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AP Calculus AB Practice Questions & Answers

Build calculus reasoning through original practice, worked explanations, concept cards and illustrated video companions.

Read 10 original sample questions with worked explanations before opening the practice studio. These examples introduce selected topics from this AP Calculus AB foundation path.

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What this practice covers

AB foundation expansion: 96 task families with unit and task crosswalks; not complete syllabus mastery

Foundation MCQ and numeric practice, 41 graph/table items and 54 original written extensions. Written work is self-assessed outside automatic scores; this is not a full AP exam simulation.

10 public examples400 question variants in the studio96 task families

Review status: Independent educator review and learner calibration are pending. Use these selected foundation examples alongside the coverage map and current official references.

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Topics in this foundation path

The examples below span 8 of 8 learning domains. Counts describe the current catalog, not the exam's official weighting.

Limits & continuity

3 lessons · 60 question variants

Differentiation

5 lessons · 84 question variants

Applications of derivatives

3 lessons · 48 question variants

Integration

5 lessons · 80 question variants

Differential equations

2 lessons · 32 question variants

Applications of integrals

1 lessons · 16 question variants

Applications of differentiation

3 lessons · 48 question variants

Applications of integration

2 lessons · 32 question variants

TRY A QUESTION. FOLLOW THE REASONING.

Sample questions with answers

Work out your answer first, then compare the explanation. This stable selection uses distinct task families across the available learning domains.

QUESTION 01Limits & continuity

Limits without guessing

Find the limit as x approaches 3 of 2x² + 4.

  1. 10
  2. 22
  3. 18
  4. 98

Answer: 22

  1. The polynomial is continuous at 3.
  2. Substitute x=3: 2(3)²+4.
  3. The limit is 22.
Compare the other answer choices
10

Use b instead of b².

18

Discard the constant term.

98

Put the constant inside the squared input.

Learn: Limits without guessing Practice this topic

Reference basis: AP Calculus AB course; Calculus Volume 1: Fundamental Theorem of Calculus; AP Calculus AB and BC Course and Exam Description. These are original learning questions, not questions reproduced from the references.

QUESTION 02Differentiation

Derivative rules and meaning

If f(x)=5x³+7, find f′(6).

    Write a numeric answer.

    Answer: 540

    1. The power rule gives f′(x)=15x².
    2. The constant term differentiates to zero.
    3. Substitute 6: f′(6)=540.
    Learn: Derivative rules and meaning Practice this topic

    Reference basis: AP Calculus AB course; Calculus Volume 1: Fundamental Theorem of Calculus; AP Calculus AB and BC Course and Exam Description. These are original learning questions, not questions reproduced from the references.

    QUESTION 03Applications of derivatives

    Position, velocity and acceleration

    An object has position s(t)=2t²+4t meters. Find its velocity at t=3 seconds.

    1. 16
    2. 30
    3. 4
    4. 12

    Answer: 16

    1. Velocity is s′(t)=4t+4.
    2. Evaluate at t=3.
    3. The velocity is 16 meters per second.
    Compare the other answer choices
    4

    Report acceleration instead of velocity.

    12

    Drop the derivative of the linear term.

    30

    Report position instead of velocity.

    Learn: Position, velocity and acceleration Practice this topic

    Reference basis: AP Calculus AB course; Calculus Volume 1: Fundamental Theorem of Calculus; AP Calculus AB and BC Course and Exam Description. These are original learning questions, not questions reproduced from the references.

    QUESTION 04Integration

    Integrals and numerical accumulation

    An antiderivative of 5x^6 is kx^7+C. Find k.

      Write a numeric answer.

      Answer: 5/7

      1. Differentiate kx^7 to obtain (7)kx^6.
      2. Match its coefficient to 5.
      3. Thus k=5/7.
      Learn: Integrals and numerical accumulation Practice this topic

      Reference basis: AP Calculus AB course; Calculus Volume 1: Fundamental Theorem of Calculus; AP Calculus AB and BC Course and Exam Description. These are original learning questions, not questions reproduced from the references.

      QUESTION 05Differential equations

      Differential equations and initial values

      A solution satisfies dy/dx=x+y. What is its slope at (2,3)?

      1. 5
      2. 6
      3. 2
      4. 3

      Answer: 5

      1. A slope field evaluates the differential equation at the specified point.
      2. Substitute x=2, y=3.
      3. The slope is 5.
      Compare the other answer choices
      2

      Ignore y in the slope law.

      3

      Ignore x in the slope law.

      6

      Multiply x and y rather than add.

      Learn: Differential equations and initial values Practice this topic

      Reference basis: AP Calculus AB course; Calculus Volume 1: Fundamental Theorem of Calculus; AP Calculus AB and BC Course and Exam Description. These are original learning questions, not questions reproduced from the references.

      QUESTION 06Applications of integrals

      Area, average value and volume

      Find the area between y=5x and y=0 from x=0 to x=6.

        Write a numeric answer.

        Answer: 90

        1. The upper function is the line throughout this nonnegative interval.
        2. Integrate 5x from 0 to 6.
        3. The area is 90 square units.
        Learn: Area, average value and volume Practice this topic

        Reference basis: AP Calculus AB course; Calculus Volume 1: Fundamental Theorem of Calculus; AP Calculus AB and BC Course and Exam Description. These are original learning questions, not questions reproduced from the references.

        QUESTION 07Applications of differentiation

        Choose and justify a theorem

        f is continuous on [0,2] and differentiable on (0,2), with −3≤f′(x)≤6. If f(0)=6, what is the smallest value f(2) could have under these constraints?

        1. 0
        2. -6
        3. 12
        4. 18

        Answer: 0

        1. The Mean Value Theorem expresses the total change as an interior derivative times interval length.
        2. The smallest allowed change is −3×2=-6.
        3. Add the initial value to obtain 0. The line f(x)=6−3x attains this bound.
        Compare the other answer choices
        12

        Replace the negative lower rate by its magnitude.

        18

        Use the upper derivative bound instead of the lower one.

        -6

        Report the minimum change without adding f(0).

        Learn: Choose and justify a theorem Practice this topic

        Reference basis: AP Calculus AB and BC Course and Exam Description; Calculus Volume 1: Fundamental Theorem of Calculus. These are original learning questions, not questions reproduced from the references.

        QUESTION 08Applications of integration

        Areas across intersections and sideways slices

        Find the total area between y=x² and y=36 for 0≤x≤12.

          Write a numeric answer.

          Answer: 432

          1. The curves cross at x=6 in the interval; the horizontal line is above before that point.
          2. Integrate (36−x²) on [0,6] and (x²−36) on [6,12].
          3. The pieces are 144 and 288; the total is 432.
          Learn: Areas across intersections and sideways slices Practice this topic

          Reference basis: AP Calculus AB and BC Course and Exam Description; Calculus Volume 1: Fundamental Theorem of Calculus. These are original learning questions, not questions reproduced from the references.

          QUESTION 09Limits & continuity

          Continuity and local change

          f(x)=x²+3 for x≠2, and f(2)=k. Which value of k makes f continuous at 2?

          1. 4
          2. 3
          3. 7
          4. 5

          Answer: 7

          1. The nearby limit equals 2²+3.
          2. Continuity requires the assigned value k to equal that limit.
          3. Set k=7.
          Compare the other answer choices
          3

          Ignore the squared input.

          4

          Ignore the constant in the nearby expression.

          5

          Treat x² as x.

          Learn: Continuity and local change Practice this topic

          Reference basis: AP Calculus AB course; Calculus Volume 1: Fundamental Theorem of Calculus; AP Calculus AB and BC Course and Exam Description. These are original learning questions, not questions reproduced from the references.

          QUESTION 10Differentiation

          Products, quotients and chains

          If f(x)=x²(x+6), find f′(5).

            Write a numeric answer.

            Answer: 135

            1. The product rule gives f′(x)=2x(x+6)+x².
            2. This simplifies to 3x²+12x.
            3. At x=5, the value is 135.
            Learn: Products, quotients and chains Practice this topic

            Reference basis: AP Calculus AB course; Calculus Volume 1: Fundamental Theorem of Calculus; AP Calculus AB and BC Course and Exam Description. These are original learning questions, not questions reproduced from the references.

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            Scope, sources and review notes

            Catalog scope record: . This records the saved content mapping, not a guarantee that every external rule is current.

            • Foundation pilot: 400 parameterized original questions across 96 task families, not 400 independently designed scenarios.
            • 54 original written extensions use self-assessment criteria; they are not official AP tasks or rubric scores.
            • Coverage remains partial: sustained calculator investigations, comprehensive slope-field interpretation and every official subtopic require additional study.
            • Difficulty is editorial and uncalibrated. Independent educator review and learner testing remain pending.
            • Foundation pilot, not full syllabus coverage or official simulation
            • Question variants are not independent question designs
            • Independent educator review pending
            • Evidence integrity and structure checks do not establish factual or audiovisual fidelity

            Check the exam provider's current information for registration, test rules and official preparation materials. The exam guide collects the relevant official references.

            OFFICIAL REFERENCEAP Calculus AB course

            Course organization and syllabus alignment

            TEXTBOOKCalculus Volume 1: Fundamental Theorem of Calculus

            Additional concept reference; examples here are original, not reproduced

            OFFICIAL REFERENCEAP Calculus AB and BC Course and Exam Description

            Unit and AB/BC scope mapping; verified topic 7.5 BC-only and 7.7 initial conditions/separation.