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
	Untitled
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
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