, Single Variable
calculus
Early Transcendentals
eighth edition
Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
, Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
, Single Variable
calculus
Early Transcendentals
eighth edition
James Stewart
M c Master University
and
University of Toronto
Australia • Brazil • Mexico • Singapore • United Kingdom • United States
Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
,This is an electronic version of the print textbook. Due to electronic rights restrictions, some third party content may be suppressed. Editorial
review has deemed that any suppressed content does not materially affect the overall learning experience. The publisher reserves the right to
remove content from this title at any time if subsequent rights restrictions require it. For valuable information on pricing, previous
editions, changes to current editions, and alternate formats, please visit www.cengage.com/highered to search by
ISBN#, author, title, or keyword for materials in your areas of interest.
Important Notice: Media content referenced within the product description or the product text may not be available in the eBook version.
Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
,Single Variable Calculus: Early Transcendentals, © 2016, 2012 Cengage Learning
Eighth Edition
WCN: 02-200-203
James Stewart
ALL RIGHTS RESERVED. No part of this work covered by the copyright
Product Manager: Neha Taleja herein may be reproduced, transmitted, stored, or used in any form or by
Senior Content Developer: Stacy Green any means graphic, electronic, or mechanical, including but not limited to
photocopying, recording, scanning, digitizing, taping, Web distribution,
Associate Content Developer: Samantha Lugtu
information networks, or information storage and retrieval systems, except
Product Assistant: Stephanie Kreuz as permitted under Section 107 or 108 of the 1976 United States Copyright
Media Developer: Guanglei Zhang Act, without the prior written permission of the publisher.
Marketing Manager: Ryan Ahern
Content Project Manager: Cheryll Linthicum For product information and technology assistance, contact us at
Art Director: Vernon Boes Cengage Learning Customer & Sales Support, 1-800-354-9706.
For permission to use material from this text or product,
Manufacturing Planner: Becky Cross submit all requests online at www.cengage.com/permissions.
Production Service: TECHarts Further permissions questions can be e-mailed to
.
Photo and Text Researcher: Lumina Datamatics
Copy Editor: Kathi Townes, TECHarts
Library of Congress Control Number: 2014951586
Illustrator: TECHarts
Text Designer: Diane Beasley ISBN: 978-1-305-27033-6
Cover Designer: Irene Morris, Morris Design
Compositor: Stephanie Kuhns, Kristina Elliott, Cengage Learning
20 Channel Center Street
and Kira Abdallah, TECHarts Boston, MA 02210
Cover Image: elisanth / 123RF, tharrison / iStock USA
Vectors / Getty Images
Cengage Learning is a leading provider of customized learning solutions
with office locations around the globe, including Singapore, the United
Kingdom, Australia, Mexico, Brazil, and Japan. Locate your local office at
www.cengage.com/global.
Cengage Learning products are represented in Canada by
Nelson Education, Ltd.
To learn more about Cengage Learning Solutions, visit www.cengage.com.
Purchase any of our products at your local college store or at our pre-
ferred online store www.cengagebrain.com.
Windows is a registered trademark of the Microsoft Corporation and used
herein under license.
Macintosh is a registered trademark of Apple Computer, Inc.
Used herein under license.
k10T14
Maple is a registered trademark of Waterloo Maple, Inc.
Mathematica is a registered trademark of Wolfram Research, Inc.
Tools for Enriching Calculus is a trademark used herein under license.
Printed in the United States of America
Print Number: 01 Print Year: 2014
Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
, Contents
Preface xi
To the Student xxii
Calculators, Computers, and other graphing devices xxiv
Diagnostic tests xxvi
A Preview of Calculus 1
1
1.1 Four Ways to Represent a Function 10
1.2 Mathematical Models: A Catalog of Essential Functions 23
1.3 New Functions from Old Functions 36
1.4 Exponential Functions 45
1.5 Inverse Functions and Logarithms 55
Review 68
Principles of Problem Solving 71
2
2.1 The Tangent and Velocity Problems 78
2.2 The Limit of a Function 83
2.3 Calculating Limits Using the Limit Laws 95
2.4 The Precise Definition of a Limit 104
2.5 Continuity 114
2.6 Limits at Infinity; Horizontal Asymptotes 126
2.7 Derivatives and Rates of Change 140
Writing Project • Early Methods for Finding Tangents 152
2.8 The Derivative as a Function 152
Review 165
Problems Plus 169
v
Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
,vi Contents
3
3.1 Derivatives of Polynomials and Exponential Functions 172
Applied Project • Building a Better Roller Coaster 182
3.2 The Product and Quotient Rules 183
3.3 Derivatives of Trigonometric Functions 190
3.4 The Chain Rule 197
Applied Project • Where Should a Pilot Start Descent? 208
3.5 Implicit Differentiation 208
Laboratory Project • Families of Implicit Curves 217
3.6 Derivatives of Logarithmic Functions 218
3.7 Rates of Change in the Natural and Social Sciences 224
3.8 Exponential Growth and Decay 237
Applied Project • Controlling Red Blood Cell Loss During Surgery 244
3.9 Related Rates 245
3.10 Linear Approximations and Differentials 251
Laboratory Project • Taylor Polynomials 258
3.11 Hyperbolic Functions 259
Review 266
Problems Plus 270
4
4.1 Maximum and Minimum Values 276
Applied Project • The Calculus of Rainbows 285
4.2 The Mean Value Theorem 287
4.3 How Derivatives Affect the Shape of a Graph 293
4.4 Indeterminate Forms and l’Hospital’s Rule 304
Writing Project • The Origins of l’Hospital’s Rule 314
4.5 Summary of Curve Sketching 315
4.6 Graphing with Calculus and Calculators 323
4.7 Optimization Problems 330
Applied Project • The Shape of a Can 343
Applied Project • Planes and Birds: Minimizing Energy 344
4.8 Newton’s Method 345
4.9 Antiderivatives 350
Review 358
Problems Plus 363
Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
, Contents vii
5
5.1 Areas and Distances 366
5.2 The Definite Integral 378
Discovery Project • Area Functions 391
5.3 The Fundamental Theorem of Calculus 392
5.4 Indefinite Integrals and the Net Change Theorem 402
Writing Project • Newton, Leibniz, and the Invention of Calculus 411
5.5 The Substitution Rule 412
Review 421
Problems Plus 425
6
6.1 Areas Between Curves 428
Applied Project • The Gini Index 436
6.2 Volumes 438
6.3 Volumes by Cylindrical Shells 449
6.4 Work 455
6.5 Average Value of a Function 461
Applied Project • Calculus and Baseball 464
Applied Project • Where To Sit at the Movies 465
Review 466
Problems Plus 468
7
7.1 Integration by Parts 472
7.2 Trigonometric Integrals 479
7.3 Trigonometric Substitution 486
7.4 Integration of Rational Functions by Partial Fractions 493
7.5 Strategy for Integration 503
7.6 Integration Using Tables and Computer Algebra Systems 508
Discovery Project • Patterns in Integrals 513
7.7 Approximate Integration 514
7.8 Improper Integrals 527
Review 537
Problems Plus 540
Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
, viii Contents
8
8.1 Arc Length 544
Discovery Project • Arc Length Contest 550
8.2 Area of a Surface of Revolution 551
Discovery Project • Rotating on a Slant 557
8.3 Applications to Physics and Engineering 558
Discovery Project • Complementary Coffee Cups 568
8.4 Applications to Economics and Biology 569
8.5 Probability 573
Review 581
Problems Plus 583
9
9.1 Modeling with Differential Equations 586
9.2 Direction Fields and Euler’s Method 591
9.3 Separable Equations 599
Applied Project • How Fast Does a Tank Drain? 608
Applied Project • Which Is Faster, Going Up or Coming Down? 609
9.4 Models for Population Growth 610
9.5 Linear Equations 620
9.6 Predator-Prey Systems 627
Review 634
Problems Plus 637
10
10.1 Curves Defined by Parametric Equations 640
Laboratory Project • Running Circles Around Circles 648
10.2 Calculus with Parametric Curves 649
Laboratory Project • Bézier Curves 657
10.3 Polar Coordinates 658
Laboratory Project • Families of Polar Curves 668
10.4 Areas and Lengths in Polar Coordinates 669
10.5 Conic Sections 674
10.6 Conic Sections in Polar Coordinates 682
Review 689
Problems Plus 692
Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
calculus
Early Transcendentals
eighth edition
Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
, Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
, Single Variable
calculus
Early Transcendentals
eighth edition
James Stewart
M c Master University
and
University of Toronto
Australia • Brazil • Mexico • Singapore • United Kingdom • United States
Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
,This is an electronic version of the print textbook. Due to electronic rights restrictions, some third party content may be suppressed. Editorial
review has deemed that any suppressed content does not materially affect the overall learning experience. The publisher reserves the right to
remove content from this title at any time if subsequent rights restrictions require it. For valuable information on pricing, previous
editions, changes to current editions, and alternate formats, please visit www.cengage.com/highered to search by
ISBN#, author, title, or keyword for materials in your areas of interest.
Important Notice: Media content referenced within the product description or the product text may not be available in the eBook version.
Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
,Single Variable Calculus: Early Transcendentals, © 2016, 2012 Cengage Learning
Eighth Edition
WCN: 02-200-203
James Stewart
ALL RIGHTS RESERVED. No part of this work covered by the copyright
Product Manager: Neha Taleja herein may be reproduced, transmitted, stored, or used in any form or by
Senior Content Developer: Stacy Green any means graphic, electronic, or mechanical, including but not limited to
photocopying, recording, scanning, digitizing, taping, Web distribution,
Associate Content Developer: Samantha Lugtu
information networks, or information storage and retrieval systems, except
Product Assistant: Stephanie Kreuz as permitted under Section 107 or 108 of the 1976 United States Copyright
Media Developer: Guanglei Zhang Act, without the prior written permission of the publisher.
Marketing Manager: Ryan Ahern
Content Project Manager: Cheryll Linthicum For product information and technology assistance, contact us at
Art Director: Vernon Boes Cengage Learning Customer & Sales Support, 1-800-354-9706.
For permission to use material from this text or product,
Manufacturing Planner: Becky Cross submit all requests online at www.cengage.com/permissions.
Production Service: TECHarts Further permissions questions can be e-mailed to
.
Photo and Text Researcher: Lumina Datamatics
Copy Editor: Kathi Townes, TECHarts
Library of Congress Control Number: 2014951586
Illustrator: TECHarts
Text Designer: Diane Beasley ISBN: 978-1-305-27033-6
Cover Designer: Irene Morris, Morris Design
Compositor: Stephanie Kuhns, Kristina Elliott, Cengage Learning
20 Channel Center Street
and Kira Abdallah, TECHarts Boston, MA 02210
Cover Image: elisanth / 123RF, tharrison / iStock USA
Vectors / Getty Images
Cengage Learning is a leading provider of customized learning solutions
with office locations around the globe, including Singapore, the United
Kingdom, Australia, Mexico, Brazil, and Japan. Locate your local office at
www.cengage.com/global.
Cengage Learning products are represented in Canada by
Nelson Education, Ltd.
To learn more about Cengage Learning Solutions, visit www.cengage.com.
Purchase any of our products at your local college store or at our pre-
ferred online store www.cengagebrain.com.
Windows is a registered trademark of the Microsoft Corporation and used
herein under license.
Macintosh is a registered trademark of Apple Computer, Inc.
Used herein under license.
k10T14
Maple is a registered trademark of Waterloo Maple, Inc.
Mathematica is a registered trademark of Wolfram Research, Inc.
Tools for Enriching Calculus is a trademark used herein under license.
Printed in the United States of America
Print Number: 01 Print Year: 2014
Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
, Contents
Preface xi
To the Student xxii
Calculators, Computers, and other graphing devices xxiv
Diagnostic tests xxvi
A Preview of Calculus 1
1
1.1 Four Ways to Represent a Function 10
1.2 Mathematical Models: A Catalog of Essential Functions 23
1.3 New Functions from Old Functions 36
1.4 Exponential Functions 45
1.5 Inverse Functions and Logarithms 55
Review 68
Principles of Problem Solving 71
2
2.1 The Tangent and Velocity Problems 78
2.2 The Limit of a Function 83
2.3 Calculating Limits Using the Limit Laws 95
2.4 The Precise Definition of a Limit 104
2.5 Continuity 114
2.6 Limits at Infinity; Horizontal Asymptotes 126
2.7 Derivatives and Rates of Change 140
Writing Project • Early Methods for Finding Tangents 152
2.8 The Derivative as a Function 152
Review 165
Problems Plus 169
v
Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
,vi Contents
3
3.1 Derivatives of Polynomials and Exponential Functions 172
Applied Project • Building a Better Roller Coaster 182
3.2 The Product and Quotient Rules 183
3.3 Derivatives of Trigonometric Functions 190
3.4 The Chain Rule 197
Applied Project • Where Should a Pilot Start Descent? 208
3.5 Implicit Differentiation 208
Laboratory Project • Families of Implicit Curves 217
3.6 Derivatives of Logarithmic Functions 218
3.7 Rates of Change in the Natural and Social Sciences 224
3.8 Exponential Growth and Decay 237
Applied Project • Controlling Red Blood Cell Loss During Surgery 244
3.9 Related Rates 245
3.10 Linear Approximations and Differentials 251
Laboratory Project • Taylor Polynomials 258
3.11 Hyperbolic Functions 259
Review 266
Problems Plus 270
4
4.1 Maximum and Minimum Values 276
Applied Project • The Calculus of Rainbows 285
4.2 The Mean Value Theorem 287
4.3 How Derivatives Affect the Shape of a Graph 293
4.4 Indeterminate Forms and l’Hospital’s Rule 304
Writing Project • The Origins of l’Hospital’s Rule 314
4.5 Summary of Curve Sketching 315
4.6 Graphing with Calculus and Calculators 323
4.7 Optimization Problems 330
Applied Project • The Shape of a Can 343
Applied Project • Planes and Birds: Minimizing Energy 344
4.8 Newton’s Method 345
4.9 Antiderivatives 350
Review 358
Problems Plus 363
Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
, Contents vii
5
5.1 Areas and Distances 366
5.2 The Definite Integral 378
Discovery Project • Area Functions 391
5.3 The Fundamental Theorem of Calculus 392
5.4 Indefinite Integrals and the Net Change Theorem 402
Writing Project • Newton, Leibniz, and the Invention of Calculus 411
5.5 The Substitution Rule 412
Review 421
Problems Plus 425
6
6.1 Areas Between Curves 428
Applied Project • The Gini Index 436
6.2 Volumes 438
6.3 Volumes by Cylindrical Shells 449
6.4 Work 455
6.5 Average Value of a Function 461
Applied Project • Calculus and Baseball 464
Applied Project • Where To Sit at the Movies 465
Review 466
Problems Plus 468
7
7.1 Integration by Parts 472
7.2 Trigonometric Integrals 479
7.3 Trigonometric Substitution 486
7.4 Integration of Rational Functions by Partial Fractions 493
7.5 Strategy for Integration 503
7.6 Integration Using Tables and Computer Algebra Systems 508
Discovery Project • Patterns in Integrals 513
7.7 Approximate Integration 514
7.8 Improper Integrals 527
Review 537
Problems Plus 540
Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
, viii Contents
8
8.1 Arc Length 544
Discovery Project • Arc Length Contest 550
8.2 Area of a Surface of Revolution 551
Discovery Project • Rotating on a Slant 557
8.3 Applications to Physics and Engineering 558
Discovery Project • Complementary Coffee Cups 568
8.4 Applications to Economics and Biology 569
8.5 Probability 573
Review 581
Problems Plus 583
9
9.1 Modeling with Differential Equations 586
9.2 Direction Fields and Euler’s Method 591
9.3 Separable Equations 599
Applied Project • How Fast Does a Tank Drain? 608
Applied Project • Which Is Faster, Going Up or Coming Down? 609
9.4 Models for Population Growth 610
9.5 Linear Equations 620
9.6 Predator-Prey Systems 627
Review 634
Problems Plus 637
10
10.1 Curves Defined by Parametric Equations 640
Laboratory Project • Running Circles Around Circles 648
10.2 Calculus with Parametric Curves 649
Laboratory Project • Bézier Curves 657
10.3 Polar Coordinates 658
Laboratory Project • Families of Polar Curves 668
10.4 Areas and Lengths in Polar Coordinates 669
10.5 Conic Sections 674
10.6 Conic Sections in Polar Coordinates 682
Review 689
Problems Plus 692
Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
