Table of Contents

1 • Functions and Models
 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
 Properties of Limits
 Limits from Tables and Graphs
 Limits by Direct Substitution
 Evaluating Limits Using Algebraic Techniques
 OneSided Limits
 Existence of Limits
 The Squeeze Theorem
 Limits in a RealWorld Context
 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
 Interpreting Graphs of 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
 Hyperbolic Functions
 Derivatives of Hyperbolic Functions
 Derivatives of Inverse Hyperbolic Functions

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
 Oblique Asymptotes
 Graphing Using Derivatives
 Optimization Using Derivatives
 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
 Indefinite Integrals: Trigonometric Functions
 Indefinite Integrals: The Power Rule
 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 of Improper Fractions
 Integration by Partial Fractions with Quadratic Factors
 Numerical Integration: The Trapezoidal Rule
 Numerical Integration: Simpson’s Rule
 Numerical Integration: Riemann Sums
 Improper Integrals: Infinite Limits of Integration
 Improper Integrals: Discontinuous Integrands
 Comparison Test for Improper Integrals
 8 • Further Application 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
 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
 Representing Sequences
 Convergent and Divergent Sequences
 Recursive Formula of a Sequence
 Monotone Convergence Theorem
 Infinite Geometric Series
 Partial Sums
 Sum of an Infinite Geometric Sequence
 Harmonic and pSeries
 Operations on Series
 Integral Test for Series
 Comparison Test for Series
 Conditional and Absolute Convergence
 Ratio Test
 Root Test
 Maclaurin Series

12 • Vectors and the Geometry of Space
 Points, Midpoints, and Distances in Space
 Equation of a Sphere
 Magnitude of a 2D Vector
 Scalars, Vectors, and Directed Line Segments
 Graphical Operations on Vectors
 Adding and Subtracting Vectors in 2D
 Dot Product
 Dot Product in 3D
 Angle between Two Vectors in Space
 The Angle between Two Vectors in the Coordinate Plane
 Direction Angles and Direction Cosines
 Vector 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
 Distances between Points and Straight Lines or Planes
 Quadratic Surfaces in Three Dimensions
 13 • Vector Functions

14 • Partial Derivatives
 Limits of Multivariable Functions
 Partial Derivatives
 Second and HigherOrder Partial Derivatives
 Tangent Planes and Linear approximation
 The Chain Rule for Multivariate Functions
 Directional Derivatives and Gradient
 Gradient in Cylindrical and Spherical Coordinates
 Extrema of a Multivariable Function
 Lagrange Multipliers
 15 • Multiple Integrals
 16 • Vector Calculus
 17 • SecondOrder Differential Equations

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
 Conversion between Radians and Degrees
 Transformation of Trigonometric Functions
 Amplitude and Period of Trigonometric Functions
 Simplifying Trigonometric Expressions Using Trigonometric Identities
 Angle Sum and Difference Identities
 DoubleAngle and HalfAngle Identities
 Sigma Notation
 Formal Definition of a Limit
 Laws of Logarithms
 Simplifying Exponential Expressions
 Introduction to Complex Numbers
 Exponential Form of a Complex Number
 Polar Form of Complex Numbers
 Pure Imaginary Numbers
 Equating, Adding, and Subtracting Complex Numbers
 Multiplying Complex Numbers
 Complex Number Conjugates
 De Moivre’s Theorem
 Quadratic Equations with Complex Coefficients