An Introduction to Analytic Geometry and Calculus

Preface1. The Coordinate System—Fundamental Relations 1-1 Introduction 1-2 Directed Lines 1-3 Cartesian Coordinates 1-4 Projections of a Line Segment on Horizontal and Vertical Lines 1-5 Midpoint of a Line Segment 1-6 Distance between Two Points 1-7 Slope of a Line 1-8 Parallel and Perpendicular Lines 1-9 Angle Formed by Intersecting Lines2. The Straight Line 2-1 Equation and Locus 2-2 The Graph of an Equation 2-3 The Graph of a Linear Equation 2-4 The Point-Slope Equation 2-5 The Two-Point Equation 2-6 The Slope-Intercept Equation 2-7 The First-Degree Equation 2-8 Intersection of Lines3. Nonlinear Equations and Graphs 3-1 Introduction 3-2 Intercepts 3-3 Symmetry 3-4 Extent 3-5 Graphing Equations 3-6 The Circle 3-7 The Parabola4. Functions and Limits 4-1 Introduction 4-2 Function 4-3 Tangent to a Curve 4-4 Limit of a Sequence 4-5 Limit of a Function 4-6 Continuous Functions5. The Derivative 5-1 Introduction 5-2 The Derivative 5-3 Some Additional Interpretations of the Derivative 5-4 Some Basic Theorems 5-5 Derivatives of Products and Quotients 5-6 The Chain Rule 5-7 Functions Defined Implicitly and Their Derivatives 5-8 Higher Derivatives6. Some Applications 6-1 The Mean Value Theorem 6-2 Sign of the First Derivative 6-3 Sign of the Second Derivative 6-4 Maxima and Minima 6-5 Applications of Maxima and Minima 6-6 Related Rates 6-7 The Differential 6-8 Applications of the Differential7. The Integral 7-1 Introduction 7-2 The Summation Notation 7-3 An Area Problem 7-4 The Definite Integral 7-5 Properties and Definitions 7-6 The Indefinite Integral 7-7 Evaluation of Indefinite Integrals 7-8 Areas of Plane Figures 7-9 More General Plane Areas 7-10 A Product Theorem 7-11 Work 7-12 Applications of the Indefinite Integral8. The Trigonometric Functions 8-1 Formulas and Definitions 8-2 An Important Limit 8-3 Derivatives of Trigonometric Functions 8-4 Inverse Trigonometric Functions 8-5 Differentiation of the Inverse Trigonometric Functions 8-6 Some Integrals9. The Exponential and Logarithm Functions 9-1 The Exponential Function 9-2 The Logarithm Function 9-3 The Number e 9-4 Derivative of the Logarithm Function 9-5 Derivative of the Exponential Function 9-6 More Integrals 9-7 Logarithmic Differentiation 9-8 Exponential Growth10. Methods of Integration 10-1 Introduction 10-2 Integration by Parts 10-3 Integrals Involving Trigonometric Functions 10-4 Algebraic Substitution 10-5 Trigonometric Substitution 10-6 Partial Fractions 10-7 Integral Tables11. More Applications 11-1 Volumes of Solids 11-2 Volumes of Solids of Revolution—Disks and Washers 11-3 Volumes of Solids of Revolution—Hollow Cylinders 11-4 Arc Length 11-5 Area of Surfaces of Revolution 11-6 Fluid Pressure12. Parametric Equations—Polar Coordinates 12-1 Parametric Representation 12-2 Derivative of a Function Defined by Parametric Equations 12-3 Polar Coordinates 12-4 Sketching Polar Curves 12-5 Intersection of Polar Curves 12-6 Plane Areas 12-7 Arc Length 12-8 Tangent Lines to Curves in Polar Coordinates13. Functions of Several Variables 13-1 Fundamental Concepts 13-2 Three-Dimensional Geometry 13-3 Partial Differentiation 13-4 Fundamental Increment Formula 13-5 Total Differential 13-6 The Chain Rule 13-7 Higher Derivatives 13-8 Implicit Differentiation14. Infinite Series 14-1 Introduction 14-2 Fundamental Concepts 14-3 General Tests 14-4 The Geometric Series 14-5 The p-Series 14-6 The Comparison Test 14-7 The Ratio Test 14-8 Alternating SeriesAppendix A Table of IntegralsAppendix B Some Function ValuesAppendix C Logarithm TablesAnswers to Odd-Numbered ProblemsIndex