Calculus of A Single Variable Book

Contents

Note: Each chapter contains Review Exercises and Problem Solving.

• P. Preparation for Calculus
  P.1 Graphs and Models
  P.2 Linear Models and Rates of Change
  P.3 Functions and Their Graphs
  P.4 Fitting Models to Data

  1. Limits and Their Properties

  1.1 A Preview of Calculus

  1.2 Finding Limits Graphically and Numerically

  1.3 Evaluating Limits Analytically

  1.4 Continuity and One-Sided Limits

  1.5 Infinite Limits

  Section Project: Graphs and Limits of Trigonometric Functions

  2. Differentiation

  2.1 The Derivative and the Tangent Line Problem

  2.2 Basic Differentiation Rules and Rates of Change

  2.3 Product and Quotient Rules and Higher-Order Derivatives

  2.4 The Chain Rule

  2.5 Implicit Differentiation

  Section Project: Optical Illusions

  2.6 Related Rates

  3. Applications of Differentiation

  3.1 Extrema on an Interval

  3.2 Rolle's Theorem and the Mean Value Theorem

  3.3 Increasing and Decreasing Functions and the First Derivative Test

  Section Project: Rainbows

  3.4 Concavity and the Second Derivative Test

  3.5 Limits at Infinity

  3.6 A Summary of Curve Sketching

  3.7 Optimization Problems

  Section Project: Connecticut River

  3.8 Newton's Method

  3.9 Differentials

  4. Integration

  4.1 Antiderivatives and Indefinite Integration

  4.2 Area

  4.3 Riemann Sums and Definite Integrals

  4.4 The Fundamental Theorem of Calculus

  Section Project: Demonstrating the Fundamental Theorem

  4.5 Integration by Substitution

  4.6 Numerical Integration

  5. Logarithmic, Exponential, and Other Transcendental Functions

  5.1 The Natural Logarithmic Function: Differentiation

  5.2 The Natural Logarithmic Function: Integration

  5.3 Inverse Functions

  5.4 Exponential Functions: Differentiation and Integration

  5.5 Bases Other Than e and Applications

  Section Project: Using Graphing Utilities to Estimate Slope

  5.6 Inverse Trigonometric Functions: Differentiation

  5.7 Inverse Trigonometric Functions: Integration

  5.8 Hyperbolic Functions

  Section Project: St. Louis Arch

  6. Differential Equations

  6.1 Slope Fields and Euler's Method

  6.2 Differential Equations: Growth and Decay

  6.3 Separation of Variables and the Logistic Equation

  6.4 First-Order Linear Differential Equations

  Section Project: Weight Loss

  7. Applications of Integration

  7.1 Area of a Region Between Two Curves

  7.2 Volume: The Disk Method

  7.3 Volume: The Shell Method

  Section Project: Saturn

  7.4 Arc Length and Surfaces of Revolution

  7.5 Work

  Section Project: Tidal Energy

  7.6 Moments, Centers of Mass, and Centroids

  7.7 Fluid Pressure and Fluid Force

  8. Integration Techniques, L'Hôpital's Rule, and Improper Integrals

  8.1 Basic Integration Rules

  8.2 Integration by Parts

  8.3 Trigonometric Integrals

  Section Project: Power Lines

  8.4 Trigonometric Substitution

  8.5 Partial Fractions

  8.6 Integration by Tables and Other Integration Techniques

  8.7 Indeterminate Forms and L'Hôpital's Rule

  8.8 Improper Integrals

  9. Infinite Series

  9.1 Sequences

  9.2 Series and Convergence

  Section Project: Cantor's Disappearing Table

  9.3 The Integral Test and p-Series

  Section Project: The Harmonic Series

  9.4 Comparisons of Series

  Section Project: Solera Method

  9.5 Alternating Series

  9.6 The Ratio and Root Tests

  9.7 Taylor Polynomials and Approximations

  9.8 Power Series

  9.9 Representation of Functions by Power Series

  9.10 Taylor and Maclaurin Series

  10. Conics, Parametric Equations, and Polar Coordinates

  10.1 Conics and Calculus

  10.2 Plane Curves and Parametric Equations

  Section Project: Cycloids

  10.3 Parametric Equations and Calculus

  10.4 Polar Coordinates and Polar Graphs

  Section Project: Anamorphic Art

  10.5 Area and Arc Length in Polar Coordinates

  10.6 Polar Equations of Conics and Kepler's Laws

  Appendices

  A. Proofs of Selected Theorems

  B. Integration Tables

  C. Additional Topics in Differential Equations (Web and CD-ROM only)

  D. Precalculus Review (Web and CD-ROM only)

  E. Rotation and the General Second-Degree Equation (Web and CD-ROM only)

  F. Complex Numbers (Web and CD-ROM only)

  G. Business and Economic Applications (Web only)