AP Calculus BC

Limits and Continuity

• Using Limits to Analyze Instantaneous Change
• Estimating Limit Values from Graphs and Tables
• Determining Limits Using Algebraic Properties and Manipulation
• Selecting Procedures for Determining Limits
• Squeeze Theorem and Representations of Limits
• Determining Continuity and Exploring Discontinuity
• Connecting Limits, Infinity, and Asymptotes
• The Intermediate Value Theorem (IVT)

Differentiation: Definition and Fundamental Properties

• Average and Instantaneous Rates of Change and the Derivative Definition
• Determining Differentiability and Estimating Derivatives
• Derivative Rules: Constant, Sum, Difference, Constant Multiple, and Power
• The Product Rule and the Quotient Rule
• Derivatives of Trigonometric Functions
• Derivatives of Exponential and Logarithmic Functions

Differentiation: Composite, Implicit, and Inverse Functions

• The Chain Rule
• Implicit Differentiation
• Differentiating Inverse Functions
• Differentiating Inverse Trigonometric Functions
• Selecting Procedures for Calculating Derivatives
• Calculating Higher-Order Derivatives

Contextual Applications of Differentiation

• Interpreting and Applying the Derivative in Motion
• Rates of Change in Applied Contexts Other Than Motion
• Related Rates
• Approximating Values of a Function Using Local Linearity and Linearization
• L’Hospital’s Rule

Analytical Applications of Differentiation

• Mean Value and Extreme Value Theorems
• Connecting Graphs of Functions and Their Derivatives
• Exploring Behaviors of Implicit Relations

Integration and Accumulation of Change

• Exploring Accumulations of Change
• Riemann Sums and the Definite Integral
• Accumulation Functions Involving Area and the Fundamental Theorem of Calculus
• Applying Properties of Definite Integrals
• Finding Antiderivatives and Indefinite Integrals
• Integrating Using Substitution
• Integrating Using Integration by Parts
• Integration Using Linear Partial Functions
• Evaluating Improper Integrals
• Integrating Functions Using Long Division and Completing the Square
• Selecting Techniques for Antidifferentiation

Differential Equations

• Solutions of Differential Equations
• Sketching and Reasoning Using Slope Fields
• Approximating Solutions Using Euler’s Method
• Finding Solutions Using Separation of Variables
• Exponential Models with Differential Equations
• Logistic Models with Differential Equations

Applications of Integration

• Average Value and Connecting Position, Velocity, and Acceleration Using Integrals
• Using Accumulation Functions and Definite Integrals in Applied Contexts
• Finding the Area Between Curves
• Finding the Area Between Curves That Intersect at More Than Two Points
• Volumes with Discs
• Volumes with Washers
• Volumes with Cross Sections
• The Arc Length of a Smooth, Planar Curve and Distance Traveled

Parametric, Polar, and Vector-Valued Equations

• Differentiating Parametric Equations and Finding Arc Length
• Differentiating and Integrating Vector-Valued Functions
• Defining Polar Coordinates and Differentiating in Polar Form
• Finding Area Bounded by Polar Curves

Infinite Sequences and Series

• Convergent and Divergent Infinite Series and Geometric Series
• Tests to Determine Convergence
• Alternating Series and Their Error Bound
• Taylor Polynomial Approximations of Functions and Evaluating Error
• Radius and Interval of Convergence of Power Series
• Finding Taylor or Maclaurin Series of a Function
• Representing Functions as a Power Series