Table of Contents

1 • Functions and Models
 Relations and Functions
 Domain and Range of a Rational Function
 Even and Odd Functions
 Graphs of Piecewise Functions
 Representing Relations
 Function Transformations: Dilation
 Combining Functions
 Composite Functions
 Function Transformations: Translations
 Exponential Functions
 Graphs of Exponential Functions
 Applications of Exponential Functions
 Inverse of a Function
 Graphs of Inverses of Functions
 Logarithmic Functions
 Graphs of Logarithmic Functions
 Solving Exponential Equations Using Logarithms
 Inverse Trigonometric Functions

2 • Limits and Derivatives
 Average Speed
 Limits from Tables and Graphs
 Limits by Direct Substitution
 Evaluating Limits Using Algebraic Techniques
 OneSided Limits
 Existence of Limits
 Formal Definition of a Limit
 Continuity at a Point
 Continuity of Functions
 Intermediate Value Theorem
 Limits at Infinity and Unbounded Limits
 Horizontal and Vertical Asymptotes of a Function
 Average and Instantaneous Rates of Change
 Rate of Change and Derivatives
 Definition of the Derivative
 Second and HigherOrder Derivatives

3 • Differentiation Rules
 Power Rule of Derivatives
 Combining the Product, Quotient, and Chain Rules
 The Product Rule
 The Quotient Rule
 Limits of Trigonometric Functions
 Differentiation of Trigonometric Functions
 The Chain Rule
 Implicit Differentiation
 Derivatives of Inverse Trigonometric Functions
 Differentiation of Logarithmic Functions
 Logarithmic Differentiation
 Exponential Growth and Decay Models
 Related Rates
 Linear Approximation

4 • Applications of Differentiation
 Critical Points and Local Extrema of a Function
 Absolute Extrema
 The Mean Value Theorem
 Increasing and Decreasing Intervals of a Function
 Increasing and Decreasing Intervals of a Function Using Derivatives
 Second Derivative Test for Local Extrema
 Concavity and Points of Inflection
 L’Hôpital’s Rule
 Indeterminate Forms and L’Hôpital’s Rule
 Graphing Using Derivatives
 Optimization: Applications on Extreme Values
 Antiderivatives

5 • Integrals
 Velocity–Time Graphs
 Riemann Sums
 Riemann Sums and Sigma Notation
 Definite Integrals as Limits of Riemann Sums
 The Fundamental Theorem of Calculus: Functions Defined by Integrals
 The Fundamental Theorem of Calculus: Evaluating Definite Integrals
 The Net Change Theorem
 Integration by Substitution: Indefinite Integrals
 Integration by Substitution: Definite Integrals
 Properties of Definite Integrals
 6 • Applications of Integration

7 • Techniques of Integration
 Integration by Parts
 Trigonometric Integrals
 Integration by Trigonometric Substitutions
 Integration by Partial Fractions with Linear Factors
 Integration by Partial Fractions with Quadratic Factors
 Integration by Partial Fractions of Improper Fractions
 Numerical Integration: The Trapezoidal Rule
 Numerical Integration: Simpson’s Rule
 Improper Integrals: Infinite Limits of Integration
 Improper Integrals: Discontinuous Integrands
 Comparison Test for Improper Integrals
 8 • Further Applications of Integration
 9 • Differential Equations

10 • Parametric Equations and Polar Coordinates
 Parametric Equations and Curves in Two Dimensions
 Second Derivatives of Parametric Equations
 Derivatives of Parametric Equations
 Area Enclosed by Parametric Curves
 Arc Length of Parametric Curves
 Surface of Revolution of Parametric Curves
 Polar Coordinates
 Conversion between Rectangular and Polar Equations
 Slope of a Polar Curve
 Graphing Polar Curves
 Area Bounded by Polar Curves
 Arc Length of a Polar Curve
 Equation of a Parabola
 Equation of an Ellipse
 Equation of a Hyperbola
 11 • Infinite Sequences and Series

Appendixes
 Real Numbers
 Intervals
 Compound Linear Inequalities
 OneVariable Absolute Value Inequalities
 OneVariable Quadratic Inequalities
 Distance on the Coordinate Plane: Horizontal and Vertical
 Midpoint on the Coordinate Plane
 Slopes of Parallel and Perpendicular Lines
 Equation of a Straight Line: Slope–Intercept Form
 Graphing Linear Functions
 Equation of a Circle
 Transformation of Trigonometric Functions
 Conversion between Radians and Degrees
 Amplitude and Period of Trigonometric Functions
 Simplifying Trigonometric Expressions Using Trigonometric Identities
 Angle Sum and Difference Identities
 DoubleAngle and HalfAngle Identities
 Sigma Notation
 Laws of Logarithms
 Simplifying Exponential Expressions with Rational Exponents
 Introduction to Complex Numbers
 Polar Form of Complex Numbers
 Exponential Form of a Complex Number
 Pure Imaginary Numbers
 Equating, Adding, and Subtracting Complex Numbers
 Multiplying Complex Numbers
 Complex Number Conjugates
 De Moivre’s Theorem
 Quadratic Equations with Complex Coefficients