INSTRUCTOR’S
SOLUTIONS MANUAL
DUANE KOUBA
University of California, Davis
T HOMAS ’ C ALCULUS
FOURTEENTH EDITION
Based on the original work by
George B. Thomas, Jr
Massachusetts Institute of Technology
as revised by
Joel Hass
University of California, Davis
Christopher Heil
Georgia Institute of Technology
Maurice D. Weir
Naval Postgraduate School
,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 © 2018, 2014, 2010 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: 978-0-13-443918-1
ISBN-10: 0-13-443918-X
, TABLE OF CONTENTS
1 Functions 1
1.1 Functions and Their Graphs 1
1.2 Combining Functions; Shifting and Scaling Graphs 9
1.3 Trigonometric Functions 19
1.4 Graphing with Software 27
Practice Exercises 32
Additional and Advanced Exercises 40
2 Limits and Continuity 45
2.1 Rates of Change and Tangents to Curves 45
2.2 Limit of a Function and Limit Laws 49
2.3 The Precise Definition of a Limit 59
2.4 One-Sided Limits 66
2.5 Continuity 72
2.6 Limits Involving Infinity; Asymptotes of Graphs 77
Practice Exercises 87
Additional and Advanced Exercises 93
3 Derivatives 101
3.1 Tangents and the Derivative at a Point 101
3.2 The Derivative as a Function 107
3.3 Differentiation Rules 118
3.4 The Derivative as a Rate of Change 123
3.5 Derivatives of Trigonometric Functions 129
3.6 The Chain Rule 138
3.7 Implicit Differentiation 148
3.8 Related Rates 156
3.9 Linearization and Differentials 161
Practice Exercises 167
Additional and Advanced Exercises 179
Copyright 2018 Pearson Education, Inc.
iii
, 4 Applications of Derivatives 185
4.1 Extreme Values of Functions 185
4.2 The Mean Value Theorem 195
4.3 Monotonic Functions and the First Derivative Test 201
4.4 Concavity and Curve Sketching 212
4.5 Applied Optimization 238
4.6 Newton's Method 253
4.7 Antiderivatives 257
Practice Exercises 266
Additional and Advanced Exercises 280
5 Integrals 287
5.1 Area and Estimating with Finite Sums 287
5.2 Sigma Notation and Limits of Finite Sums 292
5.3 The Definite Integral 298
5.4 The Fundamental Theorem of Calculus 313
5.5 Indefinite Integrals and the Substitution Method 323
5.6 Definite Integral Substitutions and the Area Between Curves 329
Practice Exercises 346
Additional and Advanced Exercises 357
6 Applications of Definite Integrals 363
6.1 Volumes Using Cross-Sections 363
6.2 Volumes Using Cylindrical Shells 375
6.3 Arc Length 386
6.4 Areas of Surfaces of Revolution 394
6.5 Work and Fluid Forces 400
6.6 Moments and Centers of Mass 410
Practice Exercises 425
Additional and Advanced Exercises 436
7 Transcendental Functions 441
7.1 Inverse Functions and Their Derivatives 441
7.2 Natural Logarithms 450
7.3 Exponential Functions 459
7.4 Exponential Change and Separable Differential Equations 473
7.5 Indeterminate Forms and L’Hôpital’s Rule 478
7.6 Inverse Trigonometric Functions 488
7.7 Hyperbolic Functions 501
7.8 Relative Rates of Growth 510
Practice Exercises 515
Additional and Advanced Exercises 529
Copyright 2018 Pearson Education, Inc.
iv
SOLUTIONS MANUAL
DUANE KOUBA
University of California, Davis
T HOMAS ’ C ALCULUS
FOURTEENTH EDITION
Based on the original work by
George B. Thomas, Jr
Massachusetts Institute of Technology
as revised by
Joel Hass
University of California, Davis
Christopher Heil
Georgia Institute of Technology
Maurice D. Weir
Naval Postgraduate School
,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 © 2018, 2014, 2010 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: 978-0-13-443918-1
ISBN-10: 0-13-443918-X
, TABLE OF CONTENTS
1 Functions 1
1.1 Functions and Their Graphs 1
1.2 Combining Functions; Shifting and Scaling Graphs 9
1.3 Trigonometric Functions 19
1.4 Graphing with Software 27
Practice Exercises 32
Additional and Advanced Exercises 40
2 Limits and Continuity 45
2.1 Rates of Change and Tangents to Curves 45
2.2 Limit of a Function and Limit Laws 49
2.3 The Precise Definition of a Limit 59
2.4 One-Sided Limits 66
2.5 Continuity 72
2.6 Limits Involving Infinity; Asymptotes of Graphs 77
Practice Exercises 87
Additional and Advanced Exercises 93
3 Derivatives 101
3.1 Tangents and the Derivative at a Point 101
3.2 The Derivative as a Function 107
3.3 Differentiation Rules 118
3.4 The Derivative as a Rate of Change 123
3.5 Derivatives of Trigonometric Functions 129
3.6 The Chain Rule 138
3.7 Implicit Differentiation 148
3.8 Related Rates 156
3.9 Linearization and Differentials 161
Practice Exercises 167
Additional and Advanced Exercises 179
Copyright 2018 Pearson Education, Inc.
iii
, 4 Applications of Derivatives 185
4.1 Extreme Values of Functions 185
4.2 The Mean Value Theorem 195
4.3 Monotonic Functions and the First Derivative Test 201
4.4 Concavity and Curve Sketching 212
4.5 Applied Optimization 238
4.6 Newton's Method 253
4.7 Antiderivatives 257
Practice Exercises 266
Additional and Advanced Exercises 280
5 Integrals 287
5.1 Area and Estimating with Finite Sums 287
5.2 Sigma Notation and Limits of Finite Sums 292
5.3 The Definite Integral 298
5.4 The Fundamental Theorem of Calculus 313
5.5 Indefinite Integrals and the Substitution Method 323
5.6 Definite Integral Substitutions and the Area Between Curves 329
Practice Exercises 346
Additional and Advanced Exercises 357
6 Applications of Definite Integrals 363
6.1 Volumes Using Cross-Sections 363
6.2 Volumes Using Cylindrical Shells 375
6.3 Arc Length 386
6.4 Areas of Surfaces of Revolution 394
6.5 Work and Fluid Forces 400
6.6 Moments and Centers of Mass 410
Practice Exercises 425
Additional and Advanced Exercises 436
7 Transcendental Functions 441
7.1 Inverse Functions and Their Derivatives 441
7.2 Natural Logarithms 450
7.3 Exponential Functions 459
7.4 Exponential Change and Separable Differential Equations 473
7.5 Indeterminate Forms and L’Hôpital’s Rule 478
7.6 Inverse Trigonometric Functions 488
7.7 Hyperbolic Functions 501
7.8 Relative Rates of Growth 510
Practice Exercises 515
Additional and Advanced Exercises 529
Copyright 2018 Pearson Education, Inc.
iv
