SOLUTIONS MANUAL
All Chapters included
vd
, INSTRUCTOR’S
SOLUTIONS MANUAL
MARK WOODARD
CALCULUS
EARLY T RANSCENDENTALS
THIRD EDITION
, Table Of 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
, 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 Di↵erential 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
11 Power Series 1003
11.1 Approximating Functions With Polynomials.......................................................................................1003
All Chapters included
vd
, INSTRUCTOR’S
SOLUTIONS MANUAL
MARK WOODARD
CALCULUS
EARLY T RANSCENDENTALS
THIRD EDITION
, Table Of 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
, 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 Di↵erential 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
11 Power Series 1003
11.1 Approximating Functions With Polynomials.......................................................................................1003
