Calculus

Front Cover
Title Page
Copyright Page
Contents
About the Authors
Preface
1 Preview of Calculus: Functions and Limits
1.1 What Is Calculus?
1.2 Preliminaries
1.3 Lines in the Plane
1.4 Functions and Their Graphs
1.5 The Limit of a Function
1.6 Properties of Limits
1.7 Continuity
1.8 Introduction to the Theory of Limits
Chapter 1 Review
Guest Essay: "Calculus was Inevitable," by John L. Troutman
Book Report: Ethnomathematics by Marcia Ascher
2 Techniques of Differentiation With Selected Applications
2.1 An Introduction to the Derivative: Tangents
2.2 Techniques of Differentiation
2.3 Derivatives of the Trigonometric Functions
2.4 Rates of Change: Rectilinear Motion
2.5 The Chain Rule
2.6 Implicit Differentiation
2.7 Related Rates
2.8 Differentials and Linear Approximations
2.9 The Newton-Raphson Method for Approximating Roots
Chapter 2 Review
Group Research Project: "Ups and Downs"
3 Additional Applications of the Derivative
3.1 Extreme Values of a Continuous Function
3.2 The Mean Value Theorem
3.3 First-Derivative Test
3.4 Concavity and the Second-Derivative Test
3.5 Infinite Limits and Asymptotes
3.6 Summary of Curve Sketching
3.7 Optimization in the Physical Sciences and Engineering
3.8 Optimization in Business, Economics, and the Life Sciences
3.9 L'Hôpital's Rule
3.10 Antiderivatives
Chapter 3 Review
Group Research Project: "Wine Barrel Capacity"
4 Integration
4.1 Area as the Limit of a Sum; Summation Notation
4.2 Riemann Sums and the Definite Integral
4.3 The Fundamental Theorems of Calculus; Integration by Substitution
4.4 Introduction to Differential Equations
4.5 The Mean Value Theorem for Integrals; Average Value
4.6 Numerical Integration: The Trapezoidal Rule and Simpson's Rule
4.7 Area Between Two Curves
Chapter 4 Review
Guest Essay: "Kinematics of Jogging," by Ralph Boas
Book Report: To Infinity and Beyond, A Cultural History of the Infinite, by Eli Maor
5 Exponential, Logarithmic, and Inverse Trigonometric Functions
5.1 Exponential Functions; The Number e
5.2 Inverse Functions; Logarithms
5.3 Derivatives Involving e^x and ln(x)
5.4 Applications Involving Derivatives of e^x and ln(x)
5.5 Integrals Involving e^x and ln(x)
5.6 The Inverse Trigonometric Functions
5.7 An Alternative Approach: The Logarithm as an Integral
Chapter 5 Review
Group Research Project: "Quality Control"
6 Additional Applications of the Integral
6.1 Volume: Disks, Washers, and Shells
6.2 Arc Length and Surface Area
6.3 Physical Applications: Work, Liquid Force, and Centroids
6.4 Growth, Decay, And First-Order Linear Differential Equations
Chapter 6 Review
Group Research Project: "Houdini's Escape"
Cumulative Review, Chapters 1-6
7 Methods of Integration
7.1 Review of Substitution and Integration by Table
7.2 Integration By Parts
7.3 The Method of Partial Fractions
7.4 Summary of Integration Techniques
7.5 Improper Integrals
7.6 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
9.5 Conic Sections: The Parabola
9.6 Conic Sections: The Ellipse and the Hyperbola
Chapter 9 Review
Group Research Project: "Security System Project"
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 Project"
11 Vector Calculus
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: "For Further Study - The Simulation of Science," by Howard Eves
Book Report: Hypatia's Heritage by Margaret Alic
Cumulative Review, 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 Integrals
14.6 Stokes' Theorem
14.7 Divergence Theorem
Chapter 14 Review
Guest Essay: "Continuous vs. Discrete Mathematics," by William F. Lucas
Book Report: The Mathematical Experience by Philip J. Davis and Reuben Hersh
Cumulative Review, Chapters 12-14
Appendices
A: Theorems by Chapter
B: Selected Proofs
C: Significant Digits
D: Short Table of Integrals
E: Answers to Selected Problems
Chapter 1
Chapter 2
Chapter 3
Chapter 4
Chapter 5
Chapter 6
Cumulative Review, Chapters 1-6
Chapter 7
Chapter 8
Chapter 9
Chapter 10
Chapter 11
Cumulative Review, Chapters 7-11
Chapter 12
Chapter 13
Chapter 14
Cumulative Review, Chapters 12-14
F: Credits
Index
Back Cover