TABLE OF CONTENTS About the Authors Preface 1 Functions, Graphs, and Models 1.1 Functions and Mathematical Modeling Investigation: Designing a Wading Pool 1.2 Graphs of Equations and Functions 1.3 Polynomials and Algebraic Functions 1.4 Transcendental Functions 1.5 Preview: What Is Calculus? REVIEW - Understanding: Concepts and Definitions Objectives: Methods and Techniques 2 Prelude to Calculus 2.1 Tangent Lines and Slope Predictors Investigation: Numerical Slope Investigations 2.2 The Limit Concept Investigation: Limits, Slopes, and Logarithms 2.3 More About Limits Investigation: Numerical Epsilon-Delta Limits 2.4 The Concept of Continuity REVIEW - Understanding: Concepts and Definitions Objectives: Methods and Techniques 3 The Derivative 3.1 The Derivative and Rates of Change 3.2 Basic Differentiation Rules 3.3 The Chain Rule 3.4 Derivatives of Algebraic Functions 3.5 Maxima and Minima of Functions on Closed Intervals Investigation: When Is Your Coffee Cup Stablest? 3.6 Applied Optimization Problems 3.7 Derivatives of Trigonometric Functions 3.8 Exponential and Logarithmic Functions Investigation: Discovering the Number e for Yourself 3.9 Implicit Differentiation and Related Rates Investigation: Constructing the Folium of Descartes 3.10 Successive Approximations and Newton's Method Investigation: How Deep Does a Floating Ball Sink? REVIEW - Understanding: Concepts, Definitions, and Formulas Objectives: Methods and Techniques 4 Additional Applications of the Derivative 4.1 Introduction 4.2 Increments, Differentials, and Linear Approximation 4.3 Increasing and Decreasing Functions and the Mean Value Theorem 4.4 The First Derivative Test and Applications Investigation: Constructing a Candy Box With Lid 4.5 Simple Curve Sketching 4.6 Higher Derivatives and Concavity 4.7 Curve Sketching and Asymptotes Investigation: Locating Special Points on Exotic Graphs 4.8 Indeterminate Forms and L'Hopital's Rule 4.9 More Indeterminate Forms REVIEW - Understanding: Concepts, Definitions, and Results Objectives: Methods and Techniques 5 The Integral 5.1 Introduction 5.2 Antiderivatives and Initial Value Problems 5.3 Elementary Area Computations 5.4 Riemann Sums and the Integral Investigation: Calculator/Computer Riemann Sums 5.5 Evaluation of Integrals 5.6 The Fundamental Theorem of Calculus 5.7 Integration by Substitution 5.8 Areas of Plane Regions 5.9 Numerical Integration Investigation: Trapezoidal and Simpson Approximations REVIEW - Understanding: Concepts, Definitions, and Results Objectives: Methods and Techniques 6 Applications of the Integral 6.1 Riemann Sum Approximations 6.2 Volumes by the Method of Cross Sections 6.3 Volumes by the Method of Cylindrical Shells Investigation: Design Your Own Ring! 6.4 Arc Length and Surface Area of Revolution 6.5 Force and Work 6.6 Centroids of Plane Regions and Curves 6.7 The Natural Logarithm as an Integral Investigation: Natural Functional Equations 6.8 Inverse Trigonometric Functions 6.9 Hyperbolic Functions REVIEW - Understanding: Concepts, Definitions, and Formulas Objectives: Methods and Techniques 7 Techniques of Integration 7.1 Introduction 7.2 Integral Tables and Simple Substitutions 7.3 Integration by Parts 7.4 Trigonometric Integrals 7.5 Rational Functions and Partial Fractions 7.6 Trigonometric Substitutions 7.7 Integrals Involving Quadratic Polynomials 7.8 Improper Integrals SUMMARY - Integration Strategies REVIEW - Understanding: Concepts and Techniques Objectives: Methods and Techniques 8 Differential Equations 8.1 Simple Equations and Models 8.2 Slope Fields and Euler's Method Investigation: Computer-Assisted Slope Fields and Euler's Method 8.3 Separable Equations and Applications 8.4 Linear Equations and Applications 8.5 Population Models Investigation: Predator-Prey Equations and Your Own Game Preserve 8.6 Linear Second-Order Equations 8.7 Mechanical Vibrations REVIEW - Understanding: Concepts, Definitions, and Methods Objectives: Methods and Techniques 9 Polar Coordinates and Parametric Curves 9.1 Analytic Geometry and the Conic Sections 9.2 Polar Coordinates 9.3 Area Computations in Polar Coordinates 9.4 Parametric Curves Investigation: Trochoids Galore 9.5 Integral Computations with Parametric Curves Investigation: Moon Orbits and Race Tracks 9.6 Conic Sections and Applications REVIEW - Understanding: Concepts, Definitions, and Formulas Objectives: Methods and Techniques 10 Infinite Series 10.1 Introduction 10.2 Infinite Sequences Investigation: Nested Radicals and Continued Fractions 10.3 Infinite Series and Convergence Investigation: Numerical Summation and Geometric Series 10.4 Taylor Series and Taylor Polynomials Investigation: Calculating Logarithms on a Deserted Island 10.5 The Integral Test Investigation: The Number p, Once and for All 10.6 Comparison Tests for Positive-Term Series 10.7 Alternating Series and Absolute Convergence 10.8 Power Series 10.9 Power Series Computations Investigation: Calculating Trigonometric Functions on a Deserted Island 10.10 Series Solutions of differential Equations REVIEW - Understanding: Concepts, and Results Objectives: Methods and Techniques 11 Vectors, Curves, and Surfaces in Space 11.1 Vectors in the Plane 11.2 Three-Dimensional Vectors 11.3 The Cross Product of Two Vectors 11.4 Lines and Planes in Space 11.5 Curves and Motion in Space Investigation: Does a Pitched Baseball Really Curve? 11.6 Curvature and Acceleration 11.7 Cylinders and Quadric Surfaces 11.8 Cylindrical and Spherical Coordinates REVIEW - Understanding: Concepts, Definitions, and Results Objectives: Methods and Techniques 12 Partial Differentiation 12.1 Introduction 12.2 Functions of Several Variables 12.3 Limits and Continuity 12.4 Partial Derivatives 12.5 Multivariable Optimization Problems 12.6 Increments and Linear Approximation 12.7 The Multivariable Chain Rule 12.8 Directional Derivatives and the Gradient Vector 12.9 Lagrange Multipliers and Constrained Optimization Investigation: Numerical Solution of Lagrange Multiplier Systems 12.10 Critical Points of Functions of Two Variables Investigation: Critical Point Investigations REVIEW - Understanding: Concepts, Definitions, and Results Objectives: Methods and Techniques 13 Multiple Integrals 13.1 Double Integrals Investigation: Midpoint Sums Approximating Double Integrals 13.2 Double Integrals over More General Regions 13.3 Area and Volume by Double Integration 13.4 Double Integrals in Polar Coordinates 13.5 Applications of Double Integrals Investigation: Optimal Design of Race Car Wheels 13.6 Triple Integrals Investigation: Archimedes' Floating Paraboloid 13.7 Integration in Cylindrical and Spherical Coordinates 13.8 Surface Area 13.9 Change of Variables in Multiple Integrals REVIEW - Understanding: Concepts, Definitions, and Results Objectives: Methods and Techniques 14 Vector Calculus 14.1 Vector Fields 14.2 Line Integrals 14.3 The Fundamental Theorem and Independence of Path 14.4 Green's Theorem 14.5 Surface Integrals Investigation: Surface Integrals and Rocket Nose Cones 14.6 The Divergence Theorem 14.7 Stokes' Theorem REVIEW - Understanding: Concepts, Definitions, and Results Objectives: Methods and Techniques Appendices A: Real Numbers and Inequalities B: The Coordinate Plane and Straight Lines C: Review of Trigonometry D: Proofs of the Limit Laws E: The Completeness of the Real Number System F: Existence of the Integral G: Approximations and Riemann Sums H: L'Hopital's Rule and Cauchy's Mean Value Theorem I: Proof of Taylor's Formula J: Conic Sections as Sections of a Cone K: Proof of the Linear Approximation Theorem L: Units of Measurement and Conversion Factors M: Formulas from Algebra, Geometry, and Trigonometry N: The Greek Alphabet Answers to Odd-Numbered Problems References for Further Study Index
