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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
8 Techniques of Integration 533
8.1 Using Basic Integration Formulas 533
8.2 Integration by Parts 546
8.3 Trigonometric Integrals 560
8.4 Trigonometric Substitutions 569
8.5 Integration of Rational Functions by Partial Fractions 578
8.6 Integral Tables and Computer Algebra Systems 589
8.7 Numerical Integration 600
8.8 Improper Integrals 611
8.9 Probability 623
Practice Exercises 632
Additional and Advanced Exercises 646
9 First-Order Differential Equations 655
9.1 Solutions, Slope Fields, and Euler's Method 655
9.2 First-Order Linear Equations 664
9.3 Applications 668
9.4 Graphical Solutions of Autonomous Equations 673
9.5 Systems of Equations and Phase Planes 680
Practice Exercises 686
Additional and Advanced Exercises 694
10 Infinite Sequences and Series 697
10.1 Sequences 697
10.2 Infinite Series 709
10.3 The Integral Test 717
10.4 Comparison Tests 726
10.5 Absolute Convergence; The Ratio and Root Tests 736
10.6 Alternating Series and Conditional Convergence 742
10.7 Power Series 752
10.8 Taylor and Maclaurin Series 765
10.9 Convergence of Taylor Series 771
10.10 The Binomial Series and Applications of Taylor Series 779
Practice Exercises 788
Additional and Advanced Exercises 799
Copyright  2018 Pearson Education, Inc.
v
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11 Parametric Equations and Polar Coordinates 805
11.1 Parametrizations of Plane Curves 805
11.2 Calculus with Parametric Curves 814
11.3 Polar Coordinates 824
11.4 Graphing Polar Coordinate Equations 829
11.5 Areas and Lengths in Polar Coordinates 837
11.6 Conic Sections 843
11.7 Conics in Polar Coordinates 854
Practice Exercises 864
Additional and Advanced Exercises 875
12 Vectors and the Geometry of Space 881
12.1 Three-Dimensional Coordinate Systems 881
12.2 Vectors 886
12.3 The Dot Product 892
12.4 The Cross Product 897
12.5 Lines and Planes in Space 904
12.6 Cylinders and Quadric Surfaces 913
Practice Exercises 918
Additional and Advanced Exercises 926
13 Vector-Valued Functions and Motion in Space 933
13.1 Curves in Space and Their Tangents 933
13.2 Integrals of Vector Functions; Projectile Motion 940
13.3 Arc Length in Space 949
13.4 Curvature and Normal Vectors of a Curve 953
13.5 Tangential and Normal Components of Acceleration 961
13.6 Velocity and Acceleration in Polar Coordinates 967
Practice Exercises 970
Additional and Advanced Exercises 977
Copyright  2018 Pearson Education, Inc.
vi
14 Partial Derivatives 981
14.1 Functions of Several Variables 981
14.2 Limits and Continuity in Higher Dimensions 991
14.3 Partial Derivatives 999
14.4 The Chain Rule 1008
14.5 Directional Derivatives and Gradient Vectors 1018
14.6 Tangent Planes and Differentials 1024
14.7 Extreme Values and Saddle Points 1033
14.8 Lagrange Multipliers 1049
14.9 Taylor's Formula for Two Variables 1061
14.10 Partial Derivatives with Constrained Variables 1064
Practice Exercises 1067
Additional and Advanced Exercises 1085
15 Multiple Integrals 1091
15.1 Double and Iterated Integrals over Rectangles 1091
15.2 Double Integrals over General Regions 1094
15.3 Area by Double Integration 1108
15.4 Double Integrals in Polar Form 1113
15.5 Triple Integrals in Rectangular Coordinates 1119
15.6 Moments and Centers of Mass 1125
15.7 Triple Integrals in Cylindrical and Spherical Coordinates 1132
15.8 Substitutions in Multiple Integrals 1146
Practice Exercises 1153
Additional and Advanced Exercises 1160
16 Integrals and Vector Fields 1167
16.1 Line Integrals 1167
16.2 Vector Fields and Line Integrals: Work, Circulation, and Flux 1173
16.3 Path Independence, Conservative Fields, and Potential Functions 1185
16.4 Green's Theorem in the Plane 1191
16.5 Surfaces and Area 1199
16.6 Surface Integrals 1209
16.7 Stokes' Theorem 1220
16.8 The Divergence Theorem and a Unified Theory 1227
Practice Exercises 1234
Additional and Advanced Exercises 1244
Copyright  2018 Pearson Education, Inc.
vii
CHAPTER 1 FUNCTIONS
1.1
FUNCTIONS AND THEIR GRAPHS
1. domain  (, ); range  [1, )
2. domain  [0, ); range  (, 1]
3. domain  [2, ); y in range and y  5 x  10  0  y can be any nonnegative real number  range  [0, ).
4. domain  (, 0]  [3, ); y in range and y  x 2  3 x  0  y can be any nonnegative real number 
range  [0, ).
5. domain  (, 3)  (3, ); y in range and y  3 4 t , now if t  3  3  t  0  3 4 t  0, or if t  3 
3  t  0  3 4 t  0  y can be any nonzero real number  range  (, 0)  (0, ).
6. domain  (,  4)  ( 4, 4)  (4, ); y in range and y  2 2
2  2
4  t  4  16  t 2  16  0   16
2
t  16
, now if t  4  t 2  16  0  2 2
t  16
2
t  16
, or if t  4  t  16  0  2 2
t  16
 0, or if
 0  y can be any nonzero
real number  range  (,  18 ]  (0, ).
7. (a) Not the graph of a function of x since it fails the vertical line test.
(b) Is the graph of a function of x since any vertical line intersects the graph at most once.
8. (a) Not the graph of a function of x since it fails the vertical line test.
(b) Not the graph of a function of x since it fails the vertical line test.
9. base  x; (height)2 
 2x   x2  height  23 x; area is a( x)  12 (base)(height)  12 ( x)  23 x   43 x2 ;
2
perimeter is p ( x)  x  x  x  3 x.
10. s  side length  s 2  s 2  d 2  s  d ; and area is a  s 2  a  12 d 2
2
11. Let D  diagonal length of a face of the cube and   the length of an edge. Then  2  D 2  d 2 and
3
  

x
2
2
D 2  2 2  3 2  d 2    d . The surface area is 6 2  6 d3  2d 2 and the volume is 3  d3

3/2
12. The coordinates of P are x, x so the slope of the line joining P to the origin is m  xx  1 ( x  0).

 
Thus, x, x 

1 , 1 .
m2 m
25
13. 2 x  4 y  5  y   12 x  54 ; L  ( x  0)2  ( y  0)2  x 2  ( 12 x  54 )2  x 2  14 x 2  54 x  16

5 x 2  5 x  25 
4
4
16
20 x 2  20 x  25

16
20 x 2  20 x  25
4
14. y  x  3  y 2  3  x; L  ( x  4) 2  ( y  0) 2  ( y 2  3  4)2  y 2  ( y 2  1)2  y 2

y4  2 y2  1  y2 
y4  y2  1
Copyright  2018 Pearson Education, Inc.
1
d3 .
3 3
2
Chapter 1 Functions
15. The domain is (, ).
16. The domain is (, ).
17. The domain is (, ).
18. The domain is (, 0].
19. The domain is (, 0)  (0, ).
20. The domain is (, 0)  (0, ).
21. The domain is (, 5)  (5, 3]  [3, 5)  (5, ) 22. The range is [2, 3).
23. Neither graph passes the vertical line test
(a)
(b)
Copyright  2018 Pearson Education, Inc.
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