A Lab Course with Microcalc® for Information About Site Licenses for Microcalc Use in Course Instructions, Please Contact Springer-Verlag

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A Lab Course with MicroCalc® For information about site licenses for MicroCalc use in course instructions, please contact Springer-Verlag:

James Chin MathematicslStatistics Product Manager 1-80o-SPRINGE(R) ext. 582 (1-800-777-4643 ext. 582) 1-212-460-1582 e-mail: [email protected]

Springer Science+Business Media, LLC A Lab Course with MicroCalc®

HARLEY FLANDERS University of Michigan, Ann Arbor

Springer Textbooks in Mathematical Sciences Series Editors

Thomas F. Banchoff Jerrold Marsden Brown University California Institute ofTechnology

Keith Devlin Stan Wagon St. Mary's College Macalester College Gaston Gonnet ETH Zentrum, Zurich

MicroCalc software copyright Harley Flanders.

Library of Congress Cataloging-in-Publication Data Flanders, Harley. Calculus : a lab course with MicroCalc / Harley Flanders . p. em. - (Textbooks in mathematical sciences) Includes index.

1. Calculus-Data processing . 2. MicroCalc. 1. Title. II. Series. QA303.5.D37F585 1996 515'.0285-dc20 95-44876

Printed on acid-free paper.

Production managed by Bill Imbornoni ; manufacturing supervised by Jacqui Ashri. Camera-ready copy provided by the author.

9 876543 2

ISBN 978-0-387-94496-8 ISBN 978-1-4757-2480-6 (eBook) DOI 10.1007/978-1-4757-2480-6 Contents v Contents

Preface IX

1 Functions and Graphs 1 1.1 Overview . 1 1.2 Polynomials . 4 1.3 Quadratic Polynomials 8 1.4 Cubics and Degree . . .11 1.5 Rational Functions . . . · 14 1.6 Degree, Algebraic Functions · 18 1.7 Zeros and Extrema . .. . 20 1.8 Trigonometric Functions .23 1.9 Exponential Functions .28 1.10 Combining Functions . . .32 1.11 What is a Function? [Optional] . 36

2 Limits 38 2.1 Introduction to Limits ...... 38 2.2 Limit Estimates by Extrapolation .. .41 2.3 What Does "Limit" Really Mean? . . .43 2.4 Properties of Limits . . . . 48 2.5 Continuous Functions . . . 54 2.6 Discontinuous Functions . 58

3 The Derivative 61 3.1 Slope . . 61 3.2 The Difference Quotient . 63 3.3 The Derivative ...... 68 3.4 Rules for Differentiating .70 3.5 Polynomials , Rational Functions . .73 3.6 Trigonometric Functions . .77 3.7 The Chain Rule . .82 3.8 Radical, Power, and Exponential Functions .85 3.9 Higher Derivatives . . . . .88 3.10 Symbolic Differentiation · 91

4 Applications of Derivatives 94 4.1 Monotonicity ...... 94 4.2 Velocity and Acceleration. . 99 vi Contents

4.3 Related Rates . 104 4.4 Tangents and Linear Approximation. 107 4.5 Concavity and Inflections . 111 4.6 Zeros-Newton's Method 116 4.7 Extrema. 122 4.8 Second Derivative Test 127 4.9 Applications of Extrema 129 4.10 Growth and Decay . 135

5 Integration 142 5.1 The Area Problem . 142 5.2 Definition of the Integral 145 5.3 Riemann Sums 147 5.4 Numerical Integration 153 5.5 Simpson's Rule 160 5.6 Properties of Integrals . 166 5.7 The Fundamental Theorem 172

6 Applications of Integration 181 6.1 Area . 181 6.2 Volume 186 6.3 Averages and Expected Values. 193 6.4 Work and Pressure. 199 6.5 Money Matters . 205

7 Functions and Integrals 208 7.1 Inverse Functions 208 7.2 The Logarithm Function 212 7.3 More on Logarithms . 217 7.4 Inverse Trigonometric Functions . 223 7.5 Hyperbolic Functions 226 7.6 Integration by Substitution 231 7.7 Integration by Parts 237 7.8 Further Integration Techniques . 241 7.9 Indefinite Integrals in This Chapter 246

8 Approximation 247 8.1 L'Hospital's Rule 247 8.2 Taylor's Formula. 254 8.3 Applications of Taylor's Formula 260 Contents vii

8.4 Sequences . 267 8.5 Infinite Series . 275 8.6 Tests for Convergence. 281 8.7 Alternating Series . . 286 8.8 Improper Integrals,I 289 8.9 Improper Integrals, II 295

9 Power Series 300 9.1 Convergence of Power Series 301 9.2 Taylor Series ...... 305 9.3 Computing Taylor Expansions 309 9.4 Differentiation and Integration . 314 9.5 The Binomial Theorem . 317

Appendix A Formulas and Derivatives 320

Appendix B Integrals 323

Index 329 Preface ix Preface Objectives of This Book • To teach calculus as a laboratory science, with the computer and software as the lab, and to use this lab as an essential tool in learning and using calculus. • To present calculus and elementary differential equations with a minimum of fuss-through practice, not theory. • To stress ideas of calculus, applications, and problem solving, rather than definitions, theorems, and proofs. • To emphasize numerical aspects: approximations, order of magnitude, concrete answers to problems. • To organize the topics consistent with the needs of students in their concurrent science and engineering courses. The subject matter of calculus courses has developed over many years, much by negotiation with the disciplines calculus serves, particularly engineering. This text covers the standard topics in their conventional order. Mostly because of commercial pressures, calculus texts have grown larger and larger, trying to include everything that anyone conceivably would cover. Calculus texts have also added more and more expensive pizzazz, up to four colors now. This text is lean; it eliminates most of the "fat" of recent calculus texts; it has a simple physical black/white format; it ignores much of current calculus "culture". The computer has forced basic changes in emphasis and how to teach calculus. The most obvious effect of computers on content is to deemphasize drill on techniques. There is no rational justification for investing large amounts of students' time drilling on what can be done by computers much better than by humans. The main contribution of computers to teaching calculus is to provide a laboratory for calculus experiments. This calculus text is written to my software package MicroCalc. MicroCalc covers almost all topics of calculus. Many topics in this book will exploit computers. As a consequence, many topics will be treated quite differently than they were in the past. I hope that courses taught with this text will be taught with microcomputer demonstrations in the classroom, and with microcomputer laboratories for students to do calculus experiments. In other words, calculus will be taught from this text as other laboratory sciences are taught. MicroCalc is a dynamic package. As I wrote this text, I made many changes in MicroCalc to make it better for teaching. It is software for teaching calculus, and was not designed as production software. MicroCalc requires the MS-DOS x Preface platform. Version 7 requires a VGA or better monitor, and is the best version to use with this text. (Version 5, which supports any graphics monitor, is adequate for much of this text.) To the Student You learn mathematics partly by listening to your instructor, partly by reading your text. But mainly you learn mathematics by doing mathematics actively, by yourself or with other students. You must work over the examples in this text, both with computer and with pen and paper. The examples are for instruction, not entertainment, and fortn an essential part of this text. To understand functions, you must see their graphs. Some graphs of functions are printed in the text; you will+ generate many more, on the computer. The exercises also are designed for instruction and should be worked. Many are easy computer experiments that go quickly. Others are for pencil, paper, and thought. Some (sequences of) exercises are substantial laboratory projects. There are not many exercises in this text, and you should work at all of them; doing so is perhaps the main tool you have for learning this course. There is substance in many of the exercises, and you will develop skills and sharpen your understanding and technique by working them. The best thing you can do to prepare for a test is to rework the examples and exercises in the sections to be tested. You will be surprised how fast this goes the second, the third, ... time you work through a batch of examples and exercises. Sometimes you will see a little box symbol: D. It is there to mark the ends of examples, etc, where the ends may not be obvious. Refer to Appendix A for summaries of useful formulas from precalculus math and differentiation . Refer to Appendix B for a short table of integrals. This textbook is a dialogue between us: me the author, referred to as "I"; you the student, referred to as "you". I challenge you to calculus! Acknowledgments I am pleased to express my gratitude to my editor at Springer-Verlag New York, Jeremiah J. Lyons, to Liesl M. B. Gibson, and to my colleagues Richard N. Barshinger and Dennis J. Gittinger, who reviewed the manuscript. The text was written with the desktop publishing package "The Publisher", from ArborText, Inc, on a SPARC workstation, first an SS-l+ from the CompuAdd Corporation, later upgraded to a micro COMPstation5 from Tatung Science and Technology, Inc. I can recommend this software-hardware combination only to the very strong of heart with considerable computer smarts. "UNIX" surely sets a world standard Preface xi for user-unfriendly operating systems. "The Publisher", although around for years and in version 3.2.2, is loaded with bugs, traps, and deficiencies. However, I am truly grateful to many technical support consultants at these companies, who averted disaster for me on numerous occasions. For anyone interested, this book's type fonts are Times Roman, Helvetica, Courier, and Computer Modem.

Harley Flanders Ann Arbor, MI September, 1995