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Verified for the 2027 AP exam

AP Calculus BC Study Guide & Review

Extend secure limits, derivatives and integrals into advanced integration, differential-equation models, parametric and polar functions, vectors and infinite series with precise AP Calculus BC justification.

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  2. 02Practise questions
  3. 03Review mistakes

Syllabus knowledge tree

Explore all ten AP Calculus BC units

Navigate 111 exact Topics across ten official Units: the complete AB foundation, BC-only additions within Units 6–8, and dedicated parametric, polar, vector and infinite-series work in Units 9–10.

10
syllabus groups
111
mapped topics

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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
6.11 Integrating Using Integration by Parts
6.12 Integrating Using Linear Partial Fractions
6.13 Evaluating Improper Integrals
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
7.5 Approximating Solutions Using Euler’s Method
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
7.9 Logistic 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
8.13 The Arc Length of a Smooth, Planar Curve and Distance Traveled
Unit 9: Parametric Equations, Polar Coordinates, and Vector-Valued Functions
9.1 Defining and Differentiating Parametric Equations
9.2 Second Derivatives of Parametric Equations
9.3 Finding Arc Lengths of Curves Given by Parametric Equations
9.4 Defining and Differentiating Vector-Valued Functions
9.5 Integrating Vector-Valued Functions
9.6 Solving Motion Problems Using Parametric and Vector-Valued Functions
9.7 Defining Polar Coordinates and Differentiating in Polar Form
9.8 Find the Area of a Polar Region or the Area Bounded by a Single Polar Curve
9.9 Finding the Area of the Region Bounded by Two Polar Curves
Unit 10: Infinite Sequences and Series
10.1 Defining Convergent and Divergent Infinite Series
10.2 Working with Geometric Series
10.3 The nth Term Test for Divergence
10.4 Integral Test for Convergence
10.5 Harmonic Series and p-Series
10.6 Comparison Tests for Convergence
10.7 Alternating Series Test for Convergence
10.8 Ratio Test for Convergence
10.9 Determining Absolute or Conditional Convergence
10.10 Alternating Series Error Bound
10.11 Finding Taylor Polynomial Approximations of Functions
10.12 Lagrange Error Bound
10.13 Radius and Interval of Convergence of Power Series
10.14 Finding Taylor or Maclaurin Series for a Function
10.15 Representing Functions as Power Series

How to study AP Calculus BC

AP Calculus BC is cumulative: BC-only methods depend on secure limits, derivatives and integrals. Use the ten-unit syllabus map to pair each advanced Topic with its shared prerequisite, then move among formulas, graphs, tables, motion or geometric contexts. Select a method from its conditions, carry notation and endpoints carefully, and justify the conclusion rather than naming a rule or convergence test alone.

Rebuild a prerequisite or BC method in Concept, retrieve definitions, series forms and test conditions in Mastery and apply them in the Question Bank. Alternate calculator and non-calculator work. After marking, classify the first failure as prerequisite, representation, method selection, condition, algebra, notation, error bound or interpretation; repair that decision and retry without notes.

Practise BC methods through connected decisions

Build advanced method selection, representation changes and rigorous conclusions while repeatedly reconnecting BC-only work to the shared calculus foundation it depends on.

Shared foundations and advanced integration

Limits, derivatives, accumulation, integration by parts, partial fractions and improper integrals

Identify whether the obstacle is a shared prerequisite or a BC integration choice. State why the technique fits, preserve bounds and convergence conditions, and verify the result by differentiation, estimation or graphical behaviour.

Practise integration

Differential-equation models

Euler's method, slope fields, separable equations, exponential and logistic models

Connect the differential equation, slope field and solution behaviour before calculating. Use initial conditions and units explicitly, distinguish approximation from exact solution and interpret carrying capacity, growth rate or long-term behaviour in context.

Practise differential equations

Parametric, polar and vector-valued functions

Motion, derivatives, speed, arc length, polar area and geometric representations

Translate the representation before applying a familiar derivative or integral. Track the parameter or angle, orientation and interval, distinguish velocity from speed and use calculator evidence to check geometry without replacing the required setup.

Practise Unit 9

Sequences, series and convergence

Convergence tests, power series, Taylor polynomials, intervals and error bounds

Identify the evidence before selecting a test, state and check its conditions and conclude absolute, conditional or divergent behaviour precisely. Test endpoints separately and connect a Taylor approximation to its interval and error control.

Practise Unit 10

BC free-response communication

Method choice, theorem conditions, calculator output, notation and handwritten justification

Plan the mathematical claim before writing, expose the setup and cite the condition that licenses it. Practise concise calculator-supported work and complete non-calculator reasoning, then repair missing endpoints, bounds, units or conclusions from official scoring evidence.

Practise BC FRQs

Where to start

Start from one known BC Topic or use a diagnostic to reveal the earliest repeated prerequisite, method-selection or justification failure.

Choose your starting point

  1. I know the weak Topic

    Open its exact Unit, identify the shared prerequisite and list the representation, conditions and method the Topic requires.

    Browse all ten Units

  2. I do not know what is weak

    Use a mixed diagnostic and stop at the first repeated prerequisite, representation, test-selection or conclusion error.

    Start a diagnostic

Choose the right starting point

  1. Reconnect the prerequisite

    Explain the shared calculus idea, then connect it to the BC representation, method or model now required.

    Review a Concept

  2. Retrieve conditions and execute

    Recall the method and its hypotheses without notes, then complete calculator and non-calculator versions when appropriate.

    Check Mastery

  3. Justify, bound and repair

    Write a precise conclusion with notation, endpoints or error bounds, then correct the first invalid decision.

    Practise BC questions

AP Calculus BC 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 fast symbolic, graphical and conceptual method selection across shared and BC-only Topics without calculator dependence.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 numerical solving, graphing, integration and parametric or polar evaluation while keeping 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: Communicate concise calculator-supported setup, relevant output, 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 conditions, convergence-test justification, symbolic reasoning, error bounds and complete notation without calculator support.60 minutes33.3%

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

AP Calculus BC questions

BC includes all Topics from the eight AB Units, then adds integration techniques, Euler's method, logistic models and arc length within Units 6–8, plus complete Units on parametric, polar and vector-valued functions and infinite sequences and series.

BC students receive a separate 1–5 AB subscore based on the approximately 60% of the exam devoted to AB Topics. College Board recommends treating it like an AP Calculus AB score, but each college sets its own credit and placement policy.

No course-content rewrite was announced. The Fall 2020 framework remains in force with Fall 2026 clarifications. The May 2027 update changes the multiple-choice count and timing, so use the current four-part exam table rather than older format summaries.

AP Calculus BC does not appear on College Board's current list of exams receiving printed reference information. Prepare to recall definitions, theorems, integration methods, convergence tests, series forms and error bounds, and recheck the official list if policy changes after 2027.