Table of Contents

1 Precalculus Review
 1.1 Real Numbers, Functions, and Graphs
 1.2 Linear and Quadratic Functions
 1.3 The Basic Classes of Functions
 1.4 Trigonometric Functions
 1.5 Inverse Functions
 1.6 Exponential and Logarithmic Functions
 1.7 Technology: Calculators and Computers

2 Limits
 2.1 Limits, Rates of Change, and Tangent Lines
 2.2 Limits: A Numerical and Graphical Approach
 2.3 Basic Limit Laws
 2.4 Limits and Continuity
 2.5 Evaluating Limits Algebraically
 2.6 Trigonometric Limits
 2.7 Limits at Infinity
 2.8 Intermediate Value Theorem
 2.9 Formal Definition of Limit

3 Differentiation
 3.1 Definition of Derivative
 3.2 The Derivative as a Function
 3.3 Product and Quotient Rules
 3.4 Rates of Change
 3.5 Higher Derivatives
 3.6 Trigonometric Functions
 3.7 The Chain Rule
 3.8 Implicit Differentiation
 3.9 Derivative of Exponential and Logarithmic Functions
 3.10 Related Rates

4 Applications of the Derivative
 4.1 Linear Approximation and Applications
 4.2 Extreme Values
 4.3 The Mean Value Theorem and Monotonicity
 4.4 The Shape of a Graph
 4.5 L’Hôpital’s Rule
 4.6 Graph Sketching and Asymptotes
 4.7 Applied Optimization
 4.8 Newton’s Method

5 The Integral
 5.1 Approximating and Computing Area
 5.2 The Definite Integral
 5.3 The Indefinite Integral
 5.4 The Fundamental Theorem of Calculus, Part I
 5.5 The Fundamental Theorem of Calculus, Part II
 5.6 Net Change as the Integral of a Rate of Change
 5.7 Substitution Method
 5.8 Further Transcendental Functions
 5.9 Exponential Growth and Decay

6 Applications of the Integral
 6.1 Area between Two Curves
 6.2 Setting up Integrals: Volume, Density, Average Value
 6.3 Volumes of Revolution
 6.4 The Method of Cylindrical Shells
 6.5 Work and Energy

7 Techniques of Integration
 7.1 Integration by Parts
 7.2 Trigonometric Integrals
 7.3 Trigonometric Substitution
 7.4 Integrals Involving Hyperbolic and Inverse Hyperbolic Functions
 7.5 The Method of Partial Fractions
 7.6 Strategies for Integration
 7.7 Improper Integrals
 7.8 Probability and Integration
 7.9 Numerical Integration

8 Further Applications of the Integral and Taylor Polynomials
 8.1 Arc Length and Surface Area
 8.2 Fluid Pressure and Force
 8.3 Center of Mass

8.4 Taylor Polynomials

Appendices
 A The Language of Mathematics
 B Properties of Real Numbers
 C Induction and the Binomial Theorem
 D Additional Proofs