you're not going anywhere. Seriously, the foundation for all higher mathematics is laid with many of the concepts that we will introduce to you here: negative numbers, absolute value, factors, multiples, decimals, and fractions to name a few. So buckle up and move your seat into the upright position. We're about to take off!
This tutorial is less about statistics and more about interpreting data--whether it is presented as a table, pictograph, bar graph or line graph. Good for someone new to these ideas. For a student in high school or college looking to learn statistics, it might make sense to skip (although it might not hurt either).
The world seldom gives you two numbers and tells you which operation to perform. More likely, you'll be presented with a bunch of information and you (yes, YOU) need to make sense of them. This tutorial gives you practice doing exactly that. When watching videos, pause and attempt it before Sal. Then work on as many problems as you want in the exercise at the end of the tutorial.
Equality is usually a good thing, but the world is not a perfect place. No matter how hard we try, we can't help but compare one thing to another and realize how unequal they may be.
This tutorial gives you the tools to do these comparisons in the mathematical world (which we call inequalities). You'll become familiar with the "greater than" and "less than symbols" and learn to use them.
You've probably been learning how to do arithmetic for some time and feel pretty good about it. This tutorial will make you feel even better once by showing you a bunch of examples of where it can be applied (using multiple skills at a time). Get through the exercises here and you really are an arithmetic rock star!
Most of us are use to using the digits 0-9 to represent numbers in the base-10 (decimal)number system. In this tutorial, we'll see that is just one of many (really infinite) number systems. In particular, we will focus on the binary (base-2) and hexadecimal (base-16) systems.
| 677.169 | 1 |
Hastings On Hudson Cal illuminate what is essential, and what should be more readily grasped once the essential material is mastered. Algebra 1 begins with learning to translate verbal phrases into symbols. This leads to the topic of formulas and equations
| 677.169 | 1 |
Single Variable Calculus : Concepts and ContextStewart's clear, direct writing style in SINGLE VARIABLE CALCULUS, VOLUME 2, 5th Edition guides you through key ideas, theorems, and problem-solving steps. Every concept is supported by thoughtfully worked examples and carefully chosen exercises. Many of the detailed examples display solutions that are presented graphically, analytically, or numerically to provide further insight into mathematical concepts. Margin notes expand on and clarify the steps of the solution.
Functions And Models
Four Ways to Represent a Function
Mathematical Models: A Catalog of Essential Functions
New Functions from Old Functions
Graphing Calculators and Computers
Review
Principles of Problem Solving
Limits
The Tangent and Velocity Problems
The Limit of a Function
Calculating Limits Using the Limit Laws
The Precise Definition of a Limit
Continuity
Review
Problems Plus
Derivatives
Derivatives and Rates of Change
Writing Project: Early Methods for Finding Tangents
The Derivative as a Function
Differentiation Formulas
Applied Project: Building a Better Roller Coaster
Derivatives of Trigonometric Functions
The Chain Rule
Applied Project: Where Should a Pilot Start Descent? Implicit Differentiation
| 677.169 | 1 |
2go: An Online Supplemental Instruction Tool Array to your Bookmark Collection or Course ePortfolio
Select this link to close drop down of your Bookmark Collection or Course ePortfolio for material Algebra2go: An Online Supplemental Instruction Tool Array
Select this link to open drop down to add material Algebra2go: An Online Supplemental Instruction Tool Array Library of Virtual Manipulatives to your Bookmark Collection or Course ePortfolio
Select this link to close drop down of your Bookmark Collection or Course ePortfolio for material National Library of Virtual Manipulatives
Select this link to open drop down to add material National Library of Virtual Manipulatives Miss Lindquist: The Tutor to your Bookmark Collection or Course ePortfolio
Select this link to close drop down of your Bookmark Collection or Course ePortfolio for material Miss Lindquist: The Tutor
Select this link to open drop down to add material Miss Lindquist: The Tutor to your Bookmark Collection or Course ePortfolio
This site is a selection of mathlets designed for "geometry classes to review basic algebra skills in such a way that they...
see more
This site is a selection of mathlets designed for "geometry classes to review basic algebra skills in such a way that they can check their answers and/or get hints as to how to proceed. The idea behind these "procedure-based" dynamic worksheets is to provide students the opportunity to review and practice algebra skills with problems they create, while at the same time providing a means for students to check their answers and to get a hint if needed. The hints will walk the student through the problem in a step by step mannerAssess Algebra Review to your Bookmark Collection or Course ePortfolio
Select this link to close drop down of your Bookmark Collection or Course ePortfolio for material Self-Assess Algebra Review
Select this link to open drop down to add material Self-Assess Algebra Review
״A System of Linear Equations" graphs the equations of a 2 x 2 system of linear equations, illustrates the geometric...
see more
״A System of Linear Equations" graphs the equations of a 2 x 2 system of linear equations, illustrates the geometric interpretation of the system, identifies the type of solution, and finds the solution when applicable Equations to your Bookmark Collection or Course ePortfolio
Select this link to close drop down of your Bookmark Collection or Course ePortfolio for material System of Equations
Select this link to open drop down to add material System of Equations Prearing Fractions to your Bookmark Collection or Course ePortfolio
Select this link to close drop down of your Bookmark Collection or Course ePortfolio for material Comparing Fractions
Select this link to open drop down to add material Comparing Fractions Dad's Math Worksheets to your Bookmark Collection or Course ePortfolio
Select this link to close drop down of your Bookmark Collection or Course ePortfolio for material Dad's Math Worksheets
Select this link to open drop down to add material Dad's Math Worksheets to your Bookmark Collection or Course ePortfolio
| 677.169 | 1 |
Algebra II
This course begins with a brief review of Algebra I and extends to include number systems, polynomials, rational expressions, linear equations and inequalities, systems of equations, elementary exponential and logarithmic functions, right-triangle trigonometry, and elementary probability and statistics. A graphing calculator is required
| 677.169 | 1 |
Editorial Reviews
Geometry's hard? Geometry's boring? No way, Math Fans! Geometry doesn't have to be hard or boring. Watch as Zero attacks each Geometry subject and problem with passion and humor to keep students interest level high. Designed to make learning Geometry fun, students will enjoy the skits, fun humor along with the solid teaching style of veteran High School math teacher – Lowell Irving. The Zero the Math Hero series is an energetic and fun series that teaches students how to learn geometry concepts, recognize and apply myriad geometry terms, functions and procedures. It incorporates two important and highly effective teaching methods - problem practice and repetition both during the lesson and with the accompanying problem practice worksheets. Each problem is presented using step-by-step instruction and should be repeated frequently to help student's retention and problem solving skills. Online Teachers Guide & Resource Guide available for download directly on our website. Each guide includes definitions, subject matter revision, practice problems and student quizzes. This program provides a discussion of the basic definition of a triangle and demonstrates how to classify triangles according to their side lengths and/or angle measures. The following Triangle types are taught: Scalene, Isosceles, Equilateral, Acute, Right, Obtuse, Equiangular.
This product is manufactured on demand using DVD-R recordable media. Amazon.com's standard return policy will apply.
| 677.169 | 1 |
There's only one small topic that SCU covers in a previous class that you may not have seen. That topic is Hyperbolic Functions, which is covered in Math 12. To get yourself up to speed on this topic, you can doEither way you go, you should do some of the basic odd numbered homework problems in your book to check that you have the essential ideas down. If you need help selecting these, just ask your current calculus instructor. For additional resources on this and other Math 12 topics, please see our
on-line help page.
For Students Who Took the AB Version of the AP Calculus Exam
Although many high school AP calculus classes cover some or all of the following five topics, they are not mandated for the AB exam: hyperbolic functions, l'Hôpital's rule, improper integration, integration by parts, and partial fractions. Here are our suggestions for learning any of these five topics if you previously missed them:
For hyperbolic functions: you can get yourself up to speed on this small topic by doingFor l'Hôpital's rule and improper integration, you can likely pick up most of what you need to know from the book. L'Hôpital's rule is Section 3.8 on pages 216-222. Improper integration is Section 7.8 on pages 491-501. If you have any questions, see your instructor.
For integration by parts and partial fractions, you'll need to do a bit more work. These are two techniques for integrating functions. Integration by parts arises a lot in probability courses like MATH 122 and AMTH 108 and also in differential equations courses like MATH 22, MATH 144 and AMTH 106. Partial fractions is the crucial tool for using Laplace transforms in MATH 22 and AMTH 106.
In other words, you need to know these two techniques. You should do either one or both of the following to get up to speed on these:
If you learn well from the book, then just read through the two book sections on these techniques. Integration by parts is Section 7.2 on pages 449-456. Partial Fractions is Section 7.7 on pages 482-491. If you have any questions, see your instructor.
If you learn better from lectures, find a Math 12 that fits into your schedule. (If you need help to do this, just talk to your current calculus instructor.) Contact the Math 12 instructor teaching that section, and they will let you know when they are lecturing on these two sections. Usually, these sections are covered somewhat late in the quarter. Do not put off attending these lectures until next quarter when you need this material!
No matter what you do, you should work through some of the basic odd numbered homework problems in your book to check that you have the essential ideas down. If you need help selecting these, just ask your current calculus instructor, or, if you attended a class, do the homework that instructor assigned. For additional resources on this and other Math 12 topics, please see our
on-line help page.
| 677.169 | 1 |
Engineering Mathematics is a comprehensive pre-degree maths text for vocational courses and foundation modules at degree level. John Bird's approach, based on numerous worked examples supported by problems, is ideal for students of a wide range of abilities, and can be worked through at the student's own pace. Theory is kept to a minimum, placing a firm emphasis on problem-solving skills, and making this a thoroughly practical introduction to the core mathematics needed for engineering studies and practice.
The third edition has been reorganised to present a logical topic progression through the book rather than following the structure of a particular syllabus. The coverage has been carefully matched to recent course specifications including AVCE and the new BTEC National.
Free Tutor Support Material including full worked solutions to the assignments featured in the book is available at Material only available to lecturers who have adopted the text as an essential purchase. In order to obtain your password to access the material please e-mail [email protected] with the following details: course title, number of students, your job title and work address.
Comprehensive coverage for introductory degree courses John Bird's 'learning by example' technique is a thoroughly practical way of gaining knowledge and understanding.
Related Subjects
Meet the Author
John Bird, the author of over 100 textbooks on engineering and mathematical subjects, is the former Head of Applied Electronics in the Faculty of Technology at Highbury College, Portsmouth, U.K. More recently, he has combined freelance lecturing at Portsmouth University, with technical writing and Chief Examiner responsibilities for City and Guilds Telecommunication Principles and Mathematics, and examining for the International Baccalaureate Organisation. John Bird is currently a Senior Training Provider at the Royal Naval School of Marine Engineering in the Defence College of Marine and Air Engineering at H.M.S. Sultan, Gosport, Hampshire, U.K. The school, which serves the Royal Navy, is one of Europe's largest engineering training establishments
| 677.169 | 1 |
This lesson from Illuminations asks students to measure the diameter and circumference of various circular objects, plot the measurements on a graph, and relate the slope of the line to ?, the ratio of circumference to ...
Created by David Smith for the Connected Curriculum Project, this module develops a graphical representation for a differential equation that reveals the nature of solutions, even when formulas for those solutions are...
A mesh based partition of unity method, known as the manifold method, is used in simulating the evolution of a slope failure. The problem configuration consists of a simple slope that has pre-existing tensile cracks...
An introduction and a guide to trigonometry, with hints and answers to exercises, and Java applets as illustrations. Contents include applications of trigonometry, angle measurement, chords, sines, cosines, tangents and
| 677.169 | 1 |
A Transition to Advanced Mathematics
A Transition to Advanced Mathematics: A Survey Course promotes the goals of a "bridge'" course in mathematics, helping students transition from courses in the calculus sequence (and other problem solving courses that involve mathematical calculations) to theoretical upper-level mathematics courses (that prove theorems and grapple with mathematical abstractions). Simultaneously, the text promotes the goals of a "survey'" course, describing intriguing questions and insights fundamental to diverse areas of mathematics, including Logic, Abstract Algebra, Number Theory, Real Analysis, Statistics, Graph Theory, and Complex Analysis.
The text's main objective is to provide a deep change in the mathematical character of students—in their thought process and in their fundamental perspective on the mathematical world. We believe this text promotes three major mathematical traits in a meaningful, transformative way: to develop an ability to communicate with precise language; to use mathematically sound reasoning; and to ask probing questions about mathematics. In short, we hope that working through A Transition to Advanced Mathematics encourages students to become mathematicians in the fullest sense of the word.
The text has a number of distinctive features that enable this transformational experience. Embedded Questions within each section and Reading Questions at each section's end illustrate and explain concepts, allowing students to test their understanding independent of the exercises. Extensive, diverse exercise sets contain an average of 70 exercises at the end of each section; in total, the text has almost 3000 distinct exercises. In addition, every chapter includes a section that explores an application of the theoretical ideas being studied. Finally, embedded reflections on the history, culture, and philosophy of mathematics are interwoven throughout the text.
Bridge. Survey. Challenging. Encouraging. Reflective. Practical. Transformational. A Transition to Advanced Mathematics: A Survey Course will make a positive difference in the learning and development of your mathematics majors.
| 677.169 | 1 |
Completely free algebra one course in animated video at Watch all units and use textbook free of charge. Includes chapter reviews and tests. Explains the nature of the Pythagorean Theorem, solves it, and provides uses and examples.
| 677.169 | 1 |
Mathway is a mathematics problem solving tool where students can select their math course - Basic Math, Pre-Algebra, Algebra,...
see more
Mathway is a mathematics problem solving tool where students can select their math course - Basic Math, Pre-Algebra, Algebra, Trigonometry, PreCalculus, Calculus or Statistics and enter a problem. The computer solves the problem and shows the steps for the solution. It also has a worksheet generatorway to your Bookmark Collection or Course ePortfolio
Select this link to close drop down of your Bookmark Collection or Course ePortfolio for material Mathway
Select this link to open drop down to add material Mathway Dad's Math Worksheets to your Bookmark Collection or Course ePortfolio
Select this link to close drop down of your Bookmark Collection or Course ePortfolio for material Dad's Math Worksheets
Select this link to open drop down to add material Dad's Math Worksheoring by the Difference of Two Perfect Squares to your Bookmark Collection or Course ePortfolio
Select this link to close drop down of your Bookmark Collection or Course ePortfolio for material Factoring by the Difference of Two Perfect Squares
Select this link to open drop down to add material Factoring by the Difference of Two Perfect Squaresraction Addition Memory Game to your Bookmark Collection or Course ePortfolio
Select this link to close drop down of your Bookmark Collection or Course ePortfolio for material Fraction Addition Memory Game
Select this link to open drop down to add material Fraction Addition Memory Mole to your Bookmark Collection or Course ePortfolio
Select this link to close drop down of your Bookmark Collection or Course ePortfolio for material Graph Mole
Select this link to open drop down to add material Graph Moleast Common Denominator to your Bookmark Collection or Course ePortfolio
Select this link to close drop down of your Bookmark Collection or Course ePortfolio for material Least Common Denominator
Select this link to open drop down to add material Least Common Denomin Equation Solver to your Bookmark Collection or Course ePortfolio
Select this link to close drop down of your Bookmark Collection or Course ePortfolio for material Linear Equation Solver
Select this link to open drop down to add material Linear Equation Solver material Problem Solving in Science and Technology
Select this link to open drop down to add material Problem Solving in Science and
| 677.169 | 1 |
McGraw -Hill's Top 50 Math Skills for GED Success34
FREE
Used Very Good(1 Copy):
Very good Ships Within 24 Hours M-F-Satisfaction Guaranteed! Unread Copy in excellent condition. Has a publisher overstock mark.
About the Book
Math skills can often be the difference between passing the GED on your first try and disappointment. But it is often hard to pinpoint those areas that you still need to master.
Let McGraw-Hill's Top 50 Math Skills for GED Success give you everything you need to ace the math questions on GED test day. Written by two experts who have years of experience with the GED, this easy-to-use book features: A pretest designed to identify your weaknesses in those math areas the GED traditionally focuses on Two-page lessons that increase your knowledge in 50 essential skills so that they become your strengths Detailed guidance on using a calculator and making estimations A full answer key with helpful explanations
Don't take chances with the GED. Brush up on the knowledge you need to know now with McGraw-Hill's Top 50 Math Skills for GED Success.
[BOX]
How to use this book to quickly--and dramatically--improve your GED math skills: 1. Take the pretest to determine where you most need help. 2.Study the two-page lesson on each skill that gave you problems. 3.Take the posttest under timed, testlike conditions. 4.Do it again, focusing only on the areas still giving you trouble. 5.PASS THE GED!
| 677.169 | 1 |
toires and a better understanding of functions. The students who used the graphing calculator for only a short period of time did no better on the posttest than the students in the control group. They merely replaced their algebraic and guess-and-test procedures with graphing methods. Unlike the students who spent more time using the graphing calculator, they were not able to enrich their conceptual understanding of functions.
The widespread availability of computer and graphing-calculator technologies has dramatically affected the kinds of representational activities that have been developed and studied since the 1980s. Today's graphing programs, curve fitters, spreadsheets, and spreadsheet-like generators of tables of values and so on have been found to provide more effective environments than pencil and paper for introducing students to variables, algebraic expressions, and equations in a problem-solving context. Research has documented that the visual and numerical supports provided for symbolic expressions by digital representations of graphs and tables help students create meaning for expressions and equations in ways difficult to manage in learning environments not supported by computers or calculators. More research is needed into the ways that computers and graphing calculators are being used and can be used effectively in the early grades.
The Transformational Activities of Algebra
What the Number-Proficient Child Brings
In the previous section, we discussed some of the perspectives brought to the study of algebra by students emerging from traditional elementary school arithmetic. These perspectives included the following:
An orientation to execute operations rather than to use them to represent relationships; which leads to
Use of the equal sign to announce a result rather than signify an equality;
Use of inverse or undoing operations to solve a problem and the corresponding absence of a notion of describing a situation with the stated operations of a problem; and
A perception of letters as representing unknowns but not variables.
In this section, we discuss additional features of arithmetic thinking that must be addressed when students encounter the transformational activities of algebra
| 677.169 | 1 |
Should College Classes Ditch the Calculator?
Should College Classes Ditch the Calculator?
According to Samuel King, postdoctoral student in the University of Pittsburgh's Learning Research and Development Center, using calculators in college math classes may be doing more harm than good. In a limited study conducted with undergraduate engineering students and published in the British Journal of Educational Technology, King has determined that our use of calculators may be serving as an alternative to an actual, deep understanding of mathematical material.
"We really can't assume that calculators are helping students," says King. "The goal is to understand the core concepts during the lecture. What we found is that use of calculators isn't necessarily helping in that regard."
King, along with co-author and director of the Mathematics Education Centre at Loughborough University, Carol Robinson, conducted the study by interviewing 10 second-year undergraduate students who were enrolled in a competitive engineering program. The students were given a number of mathematical questions dealing with sine waves, which are mathematical curves that describe a smooth repetitive oscillation. To help solve the problems, the students were given the option of using a calculator instead of completing the work entirely by hand. Over half of the students questioned opted to utilize their calculators in order to solve the problems and plot the sine waves.
"Instead of being able to accurately represent or visualize a sine wave, these students adopted a trial-and-error method by entering values into a calculator to determine which of the four answers provided was correct," says King. "It was apparent that the students who adopted this approach had limited understanding of the concept, as none of them attempted to sketch the sine wave after they worked out one or two values."
After completing the work, King and Robinson interviewed the students about how they approached the material. One student who used the calculator stated that she had trouble remembering the rules for how sine waves operate, and found it generally easier to use a calculator instead. In contrast, however, a student who opted to complete the work without a calculator stated that they couldn't see why anyone would have trouble completing the question, but did admit that it would likely be easier with a calculator.
"The limited evidence we collected about the largely procedural use of calculators as a substitute for the mathematical thinking presented indicates that there might be a need to rethink how and when calculators may be used in classes—especially at the undergraduate level," says King. "Are these tools really helping to prepare students or are the students using the tools as a way to bypass information that is difficult to understand? Our evidence suggests the latter, and we encourage more research be done in this area."
Given the small sample size used in the study, it is entirely possible that King's findings are largely anecdotal in how our usage of calculators and understanding of mathematical concepts may positively or negatively correlate. However, King does stress that while all the evidence may not be in, his study does raise important questions regarding how, when and why students choose to use calculators, and in doing so, we may develop a more holistic approach to math instruction
| 677.169 | 1 |
More About
This Textbook
Overview
AGraphical Approach to Algebra and Trigonometry continues to incorporate an open design, with helpful features and careful explanations of topics.
Related Subjects
Meet the Author
John Hornsby: When John Hornsby enrolled as an undergraduate at Louisiana State University, he was uncertain whether he wanted to study mathematics, education, or journalism. His ultimate decision was to become a teacher, but after twenty-five years of teaching at the high school and university levels and fifteen years of writing mathematics textbooks, all three of his goals have been realized; his love for teaching and for mathematics is evident in his passion for working with students and fellow teachers as well. His specific professional interests are recreational mathematics, mathematics history, and incorporating graphing calculators into the curriculum.
John's personal life is busy as he devotes time to his family (wife Gwen, and sons Chris, Jack, and Josh). He has been a rabid baseball fan all of his life. John's other hobbies include numismatics (the study of coins) and record collecting. He loves the music of the 1960s and has an extensive collection of the recorded works of Frankie Valli and the Four Seasons.
Marge Lial has always been interested in math; it was her favorite subject in the first grade! Marge's intense desire to educate both her students and herself has inspired the writing of numerous best-selling textbooks. Marge, who received Bachelor's and Master's degrees from California State University at Sacramento, is now affiliated with American River College.
Marge is an avid reader and traveler. Her travel experiences often find their way into her books as applications, exercise sets, and feature sets. She is particularly interested in archeology. Trips to various digs and ruin sites have produced some fascinating problems for her textbooks involving such topics as the building of Mayan pyramids and the acoustics of ancient ball courts in the Yucatan.
Gary Rockswold has been teaching mathematics for 33 years at all levels from seventh grade to graduate school, including junior high and high school students, talented youth, vocational, undergraduate, and graduate students, and adult education classes. He is currently employed at Minnesota State University, Mankato, where he is a full professor of mathematics. He graduated with majors in mathematics and physics from St. Olaf College in Northfield, Minnesota, where he was elected to Phi Beta Kappa. He received his Ph.D. in applied mathematics from Iowa State University. He has an interdisciplinary background and has also taught physical science, astronomy, and computer science. Outside of mathematics, he enjoys spending time with his lovely
| 677.169 | 1 |
An Introduction with Applications
0470767855
9780470767856
Details about MATLAB:
MATLAB: An Introduction with Applications 4th Edition walks readers through the ins and outs of this powerful software for technical computing. The first chapter describes basic features of the program and shows how to use it in simple arithmetic operations with scalars. The next two chapters focus on the topic of arrays (the basis of MATLAB), while the remaining text covers a wide range of other applications. MATLAB: An Introduction with Applications 4th Edition is presented gradually and in great detail, generously illustrated through computer screen shots and step-by-step tutorials, and applied in problems in mathematics, science, and engineering.
Back to top
Rent MATLAB 4th edition today, or search our site for Amos textbooks. Every textbook comes with a 21-day "Any Reason" guarantee. Published by Wiley.
| 677.169 | 1 |
Algebra 1
Saxon Algebra 1 Home Study Kit Third Edition Retail Price: $84.10 CBD Price: $71.49
( In Stock ) The Home Study Kit includes all materials necessary for one student and one teacher for a full year of study. Algebra 1 Kits include a textbook, test forms and an answer booklet. Solutions manual is also available (Stock # 98011). Algebra 1 topics range from algebra based real-world problems to functions and graphics; and from algebraic proofs to statistics and probability.
Third edition.
Algebra 1 is made up of five instructional components:
Introduction of the New Increment
Examples with Complete Solutions
Practice of the Increment
Daily Problem Sets
Cumulative Tests.
Algebra 1 covers topics typically treated in a first-year algebra course, and includes such topics as:
arithmetic and evaluation of expressions involving signed numbers, exponents, and roots
Algebra 2
Algebra 2 Homeschool Kit, 3rd Edition Retail Price: $84.10 CBD Price: $71.49
( In Stock ) The Home Study Kit includes all materials necessary for one student and one teacher for a full year of study. Algebra 2 Kits include a textbook, test forms and an answer booklet. Solutions manual must be ordered separately (Stock #91116). Algebra 2 not only covers all the topics of a second year Algebra, but also includes a considerable amount of geometry, including writing proof outlines. When your child completes this course, he will have studied the equivalent of one semester of informal geometry as well.
Algebra 2 is made up of five instructional components:
Introduction of the New Increment
Examples with Complete Solutions
Practice of the Increment
Daily Problem Sets
Cumulative Tests
Algebra 2 not only treats topics that are traditionally covered in second-year algebra, but also covers a considerable amount of geometry. Time is spent developing geometric concepts and writing proof outlines. Students completing Algebra 2 will have studied the equivalent of one semester of informal geometry. Applications to other subjects such as physics and chemistry, as well as real-world problems, are covered, including gas laws, force vectors, chemical mixtures, and percent markups. Set theory, probability and statistics, and other topics are also included.
Advanced Mathematics
Saxon Advanced Math Home Study Kit Retail Price: $87.70 CBD Price: $74.49
( In Stock ) The Home Study Kit includes all materials necessary for one student and one teacher for a full year of study. Advanced Math Kit includes a textbook, test forms and an answer booklet. Advanced Mathematics interweaves topics from algebra, geometry, trigonometry, discrete mathematics, and mathematical analysis to form a fully integrated text. A rigorous treatment of Euclidean geometry is also presented. Conceptually oriented problems that prepare students for college entrance exams (such as ACT and SAT) are included in the problem sets. Solution manual is ordered separately (Stock # 57311). Grades 9-12.
Advanced Mathematics is made up of five instructional components:
Introduction of the New Increment
Examples with Complete Solutions
Practice of the Increment
Daily Problem Sets
and Cumulative Tests.
In Advanced Mathematics, topics from algebra, geometry, trigonometry, discrete mathematics, and mathematical analysis are interwoven to form a fully integrated text. A rigorous treatment of Euclidean geometry is also presented. Word problems are developed throughout the problem sets and become progressively more elaborate. With this practice, students will be able to solve challenging problems such as rate problems and work problems involving abstract quantities. The graphing calculator is used to graph functions and perform data analysis. Conceptually oriented problems that prepare students for college entrance exams (such as the ACT and SAT) are included in the problem sets.
Saxon Physics First Edition 2001 is taught at an introductory level and does not require that the teacher have a background in physics. Anyone who has taught second-year algebra, especially Saxon's Algebra 2, can teach this book successfully. Saxon Physics was written with both average and gifted students in mind.
The entire program is based on introducing a topic to a student and then allowing them to build upon that concept as they learn new ones. Topics are never dropped but are instead increased in complexity and practiced every day, providing the time required for concepts to become totally familiar.
The subject is taught at an introductory level, allowing the average high school student to grasp the concepts of Newton's laws, statics, dynamics, thermodynamics, optics, DC circuits, waves, electromagnetics, and special relativity. The topics are covered to a depth appropriate for college students majoring in non-engineering disciplines. Consequently, gifted students who use this book will have great success with the Advanced Placement Physics examination, and average students who are willing to do the homework will also be able to pass the examination. Topics from the Advanced Placement Level B Exam can be covered before the exam is given in early May.
Calculus
Home Study Kit--Calculus, Second Edition Retail Price: $94.40 CBD Price: $79.99
( In Stock ) UnderstandingCalculus is made up of four instructional components:
Introduction of the New Increment
Examples with Complete Solutions
Daily Problem Sets
and Cumulative Tests.
Calculus treats all the topics normally covered in an Advanced Placement AB-level calculus program, as well as many from a BC-level program. The text begins with a review of the mathematical concepts and skills required for calculus. In the early problem sets, students practice setting up word problems they will later encounter as calculus problems. The problem sets contain multiple-choice and conceptually-oriented problems similar to those found on the Advanced Placement examination. Whenever possible, an intuitive introduction precedes a rigorous examination of a concept. Proofs are provided for all important theorems. For example, three proofs, one intuitive and two rigorous, are given for the Fundamental Theorem of Calculus.
Numerous applications to physics, chemistry, engineering, and business are also treated in both the lessons and the problem sets. Use of this text has allowed students to take the Advanced Placement examination and score well.
Need help with a specific homeschool question? Want to know some
additional details before you make a decision? Call our knowledgeable
homeschool specialists at 1-800-788-1221! With homeschooling experience themselves, they'll be more than glad to help you with all your curriculum questions.
| 677.169 | 1 |
Synopses & Reviews
Publisher Comments:
(back cover)
PAINLESS
Algebra
Second Edition
Really. This isn't going to hurt at all . . .
If you break out in a cold sweat at the very mention of algebra, this book will show you how to relax and master the subject without pain. The problems you now consider confusing are taken slowly, step by step. Simply follow the painless steps and soon you'll wonder why you were ever confused in the first place! You start with integers, progress to simple equations with one variable, then move forward to gain a clear understanding of inequalities, systems of equations, exponents, and roots and radicals. Finally, you graduate to mastery of quadratic equations. Along the way, you'll find fun-to-solve "brain tickler" problems in every chapter. Each "brain tickler" comes with an answer, but before you knot it, you'll be figuring out all the answers for yourself.
For Middle School and High School Students
Synopsis:
"Synopsis"
by Netread,
| 677.169 | 1 |
Textbook: Bob Jones, Life Science for Christian School .... Time may be spent to facilitate music lessons, studying for .... Grade, First Semester, 1 high school credit, 3 college credits .... Algebra II is a second year algebra course for students in grades 10-12 who have successfully ...
Source:
Well, it is definitely yes! In the early day of internet marketing, giving away ebook free was a very good list−building strategy. It worked ...
Source:
The mathematics curriculum in grade 12 is dominated by differential and integral calculus ... in their regular class. However the ..... Students were asked to apply the formula to a simple website comprising of two pages A and. B as shown in ...
Source:
Interview Questions. 6. Problem 5.1 in the text. ... Atmel microcontrollers are used quite often in the ECE curriculum at OSU and throughout industry ...
Source:
Minimum grade of C in all major/cognate courses. 3 v.052011 ...... The Biology Department offers courses designed to meet the needs of a wide ...... engage in a senior research project in which they study a relevant topic under the ...
Source:
stories of how 12d Model is being used by 12d. Modellers ... user suggestions as
possible in planning the 12d Model ... software, such as accounting packages.
Source:
Accounting Line: The accounting line to be charged will be the accounting line
set up on the fund code table for the fund code shown on the SF-1449 and the.
Source:
Accounting Line: The accounting line to be charged will be the accounting line
set up on ... Figure 12E- 18 Aviation Into – Plane Reimbursable Cards. DD-1898.
Source:
of 'C' Forms at the first instance. Dealers, who have valid Registration under the CST Act 1956, can apply for the 'C' Forms through online by login to the.
Source:
MATHEMATICS Standards Addressed Through This Lesson .... students work on this project, they'll conduct research on safe driving techniques, use formulas to estimate ... 12. Using the rubric, evaluate each skit as it is presented to the class.
Source:
| 677.169 | 1 |
Problem Solving Approach to Mathematics for Elementary School Teachers, A (11th Edition)
9780321756664
ISBN:
0321756665
Edition: 11 Pub Date: 2012 Publisher: Addison Wesley
Summary: This is the leading textbook for students learning how to teach mathematics to elementary school students, focusing on problem solving. It remains current with its discussion of standards in teaching today and it teaches students the value of professional development for their future careers. It encourages active learning and provides many exercises, study tools and opportunities for active learning. Students will ga...in valuable insight into how they can apply their mathematics and teaching knowledge in the classroom. We offer many high quality discounted mathematics textbooks to buy or rent by semester.
Rick Billstein is the author of Problem Solving Approach to Mathematics for Elementary School Teachers, A (11th Edition), published 2012 under ISBN 9780321756664 and 0321756665. Three hundred twenty Problem Solving Approach to Mathematics for Elementary School Teachers, A (11th Edition) textbooks are available for sale on ValoreBooks.com, seventy two used from the cheapest price of $29.88, or buy new starting at $173BOOK ONLY! BOOK ONLY 11th Edition. Used - Good. Used books do not include online codes or other s... [more]BOOK ONLY! BOOK ONLY 11th Edition. Used - Good. Used books do not include online codes or other supplements unless noted. Choose EXPEDITED shipping for faster delivery! h
| 677.169 | 1 |
Descriptions and Ratings (1)
The objective of this tutorial is to provide the basics of using MATLAB. Topics covered include math functions, plot curves, optimizations, manipulation of vectors and matrices, linear systems, data analysis, loops and conditions, and logical operators. Examples using real world applications, such as the deformation of a system of springs, are provided.
Material created by Professor Marc Buffat.
| 677.169 | 1 |
This popular Physical Chemistry text book is now available in electronic format. We have preserved much of the material of the former hard copy editions, making changes to improve understanding of the concepts in addition to including some of the recent discoveries in physical chemistry. Many chapters have new sections and the coverage of several chapters has been greatly expanded. The chapter on statistical mechanics, 15, has been completely rewritten.
The eBook has also been divided into smaller modules that are appropriate for specific courses in Physical Chemistry.
Easy to use Clebsch-Gordan coefficient solver for adding two angular momentums in Quantum Mechanics. This tool is created for my Quantum Mechanics II course offered by Dr. Thompson in Summer of 2007.
[Instruction]
Execute "GUI.m" script by invoking "GUI".
Inspired by a discussion with my father on how to solve sudokus, I decided to implement a GUI for MATLAB and play around with automatic solving. The result can be found here: You can use the GUI just for playing sudoku and having an online check or you may turn on the solving aids: Display tooltips showing all valid numbers so far, or have a semiautomatic or a automatic solver which evaluates the logical constraints. On top of that, a branching algorithm is implemented, which solves any arbitrary sudoku very fast.
Math Solver Free for Windows 8 is a handy tool for performing frequently used operations used for solving math problems. You can use this tool for solving quadratic equations or calculating the angles of a triangle.
The app also includes a unit converter and other useful tools for dealing with math problems by using your Windows 8 device.
Worksheet Generator for Chemistry is a handy and reliable software that helps you to easily and quickly create and customize your personal chemistry worksheets.
The application provides you with various exercise templates that allow you to adjust your worksheets. You are able to insert various chemistry exercises of different areas such as units and chemical formulae, thermochemistry, chemical kinetics, Redox reactions and organic chemistry
| 677.169 | 1 |
Synopses & Reviews
Publisher Comments:
An in-depth survey of some of the most readily applicable essentials of modern mathematics, this concise volume is geared toward undergraduates of all backgrounds as well as future math majors. By focusing on relatively few fundamental concepts, the text delves deeply enough into each subject to challenge students and to offer practical applications.
The opening chapter introduces the program of study and discusses how numbers developed. Subsequent chapters explore the natural numbers; sets, variables, and statement forms; mappings and operations; groups; relations and partitions; integers; and rational and real numbers. Prerequisites include high school courses in elementary algebra and plane geometry.
Synopsis:
| 677.169 | 1 |
Find a Pleasant Prairie PrecalculusThey include mnemonics, effective reading, concentration techniques, as well as efficient note-taking. Logic is the use and study of valid reasoning. The study of logic features most prominently in the study of mathematics
| 677.169 | 1 |
Related Products
Math Principles for Food Service Occupations
Math Principles for Food Service Occupations
Summary
"Math Principals for Food Service Occupations, 4th Edition" is an important tool for the student preparing for a career in the food service industry. The book explains that, like cooking or baking, math is sequential and a student must first master basic math skills before being able to create gourmet meals or desserts. Quotes from chefs and managers are interspersed throughout the book, relaying the relevancy of math skills to the food service professional on the job. This 4th edition contains completely updated material and presents the math problems and concepts in a simplified, logical, step-by-step process. The book offers practical and useful information including explanations relative to figuring menu and food cost procedures and teaches math skills needed to utilize a computer spreadsheet program.
| 677.169 | 1 |
Mathematics for Elementary Teachers -Text Only - 7th edition
This leading mathematics text for elementary and middle school educators helps readers quickly develop a true understanding of mathematical concepts. It integrates rich problem-solving strategies with relevant topics and extensive opportunities for hands-on experience. By progressing from the concrete to the pictorial to the abstract, Musser captures the way math is generally taught in elementary schools50 +$3.99 s/h
Good
HPB-South Center Tukwila, WA19 +$3.99 s/h
Good
HPB-Orland Park Orland
| 677.169 | 1 |
Development of appropriate plans for managing the project compexity and dynamics . ... agreeing on the project start- and the project end- event, ...
Source:
The mathematics curriculum in grade 12 is dominated by differential and integral calculus ... in their regular class. However the ..... Students were asked to apply the formula to a simple website comprising of two pages A and. B as shown in ...
Source:
New York State and the New York State Chemistry Mentors. ... The project
manager for the development of the Chemistry Core Curriculum was Diana
Kefalas ...
Source:
Textbook: Bob Jones, Life Science for Christian School .... Time may be spent to facilitate music lessons, studying for .... Grade, First Semester, 1 high school credit, 3 college credits .... Algebra II is a second year algebra course for students in grades 10-12 who have successfully ...
Source:
Coal Cake'. this cake is charged into the oven from the front side, by the .... were applied for a practical use. As science and technology progressed, ways ...
| 677.169 | 1 |
Thank you for your suggestion! We've passed this along to our developers for review. In the meantime, if you're experiencing issues with any query while using the app, we encourage user feedback by emailing us at [email protected] Algebra is a critical part of mathematics. You need to get it right!
Being successful at first-year university mathematics (or final-year secondary level), requires a lot of practice.
It also helps to recognize that there are patterns that repeat themselves over and over in the kind of problems that you encounter at this level.
This app helps in these areas. You are given a large collection of typical Linear Algebra example problems. The problems are logically grouped by topic, and are interspersed with definitions and theorems, using wording that is as simple as possible to get the ideas across.
Full solutions are given for each problem, and in many places, when you would like to see more detail about how a part of the solution was obtained, all you have to do is expand that sub-problem to see the steps that were used the free version - the "paid" version does not contain ads, has a few cosmetic changes and supports more devices (mainly large format). Are you frustrated with not fully understanding topics in linear algebra class? Do you just want to understand what inverting a matrix does in 3D development? Pocket Linear Algebra Tutor (PLAT) may be just the app you are looking for! PLAT is an app that accepts 3x3 demo matrices and walks you through five topics step-by-step, with detailed explanations along the way. If you are just wondering what the "layman's" definition is of that fancy-sounding term is (Frobenius Norm, anyone?), PLAT also includes a list of key linear algebra terms and definitions. Current topics are: -Eigenvalues and Eigenvectors -Linear Combination -Lower-Upper Decomposition -Matrix Inverse -Gauss-Jordan Elimination to Reduced Row-Echelon Format Alodynamics
| 677.169 | 1 |
This significantly revised and expanded second edition of "Mathematical Olympiad Challenges" is a rich collection of problems put together by two experienced and well-known professors and coaches of the U.S. International Mathematical Olympiad Team. Hundreds of beautiful, challenging, and instructive problems from algebra, geometry, trigonometry, combinatorics, and number theory from numerous mathematical competitions and journals have been selected and updated. The problems are clustered by topic into self-contained sections with solutions provided separately.
The International Mathematical Olympiad (IMO) is a competition for high school students. China has taken part in IMO twenty times since 1985 and has won the top ranking for countries thirteen times, with a multitude of golds for individual students. The 6 students China sent every year were selected from 20 to 30 students among approximately 130 students who take part in the China Mathematical Competition during the winter months. This volume comprises a collection of original problems with solutions that China used to train their Olympiad team in the years from 2003 to 2006.
The Handbook of Mathematical Methods in Imaging provides a comprehensive treatment of the mathematical techniques used in imaging science. The material is grouped into two central themes, namely, Inverse Problems (Algorithmic Reconstruction) and Signal and Image Processing.
Students and professionals in the fields of mathematics, physics, engineering, and economics will find this reference work invaluable. A classic resource for working with special functions, standard trig, and exponential logarithmic definitions and extensions, it features 29 sets of tables, some to as high as 20 places.
You know mathematics. You know how to write mathematics. But do you know how to produce clean, clear, well-formatted manuscripts for publication? Do you speak the language of publishers, typesetters, graphics designers, and copy editors?...
| 677.169 | 1 |
About ...
More About
This Book
About become more proficient in math and thereby do better in their jobs. Alan lives with his wife and children on the north coast of California. He spends much of his time designing and selling solar hot water systems where he is required to do a great deal of math. He also teaches for College of the Redwoods.
Book Reviews:
Editorial Reviews
Kirkus Reviews
This step-by-step guide to mastering basic construction math utilizes the building of a hypothetical bungalow as an occasion to lightheartedly relate mathematical thinking to practical application. The need for one more book in a market where a dozen books on basic math for construction workers have been published appears initially questionable. Yet Cook's book earns a place of distinction for bringing a gently humorous tone and an engaging sense of storytelling to what could be the driest of subjects. Organizing his book around the building of a hypothetical bungalow, Cook brings essential grade-school level mathematical concepts (how to solve for unknowns, distinguishing numbers and units and dealing with fundamental algebra and geometry) into play in the sequence you would need to use them if you were building your own home. This is a particularly engaging read for anyone suffering math phobia. "Solving for an unknown is like traveling to a new destination," opens the book, and, through the author's clearheaded examples of construction problems mathematically solved, math becomes a thoroughly charming form of intellectual recreation. Each chapter concludes with a helpful review of key concepts. And sprinkled through the text are Mary E. Scott's cartoons achieving a remarkable balance between adult sophistication and childlike playfulness. In Cook's chapter on reviewing how to solve fractions, decimals and percent, the author gives the all-too-gripping math problem of calculating the cost of filling up your truck's gas tank, a skyrocketing cost of doing construction work. Levity is introduced through Scott's cartoon showing a depressed consumer pondering which button to press on a gas pump with buttons labeled "Barely Afford," "Second Mortgage" and "Buy a Bike." The book's 17 compact chapters conclude with a helpful list of conversion factors and handy equations. Cook successfully establishes the mathematical foundation needed for construction with a witty, conversational tone that clarifies math while instilling confidence in a reader's capacity to practically apply
| 677.169 | 1 |
9.
CHAPTER ZERO PrefaceThis is a free and open source differential calculus book. The "free and opensource" part means you, as a student, can give digital versions of this book to any-one you want (for free). It means that if you are a teacher, you can (a) give or printor xerox copies for your students, (b) use potions for your own class notes (if theyare published then you might need to add some acknowledgement, depending onwhich parts you copied), and you can xerox even very large portions of it to yourhearts content. The "differential calculus" part means it covers derivatives andapplications but not integrals. It is heavily based on the first half of a classic text,Granville's "Elements of the Differential and Integral Calculus," quite possibly abook your great grandfather might have used when he was college age. Some ma-terial from Sean Mauch's excellent public domain text on Applied Mathematics, also included. Calculus has been around for several hundred years and the teaching of it has notchanged radically. Of course, like any topic which is taught in school, there aresome modifications, but not major ones in this case. If x(t) denotes the distance atrain has traveled in a straight line at time t then the derivative is the velocity. Ifq(t) denotes the charge on a capacitor at time t in a simple electrical circuit thenthe derivative is the current. If C(t) denotes the concentration of a solvent in achemical mixture at time t then the derivative is the reaction rate. If P (t) denotesthe population size of a country at time t then the derivative is the growth rate.If C(x) denotes the cost to manufacture x units of a production item (such as abroom, say) then the derivative is the marginal cost. ix
10.
Some of these topics, electrical circuits for example, were not studied in calcu-lus when Granville's book was first written. However, aside from some changesin grammar and terminology (which have been updated in this version), the math-ematical content of the calculus course taught today is basically the same as thattaught a hundred years ago. Terminology has changed, and no one talks about"versines" any more (they were used in navigation tables before the advent ofcomputers), but the basic techniques have not. Therefore, to make the book moreuseful to current students, some modification and rearrangement of the materialin Granville's old text is appropriate. Overall, though the rigor and detailed expla-nations are still at their same high level of quality. Here is a quote form Granville's original preface: The author has tried to write a textbook that is thoroughly modern and teachable, and the capacity and needs of the student pursuing a first course in the Calculus have been kept constantly in mind. The book contains more material than is necessary for the usual course of one hundred lessons given in our colleges and engineering schools; but this gives teachers an opportunity to choose such subjects as best suit the needs of their classes. It is believed that the volume contains all topics from which a selection naturally would be made in prepar- ing students either for elementary work in applied science or for more advanced work in pure mathematics. WILLIAM A. GRANVILLE GETTYSBURG COLLEGE Gettysburg, Pa. For further information on William Granville, please see the Wikipedia article at has a short biography and links for further information. Granville's book "Elements of the Differential and Integral Calculus" fell intothe public domain (in the United States - other countries may be different) andthen much of it (but not all, at the time of this writing) was scanned into by P. J. Hall. This wikisource document uses MathML and LTEX and Asome Greek letter fonts. x
11.
In keeping with the "free and open source" aspect of this textbook, and thetheme of updating to today's much more technologically-aware students, a freeand open sourse mathematical software package Sage was used to illustrate ex-amples throughout. You don't need to know Sage to read the book (just ignorethe Sage examples if you want) but it certainly won't hurt to learn a little aboutit. Besides, you might find that with some practice Sage is fun to "play with"and helps you with homework or other mathematical problems in some of yourother classes. It is a general purpose mathematical software program and it mayvery likely be the only mathematical software you will ever need. This LTEX'd version is due to the second-named author, who is responsible for Aformatting, the correction of any typos in the scanned version, significant revisionfor readability, and some extra material (for example, the Sage examples andgraphics). In particular, the existence of this document owes itself primarily tothree great open source projects: TEX/LTEX, Wikipedia, and Sage . All the fig- Aures were created using Sage and then edited and converted using the excellentopen source image manipulation program GIMP ( Sage code for each image can be found in the LTEX source code, available Aat information on Sage can be found at the Sage website (located at or in the Appendix (Chapter 13) below. Though the original text of Granville is public domain, the extra material addedin this version is licensed under the GNU Free Documentation License (repro-duced in an Appendix below), as is Wikipedia. Acknowledgements: I thank the following readers for careful proofreading andreporting typos: Mario Pernici, Jacob Hicks, Georg Muntingh, and Minh VanNguyen. I also thank Trevor Lipscombe for excellent stylistic advice on the pre-sentation of the book. However, any remaining errors are solely my responsibility.Please send comments, suggestions, proposed changes, or corrections by email [email protected]. xi
12.
CHAPTER ONE Variables and functions1.1 Variables and constantsA variable is a quantity to which an unlimited number of values can be assigned.Variables are denoted by the later letters of the alphabet. Thus, in the equation ofa straight line, x y + =1 a bx and y may be considered as the variable coordinates of a point moving along theline. A quantity whose value remains unchanged is called a constant.√Numerical or absolute constants retain the same values in all problems, as 2, 5, 7, π, etc. Arbitrary constants, or parameters, are constants to which any one of an unlim-ited set of numerical values may be assigned, and they are supposed to have theseassigned values throughout the investigation. They are usually denoted by the ear-lier letters of the alphabet. Thus, for every pair of values arbitrarily assigned to aand b, the equation x y + =1 a brepresents some particular straight line. 1
13.
1.2. INTERVAL OF A VARIABLE1.2 Interval of a variableVery often we confine ourselves to a portion only of the number system. Forexample, we may restrict our variable so that it shall take on only such values aslie between a and b, where a and b may be included, or either or both excluded.We shall employ the symbol [a, b], a being less than b, to represent the numbersa, b, and all the numbers between them, unless otherwise stated. This symbol[a, b] is read the interval from a to b.1.3 Continuous variationA variable x is said to vary continuously through an interval [a, b], when x startswith the value a and increases until it takes on the value b in such a manner as toassume the value of every number between a and b in the order of their magni-tudes. This may be illustrated geometrically as follows: Figure 1.1: Interval from A to B.The origin being at O, layoff on the straight line the points A and B correspondingto the numbers a and b. Also let the point P correspond to a particular value of thevariable x. Evidently the interval [a, b] is represented by the segment AB. Nowas x varies continuously from a to b inclusive, i.e. through the interval [a, b], thepoint P generates the segment AB.1.4 FunctionsA function f of the real numbers R is a well-defined rule which associated to eachx ∈ R a unique value f (x). Usually functions are described algebraically usingsome formula (such as f (x) = x2 , for all real numbers x) but it doesn't have to beso simple. For example, 2
14.
1.5. NOTATION OF FUNCTIONS x2 , if x is an integer, f (x) = 0, otherwise,is a function on R but it is given by a relatively complicated rule. Namely, the rulef tells you to associate to a number x the value 0 unless x is an integer, in whichcase you are to associate the value x2 . (In particular, f (x) is always an integer, nomatter what x is.) This type of rule defining a function of x is sometimes calleda piecewise-defined function. In this book, we shall usually focus on functionsgiven by simpler symbolic expressions. However, be aware that piecewise-definedfunctions do arise naturally in applications. For example, in electronics, when a6 volt battery-powered flashlight is powered on or off using a switch, the voltageto the lightbulb is modeled by a piecewise-defined function which has the value 0when the device is off and 6 when it is switched on. When two variables are so related that the value of the first variable depends onthe value of the second variable, then the first variable is said to be a function ofthe second variable. Nearly all scientific problems deal with quantities and relations of this sort, andin the experiences of everyday life we are continually meeting conditions illus-trating the dependence of one quantity on another. For instance, the weight a manis able to lift depends on his strength, other things being equal. Similarly, the dis-tance a boy can run may be considered as depending on the time. Or, we may saythat the area of a square is a function of the length of a side, and the volume of asphere is a function of its diameter.1.5 Notation of functionsThe symbol f (x) is used to denote a function of x, and is read "f of x". In orderto distinguish between different functions, the prefixed letter is changed, as F (x),φ(x), f ′ (x), etc. During any investigation the same functional symbol always indicates the samelaw of dependence of the function upon the variable. In the simpler cases thislaw takes the form of a series of analytical operations upon that variable. Hence,in such a case, the same functional symbol will indicate the same operations orseries of operations, even though applied to different quantities. Thus, if f (x) = x2 − 9x + 14, 3
16.
1.6. INDEPENDENT AND DEPENDENT VARIABLES1sage: f = lambda x,y: xˆ2+yˆ2sage: f(3,4)25sage: R.<x> = PolynomialRing(CC,"x")sage: f = xˆ2+2sage: f.roots()[(1.41421356237309*I, 1), (2.77555756156289e-17 - 1.41421356237309*I, 1)]1.6 Independent and dependent variablesThe second variable, to which values may be assigned at pleasure within limitsdepending on the particular problem, is called the independent variable, or argu-ment; and the first variable, whose value is determined as soon as the value of theindependent variable is fixed, is called the dependent variable, or function. Though we shall wait to introduce differentiation later, please keep in mind thatyou differentiate the dependent variable with respect to the independent variable.Example 1.6.1. In the equation of an upper half-circle of radius 1, √ y= 1 − x2 ,we typically call x the independent variable and y the dependent variable. Frequently, when we are considering two related variables, it is in our powerto fix upon whichever we please as the independent variable; but having oncemade the choice, no change of independent variable is allowed without certainprecautions and transformations. One quantity (the dependent variable) may be a function of two or more otherquantities (the independent variables, or arguments). For example, the cost ofcloth is a function of both the quality and quantity; the area of a triangle is afunction of the base and altitude; the volume of a rectangular parallelepiped is afunction of its three dimensions. In the Sage example below, t is the independent variable and f is the dependentvariable. Sagesage: t = var('t')sage: f = function('f', t)sage: f = cos 5
17.
1.7. THE DOMAIN OF A FUNCTIONsage: f(pi/2)0sage: (f(-3*pi)-2*f(1))ˆ2(-2*cos(1) - 1)ˆ21.7 The domain of a functionThe values of the independent variable for which a function f (x) is defined isoften referred to as the domain of the function, denoted domain(f ). Consider the functions x2 − 2x + 5, sin x, arctan xof the independent variable x. Denoting the dependent variable in each case by y,we may write y = x2 − 2x + 5, y = sin x, y = arctan x.In each case y (the value of the function) is known, or, as we say, defined, for allvalues of x. We write in this case, domain(f ) = R. This is not by any meanstrue of all functions, as the following examples illustrating the more commonexceptions will show. a y= (1.1) x−bHere the value of y (i.e. the function) is defined for all values of x except x = b.When x = b the divisor becomes zero and the value of y cannot be computed from(1.1). We write in this case, domain(y) = R − {b}. √ y= x. (1.2)In this case the function is defined only for positive values of x. Negative valuesof x give imaginary values for y, and these must be excluded here, where we areconfining ourselves to real numbers only. We write in this case, domain(y) ={x ∈ R | x ≥ 0}. y = loga x. a>0 (1.3) 6
20.
CHAPTER TWO Theory of limitsIn this book, a variable denotes a quantity which takes values in the real numbers.2.1 Limit of a variableIf a variable v takes on successively a series of values that approach nearer andnearer to a constant value L in such a manner that |v − L| becomes and remainsless than any assigned arbitrarily small positive quantity, then v is said to approachthe limit L, or to converge to the limit L. Symbolically this is written limv=L , ormore commonly lim . v→L The following familiar examples illustrate what is meant: 1. As the number of sides of a regular inscribed polygon is indefinitely in- creased, the limit of the area of the polygon is the area of the circle. In this case the variable is always less than its limit. 2. Similarly, the limit of the area of the circumscribed polygon is also the area of the circle, but now the variable is always greater than its limit. 3. Hold a penny exactly 1 meter above the ground and observe its motion as you release it. First it travels 1/2 the distance from the ground (at this stage its distance fallen is 1/2 meter), then it travels 1/2 that distance from the 9
22.
2.1. LIMIT OF A VARIABLE Hence, by definition of the limit of a variable, it is seen that both S2n and S2n+1 are variables approaching 2 as a limit as the number of terms in- 3 creases without limit. Summing up the first two, three, four, etc., terms of (2.1), the sums are found by ((2.2) and ((2.3) to be alternately less and greater than 2 , illustrating 3 the case when the variable, in this case the sum of the terms of ((2.1), is alternately less and greater than its limit. Sage sage: S = lambda n: add([(-1)ˆi*2ˆ(-i) for i in range(n)]) sage: RR(S(1)); RR(S(2)); RR(S(5)); RR(S(10)); RR(S(20)) 1.00000000000000 0.500000000000000 0.687500000000000 0.666015625000000 0.666666030883789 You can see from the Sage example that the limit does indeed seem to approach 2/3. In the examples shown the variable never reaches its limit. This is not by anymeans always the case, for from the definition of the limit of a variable it is clearthat the essence of the definition is simply that the absolute value of the differencebetween the variable and its limit shall ultimately become and remain less thanany positive number we may choose, however small.Example 2.1.1. As an example illustrating the fact that the variable may reachits limit, consider the following. Let a series of regular polygons be inscribed ina circle, the number of sides increasing indefinitely. Choosing anyone of these,construct. the circumscribed polygon whose sides touch the circle at the verticesof the inscribed polygon. Let pn and Pn be the perimeters of the inscribed andcircumscribed polygons of n sides, and C the circumference of the circle, andsuppose the values of a variable x to be as follows: Pn , pn+1 , C, Pn+1 , pn+2 , C, Pn+2 , etc.Then, evidently, lim x = C x→∞and the limit is reached by the variable, every third value of the variable being C. 11
23.
2.2. DIVISION BY ZERO EXCLUDED2.2 Division by zero excluded00 is indeterminate. For the quotient of two numbers is that number which multi-plied by the divisor will give the dividend. But any number whatever multipliedby zero gives zero, and the quotient is indeterminate; that is, any number whatevermay be considered as the quotient, a result which is of no value. a 0 has no meaning, a being different from zero, for there exists no number suchthat if it be multiplied by zero, the product will equal a. Therefore division by zero is not an admissible operation. Care should be taken not to divide by zero inadvertently. The following fallacyis an illustration. Assume that a = b.Then evidently ab = a2 .Subtracting b2 , ab − b2 = a2 − b2 .Factoring, b(a − b) = (a + b)(a − b).Dividing by a − b, b = a + b.But a = b, therefore b = 2b, or, 1 = 2. The result is absurd, and is caused by thefact that we divided by a − b = 0, which is illegal.2.3 InfinitesimalsDefinition 2.3.1. A variable v whose limit is zero is called an infinitesimal1 . This is written lim, or, lim , v=0 v→0and means that the successive absolute values of v ultimately become and remainless than any positive number however small. Such a variable is said to become"arbitrarily small." 1 Hence a constant, no matter how small it may be, is not an infinitesimal. 12
24.
2.4. THE CONCEPT OF INFINITY (∞) If lim v = l, then lim(v − l) = 0; that is, the difference between a variable andits limit is an infinitesimal. Conversely, if the difference between a variable and a constant is an infinitesi-mal, then the variable approaches the constant as a limit.2.4 The concept of infinity (∞)If a variable v ultimately becomes and remains greater than any assigned positivenumber, however large, we say v is "unbounded and positive " (or "increaseswithout limit"), and write lim , or, lim , or, v → +∞. v=+∞ v→+∞If a variable v ultimately becomes and remains smaller than any assigned negativenumber, we say "unbounded and negative " (or "v decreases without limit"), andwrite lim , or, lim , or, v → −∞. v=−∞ v→−∞If a variable v ultimately becomes and remains in absolute value greater than anyassigned positive number, however large, we say v, in absolute value, "increaseswithout limit", or v becomes arbitrarily large2 , and write lim , or, lim , or, v → ∞. v=∞ v→∞Infinity (∞) is not a number; it simply serves to characterize a particular mode ofvariation of a variable by virtue of which it becomes arbitrarily large. Here is a Sage example illustrating limt=∞ 1/t = limt=−∞ 1/t = 0. Sagesage: t = var('t')sage: limit(1/t, t = Infinity) 2 On account of the notation used and for the sake of uniformity, the expression v → +∞ issometimes read "v approaches the limit plus infinity". Similarly, v → −∞ is read "v approachesthe limit minus infinity", and v → ∞ is read "v, in absolute value, approaches the limit infinity".While the above notation is convenient to use in this connection, the student must not forget thatinfinity is not a limit in the sense in which we defined it in §2.2, for infinity is not a number at all. 13
25.
2.5. LIMITING VALUE OF A FUNCTION0sage: limit(1/t, t = -Infinity)02.5 Limiting value of a functionGiven a function f (x). If the independent variable x takes on any series of valuessuch that lim x = a,and at the same time the dependent variable f (x) takes on a series of correspond-ing values such that lim f (x) = A,then as a single statement this is written lim f (x) = A. x→a Here is an example of a limit using Sage : Sagesage: limit((xˆ2+1)/(2+x+3*xˆ2),x=infinity)1/3 x2 +1This tells us that limx→∞ 2+x+3∗x2 = 1. 32.6 Continuous and discontinuous functionsA function f (x) is said to be continuous for x = a if the limiting value of thefunction when x approaches the limit a in any manner is the value assigned to thefunction for x = a. In symbols, if lim f (x) = f (a), x→a 14
26.
2.6. CONTINUOUS AND DISCONTINUOUS FUNCTIONSthen f (x) is continuous for x = a. Roughly speaking, a function y = f (x) iscontinuous if you can draw its graph by hand without lifting your pencil off thepaper. In other words, the graph of a continuous function can have no "breaks."Example 2.6.1. The piecewise constant function 1, x ≥ 0, u(x) = 0, x < 0,is not continuous since its graph has a "break" at x = 0 where it "steps up" from0 to 1. This function models "on-off" switches in electrical engineering and iscalled the unit step function or the Heaviside function (after the brilliant engineerOliver Heaviside, 1850 1925). The function is said to be discontinuous for x = a if this condition is not satis-fied. For example, if lim f (x) = ∞, x→athe function is discontinuous for x = a. Sagesage: t = var('t')sage: P1 = plot(1/t, (t, -5, -0.1))sage: P2 = plot(1/t, (t, 0.1, 5))sage: show(P1+P2, aspect_ratio=1)sage: limit(1/t,t=0,dir="plus")+Infinitysage: limit(1/t,t=0,dir="minus")-InfinityThe graph in Figure 2.1 suggests that limx→0+ 1/x = +∞ and limx→0− 1/x =−∞, as the above Sage computation confirms. The attention of the student is now called to the following cases which occurfrequently.CASE I. As an example illustrating a simple case of a function continuous for aparticular value of the variable, consider the function x2 − 4 f (x) = . x−2 15
28.
2.6. CONTINUOUS AND DISCONTINUOUS FUNCTIONSCASE II. The definition of a continuous function assumes that the function isalready defined for x = a. If this is not the case, however, it is sometimes possibleto assign such a value to the function for x = a that the condition of continuityshall be satisfied. The following theorem covers these cases.Theorem 2.6.1. If f (x) is not defined for x = a, and if lim f (x) = B, x→athen f (x) will be continuous for x = a, if B is assumed as the value of f (x) forx = a. Thus the function x2 − 4 x−2is not defined for x = 2 (since then there would be division by zero). But forevery other value of x, x2 − 4 = x + 2; x+2and lim (x + 2) = 4 x→2 2therefore limx→2 x −4 = 4. Although the function is not defined for x = 2, if we x−2assign it the value 4 for x = 2, it then becomes continuous for this value. Sagesage: x = var('x')sage: limit((xˆ2-4)/(x-2), x = 2)4A function f (x) is said to be continuous in an interval when it is continuous forall values of x in this interval3 . 3 In this book we shall deal only with functions which are in general continuous, that is, con-tinuous for all values of x, with the possible exception of certain isolated values, our results in 17
29.
2.7. CONTINUITY AND DISCONTINUITY OF FUNCTIONSILLUSTRATED BY THEIR GRAPHS2.7 Continuity and discontinuity of functions illus- trated by their graphs 1. Consider the function x2 , and let y = x2 (2.4) If we assume values for x and calculate the corresponding values of y, we can plot a series of points. Drawing a smooth line free-hand through these points: a good representation of the general behavior of the function may be obtained. This picture or image of the function is called its graph. It is evidently the locus of all points satisfying equation (2.4). It is very easy to create the above plot in Sage , as the example below shows: Sage sage: P = plot(xˆ2,-2,2) sage: show(P) Such a series or assemblage of points is also called a curve. Evidently we may assume values of x so near together as to bring the values of y (and therefore the points of the curve) as near together as we please. In other words, there are no breaks in the curve, and the function x2 is continuous for all values of x. 2. The graph of the continuous function sin x, plotted by drawing the locus of y = sin x, It is seen that no break in the curve occurs anywhere. 3. The continuous function exp(x) = ex is of very frequent occurrence in the Calculus. If we plot its graph fromgeneral being understood as valid only for such values of x for which the function in question isactually continuous. Unless special attention is called thereto, we shall as a rule pay no attentionto the possibilities of such exceptional values of x for which the function is discontinuous. Thedefinition of a continuous function f(x) is sometimes roughly (but imperfectly) summed up in thestatement that a small change in x shall produce a small change in f (x). We shall not considerfunctions having an infinite number of oscillations in a limited region. 18
30.
2.7. CONTINUITY AND DISCONTINUITY OF FUNCTIONS ILLUSTRATED BY THEIR GRAPHS Figure 2.2: The parabola y = x2 . y = ex , (e = 2.718 · · · ),we get a smooth curve as shown.From this it is clearly seen that, (a) when x = 0, limx→0 y(= limx→0 ex ) = 1; (b) when x > 0, y(= ex ) is positive and increases as we pass towards the right from the origin; (c) when x < 0, y(= ex ) is still positive and decreases as we pass towards the left from the origin. 19
31.
2.7. CONTINUITY AND DISCONTINUITY OF FUNCTIONSILLUSTRATED BY THEIR GRAPHS Figure 2.3: The sine function. 4. The function ln x = loge x is closely related to the last one discussed. In fact, if we plot its graph from y = loge x, it will be seen that its graph is the reflection of the graph of y = ex about the diagonal (the x = y line). (This is because they are "inverses" of each other: loge (ex ) = x and eloge x = x.) Here we see the following facts pictured: (a) For x = 1, loge x = loge 1 = 0. (b) For x > 1, loge x is positive and increases as x increases. (c) For 1 > x > 0, loge x is negative and increases in absolute value as x, that is, limx→0 log x = −∞. (d) For x ≤ 0, loge x is not defined; hence the entire graph lies to the right of OY . 1 5. Consider the function x , and set 20
32.
2.7. CONTINUITY AND DISCONTINUITY OF FUNCTIONS ILLUSTRATED BY THEIR GRAPHS Figure 2.4: The exponential function. 1 y= x If the graph of this function be plotted, it will be seen that as x approaches the value zero from the left (negatively), the points of the curve ultimately drop down an infinitely great distance, and as x approaches the value zero from the right, the curve extends upward infinitely far. The curve then does not form a continuous branch from one side to the other of the axis of y, showing graphically that the function is discontinuous for x = 0, but continuous for all other values of x.6. From the graph (see Figure 2.7) of 2x y= 1 − x2 21
33.
2.7. CONTINUITY AND DISCONTINUITY OF FUNCTIONSILLUSTRATED BY THEIR GRAPHS Figure 2.5: The natural logarithm. 2x it is seen that the function 1−x2 is discontinuous for the two values x = ±1, but continuous for all other values of x. 22
34.
2.7. CONTINUITY AND DISCONTINUITY OF FUNCTIONS ILLUSTRATED BY THEIR GRAPHS Figure 2.6: The function y = 1/x.7. The graph of y = tan x shows that the function tan x is discontinuous for infinitely many values of the independent variable x, namely, x = nπ , where n denotes any odd 2 positive or negative integer.8. The function arctan x has infinitely many values for a given value of x, the graph of equation y = arctan x consisting of infinitely many branches. If, however, we confine ourselves to any single branch, the function is con- tinuous. For instance, if we say that y shall be the smallest angle (in radians) whose tangent is x, that is, y shall take on only values between − π and π , 2 2 then we are limited to the branch passing through the origin, and the condi- tion for continuity is satisfied. 23
35.
2.7. CONTINUITY AND DISCONTINUITY OF FUNCTIONSILLUSTRATED BY THEIR GRAPHS Figure 2.7: The function y = 2x/(1 − x2 ). 1 9. Similarly, arctan x , is found to be a many-valued function. Confining our- selves to one branch of the graph of 1 y = arctan , x we see that as x approaches zero from the left, y approaches the limit − π , 2 and as x approaches zero from the right, y approaches the limit + π . Hence 2 the function is discontinuous when x = 0. Its value for x = 0 can be assigned at pleasure. 10. As was previously mentioned, a piecewise-defined function is one which is defined by different rules on different non-overlapping intervals. For exam- ple, 24
37.
2.8. FUNDAMENTAL THEOREMS ON LIMITS Figure 2.9: The arctangent (or inverse tangent) function. sage: f Piecewise defined function with 3 parts, [[(-5, -2), -1], [(-2, 3), 3], [(3, 5), 2]] Functions exist which are discontinuous for every value of the independent vari-able within a certain range. In the ordinary applications of the Calculus, however,we deal with functions which are discontinuous (if at all) only for certain iso-lated values of the independent variable; such functions are therefore in generalcontinuous, and are the only ones considered in this book.2.8 Fundamental theorems on limitsIn problems involving limits the use of one or more of the following theorems isusually implied. It is assumed that the limit of each variable exists and is finite. 26
38.
2.8. FUNDAMENTAL THEOREMS ON LIMITS Figure 2.10: The function y = arctan(1/x).Theorem 2.8.1. The limit of the algebraic sum of a finite number of variables isequal to the algebraic sum of the limits of the several variables. In particular, lim [f (x) + g(x)] = lim f (x) + lim g(x). x→a x→a x→aTheorem 2.8.2. The limit of the product of a finite number of variables is equalto the product of the limits of the several variables. In particular, lim [f (x) · g(x)] = lim f (x) · lim g(x). x→a x→a x→a Here is a Sage example illustrating these facts in a special case. Sagesage: t = var('t')sage: f = expsage: g = sinsage: a = var('a')sage: L1 = limit(f(t)+g(t), t = a) 27
40.
2.8. FUNDAMENTAL THEOREMS ON LIMITS Figure 2.12: Another piecewise defined function. In particular, limx→a f (x) lim [f (x)/g(x)] = , x→a limx→a g(x)provided limx→a g(x) = 0. Before proving these theorems it is necessary to establish the following proper-ties of infinitesimals (Definition 2.3.1). 1. The sum of a finite number of infinitesimals is an infinitesimal. To prove this we must show that the absolute value of this sum can be made less than any small positive quantity (as ǫ) that may be assigned (§2.3). That this is possible is evident, for, the limit of each infinitesimal being zero, each ǫ one can be made less than, in absolute value, n (n being the number of infinitesimals), and therefore the absolute value of their sum can be made less than ǫ. 2. The product of a constant c = 0 and an infinitesimal is an infinitesimal. For the absolute value of the product can always be made less than any small 29
41.
2.8. FUNDAMENTAL THEOREMS ON LIMITS positive quantity (as ǫ) by making the absolute value of the infinitesimal less ǫ than |c| . 3. If v is a variable which approaches a limit L different from zero, then the quotient of an infinitesimal by v is also an infinitesimal. For if v → L, and k is any number in absolute value less than L, then, by definition of a limit, v will ultimately become and remain in absolute value greater than ǫ k. Hence the quotient v , where ǫ is an infinitesimal, will ultimately become ǫ and remain in absolute value less than k , and is therefore, by the previous item, an infinitesimal. 4. The product of any finite number of infinitesimals is an infinitesimal. For the absolute value of the product may be made less than any small positive quantity that can be assigned. If the given product contains n factors, then since each infinitesimal may be assumed less than the n − th root of ǫ, the product can be made less than ǫ itself. Proof of Theorem 2.8.1. Let v1 , v2 , v3 , . . . be the variables, and L1 , L2 , L3 , . . .their respective limits. We may then write v 1 − L1 = ǫ1 , v 2 − L2 = ǫ2 , v 3 − L3 = ǫ3 ,where ǫ1 , ǫ2 , ǫ3 , . . . are infinitesimals (i.e. variables having zero for a limit).Adding (v1 + v2 + v3 + . . . ) − (L1 + L2 + L3 + ...) = (ǫ1 + ǫ2 + ǫ3 + . . . ).Since the right-hand member is an infinitesimal by item (1) above (§2.8), we have,from the converse theorem (§2.3), lim(v1 + v2 + v3 + . . . ) = L1 + L2 + L3 + . . . ,or, lim(v1 + v2 + v3 + . . . ) = lim v1 + lim v2 + lim v3 + . . . ,which was to be proved. Proof of Theorem 2.8.2. Let v1 and v2 be the variables, L1 and L2 their respectivelimits, and ǫ1 and ǫ2 infinitesimals; then v 1 = L1 + ǫ1 30
45.
2.11. THE NUMBER E Figure 2.13: Comparing x and sin(x) on the unit circle. Although the function is not defined for x = 0, yet it is not discontinuous whenx = 0 if we define sin 0 = 1 (see Case II in §2.6). 0 Finally, we show how to use the Sage command limit to compute the limitabove. Sagesage: limit(sin(x)/x,x=0)12.11 The number eOne of the most important limits in the Calculus is 1 lim (1 + x) x = 2.71828 · · · = e x→0 34
46.
2.11. THE NUMBER E sin(x) Figure 2.14: The function x .To prove rigorously that such a limit e exists, is beyond the scope of this book.For the present we shall content ourselves by plotting the locus of the equation 1 y = (1 + x) x 1and show graphically that, as x=0, the function (1 + x) x (= y) takes on values in ˙the near neighborhood of 2.718 . . . , and therefore e = 2.718 . . . , approximately. x -.1 -.001 .001 .01 .1 1 5 10 y = (1 + x)1/x 2.8680 2.7195 2.7169 2.7048 2.5937 2.0000 1.4310 1.0096 As x → 0− from the left, y decreases and approaches e as a limit. As x → 0+from the right, y increases and also approaches e as a limit. As x → ∞, y approaches the limit 1; and as x → −1+ from the right, yincreases without limit. Natural logarithms are those which have the number e for base. These loga-rithms play a very important rle in mathematics. When the base is not indicatedexplicitly, the base e is always understood in what follows in this book. Thusloge v is written simply log v or ln v. Natural logarithms possess the following characteristic property: If x → 0 inany way whatever, 35
52.
CHAPTER THREE Differentiation3.1 IntroductionIn this chapter, we investigate the ways in which a function can change in valueas the independent variable changes. For instance, if f (t) is a function of t (time),we want to quantify what it means to talk about the "rate of change" of f (t).A fundamental problem of differential calculus is to establish a mathematicallyprecise measure of this change in the function. It was while investigating problems of this sort that Newton1 was led to thediscovery of the fundamental principles of calculus. Today, Gottfried Leibniz(1646-1716) is generally credited with independently discovering calculus aroundthe same time2 . 1 Sir Isaac Newton (1642-1727), an Englishman, was a man of the most extraordinary genius.He developed the science of calculus under the name of "Fluxions." Although Newton had discov-ered and made use of the new theory as early as 1670, his first published work in which it occurs isdated 1687, having the title Philosophiae Naturalis Principia Mathematica. This was Newton'sprincipal work. Laplace said of it, "It will always remain preeminent above all other productionsof the human mind." See frontispiece. 2 However, see v. Leibniz calculus controversyand the footnote in §3.9 below. 41
55.
3.4. DERIVATIVE OF A FUNCTION OF ONE VARIABLEIt is apparent that as ∆ x decreases, ∆ y also diminishes, but their ratio takes on ∆ythe successive values 9, 8.8, 8.6, 8.4, 8.2, 8.1, 8.01; illustrating the fact that ∆xcan be brought as near to 8 in value as we please by making ∆ x small enough.Therefore3 , ∆y lim = 8. ∆x→0 ∆x3.4 Derivative of a function of one variableThe fundamental definition of the Differential Calculus is:Definition 3.4.1. The derivative4 of a function is the limit of the ratio of the incre-ment of the function to the increment of the independent variable, when the latterincrement varies and approaches the limit zero. When the limit of this ratio exists, the function is said to be differentiable, or topossess a derivative. The above definition may be given in a more compact form symbolically asfollows: Given the function y = f (x), (3.2)and consider x to have a fixed value. Let x take on an increment ∆ x; then thefunction y takes on an increment ∆ y, the new value of the function being y + ∆ y = f (x + ∆ x). (3.3)To find the increment of the function, subtract (3.2) from (3.3), giving ∆ y = f (x + ∆ x) − f (x).Dividing by the increment of the variable, ∆ x, we get ∆y f (x + ∆x) − f (x) = . (3.4) ∆x ∆x 3 The student should guard against the common error of concluding that because the numeratorand denominator of a fraction are each approaching zero as a limit, the limit of the value of thefraction (or ratio) is zero. The limit of the ratio may take on any numerical value. In the aboveexample the limit is 8. 4 Also called the differential coefficient or the derived function. 44
56.
3.5. SYMBOLS FOR DERIVATIVESThe limit of this ratio when ∆ x approaches the limit zero is, from our definition, dythe derivative and is denoted by the symbol dx . Therefore dy f (x + ∆x) − f (x) = lim . dx ∆x→0 ∆xdefines the derivative of y [or f (x)] with respect to x. From (3.3), we also get dy ∆y = lim dx ∆x→0 ∆xThe process of finding the derivative of a function is called differentiation. It should be carefully noted that the derivative is the limit of the ratio, not the ra-tio of the limits. The latter ratio would assume the form 0 , which is indeterminate 0(§2.2).3.5 Symbols for derivativesSince ∆ y and ∆ x are always finite and have definite values, the expression ∆y ∆xis really a fraction. The symbol dy , dxhowever, is to be regarded not as a fraction but as the limiting value of a fraction.In many cases it will be seen that this symbol does possess fractional properties,and later on we shall show how meanings may be attached to dy and dx, but for dythe present the symbol dx is to be considered as a whole. Since the derivative of a function of x is in general also a function of x, thesymbol f ′ (x) is also used to denote the derivative of f (x). dy Hence, if y = f (x), we may write dx = f ′ (x), which is read "the derivative ofy with respect to x equals f prime of x." The symbol d dxwhen considered by itself is called the differentiating operator, and indicates thatany function written after it is to be differentiated with respect to x. Thus 45
57.
3.6. DIFFERENTIABLE FUNCTIONS dy d • dx or dx y indicates the derivative of y with respect to x; d • dx f (x) indicates the derivative of f (x) with respect to x; d • dx (2x2 + 5) indicates the derivative of 2x2 + 5 with respect to x; dy • y ′ is an abbreviated form of dx . dThe symbol Dx is used by some writers instead of dx . If then y = f (x),we may write the identities dy d y′ = = y = Dx f (x) = f ′ (x). dx dx3.6 Differentiable functionsFrom the theory of limits (Chapter 2), it is clear that if the derivative of a functionexists for a certain value of the independent variable, the function itself must becontinuous for that value of the variable. However, the converse is not always true. Functions have been constructed thatare continuous and yet possess no derivative. But in this book we only considerfunctions f (x) that possess a derivative for all values of the independent variable,save at most for some isolated (discrete) values of x.3.7 General rule for differentiationFrom the definition of a derivative it is seen that the process of differentiating afunction y = f (x) consists in taking the following distinct steps: General rule for differentiating5 : • FIRST STEP. In the function replace x by x + ∆ x, giving a new value of the function, y + ∆ y. • SECOND STEP. Subtract the given value of the function from the new value in order to find ∆ y (the increment of the function). 5 Also called the Four-step Rule. 46
65.
3.9. APPLICATIONS OF THE DERIVATIVE TO GEOMETRY But when we let ∆x → 0, the point Q will move along the curve and approachnearer and nearer to P , the secant will turn about P and approach the tangent as alimiting position, and we have also ∆y lim∆x→0 ∆x = lim∆x→0 tan φ = tan τ = slope of the tangent at P. dyHence , dx = slope of the tangent line P T . ThereforeTheorem 3.9.1. The value of the derivative at any point of a curve is equal to theslope of the line drawn tangent to the curve at that point. It was this tangent problem that led Leibnitz6 to the discovery of the DifferentialCalculus.Example 3.9.1. Find the slopes of the tangents to the parabola y = x2 at the 1vertex, and at the point where x = 2 . Solution. Differentiating by General Rule, (§3.7), we get dy y′ = = 2x = slope of tangent line at any point on curve. dx To find slope of tangent at vertex, substitute x = 0 in y ′ = 2x, giving dy = 0. dxTherefore the tangent at vertex has the slope zero; that is, it is parallel to the axisof x and in this case coincides with it. 1 To find slope of tangent at the point P , where x = 2 , substitute in y ′ = 2x,giving dy = 1; dxthat is, the tangent at the point P makes an angle of 45o with the axis of x. 6 Gottfried Wilhelm Leibnitz (1646-1716) was a native of Leipzig. His remarkable abilitieswere shown by original investigations in several branches of learning. He was first to publish hisdiscoveries in Calculus in a short essay appearing in the periodical Acta Eruditorum at Leipzigin 1684. It is known, however, that manuscripts on Fluxions written by Newton were already inexistence, and from these some claim Leibnitz got the new ideas. The decision of modern timesseems to be that both Newton and Leibnitz invented the Calculus independently of each other. Thenotation used today was introduced by Leibnitz. See frontispiece. 54
68.
3.10. EXERCISES12. The curves on a railway track are often made parabolic in form. Suppose that a track has the form of the parabola y = x2 (see Figure 3.2 in §3.9), the directions of the positive x-axis and positive y-axis being east and north respectively, and the unit of measurement 1 mile. If the train is going east when passing through the origin, in what direction will it be going (a) when 1 mi. east of the y-axis? 2 (Ans. Northeast.) 1 (b) when 2 mi. west of the y-axis? (Ans. Southeast.) √ (c) when 23 mi. east of the y-axis? (Ans. N. 30o E.) 1 (d) when 12 mi. north of the x-axis? (Ans. E. 30o S., or E. 30o N.)13. A street-car track has the form of the cubic y = x3 . Assume the same directions and unit as in the last example. If a car is going west when passing through the origin, in what direction will it be going 1 (a) when √ 3 mi. east of the y-axis? (Ans. Southwest.) 1 (b) when √ 3 mi. west of the y-axis? (Ans. Southwest.) 1 (c) when 2 mi. north of the x-axis? (Ans. S. 27o 43′ W.) (d) when 2 mi. south of the x-axis? (e) when equidistant from the x-axis and the y-axis? 57
70.
CHAPTER FOUR Rules for differentiating standard elementary forms4.1 Importance of General RuleThe General Rule for Differentiation, given in §3.7 of the last chapter, is funda-mental, being a step-by-step procedural implementation of the very definition ofa derivative. It should be stressed that the student should be thoroughly familiarwith this procedure. However, the process of applying the rule to examples in gen-eral is often either too tedious or too difficult. Consequently, special rules havebeen derived from the General Rule for differentiating certain standard forms offrequently occurring expressionss in order to facilitate process. It's convenient to express these special rules by means of formulas, a list ofwhich follows. The student should not only memorize each formula when de-duced, but should be able to state the corresponding rule in words. (The extratime it takes you to memorize the formulas will probably be repaid in the timesaved doing homework and exam problems correctly.) In these formulas u, v, andw denote differentiable functions of x. Formulas for differentiation dc =0 (4.1) dx dx =1 (4.2) dx 59
85.
4.12. DIFFERENTIATION OF INVERSE FUNCTIONSThe above formula is sometimes referred to as the chain rule for differentiation.If y = f (v) and v = g(x), the derivative of y with respect to x equals the productof the derivative of y with respect to v and the derivative of v with respect to x.4.12 Differentiation of inverse functionsLet y = f (x) be a given function of x. It is often possible in the case of functions considered in this book to solve thisequation for x, giving x = φ(y);that is, to consider y as the independent and x as the dependent variable. In thatcase f (x) and φ(y) are said to be inverse functions (and one often writes φ = f −1 ). When we wish to distinguish between the two it is customary to call the firstone given the direct function and the second one the inverse function. Thus, in theexamples which follow, if the second members in the first column are taken as thedirect functions, then the corresponding members in the second column will berespectively their inverse functions. √Example 4.12.1. • y = x2 + 1, x = ± y − 1. • y = ax , x = loga y. • y = sin x, x = arcsin y. The plot of the inverse function φ(y) is related to the plot of the function f (x) ina simple manner. The plot of f (x) over an interval (a, b) in which f is increasingis the same as the plot of φ(y) over (f (a), f (b)). The plot of y = f (x) is the"mirror image" of the plot of y = φ(x), reflected about the "diagonal line" y = x. √Example 4.12.2. If f (x) = x2 , for x > 0, and φ(y) = y, then the graphs are Now flip this graph about the 45o line: 74
111.
4.30. DIFFERENTIATION OF ARCCSC V Figure 4.9: The inverse secant sec−1 x using Sage .(equation (4.25) in §4.1 above).4.30 Differentiation of arccsc vLet y = arccsc v;then v = csc y.This function is defined for all values of v except those lying between −1 and +1,and is seen to be many-valued. To make the function single-valued, y is taken asthe smallest angle whose cosecant is v. This means that if v is positive, we confineourselves to points on the arc AB (Figure 4.11), y taking on values between 0 and 100
112.
4.31. EXAMPLE Figure 4.10: The standard branch of arcsec x using Sage .π2 ( π may be included); and if v is negative, we confine ourselves to points on the 2arc CD, y taking on values between −π and − π (− π may be included). 2 2 Differentiating with respect to y by 4.20 and following the method of the lastsection, we get dv d (arccscv) = − √ dx dx v v2 − 1(equation (4.26) in §4.1 above).4.31 ExampleDifferentiate the following: 1. y = arctan(ax2 ). 101
118.
4.32. IMPLICIT FUNCTIONS4.32 Implicit functionsWhen a relation between x and y is given by means of an equation not solved fory, then y is called an implicit function of x. For example, the equation x2 − 4y = 0defines y as an implicit function of x. Evidently x is also defined by means of thisequation as an implicit function of y. Similarly, x 2 + y 2 + z 2 − a2 = 0defines anyone of the three variables as an implicit function of the other two. It is sometimes possible to solve the equation defining an implicit function forone of the variables and thus change it into an explicit function. For instance, 2the above two implicit functions may be solved for y, giving y = x and y = √ 4± a2 − x2 − z 2 ; the first showing y as an explicit function of x, and the secondas an explicit function of x and z. In a given case, however, such a solution maybe either impossible or too complicated for convenient use. The two implicit functions used in this section for illustration may be respec-tively denoted by f (x, y) = 0 and F (x, y, z) = 0.4.33 Differentiation of implicit functionsWhen y is defined as an implicit function of x by means of an equation in the form f (x, y) = 0, (4.33)it was explained in the last section how it might be inconvenient to solve for y interms of x; that is, to find y as an explicit function of x so that the formulas wehave deduced in this chapter may be applied directly. Such, for instance, wouldbe the case for the equation ax6 + 2x3 y − y 7 x − 10 = 0. (4.34)We then follow the rule: Differentiate, regarding y as a function of x, and put the result equal to zero 7 .That is, 7 Only corresponding values of x and y which satisfy the given equation may be substituted inthe derivative. 107
126.
CHAPTER FIVE Simple applications of the derivative5.1 Direction of a curveIt was shown in §3.9, that if y = f (x)is the equation of a curve (see Figure 5.2), then dy = tan τ = slope of line tangent to the curve at any point P. dxExample 5.1.1. A group of hikers are climbing a hill whose height is describedby the graph of h(x) = −x4 + 29x3 − 290x2 + 1200x.Show that the hikers are climbing downhill when x = 5. This can be verified "by hand" by computing h′ (5) and checking that it is nega-tive (see also the plot in Figure 5.1), or using Sage : Sagesage: x = var("x")sage: h = -xˆ4 + 29*xˆ3 - 290*xˆ2 + 1200*xsage: Dh = h.diff(); Dh-4*xˆ3 + 87*xˆ2 - 580*x + 1200 115
127.
5.1. DIRECTION OF A CURVEsage: Dh(5)-25sage: plot(h,0,15)The output of the above plot command is in Figure 5.1. Figure 5.1: The graph of y = −x4 + 29x3 − 290x2 + 1200x. The direction of a curve at any point is defined to be the same as the direction ofthe line tangent to the curve at that point. From this it follows at once that dy = tan τ = slope of the curve at any point P. dxAt a particular point whose coordinates are known we write dy = slope of the curve (or tangent) at point (x1 , y1 ). dx x=x1 ,y=y1At points such as D or F or H where the curve (or tangent) is parallel to the x-axis, dyτ = 0o , therefore dx = 0 (see Figure 5.2 for the notation). 116
128.
5.1. DIRECTION OF A CURVE Figure 5.2: The derivative = slope of line tangent to the curve. At points such as A, B, G, where the curve (or tangent) is perpendicular to the dyx-axis, τ = 90o , therefore dx = ∞. At points such as E, where the curve is rising (moving from left to right oncurve), dy τ = an acute angle; therefore = a positive number. dxThe curve (or tangent) has a positive slope • to the left of B, • between D and F, and • to the right of G,in Figure 5.2. At points such as C, where the curve is falling, 117
132.
5.2. EXERCISES m1 − m2 tan θ = , 1 + m1 m2 − 1 +3by item 55, §12.1. Substituting, tan θ = 1+ 3 = 1; therefore θ = π/4 = 45o . 2 2This is also the angle of intersection at the point (1, −2).5.2 ExercisesThe corresponding figure should be drawn in each of the following examples: x 1. Find the slope of y = 1+x2 at the origin. Ans. 1 = tan τ . 2. What angle does the tangent to the curve x2 y 2 = a3 (x + y) at the origin make with the x-axis? Ans. τ = 135o = 3π/4. 3. What is the direction in which the point generating the graph of y = 3x2 −x tends to move at the instant when x = 1? Ans. Parallel to a line whose slope is 5. dy 4. Show that dx (or slope) is constant for a straight line. 5. Find the points where the curve y = x3 − 3x2 − 9x + 5 is parallel to the x-axis. Ans. x = 3, x = −1. 6. At what point on y 2 = 2x3 is the slope equal to 3? Ans. (2, 4). 7. At what points on the circle x2 + y 2 = r2 is the slope of the tangent line equal to − 3 ? 4 Ans. ± 3r , ± 4r 5 5 8. Where will a point moving on the parabola y = x2 − 7x + 3 be moving parallel to the line y = 5x + 2? Ans. (6, −3). 121
133.
5.2. EXERCISES 9. Find the points where a particle moving on the circle x2 + y 2 = 169 moves perpendicular to the line 5x + 12y = 60. Ans. (±12, ∓5). 10. Show that all the curves of the system y = log kx have the same slope; i.e. the slope is independent of k. 11. The path of the projectile from a mortar cannon lies on the parabola y = 2x−x2 ; the unit is 1 mile, the x-axis being horizontal and the y-axis vertical, and the origin being the point of projection. Find the direction of motion of the projectile (a) at instant of projection; 3 (b) when it strikes a vertical cliff 2 miles distant. (c) Where will the path make an inclination of 45o = π/4 with the horizon- tal? (d) Where will the projectile travel horizontally? 1 Ans. (a) arctan 2; (b) 135o = 3π/4; (c) ( 2 , 3 ); (d) (1, 1). 4 12. If the cannon in the preceding example was situated on a hillside of inclina- tion 45o = π/4, at what angle would a shot fired up strike the hillside? Ans. 45o = π/4. 13. At what angles does a road following the line 3y − 2x − 8 = 0 intersect a railway track following the parabola y 2 = 8x? Ans. arctan 5 , and arctan 1 . 1 8 14. Find the angle of intersection between the parabola y 2 = 6x and the circle x2 + y 2 = 16. √ Ans. arctan 5 3. 3 x2 y2 15. Show that the hyperbola x2 − y 2 = 5 and the ellipse 18 + 8 = 1 intersect at right angles. x3 16. Show that the circle x2 + y 2 = 8ax and the cissoid y 2 = 2a−x (a) are perpendicular at the origin; (b) intersect at an angle of 45o = π/4 at two other points. 122
134.
5.3. EQUATIONS OF TANGENT AND NORMAL LINES 17. Find the angle of intersection of the parabola x2 = 4ay and the Witch of 3 Agnesi, y = x28a 2 . +4a Ans. arctan 3 = 71o 33′ = 1.249.... For the interesting history of this curve, see for example 18. Show that the tangents to the Folium of Descartes, x3 + y 3 = 3axy at the points where it meets the parabola y 2 = ax are parallel to the y-axis. For some history of this curve, see for example 19. At how many points will a particle moving on the curve y = x3 −2x2 +x−4 be moving parallel to the x-axis? What are the points? Ans. Two; at (1, −4) and ( 3 , − 104 ). 1 27 20. Find the angle at which the parabolas y = 3x2 −1 and y = 2x2 +3 intersect. 4 Ans. arctan 97 . 21. Find the relation between the coefficients of the conics a1 x2 + b1 y 2 = 1 and a2 x2 + b2 y 2 = 1 when they intersect at right angles. 1 1 1 1 Ans. a1 − b1 = b2 − b2 .5.3 Equations of tangent and normal linesThis section will discuss equations of tangent and normal lines, lengths of subtan-gent and subnormal, and rectangular coordinates. The equation of a straight line passing through the point (x1 , y1 ) and having theslope m is y − y1 = m(x − x1 )(this is item 54, §12.1). If this line is tangent to the curve y = f (x) at the point P = (x1 , y1 ) (see Figure5.5 to visualize how these can be situated in relationship to the graph of the curve),then from §5.1, 123
137.
5.4. EXERCISES The student is advised to get the lengths of the tangent and of the normal directlyfrom the figure rather than by using these equations. When the length of subtangent or subnormal at a point on a curve is determined,the tangent and normal may be easily constructed.5.4 Exercises 1. Find the equations of tangent and normal, lengths of subtangent, subnormal x3 tangent, and normal at the point (a, a) on the cissoid y 2 = 2a−x . x3 Figure 5.6: Graph of cissoid y 2 = 2a−x with a = 1. dy 3ax2 −x3 Solution. dx = y(2a−x)2 . Hence dy1 dy 3a3 − a3 = = =2 dx1 dx x=a,y=a a(2a − a)2 126
141.
5.5. PARAMETRIC EQUATIONS OF A CURVE 13. Show that in the equilateral hyperbola 2xy = a2 the area of the triangle formed by a tangent and the coordinate axes is constant and equal to a2 . 14. Find equations of tangents and normals to the curve y 2 = 2x2 − x3 at the points where x = 1. Ans. At (1, 1), 2y = x + 1, y + 2x = 3. At (1, −1), 2y = −x − 1, y − 2x = −3. 1 1 15. Show that the sum of the intercepts of the tangent to the parabola x 2 + y 2 = 1 a2 . 16. Find the equation of tangent to the curve x2 (x+y) = a2 (x−y) at the origin. 2 2 2 17. Show that for the hypocycloid x 3 + y 3 = a 3 that portion of the tangent included between the coordinate axes is constant and equal to a. (This curve is parameterized by x = a cos(t)3 , y = a sin(t)3 , 0 ≤ t ≤ 2π. Parametric equations shall be discussed in the next section.) x 18. Show that the curve y = ae c has a constant subtangent.5.5 Parametric equations of a curveLet the equation of a curve be F (x, y) = 0. (5.7)If x is given as a function of a third variable, t say, called a parameter, then byvirtue of (5.7) y is also a function of t, and the same functional relation (5.7)between x and y may generally be expressed by means of equations in the form x = f (t), (5.8) y = g(t)each value of t giving a value of x and a value of y. Equations (5.8) are calledparametric equations of the curve. If we eliminate t between equations (5.8), it isevident that the relation (5.7) must result.Example 5.5.1. For example, take equation of circle 130
143.
5.5. PARAMETRIC EQUATIONS OF A CURVE Since from (5.8) y is given as a function of t, and t as a function of x, we have dy dy dt dx = dt · dx by (4.27) dy 1 = dt · dx by (4.28) dtthat is, dy dy dt g ′ (t) = dx = . (5.10) dx dt f ′ (t)Hence, if the parametric equations of a curve are given, we can find equations oftangent and normal, lengths of subtangent and subnormal at a given point on the dycurve, by first finding the value of dx at that point from (5.10) and then substitutingin formulas (5.1), (5.2), (5.3), (5.4) of the last section.Example 5.5.3. Find equations of tangent and normal, lengths of subtangent andsubnormal to the ellipse x = a cos φ, (5.11) y = b sin φ,at the point where φ = π . 4 As in Figure 5.7 draw the major and minor auxiliary circles of the ellipse.Through two points B and C on the same radius draw lines parallel to the axesof coordinates. These lines will intersect in a point P = (x, y) on the ellipse, be-cause x = OA = OB cos φ = a cos φ and y = AP = OD = OC sin φ = b sin φ,or, x = cos φ and y = sin φ. Now squaring and adding, we get a b x2 y 2 2 + 2 = cos2 φ + sin2 φ = 1, a bthe rectangular equation of the ellipse. φ is sometimes called the eccentric angleof the ellipse at the point P. dx dy Solution. The parameter being φ, dφ = −a sin φ, dφ = b cos φ. π a b Substituting φ = 4 in the given equations (5.11), we get √2 , √2 as the point dy bof contact. Hence dx x=x1 ,y=y1 = − a . Substituting in (5.1), b b a y−√ =− x− √ , 2 a 2 √or, bx + ay = 2ab, the equation of tangent. Substituting in (5.2), 132
151.
5.7. ANGLE BETWEEN RADIUS VECTOR AND TANGENT 21. The hyperbolic spiral x = a cos t t a y = t sin t5.7 Angle between radius vector and tangentAngle between the radius vector drawn to a point on a curve and the tangent to thecurve at that point. Let the equation of the curve in polar coordinates be ρ = f (θ). Let P be any fixed point (ρ, θ) on the curve. If θ, which we assume as the inde-pendent variable, takes on an increment ∆θ, then ρ will take on a correspondingincrement ∆ρ.Figure 5.9: Angle between the radius vector drawn to a point on a curve and thetangent to the curve at that point. Denote by Q the point (ρ+∆ρ, θ+∆θ), as in Figure 5.9, Draw PR perpendicularto OQ where R is a point at a distance of ρ cos ∆θ from the origin. Then OQ =ρ + ∆ρ, P R = ρ sin ∆θ, and OR = ρ cos ∆θ. Also, 140
155.
5.8. LENGTHS OF POLAR SUBTANGENT AND POLAR SUBNORMAL5.8 Lengths of polar subtangent and polar subnor- malDraw a line NT through the origin perpendicular to the radius vector of the pointP on the curve. If PT is the tangent and PN the normal to the curve at P, then7 Figure 5.11: The polar subtangent and polar subnormal. OT = length of polar subtangent,and ON = length of polar subnormal 7 dθ When θ increases with ρ, dρ is positive and ρ is an acute angle, as in Figure 5.11. Then thesubtangent OT is positive and is measured to the right of an observer placed at O and looking along dθOP. When dρ is negative, the subtangent is negative and is measured to the left of the observer. 144
163.
5.12. APPLICATIONS OF THE DERIVATIVE IN MECHANICS 15. Show that the condition that the equation x3 + 3qx + r = 0 shall have a double root is 4q 3 + r2 = 0. 16. Show that the condition that the equation x3 + 3px2 + r = 0 shall have a double root is r(4p3 + r) = 0.5.12 Applications of the derivative in mechanicsIncluded also are applications to velocity and rectilinear motion. Consider the motion of a point P on the straight line AB. Figure 5.13: Illustration of rectilinear motion. Let s be the distance measured from some fixed point as A to any position ofP, and let t be the corresponding elapsed time. To each value of t corresponds aposition of P and therefore a distance (or space) s. Hence s will be a function oft, and we may write s = f (t)Now let t take on an increment ∆t; then s takes on an increment8 ∆s, and 8 s being the space or distance passed over in the time ∆t. 152
164.
5.12. APPLICATIONS OF THE DERIVATIVE IN MECHANICS ∆s = the average velocity (5.20) ∆tof P during the time interval ∆t. If P moves with uniform motion, the aboveratio will have the same value for every interval of time and is the velocity at anyinstant. For the general case of any kind of motion, uniform or not, we define the velocity(or, time rate of change of s) at any instant as the limit of the ratio ∆s as ∆t ∆tapproaches the limit zero; that is, ∆s v = lim , ∆t→0 ∆tor ds v= (5.21) dtThe velocity is the derivative of the distance (= space) with respect to the time. To show that this agrees with the conception we already have of velocity, let usfind the velocity of a falling body at the end of two seconds. By experiment it has been found that a body falling freely from rest in a vacuumnear the earth's surface follows approximately the law s = 16.1t2 (5.22)where s = space fallen in feet, t = time in seconds. Apply the General Rule, §3.7,to (5.22). FIRST STEP. s + ∆s = 16.1(t + ∆t)2 = 16.1t2 + 32.2t · ∆t + 16.1(∆t)2 . SECOND STEP. ∆s = 32.2t · ∆t + 16.1(∆t)2 . THIRD STEP. ∆s = 32.2t + 16.1∆t = average velocity throughout the time ∆tinterval ∆t. Placing t = 2, ∆s = 64.4 + 16.1∆t (5.23) ∆twhich equals the average velocity throughout the time interval ∆t after two sec-onds of falling. Our notion of velocity tells us at once that (5.23) does not give usthe actual velocity at the end of two seconds; for even if we take ∆t very small, 1 1say 100 or 1000 of a second, (5.23) still gives only the average velocity during thecorresponding small interval of time. But what we do mean by the velocity at 153
165.
5.13. COMPONENT VELOCITIES. CURVILINEAR MOTIONthe end of two seconds is the limit of the average velocity when ∆t diminishestowards zero; that is, the velocity at the end of two seconds is from (5.23), 64.4 ft.per second. Thus even the everyday notion of velocity which we get from experience in-volves the idea of a limit, or in our notation ∆s v = lim = 64.4 f t./sec. ∆t→0 ∆t The above example illustrates well the notion of a limiting value. The studentshould be impressed with the idea that a limiting value is a definite, fixed value, notsomething that is only approximated. Observe that it does not make any differencehow small 16.1∆t may be taken; it is only the limiting value of 64.4 + 16.1∆t,when ∆t diminishes towards zero, that is of importance, and that value is exactly64.4.5.13 Component velocities. Curvilinear motionThe coordinates x and y of a point P moving in the xy-plane are also functionsof time, and the motion may be defined by means of two equations9 , x = f (t),y = g(t). These are the parametric equations of the path (see §5.5). The horizontal component10 vx of v is the velocity along the x-axis of the pro-jection M of P, and is therefore the time rate of change of x. Hence, from (5.21),when s is replaced by x, we get dx vx = . (5.24) dtIn the same way we get the vertical component, or time rate of change of y, dy vy = . (5.25) dtRepresenting the velocity and its components by vectors, we have at once fromthe figure v 2 = vx 2 + vy 2 , 9 The equation of the path in rectangular coordinates may often be found by eliminating tbetween their equations. 10 The direction of v is along the tangent to the path. 154
167.
5.14. ACCELERATION. RECTILINEAR MOTION5.14 Acceleration. Rectilinear motionIn general, v will be a function of t. Now let t take on an increment ∆t, then vtakes on an increment ∆v, and ∆v is the average acceleration of P during the time ∆tinterval ∆t. We define the acceleration a at any instant as the limit of the ratio ∆v ∆tas ∆t approaches the limit zero; that is, ∆v a = lim , ∆t→0 ∆tor, dv a= (5.28) dtThe acceleration is the derivative of the velocity with respect to time.5.15 Component accelerations. Curvilinear motionIn treatises on Mechanics it is shown that in curvilinear motion the accelerationis not, like the velocity, directed along the tangent, but toward the concave side,of the path of motion. It may be resolved into a tangential component, at , and anormal component, an where dv v2 at = ; an = . (5.29) dt R(R is the radius of curvature. See §11.5.) The acceleration may also be resolved into components parallel to the axes ofthe path of motion. Following the same plan used in §5.13 for finding componentvelocities, we define the component accelerations parallel to the x-axis and y-axis, dvx dvy ax = ; ay = . (5.30) dt dtAlso, 2 2 dvx dvy a= + , (5.31) dt dtwhich gives the magnitude of the acceleration at any instant. 156
168.
5.16. EXAMPLES5.16 Examples 1. By experiment it has been found that a body falling freely from rest in a vacuum near the earth's surface follows approximately the law s = 16.1t2 , where s = space (height) in feet, t = time in seconds. Find the velocity and acceleration (a) at any instant; (b) at end of the first second; (c) at end of the fifth second. Solution. We have s = 16.1t2 . (a) Differentiating, ds = 32.2t, or, from (5.21), v = 32.2t ft./sec. Differen- dt tiating again, dv = 32.2, or, from (5.28), a = 32.2 ft./(sec.)2 , which tells dt us that the acceleration of a falling body is constant; in other words, the velocity increases 32.2 ft./sec. every second it keeps on falling. (b) To find v and a at the end of the first second, substitute t = 1 to get v = 32.2 ft./sec., a = 32.2 ft./(sec.)2 . (c) To find v and a at the end of the fifth second, substitute t = 5 to get v = 161 ft./sec., a = 32.2 ft./(sec.)2 . 2. Neglecting the resistance of the air, the equations of motion for a projectile are x = v0 cos φ · t, y = v0 sin φ · t − 16.1t2 ; where v0 = initial velocity, φ = angle of projection with horizon, t = time of flight in seconds, x and y being measured in feet. Find the velocity, acceleration, component velocities, and component accelerations (a) at any instant; (b) at the end of the first second, having given v0 = 100 ft. per sec., φ = 300 = π/6; (c) find direction of motion at the end of the first second. 157
171.
5.16. EXAMPLES 7. A train left a station and in t hours was at a distance (space) of s = t3 + 2t2 + 3t miles from the starting point. Find its acceleration (a) at the end of t hours; (b) at the end of 2 hours. Ans. (a) a = 6t + 4; (b) a = 16 miles/(hour)2 . 8. In t hours a train had reached a point at the distance of 1 t4 − 4t3 + 16t2 4 miles from the starting point. (a) Find its velocity and acceleration. (b) When will the train stop to change the direction of its motion? (c) Describe the motion during the first 10 hours. Ans. (a) v = t3 − 12t2 + 32t, a = 3t2 − 24t + 32; (b) at end of fourth and eighth hours; (c) forward first 4 hours, backward the next 4 hours, forward again after 8 hours. 9. The space in feet described in t seconds by a point is expressed by the formula s = 48t − 16t2 . 3 Find the velocity and acceleration at the end of 2 seconds. Ans. v = 0,a = −32 ft./(sec.)2 . 10. Find the acceleration, having given (a) v = t2 + 2t; t = 3. Ans. a = 8. (b) v = 3t − t3 ; t = 2. Ans. a = −9. t (c) v = 4 sin 2 ; t = π . 3 √ Ans. a = 3. 160
172.
5.16. EXAMPLES (d) v = r cos 3t; t = π . 6 Ans. a = −3r. (e) v = 5e2t ; t = 1. Ans. a = 10e2 .11. At the end of t seconds a body has a velocity of 3t2 + 2t ft. per sec.; find its acceleration (a) in general; (b) at the end of 4 seconds. Ans. (a) a = 6t + 2 ft./(sec.)2 ; (b) a = 26 ft./(sec.)212. The vertical component of velocity of a point at the end of t seconds is vy = 3t2 − 2t + 6 in ft. per sec. Find the vertical component of acceleration (a) at any instant; (b) at the end of 2 seconds. Ans. (a) ay = 6t − 2; (b) 10 ft./(sec.)2 .13. If a point moves in a fixed path so that √ s= t, show that the acceleration is negative and proportional to the cube of the velocity.14. If the distance travelled at time t is given by s = c1 et + c2 e−t , for some constants c1 and c2 , show that the acceleration is always equal in magnitude to the space passed over.15. If a point referred to rectangular coordinates moves so that x = a1 + a2 cos t, y = b1 + b2 sin t, for some constants ai and bi ,show that its velocity has a constant magnitude. 161
174.
5.17. APPLICATION: NEWTON'S METHOD5.17 Application: Newton's method11 Newton's method (also known as the Newton-Raphson method) is an efficientalgorithm for finding approximations to the zeros (or roots) of a real-valued func-tion. As such, it is an example of a root-finding algorithm. It produces iterativelya sequence of approximations to the root. It can also be used to find a minimumor maximum of such a function, by finding a zero in the function's first derivative.5.17.1 Description of the methodThe idea of the method is as follows: one starts with an initial guess which isreasonably close to the true root, then the function is approximated by its tangentline (which can be computed using the tools of calculus), and one computes the x-intercept of this tangent line (which is easily done with elementary algebra). Thisx-intercept will typically be a better approximation to the function's root than theoriginal guess, and the method can be iterated. Suppose f : [a, b] → R is a differentiable function defined on the interval [a, b]with values in the real numbers R. The formula for converging on the root can beeasily derived. Suppose we have some current approximation xn . Then we canderive the formula for a better approximation, xn+1 by referring to the diagram onthe right. We know from the definition of the derivative at a given point that it isthe slope of a tangent at that point. That is rise ∆y f (xn ) − 0 0 − f (xn ) f ′ (xn ) = = = = . run ∆x xn − xn+1 (xn+1 − xn )Here, f ′ denotes the derivative of the function f . Then by simple algebra we canderive f (xn ) xn+1 = xn − . f ′ (xn )We start the process off with some arbitrary initial value x0 . (The closer to thezero, the better. But, in the absence of any intuition about where the zero mightlie, a "guess and check" method might narrow the possibilities to a reasonablysmall interval by appealing to the intermediate value theorem.) The method will 11 This section uses material modified from Wikipedia [N]. 163
175.
5.17. APPLICATION: NEWTON'S METHODusually converge, provided this initial guess is close enough to the unknown zero,and that f ′ (x0 ) = 0. Furthermore, for a zero of multiplicity 1, the convergenceis at least quadratic (see rate of convergence) in a neighbourhood of the zero,which intuitively means that the number of correct digits roughly at least doublesin every step. More details can be found in the analysis section below.Example 5.17.1. Consider the problem of finding the positive number x withcos(x) = x3 . We can rephrase that as finding the zero of f (x) = cos(x) − x3 . Wehave f ′ (x) = − sin(x) − 3x2 . Since cos(x) ≤ 1 for all x and x3 > 1 for x > 1,we know that our zero lies between 0 and 1. We try a starting value of x0 = 0.5. f (x0 ) 3 cos(0.5)−0.5 x1 = x0 − f ′ (x0 ) = 0.5 − − sin(0.5)−3×0.52 = 1.112141637097 f (x1 ) x2 = x1 − f ′ (x1 ) = 0.909672693736 f (x2 ) x3 = x2 − f ′ (x2 ) = 0.867263818209 f (x3 ) x4 = x3 − f ′ (x3 ) = 0.865477135298 f (x4 ) x5 = x4 − f ′ (x4 ) = 0.865474033111 f (x5 ) x6 = x5 − f ′ (x5 ) = 0.865474033102 The correct digits are underlined in the above example. In particular, x6 is cor-rect to the number of decimal places given. We see that the number of correctdigits after the decimal point increases from 2 (for x3 ) to 5 and 10, illustrating thequadratic convergence.5.17.2 AnalysisSuppose that the function f has a zero at a, i.e., f (a) = 0. If f is continuously differentiable and its derivative does not vanish at a, thenthere exists a neighborhood of a such that for all starting values x0 in that neigh-borhood, the sequence {xn } will converge to a. In practice this result is "local" and the neighborhood of convergence is notknown a priori, but there are also some results on "global convergence." Forinstance, given a right neighborhood U of a, if f is twice differentiable in U andif f ′ = 0, f · f ′′ > 0 in U , then, for each x0 ∈ U the sequence xk is monotonicallydecreasing to a. 164
176.
5.17. APPLICATION: NEWTON'S METHOD5.17.3 FractalsFor complex functions f : C → C, however, Newton's method can be directlyapplied to find their zeros. For many complex functions, the boundary of the set(also known as the basin of attraction) of all starting values that cause the methodto converge to a particular zero is a fractal12 For example, the function f (x) = x5 − 1, x ∈ C, has five roots, equally spacedaround the unit circle in the complex plane. If x0 is a starting point which con-verges to the root at x = 1, color x0 yellow. Repeat this using four other colors(blue, red, green, purple) for the other four roots of f . The resulting image is inFigure 5.15.Figure 5.15: Basins of attraction for x5 − 1 = 0; darker means more iterations toconverge. 12 The definition of a fractal would take us too far afield. Roughly speaking, it is a geometricalobject with certain self-similarity properties [F]. 165
188.
CHAPTER SEVEN Maxima, minima and inflection points7.1 IntroductionMany practical problems occur where we have to deal with functions that havea maximum value (or a minimum value) and it is important to know where theextreme values of the function occur.Example 7.1.1. A wooden box is to be built to contain 108 ft3 . It is to have anopen top and a square base. What must be its dimensions in order that the amountof material required shall be a minimum; that is, what dimensions will make thecost the least?Let x denote the length of side of square base in feet, and y denote the height ofbox. Since the volume of the box is given, y may be found in terms of x. Thusvolume = x2 y = 108, so y = 108 . Let M denote the number of square feet of x2lumber required as a function of x. We compute M explicitly as follows: area of base = x2 sq. ft., area of four sides = 4xy = 432 sq. ft. xHence 432 M = M (x) = x2 + xis a formula giving the number of square feet required in any such box having acapacity of 108 ft3 . Draw a graph of M (x). 177
189.
7.1. INTRODUCTIONFigure 7.1: A box with square x × x base, height y = 108/x2 , and fixed volume. 432 Figure 7.2: Sage plot of y = x2 + x , 1 < x < 10. What do we learn from the graph? (a) If the box is carefully drawn, we may measure the ordinate corresponding toany length (= x) of the side of the square base and so determine the number of 178
190.
7.1. INTRODUCTIONsquare feet of lumber required. (b) There is one horizontal tangent (RS). The ordinate from its point of contact Tis less than any other ordinate. Hence this discovery: One of the boxes evidentlytakes less lumber than any of the others. In other words, we may infer that thefunction defined by M = M (x) has a minimum value. Let us find this point onthe graph exactly, using our Calculus. Differentiating M (x) to get the slope at anypoint, we have dM 432 = 2x − 2 . dx xAt the lowest point T the slope will be zero. Hence 432 2x − = 0; x2that is, when x = 6 the least amount of lumber will be needed. Substituting in M (x), we see that this is M = 108 sq. ft. In addition to the graph, the fact that a least value of M exists can be intuitivelydeduced by the following argument. Let the base increase from a very smallsquare to a very large one. In the former case the height must be very great andtherefore the amount of lumber required will be large. In the latter case, while theheight is small, the base will take a great deal of lumber. Our intuition tells thatM therefore varies from a large value, decreases for a while, then increases againto another large value. It follows, then, that the graph of y = M (x) must have a"lowest" point corresponding to the dimensions which require the least amount oflumber, and therefore would involve the least cost. Here is how to compute the critical points of M in Sage : Sagesage: x = var("x")sage: M = xˆ2 + 432/xsage: solve(M.diff(x)==0,x)[x == 3*sqrt(3)*I - 3, x == -3*sqrt(3)*I - 3, x == 6]This says that (x2 + 432/x)′ = 0 has three roots, but only one real root - the onereported above at x = 6. 179
191.
7.1. INTRODUCTION Figure 7.3: A rectangle with circumscribed circle.Example 7.1.2. For instance, suppose that it is required to find the dimensions ofthe rectangle of greatest area that can be inscribed in a circle of radius 5 inches.Consider the circle in Figure 7.3:Inscribe any rectangle, as BCDE, where CD is the base and DE is the height. Let √CD = x, so DE = 100 − x2 , and the area of the rectangle is evidently √ A = A(x) = x 100 − x2 . 180
192.
7.1. INTRODUCTIONThat a rectangle of maximum area must exist may be seen as follows: Let the√ CD (= x) increase to 10 inches (the diameter); then the altitude DE (=base 100 − x2 ) will decrease to zero and the area will become zero. Now let the basedecrease to zero; then the altitude will increase to 10 inches and the area will againbecome zero. It is therefore intuitively evident that there exists in-between theseextremes a rectangle of greatest area. By a careful study of the figure we mightsuspect that when the rectangle becomes a square its area would be the greatest,but this would be mere guesswork1 . A better way would be to plot the graph ofthe function y = A(x) and note its behavior. To aid us in drawing the graph ofA(x), we observe that (a) from the nature of the problem it is clear that x and A(x) must both be positive; and (b) the values of x range from zero to 10 inclusive.Now draw the graph (we have used Sage in Figure 7.4). What do we learn from the graph? (a) If the rectangle is carefully drawn, we may approximate the area of the rectangle corresponding to any value x by measuring the length of the cor- responding ordinate. For example, when x = 3 inches, then the area is 9 about A(x) ≈ 28.6 inches2 ; and when x 2 inches, then the area is about A(x) ≈ 39.8 inches2 . (b) There is one horizontal tangent to the curve y = A(x). The y-coordinate at the point T there this tangent contacts the curve is greaterthan any other y-coordinate on the curve. We deduce from this that one of theinscribed rectangles has a greater area than any of the others. In other words, wemay infer from this that the function defined by y = A(x) has a maximum value.We can find this value very easily to using calculus. We observed that at T thetangent was horizontal, hence the slope will be zero at that point (Example 5.1.2).To find the x-coordinate of T we find the first derivative of A(x), set it equal tozero, and solve for x: 1 Reasoning that "by symmetry we must have base = height" happens to work in this particularexample (as we will see) but, surprisingly enough, does not hold in general. 181
193.
7.1. INTRODUCTION Figure 7.4: The area of a rectangle with fixed circumscribed circle. √ A = x 100 − x2 , dA 100−2x2 dx = √100−x2 , 100−2x2 √ 100−x2 = 0. √ √ √Solving this gives x = 5 2. Substituting back, we get DE = 100 − x2 = 5 2.Hence the rectangle √ maximum area inscribed in the circle is a square of area of √A = CD × DE = 5 2 × 5 2 = 50 square inches. The length of HT is therefore50. We will now proceed to the treatment in detail of the subject of maxima andminima. 182
194.
7.2. INCREASING AND DECREASING FUNCTIONS7.2 Increasing and decreasing functionsA function is said to be increasing when it increases as the variable increases anddecreases as the variable decreases. A function is said to be decreasing when itdecreases as the variable increases and increases as the variable decreases. The graph of a function indicates plainly whether it is increasing or decreasing.Example 7.2.1. (1) Consider the function y = ax , a > 1, whose graph is shown in Figure 7.5. Figure 7.5: Sage plot of y = 2x , −1 < x < 1. As we move along the curve from left to right the curve is rising; that is, as x increases the function y = ax always increases. Therefore ax (a > 1) is an increasing function for all values of x. (2) On the other hand, consider the function (a − x)3 whose graph (Figure 7.6) is the locus of the equation y = (a − x)3 . 183
195.
7.2. INCREASING AND DECREASING FUNCTIONS Figure 7.6: Sage plot of y = (2 − x)3 , 1 < x < 3. Now as we move along the curve from left to right the curve is falling; that is, as x increases, the function y = (a − x)3 always decreases. Hence (a − x)3 is a decreasing function for all values of x. (3) That a function may be sometimes increasing and sometimes decreasing is shown by the graph (Figure 7.7) of y = 2x3 − 9x2 + 12x − 3. Figure 7.7: Sage plot of y = 2x3 − 9x2 + 12x − 3, 0 < x < 3. As we move along the curve from left to right the curve rises until we reach 184
196.
7.3. TESTS FOR DETERMINING WHEN A FUNCTION IS INCREASING OR DECREASING the point when x = 1, then it falls from that point to the point when x = 2, and to the right of x = 2 it is always increasing. Hence (a) from x = −∞ to x = 1 the function is increasing; (b) from x = 1 to x = 2 the function is decreasing; (c) from x = 2 to x = +∞ the function is increasing. The student should study the curve carefully in order to note the behavior of the function when x = 1 and x = 2. At x = 1 the function ceases to increase and commences to decrease; at x = 2, the reverse is true. At x = 1 and at x = 2 the tangent to the curve is parallel to the x-axis, and therefore the slope is zero.7.3 Tests for determining when a function is increas- ing or decreasingIt is evident from Figure 7.7 that at a point where a function y = f (x)is increasing, the tangent in general makes an acute angle with the x-axis, so dy slope = tan τ = dx = f ′ (x) = a positive number.Similarly, at a point where a function is decreasing, the tangent in general makesan obtuse angle with the x-axis; therefore2 dy slope = tan τ = dx = f ′ (x) = a negative number.It follows from this that in order for a differentiable function to change from anincreasing to a decreasing function, or vice versa, it is a necessary and sufficientcondition that the first derivative changes sign. But this can only happen for acontinuous derivative by passing through the value zero. Thus in Figure 7.7 as 2 Conversely, for any given value of x, if f ′ (x) > 0, then f (x) is increasing; if f ′ (x) < 0,then f (x) is decreasing. When f ′ (x) = 0, we cannot decide without further investigation whetherf (x) is increasing or decreasing. 185
197.
7.4. MAXIMUM AND MINIMUM VALUES OF A FUNCTIONwe pass along the curve the derivative (= slope) changes sign at the points wherex = 1 and x = 2. In general, then, we have at these "turning points," dy = f ′ (x) = 0. dxA value of y = f (x) satisfying this condition is called a critical point of thefunction f (x).Remark 7.3.1. The derivative is continuous in nearly all our important appli-cations, but it is interesting to note the case when the derivative (= slope) of ychanges sign by "passing through ∞" (that is, its reciprocal 1/y passes throughthe value zero). This would evidently happen at the points on a curve where thetangent is perpendicular to the x-axis. At such "turning points" we have dy = f ′ (x) = inf; dxor, what amounts to the same thing, 1 = 0. f ′ (x)For example, the function y = 1/x2 has a "turning point" at x = 0, where theslope is infinite but the function changes from being increasing (for x < 0) todecreasing (for x > 0).7.4 Maximum and minimum values of a functionA maximum value of a function is one that is greater than any values immediatelypreceding or following. A minimum value of a function is one that is less than anyvalues immediately preceding or following. For example, in Figure 7.7, it is clear that the function has a maximum value(y = 2) when x = 1, and a minimum value (y = l) when x = 2. The student should observe that a maximum value is not necessarily the greatestpossible value of a function nor a minimum value the least. For in Figure 7.7 it isseen that the function (= y) has values to the right of x = 1 that are greater thanthe maximum 2, and values to the left of x = 1 that are less than the minimum 1. A function may have several maximum and minimum values. Suppose thatFigure 7.8 represents the graph of a function f (x). 186
198.
7.4. MAXIMUM AND MINIMUM VALUES OF A FUNCTION Figure 7.8: A continuous function. At B, F the function is at a local maximum, and at D, G a minimum. Thatsome particular minimum value of a function may be greater than some particularmaximum value is shown in the figure, the minimum value at D being greater thanthe maximum value at G. At the ordinary critical points D, F, H the tangent (or curve) is parallel to thex-axis; therefore dy slope = = f ′ (x) = 0. dxAt the exceptional critical points A, B, G the tangent (or curve) is perpendicularto the x-axis, giving dy slope = = f ′ (x) = ∞. dx One of these two conditions is then necessary in order that the function shallhave a maximum or a minimum value. But such a condition is not sufficient; forat H the slope is zero and at A it is infinite, and yet the function has neither amaximum nor a minimum value at either point. It is necessary for us to know,in addition, how the function behaves in the neighborhood of each point. Thusat the points of maximum value, B, F, the function changes from an increasingto a decreasing function, and at the points of minimum value, D, G, the function 187
199.
7.4. MAXIMUM AND MINIMUM VALUES OF A FUNCTIONchanges from a decreasing to an increasing function. It therefore follows from§7.3 that at maximum points dy slope = dx = f ′ (x) must change from + to -,and at minimum points dy slope = dx = f ′ (x) must change from - to +when we move along the curve from left to right. At such points as A and H where the slope is zero or infinite, but which areneither maximum nor minimum points, dy slope = dx = f ′ (x) does not change sign.We may then state the conditions in general for maximum and minimum valuesof f (x) for certain values of the variable as follows: f (x) is a maximum if f ′ (x) = 0, and f ′ (x) changes from + to − . (7.1) f (x) is a minimum if f ′ (x) = 0, and f ′ (x) changes from − to + . (7.2) The values of the variable at the turning points of a function are called criticalvalues; thus x = 1 and x = 2 are the critical values of the variable for the functionwhose graph is shown in Figure 7.7. The critical values at turning points where thetangent is parallel to the x-axis are evidently found by placing the first derivativeequal to zero and solving for real values of x, just as under §5.1. (Similarly, ifwe wish to examine a function at exceptional turning points where the tangent isperpendicular to the x-axis, we set the reciprocal of the first derivative equal tozero and solve to find critical values.) To determine the sign of the first derivative at points near a particular turningpoint, substitute in it, first, a value of the variable just a little less than the corre-sponding critical value, and then one a little greater3 . If the first gives + (as at L,Figure 7.8) and the second - (as at M), then the function (= y) has a maximum 3 In this connection the term "little less," or "trifle less," means any value between the nextsmaller root (critical value) and the one under consideration; and the term "little greater," or "triflegreater," means any value between the root under consideration and the next larger one. 188
200.
7.5. EXAMINING A FUNCTION FOR EXTREMAL VALUES: FIRST METHODvalue in that interval (as at I). If the first gives − (as at P) and the second + (as atN), then the function (= y) has a minimum value in that interval (as at C). If the sign is the same in both cases (as at Q and R), then the function (= y) hasneither a maximum nor a minimum value in that interval (as at F)4 . We shall now summarize our results into a compact working rule.7.5 Examining a function for extremal values: first methodWorking rule, sometimes referred to as the sign test of the first derivative. • FIRST STEP. Find the first derivative of the function. • SECOND STEP. Set the first derivative equal to zero5 and solve the resulting equation for real roots in order to find the critical values of the variable. • THIRD STEP. Write the derivative in factored form if possible. • FOURTH STEP. Considering one critical value at a time, test the first deriva- tive, first for a value a trifle less and then for a value a trifle greater than the critical value. If the sign of the derivative is first + and then −, the function has a maximum value for that particular critical value of the variable; but if the reverse is true, then it has a minimum value. If the sign does not change, the function has neither.Remark 7.5.1. It can be helpful to draw a sign graph for the values of the deriva-tive. This is a sketch of the real axis, with tick marks at the critical points, labelingan interval in-between successive critical points with a "+" if the derivative is pos-itive there, and labeling such an interval with a "−" otherwise.Example 7.5.1. In the problem worked out in Example 7.1.2, we showed bymeans of the graph of the function √ A = x 100 − x2 4 A similar discussion will evidently hold for the exceptional turning points B, E, and A respec-tively. 5 When the first derivative becomes infinite for a certain value of the independent variable, thenthe function should be examined for such a critical value of the variable, for it may give maximumor minimum values, as at B, E, or A (Figure 7.8). See footnote in §7.3. 189
202.
7.6. EXAMINING A FUNCTION FOR EXTREMAL VALUES: SECOND METHODand at which the area changes from increasing to decreasing. This implies that thearea is a maximum at this point.7.6 Examining a function for extremal values: sec- ond methodFrom (7.1), it is clear that in the vicinity of a maximum value of f (x), in passingalong the graph from left to right, f ′ (x) changes from + to 0 to −. Hence f ′ (x)is a decreasing function, and by §7.3 we know that its derivative, i.e. the secondderivative (= f ′′ (x)) of the function itself, is negative or zero. Similarly, we have, from (7.2), that in the vicinity of a minimum value of f (x)f ′ (x) changes from − to 0 to +. Hence f ′ (x) is an increasing function and by§7.3 it follows that f ′′ (x) is positive or zero. The student should observe that f ′′ (x) is positive not only at minimum valuesbut also at "nearby" points, P say, to the right of such a critical point. For, as dya point passes through P in moving from left to right, slope = tan τ = dx = ′f (x) is an increasing function. At such a point the curve is said to be concaveupwards. Similarly, f ′′ (x) is negative not only at maximum points but also at"nearby "points, Q say, to the left of such a critical point. For, as a point passes dythrough Q, slope = tan τ = dx = f ′ (x) is a decreasing function. At such a pointthe curve is said to be concave downwards. At a point where the curve is concave upwards we sometimes say that the curvehas a "positive bending,]] and where it is concave downwards a "negative bend-ing." We may then state the sufficient conditions for maximum and minimum valuesof f (x) for certain values of the variable as follows: f (x) is a maximum if f ′ (x) = 0 and f ′′ (x) = a negative number. (7.3) f (x) is a minimum if f ′ (x) = 0 and f ′′ (x) = a positive number. (7.4)Following is the corresponding working rule, sometimes referred to as the secondderivative test. • FIRST STEP. Find the first derivative of the function. 191
203.
7.6. EXAMINING A FUNCTION FOR EXTREMAL VALUES: SECONDMETHOD • SECOND STEP. Set the first derivative equal to zero and solve the resulting equation for real roots in order to find the critical values of the variable. • THIRD STEP. Find the second derivative. • FOURTH STEP. Substitute each critical value for the variable in the second derivative. If the result is negative, then the function is a maximum for that critical value; if the result is positive, the function is a minimum. When f ′′ (x) = 0, or does not exist, the above process fails, although there mayeven then be a maximum or a minimum; in that case the first method given inthe last section still holds, being fundamental. Usually this second method doesapply, and when the process of finding the second derivative is not too long ortedious, it is generally the shortest method.Example 7.6.1. Let us now apply the above rule to test analytically the function 432 M = x2 + xfound in Example 7.1.1. Solution. Let f (x) = x2 + 432 . x First step. Compute f ′ (x) = 2x − 432 . x2 Second step. Solve 2x − 432 = 0. (In Example 7.1.1 we got x = 6.) x2 Third step. Compute f ′′ (x) = 2 + 864 . x3 Fourth step. Use the second derivative test. f ′′ (6) = +. Hence f (6) = 108,minimum value. In Sage : Sagesage: x = var("x")sage: f(x) = xˆ2 + 432/xsage: f1(x) = diff(f(x),x); f1(x)2*x - 432/xˆ2sage: f2(x) = diff(f(x),x,2); f2(x)864/xˆ3 + 2sage: crit_pts = solve(f1(x) == 0,x); crit_pts[x == 3*sqrt(3)*I - 3, x == -3*sqrt(3)*I - 3, x == 6]sage: x0 = crit_pts[2].rhs(); x06 192
204.
7.6. EXAMINING A FUNCTION FOR EXTREMAL VALUES: SECOND METHODsage: f2(x0)6sage: f(x0)108This tells us that x0 = 6 is a critical point and that f ′′ (x0 ) > 0, so it is a minimum. The work of finding maximum and minimum values may frequently be simpli-fied by the aid of the following principles, which follow at once from our discus-sion of the subject. (a) The (local) maximum and minimum values of a continuous function must occur alternately. (In particular, you cannot have two local maximums with- out having a minimum in-between them.) (b) If c is a positive constant, c · f (x) is a maximum or a minimum for a given value of x if and only if f (x) is a maximum or a minimum at x. Consequently, in determining the critical values and testing for maxima and minima, any constant factor may be omitted. When c is negative, c · f (x) is a maximum if and only if f (x) is a minimum, and conversely. (c) If c is a constant, f (x) and c + f (x) have maximum and minimum values for the same values of x. Hence a constant term may be omitted when finding critical values of x and testing. In general we must first construct, from the conditions given in the problem,the function whose maximum and minimum values are required, as was done inthe two examples worked out in §7.1. This is sometimes a problem of consider-able difficulty. No rule applicable in all cases can be given for constructing thefunction, but in a large number of problems we may be guided by the followinggeneral directions. (a) Express the function whose maximum or minimum is involved in the prob- lem. 193
205.
7.7. PROBLEMS (b) If the resulting expression contains more than only variable, the conditions of the problem will furnish enough relations between the variables so that all may be expressed in terms of a single one. (c) To the resulting function of a single variable apply one of our two rules for finding maximum and minimum values. (d) In practical problems it is usually easy to tell which critical value will give a maximum and which a minimum value, so it is not always necessary to apply the fourth step of our rules. (e) Draw the graph of the function in order to check the work.7.7 Problems 1. It is desired to make an open-top box of greatest possible volume from a square piece of tin whose side is a, by cutting equal squares out of the corners and then folding up the tin to form the sides. What should be the length of a side of the squares cut out? Solution. Let x = side of small square = depth of box; then a − 2x = side of square forming bottom of box, and volume is V = (a − 2x)2 x, which is the function to be made a maximum by varying x. Applying rule: dV First step. dx = (a − 2x)2 − 4x(a − 2x) = a2 − 8ax + 12x2 . a Second step. Solving a2 − 8ax + 12x2 = 0 gives critical values x = 2 and a 6 . It is evident that x = a must give a minimum, for then all the tin would 2 be cut away, leaving no material out of which to make a box. By the usual 3 test, x = a is found to give a maximum volume 2a . Hence the side of the 6 27 square to be cut out is one sixth of the side of the given square. The drawing of the graph of the function in this and the following problems is left to the student. 2. Assuming that the strength of a beam with rectangular cross section varies directly as the breadth and as the square of the depth, what are the dimen- sions of the strongest beam that can be sawed out of a round log whose diameter is d? 194
206.
7.7. PROBLEMS Solution. If x = breadth and y = depth, then the beam will have maximum strength when the function xy 2 is a maximum. From the construction and the Pythagorean theorem, y 2 = d2 − x2 ; hence we should test the function f (x) = x(d2 − x2 ). First step. f ′ (x) = −2x2 + d2 − x2 = d2 − 3x2 . d Second step. d2 − 3x2 = 0. Therefore, x = √ 3 = critical value which gives a maximum. 2 Therefore, if the beam is cut so that depth = 3 of diameter of log, and 1 breadth = 3 of diameter of log, the beam will have maximum strength.3. What is the width of the rectangle of maximum area that can be inscribed in a given segment OAA′ of a parabola? Figure 7.9: An inscribed rectangle in a parabola, P = (x, y). HINT. If OC = h, BC = h − x and P P ′ = 2y; therefore the area of rectangle P DD′ P ′ is 2(h − x)y. But since P lies on the parabola y 2 = 2px, the function to be tested is √ 2(h − x) 2px 195
208.
7.7. PROBLEMS Figure 7.11: An inscribed cylinder in a cone. 7. Divide 10 into two such parts that the sum of the double of one and square of the other may be a minimum. Ans. 9 and 1. 8. Find the number that exceeds its square by the greatest possible quantity. 1 Ans. 2 . 9. What number added to its reciprocal gives the least possible sum? Ans. 1.10. Assuming that the stiffness of a beam of rectangular cross section varies 197
209.
7.7. PROBLEMS directly as the breadth and the cube of the depth, what must be the breadth of the stiffest beam that can be cut from a log 16 inches in diameter? Ans. Breadth = 8 inches. 11. A water tank is to be constructed with a square base and open top, and is to hold 64 cubic yards. If the cost of the sides is $ 1 a square yard, and of the bottom $ 2 a square yard, what are the dimensions when the cost is a minimum? What is the minimum cost? Ans. Side of base = 4 yd., height = 4 yd., cost $ 96. 12. A rectangular tract of land is to be bought for the purpose of laying out a quarter-mile track with straightaway sides and semicircular ends. In addi- tion a strip 35 yards wide along each straightaway is to be bought for grand stands, training quarters, etc. If the land costs $ 200 an acre, what will be the maximum cost of the land required? Ans. $ 856. 13. A torpedo boat is anchored 9 miles from the nearest point of a beach, and it is desired to send a messenger in the shortest possible time to a military camp situated 15 miles from that point along the shore. If he can walk 5 miles an hour but row only 4 miles an hour, required the place he must land. Ans. 3 miles from the camp. 14. A gas holder is a cylindrical vessel closed at the top and open at the bottom, where it sinks into the water. What should be its proportions for a given volume to require the least material (this would also give least weight)? Ans. Diameter = double the height. 15. What should be the dimensions and weight of a gas holder of 8, 000, 000 cubic feet capacity, built in the most economical manner out of sheet iron 1 16 of an inch thick and weighing 5 lb. per sq. ft.? 2 Ans. Height = 137 ft., diameter = 273 ft., weight = 220 tons. 16. A sheet of paper is to contain 18 sq. in. of printed matter. The margins at the top and bottom are to be 2 inches each and at the sides 1 inch each. Determine the dimensions of the sheet which will require the least amount of paper. Ans. 5 in. by 10 in. 198
210.
7.7. PROBLEMS17. A paper-box manufacturer has in stock a quantity of cardboard 30 inches by 14 inches. Out of this material he wishes to make open-top boxes by cutting equal squares out of each corner and then folding up to form the sides. Find the side of the square that should be cut out in order to give the boxes maximum volume. Ans. 3 inches.18. A roofer wishes to make an open gutter of maximum capacity whose bottom and sides are each 4 inches wide and whose sides have the same slope. What should be the width across the top? Ans. 8 inches. 419. Assuming that the energy expended in driving a steamboat through the wa- ter varies as the cube of her velocity, find her most economical rate per hour when steaming against a current running c miles per hour. HINT. Let v = most economical speed; then av 3 = energy expended each hour, a being a constant depending upon the particular conditions, and v − c av 3 = actual distance advanced per hour. Hence v−c is the energy expended per mile of distance advanced, and it is therefore the function whose minimum is wanted.20. Prove that a conical tent of a √given capacity will require the least amount of canvas when the height is 2 times the radius of the base. Show that when the canvas is laid out flat it will be a circle with a sector of 1520 9′ = 2.6555... cut out. A bell tent 10 ft. high should then have a base of diameter 14 ft. and would require 272 sq. ft. of canvas.21. A cylindrical steam boiler is to be constructed having a capacity of 1000 cu. ft. The material for the side costs $ 2 a square foot, and for the ends $ 3 a square foot. Find radius when the cost is the least. 1 Ans. √ 3 3π ft.22. In the corner of a field bounded by two perpendicular roads a spring is situated 6 rods from one road and 8 rods from the other. (a) How should a straight road be run by this spring and across the corner so as to cut off as little of the field as possible? (b) What would be the length of the shortest road that could be run across? 2 2 3 Ans. (a) 12 and 16 rods from corner. (b) (6 3 + 8 3 ) 2 rods. 199
211.
7.7. PROBLEMS 23. Show that a square is the rectangle of maximum perimeter that can be in- scribed in a given circle. 24. Two poles of height a and b feet are standing upright and are c feet apart. Find the point on the line joining their bases such that the sum of the squares of the distances from this point to the tops of the poles is a minimum. (Ans. Midway between the poles.) When will the sum of these distances be a minimum? 25. A conical tank with open top is to be built to contain V cubic feet. Deter- mine the shape if the material used is a minimum. 26. An isosceles triangle has a base 12 in. long and altitude 10 in. Find the rect- angle of maximum area that can be inscribed in it, one side of the rectangle coinciding with the base of the triangle. 27. Divide the number 4 into two such parts that the sum of the cube of one part and three times the square of the other shall have a maximum value. 28. Divide the number a into two parts such that the product of one part by the fourth power of the other part shall be a maximum. 29. A can buoy in the form of a double cone is to be made from two equal circular iron plates of radius r. Find the radius of the base of the cone when the buoy has the greatest displacement (maximum volume). 2 Ans. r 3 . 30. Into a full conical wineglass of depth a and generating angle a there is care- fully dropped a sphere of such size as to cause the greatest overflow. Show that the radius of the sphere is sinααsin α2α . cos 31. A wall 27 ft. high is 8 ft. from a house. Find the length of the shortest ladder that will reach the house if one end rests on the ground outside of the wall. √ Ans. 13 13. Here's how to solve this using Sage : Let h be the height above ground at which the ladder hits the house and let d be the distance from the wall that the ladder hits the ground on the other side of the wall. By similar triangles, 200
212.
7.7. PROBLEMS 8 h h/27 = (8 + d)/d = 1 + d , so d + 8 = 8 h−27 . The length of the ladder is, h by the Pythagorean theorem, f (h) = h2 + (8 + d)2 = h2 + (8 h−27 )2 . Sage sage: h = var("h") sage: f(h) = sqrt(hˆ2+(8*h/(h-27))ˆ2) sage: f1(h) = diff(f(h),h) sage: f2(h) = diff(f(h),h,2) sage: crit_pts = solve(f1(h) == 0,h); crit_pts [h == 21 - 6*sqrt(3)*I, h == 6*sqrt(3)*I + 21, h == 39, h == 0] sage: h0 = crit_pts[2].rhs(); h0 39 sage: f(h0) 13*sqrt(13) sage: f2(h0) 3/(4*sqrt(13)) This says f (h) has four critical points, but only one of which is meaningful, h0 = 39. At this point, f (h) is a minimum.32. A vessel is anchored 3 miles offshore, and opposite a point 5 miles further along the shore another vessel is anchored 9 miles from the shore. A boat from the first vessel is to land a passenger on the shore and then proceed to the other vessel. What is the shortest course of the boat? Ans. 13 miles.33. A steel girder 25 ft. long is moved on rollers along a passageway 12.8 ft. wide and into a corridor at right angles to the passageway. Neglecting the width of the girder, how wide must the corridor be? Ans. 5.4 ft.34. A miner wishes to dig a tunnel from a point A to a point B 300 feet below and 500 feet to the east of A. Below the level of A it is bed rock and above A is soft earth. If the cost of tunneling through earth is $ 1 and through rock $ 3 per linear foot, find the minimum cost of a tunnel. Ans. $ 1348.53. 201
213.
7.7. PROBLEMS 35. A carpenter has 108 sq. ft. of lumber with which to build a box with a square base and open top. Find the dimensions of the largest possible box he can make. Ans. 6 × 6 × 3. 36. Find the right triangle of maximum area that can be constructed on a line of length h as hypotenuse. h Ans. √ 2 = length of both legs. 37. What is the isosceles triangle of maximum area that can be inscribed in a given circle? Ans. An equilateral triangle. 38. Find the altitude of the maximum rectangle that can be inscribed in a right triangle with base b and altitude h. Ans. Altitude = h . 2 39. Find the dimensions of the rectangle of maximum area that can be inscribed in the ellipse b2 x2 + a2 y 2 = a2 b2 . √ √ Ans. a 2 × b 2; area = 2ab. 40. Find the altitude of the right cylinder of maximum volume that can be in- scribed in a sphere of radius r. 2r Ans. Altitude of cylinder = √ . 3 41. Find the altitude of the right cylinder of maximum convex (curved) surface that can be inscribed in a given sphere. √ Ans. Altitude of cylinder = r 2. 42. What are the dimensions of the right hexagonal prism of minimum surface whose volume is 36 cubic feet? √ Ans. Altitude = 2 3; side of hexagon = 2. 43. Find the altitude of the right cone of minimum volume circumscribed about a given sphere. Ans. Altitude = 4r, and volume = 2× vol. of sphere. 202
214.
7.7. PROBLEMS44. A right cone of maximum volume is inscribed in a given right cone, the vertex of the inside cone being at the center of the base of the given cone. Show that the altitude of the inside cone is one third the altitude of the given cone.45. Given a point on the axis of the parabola y 2 = 2px at a distance a from the vertex; find the abscissa of the point of the curve nearest to it. Ans. x = a − p.46. What is the length of the shortest line that can be drawn tangent to the ellipse b2 x2 + a2 y 2 = a2 b2 and meeting the coordinate axes? Ans. a + b.47. A Norman window consists of a rectangle surmounted by a semicircle. Given the perimeter, required the height and breadth of the window when the quantity of light admitted is a maximum. Ans. Radius of circle = height of rectangle.48. A tapestry 7 feet in height is hung on a wall so that its lower edge is 9 feet above an observer's eye. At what distance from the wall should he stand in order to obtain the most favorable view? (HINT. The vertical angle subtended by the tapestry in the eye of the observer must be at a maximum.) Ans. 12 feet.49. What are the most economical proportions of a tin can which shall have a given capacity, making allowance for waste? (HINT. There is no waste in cutting out tin for the side of the can, but for top and bottom a hexagon of tin circumscribing the circular pieces required is used up. NOTE 1. If no allowance is made for waste, then height = diameter. NOTE 2. We know that the shape of a bee cell is hexagonal, giving a certain capacity for honey with the greatest possible economy of wax.) √ 2 3 Ans. Height = π × diameter of base.50. An open cylindrical trough is constructed by bending a given sheet of tin at breadth 2a. Find the radius of the cylinder of which the trough forms a part when the capacity of the trough is a maximum. 2a Ans. Rad. = π ; i.e. it must be bent in the form of a semicircle. 203
215.
7.7. PROBLEMS 51. A weight W is to be raised by means of a lever with the force F at one end and the point of support at the other. If the weight is suspended from a point at a distance a from the point of support, and the weight of the beam is w pounds per linear foot, what should be the length of the lever in order that the force required to lift it shall be a minimum? 2aW Ans. x = w feet. 52. An electric arc light is to be placed directly over the center of a circular plot of grass 100 feet in diameter. Assuming that the intensity of light varies directly as the sine of the angle under which it strikes an illuminated surface, and inversely as the square of its distance from the surface, how high should the light he hung in order that the best possible light shall fall on a walk along the circumference of the plot? 50 Ans. √ 2 feet 53. The lower corner of a leaf, whose width is a, is folded over so as just to reach the inner edge of the page. (a) Find the width of the part folded over when the length of the crease is a minimum. (b) Find the width when the area folded over is a minimum. Ans. (a) 3 a; (b) 3 a. 4 2 54. A rectangular stockade is to be built which must have a certain area. If a stone wall already constructed is available for one of the sides, find the dimensions which would make the cost of construction the least. Ans. Side parallel to wall = twice the length of each end. 55. When the resistance of air is taken into account, the inclination of a pen- dulum to the vertical may be given by the formula θ = ae−kt cos (nt + η). Show that the greatest elongations occur at equal intervals π of time. n 56. It is required to measure a certain unknown magnitude x with precision. Suppose that n equally careful observations of the magnitude are made, giving the results a1 , a2 , a3 , . . . , an . The errors of these observations are evidently x − a1 , x − a2 , x − a3 , · · · , x − an , some of which are positive and some negative. It has been agreed that the most probable value of x is such that it renders the sum of the squares of the errors, namely (x − a1 )2 + (x − 204
216.
7.7. PROBLEMS Figure 7.12: A leafed page of width a. a2 )2 + (x − a3 )2 + · · · + (x − an )2 , a minimum. Show that this gives the arithmetical mean of the observations as the most probable value of x. (This is related to the method of least squares, discovered by Gauss, a com- monly used technique in statistical applications.)57. The bending moment at x of a beam of length ℓ, uniformly loaded, is given by the formula M = 1 wℓx − 2 wx2 , where w = load per unit length. Show 2 1 that the maximum bending moment is at the center of the beam. 258. If the total waste per mile in an electric conductor is W = c2 r + tr , where c = current in amperes (a constant), r = resistance in ohms per mile, and t = a constant depending on the interest on the investment and the depreciation 205
217.
7.7. PROBLEMS of the plant, what is the relation between c, r, and t when the waste is a minimum? Ans. cr = t. 59. A submarine telegraph cable consists of a core of copper wires with a cov- ering made of non-conducting material. If x denote the ratio of the radius of the core to the thickness of the covering, it is known that the speed of signaling varies as 1 x2 log . x 1 Show that the greatest speed is attained when x = √ . e 60. Assuming that the power given out by a voltaic cell is given by the formula E 2R P = , (r + R)2 when E = constant electro-motive force, r = constant internal resistance, R = external resistance, prove that P is a maximum when r = R. 61. The force exerted by a circular electric current of radius a on a small magnet whose axis coincides with the axis of the circle varies as x 5 . (a2 + x2 ) 2 where x = distance of magnet from plane of circle. Prove that the force is a maximum when x = a . 2 62. We have two sources of heat at A and B, which we visualize on the real line (with B to the right or A), with intensities a and b respectively. The total intensity of heat at a point P between A and B at a distance of x from A a b is given by the formula I = x2 + (d−x)2 . Show that the temperature at P 1/3 will be the lowest when d−x = a1/3 . that is, the distances BP and AP have x b the same ratio as the cube roots of the corresponding heat intensities. The 1 a3 d distance of P from A is x = 1 1 . a 3 +b 3 206
224.
7.8. POINTS OF INFLECTION7.8 Points of inflectionDefinition 7.8.1. Consider the graph y = f (x) is a twice continuously differen-tiable function. Points of inflection separate concave upwards sections of the graphfrom concave downwards sections. They may also be defined as points where d2 d2 (a) dxy = 0 and dxy changes sign, 2 2or 2 2 (b) d x = 0 and d x changes sign. dy 2 dy 2 Thus, if a curve y = f (x) changes from concave upwards to concave downwardsat a point, or the from concave down to concave up, then such a point is called apoint of inflection. From the discussion of §7.6, it follows at once that where the curve is concaveup, f ′′ (x) = +, and where the curve is concave down, f ′′ (x) = −. In order tochange sign it must pass through the value zero6 ; hence we have:Lemma 7.8.1. At points of inflection, f ′′ (x) = 0. Solving the equation resulting from Lemma 7.8.1 gives the x-coordinate(s) ofthe point(s) of inflection. To determine the direction of curving or direction ofbending in the vicinity of a point of inflection, test f ′′ (x) for values of x, firstslightly less and then slightly more than the x-coordinate at that point. If f ′′ (x) changes sign, we have a point of inflection, and the signs obtained deter-mine if the curve is concave upwards or concave downwards in the neighborhoodof each point of inflection. The student should observe that near a point where the curve is concave up-wards the curve lies above the tangent, and at a point where the curve is concavedownwards the curve lies below the tangent. At a point of inflection the tangentevidently crosses the curve. Following is a rule for finding points of inflection of the curve whose equationis y = f (x). This rule includes also directions for examining the direction ofcurvature of the curve in the neighborhood of each point of inflection. • FIRST STEP. Find f ′′ (x). • SECOND STEP. Set f ′′ (x) = 0, and solve the resulting equation for real roots. 6 It is assumed that f ′ (x) and f ′′ (x) are continuous. The solution of Exercise 2, §7.8, showshow to discuss a case where f ′ (x) and f ′′ (x) are both infinite. 213
226.
7.9. EXAMPLES We may therefore conclude that the tangent at (4, 2) is perpendicular to the x-axis, that to the left of (4, 2) the curve is concave upwards, and to the right of (4, 2) it is concave downwards. Therefore (4, 2) must be considered a point of inflection. 3. y = x2 . Ans. Concave upwards everywhere. 4. y = 5 − 2x − x2 . Ans. Concave downwards everywhere. 5. y = x3 . Ans. Concave downwards to the left and concave upwards to the right of (0, 0). 6. y = x3 − 3x2 − 9x + 9. Ans. Concave downwards to the left and concave upwards to the right of (1, −2). 7. y = a + (x − b)3 . Ans. Concave downwards to the left and concave upwards to the right of (b, a). x3 8. a2 y = 3 − ax2 + 2a3 . Ans. Concave downwards to the left and concave upwards to the right of (a, 4a ). 3 9. y = x4 . Ans. Concave upwards everywhere.10. y = x4 − 12x3 + 48x2 − 50. Ans. Concave upwards to the left of x = 2, concave downwards between x = 2 and x = 4, concave upwards to the right of x = 4.11. y = sin x. Ans. Points of inflection are x = nπ, n being any integer. 215
227.
7.10. CURVE PLOTTING 12. y = tan x. Ans. Points of inflection are x = n, n being any integer. 13. Show that no conic section can have a point of inflection. 14. Show that the graphs of ex and log x have no points of inflection.7.10 Curve plottingThe elementary method of plotting a curve whose equation is given in rectangularcoordinates, and one with which the student is already familiar, is to (a) solve itsequation for y (or x), (b) take several arbitrary values of x (or y), tabulate thecorresponding values of y (or x), (c) plot the respective points, and (d) draw asmooth curve through them. The result is an approximation to the required curve.This process is laborious at best, and in case the equation of the curve is of a degreehigher than the second, the solved form of such an equation may be unsuitable forthe purpose of computation, or else it may fail altogether, since it is not alwayspossible to solve the equation for y or x. The general form of a curve is usually all that is desired, and calculus furnishesus with useful methods for determining the shape of a curve with very little com-putation. The first derivative gives us the slope of the curve at any point; the second deriva-tive determines the intervals within which the curve is concave upward or concavedownward, and the points of inflection separate these intervals; the maximumpoints are the high points and the minimum points are the low points on the curve.As a guide in his work the student may follow the following rule. Rule for plotting curves in rectangular coordinates. • FIRST STEP. Find the first derivative; place it equal to zero; solving gives the abscissas of maximum and minimum points. • SECOND STEP. Find the second derivative; place it equal to zero; solving gives the abscissas of the points of inflection. • THIRD STEP. Calculate the corresponding ordinates of the points whose abscissas were found in the first two steps. Calculate as many more points as may be necessary to give a good idea of the shape of the curve. Fill out a table such as is shown in the example worked out. 216
232.
CHAPTER EIGHT Application to arclength and rates8.1 IntroductionThus far we have represented the derivative of y = f (x) by the notation dy = f ′ (x). dxWe have taken special pains to impress on the student that the symbol dy dxwas to be considered not as an ordinary fraction with dy as numerator and dx asdenominator, but as a single symbol denoting the limit of the quotient ∆y ∆xas ∆x approaches the limit zero. Problems do occur, however, where it is very convenient to be able to give ameaning to dx and dy separately, and it is especially useful in applications us-ing integral calculus. How this may be done is explained in the first part of thischapter. In the second part (starting with §8.6), we apply what we know about the deriva-tive to functions of time t. If f (t) is some quantity (for example, distance) chang-ing with time then we can regard f ′ (t) as the rate of change of f (for example, 221
233.
8.2. DEFINITIONSvelocity). The method of solving "related rates" problems will be explained in thesecond part of this chapter.8.2 DefinitionsIf f ′ (x) is the derivative of f (x) for a particular value of x, and ∆x is an arbitrarilychosen1 increment of x, then the differential of f (x), denoted by the symbol df (x),is defined by the equation df (x) = f ′ (x)∆x. (8.1)If now f (x) = x, then f ′ (x) = 1, and (8.1) reduces to dx = ∆x, showing thatwhen x is the independent variable, the differential of x (= dx) is identical with∆x. Hence, if y = f (x), (8.1) may in general be written in the form dy = f ′ (x) dx. (8.2)The differential of a function equals its derivative multiplied by the differential ofthe independent variable. Observe that, since dx may be given any arbitrary valuewhatever, dx is independent of x. Hence, dy is a function of two independentvariables x and dx. Let us illustrate what this means geometrically. Let f ′ (x) be the derivative of y = f (x) at P, then dy = f ′ (x)dx = tan τ · dx.Therefore dy, or df (x), is the increment of the y-coordinate on the tangent line tothe curve y = f (x) corresponding to replacing x by x + dx. This gives the following interpretation of the derivative as a fraction. If an arbitrarily chosen increment of the independent variable x for a point (x, y)on the curve y = f (x) is denoted by dx, then in the derivative dy = f ′ (x) = tan τ, dxdy denotes the corresponding increment of the y-coordinate drawn to the tangent. 1 The term "arbitrarily chosen" essentially means that the variable ∆x is independent from thevariable x. 222
234.
8.3. DERIVATIVE OF THE ARCLENGTH IN RECTANGULAR COORDINATES Figure 8.1: The differential of a function.8.3 Derivative of the arclength in rectangular coor- dinatesLet s be the arclength2 of the part of the curve y = f (x) from a given point A onthe curve to some 'variable" point P. Denote the increment of s (= arc PQ in Figure 8.2) by ∆s. The definition of thearclength depends on the assumption that, as Q approaches P, chordP Q lim = 1. arcP QIn the limit of the ratio of chord PQ and a second infinitesimal, chord PQ may bereplaced by arc PQ (= ∆s). From Figure 8.2, we have (chord P Q)2 = (∆x)2 + (∆y)2 . (8.3) 2 Defined in integral calculus. For now, we simply assume that there is a function s = s(x)such that if you go along the curve from a given point A (such as the point (0, f (0))) to a pointP = (x, y) then s(x) describes the arclength. 223
240.
8.6. THE DERIVATIVE CONSIDERED AS THE RATIO OF TWO RATESAt any instant the time rate of change of y (or the function) equals its derivativemultiplied by the time rate change of the independent variable. Or, write (8.11) in the form dy dy dt dx = f ′ (x) = . dt dxThe derivative measures the ratio of the time rate of change of y to that of x. Figure 8.3: Geometric visualization of the derivative the arclength.dsdt being the time rate of change of length of arc, we have from (5.26), 2 2 ds dx dt = + . (8.12) dt dt dtwhich is the relation indicated by Figure 8.3. As a guide in solving rate problems use the following rule. 229
241.
8.7. EXERCISES • FIRST STEP. Draw a figure illustrating the problem. Denote by x, y, z, etc., the quantities which vary with the time. • SECOND STEP. Obtain a relation between the variables involved which will hold true at any instant. • THIRD STEP. Differentiate with respect to the time. • FOURTH STEP. Make a list of the given and required quantities. • FIFTH STEP. Substitute the known quantities in the result found by differ- entiating (third step), and solve for the unknown.8.7 Exercises 1. A man is walking at the rate of 5 miles per hour towards the foot of a tower 60 ft. high. At what rate is he approaching the top when he is 80 ft. from the foot of the tower? Solution. Apply the above rule. First step. Draw the figure. Let x = distance of the man from the foot and y = his distance from the top of the tower at any instant. Second step. Since we have a right triangle, y 2 = x2 + 3600. Third step. Differentiating, we get 2y dy = 2x dx , or, dt dt dy dt = x dx y dt , meaning x that at any instant whatever (Rate of change of y) = y (rate of change of x). Fourth step. x = 80, dx = 5 miles/hour, dt = √× 5280f t/hour, 5 y = x2 + 3600 = 100. dy dt =? dy 80 Fifth step. Substituting back in the above dt = 100 × 5 × 5280 ft/hour = 4 miles/hour. 230
243.
8.7. EXERCISES 3. A circular plate of metal expands by heat so that its radius increases uni- formly at the rate of 0.01 inch per second. At what rate is the surface in- creasing when the radius is two inches? Solution. Let x = radius and y = area of plate. Then y = πx2 , dy = 2πx dx , dt dt That is; at any instant the area of the plate is increasing in square inches 2πx times as fast as the radius is increasing in linear inches. x = 2, dx = 0.01, dt dy dt =?. Substituting in the above, dy = 2π × 2 × 0.01 = 0.04π sq. in. per dt sec. 4. A street light is hung 12 ft. directly above a straight horizontal walk on which a boy 5 ft. in height is walking. How fast is the boy's shadow length- ening when he is walking away from the light at the rate of 168 ft. per minute? Solution. Let x = distance of boy from a point directly under light L, and y = length of boy's shadow. By similar triangle, y/(y +x) = 5/12, or y = 5 x. 7 Differentiating, dy = 5 dx ; i.e. the shadow is lengthening 5 as fast as the dt 7 dt 7 boy is walking, or 120 ft. per minute. 5. In a parabola y 2 = 12x, if x increases uniformly at the rate of 2 in. per second, at what rate is y increasing when x = 3 in. ? Ans. 2 in. per sec. 6. At what point on the parabola of the last example do the x-coordinate and y-coordinate increase at the same rate? Ans. (3, 6). 7. In the function y = 2x3 + 6, what is the value of x at the point where y increases 24 times as fast as x? Ans. x = ±2. 8. The y-coordinate of a point describing the curve x2 + y 2 = 25 is decreasing at the rate of 3/2 in. per second. How rapidly is the x-coordinate changing when the y-coordinate is 4 inches? dx Ans. dt = 2 in. per sec. 9. Find the values of x at the points where the rate of change of x3 − 12x2 + 45x − 13 is zero. Ans. x = 3 and 5. 232
244.
8.7. EXERCISES 10. At what point on the ellipse 16x2 + 9y 2 = 400 does y decrease at the same rate that x increases? Ans. (3, 16 ). 3 11. Where in the first quadrant does the arclength increase twice as fast as the y-coordinate? Ans. At 60o = π/3. A point generates each of the following curves (problems 12-16). Find the rateat which the arclength is increasing in each case: dx 12. y 2 = 2x; = 2, x = 2. dt ds √ Ans. dt = 5. dy 13. xy = 6; dt = 2, y = 3. ds √ Ans. dt = 2 13. 3 dx 14. x2 + 4y 2 = 20; dt = −1, y = 1. √ Ans. ds = 2. dt dx 15. y = x3 ; dt = 3, x = −3. dy 16. y 2 = x3 ; dt = 4, y = 8. 17. The side of an equilateral triangle is 24 inches long, and is increasing at the rate of 3 inches per hour. How fast is the area increasing? √ Ans. 36 3 sq. in. per hour. 18. Find the rate of change of the area of a square when the side b is increasing at the rate of a units per second. Ans. 2ab sq. units per sec. 19. (a) The,volume of a spherical soap bubble increases how many times as fast as the radius? (b) When its radius is 4 in. and increasing at the rate of 1/2 in. per second, how fast is the volume increasing? Ans. (a) 4πr2 times as fast; (b) 32π cu. in. per sec. How fast is the surface increasing in the last case? 233
245.
8.7. EXERCISES 20. One end of a ladder 50 ft. long is leaning against a perpendicular wall standing on a horizontal plane. Supposing the foot of the ladder to be pulled away from the wall at the rate of 3 ft. per minute; (a) how fast is the top of the ladder descending when the foot is 14 ft. from the wall? (b) when will the top and bottom of the ladder move at the same rate? (c) when is the top of the ladder descending at the rate of 4 ft. per minute? 7 √ Ans. (a) 78 ft. per min.; (b) when 25 2 ft. from wall; (c) when 40 ft. from wall. 21. A barge whose deck is 12 ft. below the level of a dock is drawn up to it by means of a cable attached to a ring in the floor of the dock, the cable being hauled in by a windlass on deck at the rate of 8 ft. per minute. How fast is the barge moving towards the dock when 16 ft. away? Ans. 10 ft. per minute. 22. An elevated car is 40 ft. immediately above a surface car, their tracks inter- secting at right angles. If the speed of the elevated car is 16 miles per hour and of the surface car 8 miles per hour, at what rate are the cars separating 5 minutes after they meet? Ans. 17.9 miles per hour. 23. One ship was sailing south at the rate of 6 miles per hour; another east at the rate of 8 miles per hour. At 4 P.M. the second crossed the track of the first where the first was two hours before; (a) how was the distance between the ships changing at 3 P.M.? (b) how at 5 P.M.? (c) when was the distance between them not changing? Ans. (a) Diminishing 2.8 miles per hour; (b) increasing 8.73 miles per hour; (c) 3 : 17 P.M. 24. Assuming the volume of the wood in a tree to be proportional to the cube of its diameter, and that the latter increases uniformly year by year when growing, show that the rate of growth when the diameter is 3 ft. is 36 times as great as when the diameter is 6 inches. 25. A railroad train is running 15 miles an hour past a station 800 ft. long, the track having the form of the parabola y 2 = 600x, and situated as shown in Figure 8.4. 234
246.
8.7. EXERCISES Figure 8.4: Train station and the train's trajectory. If the sun is just rising in the east, find how fast the shadow S of the loco- motive L is moving along the wall of the station at the instant it reaches the end of the wall. Solution. y 2 = 600x, 2y dy = 600 dx , or dt dt dx dt y dy = 300 dt . Substituting this value dx 2 2 2 2 2 of dx in ds = dt dt dt + dy , we get ds = 300 dy + dy . Now dt dt y dt dt ds dt = 15 miles per hour = 22 ft. per sec., y = 400 and dy =?. Substituting dt dy 2 back in the above, we get (22) = 9 + 1 dt , or, dy = 13 5 ft. per 2 16 dt 1 second.26. An express train and a balloon start from the same point at the same instant. The former travels 50 miles an hour and the latter rises at the rate of 10 miles an hour. How fast are they separating? Ans. 51 miles an hour.27. A man 6 ft. tall walks away from a lamp-post 10 ft. high at the rate of 4 miles an hour. How fast does the shadow of his head move? Ans. 10 miles an hour. 235
247.
8.7. EXERCISES 28. The rays of the sun make an angle of 30o = π/6 with the horizon. A ball is thrown vertically upward to a height of 64 ft. How fast is the shadow of the ball moving along the ground just before it strikes the ground? Ans. 110.8 ft. per sec. 29. A ship is anchored in 18 ft. of water. The cable passes over a sheave on the bow 6 ft. above the surface of the water. If the cable is taken in at the rate of 1 ft. a second, how fast is the ship moving when there are 30 ft. of cable out? 5 Ans. 3 ft. per sec. 30. A man is hoisting a chest to a window 50 ft. up by means of a block and tackle. If he pulls in the rope at the rate of 10 ft. a minute while walking away from the building at the rate of 5 ft. a minute, how fast is the chest rising at the end of the second minute? Ans. 10.98 ft. per min. 31. Water flows from a faucet into a hemispherical basin of diameter 14 inches at the rate of 2 cu. in. per second. How fast is the water rising (a) when the water is halfway to the top? (b) just as it runs over? (The volume of a spherical segment = 1 πr2 h + 6 πh3 , where h = altitude of segment.) 2 1 32. Sand is being poured on the ground from the orifice of an elevated pipe, and forms a pile which has always the shape of a right circular cone whose height is equal to the radius of the base. If sand is falling at the rate of 6 cu. ft. per sec., how fast is the height of the pile increasing when the height is 5 ft.? 33. An aeroplane is 528 ft. directly above an automobile and starts east at the rate of 20 miles an hour at the same instant the automobile starts east at the rate of 40 miles an hour. How fast are they separating? 34. A revolving light sending out a bundle of parallel rays is at a distance of t a mile from the shore and makes 1 revolution a minute. Find how fast the light is traveling along the straight beach when at a distance of 1 mile from the nearest point of the shore. Ans. 15.7 miles per min. 236
248.
8.7. EXERCISES35. A kite is 150 ft. high and 200 ft. of string are out. If the kite starts drifting away horizontally at the rate of 4 miles an hour, how fast is the string being paid out at the start? Ans. 2.64 miles an hour.36. A solution is poured into a conical filter of base radius 6 cm. and height 24 cm. at the rate of 2 cu. cm. a second, and filters out at the rate of 1 cu. cm. a second. How fast is the level of the solution rising when (a) one third of the way up? (b) at the top? Ans. (a) 0.079 cm. per sec.; (b) 0.009 cm. per sec.37. A horse runs 10 miles per hour on a circular track in the center of which is a street light. How fast will his shadow move along a straight board fence (tangent to the track at the starting point) when he has completed one eighth of the circuit? Ans. 20 miles per hour.38. The edges of a cube are 24 inches and are increasing at the rate of 0.02 in. per minute. At what rate is (a) the volume increasing? (b) the area increasing?39. The edges of a regular tetrahedron are 10 inches and are increasing at the rate of 0.3 in. per hour. At what rate is (a) the volume increasing? (b) the area increasing?40. An electric light hangs 40 ft. from a stone wall. A man is walking 12 ft. per second on a straight path 10 ft. from the light and perpendicular to the wall. How fast is the man's shadow moving when he is 30 ft. from the wall? Ans. 48 ft. per sec.41. The approach to a drawbridge has a gate whose two arms rotate about the same axis as shown in the figure. The arm over the driveway is 4 yards long and the arm over the footwalk is 3 yards long. Both arms rotate at the rate of 5 radians per minute. At what rate is the distance between the extremities of the arms changing when they make an angle of 45o = π/4 with the horizontal? Ans. 24 yd. per min. 237
249.
8.7. EXERCISES 42. A conical funnel of radius 3 inches and of the same depth is filled with a solution which filters at the rate of 1 cu. in. per minute. How fast is the surface falling when it is 1 inch from the top of the funnel? 1 Ans. 4π in. per mm. 43. An angle is increasing at a constant rate. Show that the tangent and sine are increasing at the same rate when the angle is zero, and that the tangent increases eight times as fast as the sine when the angle is 60o = π/3. 238
250.
CHAPTER NINE Change of variable dyIf y = f (x) is a function of x and x is a function of some other variable t then dx ,d2y 2ydx2 , etc., can be expressed in terms of dy , dx , d 2 , etc.. This chapter is devoted to dt dt dtexplaining the techniques to find the formulas necessary for making such a changeof variables.9.1 Interchange of dependent and independent vari- ablesIf y = f (x) is a one-to-one function of x then it can be "inverted" so that x =f −1 (y) is a function of y. It is sometimes desirable to transform an expressioninvolving derivatives of y with respect to x into an equivalent expression involvingderivatives of x with respect to y. Our examples will show that in many cases sucha change transforms the given expression into a much simpler one. Or perhaps xis given as an explicit function of y in a problem, and it is found more convenient 2 dy d2to use a formula involving dx , d x , etc., than one involving dx , dxy , etc. We shall dy dy 2 2now find the formulas necessary for making such transformations. Given y = f (x), then from item 4.28 in §4.1, we have dy 1 dx = dx , =0 (9.1) dx dy dy dy dxgiving dx in terms of dy . Also, by 4.27 in §4.1, 239
260.
CHAPTER TEN Applications of higher derivativesWe have seen how the first derivative can be applied to solving max-min problemsand related rate problems. In this chapter, we present some applications of higherderivatives. Below, we introduce the mean value theorem, L'Hˆ pital's rule for olimits of "indeterminant forms," and Taylor series approximations.10.1 Rolle's TheoremLet y = f (x) be a continuous single-valued function of x, vanishing for x = a andx = b, and suppose that f ′ (x) changes continuously when x varies from a to b.The function will then be represented graphically by a continuous curve startingat a point on the x-axis and ending at another point on the x-axis, as in Figure10.1. Geometric intuition tells us that for at least one value of x between a and bthe tangent is parallel to the x-axis (as at P); that is, the slope is zero.This illustrates Rolle's Theorem: If f (x) vanishes when x = a and x = b, and f (x) and f ′ (x)are continuous for all values of x from x = a to x = b, then f ′ (x) will be zero forat least one value of x between a and b. This theorem is obviously true, because as x increases from a to b, f (x) cannotalways increase or always decrease as x increases, since f (a) = 0 and f (b) = 0.Hence for at least one value of x between a and b, f (x) must cease to increaseand begin to decrease, or else cease to decrease and begin to increase; and for that 249
262.
10.2. THE MEAN VALUE THEOREM (a) the graph of a function which is discontinuous (= ∞) for x = c, a value lying between a and b, and (b) a continuous function whose first derivative is discontinuous (= ∞) for such an intermediate value x = c.In either case it is seen that at no point on the graph between x = a and x = bdoes the tangent (or curve) be,come parallel to the x-axis.10.2 The mean value theoremConsider the quantity Q defined by the equation f (b) − f (a) = Q, (10.1) b−aor f (b) − f (a) − (b − a)Q = 0. (10.2)Let F (x) be a function formed by replacing b by x in the left-hand member of(10.2); that is, F (x) = f (x) − f (a) − (x − a)Q. (10.3)From (10.2), F (b) = 0, and from (10.3), F (a) = 0; therefore, by Rolle's Theorem(see §10.1), F ′ (x) must be zero for at least one value of x between a and b, sayfor x1 . But by differentiating (10.3) we get F ′ (x) = f ′ (x) − Q.Therefore, since F ′ (x1 ) = 0, then also f ′ (x1 ) − Q = 0, and Q = f ′ (x1 ). Substi-tuting this value of Q in (10.1), we get the ean value theorem, f (b) − f (a) = f ′ (x1 ), a < x1 < b (10.4) b−awhere in general all we know about x1 is that it lies between a and b.The mean value theorem interpreted geometrically. Let the curve in the figure be the locus of y = f (x). 251
266.
10.5. MAXIMA AND MINIMA TREATED ANALYTICALLY 2. y = x3 − 6x2 + 12x + 48 Ans. x = 2 gives neither. 3. y = (x − 1)2 (x + 1)3 Ans. x = 1 is a min., y = 0; x = 1/5 is a max; x = −1 gives neither. 4. Investigate y = x5 − 5x4 + 5x3 − 1 at x = 1 and x = 3. 5. Investigate y = x3 − 3x2 + 3x + 7 at x = 1. 6. Show the if the first derivative of f (x) which does not vanish at x = a is of odd order n then f (x) is increasing or decreasing at x = a, according to whether f (n) (a) is positive or negative.10.5 Maxima and minima treated analyticallyBy making use of the results of the last two sections we can now give a generaldiscussion of maxima and minima of functions of a single independent variable. Given the function f (x). Let h be a positive number as small as we please;then the definitions given in §7.4, may be stated as follows: If, for all values of xdifferent from a in the interval [a − h, a + h], f (x) − f (a) = a negative number, (10.10)then f (x) is said to be a maximum when x = a. If, on the other hand, f (x) − f (a) = a positive number, (10.11)then f (x) is said to be a minimum when x = a. Consider the following cases: I Let f ′ (a) = 0. From (10.5), [§10.2], replacing b by x and transposing f(a), f (x) − f (a) = (x − a)f ′ (x1 ), a < x1 < x, (10.12) Since f ′ (a) = 0, and f ′ (x) is assumed as continuous, h may be chosen so small that f ′ (x) will have the same sign as f ′ (a) for all values of x in the interval [a−h, a+h]. Therefore f ′ (x1 ) has the same sign as f ′ (a) (Chap. 2). But x − a changes sign according as x is less or greater than a. Therefore, from (10.12), the difference f (x) − f (a) will also change sign, and, by 255
267.
10.5. MAXIMA AND MINIMA TREATED ANALYTICALLY (10.10) and (10.11), f (a) will be neither a maximum nor a minimum. This result agrees with the discussion in §7.4, where it was shown that for all values of x for which f (x) is a maximum or a minimum, the first derivative f ′ (x) must vanish. II Let f ′ (a) = 0, and f ′′ (a) = 0. From (10.12), replacing b by x and transpos- ing f (a), (x − a)2 ′′ f (x) − f (a) = f (x2 ), a < x2 < x. (10.13) 2! Since f ′′ (a) = 0, and f ′′ (x) is assumed as continuous, we may choose our interval [a − h, a + h] so small that f ′′ (x2 ) will have the same sign as f ′′ (a) (Chap. 2). Also (x − a)2 does not change sign. Therefore the second member of (10.13) will not change sign, and the difference f (x) − f (a) will have the same sign for all values of x in the interval [a − h, a + h], and, moreover, this sign will be the same as the sign of f ′′ (a). It therefore follows from our definitions (10.10) and (10.11) that f (a) is a maximum if f ′ (a) = 0 and f ′′ (a) = a negative number; (10.14) ′ ′′ f (a) is a minimum if f (a) = 0 and f (a) = a positive number (10.15) These conditions are the same as (7.3) and (7.4), [§7.6]. III Let f ′ (a) = f ′′ (a) = 0, and f ′′′ (a) = 0. From (10.9), [§10.3], replacing b by x and transposing f (a), 1 f (x) − f (a) = (x − a)3 f ′′′ (x3 ), a < x3 < x. (10.16) 3! As before, f ′′′ (x3 ) will have the same sign as f ′′′ (a). But (x−a)3 changes its sign from − to + as x increases through a. Therefore the difference f (x) − f (a) must change sign, and f (a) is neither a maximum nor a minimum. IV Let f ′ (a) = f ′′ (a) = · · · = f (n−l) (a) = 0, and f (n) (a) = 0. By continuing the process as illustrated in I, II, and III, it is seen that if the first derivative of f (x) which does not vanish for x = a is of even order (= n), then2 2 As in §7.4, a critical value x = a is found by placing the first derivative equal to zero andsolving the resulting equation for real roots. 256
269.
10.7. INDETERMINATE FORMS 6. Show that if the first derivative of f (x) which does not vanish for x = a is of odd order (= n), then f (x) is an increasing or decreasing function when x = a, according as f (n) (a) is positive or negative.10.7 Indeterminate formsSome singularities are easy to diagnose. Consider the function cos x at the point x 1x = 0 (see Figure 10.4). The function evaluates to 0 and is thus discontinuousat that point. Since the numerator and denominator are continuous functions andthe denominator vanishes while the numerator does not, the left and right limitsas x → 0 do not exist. Thus the function has an infinite discontinuity at the pointx = 0. cos(x) Figure 10.4: x .More generally, a function which is composed of continuous functions and evalu-ates to a at a point where a = 0 must have an infinite discontinuity there. 0 Other singularities require more analysis to diagnose. Consider the functionssin x sin x sin x 0 x , |x| and 1−cos x at the point x = 0. All three functions evaluate to 0 atthat point, but have different kinds of singularities. The first has a removablediscontinuity, the second has a finite discontinuity and the third has an infinitediscontinuity. See Figure 10.5. An expression that evaluates (for a particular value of the independent variable)to 0 , ∞ , 0 · ∞, ∞ − ∞, 1∞ , 00 or ∞0 is called an indeterminate form. A function 0 ∞ 258
270.
10.8. EVALUATION OF A FUNCTION TAKING ON AN INDETERMINATE FORM sin x sin x sin x Figure 10.5: The functions x , |x| , 1−cos x , resp..h(x) which takes an indeterminate form at x = a is not defined for x = a by thegiven analytical expression. For example, suppose we have f (x) h(x) = , g(x)where at x = a, f (a) = 0, and g(a) = 0.For this value of x our function is not defined and we may therefore assign to itany value we please. It is usually desirable to assign to the function a value thatwill make it continuous when x = a whenever it is possible to do so. L'Hˆ pital's orule, given in (10.19) below, helps us determine this value of h(a) which makes hcontinuous at x = a.10.8 Evaluation of a function taking on an indeter- minate formIf when x = a the function f (x) assumes an indeterminate form, then lim f (x) x=ais taken as the value of f (x) for x = a. The calculation of this limiting value iscalled evaluating the indeterminate form. The assumption of this limiting value makes f (x) continuous for x = a. Thisagrees with the theorem under Case II [§2.6], and also with our practice in Chapter 02, where several functions assuming the indeterminate form 0 were evaluated. 259
271.
010.9. EVALUATION OF THE INDETERMINATE FORM 0 x2 −4 0Example 10.8.1. For x = 2 the function x−2 assumes the form 0 but x2 − 4 lim = 4. x→2 x − 2Hence 4 is taken as the value of the function for x = 2. Let us now illustrategraphically the fact that if we assume 4 as the value of the function for x = 2, 2then the function is continuous for x = 2. Let y = x −4 This equation may also x−2be written in the form y(x − 2) = (x − 2)(x + 2); or (x − 2)(y − x − 2) = 0.Placing each factor separately equal to zero, we have x = 2, and y = x + 2. Also,when x = 2, we get y = 4. In plotting, the loci of these equations are found to be two lines. Since thereare infinitely many points on a line, it is clear that when x = 2, the value ofy (or the function) may be taken as any number whatever. When x is differentfrom 2, it is seen from the graph of the function that the corresponding value ofy (or the function) is always found from y = x + 2, which we saw was also thelimiting value of y (or the function) for x = 2. It is evident from geometricalconsiderations that if we assume 4 as the value of the function for x = 2, then thefunction is continuous for x = 2. Similarly, several of the examples given in Chapter 2 illustrate how the limitingvalues of many functions assuming indeterminate forms may be found by em-ploying suitable algebraic or trigonometric transformations, and how in generalthese limiting values make the corresponding functions continuous at the pointsin question. The most general methods, however, for evaluating indeterminateforms depend on differentiation. 010.9 Evaluation of the indeterminate form 0Given a function of the form f (x) such that f (a) = 0 and g(a) = 0; that is, the g(x) 0function takes on the indeterminate form 0 when a is substituted for x. It is thenrequired to find f (x) lim . x→a g(x)(See Figure 10.6.) Since, by hypothesis, f (a) = 0 and g(a) = 0, these graphsintersect at (a, 0). 260
296.
CHAPTER ELEVEN CurvatureThis is a chapter of advanced topics devoted to the elementary differential geom-etry of curves. Given a curve y = f (x) in the plane, we have studied how well thetangent line at a point P0 = (x0 , y0 ) on the curve approximates the graph near P0 .Analgously, we can study how well the a "tangent circle" at a point P0 = (x0 , y0 )on the curve approximates the graph near P0 . This "tangent circle" is called the"circle of curvature," its radius the "radius of curvature," and its center the "cen-ter of curvature." The topics covered include: the radius of curvature, curvature(which is the inverse of the radius of curvature), circle of curvature, and center ofcurvature.11.1 CurvatureThe shape of a curve depends very largely upon the rate at which the directionof the tangent changes as the point of contact describes the curve. This rate ofchange of direction is called curvature and is denoted by K. We now proceed tofind its analytical expression, first for the simple case of the circle, and then forcurves in general.11.2 Curvature of a circleConsider a circle of radius R. In the notation of Figure 11.1, let 285
297.
11.2. CURVATURE OF A CIRCLE Figure 11.1: The curvature of a circle. τ = angle that the tangent at P makes with the x-axis,and τ + ∆τ = angle made by the tangent at a neighboring point P′ .Then we say ∆τ = total curvature of arc PP′ . If the point P with its tangent besupposed to move along the curve to P′ , the total curvature (= ∆τ ) would measurethe total change in direction, or rotation, of the tangent; or, what is the same thing,the total change in direction of the arc itself. Denoting by s the length of the arcof the curve measured from some fixed point (as A) to P, and by ∆s the length ofthe arc P P′ , then the ratio ∆τ measures the average change in direction per unit ∆slength of arc1 . Since, from Figure 11.1, ∆s = R · ∆τ , or ∆τ = R , it is evident ∆s 1that this ratio is constant everywhere on the circle. This ratio is, by definition, thecurvature of the circle, and we have 1 K= . (11.1) RThe curvature of a circle equals the reciprocal of its radius. ∆τ 1 Thus, if ∆τ = π radians (= 30o ), and ∆s = 3 centimeters, then 6 ∆s = π 18 radians percentimeter = 10o per centimeter = average rate of change of direction. 286
298.
11.3. CURVATURE AT A POINT11.3 Curvature at a pointConsider any curve. As in the last section, ∆τ = total curvature of the arc PP',and ∆τ = average curvature of the arc PP'. ∆s Figure 11.2: Geometry of the curvature at a point. More important, however, than the notion of the average curvature of an arc isthat of curvature at a point. This is obtained as follows. Imagine P to approachP along the curve; then the limiting value of the average curvature = ∆τ as P′ ∆sapproaches P along the curve is defined as the curvature at P, that is, ∆τ dτ Curvature at a point = lim∆s→0 ∆s = ds .Therefore, dτ K= = curvature. (11.2) dsSince the angle ∆τ is measured in radians and the length of arc ∆s in units oflength, it follows that the unit of curvature at a point is one radian per unit oflength. 287
299.
11.4. FORMULAS FOR CURVATURE11.4 Formulas for curvatureIt is evident that if, in the last section, instead of measuring the angles which thetangents made with the x-axis, we had denoted by τ and τ + ∆τ the angles madeby the tangents with any arbitrarily fixed line, the different steps would in no wisehave been changed, and consequently the results are entirely independent of thesystem of coordinates used. However, since the equations of the curves we shallconsider are all given in either rectangular or polar coordinates, it is necessary dyto deduce formulas for K in terms of both. We have tan τ = dx by §3.9, or dyτ = arctan dx . Differentiating with respect to x, using (4.23) in §4.1, d y 2 dτ dx2 = 2. dx dy 1 + dxAlso 1 2 2 ds dy = 1+ , dx dxby (8.4). Dividing one equation into the other gives dτ d2 y dx dx2 ds = 3 . dx dy 2 2 1+ dxBut dτ dx dτ ds = = K. dx dsHence d2 y dx2 K= 3 . (11.3) dy 2 2 1+ dxIf the equation of the curve be given in polar coordinates, K may be found asfollows: From (5.13), τ = θ + ψ. 288
| 677.169 | 1 |
Hi prosperous,
Why don't you let us know your grade level, the math courses that you have taken, and some examples of math topics that interest you. That will help us find a suitable project for you. If there are any projects that you've seen that sound interesting, but are too simple, please let us know. It is generally not too hard to build more complexity into a simple project.
| 677.169 | 1 |
Editorial Reviews
From the Publisher
"A brilliant and engrossing view of the development of mathematics...wonderful at communicating its beauty and excitement to the general reader." The New York Times
"A perfectly marvelous book." The New Yorker
"A true gem, one of the masterpieces of our age." American Monthly
New York Times Book Review
"A brilliant and engrossing view of the development of mathematics...wonderful at communicating its beauty and excitement to the general reader." --New York Times
Booknews
The addition of exercises and problems converts the 1981 edition into a textbook for math courses for liberal arts students and future secondary school math teachers, and courses in the appreciation of mathematics. Among the topics are the mathematical landscape, why math works and what it is good for, teaching and learning, certainty and fallibility, and mathematical reality. Includes a glossary without pronunciation
| 677.169 | 1 |
0618218785
9780618218783
Details about Intermediate Algebra:
Intermediate Algebra: Graphs and Functions 3e offers proven pedagogy, innovative features, real-life applications and flexible technology with comprehensive coverage for a solid course in intermediate algebra. In general, Larson's early functions approach along with early polynomials makes the topical organisation very appealing, and highly effective method of teaching / learning for teacher and pupil alike. Highlights of this third edition include: New Side-by-side Algebraic, Graphical, and Numerical Solutions New Chapter Opener - each chapter now opens with an objective based overview of the chapter concepts and Key Terms, a list of the mathematical vocabulary New Collaborate Appearing at the end of selected sections these activities can be assigned for small group work or for whole class discussions Revised Exercise Sets -now grouped into four categories: Developing Skills, Solving Problems, Explaining Concepts, and Ongoing Review they offer computational, applied, and conceptual problems
Back to top
Rent Intermediate Algebra 3rd edition today, or search our site for Larson textbooks. Every textbook comes with a 21-day "Any Reason" guarantee. Published by CENGAGE Learning.
| 677.169 | 1 |
Now students have nothing to fear!
Math textbooks can be as baffling as the subject they're teaching. Not anymore. The best-selling author of The Complete Idiot's Guide® to Calculus has taken what appears to be a typical calculus workbook, chock full of solved calculus problems, and made legible notes in the margins, adding missing steps
This concise, self-contained textbook gives an in-depth look at problem-solving from a mathematician's point-of-view. Each chapter builds off the previous one, while introducing a variety of methods that could be used when approaching any given problem. Creative thinking is the key to solving mathematical problems, and this book outlines the tools... more...
Slay the calculus monster with this user-friendly guide Calculus For Dummies, 2 nd Edition makescalculus manageable—even if you're one of the many studentswho sweat at the thought of it. By breaking down differentiationand integration into digestible concepts, this guide helps youbuild a stronger foundation with a solid understanding of... more...
| 677.169 | 1 |
Math Mammoth Ratios, Proportions & Problem Solving is a worktext that concentrates, first of all, on two important concepts: ratios and proportions, and then on problem solving. The book suits... More > best grade levels 6 and 7.
The book starts out with the basic concepts of ratio, rate, and unit rate. The lessons on equivalent rates allow students to solve a variety of problems involving ratios and rates. We connect the tables of equivalent rates with ordered pairs, use equations (such as y = 3x) to describe these tables, and plot the ordered pairs. The lessons about proportions show how to solve proportions using cross-multiplying and how to set up proportions the correct way.
The concept of direct variation is introduced in the lesson Proportional Relationships.
The lessons Scaling Figures, Floor Plans, and Maps give useful applications and more practice to master the concepts of proportions.
Lastly, we study various kinds of word problems involving ratios and use a bar model to solve these problems everything you wanted to know about ratios. Topics include rates and unit rates, equivalent ratios, groupings, combining ratios, similar figures, solving proportions, setting up proportions to solve word problems, and much more!Descriptive and Inferential Statistics, is a book that is intended for university students of any college. You'll find theory as summaries, and exercises solved, on the following topics: Descriptive... More > Statistics, Confidence Intervals and Test Hypothesis for means, proportions and variances for one sample, Chi Square Test, Test Hypothesis for means, proportions and variances, for two or more samples, and Regression line. Statistical software such as SPSS, Minitab, programs have been used in the resolution of problems and in some cases have been resolved by using the Excel and also manually.< Less
The key to doing well on the SAT Math is knowing how to set up and solve word problems.
The SAT Math Review Book for People Who Hate Math differs from the other books on the market because it gives... More > you in-depth teaching on word problems. By studying this book, you will learn how to set up and solve different kinds of word problems: distance, rate of work, mixture, age, money, Pythagorean Theorem problems and many more.
In addition to word problems, the book contains a complete review of arithmetic, algebra, and geometry
Instead of spending four years at your "safety school," get into the college of your dreams by scoring well on the SAT.< Less
A tale of epic proportions written in the style of a ballad poem. Young Harold Sprack has problems and the only way to solve them is by slaying forty dragons. Can he do it? Or will the dragons get... More > the better of him
| 677.169 | 1 |
Essentials of Geometry for College Students
Browse related Subjects
...Read More deductive proof. To make learning interactive and enjoyable, this new edition includes exciting new features such as Technology Connections and Hands-on Activities. Knowledge of beginning algebra and a scientific calculator are required for this text
| 677.169 | 1 |
Synopses & Reviews
Publisher Comments:
Quantitative Reasoning: Tools for Today's Informed Citizen helps readers connect mathematics in the real world. Through a series of hands-on activities and explorations, the text empowers readers by teaching them to apply quantitative reasoning skills to make informed decisions in their daily lives. Technology is an essential component of the text, and it is integrated in Activities that use Microsoft? Excel or, alternatively, the TI-83/84 Plus graphing calculator. The use of technology helps readers concentrate on ideas rather than computational details--allowing them to investigate significant and more realistic
| 677.169 | 1 |
Numericon tells the stories of the numbers, mathematical discoveries, oddities and personalities that have shaped the way we understand the world around us. Each chapter is its own story about a number: why 12 is a sublime number, why 13 is unlucky and 7 lucky, and how imaginary numbers hold up buildings. The book tells the stories of ancient mathematicians,... more...
book?s simple approach easy to understand while... more...
Introduction to Theory of Control in Organizations explains how methodologies from systems analysis and control theory, including game and graph theory, can be applied to improve organizational management. The theory presented extends the traditional approach to management science by introducing the optimization and game-theoretical tools required... more...
A considerable number of papers on Pareto distributions and related topics have appeared since the first edition of this hadbook. This book updates the popular first edition by interleafing developments inequality indices, parametric families of Lorenz curves, Pareto processes, hidden truncation, and more.Praise for the First Edition "…[t]he book is great for readers who need to apply the methods and models presented but have little background in mathematics and statistics." - MAA Reviews Thoroughly updated throughout, Introduction to Time Series Analysis and Forecasting, Second Edition presents the underlying theories of time series... more...
What goes on inside the mind of a rock-star mathematician? Where does inspiration come from?
With a storyteller?s gift, CÚdric Villani takes us on a mesmerising journey as he wrestles with a new theorem that will win him the most coveted prize in mathematics. Along the way he encounters obstacles and setbacks, losses of faith and even... more...
A concise yet rigorous introduction to logic and discrete mathematics. This book features a unique combination of comprehensive coverage of logic with a solid exposition of the most important fields of discrete mathematics, presenting material that has been tested and refined by the authors in university courses taught over more than a decade. ... more...
Solutions manual to accompany Logic and Discrete Mathematics: A Concise Introduction This book features a unique combination of comprehensive coverage of logic with a solid exposition of the most important fields of discrete mathematics, presenting material that has been tested and refined by the authors in university courses taught over more... more...
| 677.169 | 1 |
Curriculum
Courses offered in the Mathematics department:
Prerequisites: All prerequisites must have been completed or assessments taken within two years
of the course start date. Students must be sure that they meet the prerequisite in
order to be successful in a mathematics course. The College reserves the right to
drop a student from a class without refund if the prerequisite has not been met.
Course Descriptions
MATH 0500 Arithmetic & Algebra Review 1 cr (formerly Math 50)
Provides an intensive arithmetic and elementary algebra review that may be taken before
taking the IHCC assessment inventory. The student may also take this review and then
re-assess. The course is designed only for students who have previously learned arithmetic
and elementary algebra but need a refresher. Each class will consist of a fast-paced
review of specific topics.
MATH 0640 Essential Mathematics 2 cr (formerly Math 90)
Covers basic arithmetic operations by both hand computation and calculator of whole
numbers, fractions, mixed numbers, decimals and percents; comparisons and conversions
among different forms of whole numbers, fractions, mixed numbers, decimals, and percents;
estimation; numbers properties; natural number exponents; order of operations; and
applications. Prereq: Recommendation based on the results of the Inver Hills Assessment.
MATH 0740 Prealgebra 4 cr (formerly Math 92)
Integrates topics of algebra throughout, including simplifying expressions and solving
equations. Covers: operations with and properties of signed rational numbers; conversion
between fractions, decimals, and percents; estimation; exponents; order of operations;
ratios, rates, and proportions; descriptive statistics; interpretation of data from
charts, tables, and graphs; US & metric measures and conversions; square roots; similarity;
perimeter, area, and volume of geometric shapes; applications; graphing linear equations
by point-plotting; and positive and negative slope. Prereq: Recommendation based on
the results of the Inver Hills Assessment Inventory or a grade of "C" or higher in
Math 0640.
Is a combined course that allows students to complete both Math 0740 and Math 08 0740: Integrates topics of
algebra throughout, including simplifying expressions and solving equations. Covers:
operations with and properties of signed rational numbers; conversion between fractions,
decimals, and percents; estimation; exponents; order of operations; ratios, rates,
and proportions; descriptive statistics; interpretation of data from charts, tables,
and graphs; US & metric measures and conversions; square roots; similarity; perimeter,
area, and volume of geometric shapes; applications; graphing linear equations by point-plotting;
and positive and negative slope. MATH 0840: Course topics include real number operations
and properties; integer exponents; radicals; polynomial operations; reduction, multiplication,
and division of rational expressions; solving linear equations and inequalities; solving
quadratic equations by factoring; solving square root equations; graphing equations
in two variables; and applications of the above topics. This course will not fulfill
any degree requirements. Prereq: Recommendation based on the results of the Inver
Hills Assessment Inventory or grade of "C" or higher in Math 0640.
MATH 0820 Algebra Fundamentals 4 cr
Is a course designed to cover algebra topics that are prerequisite to Math 1101, 1103,
and 1107. It will not fulfill the prerequisite for Math 1118 or 1127. Topics include:
order of operations; exponents; scientific notation; significant digits; variables;
expressions; linear equations and inequalities; linear problem solving; proportions;
linear graphs; intro to functions; 2x2 systems of equations; quadratic, square root,
and exponential functions and their graphs; regression and modeling; summation notation;
applications integrated throughout; and use of a graphing utility or Excel spreadsheets
integrated throughout.
MATH 0840 Introductory Algebra 4 cr (formerly Math 94)
Is the first of a two-semester pre-college algebra sequence. The course is for students
needing work in beginning algebra who are preparing for intermediate algebra. The
topics in this course include real number operations and properties; integer exponents;
radicals; polynomial operations; reduction, multiplication, and division of rational
expressions; solving linear equations and inequalities; solving quadratic equations
by factoring; solving square root equations; graphing equations in two variables;
and applications of the above topics. This course will not fulfill any degree requirements.
Prereq: Recommendation based on the results of the Inver Hills Assessment Inventory
or grade of "C" or higher in MATH 0740.
Can be taken in place of Math 0940 to fulfill the intermediate algebra prerequisite
for Math 1101 and Math 1103. This course will NOT fulfill the intermediate algebra
prerequisite requirement for Math 1105, Math 1106, Math 1118, or Math 1127. Topics
include functions, linear functions and slope, linear inequalities in one and two
variables, compound inequalities, linear systems in two and three variables, matrices,
integer exponents, radicals and radical functions, rational exponents, complex numbers,
polynomial functions, the quadratic function and its graph, solving quadratic equations,
quadratic formula, exponential functions, logarithmic expressions and functions, summation
notation, and sequences and series. Applications will be integrated throughout. Prereq:
Recommendation based on the results of the Inver Hills Assessment Inventory or a grade
of "C" or higher in MATH 0840.
Is a combined course that allows students to complete both Math 0840 and Math 09 0840: Course topics include
real number operations and properties; integer exponents; radicals; polynomial operations;
reduction, multiplication, and division of rational expressions; solving linear equations
and inequalities; solving quadratic equations by factoring; solving square root equations;
graphing equations in two variables; and applications of the above topics. MATH 0940: This course will not fulfill any degree
requirements. Prereq: Recommendation based on the results of the Inver Hills Assessment
Inventory or a grade of "C" or higher in MATH 0740.
MATH 0940 Intermediate Algebra 5 cr (formerly Math 99)
Is the second semester of a two-semester sequence intended for students preparing
for course work in Math for Elementary Education, College Algebra, or Pre-Calculus. Prereq: Recommendation based on the results
of the Inver Hills Assessment Inventory or a grade of C or higher in MATH 0840.
MATH 1101 Math for Liberal Arts 3 cr
Is designed to give non-mathematicians an appreciation of mathematical ideas and the
power and utility of mathematical skills in the modern world. Topics will be selected
from: voting and weighted voting systems, fair division and apportionment; routing,
minimum network and scheduling problems; mathematical growth and math of finance;
descriptive statistics and data analysis; basic probability and normal distributions.
Prereq: Recommendation based on the results of the Inver Hills/Assessment Inventory
or a grade of "C" or higher in MATH 0096/0820/0880/0940.
MATH 1103 Introduction to Statistics 4 cr
Presents basic statistics for business management, economics, education, psychology,
science or sociology majors. Topics include descriptive and graphical statistics,
basic probability, probability distributions, sampling distributions, confidence intervals,
hypotheses testing, linear regression, chi-square tests, and ANOVA. Approximately
20 hours of computer time outside of class is required for Minitab proficiency. Prereq.:
Recommendation based on the results of the Inver Hills Assessment Inventory or a grade
of "C" or higher in MATH 0096/0820/0880/0940.
MATH 1107 Mathematical Foundations 4 cr
Includes the topics: analysis of the problem solving process; sets and number systems;
operations, properties, and computation with the real numbers and their classic subsets;
number theory; decimals, ratio, proportion, and percents; statistics; probability;
measurement; relations, functions and patterns; algebra models; geometric shapes;
and applications. This course can be used for prospective elementary education majors
and liberal arts students. Prereq: Recommendation based on the results of the Inver
Hills Assessment Inventory or grade of "C" or higher in Math 0820/0880/0940, or completion
of a college level math course.
MATH 1118 College Algebra I 4 cr
Prepares students for Math 1120. It may also be taken as a first course in a sequence
with Math 1119 to prepare students for the Calculus sequence (MATH 1133). Topics include
algebraic, exponential, and logarithmic functions; graphing absolute value, reciprocal,
square root, piecewise defined, polynomial, rational, logarithmic, and exponential
functions; transformations, inverse functions, modeling problems, zeros of polynomials,
systems of linear and non-linear equations, and matrices. Time permitting, additional
topics selected from data analysis, determinants, arithmetic and geometric sequences,
linear programming, and binomial theorem. Graphing calculator required. Prerequisite:
Recommendation based on the results of the Inver Hills Assessment Inventory or grade
of C or higher in Math 0940.
MATH 1119 College Algebra II 3 cr
Is the second course in a two-semester sequence preparing students for the Calculus
sequence. Topics include trig functions, identities and applications, law of sines
and cosines, graphing, and solving trig equations, parametric equations, conics and
polar coordinates. Graphing calculator required. Prereq: Recommendation based on the
results of the Inver Hills Assessment Inventory or a grade of C or higher in MATH
1118.
MATH 1120 Survey of Calculus 3 cr
Prepares students who may be required to have an introductory course in calculus for
business, forestry, and other majors. This course is not intended as a prerequisite
for other courses in calculus. Topics include functions and limits; differentiation
of algebraic, exponential, and logarithmic functions; application of differentiation
to optimization problems, especially business-related problems; integration of the
basic forms of the previously mentioned functions, with applications to area and economics.
Graphing calculator is required. Prereq: Recommendation based on the Inver Hills Assessment
Inventory results or grade of "C" or higher in MATH 1118 or MATH 1127.
MATH 1127 PreCalculus 5 cr
Is a preparation for the calculus sequence. Topics include the study of algebraic,
exponential, logarithmic, and trigonometric functions; graphs of absolute value, radical,
polynomial, rational, logarithmic, exponential, and trigonometric functions; graph
reading, transformations; additional topics include solving equations, inequalities,
trigonometric identities, right triangle trig., polar coordinate and binomial theorem.
Graphing calculator is required. Prereq: Recommendation based on the results of the
Inver Hills Assessment Inventory or a grade of B or higher in MATH 0940.
MATH 1133 Calculus I 5 cr
Is intended for students majoring in math, computer science, engineering, science,
pharmacy, architecture or pre-med. The topics include limits, derivatives (including
trigonometric, exponential and logarithmic functions), continuity, applications of
derivatives to related rates, max/min and graphing, differential equations, Riemann
Sums, basic integration techniques, definite integrals, Fundamental Theorem of Calculus,
Trapezoidal and Simpson's Rule. Students will become proficient with a graphing calculator
and a computer algebra system such as Maple. Graphing calculator is required. Prereq:
Recommendation based on the Inver Hills Assessment Inventory results or grade of "C"
or higher in MATH 1127 or MATH 1119.
MATH 1134 Calculus II 5 cr
Is a continuation of Math 1133. Topics include applications of the integral-area,
volume, center of mass, work, liquid pressure and arc length; techniques of integration,
conics, polar coordinates, improper integrals, and infinite series. Prereq: Grade
of C or higher in MATH 1133.
MATH 2219 Multivariable Calculus 5 cr
Is intended for students majoring in math, science, computer science or engineering.
The topics include vectors in three-dimensional space, quadric surfaces, limits in
two and three dimensions, partial derivatives, gradients, extreme value problems,
multiple integration, space curves, curvature, the Frenet frame, divergence, curl,
line integrals, conservative vector fields and potential functions, surface and volume
integrals, Green's, Stokes' and the Divergence Theorems. Knowledge of a computer algebra
system such as DERIVE is expected. Prereq: Grade of C or higher in MATH 1134.
MATH 2223 Linear Algebra and Differential Equations 5 cr
Is intended for students majoring in math, science, computer science or engineering.
Topics include systems of linear equations, matrices, determinants, vector spaces,
linear transformations and eigenvalues and eigenvectors, basic definitions, ideas,
and terminology of ordinary differential equations. Emphasis on the techniques of
problem solving. Specific topics covered include solutions and applications of first-order
differential equations, solutions of linear differential equations of higher order,
applications of second-order linear differential equations, power series solutions,
the Laplace transform, and systems of linear differential equations. There is also
an introduction to numerical and graphical methods of solution. Selected applications
and use of technology will be included. Graphing calculator required and knowledge
of a computer algebra system such as Derive, Maple, or Mathematica expected. Prereq:
Grade of "C" or higher in Math 2219.
| 677.169 | 1 |
Store Search search Title, ISBN and Author Schaum's Easy Outline of Linear Algebra by Seymour Lipschutz Estimated delivery 3-12 business days Format Paperback Condition Brand New Boiled-down essentials of the top-selling Schaum #039;s Outline series for the student with limited time What could be better than the bestselling Schaum #039;s Outline series? For students looking for a quick nuts-and-bolts overview, it would have to be Schaum #039;s Easy Outline series. Every book in this series is a
Store Search search Title, ISBN and Author Schaum's Outline of Abstract Algebra by Lloyd R. Jaisingh, Lloyd R. Jasingh Estimated delivery 3-12 business days Format Paperback Condition Brand New Confusing Textbooks?Missed Lectures?Tough Test Questions? Fortunately for you, there #039;s Schaum #039;s Outlines. More than 40 million students have trusted Schaum #039;s to help them succeed in the classroom and on exams. Schaum #039;s is the key to faster learning and higher grades in every subject. Eand clear information make this bulletin board set doubly appealing! Vibrant color and contemporary 3-D graphics enhance this well-organized collection of algebra terms, which is comprised of (4) 17 x 24 inch charts. Each algebra term is defined using straightforward language and examples as well as any associated symbols and formulas.
make the transition from Algebra to Algebra II! Written for teachers to use as a full unit of study or as a supplement to their curriculum, this book helps simplify algebraic concepts. Parents and students can also use this resource at home as a tutorial or to enhance what is being taught in the classroom. Each book includes: simple step-by-step instructions with examples, practice problems using the concepts, real-life applications, a list of symbols and terms, tips, answer keys, and references. This book also meets NCTM Standards and Expectations.Key To Algebra offers a unique, proven way to introduce algebra to your students. New concepts are explained in simple language and examples are easy to follow. Word problems relate algebra to familiar situations, helping students understand abstract concepts. Students develop understanding by solving equations and inequalities intuitively before formal solutions are introduced. Students begin their study of algebra in Books 1-4 using only integers. Books 5-7 introduce rational numbers and expressions. Books 8-10 extend coverage to the real number system. This kit contains only Books 1-10. Answers Notes for Books 1-4 Books 5-7 and Books 8-10 are available separately, as well as the Key to Algebra Reproducible Tests
| 677.169 | 1 |
GeoGebra is a dynamic mathematics software for education in secondary schools that joins geometry, algebra and calculus. On...
see more
GeoGebra is a dynamic mathematics software for education in secondary schools that joins geometry, algebra and calculus. On the one hand, GeoGebra is a dynamic geometry system. You can do constructions with points, vectors, segments, lines, conic sections as well as functions and change them dynamically afterwards.On the other hand, equations and coordinates can be entered directly. Thus, GeoGebra has the ability to deal with variables for numbers, vectors and points, finds derivatives and integrals of functions and offers commands like Root or Extremum.The GeoGebraWiki is a free pool of educational materials for GeoGebra. Everyone can contribute and upload materials there: International GeoGebraWiki - pool of educational materials for GeoGebra and the German GeoGebraWiki The Dynamic Worksheets GeoGebra can also be used to create dynamic worksheets:Pythagorasvisualisation of Pythagoras' theoremLadder against the Wallapplication of Pythagoras' theorem Circle and its Equationconnection between a circle's center, radius and equation Slope and Derivative of a Function (3 sheets)relation between slope, derivative and local extrema of a functionDerivative of a Polynomial interactive exercise to practice finding the derivative of a cubic polynomialUpper- and Lower Sums of a Functionvisualisation of the backgrounds of Riemann's IntegralGebra: Educational Materials & Dynamic Worksheets to your Bookmark Collection or Course ePortfolio
Select this link to close drop down of your Bookmark Collection or Course ePortfolio for material GeoGebra: Educational Materials & Dynamic Worksheets
Select this link to open drop down to add material GeoGebra: Educational Materials & Dynamic Workshe Geriatric Jeopardy to your Bookmark Collection or Course ePortfolio
Select this link to close drop down of your Bookmark Collection or Course ePortfolio for material Geriatric Jeopardy
Select this link to open drop down to add material Geriatric Jeopardy to your Bookmark Collection or Course ePortfolio
"German for Music Lovers" is a collection of language-learning resources for students, teachers and anyone who...
see more
"German for Music Lovers" is a collection of language-learning resources for students, teachers and anyone who loves music. Ranging in style from classical to hiphop, from rock to rap, each song is furnished with a German/English vocabulary and supplemented with interestingGerman History in Documents and Images (GHDI) is a comprehensive collection of primary source materials documenting Germany's...
see more
German History in Documents and Images (GHDI) is a comprehensive collection of primary source materials documenting Germany's political, social, and cultural history from 1500 to the present. It comprises original German texts, all of which are accompanied by new English translations, and a wide range of visual imagery. The materials are presented in ten sections, which have been compiled by leading scholars. All of the materials can be downloaded free of charge for teaching, research, and related purposes; the site is strictly intended for individual, non-commercial History in Documents and Images to your Bookmark Collection or Course ePortfolio
Select this link to close drop down of your Bookmark Collection or Course ePortfolio for material German History in Documents and Images
Select this link to open drop down to add material German History in Documents and Images use of discussion boards to your Bookmark Collection or Course ePortfolio
Select this link to close drop down of your Bookmark Collection or Course ePortfolio for material Improving the use of discussion boards
Select this link to open drop down to add material Improving the use of discussion boards Tutorial to your Bookmark Collection or Course ePortfolio
Select this link to close drop down of your Bookmark Collection or Course ePortfolio for material Information Literacy Tutorial
Select this link to open drop down to add material Information Literacy Tutorial to your Bookmark Collection or Course ePortfolio
In this workshop you will study principles of pedagogy and instructional design and apply these theories to one of your...
see more Japanese Religion to your Bookmark Collection or Course ePortfolio
Select this link to close drop down of your Bookmark Collection or Course ePortfolio for material Intro to Japanese Religion
Select this link to open drop down to add material Intro to Japanese Religion Podcasting to your Bookmark Collection or Course ePortfolio
Select this link to close drop down of your Bookmark Collection or Course ePortfolio for material Introduction to Podcasting
Select this link to open drop down to add material Introduction Lightness Perception and Lightness Illusions to your Bookmark Collection or Course ePortfolio
Select this link to close drop down of your Bookmark Collection or Course ePortfolio for material Lightness Perception and Lightness Illusions
Select this link to open drop down to add material Lightness Perception and Lightness Illusions to your Bookmark Collection or Course ePortfolio
| 677.169 | 1 |
Note: Citations are based on reference standards. However, formatting rules can vary widely between applications and fields of interest or study. The specific requirements or preferences of your reviewing publisher, classroom teacher, institution or organization should be applied.
Discourses on algebra
Euclid's classic geometry attracts many by its beauty, elegance and logical cohesion. In this book, Russian algebraist I.R. Shafarevich argues with examples that algebra is no less beautiful, elegant and logically cohesive than geometry. The book presupposes very limited mathematical knowledge.Read more...
Reviews
Editorial reviews
Publisher Synopsis
From the reviews: "... What I found particularly attractive about this book are the historical notes, the references to many mathematicians and their work, as well as many original proofs included. In closing: I think that any student and any teacher interested in a deeper study of elementary (and maybe not so elementry) study of such topics as sets, polynomials, and numbers should read (pencil in hand!) this book. It may be particulary valuable for future teachers. The book is very well written, and it has detailed proofs and many exercises. Above all, this book will be remembered for its beauty and elegance." (M. Poplicher, Read This! The MAA Online book review column (for complete review see "... As the book progresses it becomes increasingly clear that the author has planned the development of ideas meticulously, not only so that he can constantly surprise the reader with the next turn of events, but also so that he can demonstrate unexpected links between results in algebra, number theory and probability. There are numerous ideas here for providing stimulating lessons for able pupils as well as for self-study, and the book would make a valuable addition to the school or department library." (G. Leversha, The Mathematical Gazette, 88:511, 2004) "Since it happens so rarely, one must take notice when a great master of mathematics writes an elementary text. ... Shafarevich takes the subject much farther than any high school text. ... This book could form the basis for a great "transition to abstraction" course and should certainly have a role in programs training high school mathematics teachers. Summing up: Recommended. General readers; lower- and upper-division undergraduates; professionals." (D.V. Feldman, Choice, June 2003) "... While written for students at the secondary level, the text goes quite thoroughly into foundational matters normally encountered at university ... This book is highly recommended for teenagers with a strong desire to study mathematics and for secondary teachers seeking to contextualize what is on the school syllabus and to appreciate what some of their students will meeet in university mathematics." (Edward J. Barbeau, Zentralblatt fur Mathematik 1004.00001) "Discourses on Algebra is an advanced elementary algebra book intended to supplement the content of algebra courses ... . All the sections in the book are followed by exercises intended to test both the understanding of the material presented and proved in the book, and the creativity and mathematical skills of the readers. ... The book is very well written, and it has detailed proofs and many exercises. Above all, this book will be remembered for its beauty and elegance." (Mihaela Poplicher, MAA Online, September, 2004) "In this introduction to algebra, the author aims to show that the subject is no less beautiful, elegant and logically coherent than Euclidean geometry. ... it should appeal to able students and undergraduates who are looking for a more sustained and challenging course in elementary algebra. ... There are numerous ideas here for providing stimulating lessons for able pupils as well as for self-study, and the book would make a valuable addition to the school or department library." (Gerry Leversha, The Mathematical Gazette, 88:511), 2004) "In this book, the author - a famous Russian algebraist - argues with examples that algebra is no less beautiful, elegant, and logically cohesive than geometry. It contains an exposition of some rudiments of algebra, number theory, set theory and probability presupposing very limited knowledge of mathematics." (Zentralblatt fur Didaktik der Mathematik, June 2002) "In this book, 'the elements of algebra as a field of contemporary mathematics are laid out, based on material bordering the school program as closely as possible'. The author would like algebra to appeal to students in the way that Euclidean geometry does. ... the text goes quite thoroughly into foundational matters normally encountered at university. ... This book is highly recommended for teenagers with a strong desire to study mathematics and for secondary teachers seeking to contextualize what is on the school syllabus and to appreciate what some their students will meet in university mathematics." (Edward J. Barbeau, Zentralblatt MATH, 1004:4, 2003)Read more...
| 677.169 | 1 |
Have you ever been frustrated at calculation problems? Have you ever thought that algebra might be impossible? If you can read, you can learn algebra. Believe it or not, math is just a language--a language of patterns and numbers. Here's where you start.
Ad
Steps
1
Review every time your teacher teaches you. If you do not understand something, ask your parents, your friends, or even your older siblings. They would definitely be glad to help you. You should review the material very day that the teacher teaches you because who knows? Maybe the next day when you try to review, you realize that you have forgotten an important part. Review while it is still fresh in your mind.
Ad
2
Do a prep study. Preparation studies help you. If you prepare yourself to "think in math language", you will know what the teacher is talking about the next time she teaches you a lesson.
3
Focus and ask questions during class. If you don't focus, you won't get much information out of what the teacher says. If you have questions about the lesson, you can ask the teacher, or, if you are a shy person, ask the person sitting next to you without distracting the class or the teacher. Or better yet, maybe you can ask your friends after class. It has to be as soon as the class is over so you don't forget. You can also ask your teacher after school, if he/she is free.
4
Study, study, study! But never cram. Some people think that cramming is good because it makes them study more. However, cramming will make you forget the material the next day when you have a test. Too many new things at once just won't stay in your mind. Two weeks before the test, study somewhere between 30 minutes to 1 hour everyday.
5
The more problems you solve, the better. Practicing is always the best way to study algebra. The point in solving problems is to get used to using the formulas! That way, you can solve the problems faster and you can think faster. Practice, practice, practice. This applies to anything, really. If you practice you will get better at it and you will succeed!
6
Never cheat! Never cheat on a test. Cheating can get you a big old goose egg, "0". You don't want that to happen do you? Cheating can lead to suspension and even expulsion, so if you ever feel the need to glance over and compare answers control yourself, you never know what the consequence can be..but the worst part is that when you cheat, you don't learn. If you go to school and you don't learn, the whole thing seems like a waste of time. Cheating cheats YOU out of an education.
7
Do not panic during a test. Your mind won't be able to think clearly and you will forget the formulas. If you have studied and can solve the problems you do for homework, you will probably do fine on the test. Do not panic.
8
Accept your grades. That's right, accept it. When you get your test back and find out that you have gotten a bad grade, accept it as your own responsibility. That will make you want to figure out how to learn the material. Most importantly, make sure to put in extra effort the next time you study for a test Did you get a bad grade because you didn't study enough? Did you make careless errors? Whatever the reason, locate it and fix it as soon as possible without getting too down on yourself.
Look at your score. Do you want to make it better? Never think, "Hurray! I got 100! Now I don't have to study anymore!" If you think that, then you will get a bad grade next time because you will feel that there is no need to study. If you get a 100, just think, "I'll have to keep it up!"
Make corrections. Look at the problems you got wrong and find out where you went wrong. Practice them again and again until you get it right. Review each day. The whole point of the tests you take in school is to figure out what things you still need to learn, then to work on those things.
Reflect on your scores.
9
You will be better in algebra if you develop and keep up good efforts so that you can build your confidence by establishing good study habits!
Ad
We could really use your help!
Can you tell us about currency exchange?
Yes
No
currency exchange
how to buy Euros
Can you tell us about Minecraft?
Yes
No
Minecraft
how to get fish in Minecraft
Can you tell us about dogs?
Yes
No
dogs
how to hold a Dachshund properly like most people, the best time to study is when you are fresh. If you are too tired, your mind can go blurry. Try a quick review of your studies in the morning as soon as you wake up, or maybe after breakfast. Go through simple calculations, review steps, techniques, and rules that you have already begun to understand before moving on to the parts that you don't yet understand. That will keep you from beating yourself up about something you don't understand. Studying right can clear your mind and might even help you have greater confidence toward other subjects.
Practice daily. Not only it will help to memorize methods and steps, but also helps to increase your speed at doing math!
Try to understand, rather than plainly memorizing; the former leads you to appreciate algebra (and mathematics as a whole), and will help create an enjoyable environment for you to continue your journey into the world of mathematics.
Always be sure to ask a friend, teacher, or family member if you need help. They will most likely be very happy to help you.
Warnings
Don't over study (called cramming for a test) so much that you get no sleep, get so tired that you can't see straight, or get too nervous to think. Take breaks and don't overdo it. Over-studying won't make you learn more, but it could give you a bad headache
| 677.169 | 1 |
Mathematical Sciences Department
Leonid Kalachev, Chair
Mathematics is studied both as a tool and for its own sake. Its usefulness in the sciences - physical, biological, social, behavioral, and environmental - and in decision-making processes is so established that it is an indispensable part of many curricula.
Mathematics is chosen as a major area of study by individuals who find it challenging, fascinating, and beautiful. It is also appreciated by many who seek primarily to use mathematics as a tool.
A career in mathematics, except for teaching at the secondary level, generally requires a graduate degree as preparation. Careers include teaching, research, and the application of mathematics to diverse problems in institutions of higher learning, business, industry, and government.
The Bachelor of Arts, Master of Arts, and Doctor of Philosophy degrees are offered as well as a Bachelor of Science in Mathematical Sciences–Computer Science.
High School Preparation: For studying mathematics at the University level, it is recommended that the high school course work consist of four years of college-preparatory mathematics, including geometry, trigonometry, and college algebra or precalculus. A course in calculus or statistics is helpful, but not necessary. It is unusual to complete an undergraduate degree in mathematics in four years without the necessary background to take Calculus I (M 171) during the freshman year (preferably during the first semester at the university).
Lecturers
Lauren Fern, Lecturer
Cindy Leary, Lecturer
Regina Souza, Lecturer
Emeritus Professors
Richard Billstein, Emeritus
Course Descriptions
Mathematics
M 104 - Numbers as News
Credits: 3. Level: Undergraduate.
Offered spring. Prereq., M 090 with a grade of B- or better, or M 095, or ALEKS placement >= 3. An exploration of mathematics and statistics as used in the popular media. For students in the School of Journalism only.
M 105 - Contemporary Mathematics
Credits: 3. Level: Undergraduate.
Offered every term. Prereq., M 090 with a grade of B- or better, or M 095, or ALEKS placement >= 3. An introduction to mathematical ideas and their impact on society. Intended for students wishing to satisfy the general education mathematics requirement.
M 115 - Probability and Linear Math
Credits: 3. Level: Undergraduate.
Offered every term. Prereq., M 090 with a grade of B- or better, or M 095, or ALEKS placement >= 3. Systems of linear equations and matrix algebra. Introduction to probability with emphasis on models and probabilistic reasoning. Examples of applications of the material in many fields.
M 118 - Math for Music Enthusiasts
Credits: 3. Level: Undergraduate.
Offered autumn and/or spring. Prereq. M 090 with a grade of B- or better, or M 095, or ALEKS placement >= 3; and elementary music background. An introduction to the interplay between mathematics and music. Course intended for Music majors/minors, and others with musical backgrounds/interests, who wish to satisfy the general education mathematics requirement.
M 122 - College Trigonometry
Credits: 3. Level: Undergraduate.
Offered autumn and spring. Prereq., M 121 or ALEKS placement >= 4. Preparation for calculus based on college algebra. Review of functions and their inverses. Trigonometric functions and identities, polar coordinates and an optional topic such as complex numbers, vectors or parametric equations. Credit not allowed for both M 122 and M 151.
M 151 - Precalculus
Credits: 4. Level: Undergraduate.
Offered autumn and spring. Prereq., ALEKS placement >= 4. A one semester preparation for calculus (as an alternative to M 121-122. Functions of one real variable are introduced in general and then applied to the usual elementary functions, namely polynomial and rational functions, exponential and logarithmic functions, trigonometric functions, and miscellaneous others. Inverse functions, polar coordinates and trigonometric identities are included. Credit not allowed for both M 151 and M 121 or 122.
M 162 - Applied Calculus
Credits: 4. Level: Undergraduate.
Offered autumn and spring. Prereq., ALEKS placement >= 5 or one of M 121, 122 or 151. Introductory course surveying the principal ideas of differential and integral calculus with emphasis on applications and computer software. Mathematical modeling in discrete and continuous settings. Intended primarily for students who do not plan to take higher calculus.
M 182 - Honors Calculus II
Course Attributes:
Honors Course
M 191 - Special Topics
Credits: 1 TO 6. Level: Undergraduate.
(R 6) Offered autumn and spring. Prereq., consent of instr. Experimental offerings of visiting professors, experimental offerings of new courses, or one time offerings of current topics.
M 192 - Independent Study
M 210 - Intro to Mathematical Software
Credits: 3. Level: Undergraduate.
Offered spring. Prereq., one of M 162, 171, or 181, or consent of instr. Software packages useful for doing and writing mathematics. Introduction to a computer algebra system (such as Maple or Mathematica), a numerical package (such as MATLAB or R), and elementary programming. Writing and communicating mathematics using the mathematical typesetting system LaTeX.
M 301 - Math Technology for Teachers
M 307 - Intro to Abstract Mathematics
Credits: 3. Level: Undergraduate.
Offered autumn and spring. Prereq., M 172 or 182. Designed to prepare students for upper–division proof–based mathematics courses. Topics include proof techniques, logic, sets, relations, functions and axiomatic methods. Students planning to take both M 221 and 307 are encouraged to take M 221 first.
M 490 - Undergraduate Research
Credits: 1 TO 4. Level: Undergraduate, Graduate.
(R-12) Offered every term. Prereq., consent of instr. Undergraduate research in the mathematical sciences under the direction of a faculty member. Graded credit/no credit.
M 501 - Tech Math for Tchrs
Credits: 3. Level: Undergraduate, Graduate.
Offered intermittently. Prereq., teacher certification or consent of instructor. Technology usage when it is appropriate and when it is not. Experience is provided with scientific calculators, graphing utilities, computers, and identification of exemplary software.
M 504 - Topics in Math Education
Credits: 1 TO 12. Level: Undergraduate, Graduate.
(R–12) Offered intermittently. Prereq., teacher certification. Topics of current interest which may include calculus, number theory, probability and statistics, geometry, or algebra, at a level suitable for teachers.
M 532 - Algebraic Topology
Credits: 3. Level: Undergraduate, Graduate.
Offered spring alternate years. Prereq., M 431 and M 531 or consent of instr.Introduction to algebraic topology through one or more topics chosen from the fundamental group and higher homotopy groups, singular homology, and simplicial homology.
M 573 - Geometry Middle Sch Tchrs
Credits: 3. Level: Undergraduate, Graduate.
Offered intermittently in summer. Prereq., teacher certification or consent of instr. Introduction to synthetic, analytic, vector, and transformational approaches to geometry. Includes topics in 2- and 3-dimensional geometry and measurement appropriate for teachers of middle school mathematics.
M 574 - Prob & Stat Mdl Sch Tchrs
Credits: 3. Level: Undergraduate, Graduate.
Offered intermittently in summer. Prereq., teacher certification or consent of instr. A survey of topics in probability and statistics appropriate for teachers of middle school mathematics.
M 584 - Topics in Combin and Optim
Credits: 3. Level: Undergraduate, Graduate.
(R–12) Offered spring odd–numbered years. Prereq., consent of instr. Topics chosen from the areas of combinatorics and optimization. May include classical problems, current trends, research interests or other topics chosen by the instructor.
M 593 - Professional Project
Credits: 1 TO 6. Level: Undergraduate, Graduate.
(R–6) Offered autumn and spring. Prereq., consent of advisor. Preparation of a professional paper appropriate to the needs and objectives of the individual student.
M 596 - Independent Study
Course Attributes:
Service Learning/Volunteer
M 597 - Research
Credits: 1 TO 12. Level: Undergraduate, Graduate.
(R–12) Offered autumn and spring. Prereq., consent of instr. Directed individual research and study appropriate to the back ground and objectives of the student.
Course Attributes:
Internships/Practicums
M 599 - Thesis
Credits: 1 TO 6. Level: Undergraduate, Graduate.
(R–6) Offered autumn and spring. Prereq., consent of instr. Preparation of a thesis or manuscript based on research for presentation and/or publication.
M 600 - Math Colloquium
M 602 - Teach College Math
Credits: 3. Level: Undergraduate, Graduate.
Prereq., second year standing in graduate school. Topics include publishing, grant writing, writing in mathematics classes, media use in mathematics, evaluation and assessment of curricular materials and programs, instructional methods in university mathematics courses, and other selected topics.
M 606 - Math History Topics
Credits: 3. Level: Undergraduate, Graduate.
Examination of mathematical history topics from the latter part of the 20th century. Discussions may focus on the impact of Hilbert's Problems. Research on current mathematics.
STAT 421 - Probability Theory
Credits: 3. Level: Undergraduate, Graduate.
Offered autumn. Prereq., M 273 and STAT 341 or consent of instr. An introduction to probability, random variables and their probability distributions, estimation and hypothesis testing. This course is the foundation on which more advanced statistics courses build.
STAT 422 - Mathematical Statistics
STAT 451 - Statistical Methods I
Credits: 3. Level: Undergraduate, Graduate.
Offered autumn. Prereq., one year of college mathematics including M 115 or equiv. course in probability or consent of instr. May not be counted toward a major in mathematics. Intended primarily for non-mathematics majors who will be analyzing data. Graphical and numerical summaries of data, elementary sampling, designing experiments, probability as a model for random phenomena and as a tool for making statistical inferences, random variables, basic ideas of inference and hypothesis testing.
STAT 458 - Computer Data Analysis II
STAT 491 - Special Topics
Credits: 1 TO 9. Level: Undergraduate, Graduate.
(R 9) Offered autumn and spring. Prereq., consent of instr. Experimental offerings of visiting professors, experimental offerings of new courses, or one time offerings of current topics.
STAT 540 - Probability &Stats Teachers
Credits: 3. Level: Undergraduate, Graduate.
Offered intermittently in summer. Prereq., STAT 341 or equiv. A survey of modern topics in probability and statistics. Emphasis will be on applications of statistics in real situations.
STAT 545 - Theory of Linear Models
Credits: 3. Level: Undergraduate, Graduate.
Offered autumn odd-numbered years. Prereq., STAT 422. Multivariate normal distribution, distribution of quadratic forms, estimation and hypothesis testing in the full rank and less than full rank general linear models.
STAT 547 - Applied Nonparametric Stats
Credits: 3. Level: Undergraduate, Graduate.
Offered autumn odd-numbered years. Prereq., STAT 421 or 452 or consent of instr. Statistical estimation and inference based on ranks and elementary counting methods. Applications to a variety of situations including one- and two-sample, correlation, regression, analysis of variance, and goodness-of-fit problems. Use of the computer and real data sets integrated throughout. Intended for students in mathematics and in other fields.
STAT 549 - Applied Sampling
Credits: 3. Level: Undergraduate, Graduate.
Offered autumn even-numbered years. Theory and application of methods for selecting samples from populations in order to efficiently estimate parameters of interest. Includes simple random, systematic, cluster, stratified, multistage, line transect, distance and adaptive sampling. Use of the computer and real data sets integrated throughout. Intended for students in mathematics and in other fields.
STAT 640 - Gr Sem Prob & Stats
Credits: 1 TO 12. Level: Undergraduate, Graduate.
(R-12) Offered autumn and spring. Prereq., consent of instr. A review and discussion of current research.
| 677.169 | 1 |
This chapter introduces exponents and exponential functions. Properties of exponents, exponential equations, and applications of exponential functions are also explored.
Share this:
Description
CK-12 Foundation's Algebra I Concepts - Honors is a high-level complete high school algebra course, that cover topics rigorously. This preview version is being released in stages several chapters at a time.
| 677.169 | 1 |
More About
This Textbook
Overview
The most successful calculus book of its generation, Jon Rogawski's Calculus offers an ideal balance of formal precision and dedicated conceptual focus, helping students build strong computational skills while continually reinforcing the relevance of calculus to their future studies and their lives.
Guided by new author Colin Adams, the new edition stays true to the late Jon Rogawski's refreshing and highly effective approach, while drawing on extensive instructor and student feedback, and Adams' three decades as a calculus teacher and author of math books for general audiences.
The new edition is also a fully integrated text/media package, with its own dedicated version of LaunchPad, W. H. Freeman's breakthrough online course space.
Product Details
Meet the Author
About Jon Rogawski
Jon Rogawski received his undergraduate degree (and simultaneously a master's degree in mathematics) at Yale, and a Ph.D. in mathematics from Princeton University, where he studied under Robert Langlands. Prior to joining the Department of Mathematics at UCLA, where he is currently Full Professor, he held teaching positions at Yale and the University of Chicago, and research positions at the Institute for Advanced Study and University of Bonn.
Jon's areas of interest are number theory, automorphic forms, and harmonic analysis on semisimple groups. He has published numerous research articles in leading mathematical journals, including a research monograph entitled "Automorphic Representations of Unitary Groups in Three Variables" (Princeton University Press). He is the recipient of a Sloan Fellowship and an editor of The Pacific Journal of Mathematics.
Jon and his wife Julie, a physician in family practice, have four children. They run a busy household and, whenever possible, enjoy family vacations in the mountains of California. Jon is a passionate classical music lover and plays the violin and classical guitar.
Chapter 5: The Integral 5.1 Approximating and Computing Area
5.2 The Definite Integral
5.3 The Indefinite Integral
5.4 The Fundamental Theorem of Calculus, Part I
5.5 The Fundamental Theorem of Calculus, Part II
5.6 Net Change as the Integral of a Rate
5.7 Substitution Method
5.8 Further Transcendental Functions
5.9 Exponential Growth and Decay
Chapter Review Exercises
Chapter 6: Applications of the Integral 6.1 Area Between Two Curves
6.2 Setting Up Integrals: Volume, Density, Average Value
6.3 Volumes of Revolution
6.4 The Method of Cylindrical Shells
6.5 Work and Energy
Chapter Review Exercises
Chapter 14: Differentiation in Several Variables 14.1 Functions of Two or More Variables
14.2 Limits and Continuity in Several Variables
14.3 Partial Derivatives
14.4 Differentiability and Tangent Planes
14.5 The Gradient and Directional Derivatives
14.6 The Chain Rule
14.7 Optimization in Several Variables
14.8 Lagrange Multipliers: Optimizing with a Constraint
Chapter Review Exercises
| 677.169 | 1 |
1
00:00:05 --> 00:00:06
OK.
2
00:00:06 --> 00:00:06
Good.
3
00:00:06 --> 00:00:12
The final class in linear
algebra at MIT this Fall is to
4
00:00:12 --> 00:00:14
review the whole course.
5
00:00:14 --> 00:00:20
And, you know the best way I
know how to review is to take
6
00:00:20 --> 00:00:25
old exams and just think through
the problems.
7
00:00:25 --> 00:00:31
So it will be a three-hour exam
next Thursday.
8
00:00:31 --> 00:00:36
Nobody will be able to take an
exam before Thursday,
9
00:00:36 --> 00:00:41
anybody who needs to take it in
some different way after
10
00:00:41 --> 00:00:44
Thursday should see me next
Monday.
11
00:00:44 --> 00:00:47
I'll be in my office Monday.
12
00:00:47 --> 00:00:47.83
OK.
13
00:00:47.83 --> 00:00:51
May I just read out some
problems and,
14
00:00:51 --> 00:00:56
let me bring the board down,
and let's start.
15
00:00:56 --> 00:00:57.43
OK.
16
00:00:57.43 --> 00:01:00
Here's a question.
17
00:01:00 --> 00:01:06
This is about a 3-by-n matrix.
18
00:01:06 --> 00:01:18
And we're given -- so we're
given -- given -- A x equals 1 0
19
00:01:18 --> 00:01:21
0 has no solution.
20
00:01:21 --> 00:01:35
And we're also given A x equals
0 1 0 has exactly one solution.
21
00:01:35 --> 00:01:35
OK.
22
00:01:35 --> 00:01:41
So you can probably anticipate
my first question,
23
00:01:41 --> 00:01:44
what can you tell me about m?
24
00:01:44 --> 00:01:49
It's an m-by-n matrix of rank
r, as always,
25
00:01:49 --> 00:01:54
what can you tell me about
those three numbers?
26
00:01:54 --> 00:02:00
So what can you tell me about
m, the number of rows,
27
00:02:00 --> 00:02:06
n, the number of columns,
and r, the rank?
28
00:02:06 --> 00:02:07
OK.
29
00:02:07 --> 00:02:14
See, do you want to tell me
first what m is?
30
00:02:14 --> 00:02:18
How many rows in this matrix?
31
00:02:18 --> 00:02:22
Must be three,
right?
32
00:02:22 --> 00:02:30
We can't tell what n is,
but we can certainly tell that
33
00:02:30 --> 00:02:32
m is three.
34
00:02:32 --> 00:02:32.84
OK.
35
00:02:32.84 --> 00:02:40
And, what do these things tell
us?
36
00:02:40 --> 00:02:43
Let's take them one at a time.
37
00:02:43 --> 00:02:48
When I discover that some
equation has no solution,
38
00:02:48 --> 00:02:53
that there's some right-hand
side with no answer,
39
00:02:53 --> 00:02:59.68
what does that tell me about
the rank of the matrix?
40
00:02:59.68 --> 00:03:02
It's smaller m.
41
00:03:02 --> 00:03:04
Is that right?
42
00:03:04 --> 00:03:13.78
If there is no solution,
that tells me that some rows of
43
00:03:13.78 --> 00:03:21
the matrix are combinations of
other rows.
44
00:03:21 --> 00:03:25
Because if I had a pivot in
every row, then I would
45
00:03:25 --> 00:03:28
certainly be able to solve the
system.
46
00:03:28 --> 00:03:32.14
I would have particular
solutions and all the good
47
00:03:32.14 --> 00:03:32
stuff.
48
00:03:32 --> 00:03:36
So any time that there's a
system with no solutions,
49
00:03:36 --> 00:03:40
that tells me that r must be
below m.
50
00:03:40 --> 00:03:45
What about the fact that if,
when there is a solution,
51
00:03:45 --> 00:03:46
there's only one?
52
00:03:46 --> 00:03:48
What does that tell me?
53
00:03:48 --> 00:03:52.93
Well, normally there would be
one solution,
54
00:03:52.93 --> 00:03:58
and then we could add in
anything in the null space.
55
00:03:58 --> 00:04:04
So this is telling me the null
space only has the 0 vector in
56
00:04:04 --> 00:04:04
it.
57
00:04:04 --> 00:04:09
There's just one solution,
period, so what does that tell
58
00:04:09 --> 00:04:10
me?
59
00:04:10 --> 00:04:15
The null space has only the
zero vector in it?
60
00:04:15 --> 00:04:18
What does that tell me about
the relation of r to n?
61
00:04:18 --> 00:04:22
So this one solution only,
that means the null space of
62
00:04:22 --> 00:04:26.17
the matrix must be just the zero
vector, and what does that tell
63
00:04:26.17 --> 00:04:27
me about r and n?
64
00:04:27 --> 00:04:28
They're equal.
65
00:04:28 --> 00:04:30
The columns are independent.
66
00:04:30 --> 00:04:31
So I've got,
now, r equals n,
67
00:04:31 --> 00:04:35
and r less than m,
and now I also know m is three.
68
00:04:35 --> 00:04:40
So those are really the facts I
know.
69
00:04:40 --> 00:04:44
n=r and those numbers are
smaller than three.
70
00:04:44 --> 00:04:48
Sorry, yes, yes.
r is smaller than m,
71
00:04:48 --> 00:04:51
and n, of course,
is also.
72
00:04:51 --> 00:04:57
So I guess this summarizes what
we can tell.
73
00:04:57 --> 00:05:02
In fact, why not give me a
matrix -- because I would often
74
00:05:02 --> 00:05:06
ask for an example of such a
matrix -- can you give me a
75
00:05:06 --> 00:05:08
matrix A that's an example?
76
00:05:08 --> 00:05:10
That shows this possibility?
77
00:05:10 --> 00:05:14
Exactly, that there's no
solution with that right-hand
78
00:05:14 --> 00:05:19
side, but there's exactly one
solution with this right-hand
79
00:05:19 --> 00:05:20.49
side.
80
00:05:20.49 --> 00:05:25
Anybody want to suggest a
matrix that does that?
81
00:05:25 --> 00:05:26.14
Let's see.
82
00:05:26.14 --> 00:05:31
What do I -- what vector do I
want in the column space?
83
00:05:31 --> 00:05:35
I want zero,
one, zero, to be in the column
84
00:05:35 --> 00:05:40
space, because I'm able to solve
for that.
85
00:05:40 --> 00:05:45
So let's put zero,
one, zero in the column space.
86
00:05:45 --> 00:05:48
Actually, I could stop right
there.
87
00:05:48 --> 00:05:53
That would be a matrix with m
equal three, three rows,
88
00:05:53 --> 00:05:58
and n and r are both one,
rank one, one column,
89
00:05:58 --> 00:06:02
and, of course,
there's no solution to that
90
00:06:02 --> 00:06:04
one.
91
00:06:04 --> 00:06:06
So that's perfectly good as it
is.
92
00:06:06 --> 00:06:10
Or if you, kind of,
have a prejudice against
93
00:06:10 --> 00:06:15
matrices that only have one
column, I'll accept a second
94
00:06:15 --> 00:06:15
column.
95
00:06:15 --> 00:06:20
So what could I include as a
second column that would just be
96
00:06:20 --> 00:06:24
a different answer but equally
good?
97
00:06:24 --> 00:06:27.98
I could put this vector in the
column space,
98
00:06:27.98 --> 00:06:29.29
too, if I wanted.
99
00:06:29.29 --> 00:06:33
That would now be a case with
r=n=2, but, of course,
100
00:06:33 --> 00:06:37
three m eq- m is still three,
and this vector is not in the
101
00:06:37 --> 00:06:38
column space.
102
00:06:38 --> 00:06:43
So you're -- this is just like
prompting us to remember all
103
00:06:43 --> 00:06:45
those things,
column space,
104
00:06:45 --> 00:06:48
null space, all that stuff.
105
00:06:48 --> 00:06:54
Now, I probably asked a second
question about this type of
106
00:06:54 --> 00:06:55.15
thing.
107
00:06:55.15 --> 00:06:55
Ah.
108
00:06:55 --> 00:06:55
OK.
109
00:06:55 --> 00:07:00
Oh, I even asked,
write down an example of a
110
00:07:00 --> 00:07:04
matrix that fits the
description.
111
00:07:04 --> 00:07:04
Hm.
112
00:07:04 --> 00:07:11
I guess I haven't learned
anything in twenty-six years.
113
00:07:11 --> 00:07:11.8
CK.
114
00:07:11.8 --> 00:07:16
Cross out all statements that
are false about any matrix with
115
00:07:16 --> 00:07:19
these -- so again,
these are -- this is the
116
00:07:19 --> 00:07:22
preliminary sta- these are the
facts about my matrix,
117
00:07:22 --> 00:07:24
this is one example.
118
00:07:24 --> 00:07:27
But, of course,
by having an example,
119
00:07:27 --> 00:07:30.2
it will be easy to check some
of these facts,
120
00:07:30.2 --> 00:07:32
or non-facts.
121
00:07:32 --> 00:07:36
Let me, let me write down some,
facts.
122
00:07:36 --> 00:07:38
Some possible facts.
123
00:07:38 --> 00:07:42
So this is really true or
false.
124
00:07:42 --> 00:07:49.24
The determinant -- this is part
one, the determinant of A
125
00:07:49.24 --> 00:07:57
transpose A is the same as the
determinant of A A transpose.
126
00:07:57 --> 00:08:00
Is that true or not?
127
00:08:00 --> 00:08:05
Second one, A transpose A,
is invertible.
128
00:08:05 --> 00:08:07.85
Is invertible.
129
00:08:07.85 --> 00:08:14.24
Third possible fact,
A A transpose is positive
130
00:08:14.24 --> 00:08:15
definite.
131
00:08:15 --> 00:08:20
So you see how,
on an exam question,
132
00:08:20 --> 00:08:29
I try to connect the different
parts of the course.
133
00:08:29 --> 00:08:34
So, well, I mean,
the simplest way would be to
134
00:08:34 --> 00:08:38.51
try it with that matrix as a
good example,
135
00:08:38.51 --> 00:08:42
but maybe we can answer,
even directly.
136
00:08:42 --> 00:08:45
Let me take number two first.
137
00:08:45 --> 00:08:51
Because I'm -- you know,
I'm very, very fond of that
138
00:08:51 --> 00:08:54
matrix, A transpose A.
139
00:08:54 --> 00:08:56.91
And when is it invertible?
140
00:08:56.91 --> 00:09:00
When is the matrix A transpose
A, invertible?
141
00:09:00 --> 00:09:05.13
The great thing is that I can
tell from the rank of A that I
142
00:09:05.13 --> 00:09:08
don't have to multiply out A
transpose A.
143
00:09:08 --> 00:09:11
A transpose A,
is invertible -- well,
144
00:09:11 --> 00:09:15
if A has a null space other
than the zero vector,
145
00:09:15 --> 00:09:20
then it -- it's -- no way it's
going to be invertible.
146
00:09:20 --> 00:09:25
But the beauty is,
if the null space of A is just
147
00:09:25 --> 00:09:31
the zero vector,
so the fact -- the key fact is,
148
00:09:31 --> 00:09:36
this is invertible if r=n,
by which I mean,
149
00:09:36 --> 00:09:39
independent columns of A.
150
00:09:39 --> 00:09:40
In A.
151
00:09:40 --> 00:09:42.18
In the matrix A.
152
00:09:42.18 --> 00:09:47.47
If r=n -- if the matrix A has
independent columns,
153
00:09:47.47 --> 00:09:51
then this combination,
A transpose A,
154
00:09:51 --> 00:09:57
is square and still that same
null space, only the zero
155
00:09:57 --> 00:10:01
vector, independent columns all
good, and so,
156
00:10:01 --> 00:10:05
what's the true/false?
157
00:10:05 --> 00:10:08
Is it -- is this middle one T
or F for this,
158
00:10:08 --> 00:10:09
in this setup?
159
00:10:09 --> 00:10:14
Well, we discovered that -- we
discovered that -- that r was n,
160
00:10:14 --> 00:10:15
from that second fact.
161
00:10:15 --> 00:10:16
So this is a true.
162
00:10:16 --> 00:10:17
That's a true.
163
00:10:17 --> 00:10:20
And, of course,
A transpose A,
164
00:10:20 --> 00:10:23
in this example,
would probably be -- what would
165
00:10:23 --> 00:10:26
A transpose A,
be, for that matrix?
166
00:10:26 --> 00:10:34.46
Can you multiply A transpose A,
and see what it looks like for
167
00:10:34.46 --> 00:10:35
that matrix?
168
00:10:35 --> 00:10:38.83
What shape would it be?
169
00:10:38.83 --> 00:10:41
It will be two by two.
170
00:10:41 --> 00:10:44
And what matrix will it be?
171
00:10:44 --> 00:10:46
The identity.
172
00:10:46 --> 00:10:49
So, it checks out.
173
00:10:49 --> 00:10:53
OK, what about A A transpose?
174
00:10:53 --> 00:10:59
Well, depending on the shape of
A, it could be good or not so
175
00:10:59 --> 00:11:00
good.
176
00:11:00 --> 00:11:04
It's always symmetric,
it's always square,
177
00:11:04 --> 00:11:07
but what's the size,
now?
178
00:11:07 --> 00:11:12
This is three by n,
and this is n by three,
179
00:11:12 --> 00:11:15
so the result is three by
three.
180
00:11:15 --> 00:11:19
Is it positive definite?
181
00:11:19 --> 00:11:20
I don't think so.
182
00:11:20 --> 00:11:21
False.
183
00:11:21 --> 00:11:25.2
If I multiply that by A
transpose, A A transpose,
184
00:11:25.2 --> 00:11:26
what would the rank be?
185
00:11:26 --> 00:11:31
It would be the same as the
rank of A, that's -- it would be
186
00:11:31 --> 00:11:32
just rank two.
187
00:11:32 --> 00:11:36
And if it's three-by-three,
and it's only rank two,
188
00:11:36 --> 00:11:40
it's certainly not positive
definite.
189
00:11:40 --> 00:11:46
So what could I say about A A
transpose, if I wanted to,
190
00:11:46 --> 00:11:49
like, say something true about
it?
191
00:11:49 --> 00:11:54
It's true that it is positive
semi-definite.
192
00:11:54 --> 00:12:00
If I made this semi-definite,
it would always be true,
193
00:12:00 --> 00:12:02
always.
194
00:12:02 --> 00:12:06
But if I'm looking for positive
definite, then I'm looking at
195
00:12:06 --> 00:12:09
the null space of whatever's
here, and, in this case,
196
00:12:09 --> 00:12:11.33
it's got a null space.
197
00:12:11.33 --> 00:12:14
So A, A -- eh,
shall we just figure it out,
198
00:12:14 --> 00:12:14
here?
199
00:12:14 --> 00:12:16
A A transpose,
for that matrix,
200
00:12:16 --> 00:12:18
will be three-by-three.
201
00:12:18 --> 00:12:21.95
If I multiplied A by A
transpose, what would the first
202
00:12:21.95 --> 00:12:23
row be?
203
00:12:23 --> 00:12:25
All zeroes, right?
204
00:12:25 --> 00:12:30
First row of A A transpose,
could only be all zeroes,
205
00:12:30 --> 00:12:34
so it's probably a one there
and a one there,
206
00:12:34 --> 00:12:36
or something like that.
207
00:12:36 --> 00:12:40
But, I don't even know if
that's right.
208
00:12:40 --> 00:12:45
But it's all zeroes there,
so it's certainly not positive
209
00:12:45 --> 00:12:47
definite.
210
00:12:47 --> 00:12:51.16
Let me not put anything up I'm
not sh- don't check.
211
00:12:51.16 --> 00:12:53
What about this determinant?
212
00:12:53 --> 00:12:56
Oh, well, I guess -- that's a
sort of tricky question.
213
00:12:56 --> 00:12:58.91
Is it true or false in this
case?
214
00:12:58.91 --> 00:13:01
It's false, apparently,
because A transpose A,
215
00:13:01 --> 00:13:04
is invertible,
we just got a true for this
216
00:13:04 --> 00:13:07
one, and we got a false,
we got a z- we got a
217
00:13:07 --> 00:13:10.98
non-invertible one for this one.
218
00:13:10.98 --> 00:13:14
So actually,
this one is false,
219
00:13:14 --> 00:13:15
number one.
220
00:13:15 --> 00:13:19
That surprises us,
actually, because it's,
221
00:13:19 --> 00:13:22.02
I mean, why was it tricky?
222
00:13:22.02 --> 00:13:26
Because what is true about
determinants?
223
00:13:26 --> 00:13:31
This would be true if those
matrices were square.
224
00:13:31 --> 00:13:35
If I have two square matrices,
A and any other matrix B,
225
00:13:35 --> 00:13:37
could be A transpose,
could be somebody else's
226
00:13:37 --> 00:13:38
matrix.
227
00:13:38 --> 00:13:41
Then it would be true that the
determinant of B A would equal
228
00:13:41 --> 00:13:42
the determinant of A B.
229
00:13:42 --> 00:13:46
But if the matrices are not
square and it would actually be
230
00:13:46 --> 00:13:49
true that it would be equal --
that this would equal the
231
00:13:49 --> 00:13:53
determinant of A times the
determinant of A transpose.
232
00:13:53 --> 00:13:57
We could even split up those
two separate determinants.
233
00:13:57 --> 00:14:00.77
And, of course,
those would be equal.
234
00:14:00.77 --> 00:14:02.92
But only when A is square.
235
00:14:02.92 --> 00:14:06
So that's just,
that's a question that rests on
236
00:14:06 --> 00:14:11
the, the falseness rests on the
fact that the matrix isn't
237
00:14:11 --> 00:14:14
square in the first place.
238
00:14:14 --> 00:14:15
OK, good.
239
00:14:15 --> 00:14:16
Let's see.
240
00:14:16 --> 00:14:20
Oh, now, even asks more.
241
00:14:20 --> 00:14:26
Prove that A transpose y equals
c -- hah-God,
242
00:14:26 --> 00:14:30
it's -- this question goes on
and on.
243
00:14:30 --> 00:14:35
now I ask you about A transpose
y=c.
244
00:14:35 --> 00:14:43
So I'm asking you about the
equation -- about the matrix A
245
00:14:43 --> 00:14:46.02
transpose.
246
00:14:46.02 --> 00:14:54
And I want you to prove that it
has at least one solution -- one
247
00:14:54 --> 00:15:00
solution for every c,
every right-hand side c,
248
00:15:00 --> 00:15:07
and, in fact -- in fact,
infinitely many solutions for
249
00:15:07 --> 00:15:09
every c.
250
00:15:09 --> 00:15:09.46
OK.
251
00:15:09.46 --> 00:15:15
Well, none -- none of this is
difficult, but,
252
00:15:15 --> 00:15:20
it's been a little while.
253
00:15:20 --> 00:15:23
So we just have to think again.
254
00:15:23 --> 00:15:28
When I have a system of
equations -- this is -- this
255
00:15:28 --> 00:15:33
matrix A transpose is now,
instead of being three by n,
256
00:15:33 --> 00:15:35
it's n by three,
it's n by m.
257
00:15:35 --> 00:15:36
Of course.
258
00:15:36 --> 00:15:41
To show that a system has at
least one solution,
259
00:15:41 --> 00:15:45
when does this,
when does this system -- when
260
00:15:45 --> 00:15:49
is the system always solvable?
261
00:15:49 --> 00:15:58
When it has full row rank,
when the rows are independent.
262
00:15:58 --> 00:16:04
Here, we have n rows,
and that's the rank.
263
00:16:04 --> 00:16:12
So at least one solution,
because the number of rows,
264
00:16:12 --> 00:16:19
which is n, for the transpose,
is equal to r,
265
00:16:19 --> 00:16:22
the rank.
266
00:16:22 --> 00:16:27
This A transpose had
independent rows because A had
267
00:16:27 --> 00:16:29
independent columns,
right?
268
00:16:29 --> 00:16:34
The original A had independent
columns, when we transpose it,
269
00:16:34 --> 00:16:38
it has independent rows,
so there's at least one
270
00:16:38 --> 00:16:39
solution.
271
00:16:39 --> 00:16:44
But now, how do I even know
that there are infinitely many
272
00:16:44 --> 00:16:46
solutions?
273
00:16:46 --> 00:16:52
Oh, what do I -- I want to know
something about the null space.
274
00:16:52 --> 00:16:57.34
What's the dimension of the
null space of A transpose?
275
00:16:57.34 --> 00:17:03
So the answer has got to be the
dimension of the null space of A
276
00:17:03 --> 00:17:07.28
transpose, what's the general
fact?
277
00:17:07.28 --> 00:17:13
If A is an m by n matrix of
rank r, what's the dimension of
278
00:17:13 --> 00:17:14
A transpose?
279
00:17:14 --> 00:17:17
The null space of A transpose?
280
00:17:17 --> 00:17:23
Do you remember that little
fourth subspace that's tagging
281
00:17:23 --> 00:17:26
along down in our big picture?
282
00:17:26 --> 00:17:28
It's dimension was m-r.
283
00:17:28 --> 00:17:34
And, that's bigger than zero.
m is bigger than r.
284
00:17:34 --> 00:17:38
So there's a lot in that null
space.
285
00:17:38 --> 00:17:44
So there's always one solution
because n i- this is speaking
286
00:17:44 --> 00:17:46
about A transpose.
287
00:17:46 --> 00:17:51
So for A transpose,
the roles of m and n are
288
00:17:51 --> 00:17:56
reversed, of course,
so I'm -- keep in mind that
289
00:17:56 --> 00:18:02
this board was about A
transpose, so the roles -- so
290
00:18:02 --> 00:18:07
it's the null space of a
transpose, and there are m-r
291
00:18:07 --> 00:18:09
free variables.
292
00:18:09 --> 00:18:12
OK, that's, like,
just some, review.
293
00:18:12 --> 00:18:18
Can I take another problem
that's also sort of
294
00:18:18 --> 00:18:26
-- suppose the matrix A has
three columns,
295
00:18:26 --> 00:18:28.43
v1, v2, v3.
296
00:18:28.43 --> 00:18:35
Those are the columns of the
matrix.
297
00:18:35 --> 00:18:37
All right.
298
00:18:37 --> 00:18:39
Question A.
299
00:18:39 --> 00:18:42
Solve Ax=v1-v2+v3.
300
00:18:42 --> 00:18:46
Tell me what x is.
301
00:18:46 --> 00:18:55
Well, there,
you're seeing the most
302
00:18:55 --> 00:18:59
-- the one absolutely essential
fact about matrix
303
00:18:59 --> 00:19:01
multiplication,
how does it work,
304
00:19:01 --> 00:19:05
when we do it a column at a
time, the very,
305
00:19:05 --> 00:19:08.69
very first day,
way back in September,
306
00:19:08.69 --> 00:19:12
we did multiplication a column
at a time.
307
00:19:12 --> 00:19:13
So what's x?
308
00:19:13 --> 00:19:14
Just tell me?
309
00:19:14 --> 00:19:15
One minus one,
one.
310
00:19:15 --> 00:19:17
Thanks.
311
00:19:17 --> 00:19:17
OK.
312
00:19:17 --> 00:19:20
Everybody's got that.
313
00:19:20 --> 00:19:20
OK?
314
00:19:20 --> 00:19:26
Then the next question is,
suppose that combination is
315
00:19:26 --> 00:19:31
zero -- oh, yes,
OK, so question (b) says --
316
00:19:31 --> 00:19:36
part (b) says,
suppose this thing is zero.
317
00:19:36 --> 00:19:39
Suppose that's zero.
318
00:19:39 --> 00:19:42
Then the solution is not
unique.
319
00:19:42 --> 00:19:49
Suppose I want true or false.
-- and a reason.
320
00:19:49 --> 00:19:52
Suppose this combination is
zero.
321
00:19:52 --> 00:19:53
v1-v2+v3.
322
00:19:53 --> 00:19:57
Show that -- what does that
tell me?
323
00:19:57 --> 00:20:02
So it's a separate question,
maybe I sort of saved time by
324
00:20:02 --> 00:20:07
writing it that way,
but it's a totally separate
325
00:20:07 --> 00:20:09
question.
326
00:20:09 --> 00:20:15
If I have a matrix,
and I know that column one
327
00:20:15 --> 00:20:21
minus column two plus column
three is zero,
328
00:20:21 --> 00:20:28
what does that tell me about
whether the solution is unique
329
00:20:28 --> 00:20:30
or not?
330
00:20:30 --> 00:20:33.94
Is there more than one
solution?
331
00:20:33.94 --> 00:20:36
What's uniqueness about?
332
00:20:36 --> 00:20:42
Uniqueness is about,
is there anything in the null
333
00:20:42 --> 00:20:43.5
space, right?
334
00:20:43.5 --> 00:20:50
The solution is unique when
there's nobody in the null space
335
00:20:50 --> 00:20:53
except the zero vector.
336
00:20:53 --> 00:21:00
And, if that's zero,
then this guy would be in the
337
00:21:00 --> 00:21:01
null space.
338
00:21:01 --> 00:21:07
So if this were zero,
then this x is in the null
339
00:21:07 --> 00:21:09
space of A.
340
00:21:09 --> 00:21:16
So solutions are never unique,
because I could always add that
341
00:21:16 --> 00:21:22
to any solution,
and Ax wouldn't change.
342
00:21:22 --> 00:21:27
So it's always that question.
343
00:21:27 --> 00:21:31
Is there somebody in the null
space?
344
00:21:31 --> 00:21:32
OK.
345
00:21:32 --> 00:21:37
Oh, now, here's a totally
different question.
346
00:21:37 --> 00:21:44
Suppose those three vectors,
v1, v2, v3, are orthonormal.
347
00:21:44 --> 00:21:52
So this isn't going to happen
for orthonormal vectors.
348
00:21:52 --> 00:21:55
OK, so part (c),
forget part (b).
349
00:21:55 --> 00:21:55
c.
350
00:21:55 --> 00:22:00.18
If v1, v2, v3,
are orthonormal -- so that I
351
00:22:00.18 --> 00:22:04
would usually have called them
q1, q2, q3.
352
00:22:04 --> 00:22:09
Now, what combination -- oh,
here's a nice question,
353
00:22:09 --> 00:22:14
if I say so myself -- what
combination of v1 and v2 is
354
00:22:14 --> 00:22:17
closest to v3?
355
00:22:17 --> 00:22:23
What point on the plane of v1
and v2 is the closest point to
356
00:22:23 --> 00:22:27
v3 if these vectors are
orthonormal?
357
00:22:27 --> 00:22:33
So let me -- I'll start the
sentence -- then the combination
358
00:22:33 --> 00:22:39
something times v1 plus
something times v2 is the
359
00:22:39 --> 00:22:42
closest combination to v3?
360
00:22:42 --> 00:22:45
And what's the answer?
361
00:22:45 --> 00:22:49.75
What's the closest vector on
that plane to v3?
362
00:22:49.75 --> 00:22:50
Zeroes.
363
00:22:50 --> 00:22:51
Right.
364
00:22:51 --> 00:22:55
We just imagine the x,
y, z axes, the v1,
365
00:22:55 --> 00:22:59.17
v2, th- v3 could be the
standard basis,
366
00:22:59.17 --> 00:23:02
the x, y, z vectors,
and, of course,
367
00:23:02 --> 00:23:08
the point on the xy plane
that's closest to v3 on the z
368
00:23:08 --> 00:23:11
axis is zero.
369
00:23:11 --> 00:23:18
So if we're orthonormal,
then the projection of v3 onto
370
00:23:18 --> 00:23:25
that plane is perpendicular,
it hits right at zero.
371
00:23:25 --> 00:23:33
OK, so that's like a quick --
you know, an easy question,
372
00:23:33 --> 00:23:37
but still brings it out.
373
00:23:37 --> 00:23:38
OK.
374
00:23:38 --> 00:23:46
Let me see what,
shall I write down a Markov
375
00:23:46 --> 00:23:54
matrix, and I'll ask you for its
eigenvalues.
376
00:23:54 --> 00:23:55
OK.
377
00:23:55 --> 00:24:08
Here's a Markov matrix -- this
-- and, tell me its eigenvalues.
378
00:24:08 --> 00:24:16
So here -- I'll call the matrix
A, and I'll call this as point
379
00:24:16 --> 00:24:21
two, point four,
point four, point four,
380
00:24:21 --> 00:24:26
point four, point two,
point four, point three,
381
00:24:26 --> 00:24:29
point three,
point four.
382
00:24:29 --> 00:24:30
OK.
383
00:24:30 --> 00:24:36.99
Let's see -- it helps out to
notice that column one plus
384
00:24:36.99 --> 00:24:41
column two
-- what's interesting about
385
00:24:41 --> 00:24:44
column one plus column two?
386
00:24:44 --> 00:24:46
It's twice as much as column
three.
387
00:24:46 --> 00:24:51
So column one plus column two
equals two times column three.
388
00:24:51 --> 00:24:55
I put that in there,
column one plus column two
389
00:24:55 --> 00:24:57
equals twice column three.
390
00:24:57 --> 00:25:00
That's observation.
391
00:25:00 --> 00:25:00
OK.
392
00:25:00 --> 00:25:04
Tell me the eigenvalues of the
matrix.
393
00:25:04 --> 0.
OK, tell me one eigenvalue?
394
0. --> 00:25:06.93
395
00:25:06.93 --> 00:25:09
Because the matrix is singular.
396
00:25:09 --> 00:25:12
Tell me another eigenvalue?
397
00:25:12 --> 00:25:18
One, because it's a Markov
matrix, the columns add to the
398
00:25:18 --> 00:25:22
all ones vector,
and that will be an eigenvector
399
00:25:22 --> 00:25:25
of A transpose.
400
00:25:25 --> 00:25:28
And tell me the third
eigenvalue?
401
00:25:28 --> 00:25:33
Let's see, to make the trace
come out right,
402
00:25:33 --> 00:25:37
which is point eight,
we need minus point two.
403
00:25:37 --> 00:25:38
OK.
404
00:25:38 --> 00:25:42
And now, suppose I start the
Markov process.
405
00:25:42 --> 00:25:49
Suppose I start with u(0) -- so
I'm going to look at the powers
406
00:25:49 --> 00:25:52
of A applied to u(0).
407
00:25:52 --> 00:25:54
This is uk.
408
00:25:54 --> 00:26:04
And there's my matrix,
and I'm going to let u(0) be --
409
00:26:04 --> 00:26:10
this is going to be zero,
ten, zero.
410
00:26:10 --> 00:26:19
And my question is,
what does that approach?
411
00:26:19 --> 00:26:24
If u(0) is equal to this --
there is u(0).
412
00:26:24 --> 00:26:27
Shall I write it in?
413
00:26:27 --> 00:26:30
Maybe I'll just write in u(0).
414
00:26:30 --> 00:26:37
A to the k, starting with ten
people in state two,
415
00:26:37 --> 00:26:43
and every step follows the
Markov rule, what does the
416
00:26:43 --> 00:26:49
solution look like after k
steps?
417
00:26:49 --> 00:26:51
Let me just ask you that.
418
00:26:51 --> 00:26:54.85
And then, what happens as k
goes to infinity?
419
00:26:54.85 --> 00:26:57
This is a steady-state
question, right?
420
00:26:57 --> 00:27:00.4
I'm looking for the steady
state.
421
00:27:00.4 --> 00:27:04
Actually, the question doesn't
ask for the k step answer,
422
00:27:04 --> 00:27:09
it just jumps right away to
infinity -- but how would I
423
00:27:09 --> 00:27:12
express the solution after k
steps?
424
00:27:12 --> 00:27:19
It would be some multiple of
the first eigenvalue to the k-th
425
00:27:19 --> 00:27:24
power -- times the first
eigenvector, plus some other
426
00:27:24 --> 00:27:28
multiple of the second
eigenvalue, times its
427
00:27:28 --> 00:27:34
eigenvector, and some multiple
of the third eigenvalue,
428
00:27:34 --> 00:27:37
times its eigenvector.
429
00:27:37 --> 00:27:38
OK.
430
00:27:38 --> 00:27:38
Good.
431
00:27:38 --> 00:27:45
And these eigenvalues are zero,
one, and minus point two.
432
00:27:45 --> 00:27:49
So what happens as k goes to
infinity?
433
00:27:49 --> 00:27:55
The only thing that survives
the steady state -- so at u
434
00:27:55 --> 00:28:00
infinity, this is gone,
this is gone,
435
00:28:00 --> 00:28:02
all that's left is c2x2.
436
00:28:02 --> 00:28:05
So I'd better find x2.
437
00:28:05 --> 00:28:11
I've got to find that
eigenvector to complete the
438
00:28:11 --> 00:28:13
answer.
439
00:28:13 --> 00:28:18
What's the eigenvector that
corresponds to lambda equal one?
440
00:28:18 --> 00:28:22
That's the key eigenvector in
any Markov process,
441
00:28:22 --> 00:28:24
is that eigenvector.
442
00:28:24 --> 00:28:28
Lambda equal one is an
eigenvalue, I need its
443
00:28:28 --> 00:28:31
eigenvector x2,
and then I need to know how
444
00:28:31 --> 00:28:36
much of it is in the starting
vector u0.
445
00:28:36 --> 00:28:36.6
OK.
446
00:28:36.6 --> 00:28:40
So, how do I find that
eigenvector?
447
00:28:40 --> 00:28:45.86
I guess I subtract one from the
diagonal, right?
448
00:28:45.86 --> 00:28:51
So I have minus point eight,
minus point eight,
449
00:28:51 --> 00:28:54
minus point six,
and the rest,
450
00:28:54 --> 00:28:59
of course, is just
-- still point four,
451
00:28:59 --> 00:29:04
point four, point four,
point four, point three,
452
00:29:04 --> 00:29:07
point three,
and hopefully,
453
00:29:07 --> 00:29:12
that's a singular matrix,
so I'm looking to solve A minus
454
00:29:12 --> 00:29:14
Ix equal zero.
455
00:29:14 --> 00:29:18
Let's see -- can anybody spot
the solution here?
456
00:29:18 --> 00:29:22
I don't know,
I didn't make it easy for
457
00:29:22 --> 00:29:24
myself.
458
00:29:24 --> 00:29:27
What do you think there?
459
00:29:27 --> 00:29:33
Maybe those first two entries
might be -- oh,
460
00:29:33 --> 00:29:36
no, what do you think?
461
00:29:36 --> 00:29:37
Anybody see it?
462
00:29:37 --> 00:29:43
We could use elimination if we
were desperate.
463
00:29:43 --> 00:29:47
Are we that desperate?
464
00:29:47 --> 00:29:53
Anybody just call out if you
see the vector that's in that
465
00:29:53 --> 00:29:54.41
null space.
466
00:29:54.41 --> 00:29:59
Eh, there better be a vector in
that null space,
467
00:29:59 --> 00:30:00
or I'm quitting.
468
00:30:00 --> 00:30:04
Uh, ha- OK, well,
I guess we could use
469
00:30:04 --> 00:30:05
elimination.
470
00:30:05 --> 00:30:12
I thought maybe somebody might
see it from further away.
471
00:30:12 --> 00:30:18
Is there a chance that these
guys are -- could it be that
472
00:30:18 --> 00:30:24
these two are equal and this is
whatever it takes,
473
00:30:24 --> 00:30:28
like, something like three,
three, two?
474
00:30:28 --> 00:30:31
Would that possibly work?
475
00:30:31 --> 00:30:36
I mean, that's great for this
-- no, it's not that great.
476
00:30:36 --> 00:30:39
Three, three,
four -- this is,
477
00:30:39 --> 00:30:42
deeper mathematics you're
watching now.
478
00:30:42 --> 00:30:46
Three, three,
four, is that -- it works!
479
00:30:46 --> 00:30:48
Don't mess with it!
480
00:30:48 --> 00:30:49
It works!
481
00:30:49 --> 00:30:50
Uh, yes.
482
00:30:50 --> 00:30:53
OK, it works,
all right.
483
00:30:53 --> 00:30:57
And, yes, OK,
and, so that's x2,
484
00:30:57 --> 00:31:01
three, three,
four, and, how much of that
485
00:31:01 --> 00:31:05
vector is in the starting
vector?
486
00:31:05 --> 00:31:10
Well, we could go through a
complicated process.
487
00:31:10 --> 00:31:16
But what's the beauty of Markov
things?
488
00:31:16 --> 00:31:21
That the total number of the
total population,
489
00:31:21 --> 00:31:24
the sum of these doesn't
change.
490
00:31:24 --> 00:31:30
That the total number of
people, they're moving around,
491
00:31:30 --> 00:31:35
but they don't get born or die
or get dead.
492
00:31:35 --> 00:31:39
So there's ten of them at the
start, so there's ten of them
493
00:31:39 --> 00:31:42.35
there, so c2 is actually one,
yes.
494
00:31:42.35 --> 00:31:45
So that would be the correct
solution.
495
00:31:45 --> 00:31:45
OK.
496
00:31:45 --> 00:31:47
That would be the u infinity.
497
00:31:47 --> 00:31:48
OK.
498
00:31:48 --> 00:31:50
So I used there,
in that process,
499
00:31:50 --> 00:31:54
sort of, the main facts about
Markov matrices to,
500
00:31:54 --> 00:31:57
to get a jump on the answer.
501
00:31:57 --> 00:31:58
OK.
let's see.
502
00:31:58 --> 00:32:01
OK, here's some,
kind of quick,
503
00:32:01 --> 00:32:02
short questions.
504
00:32:02 --> 00:32:07
Uh, maybe I'll move over to
this board, and leave that for
505
00:32:07 --> 00:32:07
the moment.
506
00:32:07 --> 00:32:10
I'm looking for two-by-two
matrices.
507
00:32:10 --> 00:32:15
And I'll read out the property
I want, and you give me an
508
00:32:15 --> 00:32:20
example, or tell me there isn't
such a matrix.
509
00:32:20 --> 00:32:21
All right.
510
00:32:21 --> 00:32:23
Here we go.
511
00:32:23 --> 00:32:27
First -- so two-by-twos.
512
00:32:27 --> 00:32:37
First, I want the projection
onto the line through A equals
513
00:32:37 --> 00:32:40
four minus three.
514
00:32:40 --> 00:32:49
So it's a one-dimensional
projection matrix I'm looking
515
00:32:49 --> 00:32:51
for.
516
00:32:51 --> 00:32:53
And what's the formula for it?
517
00:32:53 --> 00:32:57
What's the formula for the
projection matrix P onto a line
518
00:32:57 --> 00:32:57
through A.
519
00:32:57 --> 00:33:00
And then we'd just plug in this
particular A.
520
00:33:00 --> 00:33:02
Do you remember that formula?
521
00:33:02 --> 00:33:06
There's an A and an A
transpose, and normally we would
522
00:33:06 --> 00:33:10
have an A transpose A inverse in
the middle, but here we've just
523
00:33:10 --> 00:33:14
got numbers, so we just divide
by it.
524
00:33:14 --> 00:33:20.09
And then plug in A and we've
got it.
525
00:33:20.09 --> 00:33:20
OK.
526
00:33:20 --> 00:33:22
So, equals.
527
00:33:22 --> 00:33:26
You can put in the numbers.
528
00:33:26 --> 00:33:29
Trivial, right.
529
00:33:29 --> 00:33:29
OK.
530
00:33:29 --> 00:33:31
Number two.
531
00:33:31 --> 00:33:37
So this is a new problem.
532
00:33:37 --> 00:33:43
The matrix with eigenvalue zero
and three and eigenvectors --
533
00:33:43 --> 00:33:47
well, let me write these down.
eigenvalue zero,
534
00:33:47 --> 00:33:51
eigenvector one,
two, eigenvalue three,
535
00:33:51 --> 00:33:53
eigenvector two,
one.
536
00:33:53 --> 00:33:59
I'm giving you the eigenvalues
and eigenvectors instead of
537
00:33:59 --> 00:34:00
asking for them.
538
00:34:00 --> 00:34:05
Now I'm asking for the matrix.
539
00:34:05 --> 00:34:08
What's the matrix,
then?
540
00:34:08 --> 00:34:09
What's A?
541
00:34:09 --> 00:34:15
Here was a formula,
then we just put in some
542
00:34:15 --> 00:34:23
numbers, what's the formula
here, into which we'll just put
543
00:34:23 --> 00:34:25
the given numbers?
544
00:34:25 --> 00:34:31
It's the S lambda S inverse,
right?
545
00:34:31 --> 00:34:35.57
So it's S, which is this
eigenvector matrix,
546
00:34:35.57 --> 00:34:39
it's the lambda,
which is the eigenvalue matrix,
547
00:34:39 --> 00:34:43
it's the S inverse,
whatever that turns out to be,
548
00:34:43 --> 00:34:46
let me just leave it as
inverse.
549
00:34:46 --> 00:34:48
That has to be it,
right?
550
00:34:48 --> 00:34:54
Because if we went in the other
direction, that matrix S would
551
00:34:54 --> 00:34:56
diagonalize A to produce lambda.
552
00:34:56 --> 00:35:00
So it's S lambda S inverse.
553
00:35:00 --> 00:35:00
Good.
554
00:35:00 --> 00:35:03
OK, ready for number three.
555
00:35:03 --> 00:35:09
A real matrix that cannot be
factored into A -- I'm looking
556
00:35:09 --> 00:35:15
for a matrix A that never could
equal B transpose B,
557
00:35:15 --> 00:35:16.06
for any B.
558
00:35:16.06 --> 00:35:22
A two-by-two matrix that could
not be factored in the form B
559
00:35:22 --> 00:35:24
transpose B.
560
00:35:24 --> 00:35:30
So all you have to do is think,
well, what does B transpose B,
561
00:35:30 --> 00:35:34
look like, and then pick
something different.
562
00:35:34 --> 00:35:36
What do you suggest?
563
00:35:36 --> 00:35:37
Let's see.
564
00:35:37 --> 00:35:43
What shall we take for a matrix
that could not have this form,
565
00:35:43 --> 00:35:46
B transpose B.
566
00:35:46 --> 00:35:48
Well, what do we know about B
transpose B?
567
00:35:48 --> 00:35:50
It's always symmetric.
568
00:35:50 --> 00:35:52
So just give me any
non-symmetric matrix,
569
00:35:52 --> 00:35:54
it couldn't possibly have that
form.
570
00:35:54 --> 00:35:55
OK.
571
00:35:55 --> 00:35:58
And let me ask the fourth part
of this question -- a matrix
572
00:35:58 --> 00:36:00
that has orthogonal
eigenvectors,
573
00:36:00 --> 00:36:03
but it's not symmetric.
574
00:36:03 --> 00:36:10
What matrices have orthogonal
eigenvectors,
575
00:36:10 --> 00:36:15
but they're not symmetric
matrices?
576
00:36:15 --> 00:36:27
What other families of matrices
have orthogonal eigenvectors?
577
00:36:27 --> 00:36:31
We know symmetric matrices do,
but others, also.
578
00:36:31 --> 00:36:35
So I'm looking for orthogonal
eigenvectors,
579
00:36:35 --> 00:36:37
and, what do you suggest?
580
00:36:37 --> 00:36:40
The matrix could be
skew-symmetric.
581
00:36:40 --> 00:36:43
It could be an orthogonal
matrix.
582
00:36:43 --> 00:36:47.67
It could be symmetric,
but that was too easy,
583
00:36:47.67 --> 00:36:50
so I ruled that out.
584
00:36:50 --> 00:37:00
It could be skew-symmetric like
one minus one,
585
00:37:00 --> 00:37:02
like that.
586
00:37:02 --> 00:37:14
Or it could be an orthogonal
matrix like cosine sine,
587
00:37:14 --> 00:37:19
minus sine, cosine.
588
00:37:19 --> 00:37:27
All those matrices would have
complex orthogonal eigenvectors.
589
00:37:27 --> 00:37:34
But they would be orthogonal,
and so those examples are fine.
590
00:37:34 --> 00:37:34
OK.
591
00:37:34 --> 00:37:41
We can continue a little longer
if you would like to,
592
00:37:41 --> 00:37:44
with these -- from this exam.
593
00:37:44 --> 00:37:47
From these exams.
594
00:37:47 --> 00:37:49
Least squares?
595
00:37:49 --> 00:37:53
OK, here's a least squares
problem in which,
596
00:37:53 --> 00:37:58
to make life quick,
I've given the answer -- it's
597
00:37:58 --> 00:38:01
like Jeopardy!,
right?
598
00:38:01 --> 00:38:06
I just give the answer,
and you give the question.
599
00:38:06 --> 00:38:06
OK.
600
00:38:06 --> 00:38:08
Whoops, sorry.
601
00:38:08 --> 00:38:14
Let's see, can I stay over here
for the next question?
602
00:38:14 --> 00:38:17.44
OK.
least squares.
603
00:38:17.44 --> 00:38:23
So I'm giving you the problem,
one, one, one,
604
00:38:23 --> 00:38:28
zero, one, two,
c d equals three,
605
00:38:28 --> 00:38:35
four, one, and that's b,
of course, this is Ax=b.
606
00:38:35 --> 00:38:41
And the least squares solution
--
607
00:38:41 --> 00:38:45
Maybe I put c hat d hat to
emphasize it's not the true
608
00:38:45 --> 00:38:46
solution.
609
00:38:46 --> 00:38:51
So the least square solution --
the hats really go here -- is
610
00:38:51 --> 00:38:53
eleven-thirds and minus one.
611
00:38:53 --> 00:38:57
Of course, you could have
figured that out in no time.
612
00:38:57 --> 00:39:00
So this year,
I'll ask you to do it,
613
00:39:00 --> 00:39:02
probably.
614
00:39:02 --> 00:39:08
But, suppose we're given the
answer, then let's just remember
615
00:39:08 --> 00:39:10
what happened.
616
00:39:10 --> 00:39:12
OK, good question.
617
00:39:12 --> 00:39:19
What's the projection P of this
vector onto the column space of
618
00:39:19 --> 00:39:20
that matrix?
619
00:39:20 --> 00:39:26.17
So I'll write that question
down, one.
620
00:39:26.17 --> 00:39:27
What is P?
621
00:39:27 --> 00:39:28
The projection.
622
00:39:28 --> 00:39:35
The projection of b onto the
column space of A is what?
623
00:39:35 --> 00:39:41
Hopefully, that's what the
least squares problem solved.
624
00:39:41 --> 00:39:42
What is it?
625
00:39:42 --> 00:39:48
This was the best solution,
it's eleven-thirds times column
626
00:39:48 --> 00:39:55
one, plus -- or rather,
minus one times column two.
627
00:39:55 --> 00:39:56
Right?
628
00:39:56 --> 00:40:00
That's what least squares did.
629
00:40:00 --> 00:40:08
It found the combination of the
columns that was as close as
630
00:40:08 --> 00:40:10
possible to b.
631
00:40:10 --> 00:40:15
That's what least squares was
doing.
632
00:40:15 --> 00:40:18
It found the projection.
633
00:40:18 --> 00:40:20
OK?
634
00:40:20 --> 00:40:23
Secondly, draw the straight
line problem that corresponds to
635
00:40:23 --> 00:40:23
this system.
636
00:40:23 --> 00:40:26
So I guess that the straight
line fitting a straight line
637
00:40:26 --> 00:40:28
problem, we kind of recognize.
638
00:40:28 --> 00:40:30
So we recognize,
these are the heights,
639
00:40:30 --> 00:40:32
and these are the points,
and so at zero,
640
00:40:32 --> 00:40:34
one, two, the heights are
three, and at t equal to one,
641
00:40:34 --> 00:40:36
the height is four,
one, two, three,
642
00:40:36 --> 00:40:39
four, and at t equal to two,
the height is one.
643
00:40:39 --> 00:40:48
So I'm trying to fit the best
straight line through those
644
00:40:48 --> 00:40:50
points.
645
00:40:50 --> 00:40:50.72
God.
646
00:40:50.72 --> 00:40:59
I could fit a triangle very
well, but, I don't even know
647
00:40:59 --> 00:41:06
which way the best straight line
goes.
648
00:41:06 --> 00:41:14
Oh, I do know how it goes,
because there's the answer,yes.
649
00:41:14 --> 00:41:21
It has a height eleven-thirds,
and it has slope minus one,
650
00:41:21 --> 00:41:26.17
so it's something like that.
651
00:41:26.17 --> 00:41:26
OK.
652
00:41:26 --> 00:41:27.21
Great.
653
00:41:27.21 --> 00:41:33.45
Now, finally -- and this
completes the course -- find a
654
00:41:33.45 --> 00:41:37.38
different vector b,
not all zeroes,
655
00:41:37.38 --> 00:41:43
for which the least square
solution would be zero.
656
00:41:43 --> 00:41:49
So I want you to find a
different B so that the least
657
00:41:49 --> 00:41:53
square solution changes to all
zeroes.
658
00:41:53 --> 00:41:59
So tell me what I'm really
looking for here.
659
00:41:59 --> 00:42:04
I'm looking for a b where the
best combination of these two
660
00:42:04 --> 00:42:07
columns is the zero combination.
661
00:42:07 --> 00:42:10
So what kind of a vector b I
looking for?
662
00:42:10 --> 00:42:14
I'm looking for a vector b
that's orthogonal to those
663
00:42:14 --> 00:42:15
columns.
664
00:42:15 --> 00:42:19
It's orthogonal to those
columns, it's orthogonal to the
665
00:42:19 --> 00:42:24
column space,
the best possible answer is
666
00:42:24 --> 00:42:24
zero.
667
00:42:24 --> 00:42:30
So a vector b that's orthogonal
to those columns -- let's see,
668
00:42:30 --> 00:42:34
maybe one of those minus two of
those, and one of those?
669
00:42:34 --> 00:42:38
That would be orthogonal to
those columns,
670
00:42:38 --> 00:42:41
and the best vector would be
zero, zero.
671
00:42:41 --> 00:42:43
OK.
672
00:42:43 --> 00:42:47
So that's as many questions as
I can do in an hour,
673
00:42:47 --> 00:42:50
but you get three hours,
and, let me just say,
674
00:42:50 --> 00:42:54
as I've said by e-mail,
thanks very much for your
675
00:42:54 --> 00:42:58.38
patience as this series of
lectures was videotaped,
676
00:42:58.38 --> 00:43:02
and, thanks for filling out
these forms, maybe just leave
677
00:43:02 --> 00:43:06.8
them on the table up there as
you go out
678
00:43:06.8 --> 00:43:10
-- and above all,
thanks for taking the course.
679
00:43:10 --> 00:43:11
Thank you.
680
00:43:11 --> 00:43:14
Thanks.
| 677.169 | 1 |
Course Objectives: After completion of the course,
the student will demonstrate the following:
an understanding of standard vocabulary and symbols associated
with trigonometry and calculus;
a better understanding of fundamental concepts in trignometry,
including angle measure (degree and radian), trig ratios, identities,
Law of Sines, Law of Cosines, solving triangles, and graphing trigonometric
functions;
a better understanding of fundamental concepts in calculus, including
limits, continuity, derivatives and their applications; and
an understanding of the scope and sequence of the P-12 mathematics
curriculum.
*Non-credit for M.Ed. or Ed.S. in Secondary Education with concentration
in Mathematics.
Late Assignments Policy: Hard copies of all assignments, journal
entries, etc. are due at the beginning of class on the specified date (see
course schedule). Late assignments will not be accepted unless prior approval
has been given by the instructor. In the event that a late assignment is
accepted, the grade on the assignment will be lowered.
Attendance Policy: Students are expected to attend all
classes. This term a student may withdraw with a grade of W through March 2nd,
regardless of grades, absences, etc. This deadline has been established by the
University. After this deadline, if a student has accumulated more than two
absences throughout the semester, he/she will normally receive a grade of WF. (A
grade of WF counts as an F.) The two absences should be saved for sickness and
other emergencies. Late arrivals and early exits count one-half of an absence.
If a student is absent for a test and has an excuse from someone in authority,
then the student will be expected to complete a make-up exam within one week of
the original test date. Students who maintain a perfect attendance record (i.e.
no excused or unexcused absences) will have 5 points added to their Total Points
at the end of the semester.
Suggested Problems: For each section covered in class there
will be a set of problems provided. These are not homework problems
in the sense that they will be taken up and graded. Instead,
these are problems that are recommended for you to work in order to
be successful in the class. If you have questions concerning the
suggested problems, you should address these questions to the instructor
during office hours, before or after class, or during the review session
prior to the test.
Conferences: Conferences can be beneficial and are
encouraged. All conferences should occur during the instructor's office
hours, whenever possible. If these hours conflict with a student's schedule,
then appointments should be made. The conference time is not to be used
for duplication of lectures that were missed; it is the student's responsibility
to obtain and review lecture notes before consulting with the instructor.
The instructor is very concerned about the student's achievement and well-being
and encourages anyone having difficulties with the course to come by the
office for extra help. Grades will be based on coursework, not on Hope Grant
needs, GPA, or any other factors outside the realm of coursework.
| 677.169 | 1 |
Our elementary math curriculum is designed to meet the Common Core Learning Standards (CCLS) as adopted by New York State. The primary resource through which we deliver instruction is enVision Math Common Core (Pearson, 2012), commonly called Envisions. For more information on this program see PearsonSchool.com/envisionmathcommoncore. This program was overwhelming selected as the primary mathematics resource for the district by the Elementary Math Curriculum Advisory Committee. This committee, comprised of teachers and leaders representing all 5 elementary schools and all grades K - 5, met throughout the 2011-12 school year. After studying the CCLS, examining a variety of resources and piloting 2 resources, the committee selected Envsions as the resource that would best meet the needs of all our students.
Middle School, Grades 6-8
The mathematics curriculum in grades 6 - 8 is aligned with the Common Core Learning Standards (CCLS) as adopted by New Your State. To see these standards, click here. The primary resource for the mathematics curriculum at the middle school level is Mathematics: Applications and Concepts, Course 2 (6th), Course 3 (7th) and Pre-Algebra (8th), published by Glencoe. Additional resources include Connected Math,HoltAlgebra 1 (8 Accelerated). For more information about Connected Math, click here.
High School, Grades 9-12
At the high school level, New York State administers 3 Regents exams in Mathematics; Algebra, Geometry and Algebra 2 Trigonometry. The Algebra Regents is required for all students to earn a Regents diploma. All 3 Regents exams are required for students to earn an Advanced Regents Diploma. Please note that White Plains High School offers the study of Algebra in one year (Algebra) and in two years (Algebra 9 and Algebra 10).
The district has adopted the following textbooks which are aligned with the 2005 curriculum: Algebra 1 and Geometry published by Holt and Algebra 2 Trigonometry published by AMSCO. We continue to use a variety of resources for our upper-level courses.
The high school continues to offer a number of courses outside those previously mentioned, including Contemporary Math, College Algebra with Trigonometry, Precalculus (honors and regular), Calculus (ACE, AP AB and AP BC) and Statistics AP. Contemporary Math, College Algebra with Trigonometry, Calculus and Precalculus are offered for dual enrollment with Westchester Community College through their Advanced College Experience(ACE) program.
To view a flow chart of the math course offerings at White Plains High School, please click here.
| 677.169 | 1 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.