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

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

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

Unit 1: Limits and Continuity

Open unit questions
1.1 Introducing Calculus: Can Change Occur at an Instant?1.2 Defining Limits and Using Limit Notation1.3 Estimating Limit Values from Graphs1.4 Estimating Limit Values from Tables1.5 Determining Limits Using Algebraic Properties of Limits1.6 Determining Limits Using Algebraic Manipulation1.7 Selecting Procedures for Determining Limits1.8 Determining Limits Using the Squeeze Theorem1.9 Connecting Multiple Representations of Limits1.10 Exploring Types of Discontinuities1.11 Defining Continuity at a Point1.12 Confirming Continuity over an Interval1.13 Removing Discontinuities1.14 Connecting Infinite Limits and Vertical Asymptotes1.15 Connecting Limits at Infinity and Horizontal Asymptotes1.16 Working with the Intermediate Value Theorem (IVT)

Unit 2: Differentiation: Definition and Fundamental Properties

Open unit questions
2.1 Defining Average and Instantaneous Rates of Change at a Point2.2 Defining the Derivative of a Function and Using Derivative Notation2.3 Estimating Derivatives of a Function at a Point2.4 Connecting Differentiability and Continuity: Determining When Derivatives Do and Do Not Exist2.5 Applying the Power Rule2.6 Derivative Rules: Constant, Sum, Difference, and Constant Multiple2.7 Derivatives of cos x, sin x, eˣ, and ln x2.8 The Product Rule2.9 The Quotient Rule2.10 Finding the Derivatives of Tangent, Cotangent, Secant, and/or Cosecant Functions

Unit 3: Differentiation: Composite, Implicit, and Inverse Functions

Open unit questions
3.1 The Chain Rule3.2 Implicit Differentiation3.3 Differentiating Inverse Functions3.4 Differentiating Inverse Trigonometric Functions3.5 Selecting Procedures for Calculating Derivatives3.6 Calculating Higher-Order Derivatives

Unit 4: Contextual Applications of Differentiation

Open unit questions
4.1 Interpreting the Meaning of the Derivative in Context4.2 Straight-Line Motion: Connecting Position, Velocity, and Acceleration4.3 Rates of Change in Applied Contexts Other Than Motion4.4 Introduction to Related Rates4.5 Solving Related Rates Problems4.6 Approximating Values of a Function Using Local Linearity and Linearization4.7 Using L’Hospital’s Rule for Determining Limits of Indeterminate Forms

Unit 5: Analytical Applications of Differentiation

Open unit questions
5.1 Using the Mean Value Theorem5.2 Extreme Value Theorem, Global Versus Local Extrema, and Critical Points5.3 Determining Intervals on Which a Function Is Increasing or Decreasing5.4 Using the First Derivative Test to Determine Relative (Local) Extrema5.5 Using the Candidates Test to Determine Absolute (Global) Extrema5.6 Determining Concavity of Functions over Their Domains5.7 Using the Second Derivative Test to Determine Extrema5.8 Sketching Graphs of Functions and Their Derivatives5.9 Connecting a Function, Its First Derivative, and Its Second Derivative5.10 Introduction to Optimization Problems5.11 Solving Optimization Problems5.12 Exploring Behaviors of Implicit Relations

Unit 6: Integration and Accumulation of Change

Open unit questions
6.1 Exploring Accumulations of Change6.2 Approximating Areas with Riemann Sums6.3 Riemann Sums, Summation Notation, and Definite Integral Notation6.4 The Fundamental Theorem of Calculus and Accumulation Functions6.5 Interpreting the Behavior of Accumulation Functions Involving Area6.6 Applying Properties of Definite Integrals6.7 The Fundamental Theorem of Calculus and Definite Integrals6.8 Finding Antiderivatives and Indefinite Integrals: Basic Rules and Notation6.9 Integrating Using Substitution6.10 Integrating Functions Using Long Division and Completing the Square6.11 Integrating Using Integration by Parts6.12 Integrating Using Linear Partial Fractions6.13 Evaluating Improper Integrals6.14 Selecting Techniques for Antidifferentiation

Unit 7: Differential Equations

Open unit questions
7.1 Modeling Situations with Differential Equations7.2 Verifying Solutions for Differential Equations7.3 Sketching Slope Fields7.4 Reasoning Using Slope Fields7.5 Approximating Solutions Using Euler’s Method7.6 Finding General Solutions Using Separation of Variables7.7 Finding Particular Solutions Using Initial Conditions and Separation of Variables7.8 Exponential Models with Differential Equations7.9 Logistic Models with Differential Equations

Unit 8: Applications of Integration

Open unit questions
8.1 Finding the Average Value of a Function on an Interval8.2 Connecting Position, Velocity, and Acceleration of Functions Using Integrals8.3 Using Accumulation Functions and Definite Integrals in Applied Contexts8.4 Finding the Area Between Curves Expressed as Functions of x8.5 Finding the Area Between Curves Expressed as Functions of y8.6 Finding the Area Between Curves That Intersect at More Than Two Points8.7 Volumes with Cross Sections: Squares and Rectangles8.8 Volumes with Cross Sections: Triangles and Semicircles8.9 Volume with Disc Method: Revolving Around the x- or y-Axis8.10 Volume with Disc Method: Revolving Around Other Axes8.11 Volume with Washer Method: Revolving Around the x- or y-Axis8.12 Volume with Washer Method: Revolving Around Other Axes8.13 The Arc Length of a Smooth, Planar Curve and Distance Traveled

Unit 9: Parametric Equations, Polar Coordinates, and Vector-Valued Functions

Open unit questions
9.1 Defining and Differentiating Parametric Equations9.2 Second Derivatives of Parametric Equations9.3 Finding Arc Lengths of Curves Given by Parametric Equations9.4 Defining and Differentiating Vector-Valued Functions9.5 Integrating Vector-Valued Functions9.6 Solving Motion Problems Using Parametric and Vector-Valued Functions9.7 Defining Polar Coordinates and Differentiating in Polar Form9.8 Find the Area of a Polar Region or the Area Bounded by a Single Polar Curve9.9 Finding the Area of the Region Bounded by Two Polar Curves

Unit 10: Infinite Sequences and Series

Open unit questions
10.1 Defining Convergent and Divergent Infinite Series10.2 Working with Geometric Series10.3 The nth Term Test for Divergence10.4 Integral Test for Convergence10.5 Harmonic Series and p-Series10.6 Comparison Tests for Convergence10.7 Alternating Series Test for Convergence10.8 Ratio Test for Convergence10.9 Determining Absolute or Conditional Convergence10.10 Alternating Series Error Bound10.11 Finding Taylor Polynomial Approximations of Functions10.12 Lagrange Error Bound10.13 Radius and Interval of Convergence of Power Series10.14 Finding Taylor or Maclaurin Series for a Function10.15 Representing Functions as Power Series

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.