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Complete Solution Manual Calculus Early Transcendentals 3rd Edition Questions & Answers with rationales

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INSTRUCTOR’S
SOLUTIONS MANUAL
MARK WOODARD
Furman University

E
C
C ALCULUS

N
E ARLY T RANSCENDENTALS
LE
THIRD EDITION
EL
William Briggs
C

Lyle Cochran
EX

Bernhard Gillett
ED

** Immediate Download
M

** Swift Response
** All Chapters included
** Solutions to Guided Projects

,Contents

1 Functions 5

E
1.1 Review of Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.2 Representing Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

C
1.3 Inverse, Exponential and Logarithmic Functions . . . . . . . . . . . . . . . . . . . . . . . . . 32
1.4 Trigonometric Functions and Their Inverses . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
Chapter One Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

N
2 Limits 67
2.1 The Idea of Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

LE
2.2 Deﬁnition of a Limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
2.3 Techniques for Computing Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
2.4 Inﬁnite Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
2.5 Limits at Inﬁnity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
EL
2.6 Continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
2.7 Precise Deﬁnitions of Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
Chapter Two Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
C

3 Derivatives 155
3.1 Introducing the Derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
3.2 The Derivative as a Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164
EX

3.3 Rules of Diﬀerentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182
3.4 The Product and Quotient Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190
3.5 Derivatives of Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202
3.6 Derivatives as Rates of Change . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211
ED

3.7 The Chain Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226
3.8 Implicit Diﬀerentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240
3.9 Derivatives of Logarithmic and Exponential Functions . . . . . . . . . . . . . . . . . . . . . . 258
3.10 Derivatives of Inverse Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . . . . . 270
3.11 Related Rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279
M

Chapter Three Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291

4 Applications of the Derivative 307
4.1 Maxima and Minima . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307
4.2 Mean Value Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322
4.3 What Derivatives Tell Us . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 330
4.4 Graphing Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 348
4.5 Optimization Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 382
4.6 Linear Approximation and Diﬀerentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 405
4.7 L’Hôpital’s Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 415
4.8 Newton’s Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 429
4.9 Antiderivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 445
Chapter Four Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 456

1

,2 Contents

5 Integration 479
5.1 Approximating Areas under Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 479
5.2 Deﬁnite Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 499
5.3 Fundamental Theorem of Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 519
5.4 Working with Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 536
5.5 Substitution Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 546
Chapter Five Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 557

6 Applications of Integration 573
6.1 Velocity and Net Change . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 573
6.2 Regions Between Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 587
6.3 Volume by Slicing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 602
6.4 Volume by Shells . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 610

E
6.5 Length of Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 620
6.6 Surface Area . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 626

C
6.7 Physical Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 634
Chapter Six Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 644

N
7 Logarithmic, Exponential, and Hyperbolic Functions 661
7.1 Logarithmic and Exponential Functions Revisited . . . . . . . . . . . . . . . . . . . . . . . . 661

LE
7.2 Exponential Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 669
7.3 Hyperbolic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 675
Chapter Seven Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 687

8 Integration Techniques 693
EL
8.1 Basic Approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 693
8.2 Integration by Parts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 703
8.3 Trigonometric Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 722
8.4 Trigonometric Substitutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 731
C

8.5 Partial Fractions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 748
8.6 Integration Strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 763
8.7 Other Methods of Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 798
EX

8.8 Numerical Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 807
8.9 Improper Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 818
Chapter Eight Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 833
ED

9 Diﬀerential Equations 855
9.1 Basic Ideas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 855
9.2 Direction Fields and Euler’s Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 861
9.3 Separable Diﬀerential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 873
9.4 Special First-Order Linear Diﬀerential Equations . . . . . . . . . . . . . . . . . . . . . . . . . 886
M

9.5 Modeling with Diﬀerential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 894
Chapter Nine Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 903

10 Sequences and Inﬁnite Series 911
10.1 An Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 911
10.2 Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 919
10.3 Inﬁnite Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 933
10.4 The Divergence and Integral Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 944
10.5 Comparison Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 955
10.6 Alternating Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 963
10.7 The Ratio and Root Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 970
10.8 Choosing a Convergence Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 976
Chapter Ten Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 993

, Contents 3

11 Power Series 1005
11.1 Approximating Functions With Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . 1005
11.2 Properties of Power Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1023
11.3 Taylor Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1032
11.4 Working with Taylor Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1047
Chapter Eleven Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1061

12 Parametric and Polar Curves 1071
12.1 Parametric Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1071
12.2 Polar Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1088
12.3 Calculus in Polar Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1108
12.4 Conic Sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1123

E
Chapter Twelve Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1143

13 Vectors and the Geometry of Space 1161

C
13.1 Vectors in the Plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1161
13.2 Vectors in Three Dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1169

N
13.3 Dot Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1179
13.4 Cross Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1187

LE
13.5 Lines and Planes in Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1196
13.6 Cylinders and Quadric Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1204
Chapter Thirteen Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1219
EL
14 Vector-Valued Functions 1233
14.1 Vector-Valued Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1233
14.2 Calculus of Vector-Valued Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1241
14.3 Motion in Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1247
C

14.4 Length of Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1266
14.5 Curvature and Normal Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1272
Chapter Fourteen Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1283
EX

15 Functions of Several Variables 1299
15.1 Graphs and Level Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1299
15.2 Limits and Continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1311
ED

15.3 Partial Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1317
15.4 The Chain Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1329
15.5 Directional Derivatives and the Gradient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1340
15.6 Tangent Planes and Linear Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1354
15.7 Maximum/Minimum Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1362
M

15.8 Lagrange Multipliers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1373
Chapter Fifteen Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1382

16 Multiple Integration 1393
16.1 Double Integrals over Rectangular Regions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1393
16.2 Double Integrals over General Regions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1400
16.3 Double Integrals in Polar Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1416
16.4 Triple Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1430
16.5 Triple Integrals in Cylindrical and Spherical Coordinates . . . . . . . . . . . . . . . . . . . . 1442
16.6 Integrals for Mass Calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1454
16.7 Change of Variables in Multiple Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1465
Chapter Sixteen Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1477

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Exam (elaborations)

Complete Solution Manual Calculus Early Transcendentals 3rd Edition Questions & Answers with rationales

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