Table of Contents

1 • Functions
 Graphing Linear Functions
 Graphing Quadratic Functions
 Domain and Range of a Piecewise Function
 Piecewise Functions
 Graphs of Piecewise Functions
 Increasing and Decreasing Intervals of a Function
 Even and Odd Functions
 Composite Functions
 Radical Functions
 Conversion between Radians and Degrees
 Angles in Standard Position
 Evaluating Trigonometric Function Values with Angles 30, 45, and 60
 Simplifying Trigonometric Expressions Using Trigonometric Identities
 Law of Sines
 Law of Cosines
 Exponential Growth and Decay
 Inverse of a Function
 Laws of Logarithms

2 • Limits and Continuity
 Average and Instantaneous Rates of Change
 Rate of Change and Derivatives
 Limits from Tables and Graphs
 Limits by Direct Substitution
 Evaluating Limits Using Algebraic Techniques
 Formal Definition of Infinite Limits and Limits at Infinity
 OneSided Limits
 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 • Derivatives
 Definition of the Derivative
 Interpreting Graphs of Derivatives
 Power Rule of Derivatives
 Combining the Product, Quotient, and Chain Rules
 The Product Rule
 The Quotient Rule
 Differentiation of Exponential Functions
 Second and HigherOrder Derivatives
 Differentiation of Reciprocal Trigonometric Functions
 The Chain Rule
 Implicit Differentiation
 Differentiation of Inverse Functions
 Differentiation of Logarithmic Functions
 Logarithmic Differentiation
 Derivatives of Inverse Trigonometric Functions
 Related Rates
 Linear Approximation

4 • Applications of Derivatives
 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
 Indeterminate Forms and L’Hôpital’s Rule
 Optimization: Applications on Extreme Values
 Antiderivatives

5 • Integrals
 Riemann Sums
 Riemann Sums and Sigma Notation
 Sigma Notation
 Definite Integrals as Limits of Riemann Sums
 Properties of Definite Integrals
 The Fundamental Theorem of Calculus: Evaluating Definite Integrals
 Average Value of a Function
 The Fundamental Theorem of Calculus: Functions Defined by Integrals
 The Net Change Theorem
 Integration by Substitution: Indefinite Integrals
 Integration by Substitution: Definite Integrals
 6 • Applications of Definite Integrals
 7 • Integrals and Transcendental Functions

8 • Techniques of Integration
 Integration by Parts
 Indefinite Integrals: Trigonometric Functions
 Trigonometric Integrals
 Integration by Trigonometric Substitutions
 Reduction Formulae for Integration
 Integration by Partial Fractions with Linear Factors
 Integration by Partial Fractions with Quadratic Factors
 Integration by Partial Fractions of Improper Fractions
 Numerical Integration: Riemann Sums
 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 • FirstOrder Differential Equations

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

11 • Parametric Equations and Polar Coordinates
 Conversion between Parametric and Rectangular Equations
 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
 Graphing Polar Curves
 Conversion between Rectangular and Polar Equations
 Slope of a Polar Curve
 Area Bounded by Polar Curves
 Identifying Conic Sections
 Rotation of Conics

12 • Vectors and the Geometry of Space
 Points, Midpoints, and Distances in Space
 Equation of a Sphere
 Polar Form of a Vector
 Adding and Subtracting Vectors in 2D
 Vectors in Terms of Fundamental Unit Vectors
 Dot Product in 2D
 Dot Product in 3D
 Cross Product in 3D
 Vector Triple Product
 Equation of a Straight Line: Vector Form
 Equation of a Straight Line in Space: Cartesian and Vector Forms
 Equation of a Plane: Vector, Scalar, and General Forms
 13 • VectorValued Functions and Motion in Space
 14 • Partial Derivatives
 15 • Multiple Integrals
 16 • Integrals and Vector Fields
 17 • SecondOrder Differential Equations