AP Calculus: AB and BC

Limits and Continuity

Differentiation: Definition and Fundamental Properties

Differentiation: Composite, Implicit, and Inverse Functions

Contextual Applications of Differentiation

Analytical Applications of Differentiation

Integration and Accumulation of Change

Differential Equations

Applications of Integration

Parametric Equations, Polar Coordinates, and Vector-Valued Functions (BC ONLY) 

Infinite Sequences and Series (BC ONLY)

Introducing Calculus: Can Change Occur at an Instant?

Defining Limits and Using Limit Notation

Estimating Limit Values from Graphs

Estimating Limit Values from Tables

Determining Limits Using Algebraic Properties of Limits

Determining Limits Using Algebraic Manipulation

Selecting Procedures for Determining Limits

Determining Limits Using the Squeeze Theorem

Connecting Multiple Representations of Limits

Exploring Types of Discontinuities

Defining Continuity at a Point

Confirming Continuity over an Interval

Removing Discontinuities

Connecting Infinite Limits and Vertical Asymptotes

Connecting Limits at Infinity and Horizontal Asymptotes

Working with the Intermediate Value Theorem (IVT)

Defining Average and Instantaneous Rates of Change at a Point

Defining the Derivative of a Function and Using Derivative Notation

Estimating Derivatives of a Function at a Point

Connecting Differentiability and Continuity: Determining When Derivatives Do and Do Not Exist

Applying the Power Rule

Derivative Rules: Constant, Sum, Difference, and Constant Multiple

Derivatives of cos x, sin x, ex LIM , and ln x

The Product Rule

The Quotient Rule

Finding the Derivatives of Tangent, Cotangent, Secant, and/or Cosecant Functions

The Chain Rule

Implicit Differentiation

Differentiating Inverse Functions

Differentiating Inverse Trigonometric Functions

Selecting Procedures for Calculating Derivatives

Calculating Higher Order Derivatives

Interpreting the Meaning of the Derivative in Context

Straight-Line Motion: Connecting Position, Velocity, and Acceleration

Rates of Change in Applied Contexts Other Than Motion

Introduction to Related Rates

Solving Related Rates Problems

Approximating Values of a Function Using Local Linearity and Linearization

Using L’Hospital’s Rule for Determining Limits of Indeterminate Forms

Using the Mean Value Theorem

Extreme Value Theorem, Global Versus Local Extrema, and Critical Points

Determining Intervals on Which a Function Is Increasing or Decreasing

Using the First Derivative Test to Determine Relative (Local) Extrema

Using the Candidates Test to Determine Absolute (Global) Extrema

Determining Concavity of Functions over Their Domains

Using the Second Derivative Test to Determine Extrema

Sketching Graphs of Functions and Their Derivatives

Connecting a Function, Its First Derivative, and Its Second Derivative

Introduction to Optimization Problems

Solving Optimization Problems

Exploring Behaviors of Implicit Relations

Exploring Accumulations of Change

Approximating Areas with Riemann Sums

Riemann Sums, Summation Notation, and Definite Integral Notation

The Fundamental Theorem of Calculus and Accumulation Functions

Interpreting the Behavior of Accumulation Functions Involving Area

Applying Properties of Definite Integrals

The Fundamental Theorem of Calculus and Definite Integrals

Finding Antiderivatives and Indefinite Integrals: Basic Rules and Notation

Integrating Using Substitution

Integrating Functions Using Long Division and Completing the Square

Integrating Using Integration by Parts BC ONLY

Using Linear Partial Fractions BC ONLY

Evaluating Improper Integrals BC ONLY

Selecting Techniques for Antidifferentiation

Modeling Situations with Differential Equations

Verifying Solutions for Differential Equations

Sketching Slope Fields

Reasoning Using Slope Fields

Approximating Solutions Using Euler’s Method BC ONLY

Finding General Solutions Using Separation of Variables

Finding Particular Solutions Using Initial Conditions and Separation of Variables

Exponential Models with Differential Equations

Logistic Models with Differential Equations BC ONLY

Finding the Average Value of a Function on an Interval

Connecting Position, Velocity, and Acceleration of Functions Using Integrals

Using Accumulation Functions and Definite Integrals in Applied Contexts

Finding the Area Between Curves Expressed as Functions of x

Finding the Area Between Curves Expressed as Functions of y

Finding the Area Between Curves That Intersect at More Than Two Points

Volumes with Cross Sections: Squares and Rectangles

Volumes with Cross Sections: Triangles and Semicircles

Volume with Disc Method: Revolving Around the x- or y-Axis

Volume with Disc Method: Revolving Around Other Axes

Volume with Washer Method: Revolving Around the x- or y-Axis

Volume with Washer Method: Revolving Around Other Axes

The Arc Length of a Smooth, Planar Curve and Distance Traveled BC ONLY

Defining and Differentiating Parametric Equations

Second Derivatives of Parametric Equations

Finding Arc Lengths of Curves Given by Parametric Equations

Defining and Differentiating Vector Valued Functions

Integrating Vector Valued Functions

Solving Motion Problems Using Parametric and Vector Valued Functions

Defining Polar Coordinates and Differentiating in Polar Form

Find the Area of a Polar Region or the Area Bounded by a Single Polar Curve

Finding the Area of the Region Bounded by Two Polar Curves 

Defining Convergent and Divergent Infinite Series

Working with Geometric Series

The nth Term Test for Divergence

Integral Test for Convergence

Harmonic Series and p-Series

Comparison Tests for Convergence

Alternating Series Test for Convergence

Ratio Test for Convergence

Determining Absolute or Conditional Convergence

Alternating Series Error Bound

Finding Taylor Polynomial Approximations of Functions

Lagrange Error Bound

Radius and Interval of Convergence of Power Series

Finding Taylor or Maclaurin Series for a Function

Representing Functions as Power Series