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AP Calculus AB

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AP Calculus AB covers limits and continuity, differentiation and its applications, integration and accumulation of change, differential equations, and applications of integration including areas and volumes. It is roughly one semester of university calculus.

The exam is about meaning as much as method. You will be asked what a derivative means in context — with units — and to justify a maximum or a point of inflection by the sign of a derivative, not by inspection of a graph.

The Fundamental Theorem of Calculus is the spine of the course. Understanding it as "the integral of a rate gives the total change" is what makes accumulation questions readable.

Notes take each unit in order, with the theorems stated the way the exam expects them to be cited. Past FRQs with scoring guidelines are in the library — AB marks often go to the justification in words, so the worked solutions write the reasoning out instead of stopping at the answer.

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Past Papers
  • 1

    Limits and Continuity

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    • 1.1 Introducing Calculus: Can Change Occur at an Instant?
    • 1.2 Defining Limits and Using Limit Notation
    • 1.3 Estimating Limit Values from Graphs
    • 1.4 Estimating Limit Values from Tables
    • 1.5 Determining Limits Using Algebraic Properties of Limits
    • 1.6 Determining Limits Using Algebraic Manipulation
    • 1.7 Selecting Procedures for Determining Limits
    • 1.8 Determining Limits Using the Squeeze Theorem
    • 1.9 Connecting Multiple Representations of Limits
    • 1.10 Exploring Types of Discontinuities
    • 1.11 Defining Continuity at a Point
    • 1.12 Confirming Continuity over an Interval
    • 1.13 Removing Discontinuities
    • 1.14 Connecting Infinite Limits and Vertical Asymptotes
    • 1.15 Connecting Limits at Infinity and Horizontal Asymptotes
    • 1.16 Working with the Intermediate Value Theorem (IVT)
    • 1.16 Exam tips
  • 2

    Differentiation: Definition and Fundamental Properties

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    • 2.1 Average and Instantaneous Rates of Change
    • 2.2 Defining the Derivative and Its Notation
    • 2.3 Estimating a Derivative at a Point
    • 2.4 When Does a Derivative Exist?
    • 2.5 The Power Rule
    • 2.6 Constant, Sum, Difference, and Constant Multiple Rules
    • 2.7 Derivatives of sin x, cos x, e^x, and ln x
    • 2.8 The Product Rule
    • 2.9 The Quotient Rule
    • 2.10 Derivatives of tan x, cot x, sec x, and csc x
    • 2.10 Exam tips
  • 3

    Differentiation: Composite, Implicit, and Inverse Functions

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    • 3.1 The Chain Rule
    • 3.2 Implicit Differentiation
    • 3.3 Differentiating Inverse Functions
    • 3.4 Differentiating Inverse Trigonometric Functions
    • 3.5 Selecting Procedures for Calculating Derivatives
    • 3.6 Calculating Higher-Order Derivatives
    • 3.6 Exam tips
  • 4

    Contextual Applications of Differentiation

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    • 4.1 Interpreting the Derivative in Context
    • 4.2 Straight-Line Motion: Position, Velocity, and Acceleration
    • 4.3 Rates of Change Beyond Motion
    • 4.4 Introduction to Related Rates
    • 4.5 Solving Related Rates Problems
    • 4.6 Local Linearity and Linearization
    • 4.7 L'Hospital's Rule for Indeterminate Forms
    • 4.7 Exam tips
  • 5

    Analytical Applications of Differentiation

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    • 5.1 The Mean Value Theorem
    • 5.2 Extreme Values and Critical Points
    • 5.3 Increasing and Decreasing Intervals
    • 5.4 The First Derivative Test
    • 5.5 The Candidates Test for Absolute Extrema
    • 5.6 Concavity and Points of Inflection
    • 5.7 The Second Derivative Test
    • 5.8 Sketching a Function and Its Derivatives
    • 5.9 Connecting f, f-prime, and f-double-prime
    • 5.10 Introduction to Optimization
    • 5.11 Solving Optimization Problems
    • 5.12 Behaviors of Implicit Relations
    • 5.12 Exam tips
  • 6

    Integration and Accumulation of Change

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    • 6.1 Exploring Accumulations of Change
    • 6.2 Approximating Areas with Riemann Sums
    • 6.3 Riemann Sums and Integral Notation
    • 6.4 The Fundamental Theorem of Calculus and Accumulation Functions
    • 6.5 Behavior of Accumulation Functions
    • 6.6 Properties of Definite Integrals
    • 6.7 The Fundamental Theorem and Evaluating Integrals
    • 6.8 Antiderivatives and Indefinite Integrals
    • 6.9 Integrating Using Substitution
    • 6.10 Long Division and Completing the Square
    • 6.14 Selecting Techniques for Antidifferentiation
    • 6.14 Exam tips
  • 7

    Differential Equations

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    • 7.1 Modeling Situations with Differential Equations
    • 7.2 Verifying Solutions for Differential Equations
    • 7.3 Sketching Slope Fields
    • 7.4 Reasoning Using Slope Fields
    • 7.6 Finding General Solutions Using Separation of Variables
    • 7.7 Finding Particular Solutions Using Initial Conditions and Separation of Variables
    • 7.8 Exponential Models with Differential Equations
    • 7.8 Exam tips
  • 8

    Applications of Integration

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    • 8.1 Average Value of a Function
    • 8.2 Position, Velocity, and Acceleration Using Integrals
    • 8.3 Accumulation Functions and Definite Integrals in Applied Contexts
    • 8.4 Finding the Area Between Curves (Functions of x)
    • 8.5 Finding the Area Between Curves (Functions of y)
    • 8.6 Area Between Curves With More Than Two Intersections
    • 8.7 Volumes with Cross Sections: Squares and Rectangles
    • 8.8 Volumes with Cross Sections: Triangles and Semicircles
    • 8.9 Volume with the Disc Method (About the x- or y-Axis)
    • 8.10 Volume with the Disc Method (About Other Axes)
    • 8.11 8.12 Volume with the Washer Method
    • 8.11 8.12 Exam tips

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