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Tables of Contents for Single Variable Calculus
Chapter/Section Title
Page #
Page Count
About the Authors
xi
Preface
xiii
Functions, Graphs, and Models
1
52
Functions and Mathematical Modeling
2
10
Project: A Square Wading Pool
11
1
Graphs of Equations and Functions
12
12
Project: A Broken Tree
23
1
Polynomials and Algebraic Functions
24
9
Project: A Leaning Ladder
33
1
Transcendental Functions
33
12
Project: A Spherical Asteroid
45
1
Preview: What Is Calculus?
45
8
Review: Definitions and Concepts
48
5
Prelude to Calculus
53
48
Tangent Lines and Slope Predictors
54
9
Project: Numerical Slope Investigations
63
1
The Limit Concept
63
12
Project: Limits, Slopes, and Logarithms
74
1
More About Limits
75
13
Project: Numerical Epsilon-Delta Limit Investigations
87
1
The Concept of Continuity
88
13
Review: Definitions, Concepts, Results
99
2
The Derivative
101
92
The Derivative and Rates of Change
102
13
Basic Differentiation Rules
115
11
The Chain Rule
126
7
Derivatives of Algebraic Functions
133
9
Maxima and Minima of Functions on Closed Intervals
142
10
Project: When Is Your Coffee Cup Stablest?
150
2
Applied Optimization Problems
152
13
Derivatives of Trigonometric Functions
165
11
Successive Approximations and Newton's Method
176
17
Project: How Deep Does a Floating Ball Sink?
188
1
Review: Formulas, Concepts, Definitions
189
4
Additional Applications of the Derivative
193
78
Implicit Functions and Related Rates
194
10
Project: Investigating the Folium of Descartes
203
1
Increments, Differentials, and Linear Approximation
204
8
Increasing and Decreasing Functions and the Mean Value Theorem
212
10
The First Derivative Test and Applications
222
10
Project: Making a Candy Box With Lid
232
1
Simple Curve Sketching
232
10
Higher Derivatives and Concavity
242
14
Curve Sketching and Asymptotes
256
15
Project: Locating Special Points on Exotic Graphs
267
1
Review: Definitions, Concepts, Results
267
4
The Integral
271
94
Introduction
272
1
Antiderivatives and Initial Value Problems
272
14
Elementary Area Computations
286
12
Riemann Sums and the Integral
298
10
Project: Calculator/Computer Riemann Sums
307
1
Evaluation of Integrals
308
10
The Fundamental Theorem of Calculus
318
10
Integration by Substitution
328
8
Areas of Plane Regions
336
10
Numerical Integration
346
19
Project: Trapezoidal and Simpson Approximations
359
2
Review: Definitions, Concepts, Results
361
4
Applications of the Integral
365
62
Riemann Sum Approximations
366
10
Volumes by the Method of Cross Sections
376
11
Volumes by the Method of Cylindrical Shells
387
9
Project: Design Your Own Ring!
395
1
Arc Length and Surface Area of Revolution
396
9
Force and Work
405
11
Centroids of Plane Regions and Curves
416
11
Review: Definitions, Concepts, Results
423
4
Calculus of Transcendental Functions
427
62
Exponential and Logarithmic Functions
428
13
Project: Discovering the Number e for Yourself
441
1
Indeterminate Forms and L'Hopital's Rule
441
8
More Indeterminate Forms
449
6
The Natural Logarithm as an Integral
455
12
Project: Natural Functional Equations
466
1
Inverse Trigonometric Functions
467
10
Hyperbolic Functions
477
12
Review: Formulas, Concepts, Definitions
485
4
Techniques of Integration
489
56
Introduction
490
1
Integral Tables and Simple Substitutions
490
4
Integration by Parts
494
7
Trigonometric Integrals
501
7
Rational Functions and Partial Fractions
508
7
Trigonometric Substitution
515
6
Integrals Involving Quadratic Polynomials
521
5
Improper Integrals
526
19
Summary
539
6
Differential Equations
545
78
Simple Equations and Models
546
12
Slope Fields and Euler's Method
558
10
Project: Computer-Assisted Slope Fields and Euler's Method
567
1
Separable Equations and Applications
568
7
Linear Equations and Applications
575
12
Population Models
587
11
Project: Predator-Prey Equations and Your Own Game Preserve
597
1
Linear Second-Order Equations
598
9
Mechanical Vibrations
607
16
Review: Definitions, Concepts, Results
618
5
Polar Coordinates and Parametric Curves
623
58
Analytic Geometry and the Conic Sections
624
5
Polar Coordinates
629
9
Area Computations in Polar Coordinates
638
5
Parametric Curves
643
10
Project: Trochoid Investigations
652
1
Integral Computations with Parametric Curves
653
8
Project: Moon Orbits and Race Tracks
660
1
Conic Sections and Applications
661
20
Review: Concepts and Definitions
679
2
Infinite Series
681
90
Introduction
682
1
Infinite Sequences
682
9
Project: Nested Radicals and Continued Fractions
691
1
Infinite Series and Convergence
691
11
Project: Numerical Summation and Geometric Series
701
1
Taylor Series and Taylor Polynomials
702
13
Project: Calculating Logarithms on a Deserted Island
715
1
The Integral Test
715
7
Project: The Number π, Once and for All
722
1
Comparison Tests for Positive-Term Series
722
6
Alternating Series and Absolute Convergence
728
9
Power Series
737
13
Power Series Computations
750
8
Project: Calculating Trigonometric Functions on a Deserted Island
758
1
Series Solutions of Differential Equations
758
13
Review: Definitions, Concepts, Results
767
4
Vectors, Curves, and Surfaces in Space
771
78
Vectors in the Plane
772
6
Three-Dimensional Vectors
778
10
The Cross Product of Vectors
788
8
Lines and Planes in Space
796
7
Curves and Motion in Space
803
14
Project: Does a Pitched Baseball Really Curve?
816
1
Curvature and Acceleration
817
13
Cylinders and Quadric Surfaces
830
8
Cylindrical and Spherical Coordinates
838
11
Review: Definitions, Concepts, Results
845
4
Partial Differentiation
849
90
Introduction
850
1
Functions of Several Variables
850
10
Limits and Continuity
860
8
Partial Derivatives
868
10
Multivariable Optimization Problems
878
11
Increments and Linear Approximation
889
7
The Multivariable Chain Rule
896
11
Directional Derivatives and the Gradient Vector
907
11
Lagrange Multipliers and Constrained Optimization
918
9
Project: Numerical Solution of Lagrange Multiplier Systems
927
1
Critical Points of Functions of Two Variables
927
12
Project: Critical Point Investigations
935
1
Review: Definitions, Concepts, Results
936
3
Multiple Integrals
939
74
Double Integrals
940
7
Project: Midpoint Sums Approximating Double Integrals
947
1
Double Integrals over More General Regions
947
7
Area and Volume by Double Integration
954
7
Double Integrals in Polar Coordinates
961
7
Applications of Double Integrals
968
11
Project: Optimal Design of Downhill Race-Car Wheels
978
1
Triple Integrals
979
9
Project: Archimedes' Floating Paraboloid
987
1
Integration in Cylindrical and Spherical Coordinates
988
8
Surface Area
996
5
Change of Variables in Multiple Integrals
1001
12
Review: Definitions, Concepts, Results
1009
4
Vector Calculus
1013
1
Vector Fields
1014
5
Line Integrals
1019
11
The Fundamental Theorem and Independence of Path
1030
7
Green's Theorem
1037
10
Surface Integrals
1047
10
Project: Surface Integrals and Rocket Nose Cones
1057
1
The Divergence Theorem
1057
8
Stokes' Theorem
1065
7
Review: Definitions, Concepts, Results
1072
APPENDICES
1
44
A: Real Numbers and Inequalities
1
5
B: The Coordinate Plane and Straight Lines
6
7
C: Review of Trigonometry
13
6
D: Proofs of the Limit Laws
19
4
E: The Completeness of the Real Number System
23
5
F: Existence of the Integral
28
5
G: Approximations and Riemann Sums
33
3
H: L'Hopital's Rule and Cauchy's Mean Value Theorem
36
2
I: Proof of Taylor's Formula
38
1
J: Conic Sections as Sections of a Cone
39
1
K: Proof of the Linear Approximation Theorem
40
1
L: Units of Measurement and Conversion Factors
41
1
M: Formulas from Algebra, Geometry, and Trigonometry
42
2
N: The Greek Alphabet
44
1
Answers to Odd-Numbered Problems
45
40
References for Further Study
85
Index
1
1
Table of Integrals
1