Master Math: Calculus

352 Pages·1998·11 MB·English

by  Debra Anne Ross

Master Math
&
Calculus
BY
Debra Anne Ross
CAREER PRESS
3 Tice Road
P.O. Box 687
Frankh Lakes, NJ 07417
1 -800-CAREER-1
201-848-0310( NJ and outside U.S.)
Fax: [phone]
Copyright 0 1998 by Debra Anne Ross
AU rights reserved under the Pan-American and Interna-
tional Copyright Conventions. This book may not be reproduced,
in whole or in part, in any form or by any means electronic or
mechanical, includmg photocopying, recording, or by any infor-
mation storage and retrieval system now known or hereafter
invented, without written permission from the pubhsher, The
Career Press.
MASTERM ATH:C ALCULUS
Cover design by The Visual Group
Printed in the U.S.A. by Book-mart Press
To order this title, please call toll-free 1-800-CAREER4
(NJ and Canada: 201-848-0310) to order using VISA or
Mastercard, or for further information on books from Career
Press.
Library of Congress Cataloging-in-Publication Data
Ross, Debra, 1958-
Master math : calculus /by Debra Anne Ross.
p. cm.
Includes index.
ISBN 1-56414-337-6O pbk.)
1. Calculus. I. Title.
QA303.R725
5 15--dc21 98-257 69
Ac know ledgrne nt
I am indebted to Cyndy Lakowske for reading this
book for accuracy and for all of her helpful comments. I am
also indebted to Dr. Melanie McNeil for reading the first
three books in the Muster Muth series for accuracy and for
all of her helpful comments. I am grateful to Dr. Channing
Robertson for reviewing all four books of the series and, in
general, for all his guidance.
I would especially like to thank Dr. Sidney Kramer,
my agent, and the staff of Mews Books. Without Sidney,
the Master Muth series would not have been published. I
would like to thank the entire staff at Career Press, and
especially Betsy Sheldon, chief editor, who worked closely
with me. I would also like to thank my mother for looking
over the final prints of all four books and for all of her
helpful com ment s.
Finally, I would like to thank both of my parents for
their moral support and my aunt and uncle for helping me
with my education.
i
This page intentionally left blank
Introduction 1
Chapter 1. Functions
1.1. Functions: types, properties and definitions 3
1.2. Exponents and logarithms 9
1.3. Trigonometric functions 11
1.4. Circular motion 21
1.5. Relationship between trigonometric and
exponential functions 26
1.6. Hyperbolic functions 27
1.7. Polynomial functions 29
1.8. Functions of more than one variable and
contour diagrams 30
1.9. Coordinate systems 34
1-10. Complex numbers 38
1.11 . Parabolas, circles, ellipses and hyperbolas 40
Chapter 2. The Derivative
2.1. The limit 49
2.2. Continuity 52
2.3. Differentia b ilit y 55
2.4. The definition of the derivative and rate
of change 55
2.5. A (delta) notation and the definition of
the derivative 59
2.6. Slope of a tangent line and the definition of
the derivative 61
2.7. Velocity, distance, slope, area and the
definition of the derivative 63
2.8. Evaluating derivatives of constants and
linear functions 67
...
Ill
2.9. Evaluating derivatives using the derivative
formula 68
2.10. The derivatives of a variable, a constant
with a variable, a constant with a function
and a variable raised to a power 69
2.11. Examples of differentiating using the
derivative formula 71
2.12. Derivatives of powers of functions 73
2.13. Derivatives of ax,e ~ and In x 74
2.14. Applications of exponential equation 80
2.15. Differentiating sums, differences and
p oly no m ia 1s 83
2.16. Taking second derivatives 84
2.17. Derivatives of products: the product rule 86
2.18. Derivatives of quotients: the quotient rule 88
2.19. The chain rule for differentiating
corn plica ted functions 89
2.20. Rate problem examples 93
2.21 . Differentiating trigonometric functions 94
2.22. Inverse functions and inverse trigonometric
functions and their derivatives 98
2.23. Differentiating hyperbolic functions 102
2 24. Differentiating mult ivariable functions 103
2.25. Differentiation of implicit vs. explicit functions 104
2.26. Selected rules of differentiation 105
2.27. Minimum, maximum and the first and
second derivatives 105
2.28. Notes on local linearity, approximating
slope of curve and numerical methods 111
Chapter 3. The Integral
3.1. Introduction 115
3.2. Sums and sigma notation 115
3.3. The antiderivative or indefinite integral
and the integral formula 119
3.4. The definite integral and the Fundamental
Theorem of Calculus 122
3.5. Improper integrals 125
iv
3.6. The integral and the area under a curve 127
3.7. Estimating integrals using sums and
associated error 131
3.8. The integral and the average value 134
3.9. Area below the X-axis, even and odd
functions and their integrals 135
3.10. Integrating a function and a constant,
the sum of functions, a polynomial, and
properties of integrals 137
3.1 1. Multiple integrals 140
3.12. Examples of common integrals 142
3.13. Integrals describing length 143
3.14. Integrals describing area 144
3.15. Integrals describing volume 149
3.16. Changing coordinates and variables 157
3.17. Applications of the integral 162
3.18. Evaluating integrals using integration by parts 167
3.19. Evaluating integrals using substitution 169
3.20. Evaluating integrals using partial fractions 177
3.2 1. Evaluating integrals using tables 182
Chapter 4. Series and Approximations
4.1. Sequences, progressions and series 183
4.2. Infinite series and tests for convergence 186
4.3. Expanding functions into series, the
power series, Taylor series, Maclaurin
series, and the binomial expansion 192
Chapter 5. Vectors, Matrices, Curves,
Surfaces and Motion
5.1. Introduction to vectors 199
5.2. Introduction to matrices 207
5.3. Multiplication of vectors and matrices 210
5.4. Dot or scalar products 213
5.5. Vector or cross product 216
5.6. Summary of determinants 220
5.7. Matrices and linear algebra 222
V
5.8. The position vector, parametric equations,
curves and surfaces 228
5.9. Motion, velocity and acceleration 234
Chapter 6. Partial Derivatives
6.1. Partial derivatives: representation and
evaluation 247
6.2. The chain rule 250
6.3. Representation on a graph 251
6.4. Local linearity, linear approximations,
quadratic approximations and differentials 254
6.5. Directional derivative and gradient 259
6.6. Minima, maxima and optimization 264
Chapter 7. Vector Calculus
7.1. Summary of scalars, vectors, the directional
derivative and the gradient 271
7.2. Vector fields and field lines 275
7.3. Line integrals and conservative vector fields 281
7.4. Green’s Theorem: tangent and normal
(flux) forms 287
7.5. Surface integrals and flux 292
7.6. Divergence 300
7.7. Curl 305
7.8. Stokes’ Theorem 309
Chapter 8. Introduction to Differential Equations
8.1. First-order differential equations 313
8.2. Second-order linear differential equations 318
8.3. Higher-order linear differential equations 32 1
8.4. Series solutions to differential equations 323
8.5. Systems of differential equations 325
8.6. Laplace transform method 327
8.7. Numerical methods for solving differential
equations 328
8.8. Partial differential equations 330
Index 333
vi
Introduction
Master Muth: CuZcuZus is a comprehensive reference
book for advanced high school and college students that
explains and clarifies the key principles of calculus. The
purpose of the book is to provide an easy-to-access reference
source for locating specific calculus topics. This book is de-
signed so that a student can quickly look up a topic and, by
reading the explanation and the information in its section,
find the relevant facts and formulas. This book can also be
used to obtain a general knowledge and understanding of
calculus and it provides a complete breadth of material so
that most topics related to calculus are explained.
Master Math: Calculus reviews functions and ex-
plains the principles and operations of the derivative, the
integral, series and approximations, vectors, matrices,
curves, surfaces, motion, partial derivatives, vector calculus
and introductory differential equations. The chapters in
this book are divided into major sections containing
independent topics housed within the context of where they
fit into the discipline of calculus.
This is the fourth book in the Muster Math series.
The first three books are Basic Math and Pre-Algebra,
Algebra and Pre-Calculus and Geometry. The Muster Math
series presents the general principles of mathematics from
grade school through college including arithmetic, algebra,
geometry, trigonometry, pre-calculus and calculus.