Table of Contents

1 Functions
 A Definitions
 B Special Functions
 C Polynomial and Other Rational Functions
 D Trigonometric Functions
 E Exponential and Logarithmic Functions (BC Only)
 F Parametrically Defined Functions (BC Only)
 G Polar Functions

2 Limits and Continuity
 A Definitions and Examples
 B Asymptotes
 C Theorems on Limits
 D Limit of a Quotient of Polynomials
 E Other Basic Limits
 F Continuity

3 Differentiation
 A Definition of Derivative

B Formulas
 Power Rule of Derivatives
 Differentiation of Trigonometric Functions
 Differentiation of Reciprocal Trigonometric Functions
 Differentiation of Exponential Functions
 Differentiation of Logarithmic Functions
 Derivatives of Inverse Trigonometric Functions
 Combining the Product, Quotient, and Chain Rules
 Second and HigherOrder Derivatives
 C The Chain Rule; the Derivative of a Composite Function
 D Differentiability and Continuity

E Estimating a Derivative
 E1 Numerically
 E2 Graphically (BC Only)
 F Derivatives of Parametrically Defined Functions
 G Implicit Differentiation
 H Derivative of the Inverse of a Function
 I The Mean Value Theorem
 J Indeterminate Forms and L’Hôpital’s Rule
 K Recognizing a Given Limit as a Derivative

4 Applications of Differential Calculus
 A Slope; Critical Points
 B Tangents to a Curve

C Increasing and Decreasing Functions
 Case I Functions with Continuous Derivatives
 Case II Functions Whose Derivatives Have Discontinuities
 D Maximum, Minimum, Concavity, and Inflection Points: Definitions

E Maximum, Minimum, and Inflection Points: Curve Sketching
 Case I Functions That Are Everywhere Differentiable
 Case II Functions Whose Derivatives May Not Exist Everywhere

F Global Maximum or Minimum
 Case I Differentiable Functions
 Case II Functions That Are Not Everywhere Differentiable
 G Further Aids in Sketching
 H Optimization: Problems Involving Maxima and Minima
 I Relating a Function and Its Derivatives Graphically
 J Motion along a Line (BC Only)
 K Motion along a Curve: Velocity and Acceleration Vectors
 L TangentLine Approximations
 M Related Rates (BC Only)
 N Slope of a Polar Curve

5 Antidifferentiation
 A Antiderivatives

B Basic Formulas (BC Only)
 Indefinite Integrals: The Power Rule
 Indefinite Integrals: Trigonometric Functions
 Indefinite Integrals: Exponential and Reciprocal Functions
 Integration by Substitution: Indefinite Integrals
 Integration by Substitution: Definite Integrals
 Integrals Resulting in Logarithmic Functions
 Integrals Resulting in Inverse Trigonometric Functions
 C Integration by Partial Fractions (BC Only)
 D Integration by Parts
 E Applications of Antiderivatives; Differential Equations

6 Definite Integrals
 A Fundamental Theorem of Calculus (FTC); Evaluation of Definite Integral
 B Properties of Definite Integrals
 C Definition of Definite Integral as the Limit of a Riemann Sum
 D The Fundamental Theorem Again

E Approximations of the Definite Integral; Riemann Sums
 E1 Using Rectangles
 E2 Using Trapezoids
 E3 Comparing Approximating Sums
 F Graphing a Function from Its Derivative; Another Look
 G Interpreting ln 𝑥 as an Area
 H Average Value

7 Applications of Integration to Geometry

A Area
 A1 Area between Curves
 A2 Using Symmetry

B Volume
 B1 Solids with Known Cross Sections
 B2 Solids of Revolution (BC Only)
 C Arc Length (BC Only)
 D Improper Integrals

A Area

8 Further Applications of Integration
 A Motion along a Straight Line (BC Only)
 B Motion along a Plane Curve
 C Other Applications of Riemann Sums
 D FTC: Definite Integral of a Rate Is Net Change

9 Differential Equations
 A Basic Definitions
 B Slope Fields (BC Only)
 C Euler’s Method
 D Solving FirstOrder Differential Equations Analytically

E Exponential Growth and Decay
 Case I Exponential Growth
 Case II Restricted Growth (BC Only)
 Case III Logistic Growth

10 Sequences and Series
 A Sequences of Real Numbers (BC Only)

B Infinite Series
 B1 Definitions
 B2 Theorems about Convergence or Divergence of Infinite Series
 B3 Tests for Convergence of Infinite Series
 B4 Tests for Convergence of Nonnegative Series
 B5 Alternating Series and Absolute Convergence (BC Only)

C Power Series
 C1 Definitions; Convergence
 C2 Functions Defined by Power Series
 C3 Finding a Power Series for a Function: Taylor and Maclaurin Series
 C4 Approximating Functions with Taylor and Maclaurin Polynomials
 C5 Taylor’s Formula with Remainder; Lagrange Error Bound
 C6 Computations with Power Series (BC Only)
 C7 Power Series over Complex Numbers