Table of Contents

1 • Preparation for Calculus
 Graphing Linear Functions
 Slopes and Intercepts of Linear Functions
 Equation of a Straight Line: Slope–Intercept Form
 Equation of a Straight Line: Standard and Point–Slope Forms
 Slope and Rate of Change
 Slopes of Parallel and Perpendicular Lines
 Evaluating Functions
 Radical Functions
 Graphs of Piecewise Functions
 Composite Functions
 Even and Odd Functions
 Graphs of Inverses of Functions
 Inverse of a Function
 Solving Equations Using Inverse Trigonometric Functions
 Graphs of Exponential Functions
 Simplifying Natural Logarithmic Expressions
 Natural Logarithmic Equations
 Solving Exponential Equations Using Logarithms

2 • Limits and Their Properties
 Average and Instantaneous Rates of Change
 Limits from Tables and Graphs
 Formal Definition of Infinite Limits and Limits at Infinity
 Limits by Direct Substitution
 Limits of Trigonometric Functions
 Evaluating Limits Using Algebraic Techniques
 The Squeeze Theorem
 Limits in a RealWorld Context
 Continuity at a Point
 Classifying Discontinuities
 Continuity of Functions
 Intermediate Value Theorem
 Limits at Infinity and Unbounded Limits
 Horizontal and Vertical Asymptotes of a Function

3 • Differentiation
 Rate of Change and Derivatives
 Definition of the Derivative
 The Differentiability of a Function
 Power Rule of Derivatives
 Combining the Product, Quotient, and Chain Rules
 Differentiation of Trigonometric Functions
 Differentiation of Exponential Functions
 The Product Rule
 The Quotient Rule
 Second and HigherOrder Derivatives
 The Chain Rule
 Implicit Differentiation
 Derivatives of Inverse Trigonometric Functions
 Related Rates

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 Using Derivatives
 Concavity and Points of Inflection
 Second Derivative Test for Local Extrema
 Graphing Using Derivatives
 Optimization: Applications on Extreme Values

5 • Integration
 Antiderivatives
 Indefinite Integrals: The Power Rule
 Riemann Sums
 Riemann Sums and Sigma Notation
 Definite Integrals as Limits of Riemann Sums
 The Fundamental Theorem of Calculus: Functions Defined by Integrals
 Integration by Substitution: Indefinite Integrals
 Integration by Substitution: Definite Integrals
 Integration by Trigonometric Substitutions
 Numerical Integration: Riemann Sums
 Numerical Integration: The Trapezoidal Rule
 Numerical Integration: Simpson’s Rule
 Integrals Resulting in Inverse Trigonometric Functions
 Derivatives of Hyperbolic Functions
 Derivatives of Inverse Hyperbolic Functions
 Hyperbolic Functions
 6 • Differential Equations
 7 • Applications of Integration

8 • Integration Techniques, L’Hôpital’s Rule, and Improper Integrals
 Integration by Parts
 Indefinite Integrals: Trigonometric Functions
 Trigonometric Integrals
 Integration by Partial Fractions with Linear Factors
 Integration by Partial Fractions with Quadratic Factors
 Integration by Partial Fractions of Improper Fractions
 L’Hôpital’s Rule
 Improper Integrals: Infinite Limits of Integration
 Improper Integrals: Discontinuous Integrands
 Comparison Test for Improper Integrals

9 • Infinite Series
 Representing Sequences
 Convergent and Divergent Sequences
 Recursive Formula of a Sequence
 Monotone Convergence Theorem
 Infinite Geometric Series
 The nth Term Divergence Test
 Operations on Series
 Partial Sums
 Integral Test for Series
 Harmonic and pSeries
 Comparison Test for Series
 Limit Comparison Test
 Alternating Series Test
 Ratio Test
 Root Test
 Taylor Polynomials Approximation to a Function
 Power Series and Radius of Convergence
 Operations on Power Series
 Differentiating and Integrating Power Series
 Maclaurin Series
 Taylor Series
 10 • Conics, Parametric Equations, and Polar Coordinates

11 • Vectors and the Geometry of Space
 Scalar Projection
 Adding and Subtracting Vectors in 2D
 Magnitude of a 2D Vector
 Scalar Multiplication and Unit Vectors
 Points, Midpoints, and Distances in Space
 Equation of a Sphere
 Dot Product in 2D
 Dot Product in 3D
 Cross Product in 3D
 Vector Triple Product
 Scalar Triple Product
 The Perpendicular Distance between Points and Planes
 12 • VectorValued Functions
 13 • Functions of Several Variables
 14 • Multiple Integration
 15 • Vector Analysis
 Appendices