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Calculus BRIAN E. BLANK STEVEN G. KRANTZ Washington University in St. Louis

� WILEY

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This book was set in 10/12 TimesTen-Roman at MPS Limited, a Macmillan Company, and printed and bound by R. R. Donnelley (Jefferson City). The cover was printed by R. R. Donnelley (Jefferson City). Founded in 1807. John Wiley & Sons, Inc. has been a valued source of knowledge and understanding for more than 200 years, helping people around the world meet their needs and fulfill their aspirations. Our company is built on a foundatin of principles that include responsibility to the communities we serve and where we live and work. In 2008, we launched a Corporate Citizenship Initiative, a global effort to address the environmental, social, economic, and ethical challenges we face in our business. Among the issues we are addressing are carbon impact, paper specificatinos and procurement, ethical conduct within our business and among our vendors, and community and charitable support. For more information, please visit our website:

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This book is printed on acid free paper. 8 Copyright © 2011, 2006 by John Wiley & Sons, Inc. All rights reserved. No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning or otherwise, except as permitted under Sections 107 or 108 of the 1976 United States Copyright Act, without either the prior written permission of the Publisher, or authorization through payment of the appropriate per-copy fee to the Copyright Clearance Center, Inc. 222 Rosewood Drive, Danvers, MA 01923, (978) 750-8400, fax (978) 646-8600. Requests to the Publisher for permission should be addressed to the Permissions Department, John Wiley & Sons, Inc., 111 River Street, Hoboken, NJ 07030-5774, (201) 748-6011, fax (201) 748-6008. Evaluation copies are provided to qualified academics and professionals for review purposes only, for use in their courses during the next academic year. These copies are licensed and may not be sold or transferred to a third party. Upon completion of the review period, please return the evaluation copy to Wiley. Return instructions and a free of charge return shipping label are available at www.wiley.com/go/returnlabel. Outside of the United States, please contact your local representative. ISBN 13

978-0470-45359-9 (Multivariable)

ISBN 13

978-0470-45360-5 (Single & Multivariable)

ISBN 13

978-0470-60198-3 (Single Variable)

Printed in the United States of America 10 9 8 7 6 5 4 3 2 1

A pebble for Louis.

BEB

This book is for Hypatia, the love of my life.

SGK

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Preface

ix

Supplementary Resources

c H A pT E R 9

Acknowledgments

xiv

About the Authors

xvi

Vectors Preview

721 721

1

Vectors in the Plane

2

Vectors in Three-Dimensional Space

3

The Dot Product and Applications

4

The Cross Product and Triple Product

5

Lines and Planes in Space

722

Summary of Key Topics Review Exercises

782 787

191

791

1

Vector-Valued Functions-Limits, Derivatives, and Continuity

2

Velocity and Acceleration

3

Tangent Vectors and Arc Length

4

Curvature

5

Applications of Vector-Valued Functions to Motion

Review Exercises

813 834

850

853

Genesis & Development 10

856

Functions of Several Variables Preview

792

803

824

Summary of Key Topics

C H A PT E R 1 1

753

766

Vector-Valued Functions Preview

732 741

785

Genesis & Development 9

C H A PT E R 1 o

xiii

861

861

1

Functions of Several Variables

2

Cylinders and Quadric Surfaces

3

Limits and Continuity

4

Partial Derivatives

5

Differentiability and the Chain Rule

6

Gradients and Directional Derivatives

7

Tangent Planes

8

Maximum-Minimum Problems

863 874

882

890 900 913

921 932

vii

viii

Contents 9

Lagrange Multipliers

Summary of Key Topics Review Exercises

946 957

960

Genesis & Development 11

CH APTER 1 2

Multiple Integrals 967 Preview

1 2 3 4 5 6 7 8

967

Double Integrals Over Rectangular Regions Integration Over More General Regions Polar Coordinates

976

990

Integrating in Polar Coordinates

999

1014

Triple Integrals

Physical Applications

1020

Other Coordinate Systems

Review Exercises

968

984

Calculation of Volumes of Solids

Summary of Key Topics

1029

1037

1042

Genesis & Development 12

CH APTER 1 3

963

1045

Vector Calculus 1049 Preview

1 2 3 4 5 6 7 8

1049

Vector Fields Line Integrals

1050 1061

Conservative Vector Fields and Path Independence Divergence, Gradient, and Curl

1085

1094 1104 Stokes's Theorem 1116 Green's Theorem

Surface Integrals

1131

The Divergence Theorem

Summary of Key Topics Review Exercises

1139

1143

Genesis & Development 13

1146

Table of Integrals T-1 Formulas from Calculus: Single Variable T-14 Answers to Selected Exercises A·1 Index 1-1

1071

Calculus is one of the milestones of human thought. In addition to its longstanding role as the gateway to science and engineering, calculus is now found in a diverse array of applications in business, economics, medicine, biology, and the social sciences. In today's technological world, in which more and more ideas are quan­ tified, knowledge of calculus has become essential to an increasingly broad cross­ section of the population. Today's students, more than ever, comprise a highly heterogeneous group. Calculus students come from a wide variety of disciplines and backgrounds. Some study the subject because it is required, and others do so because it will widen their career options. Mathematics majors are going into law, medicine, genome research, the technology sector, government agencies, and many other professions. As the teaching and learning of calculus is rethought, we must keep our students' back­ grounds and futures in mind. In our text, we seek to offer the best in current calculus teaching. Starting in the 1980s, a vigorous discussion began about the approaches to and the methods of teaching calculus. Although we have not abandoned the basic framework of calculus instruction that has resulted from decades of experience, we have incorporated a number of the newer ideas that have been developed in recent years. We have worked hard to address the needs of today's students, bringing together time-tested as well as innovative pedagogy and exposition. Our goal is to enhance the critical thinking skills of the students who use our text so that they may proceed successfully in whatever major or discipline that they ultimately choose to study. Many resources are available to instructors and students today, from Web sites to interactive tutorials. A calculus textbook must be a tool that the instructor can use to augment and bolster his or her lectures, classroom activities, and resources. It must speak compellingly to students and enhance their classroom experience. It must be carefully written in the accepted language of mathematics but at a level that is appropriate for students who are still learning that language. It must be lively and inviting. It must have useful and fascinating applications. It will acquaint students with the history of calculus and with a sense of what mathematics is all about. It will teach its readers technique but also teach them concepts. It will show students how to discover and build their own ideas and viewpoints in a scientific subject. Particularly important in today's world is that it will illustrate ideas using computer modeling and calculation. We have made every effort to insure that ours is such a calculus book. To attain this goal, we have focused on offering our readers the following: •

A writing style that is lucid and readable

•

Motivation for important topics that is crisp and clean

•

Examples that showcase all key ideas

•

Seamless links between theory and applications

•

Applications from diverse disciplines, including biology, economics, physics, and engineering

ix

x

Preface •

Graphical interpretations that reinforce concepts

•

Numerical investigations that make abstract ideas more concrete

Content We present topics in a sequence that is fairly close to what has become a standard order for multivariable calculus. This volume refers to only ten formulas from Calculus: Single Variable. To facilitate the use of this book for students who own a different single variable text, we have collected these equations and presented them on p. T-14. In the outline that follows, we draw attention to several sections that may be regarded as optional. Chapter numbering follows that of our eight chapter single variable text. Chapter 9, the first chapter of multivariable calculus, begins with a section that introduces vectors in the plane. It is followed by a section that covers the same material in space. After sections on the dot and cross product, Chapter 9 concludes with a comprehensive account of lines and planes in space. Chapter 10 is devoted to the differential calculus and geometry of space curves. The final two sections are concerned with curvature and associated concepts, including applications to motion and derivations of Kepler's Laws. Because these two sections are not used later in the text, they may be omitted by instructors who need additional time for other topics. The differential calculus of functions of two and three variables is taken up in Chapter 11. All topics in the standard curriculum are discussed. One less con­ ventional topic that we treat is the development of order 2 and order 3 Taylor polynomials for functions of two variables. We use these ideas in the discriminant test for saddle points and local extrema but nowhere else. It is therefore feasible to omit the discussion of multivariable Taylor polynomials. Chapter 12 is devoted to multiple integrals and their applications. In a section that precedes cylindrical and spherical coordinates, we develop polar coordinates ab initio. This section may be omitted, of course, if polar coordinates have been introduced earlier in the calculus curriculum. The concluding chapter on vector calculus, Chapter 13, covers vector fields, line and surface integrals, divergence, curl, flux, Green's Theorem, Stokes's The­ orem, and the Divergence Theorem.

Structural Elements We start each chapter with a preview of the topics that will be covered. This short initial discussion gives an overview and provides motivation for the chapter. Each section of the chapter concludes with three or four Quick Quiz questions located before the exercises. Some of these questions are true/false tests of the theory. Most are quick checks of the basic computations of the section. The final section of every chapter is followed by a summary of the important formulas, theorems, definitions, and concepts that have been learned. This end-of-chapter summary is, in tum, followed by a large collection of review which are similar to the worked examples found in the chapter. Each chapter ends with a section called Genesis &

Preface

xi

Development, in which we discuss the history and evolution of the material of the chapter. We hope that students and instructors will find these supplementary dis­ cussions to be enlightening. Occasionally within the prose, we remind students of concepts that have been learned earlier in the text. Sometimes we offer previews of material still to come. These discussions are tagged A Look Back or A Look Forward (and sometimes both). Calculus instructors frequently offer their insights at the blackboard. We have included discussions of this nature in our text and have tagged them Insights.

Proofs During the reviewing of our text, and after the first edition, we received every possible opinion concerning the issue of proofs-from the passionate Every proof must be included to the equally fervent No proof should be presented, to the see­ mingly cynical It does not matter because students will skip over them anyway. In fact, mathematicians reading research articles often do skip over proofs, returning later to those that are necessary for a deeper understanding of the material. Such an approach is often a good idea for the calculus student: Read the statement of a theorem, proceed immediately to the examples that illustrate how the theorem is used, and only then, when you know what the theorem is really saying, turn back to the proof (or sketch of a proof, because, in some cases, we have chosen to omit details that seem more likely to confuse than to enlighten, preferring instead to concentrate on a key, illuminating idea).

Exercises There is a mantra among mathematicians that calculus is learned by doing, not by watching. Exercises therefore constitute a crucial component of a calculus book. We have divided our end-of-section exercises into three types: Problems for Practice, Further Theory and Practice, and Calculator/Computer Exercises. In general, exercises of the first type follow the worked examples of the text fairly closely. We have provided an ample supply, often organized into groups that are linked to particular examples. Instructors may easily choose from these for creating assignments. Students will find plenty of unassigned problems for additional practice, if needed. The Further Theory and Practice exercises are intended as a supplement that the instructor may use as desired. Many of these are thought problems or open­ ended problems. In our own courses, we often have used them sparingly and sometimes not at all. These exercises are a mixed group. Computational exercises that have been placed in this subsection are not necessarily more difficult than those in the Problems for Practice exercises. They may have been excluded from that group because they do not closely follow a worked example. Or their solutions may involve techniques from earlier sections. Or, on occasion, they may indeed be challenging.

xii

Preface The Calculator/Computer exercises give the students (and the instructor) an opportunity to see how technology can help us to see and to perceive. These are problems for exploration, but they are problems with a point. Each one teaches a lesson.

Notation Throughout our text, we use notation that is consistent with the requirements of technology. Because square brackets, brace brackets, and parentheses mean dif­ ferent things to computer algebra systems, we do not use them interchangeably. For example, we use only parentheses to group terms. Without exception, we enclose all functional arguments in parentheses. Thus, we write sin(x) and not sin x. With this convention, an expression such as cos(x)2 is unambiguously defined: It must mean the square of cos(x); it cannot mean the cosine of (x)2 because such an interpretation would understand (x)2 to be the argument of the cosine, which is impossible given that (x)2 is not found inside parentheses. Our experience is that students quickly adjust to this notation because it is logical and adheres to strict, exceptionless rules. Occasionally, we use exp(x) to denote the exponential function. That is, we sometimes write exp(x) instead of