Calculus of a Single Variable

Chapter P: Preparation for Calculus
Graphs and Models
Linear Models and Rates of Change
Functions and Their Graphs
Fitting Models to Data
Review Exercises
Chapter 1: Limits and Their Properties
A Preview of Calculus
Finding Limits Graphically and Numerically
Evaluating Limits Analytically
Continuity and One-Sided Limits
Infinite Limits
Review Exercises
Chapter 2: Differentiation
The Derivative and the Tangent Line Problem
Basic Differentiation Rules and Rates of Change
The Product and Quotient Rules and Higher-Order Derivatives
The Chain Rule
Implicit Differentiation
Related Rates
Review Exercises
Chapter 3: Applications of Differentiation
Extrema on an Interval
Rolles Theorem and the Mean Value Theorem
Increasing and Decreasing Functions and the First Derivative Test
Concavity and the Second Derivative Test
Limits at Infinity
A Summary of Curve Sketching
Optimization Problems
Newton’s Method
Differentials
Business and Economics Applications
Review Exercises
Chapter 4: Integration
Antiderivatives and Indefinite Integration
Area
Riemann Sums and Definite Integrals
The Fundamental Theorem of Calculus
Integration by Substitution
Numerical Integration
Review Exercises
Chapter 5: Logarithmic, Exponential, and Other Transcendental Functions
The Natural Logarithmic Function and Differentiation
The Natural Logarithmic Function and Integration
Inverse Functions
Exponential Functions: Differentiation and Integration
Bases Other than e and Applications
Differential Equations: Growth and Decay
Differential Equations: Separation of Variables
Inverse Trigonometric Functions and Differentiation
Inverse Trigonometric Functions and Integration
Hyperbolic Functions
Review Exercises
Chapter 6: Applications of Integration
Area of a Region Between Two Curves
Volume: The Disc Method
Volume: The Shell Method
Arc Length and the Surfaces of Revolution
Work
Moments, Centers of Mass, and Centroids
Fluid Pressure and Fluid Force
Review Exercises
Chapter 7: Integration Techniques, Lhopitals Rule, and Improper Integrals
Basic Integration Rules
Integration by Parts
Trigonometric Integrals
Trigonometric Substitution
Partial Fractions
Integration by Tables and Other Integration Techniques
Indeterminate Forms and Lhopitals Rule
Improper Integrals
Review Exercises
Chapter 8: Infinite Series
Sequences
Series and Convergence
The Integral Test and p-Series
Comparisons of Series
Alternating Series
The Ratio and Root Tests
Taylor Polynomials and Approximations
Power Series
Representation of Functions by Power Series
Taylor and Maclaurin Series
Review Exercises
Chapter 9: Conics, Parametric Equations, and Polar Coordinates
Conics and Calculus
Plane Curves and Parametric Equations
Parametric Equations and Calculus
Polar Coordinates and Polar Graphs
Area and Arc Length in Polar Coordinates
Polar Equations of Conics and Keplers Laws
Review Exercises
Appendix A: Precalculus Review
Real Numbers and the Real Line
The Cartesian Plane
Review of Trigonometric Functions
Appendix B: Proofs of Selected Theorems
Appendix C: Basic Differentiation Rules for Elementary Functions
Appendix D: Integration Tables
Appendix E: Rotation and the General Second-Degree Equation
Appendix F: Complex Numbers
Answers to Odd-Numbered Exercises
Index