AP Calculus ✏ In a Nutshell



1. Functions

Definitions
Special Functions
Polynomial and Other Rational Functions
Trigonometric Functions
Exponential and Logarithmic Functions
Parametrically Defined Functions (BC)
Polar Functions (BC)

2. Limits and Continuity

Definitions and Examples
Asymptotes
Theorems on Limits
Limit of a Quotient of Polynomials
Other Basic Limits
Continuity

3. Differentiation

Definition of Derivative
Formulas
The Chain Rule: The Derivative of a Composite Function
Differentiability and Continuity
Estimating a Derivative
Derivatives of Parametrically Defined Functions (BC)
Implicit Differentiation
Derivative of the Inverse of a Function
The Mean Value Theorem
Indeterminate Forms and L’Hospital’s Rule
Recognizing a Given Limit as a Derivative

4. Applications of Differential Calculus

Slope; Critical Points
Tangents to a Curve
Increasing and Decreasing Functions
Maximum, Minimum, Concavity, and Inflection Points
Maximum, Minimum, and Inflection Points: Curve Sketching
Global Maximum or Minimum
Further Aids in Sketching
Optimization
Relating a Function and Its Derivatives Graphically
Motion Along a Line
Motion Along a Curve: Velocity and Acceleration Vectors (BC)
Tangent-Line Approximations
Related Rates
Slope of a Polar Curve (BC)

5. Antidifferentiation

Antiderivatives
Basic Formulas
Integration by Partial Fractions (BC)
Integration by Parts (BC)
Applications of Antiderivatives; Differential Equations

6. Definite Integrals

Fundamental Theorem of Calculus (FTC)
Properties of Definite Integrals
Definition of Definite Integral as the Limit of a Riemann Sum
The Fundamental Theorem Again
Approximations of the Definite Integral; Riemann Sums
Graphing a Function from Its Derivative
Interpreting ln x as an Area
Average Value

7. Applications of Integration to Geometry

Area

  • Area Between Curves

  • Using Symmetry

  • Region Bounded by Polar Curve (BC)
    Volume

  • Solids with Known Cross Sections

  • Solids of Revolution
    Length of Curve (Arc Length) (BC)
    Improper Integrals (BC)

8. Further Applications of Integration

Motion Along a Straight Line
Motion Along a Plane Curve (BC)
Other Applications of Riemann Sums
FTC: Definite Integral of a Rate Is Net Change

9. Differential Equations

Basic Definitions
Slope Fields
Euler’s Method (BC)
Solving First-Order Differential Equations Analytically
Exponential Growth and Decay

  • Exponential Growth

  • Restricted Growth

  • Logistic Growth (BC)

10. Sequences and Series (BC)

Sequences of Real Numbers
Infinite Series

  • Definitions

  • Theorems About Convergence or Divergence

  • Tests for Convergence (General and Nonnegative)

  • Alternating Series and Absolute Convergence
    Power Series

  • Definitions and Convergence

  • Functions Defined by Power Series

  • Taylor and Maclaurin Series

  • Taylor’s Formula with Remainder; Lagrange Error Bound

  • Computations with Power Series

  • Power Series over Complex Numbers