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Master Math: Calculus PDF

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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: 201-848-1 727 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.

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