Idea Transcript
THOMAS'
CALCULUS Twelfth Edition
Multivariable Based on the original work by
George B. Thomas, Jr. Massachusetts Institute of Technology as revised by
Maurice D. Weir Naval Postgraduate School Joel Hass University of California, Davis
AddisonWesley Boston Columbus Indianapolis New York San Francisco Upper Saddle River Amsterdam Cape Town Dubai London Madrid Milan Munich Paris Montreal Toronto Delhi Mexico City Sao Paulo Sydney Hong Kong Seoul Singapore Taipei Tokyo
EdltorInCblef: Deirdre Lynch 8omor AcqailltiODl Editor: William Hoffirum 8omor Project Editor: Rachel S. Reeve As.ociate Editor: Caroline Celano As.ociate Project Editor: Leah Goldberg 8omor Managing Editor: Karen Wernhohn 8omor Prodnctlon Snpen1..r: Sheila Spinoey Senior Design Supervisor: Andrea Nix Digital A"ets Manager: Mariaone Groth Media Producer: Lin Mahoney Software Development: Mary Dumwa1d aod Bob Carroll EIecutive Marketing Manager: Jeff Weidenaar MarketingAssilltant: Kendra Bas.i Senior Author Supportlfec:hnology Specialist: Joe Vetere Senior Prepress Supervisor: Caroline Fell Manufaeloring Manager: Evelyn Beaton Production Coordinator: Kathy Diamond Composition: Nesbitt Graphics, Inc. 1lln.tratioDl: Karen Hey!, lllustraThch Cover Design: Rokusek Design Cover image: Forest Edge, Hokuto, Hokkaido, Japao 2004 © Michael Eenoa About the cover: The cover image of a tree line on a snowswept landscape, by the photographer Michael Kenna, was taken in Hokkaido, Japan. The artist was not thinking of calculus when he composed the image, but rather, of a visual haiku CODIisting of a few elemeots that would spaIk the viewer's imagination. Similarly, the minima1 design of litis text allows the central ideas of calcolu. developed in litis book to uofold to igoite the leamer's imagination.
For permission to use copyrighted matetial, grateful acknowledgment is made to the copyright bolders on page CI, which is hereby made part of this copyright page. Many ofthe desil!lllltions used by manufacturers and sellers to distiollUish their products are claimed as trsdemarks. ~
•• l l.... .... LUU"....
'"'''~!)llAuu~...
app"....
.u~ u..u..
uuu..............u
.1
.n.UUli>UU ......;»....
""g" ............... U~
a. ... ...Jetnark
claim, the designa
tions have been printed in initial caps or all caps.
Ubrary of Congress CataloginginPublication Data Weir, Maurice D. Thomas' Calculus I Maurice D. Weir, Joel Hass, George B. Thomas.12th ed. p.em ISBN 9780321643698 I. CaicolUllTextbooks. I. Hass, Joel. II. Thomas, George B. (George Brinton), 19142006. III. Thoroas, George B. (George Brinton), 19142006. Calcolus. Iv. Title V. Title: Calcolus. QA303.2.W452009b 515dc22
2009023069
Copyright 0 2010, 2005, 2001 Pearson Education, Inc. All rights reserved. No part oflitis publication may be reproduced, stated in a retrieval system, or transnti!ted, in any form or by any means, electrortic, mecbaeical, photocopying, recording, or otherwise, without the prior written permission of the publisher. Printed in the United
States ofAmerica. For information on obtaining permission for use ofmaterial in this work, please submit a written request to PearnJn Edoestion, Inc., Rights aod Contracts Departmeot, SOl Boylston Street, Suite 900, Boston, MA 02116, fax your request to 6178487047, or email at http://www.pearsoned.comllegallpermissions.htro. 12345678 91oCRK12 11 10 09
AddisonWesley is an imprint of

PEARSON www.pearsoned.com
ISBNl0' 0321643690 ISBN13' 9780321643698
CONTENTS Preface
1
I
ix
1
Functions 1.1 1.2 1.3 1.4
Functions and Their Graphs I Combining Functions; Shifting and Scaling Graphs Trigonometric Functions 22 Graphing with Calco1ators and Computers 30 QuEsTIONS TO GUIDE YOUR REVIEW PRACTICE EXERCISES
34
35 37
AoDffiONAL AND ADvANCED EXERCISES
2
Limits and Continuity 2.1 2.2 2.3 2.4 2.5 2.6
39
Rates of Change and Tangents to Curves 39 Limit of a Function and Limit Laws 46 The Precise Definition of a Limit 57 OneSided Limits 66 Continuity 73 Limits Involving Infinity; Asymptotes of Graphs QuEsTIONS TO GUIDE YOUR REVIEW PRACTICE EXERCISES
98
Differentiation 3.1 3.2 3.3 3.4 3.5 3.6
84
96
97
AoDffiONAL AND ADvANCED EXERCISES
3
14
102 Tangents and the Derivative at a Point 102 The Derivative as a Function 106 Differentiation Rules 115 The Derivative as a Rate of Change 124 Derivatives ofTrigonometric Functions 135 The Chain Rule 142
iii
iv
Contents
3.7 3.8 3.9
Implicit Differentiation 149 Related Rates 155 Linearization and Differentials
164
QuESTIONS TO GUIDE YOUR REVIEW PRACTICE EXERCISES
ADDITIONAL AND ADvANCED EXERCISES
4
184
Extreme Values of Functions 184 The Mean Value Theorem 192 Monotonic Functions and the First Derivative Test Concavity and Curve Sketching 203 Applied Optimization 214 Newton's Method 225 Antiderivatives 230 QuESTIONS TO GUIDE YOUR REVIEW 239 PRACTICE EXERCISES 240 ADDITIONAL AND ADvANCED EXERCISES 243
198
I Integration
246 5.1 5.2 5.3 5.4 5.5 5.6
Area and Estimating with Finite Sums 246 Sigma Notation and Limits of Finite Sums 256 The Definite Integral 262 The Fundamental Theorem of Calculus 274 Indefinite Integrals and the Substitution Method 284 Substitution and Area Between Curves 291 QuESTIONS TO GUIDE YOUR REVIEW 300 PRACTICE EXERCISES 301 ADDITIONAL AND ADvANCED EXERCISES
6
180
Applications of Derivatives 4.1 4.2 4.3 4.4 4.5 4.6 4.7
5
175
176
304
Applications of Definite Integrals 6.1 6.2 6.3 6.4 6.5 6.6
Volumes Using CrossSections 308 Volumes Using Cylindrical Shells 319 Arc Length 326 Areas of Surfaces of Revolution 332 Work and Fluid Forces 337 Moments and Centers of Mass 346 QuESTIONS TO GUIDE YOUR REVIEW 357 PRACTICE EXERCISES 357 ADDITIONAL AND ADvANCED EXERCISES 359
308
Contents
7
Transcendental Functions 7.1 7.2 7.3 7.4 7.5 7.6 7.7 7.8
8
9
Inverse Functions and Their Derivatives 361 Natural Logarithms 369 Exponential Functions 377 Exponential Change and Separable Differential Equations Indeterminate Forms and VHopital's Rule 396 Inverse Trigonometric Functions 404 Hyperbolic Functions 416 Relative Rates of Growth 424 QuEsTIONS TO GUIDE YOUR REVIEW 429 PRACTICE EXERCISES 430 ADOffiONAL AND ADvANCED EXERCISES 433
Integration by Parts 436 Trigonometric Integrals 444 Trigonometric Substitutions 449 Integration of Rational Functions by Partial Fractions Integral Tables and Computer Algebra Systems 463 Numerical Integration 468 Improper Integrals 478 QuEsTIONS TO GUIDE YOUR REVIEW 489 PRACTICE EXERCISES 489 ADOffiONAL AND ADvANCED EXERCISES 491
453
496
Solutions, Slope Fields, and Euler's Method 496 FirstOrder Linear Equations 504 Applications 510 Graphical Solutions ofAutonomous Equations 516 Systems of Equations and Phase Planes 523 QuEsTIONS TO GUIDE YOUR REVIEW 529 PRACTICE EXERCISES 529 ADOffiONAL AND ADVANCED EXERCISES 530
Infinite Sequences and Series 10.1 10.2 10.3 10.4 10.5
387
435
FirstOrder Differential Equations 9.1 9.2 9.3 9.4 9.5
10
361
Techniques of Integration 8.1 8.2 8.3 8.4 8.5 8.6 8.7
V
Sequences 532 Infinite Series 544 The Integral Test 553 Comparison Tests 558 The Ratio and Root '!bsts
532
563
vi
Contents
10.6 10.7 10.8 10.9 10.10
11
Parametric Equations and Polar Coordinates 11.1 11.2 11.3 11.4 11.5 11.6 11.7
12
660
ThreeDimensional Coordinate Systems 660 Vectors 665 The Dot Product 674 The Cross Product 682 Lines and Planes in Space 688 Cylinders and Quadric Surfaces 696 QuESTIONS TO GUIDE YOUR REVIEW 701 PRACTICE EXERCISES 702 ADDITIONAL AND ADvANCED EXERCISES 704
VectorValued Functions and Motion in Space 13.1 13.2 13.3 13.4 13.5 13.6
610
Parametrizations ofPlane Curves 610 Calculus with Parametric Curves 618 Polar Coordinates 627 Graphing in Polar Coordinates 631 Areas and Lengths in Polar Coordinates 635 Conic Sections 639 Conics in Polar Coordinates 648 QuESTIONS TO GUIDE YOUR REVIEW 654 PRACTICE EXERCISES 655 ADDITIONAL AND ADvANCED EXERCISES 657
Vectors and the Geometry of Space 12.1 12.2 12.3 12.4 12.5 12.6
13
Alternating Series, Absolute and Conditional Convergence 568 Power Series 575 Taylor and Maclaurin Series 584 Convergence ofTaylor Series 589 The Binomial Series and Applications ofTaylor Series 596 QuESTIONS TO GUIDE YOUR REVIEW 605 PRACTICE EXERCISES 605 ADDITIONAL AND ADvANCED EXERCISES 607
Curves in Space and Their Tangents 707 Integrals ofVector Functions; Projectile Motion 715 Arc Length in Space 724 Curvatore andNorma1 Vectors ofa Curve 728 Tangential and Normal Components ofAcceleration 734 Velocity and Acceleration in Polar Coordinates 739 QuESTIONS TO GUIDE YOUR REVIEW 742 PRACTICE EXERCISES 743 ADDITIONAL AND ADvANCED EXERCISES 745
707
Contents
14
Partial Derivatives 14.1 14.2 14.3 14.4 14.5 14.6 14.7 14.8 14.9 14.10
15
16
747 Functions of Several Variables 747 Limits and Continuity in Higher Dimensions 755 Partial Derivatives 764 The Chain Rule 775 Directional Derivatives and Gradient Vectors 784 Tangent Planes and Differentials 791 Extreme Values and Saddle Points 802 Lagrange Multipliers 811 Taylor's Formula for Two Variables 820 Partial Derivatives with Constrained Variables 824 QUESTIONS TO GUIDE YOUR REVIEW 829 PRACTICE ExERCISES 829 ADomONAL AND ADvANCED EXERCISES 833
Multiple Integrals 15.1 15.2 15.3 15.4 15.5 15.6 15.7 15.8
836 Double and Iterated Integrals over Rectangles 836 Double Integrals over General Regions 841 Area by Double Integration 850 Double Integrals in Polar Form 853 Triple Integrals in Rectangular Coordinates 859 Moments and Centers ofMass 868 Triple Integrals in Cylindrical and Spherical Coordinates Substitutions in Multiple Integrals 887 QUESTIONS TO GUIDE YOUR REVIEW 896 PRACTICE ExERCISES 896 ADomONAL AND ADvANCED EXERCISES 898
875
Integration in Vector Fields 16.1 16.2 16.3 16.4 16.5 16.6 16.7 16.8
vii
Line Integrals 901 Vector Fields and Line Integrals: Work, Circulation, and Flux 907 Path Independence, Conservative Fields, and Potential Functions 920 Green's Theorem in the Plane 931 Surfaces and A...