10/7/23, 1:37 PM Thomas - Calculus Early Transcendentals 13th - 2014
THOMAS’
CALCULUS
EARLY TRANSCENDENTALS
Thirteenth Edition
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
with the assistance of
Christopher Heil
Georgia Institute of Technology
Boston Columbus Indianapolis New York San Francisco Upper Saddle River
Amsterdam Cape Town Dubai London Madrid Milan Munich Paris Montréal Toronto
Delhi Mexico City São Paulo Sydney Hong Kong Seoul Singapore Taipei Tokyo
Editor-in-Chief: Deirdre Lynch
Senior Acquisitions Editor: William Hoffman
Senior Content Editor: Rachel S. Reeve
Senior Managing Editor: Karen Wernholm
Associate Managing Editor: Tamela Ambush
Senior Production Project Manager: Sheila Spinney; Sherry Berg
Associate Design Director, USHE EMSS, TED and HSC: Andrea Nix
Art Director and Cover Design: Beth Paquin
Digital Assets Manager: Marianne Groth
Associate Producer Multimedia: Nicholas Sweeny
Software Development: John Flanagan and Kristina Evans
Executive Marketing Manager: Jeff Weidenaar
about:blank 1/1206
,10/7/23, 1:37 PM Thomas - Calculus Early Transcendentals 13th - 2014
Delhi Mexico City São Paulo Sydney Hong Kong Seoul Singapore Taipei Tokyo
Editor-in-Chief: Deirdre Lynch
Senior Acquisitions Editor: William Hoffman
Senior Content Editor: Rachel S. Reeve
Senior Managing Editor: Karen Wernholm
Associate Managing Editor: Tamela Ambush
Senior Production Project Manager: Sheila Spinney; Sherry Berg
Associate Design Director, USHE EMSS, TED and HSC: Andrea Nix
Art Director and Cover Design: Beth Paquin
Digital Assets Manager: Marianne Groth
Associate Producer Multimedia: Nicholas Sweeny
Software Development: John Flanagan and Kristina Evans
Executive Marketing Manager: Jeff Weidenaar
Marketing Assistant: Caitlin Crain
Senior Author Support/Technology Specialist: Joe Vetere
Manufacturing Manager: Carol Melville
Text Design, Production Coordination, Composition: Cenveo® Publisher Services
Illustrations: Karen Hartpence, IlustraTech; Cenveo® Publisher Services
Cover image: Art on File/Corbis
For permission to use copyrighted material, grateful acknowledgment is made to the copyright holders on
page C-1, which is hereby made part of this copyright page.
Many of the designations used by manufacturers and sellers to distinguish their products are claimed as trademarks.
Where those designations appear in this book, and Pearson Education was aware of a trademark claim, the designa-
tions have been printed in initial caps or all caps.
Library of Congress Cataloging-in-Publication Data
Weir, Maurice D.
Thomas’ calculus : early transcendentals : based on the original work by George B. Thomas, Jr., Massachusetts
Institute of Technology.—Thirteenth edition / as revised by Maurice D. Weir, Naval Postgraduate School, Joel
Hass, University of California, Davis.
pages cm
ISBN 978-0-321-88407-7 (hardcover)
I. Hass, Joel.II. Thomas, George B. (George Brinton), Jr., 1914–2006. Calculus. Based on (Work):III.
Title. IV. Title: Calculus.
QA303.2.W45 2014
515–dc23 2013023096
Copyright © 2014, 2010, 2008 Pearson Education, 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, or otherwise, without the prior written permission of the publisher. Printed in the United
States of America. For information on obtaining permission for use of material in this work, please submit a written
request to Pearson Education, Inc., Rights and Contracts Department, 501 Boylston Street, Suite 900, Boston, MA 02116,
fax your request to 617-848-7047, or e-mail at http://www.pearsoned.com/legal/permissions.htm.
1 2 3 4 5 6 7 8 9 10—CRK—17 16 15 14 13
ISBN-10: 0-321-88407-8
ISBN-13: 978-0-321-88407-7
www.pearsonhighered.com
Contents
Preface ix
1 Functions 1
about:blank 2/1206
,10/7/23, 1:37 PM Thomas - Calculus Early Transcendentals 13th - 2014
ISBN 10: 0 321 88407 8
ISBN-13: 978-0-321-88407-7
www.pearsonhighered.com
Contents
Preface ix
1 Functions 1
1.1 Functions and Their Graphs 1
1.2 Combining Functions; Shifting and Scaling Graphs 14
1.3 Trigonometric Functions 21
1.4 Graphing with Software 29
1.5 Exponential Functions 36
1.6 Inverse Functions and Logarithms 41
Questions to Guide Your Review 54
Practice Exercises 54
Additional and Advanced Exercises 57
2 Limits and Continuity 59
2.1 Rates of Change and Tangents to Curves 59
2.2 Limit of a Function and Limit Laws 66
2.3 The Precise Definition of a Limit 77
2.4 One-Sided Limits 86
2.5 Continuity 93
2.6 Limits Involving Infinity; Asymptotes of Graphs 104
Questions to Guide Your Review 118
Practice Exercises 118
Additional and Advanced Exercises 120
3 Derivatives 123
3.1 Tangents and the Derivative at a Point 123
3.2 The Derivative as a Function 128
3.3 Differentiation Rules 136
3.4 The Derivative as a Rate of Change 146
3.5 Derivatives of Trigonometric Functions 156
3.6 The Chain Rule 163
3.7 Implicit Differentiation 171
3.8 Derivatives of Inverse Functions and Logarithms 177
3.9 Inverse Trigonometric Functions 187
3.10 Related Rates 193
3.11 Linearization and Differentials 202
Questions to Guide Your Review 214
Practice Exercises 215
Additional and Advanced Exercises 219
iii
iv Contents
4 Applications of Derivatives 223
4.1 Extreme Values of Functions 223
4.2 The Mean Value Theorem 231
4.3 Monotonic Functions and the First Derivative Test 239
4.4 Concavity and Curve Sketching 244
4.5 Indeterminate Forms and L’Hôpital’s Rule 255
4.6 Applied Optimization 264
4.7 Newton’s Method 276
48 A tid i ti 281
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, 10/7/23, 1:37 PM Thomas - Calculus Early Transcendentals 13th - 2014
iii
iv Contents
4 Applications of Derivatives 223
4.1 Extreme Values of Functions 223
4.2 The Mean Value Theorem 231
4.3 Monotonic Functions and the First Derivative Test 239
4.4 Concavity and Curve Sketching 244
4.5 Indeterminate Forms and L’Hôpital’s Rule 255
4.6 Applied Optimization 264
4.7 Newton’s Method 276
4.8 Antiderivatives 281
Questions to Guide Your Review 291
Practice Exercises 291
Additional and Advanced Exercises 295
5 Integrals 299
5.1 Area and Estimating with Finite Sums 299
5.2 Sigma Notation and Limits of Finite Sums 309
5.3 The Definite Integral 316
5.4 The Fundamental Theorem of Calculus 328
5.5 Indefinite Integrals and the Substitution Method 339
5.6 Definite Integral Substitutions and the Area Between Curves 347
Questions to Guide Your Review 357
Practice Exercises 357
Additional and Advanced Exercises 361
6 Applications of Definite Integrals 365
6.1 Volumes Using Cross-Sections 365
6.2 Volumes Using Cylindrical Shells 376
6.3 Arc Length 384
6.4 Areas of Surfaces of Revolution 390
6.5 Work and Fluid Forces 395
6.6 Moments and Centers of Mass 404
Questions to Guide Your Review 415
Practice Exercises 416
Additional and Advanced Exercises 417
7 Integrals and Transcendental Functions 420
7.1 The Logarithm Defined as an Integral 420
7.2 Exponential Change and Separable Differential Equations 430
7.3 Hyperbolic Functions 439
7.4 Relative Rates of Growth 448
Questions to Guide Your Review 453
Practice Exercises 453
Additional and Advanced Exercises 455
Contents v
8 Techniques of Integration 456
8.1 Using Basic Integration Formulas 456
8.2 Integration by Parts 461
8.3 Trigonometric Integrals 469
8.4 Trigonometric Substitutions 475
8.5 Integration of Rational Functions by Partial Fractions 480
8.6 Integral Tables and Computer Algebra Systems 489
8.7 Numerical Integration 494
88 I I t l 504
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THOMAS’
CALCULUS
EARLY TRANSCENDENTALS
Thirteenth Edition
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
with the assistance of
Christopher Heil
Georgia Institute of Technology
Boston Columbus Indianapolis New York San Francisco Upper Saddle River
Amsterdam Cape Town Dubai London Madrid Milan Munich Paris Montréal Toronto
Delhi Mexico City São Paulo Sydney Hong Kong Seoul Singapore Taipei Tokyo
Editor-in-Chief: Deirdre Lynch
Senior Acquisitions Editor: William Hoffman
Senior Content Editor: Rachel S. Reeve
Senior Managing Editor: Karen Wernholm
Associate Managing Editor: Tamela Ambush
Senior Production Project Manager: Sheila Spinney; Sherry Berg
Associate Design Director, USHE EMSS, TED and HSC: Andrea Nix
Art Director and Cover Design: Beth Paquin
Digital Assets Manager: Marianne Groth
Associate Producer Multimedia: Nicholas Sweeny
Software Development: John Flanagan and Kristina Evans
Executive Marketing Manager: Jeff Weidenaar
about:blank 1/1206
,10/7/23, 1:37 PM Thomas - Calculus Early Transcendentals 13th - 2014
Delhi Mexico City São Paulo Sydney Hong Kong Seoul Singapore Taipei Tokyo
Editor-in-Chief: Deirdre Lynch
Senior Acquisitions Editor: William Hoffman
Senior Content Editor: Rachel S. Reeve
Senior Managing Editor: Karen Wernholm
Associate Managing Editor: Tamela Ambush
Senior Production Project Manager: Sheila Spinney; Sherry Berg
Associate Design Director, USHE EMSS, TED and HSC: Andrea Nix
Art Director and Cover Design: Beth Paquin
Digital Assets Manager: Marianne Groth
Associate Producer Multimedia: Nicholas Sweeny
Software Development: John Flanagan and Kristina Evans
Executive Marketing Manager: Jeff Weidenaar
Marketing Assistant: Caitlin Crain
Senior Author Support/Technology Specialist: Joe Vetere
Manufacturing Manager: Carol Melville
Text Design, Production Coordination, Composition: Cenveo® Publisher Services
Illustrations: Karen Hartpence, IlustraTech; Cenveo® Publisher Services
Cover image: Art on File/Corbis
For permission to use copyrighted material, grateful acknowledgment is made to the copyright holders on
page C-1, which is hereby made part of this copyright page.
Many of the designations used by manufacturers and sellers to distinguish their products are claimed as trademarks.
Where those designations appear in this book, and Pearson Education was aware of a trademark claim, the designa-
tions have been printed in initial caps or all caps.
Library of Congress Cataloging-in-Publication Data
Weir, Maurice D.
Thomas’ calculus : early transcendentals : based on the original work by George B. Thomas, Jr., Massachusetts
Institute of Technology.—Thirteenth edition / as revised by Maurice D. Weir, Naval Postgraduate School, Joel
Hass, University of California, Davis.
pages cm
ISBN 978-0-321-88407-7 (hardcover)
I. Hass, Joel.II. Thomas, George B. (George Brinton), Jr., 1914–2006. Calculus. Based on (Work):III.
Title. IV. Title: Calculus.
QA303.2.W45 2014
515–dc23 2013023096
Copyright © 2014, 2010, 2008 Pearson Education, 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, or otherwise, without the prior written permission of the publisher. Printed in the United
States of America. For information on obtaining permission for use of material in this work, please submit a written
request to Pearson Education, Inc., Rights and Contracts Department, 501 Boylston Street, Suite 900, Boston, MA 02116,
fax your request to 617-848-7047, or e-mail at http://www.pearsoned.com/legal/permissions.htm.
1 2 3 4 5 6 7 8 9 10—CRK—17 16 15 14 13
ISBN-10: 0-321-88407-8
ISBN-13: 978-0-321-88407-7
www.pearsonhighered.com
Contents
Preface ix
1 Functions 1
about:blank 2/1206
,10/7/23, 1:37 PM Thomas - Calculus Early Transcendentals 13th - 2014
ISBN 10: 0 321 88407 8
ISBN-13: 978-0-321-88407-7
www.pearsonhighered.com
Contents
Preface ix
1 Functions 1
1.1 Functions and Their Graphs 1
1.2 Combining Functions; Shifting and Scaling Graphs 14
1.3 Trigonometric Functions 21
1.4 Graphing with Software 29
1.5 Exponential Functions 36
1.6 Inverse Functions and Logarithms 41
Questions to Guide Your Review 54
Practice Exercises 54
Additional and Advanced Exercises 57
2 Limits and Continuity 59
2.1 Rates of Change and Tangents to Curves 59
2.2 Limit of a Function and Limit Laws 66
2.3 The Precise Definition of a Limit 77
2.4 One-Sided Limits 86
2.5 Continuity 93
2.6 Limits Involving Infinity; Asymptotes of Graphs 104
Questions to Guide Your Review 118
Practice Exercises 118
Additional and Advanced Exercises 120
3 Derivatives 123
3.1 Tangents and the Derivative at a Point 123
3.2 The Derivative as a Function 128
3.3 Differentiation Rules 136
3.4 The Derivative as a Rate of Change 146
3.5 Derivatives of Trigonometric Functions 156
3.6 The Chain Rule 163
3.7 Implicit Differentiation 171
3.8 Derivatives of Inverse Functions and Logarithms 177
3.9 Inverse Trigonometric Functions 187
3.10 Related Rates 193
3.11 Linearization and Differentials 202
Questions to Guide Your Review 214
Practice Exercises 215
Additional and Advanced Exercises 219
iii
iv Contents
4 Applications of Derivatives 223
4.1 Extreme Values of Functions 223
4.2 The Mean Value Theorem 231
4.3 Monotonic Functions and the First Derivative Test 239
4.4 Concavity and Curve Sketching 244
4.5 Indeterminate Forms and L’Hôpital’s Rule 255
4.6 Applied Optimization 264
4.7 Newton’s Method 276
48 A tid i ti 281
about:blank 3/1206
, 10/7/23, 1:37 PM Thomas - Calculus Early Transcendentals 13th - 2014
iii
iv Contents
4 Applications of Derivatives 223
4.1 Extreme Values of Functions 223
4.2 The Mean Value Theorem 231
4.3 Monotonic Functions and the First Derivative Test 239
4.4 Concavity and Curve Sketching 244
4.5 Indeterminate Forms and L’Hôpital’s Rule 255
4.6 Applied Optimization 264
4.7 Newton’s Method 276
4.8 Antiderivatives 281
Questions to Guide Your Review 291
Practice Exercises 291
Additional and Advanced Exercises 295
5 Integrals 299
5.1 Area and Estimating with Finite Sums 299
5.2 Sigma Notation and Limits of Finite Sums 309
5.3 The Definite Integral 316
5.4 The Fundamental Theorem of Calculus 328
5.5 Indefinite Integrals and the Substitution Method 339
5.6 Definite Integral Substitutions and the Area Between Curves 347
Questions to Guide Your Review 357
Practice Exercises 357
Additional and Advanced Exercises 361
6 Applications of Definite Integrals 365
6.1 Volumes Using Cross-Sections 365
6.2 Volumes Using Cylindrical Shells 376
6.3 Arc Length 384
6.4 Areas of Surfaces of Revolution 390
6.5 Work and Fluid Forces 395
6.6 Moments and Centers of Mass 404
Questions to Guide Your Review 415
Practice Exercises 416
Additional and Advanced Exercises 417
7 Integrals and Transcendental Functions 420
7.1 The Logarithm Defined as an Integral 420
7.2 Exponential Change and Separable Differential Equations 430
7.3 Hyperbolic Functions 439
7.4 Relative Rates of Growth 448
Questions to Guide Your Review 453
Practice Exercises 453
Additional and Advanced Exercises 455
Contents v
8 Techniques of Integration 456
8.1 Using Basic Integration Formulas 456
8.2 Integration by Parts 461
8.3 Trigonometric Integrals 469
8.4 Trigonometric Substitutions 475
8.5 Integration of Rational Functions by Partial Fractions 480
8.6 Integral Tables and Computer Algebra Systems 489
8.7 Numerical Integration 494
88 I I t l 504
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