# ɷ[eli passow] schaum's outline of understanding cal

Post on 07-Apr-2016

223 views

Embed Size (px)

DESCRIPTION

ɷ[eli passow] schaum's outline of understanding calTRANSCRIPT

SCHAUMS OUTLINE OF

THEORY AND PROBLEMS

OF

UNDERSTANDING

CALCULUS CONCEPTS

EL1 PASSOW, Ph.D. Professor of Mathematics

Temple University Philadelphia, Pennsylvania

SCHAUMS OUTLINE SERIES McGRAW-HILL

New York San Francisco Washington, D.C. Auckland Bogotd Caracas Lisbon London Madrid Mexico Ci ty Milan Montreal New Dehli

San Juan Singapore Sydney Tokyo Toronto

SCHAUMS OUTLINE OF

THEORY AND PROBLEMS

OF

UNDERSTANDING

CALCULUS CONCEPTS

EL1 PASSOW, Ph.D. Professor of Mathematics

Temple University Philadelphia, Pennsylvania

SCHAUMS OUTLINE SERIES McGRAW-HILL

New York San Francisco Washington, D.C. Auckland Bogotd Caracas Lisbon London Madrid Mexico Ci ty Milan Montreal New Dehli

San Juan Singapore Sydney Tokyo Toronto

Ell Passow is a Professor of Mathematics at Temple University, having received a B.S. in Mathematics from the Massachusetts Institute of Technology, and M.A. and Ph.D. degrees from Yeshiva University. He has taught at Yeshiva University, Bar-Ilan University, and The Technion, and has published over 35 papers in Approximation and Interpolation Theory. He is a member of the Mathematical Association of America.

Schaums Outline of Theory and Problems of UNDERSTANDING CALCULUS CONCEPTS

Copyright (0 19% by The McGraw-Hill Companies, Inc.All rights teserved. Rinted in the United States of America. Except as permitted under the United States Copyright Act of 1976, no part of this publication may be reproduced or distributed in any form or by any means, or stored in a data base or retrieval system, without the prior written permission of the publisher.

4 5 6 7 8 9 10 1 1 12 13 14 15 16 17 18 19 20 PRS/PRS 9 0 I 0 9

ISBN 0-07-048738-3

Sponsoring Editor: Arthur Biderman Production Supervisor: Donald E Schmidt Editing Supervisor: Maureen Walker

Library of CongressCataloging-in-PublicationData

Passow, Eli. Schaums outline of theory and problems of understanding calculus

concepts / Eli Passow. p. cm. - (Schaums outline series)

Includes index. ISBN 0-07-048738-3 1. Calculus-Outlines, syllabi, etc. 2. Calculus-Problems,

exercises, etc. 1. Title. QA303.P284 1996 5 I Y . 0 7 6 6 ~ 2 0 95-47937

CIP

McGraw-Hill iz ,A\ Diiwion of TheMcGmwHiU Companies

Dedicated to

Chaya

Shuli, Nati and Dani

and

In Memory of

my parents

Meyer and Clara Passow

Acknowledgements

I owe a great deal to two very special people. Rachel Ebner did a magnificent job of editing my manuscript, which substantially enhanced its clarity. Lev Brutman taught me the intricacies of and provided important advice and encouragement throughout the writing of this book. Thanks also to my Sponsoring Editor, Arthur Biderman, for his patience and help, the staff at Schaums, and to the reviewers, for their many useful suggestions.

Preface

Dear Student,

Feeling butterflies at the thought of. beginning your Calculus course? Or worse? Well youre not alone. Theres a long tradition of students suffering from Calculus Syndrome. I myself took Calculus in 1968 and hated every minute of it. I somehow survived the agony, passed the course barely respectably, and sold that fat, heavy textbook right away -who needed memories of humiliation? I went on to major in languages so that I could never again be traumatized by numbers.

And so, when 25 years later Dr. Eli Passow approached me to edit a companion volume to Calculus texts, I laughed and told him straightforwardly: Youve come to the wrong address. Im allergic to mathematics.

But Dr. Passow convinced me that I was just the person he was looking for. He had developed a simple and clear conceptual approach to Calculus and wanted to test out his ideas on someone who was absolutely certain that Calculus could never penetrate her brain. Okay, so he had the right address after all.

I approached Understanding Calculus Concepts with all the skepticism Dr. Pas-sow could have wished for - it came quite naturally - but I was quickly surprised. Lightbulbs of understanding started going on in my head that should have lit up a quarter century ago! Dr. Passows d-R-C approach (Approximation, Refinement, Limit - Ill let him explain it) to all the major concepts of Calculus quickly led me to the crucial awareness of the unity of the subject, a soothing substitution for my past experience of Calculus as a zillion different kinds of problems, each demanding a different technique for solution. With the d-R-L method, the approach to a wide variety of problems is the same; learning takes place in a comprehensive way instead of in unrelated fragments.

And most important, in Dr. Passows writing I heard the voice of a real teacher, the teacher we all wish we had, especially for Calculus. He has a gift for simplifying and a knack for presenting illuminating illustrations and examples that help each concept click into place, even in a mind that was determined not to understand.

So, dont panic. Youre in good hands. Calculus Syndrome has a cure and youre holding it right now.

Good luck!

Rachel Ebner

This page intentionally left blank

Contents

Chapter 2

Chapter 3

Chapter 1

Chapter 4

WHAT ITS ALL ABOUT .................................. 1

1.1 Introduction .............................................. 1

1.2 Functions ................................................. 2

1.3 Calculus: One Basic Idea .................................. 4

1.4 Conceptual Development .................................. 5

1.5 About this Book .......................................... 7

THE DERIVATIVE ........................................ 10

2.1 Motivation ............................................... 10

2.1.1 Slopes and Tangent Lines ......................... 10

2.1.2 Finding the Instantaneous Velocity . . . . . . . . . . . . . . . . 22

2.2 Definition of the Derivative ............................... 26

2.3 Notation for the Derivative ............................... 27

2.4 Calculating Derivatives ................................... 29

2.5 Applications of the Derivative ............................ 36

Solved Problems ............................................... 36

APPLICATIONS OF THE DERIVATIVE . . . . . . . . . . . . . . . 56

3.1 Newtons Method ........................................ 56

3.1.1 Introduction ...................................... 56

3.1.2 The Method ...................................... 57

3.2 Linear Approximation and Taylor Polynomials . . . . . . . . . . . . 61

Solved Problems ............................................... 77

THE INTEGRAL ........................................... 89

4.1 Motivation ............................................... 89

4.1.1 Velocity and Distance ............................. 90

4.1.2 Work ............................................. 98

4.1.3 Area ............................................ 102

4.2 Definition of the Integral ................................ 103

4.3 Notation for the Integral ................................ 104

4.3.1 Riemann Sums .................................. 105

4.4 Computational Techniques .............................. 109

4.5 Applications of the Integral ............................. 115

Solved Problems .............................................. 115

Chapter 5

Chapter 6

Chapter 7

APPLICATIONS OF THE INTEGRAL . . . . . . . . . . . . . . . . 128

5.1 Arc Length ............................................. 128

5.2 Volume of a Solid of Revolution ......................... 134

Solved Problems .............................................. 139

TOPICS IN INTEGRATION ............................ 150

6.1 Improper Integrals ...................................... 150

6.2 Numerical Integration ................................... 158

6.2.1 Trapezoidal Rule ................................ 158

6.2.2 Simpsons Rule .................................. 164

Solved Problems .............................................. 167

INFINITE SERIES ........................................ 178

7.1 Motivation ............................................. 178

7.2 Definition .............................................. 182

7.3 Notation ................................................ 183

7.4 Computational Techniques .............................. 183

7.5 Applications of Series ................................... 192

Solved Problems .............................................. 202

INDEX ...................................................................... 214

ChaDter I

What Its All About

1.1 Introduction

For many students, calculus is a frightening subject, frightening even before the course begins! For one thing, you are handed (or, more correctly, you buy at a handsome price) an intimidating 1000-page volume chock-full of complicated material for which you will be responsible over the next two or three semesters. The material itself has the reputation of being far more difficult than any mathematics youve previously studied, filled with a bewildering assortment of apparently unrelated techniques and methods. Last but not least, the high failure rate in calculus is notorious; students repeating the course are clearly in evidence around you. So youre probably asking yourself, What am I doing here?)

Yet, as well see in section 1.3, calculus need not b