Exam (elaborations)

Solutions Manual for Calculus: Early Transcendentals, 3rd edition by William L. Briggs, Lyle Cochran, Bernard Gillett, Eric Schulz

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INSTRUCTOR’S
SOLUTIONS MANUAL
MARK WOODARD
R
Furman University
E U
C ALCULUS
S S
THIRD EDITION
I
O
N
William Briggs
N
University of Colorado at Denver
O
Lyle Cochran
C
Whitworth University
D
Bernhard Gillett
E
University of Colorado, Boulder
M
Eric Schulz
Walla Walla Community College

,Contents
R
U
1 Functions 5
1.1 Review of Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
E
1.2 Representing Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.3 Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
S
Chapter One Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
S
2 Limits 51
2.1 The Idea of Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
I
2.2 Definitions of Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
O
2.3 Techniques for Computing Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
2.4 Infinite Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
2.5 Limits at Infinity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
N
2.6 Continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
2.7 Precise Definitions of Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
N
Chapter Two Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

3 Derivatives 133
O
3.1 Introducing the Derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
3.2 The Derivative as a Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
C
3.3 Rules of Di↵erentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
3.4 The Product and Quotient Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
3.5 Derivatives of Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179
D
3.6 Derivatives as Rates of Change . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189
3.7 The Chain Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204
E
3.8 Implicit Di↵erentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216
3.9 Related Rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234
M
Chapter Three Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247

4 Applications of the Derivative 259
4.1 Maxima and Minima . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259
4.2 Mean Value Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273
4.3 What Derivatives Tell Us . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 280
4.4 Graphing Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297
4.5 Optimization Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 329
4.6 Linear Approximation and Di↵erentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 351
4.7 L’Hôpital’s Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 361
4.8 Newton’s Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 367
4.9 Antiderivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 382
Chapter Four Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 392

1

,2 Contents

5 Integration 409
5.1 Approximating Areas under Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 409
5.2 Definite Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 429
5.3 Fundamental Theorem of Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 448
5.4 Working with Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 465
5.5 Substitution Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 475
Chapter Five Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 487

6 Applications of Integration 503
R
6.1 Velocity and Net Change . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 503
6.2 Regions Between Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 516
6.3 Volume by Slicing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 532
U
6.4 Volume by Shells . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 539
6.5 Length of Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 549
E
6.6 Surface Area . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 555
6.7 Physical Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 562
Chapter Six Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 572
S
7 Logarithmic and Exponential Functions 587
S
7.1 Inverse Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 587
7.2 The Natural Logarithmic and Exponential Functions . . . . . . . . . . . . . . . . . . . . . . 597
I
7.3 Logarithmic and Exponential Functions with Other Bases . . . . . . . . . . . . . . . . . . . . 610
O
7.4 Exponential Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 618
7.5 Inverse Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 625
7.6 L’Höpital’s Rule and Growth Rates of Functions . . . . . . . . . . . . . . . . . . . . . . . . . 638
N
7.7 Hyperbolic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 646
Chapter Seven Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 658
N
8 Integration Techniques 675
8.1 Basic Approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 675
O
8.2 Integration by Parts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 685
8.3 Trigonometric Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 703
8.4 Trigonometric Substitutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 712
C
8.5 Partial Fractions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 729
8.6 Integration Strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 744
8.7 Other Methods of Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 779
D
8.8 Numerical Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 788
8.9 Improper Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 799
E
Chapter Eight Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 814

9 Di↵erential Equations 837
M
9.1 Basic Ideas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 837
9.2 Direction Fields and Euler’s Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 843
9.3 Separable Di↵erential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 855
9.4 Special First-Order Linear Di↵erential Equations . . . . . . . . . . . . . . . . . . . . . . . . . 868
9.5 Modeling with Di↵erential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 876
Chapter Nine Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 885

10 Sequences and Infinite Series 893
10.1 An Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 893
10.2 Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 900
10.3 Infinite Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 914
10.4 The Divergence and Integral Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 925
10.5 Comparison Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 936
10.6 Alternating Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 944

Copyright c 2019 Pearson Education, Inc.

, Contents 3

10.7 The Ratio and Root Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 951
10.8 Choosing a Convergence Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 957
Chapter Ten Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 974

11 Power Series 987
11.1 Approximating Functions With Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . 987
11.2 Properties of Power Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1005
11.3 Taylor Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1014
11.4 Working with Taylor Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1029
R
Chapter Eleven Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1043

12 Parametric and Polar Curves 1053
U
12.1 Parametric Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1053
12.2 Polar Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1070
E
12.3 Calculus in Polar Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1090
12.4 Conic Sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1105
Chapter Twelve Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1125
S
13 Vectors and the Geometry of Space 1143
S
13.1 Vectors in the Plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1143
13.2 Vectors in Three Dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1151
13.3 Dot Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1161
I
13.4 Cross Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1169
O
13.5 Lines and Planes in Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1178
13.6 Cylinders and Quadric Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1186
N
Chapter Thirteen Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1201

14 Vector-Valued Functions 1215
N
14.1 Vector-Valued Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1215
14.2 Calculus of Vector-Valued Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1223
14.3 Motion in Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1229
O
14.4 Length of Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1248
14.5 Curvature and Normal Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1254
C
Chapter Fourteen Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1265

15 Functions of Several Variables 1279
D
15.1 Graphs and Level Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1279
15.2 Limits and Continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1291
15.3 Partial Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1297
E
15.4 The Chain Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1309
15.5 Directional Derivatives and the Gradient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1320
M
15.6 Tangent Planes and Linear Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1334
15.7 Maximum/Minimum Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1342
15.8 Lagrange Multipliers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1353
Chapter Fifteen Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1362

16 Multiple Integration 1373
16.1 Double Integrals over Rectangular Regions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1373
16.2 Double Integrals over General Regions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1380
16.3 Double Integrals in Polar Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1396
16.4 Triple Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1410
16.5 Triple Integrals in Cylindrical and Spherical Coordinates . . . . . . . . . . . . . . . . . . . . 1422
16.6 Integrals for Mass Calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1434
16.7 Change of Variables in Multiple Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1445
Chapter Sixteen Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1457

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Exam (elaborations)

Solutions Manual for Calculus: Early Transcendentals, 3rd edition by William L. Briggs, Lyle Cochran, Bernard Gillett, Eric Schulz

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