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INSTRUCTOR’S
SOLUTIONS MANUAL
MARK WOODARD
Furman University
C ALCULUS
E ARLY T RANSCENDENTALS
THIRD EDITION
William Briggs
University of Colorado at Denver
Lyle Cochran
Whitworth University
Bernhard Gillett
University of Colorado, Boulder
Eric Schulz
Walla Walla Community College
Briggs.pdf&&&&&&&&&&&&&&&&&&@#%%%%%%%%%%%%%%%%%%%%%%%*!^(((((((((((((((((((((((((((((((((((((#&*^)(((((((((((((
~~~~~~~~~~~~~~~~~~~~~~~~_OceanofPDF.com_Calculus_Early_Transcendentals_-_Instructors_Solutions_Manual_-_William_Briggs.pd
_OceanofPDF.com_Calculus_Early_Transcendentals_-_Instructors_Solutions_Manual_-_William_Briggs.pdf!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
,Briggs.pdf&&&&&&&&&&&&&&&&&&@#%%%%%%%%%%%%%%%%%%%%%%%*!^(((((((((((((((((((((((((((((((((((((#&*^)(((((((((((((
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~_OceanofPDF.com_Calculus_Early_Transcende
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
The author and publisher of this book have used their best efforts in preparing this book. These efforts include the
development, research, and testing of the theories and programs to determine their effectiveness. The author and publisher
make no warranty of any kind, expressed or implied, with regard to these programs or the documentation contained in this
book. The author and publisher shall not be liable in any event for incidental or consequential damages in connection with, or
arising out of, the furnishing, performance, or use of these programs.
Reproduced by Pearson from electronic files supplied by the author.
Copyright © 2019, 2015, 2011 Pearson Education, Inc.
Publishing as Pearson, 330 Hudson Street, NY NY 10013
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.
ISBN-13: 978-0-13-476681-2
ISBN-10: 0-13-476681-4
Briggs.pdf&&&&&&&&&&&&&&&&&&@#%%%%%%%%%%%%%%%%%%%%%%%*!^(((((((((((((((((((((((((((((((((((((#&*^)(((((((((((((
~~~~~~~~~~~~~~~~~~~~~~~~_OceanofPDF.com_Calculus_Early_Transcendentals_-_Instructors_Solutions_Manual_-_William_Briggs.pd
_OceanofPDF.com_Calculus_Early_Transcendentals_-_Instructors_Solutions_Manual_-_William_Briggs.pdf!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
,Briggs.pdf&&&&&&&&&&&&&&&&&&@#%%%%%%%%%%%%%%%%%%%%%%%*!^(((((((((((((((((((((((((((((((((((((#&*^)(((((((((((((
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~_OceanofPDF.com_Calculus_Early_Transcende
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Contents
1 Functions 3
1.1 Review of Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 Representing Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.3 Inverse, Exponential and Logarithmic Functions . . . . . . . . . . . . . . . . . . . . . . . . . 30
1.4 Trigonometric Functions and Their Inverses . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
Chapter One Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
2 Limits 65
2.1 The Idea of Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
2.2 Definition of a Limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
2.3 Techniques for Computing Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
2.4 Infinite Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
2.5 Limits at Infinity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
2.6 Continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
2.7 Precise Definitions of Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
Chapter Two Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
3 Derivatives 153
3.1 Introducing the Derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
3.2 The Derivative as a Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
3.3 Rules of Differentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180
3.4 The Product and Quotient Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188
3.5 Derivatives of Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200
3.6 Derivatives as Rates of Change . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209
3.7 The Chain Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224
3.8 Implicit Differentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238
3.9 Derivatives of Logarithmic and Exponential Functions . . . . . . . . . . . . . . . . . . . . . . 256
3.10 Derivatives of Inverse Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . . . . . 268
3.11 Related Rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277
Chapter Three Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289
4 Applications of the Derivative 305
4.1 Maxima and Minima . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305
4.2 Mean Value Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 320
4.3 What Derivatives Tell Us . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 328
4.4 Graphing Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 346
4.5 Optimization Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 380
4.6 Linear Approximation and Differentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 403
4.7 L’Hôpital’s Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413
4.8 Newton’s Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 427
4.9 Antiderivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 443
Chapter Four Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 454
1
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2 Contents
5 Integration 477
5.1 Approximating Areas under Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 477
5.2 Definite Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 497
5.3 Fundamental Theorem of Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 517
5.4 Working with Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 534
5.5 Substitution Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 544
Chapter Five Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 555
6 Applications of Integration 571
6.1 Velocity and Net Change . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 571
6.2 Regions Between Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 585
6.3 Volume by Slicing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 600
6.4 Volume by Shells . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 608
6.5 Length of Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 618
6.6 Surface Area . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 624
6.7 Physical Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 632
Chapter Six Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 642
7 Logarithmic, Exponential, and Hyperbolic Functions 659
7.1 Logarithmic and Exponential Functions Revisited . . . . . . . . . . . . . . . . . . . . . . . . 659
7.2 Exponential Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 667
7.3 Hyperbolic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 673
Chapter Seven Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 685
8 Integration Techniques 691
8.1 Basic Approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 691
8.2 Integration by Parts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 701
8.3 Trigonometric Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 720
8.4 Trigonometric Substitutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 729
8.5 Partial Fractions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 746
8.6 Integration Strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 761
8.7 Other Methods of Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 796
8.8 Numerical Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 805
8.9 Improper Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 816
Chapter Eight Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 831
9 Differential Equations 853
9.1 Basic Ideas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 853
9.2 Direction Fields and Euler’s Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 859
9.3 Separable Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 871
9.4 Special First-Order Linear Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . 884
9.5 Modeling with Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 892
Chapter Nine Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 901
10 Sequences and Infinite Series 909
10.1 An Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 909
10.2 Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 917
10.3 Infinite Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 931
10.4 The Divergence and Integral Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 942
10.5 Comparison Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 953
10.6 Alternating Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 961
10.7 The Ratio and Root Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 968
10.8 Choosing a Convergence Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 974
Chapter Ten Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 991
Copyright c 2019 Pearson Education, Inc.
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~~~~~~~~~~~~~~~~~~~~~~~~_OceanofPDF.com_Calculus_Early_Transcendentals_-_Instructors_Solutions_Manual_-_William_Briggs.pd
_OceanofPDF.com_Calculus_Early_Transcendentals_-_Instructors_Solutions_Manual_-_William_Briggs.pdf!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~_OceanofPDF.com_Calculus_Early_Transcende
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
INSTRUCTOR’S
SOLUTIONS MANUAL
MARK WOODARD
Furman University
C ALCULUS
E ARLY T RANSCENDENTALS
THIRD EDITION
William Briggs
University of Colorado at Denver
Lyle Cochran
Whitworth University
Bernhard Gillett
University of Colorado, Boulder
Eric Schulz
Walla Walla Community College
Briggs.pdf&&&&&&&&&&&&&&&&&&@#%%%%%%%%%%%%%%%%%%%%%%%*!^(((((((((((((((((((((((((((((((((((((#&*^)(((((((((((((
~~~~~~~~~~~~~~~~~~~~~~~~_OceanofPDF.com_Calculus_Early_Transcendentals_-_Instructors_Solutions_Manual_-_William_Briggs.pd
_OceanofPDF.com_Calculus_Early_Transcendentals_-_Instructors_Solutions_Manual_-_William_Briggs.pdf!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
,Briggs.pdf&&&&&&&&&&&&&&&&&&@#%%%%%%%%%%%%%%%%%%%%%%%*!^(((((((((((((((((((((((((((((((((((((#&*^)(((((((((((((
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~_OceanofPDF.com_Calculus_Early_Transcende
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
The author and publisher of this book have used their best efforts in preparing this book. These efforts include the
development, research, and testing of the theories and programs to determine their effectiveness. The author and publisher
make no warranty of any kind, expressed or implied, with regard to these programs or the documentation contained in this
book. The author and publisher shall not be liable in any event for incidental or consequential damages in connection with, or
arising out of, the furnishing, performance, or use of these programs.
Reproduced by Pearson from electronic files supplied by the author.
Copyright © 2019, 2015, 2011 Pearson Education, Inc.
Publishing as Pearson, 330 Hudson Street, NY NY 10013
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.
ISBN-13: 978-0-13-476681-2
ISBN-10: 0-13-476681-4
Briggs.pdf&&&&&&&&&&&&&&&&&&@#%%%%%%%%%%%%%%%%%%%%%%%*!^(((((((((((((((((((((((((((((((((((((#&*^)(((((((((((((
~~~~~~~~~~~~~~~~~~~~~~~~_OceanofPDF.com_Calculus_Early_Transcendentals_-_Instructors_Solutions_Manual_-_William_Briggs.pd
_OceanofPDF.com_Calculus_Early_Transcendentals_-_Instructors_Solutions_Manual_-_William_Briggs.pdf!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
,Briggs.pdf&&&&&&&&&&&&&&&&&&@#%%%%%%%%%%%%%%%%%%%%%%%*!^(((((((((((((((((((((((((((((((((((((#&*^)(((((((((((((
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~_OceanofPDF.com_Calculus_Early_Transcende
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Contents
1 Functions 3
1.1 Review of Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 Representing Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.3 Inverse, Exponential and Logarithmic Functions . . . . . . . . . . . . . . . . . . . . . . . . . 30
1.4 Trigonometric Functions and Their Inverses . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
Chapter One Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
2 Limits 65
2.1 The Idea of Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
2.2 Definition of a Limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
2.3 Techniques for Computing Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
2.4 Infinite Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
2.5 Limits at Infinity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
2.6 Continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
2.7 Precise Definitions of Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
Chapter Two Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
3 Derivatives 153
3.1 Introducing the Derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
3.2 The Derivative as a Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
3.3 Rules of Differentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180
3.4 The Product and Quotient Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188
3.5 Derivatives of Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200
3.6 Derivatives as Rates of Change . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209
3.7 The Chain Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224
3.8 Implicit Differentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238
3.9 Derivatives of Logarithmic and Exponential Functions . . . . . . . . . . . . . . . . . . . . . . 256
3.10 Derivatives of Inverse Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . . . . . 268
3.11 Related Rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277
Chapter Three Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289
4 Applications of the Derivative 305
4.1 Maxima and Minima . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305
4.2 Mean Value Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 320
4.3 What Derivatives Tell Us . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 328
4.4 Graphing Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 346
4.5 Optimization Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 380
4.6 Linear Approximation and Differentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 403
4.7 L’Hôpital’s Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413
4.8 Newton’s Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 427
4.9 Antiderivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 443
Chapter Four Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 454
1
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2 Contents
5 Integration 477
5.1 Approximating Areas under Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 477
5.2 Definite Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 497
5.3 Fundamental Theorem of Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 517
5.4 Working with Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 534
5.5 Substitution Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 544
Chapter Five Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 555
6 Applications of Integration 571
6.1 Velocity and Net Change . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 571
6.2 Regions Between Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 585
6.3 Volume by Slicing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 600
6.4 Volume by Shells . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 608
6.5 Length of Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 618
6.6 Surface Area . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 624
6.7 Physical Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 632
Chapter Six Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 642
7 Logarithmic, Exponential, and Hyperbolic Functions 659
7.1 Logarithmic and Exponential Functions Revisited . . . . . . . . . . . . . . . . . . . . . . . . 659
7.2 Exponential Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 667
7.3 Hyperbolic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 673
Chapter Seven Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 685
8 Integration Techniques 691
8.1 Basic Approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 691
8.2 Integration by Parts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 701
8.3 Trigonometric Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 720
8.4 Trigonometric Substitutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 729
8.5 Partial Fractions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 746
8.6 Integration Strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 761
8.7 Other Methods of Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 796
8.8 Numerical Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 805
8.9 Improper Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 816
Chapter Eight Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 831
9 Differential Equations 853
9.1 Basic Ideas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 853
9.2 Direction Fields and Euler’s Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 859
9.3 Separable Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 871
9.4 Special First-Order Linear Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . 884
9.5 Modeling with Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 892
Chapter Nine Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 901
10 Sequences and Infinite Series 909
10.1 An Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 909
10.2 Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 917
10.3 Infinite Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 931
10.4 The Divergence and Integral Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 942
10.5 Comparison Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 953
10.6 Alternating Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 961
10.7 The Ratio and Root Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 968
10.8 Choosing a Convergence Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 974
Chapter Ten Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 991
Copyright c 2019 Pearson Education, Inc.
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