Table of Contents

1 • Functions and Limits
 Domain and Range of a Rational Function
 Even and Odd Functions
 Graphs of Piecewise Functions
 Relations and Functions
 Linear Relationships
 Identifying Polynomials
 Graphs of Trigonometric Functions
 Function Transformations: Translations
 Composite Functions
 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
 Formal Definition of a Limit
 Limits in a RealWorld Context
 Continuity of Functions
 Intermediate Value Theorem
 2 • Derivatives

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
 Concavity and Points of Inflection
 Second Derivative Test for Local Extrema
 Limits at Infinity and Unbounded Limits
 Horizontal and Vertical Asymptotes of a Function
 Oblique Asymptotes
 Graphing Using Derivatives
 Optimization Using Derivatives
 Indefinite Integrals: The Power Rule
 Antiderivatives

4 • 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
 The Net Change Theorem
 Integration by Substitution: Indefinite Integrals
 Integration by Substitution: Definite Integrals
 Properties of Definite Integrals
 5 • Applications of Integration

6 • Inverse Functions: Exponential, Logarithmic, Inverse Trigonometric Functions
 Inverse of a Function
 Graphs of Inverses of Functions
 Exponential Functions
 Graphs of Exponential Functions
 Differentiation of Logarithmic Functions
 Logarithmic Functions
 Graphs of Logarithmic Functions
 Indefinite Integrals: Exponential and Reciprocal Functions
 Differentiation of Exponential Functions
 Logarithmic Differentiation
 Integrals Resulting in Logarithmic Functions
 Euler’s Number as a Limit
 Exponential Growth and Decay Models
 Compound Interest
 Inverse Trigonometric Functions
 Derivatives of Inverse Trigonometric Functions
 Hyperbolic Functions
 Derivatives of Hyperbolic Functions
 Derivatives of Inverse Hyperbolic Functions
 Indeterminate Forms and L’Hôpital’s Rule
 L’Hôpital’s Rule

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 Applications of Integration
 9 • Differential Equations

10 • Parametric Equations and Polar Coordinates
 Parametric Equations and Curves in Two Dimensions
 Conversion between Parametric and Rectangular Equations
 Derivatives of Parametric Equations
 Area Enclosed by Parametric Curves
 Arc Length of Parametric Curves
 Second Derivatives of Parametric Equations
 Polar Coordinates
 Graphing Polar Curves
 Slope of a Polar Curve
 Conversion between Rectangular and Polar Equations
 Area Bounded by Polar Curves
 Arc Length of a Polar Curve
 Equation of a Parabola
 Equation of an Ellipse
 Equation of a Hyperbola
 Identifying Conic Sections
 Polar Equation of a Conic

11 • Infinite Sequences and Series
 Representing Sequences
 Finding the Terms of a Sequence Given Its General Term
 Recursive Formula of a Sequence
 Convergent and Divergent Sequences
 Monotone Convergence Theorem
 Partial Sums
 Sum of a Finite Geometric Sequence
 Infinite Geometric Series
 Operations on Series
 Sum of an Infinite Geometric Sequence
 The nth Term Divergence Test
 Integral Test for Series
 Comparison Test for Series
 Limit Comparison Test
 Alternating Series Test
 Remainder of an Alternating Series
 Conditional and Absolute Convergence
 Ratio Test
 Root Test
 Power Series and Radius of Convergence
 Representing Rational Functions Using Power Series
 Differentiating and Integrating Power Series
 Taylor Series
 Maclaurin Series
 Maclaurin and Taylor Series of Common Functions
 Taylor Polynomials Approximation to a Function

12 • Vectors and the Geometry of Space
 Points, Midpoints, and Distances in Space
 Equation of a Sphere
 Vector Operations in 2D
 Magnitude of a Vector in 3D
 Magnitude of a 2D Vector
 Adding and Subtracting Vectors in 2D
 Graphical Operations on Vectors
 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 2D
 Cross Product in 3D
 Scalar Triple Product
 Equation of a Straight Line: Vector Form
 Equation of a Straight Line in Space: Parametric Form
 Equation of a Straight Line in Space: Cartesian and Vector Forms
 Equation of a Plane: Vector, Scalar, and General Forms
 Equation of a Plane: Intercept and Parametric Forms
 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
 Partial Derivatives and the Fundamental Theorem of Calculus
 Second and HigherOrder Partial Derivatives
 Tangent Planes and Linear approximation
 The Chain Rule for Multivariate Functions
 Implicit Differentiation and Partial derivatives
 Directional Derivatives and Gradient
 Lagrange Multipliers
 15 • Multiple Integrals
 16 • Vector Calculus
 17 • SecondOrder Differential Equations
 Appendixes