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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