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Verified for the 2027 AP exam

AP Calculus AB Study Guide & Review

Connect limits, derivatives and integrals across formulas, graphs, tables and real contexts while building the justification, notation and calculator judgment required throughout AP Calculus AB.

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  1. 01Learn concepts
  2. 02Practise questions
  3. 03Review mistakes

Syllabus knowledge tree

Explore all eight AP Calculus AB units

Navigate 81 exact Topics across the eight official AB Units. Use current College Board MCQ weighting ranges to plan revision while keeping representation, process and justification practice active across every Unit.

8
syllabus groups
81
mapped topics

Preview the course map

Explore the course units and topics. Sign in to open the full mastery workspace and see your progress.

Unit 1: Limits and Continuity
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)
Unit 2: Differentiation: Definition and Fundamental Properties
2.1 Defining Average and Instantaneous Rates of Change at a Point
2.2 Defining the Derivative of a Function and Using Derivative Notation
2.3 Estimating Derivatives of a Function at a Point
2.4 Connecting Differentiability and Continuity: Determining When Derivatives Do and Do Not Exist
2.5 Applying the Power Rule
2.6 Derivative Rules: Constant, Sum, Difference, and Constant Multiple
2.7 Derivatives of cos x, sin x, eˣ, and ln x
2.8 The Product Rule
2.9 The Quotient Rule
2.10 Finding the Derivatives of Tangent, Cotangent, Secant, and/or Cosecant Functions
Unit 3: Differentiation: Composite, Implicit, and Inverse Functions
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
Unit 4: Contextual Applications of Differentiation
4.1 Interpreting the Meaning of the Derivative in Context
4.2 Straight-Line Motion: Connecting Position, Velocity, and Acceleration
4.3 Rates of Change in Applied Contexts Other Than Motion
4.4 Introduction to Related Rates
4.5 Solving Related Rates Problems
4.6 Approximating Values of a Function Using Local Linearity and Linearization
4.7 Using L’Hospital’s Rule for Determining Limits of Indeterminate Forms
Unit 5: Analytical Applications of Differentiation
5.1 Using the Mean Value Theorem
5.2 Extreme Value Theorem, Global Versus Local Extrema, and Critical Points
5.3 Determining Intervals on Which a Function Is Increasing or Decreasing
5.4 Using the First Derivative Test to Determine Relative (Local) Extrema
5.5 Using the Candidates Test to Determine Absolute (Global) Extrema
5.6 Determining Concavity of Functions over Their Domains
5.7 Using the Second Derivative Test to Determine Extrema
5.8 Sketching Graphs of Functions and Their Derivatives
5.9 Connecting a Function, Its First Derivative, and Its Second Derivative
5.10 Introduction to Optimization Problems
5.11 Solving Optimization Problems
5.12 Exploring Behaviors of Implicit Relations
Unit 6: Integration and Accumulation of Change
6.1 Exploring Accumulations of Change
6.2 Approximating Areas with Riemann Sums
6.3 Riemann Sums, Summation Notation, and Definite Integral Notation
6.4 The Fundamental Theorem of Calculus and Accumulation Functions
6.5 Interpreting the Behavior of Accumulation Functions Involving Area
6.6 Applying Properties of Definite Integrals
6.7 The Fundamental Theorem of Calculus and Definite Integrals
6.8 Finding Antiderivatives and Indefinite Integrals: Basic Rules and Notation
6.9 Integrating Using Substitution
6.10 Integrating Functions Using Long Division and Completing the Square
6.14 Selecting Techniques for Antidifferentiation
Unit 7: Differential Equations
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
Unit 8: Applications of Integration
8.1 Finding the Average Value of a Function on an Interval
8.2 Connecting Position, Velocity, and Acceleration of Functions Using Integrals
8.3 Using Accumulation Functions and Definite Integrals in Applied Contexts
8.4 Finding the Area Between Curves Expressed as Functions of x
8.5 Finding the Area Between Curves Expressed as Functions of y
8.6 Finding the Area Between Curves That Intersect at More Than Two Points
8.7 Volumes with Cross Sections: Squares and Rectangles
8.8 Volumes with Cross Sections: Triangles and Semicircles
8.9 Volume with Disc Method: Revolving Around the x- or y-Axis
8.10 Volume with Disc Method: Revolving Around Other Axes
8.11 Volume with Washer Method: Revolving Around the x- or y-Axis
8.12 Volume with Washer Method: Revolving Around Other Axes

How to study AP Calculus AB

AP Calculus AB improves when every method is connected to meaning. Use the eight-unit syllabus map to choose one exact Topic, then represent it analytically, graphically, numerically and verbally. State the theorem or definition that permits a step, check its conditions, carry notation and units correctly, and explain what the derivative, integral or limit says about the function or context.

Rebuild weak reasoning in Concept, retrieve definitions and methods in Mastery and apply them through the Question Bank. For the same idea, practise once without a calculator and once with graphing or numerical support. After marking, classify the first failure as concept, representation, process, theorem condition, notation, calculator use or interpretation; repair it and retry without notes.

Practise calculus as connected reasoning

Use the four mathematical practices to connect processes, representations, justification and notation across each major AP Calculus AB problem family.

Limits and continuity

Estimating limits, limit laws, asymptotes, continuity and the Intermediate Value Theorem

Move among graphs, tables and expressions before calculating. Distinguish a function value from a limit, state one-sided behaviour precisely and check every theorem condition before claiming existence, continuity or an intermediate value.

Practise limits

Derivative methods

Definitions, rules, composite functions, implicit differentiation and inverse functions

Identify the function structure before choosing a rule, keep derivative notation attached to the correct variable and verify the result from a graph or local rate. Mix routine execution with explanation of what the derivative represents.

Practise differentiation

Derivative applications

Related rates, linearization, extrema, monotonicity, concavity and optimization

Define quantities and units, translate the context into a relationship and differentiate before substituting values. For analytical problems, build a sign argument and cite the relevant condition rather than reading a conclusion from an unsupported sketch.

Practise applications

Integration and accumulation

Riemann sums, definite integrals, antiderivatives and the Fundamental Theorem of Calculus

Connect signed area, net change and accumulation without treating them as interchangeable phrases. Track bounds, units and constants, interpret the integrand first and verify whether the final value represents an amount, change or average.

Practise integration

Differential equations and integral applications

Slope fields, separation, exponential models, area, volume and average value

Translate the model or geometry before applying a procedure. Check initial conditions, domain and units, connect solutions to slope-field behaviour and make the washer, disk or cross-section geometry visible before writing an integral.

Practise modelling

Where to start

Start from one known calculus Topic or use a diagnostic to reveal the first repeated representation, process or justification failure.

Choose your starting point

  1. I know the weak Topic

    Open the exact Unit and Topic, then identify its prerequisite skills, representations and theorem conditions before solving.

    Browse all eight Units

  2. I do not know what is weak

    Use a mixed diagnostic and stop at the first repeated error in setup, process, representation or justification.

    Start a diagnostic

Choose the right starting point

  1. Understand and connect

    Explain the definition or theorem, then connect its formula, graph, table and contextual meaning.

    Review a Concept

  2. Retrieve and execute

    Reproduce the method without notes and solve both a non-calculator and calculator-supported version when appropriate.

    Check Mastery

  3. Justify, check and repair

    Write a complete conclusion with conditions, notation and units, then correct the first failed reasoning step.

    Practise questions

AP Calculus AB exam format for 2027

The verified May 2027 hybrid exam lasts 3 hours 10 minutes: 42 Bluebook multiple-choice questions and six free-response prompts answered by hand, with calculator access limited to designated parts.

Paper / componentQuestionsTime% of grade
Section I, Part A: Multiple Choice29 questionsCompleted in Bluebook without a graphing calculator.How to prepare: Build fluent symbolic, numerical, graphical and conceptual reasoning without depending on calculator output.62 minutes35%
Section I, Part B: Multiple Choice13 questionsCompleted in Bluebook with a required graphing calculator; built-in Desmos is available.How to prepare: Practise graphing, numerical solving, derivatives and definite integrals while keeping the calculus setup and interpretation explicit.38 minutes15%
Section II, Part A: Free Response2 questionsCalculator-required prompts are viewed in Bluebook and responses are handwritten.How to prepare: Plan concise calculator-supported work and communicate setup, interpretation, units and conclusions clearly on paper.30 minutes16.7%
Section II, Part B: Free Response4 questionsNon-calculator prompts are viewed in Bluebook and responses are handwritten.How to prepare: Practise theorem conditions, symbolic reasoning, representation links and complete handwritten justification without calculator support.60 minutes33.3%

SourceCollege Board · AP Calculus AB ExamAP Calculus AB · May 2027 exam

AP Calculus AB questions

No course-content rewrite was announced. College Board continues to use the Fall 2020 course framework with clarifications implemented Fall 2026. The May 2027 change affects the multiple-choice count and timing, which is why older 45-question exam summaries are now stale.

This Hub contains only the eight AP Calculus AB Units and its 81 mapped Topics. BC adds further content, including parametric, polar and vector-valued functions and infinite sequences and series. Numbering gaps within AB Units 6 and 7 preserve the shared official framework rather than indicating missing content.

AP Calculus AB does not appear on College Board's current list of exams receiving printed reference information. Prepare to know and apply the required definitions, theorems, notation and methods, and recheck the official reference-information page if policies change after the 2027 administration.

College Board expects strong algebra, geometry, trigonometry, analytic geometry and elementary-function skills, including graph interpretation, composition and inverses. If a calculus error begins with equation solving, function notation, domain or trigonometry, repair that prerequisite directly instead of repeating the later derivative or integral procedure.