Calculus

Front Cover
Title Page
Copyright Page
Contents
Preface
A Guide to Using this Text
1 Functions and Graphs
1.1 Preliminaries
1.2 Lines in the Plane
1.3 Functions and Graphs
1.4 Inverse Functions; Inverse Trigonometric Functions
Chapter 1 Review
Guest Essay: Calculus Was Inevitable, John Troutman
Mathematical Essays
2 Limits and Continuity
2.1 The Limit of a Function
2.2 Algebraic Computation of Limits
2.3 Continuity
2.4 Exponential and Logarithmic Functions
Chapter 2 Review
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: Modeling 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 Using Derivatives to Sketch the Graph of a Function
4.4 Curve Sketching with Asymptotes: Limits Involving Infinity
4.5 l'Hôpital's Rule
4.6 Optimization in the Physical Sciences and Engineering
4.7 Optimization in Business, Economics, and the Life Sciences
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
5.8 Numerical Integration: The Trapezoidal Rule and Simpson's Rule
5.9 An Alternative Approach: The Logarithm as an Integral
Chapter 5 Review
Guest Essay: Kinematics of Jogging, Ralph Boas
Mathematical Essays
Chapters 1-5 Cumulative Review
6 Additional Applications of the Integral
6.1 Area Between Two Curves
6.2 Volume
6.3 Polar Forms and Area
6.4 Arc Length and Surface Area
6.5 Physical Applications: Work, Liquid Force, and Centroids
6.6 Applications to Business, Economics, and Life Sciences
Chapter 6 Review
Group Research Project: "Houdini's Escape"
7 Methods of Integration
7.1 Review of Substitution and Integration by Table
7.2 Integration by Parts
7.3 Trigonometric Methods
7.4 Method of Partial Fractions
7.5 Summary of Integration Techniques
7.6 First-Order Differential Equations
7.7 Improper Integrals
7.8 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
Chapters 6-8 Cumulative Review
9 Vectors in the Plane and in Space
9.1 Vectors in R^2
9.2 Coordinates and Vectors in R^3
9.3 The Dot Product
9.4 The Cross Product
9.5 Parametric Representation of Curves; Lines in R^3
9.6 Planes in R^3
9.7 Quadric Surfaces
Chapter 9 Review
Group Research Project: Star Trek
10 Vector-Valued Functions
10.1 Introduction to Vector Functions
10.2 Differentiation and Integration of Vector Functions
10.3 Modeling Ballistics and Planetary Motion
10.4 Unit Tangent and Principal Unit Normal Vectors; Curvature
10.5 Tangential and Normal Components of Acceleration
Chapter 10 Review
Guest Essay: The Stimulation of Science, Howard Eves
Mathematical Essays
Chapters 1-10 Cumulative Review
11 Partial Differentiation
11.1 Functions of Several Variables
11.2 Limits and Continuity
11.3 Partial Derivatives
11.4 Tangent Planes, Approximations, and Differentiability
11.5 Chain Rules
11.6 Directional Derivatives and the Gradient
11.7 Extrema of Functions of Two Variables
11.8 Lagrange Multipliers
Chapter 11 Review
Group Research Project: Desertification
12 Multiple Integration
12.1 Double Integration over Rectangular Regions
12.2 Double Integration over Nonrectangular Regions
12.3 Double Integrals in Polar Coordinates
12.4 Surface Area
12.5 Triple Integrals
12.6 Mass, Moments, and Probability Density Functions
12.7 Cylindrical and Spherical Coordinates
12.8 Jacobians: Change of Variables
Chapter 12 Review
Group Research Project: Space-Capsule Design
13 Vector Analysis
13.1 Properties of a Vector Field: Divergence and Curl
13.2 Line Integrals
13.3 The Fundamental Theorem and Path Independence
13.4 Green's Theorem
13.5 Surface Integrals
13.6 Stokes' Theorem
13.7 The Divergence Theorem
Chapter 13 Review
Guest Essay: Continuous vs. Discrete Mathematics, William F. Lucas
Mathematical Essays
Chapters 11-13 Cumulative Review
14 Introduction to Differential Equations
14.1 First-Order Differential Equations
14.2 Second-Order Homogeneous Linear Differential Equations
14.3 Second-Order Nonhomogeneous Linear Differential Equations
Chapter 14 Review
Group Research Project: Save the Perch Project
Appendices
A: Introduction to the Theory of Limits
B: Selected Proofs
C: Significant Digits
D: Short Table of Integrals
E: Trigonometric Formulas
F: Answers to Selected Problems
Chapter 1: Functions and Graphs
Chapter 2: Limits and Continuity
Chapter 3: Differentiation
Chapter 4: Additional Applications of the Derivative
Chapter 5: Integration
Chapter 6: Additional Applications of the Integral
Chapter 7: Methods of Integration
Chapter 8: Infinite Series
Chapter 9: Vectors in the Plane and in Space
Chapter 10: Vector-Valued Functions
Chapter 11: Partial Differentiation
Chapter 12: Multiple Integration
Chapter 13: Vector Analysis
Chapter 14: Introduction to Differential Equations
G: Credits
Index
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