Calculus

Front Cover
Title Page
Copyright Page
Contents
About the Authors
Foreword
Preface
1 Functions and Graphs
1.1 Preliminaries
1.2 Lines in the Plane
1.3 Functions
1.3 Graphs of Functions
1.5 Inverse Functions; Inverse Trigonometric Functions
1.6 Exponential and Logarithmic Functions
Chapter 1 Review
2 Limits and Continuity
2.1 What Is Calculus?
2.2 The Limit of a Function
2.3 Properties of Limits
2.4 Continuity
Chapter 2 Review
Guest Essay: "Calculus Was Inevitable," John Troutman
3 Differentiation
3.1 An Introduction to the Derivative: Tangents
3.2 Techniques of Differentiation
3.3 Derivatives of Trigonometric, Exponential, and Logarithmic Functions
3.4 Rates of Change: Rectilinear Motion
3.5 The Chain Rule
3.6 Implicit Differentiation
3.7 Related Rates and Applications
3.8 Linear Approximation and Differentials
Chapter 3 Review
Group Research Project: Chaos
4 Additional Applications of the Derivative
4.1 Extreme Values of a Continuous Function
4.2 The Mean Value Theorem
4.3 First-Derivative Test
4.4 Concavity and the Second-Derivative Test
4.5 Curve Sketching: Limits Involving Infinity and Asymptotes
4.6 Optimization in the Physical Sciences and Engineering
4.7 Optimization in Business, Economics, and the Life Sciences
4.8 l'Hôpital's Rule
Chapter 4 Review
Group Research Project: Wine Barrel Capacity
5 Integration
5.1 Antidifferentiation
5.2 Area as the Limit of a Sum
5.3 Riemann Sums and the Definite Integral
5.4 The Fundamental Theorems of Calculus
5.5 Integration by Substitution
5.6 Introduction to Differential Equations
5.7 The Mean Value Theorem for Integrals; Average Value
5.9 An Alternative Approach: The Logarithm as an Integral
Chapter 5 Review
Guest Essay: "Kinematics of Jogging," Ralph Boas
6 Additional Applications of the Integral
6.1 Area Between Two Curves
6.2 Volume by Disks and Washers
6.3 Volume by Shells
6.4 Arc Length and Surface Area
6.5 Physical Applications: Work, Liquid Force, and Centroids
Chapter 6 Review
Group Research Project: "Houdini's Escape"
Cumulative Review Problems for Chapters 1-6
7 Methods of Integration
7.1 Review of Substitution and Integration by Table
7.2 Integration by Ports
7.3 Trigonometric Methods
7.4 The Method of Partial Fractions
7.5 Summary of Integration Techniques
7.6 First-Order Differential Equations
7.7 Improper Integrals
7.8 The Hyperbolic and Inverse Hyperbolic Functions
Chapter 7 Review
Group Research Project: Buoy Design
8 Infinite Series
8.1 Sequences and Their Limits
8.2 Introduction to Infinite Series: Geometric Series
8.3 The Integral Test: p-Series
8.4 Comparison Tests
8.5 The Ratio Test and the Root Test
8.6 Alternating Series; Absolute and Conditional Convergence
8.7 Power Series
8.8 Taylor and Maclaurin Series
Chapter 8 Review
Group Research Project: Elastic Tightrope Project
9 Polar Coordinates and Parametric Forms
9.1 The Polar Coordinate System
9.2 Graphing in Polar Coordinates
9.3 Area and Tangent Lines in Polar Coordinates
9.4 Parametric Representation of Curves
Chapter 9 Review
Group Research Project: Security Systems
10 Vectors in the Plane and in Space
10.1 Vectors in the Plane
10.2 Quadric Surfaces and Graphing in Three Dimensions
10.3 The Dot Product
10.4 The Cross Product
10.5 Lines and Planes in Space
10.6 Vector Methods for Measuring Distance in R^3
Chapter 10 Review
Group Research Project: Star Trek
11 Vector-Valued Functions
11.1 Introduction to Vector Functions
11.2 Differentiation and Integration of Vector Functions
11.3 Modeling Ballistics and Planetary Motion
11.4 Unit Tangent and Normal Vectors; Curvature
11.5 Tangential and Normal Components of Acceleration
Chapter 11 Review
Guest Essay: "The Stimulation of Science," Howard Eves
Cumulative Review Problems for Chapters 7-11
12 Partial Differentiation
12.1 Functions of Several Variables
12.2 Limits and Continuity
12.3 Partial Derivatives
12.4 Tangent Planes, Approximations, and Differentiability
12.5 Chain Rules
12.6 Directional Derivatives and the Gradient
12.7 Extrema of Functions of Two Variables
12.8 Lagrange Multipliers
Chapter 12 Review
Group Research Project: Desertification
13 Multiple Integration
13.1 Double Integration over Rectangular Regions
13.2 Double Integration over Nonrectangular Regions
13.3 Double Integrals in Polar Coordinates
13.4 Surface Area
13.5 Triple Integrals
13.6 Mass, Moments, and Probability Density Functions
13.7 Cylindrical and Spherical Coordinates
13.8 Jacobians: Change of Variables
Chapter 13 Review
Group Research Project: Space-Capsule Design
14 Vector Analysis
14.1 Properties of a Vector Field: Divergence and Curl
14.2 Line Integrals
14.3 Independence of Path
14.4 Green's Theorem
14.5 Surface Integration
14.6 Stokes' Theorem
14.7 Divergence Theorem
Chapter 14 Review
Guest Essay: "Continuous vs. Discrete Mathematics," William F. Lucas
15 Introduction to Differential Equations
15.1 First-Order Differential Equations
15.2 Second-Order Homogeneous Linear Differential Equations
15.3 Second-Order Nonhomogeneous Linear Differential Equations
Chapter 15 Review
Group Research Project: Save the Perch Project
Cumulative Review Problems for Chapters 12-15
Appendices
A: Introduction to the Theory of Limits
B: Selected Proofs
C: Significant Digits
D: Short Table of Integrals
E: Answers to Selected Problems
1 Functions and Graphs
2 Limits and Continuity
3 Differentiation
4 Additional Applications of the Derivative
5 Integration
6 Additional Applications of the Integral
Cumulative Review Problems for Chapters 1-6
7 Methods of Integration
8 Infinite Series
9 Polar Coordinates and Parametric Forms
10 Vectors in the Plane and in Space
11 Vector-Valued Functions
Cumulative Review Problems for Chapters 7-11
12 Partial Differentiation
13 Multiple Integration
14 Vector Analysis
15 Introduction to Differential Equations
Cumulative Review Problems for Chapters 12-15
F: Credits
Index
Back Cover