Table of Contents

P • Preparation for Calculus
 Graphing Linear Functions
 Graphing Quadratic Functions
 Symmetry of Graphs and Functions
 Intersection Points of Two Functions
 Slope of a Line through Two Points
 Slope of a Line from a Graph or a Table
 Equation of a Straight Line: Slope–Intercept Form
 Slope and Rate of Change
 Slopes of Parallel and Perpendicular Lines
 Equations of Parallel and Perpendicular Lines
 Relations and Functions
 Domain and Range of a Rational Function
 Radical Functions
 Graphs of Piecewise Functions
 Function Transformations: Dilation
 Function Transformations: Translations
 Combining Functions
 Composite Functions
 Even and Odd Functions
 Conversion between Radians and Degrees
 Trigonometric Ratios on the Unit Circle
 Exact Values of Trigonometric Ratios
 Simple Trigonometric Equations
 Amplitude and Period of Trigonometric Functions
 Transformation of Trigonometric Functions
 Graphs of Trigonometric Functions
 The Graphs of Reciprocal Trigonometric Functions

1 • Limits and Their Properties
 Limits from Tables and Graphs
 Formal Definition of a Limit
 Formal Definition of Infinite Limits and Limits at Infinity
 Limits by Direct Substitution
 Evaluating Limits Using Algebraic Techniques
 Limits of Trigonometric Functions
 Continuity at a Point
 Continuity of Functions
 OneSided Limits
 Existence of Limits
 Intermediate Value Theorem

2 • Differentiation
 Average and Instantaneous Rates of Change
 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
 The Product Rule
 The Quotient Rule
 Differentiation of Trigonometric Functions
 Second and HigherOrder Derivatives
 The Chain Rule
 Implicit Differentiation
 Related Rates

3 • 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
 Limits at Infinity and Unbounded Limits
 Horizontal and Vertical Asymptotes of a Function
 Graphing Using Derivatives
 Optimization: Applications on Extreme Values
 Linear Approximation

4 • Integration
 Antiderivatives
 Indefinite Integrals: The Power Rule
 Indefinite Integrals and Initial Value Problems
 Riemann Sums
 Riemann Sums and Sigma Notation
 Definite Integrals as Limits of Riemann Sums
 The Fundamental Theorem of Calculus: Evaluating Definite Integrals
 The Fundamental Theorem of Calculus: Functions Defined by Integrals
 The Net Change Theorem
 Average Value of a Function
 Integration by Substitution: Definite Integrals
 Properties of Definite Integrals

5 • Logarithmic, Exponential, and Other Transcendental Functions
 Differentiation of Logarithmic Functions
 Integrals Resulting in Logarithmic Functions
 Indefinite Integrals: Trigonometric Functions
 Trigonometric Integrals
 Inverse of a Function
 Graphs of Inverses of Functions
 Differentiation of Inverse Functions
 Differentiation of Exponential Functions
 Indefinite Integrals: Exponential and Reciprocal Functions
 Applications of Exponential Functions
 L’Hôpital’s Rule
 Indeterminate Forms and L’Hôpital’s Rule
 Inverse Trigonometric Functions
 Derivatives of Inverse Trigonometric Functions
 Integration by Trigonometric Substitutions
 Integrals Resulting in Inverse Trigonometric Functions
 Hyperbolic Functions
 Derivatives of Hyperbolic Functions
 Derivatives of Inverse Hyperbolic Functions
 6 • Differential Equations
 7 • Applications of Integration

8 • Integration Techniques and Improper Integrals
 Integration by Substitution: Indefinite Integrals
 Integration by Parts
 Integration by Partial Fractions with Linear Factors
 Integration by Partial Fractions of Improper Fractions
 Integration by Partial Fractions with Quadratic Factors
 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

9 • Infinite Series
 Representing Sequences
 Convergent and Divergent Sequences
 Recursive Formula of a Sequence
 Infinite Geometric Series
 Sum of an Infinite Geometric Sequence
 The nth Term Divergence Test
 Integral Test for Series
 Comparison Test for Series
 Alternating Series Test
 Conditional and Absolute Convergence
 Ratio Test
 Root Test
 Taylor Polynomials Approximation to a Function
 Lagrange Error Bound
 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
 Equation of a Parabola
 Equation of an Ellipse
 Equation of a Hyperbola
 Parametric Equations and Curves in Two Dimensions
 Conversion between Parametric and Rectangular Equations
 Second Derivatives of Parametric Equations
 Derivatives of Parametric Equations
 Area Enclosed by 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

11 • Vectors and the Geometry of Space
 Magnitude of a 2D Vector
 Scalars, Vectors, and Directed Line Segments
 Graphical Operations on Vectors
 Adding and Subtracting Vectors in 2D
 Points, Midpoints, and Distances in Space
 Equation of a Sphere
 Dot Product in 2D
 The Angle between Two Vectors in the Coordinate Plane
 Direction Angles and Direction Cosines
 Scalar Projection
 Cross Product in 3D
 Scalar Triple Product
 Equation of a Straight Line in Space: Cartesian and Vector Forms
 Equation of a Plane: Intercept and Parametric Forms
 Parallel and Perpendicular Vectors in Space
 Intersection of Planes
 The Perpendicular Distance between Points and Planes
 12 • VectorValued Functions
 13 • Functions of Several Variables
 14 • Multiple Integration
 15 • Vector Analysis
 16 • Additional Topics in Differential Equations (Online)*