Free AP Calculus AB Practice Exam (PDF)

This free AP Calculus AB practice exam mirrors the real test: multiple choice across limits, derivatives, and integrals, then three free-response questions in the classic genres — particle motion, area and volume of a region, and a rate-in/rate-out accumulation problem. Every solution shows the working and the justification language AP graders award points for. View it online below or download the print-ready PDF; no account needed.

1. Question 12 marks · Limits
   Evaluate $\lim_{x \to 3} \dfrac{x^2 - 9}{x - 3}$.
   • A.
     $0$
   • B.
     $3$
   • C.
     $6$
   • D.
     The limit does not exist
2. Question 22 marks · Limit definition of the derivative
   $\lim_{h \to 0} \dfrac{\sqrt{9 + h} - 3}{h}$ is:
   • A.
     $0$
   • B.
     $\frac{1}{6}$
   • C.
     $\frac{1}{3}$
   • D.
     nonexistent
3. Question 32 marks · Continuity & IVT
   A function $f$ is continuous on $[1, 4]$ with $f(1) = -2$ and $f(4) = 5$. Which of the following must be true?
   • A.
     f$ has a zero in the open interval $(1, 4)

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   • B.
     f$ is increasing on $[1, 4]

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   • C.
     f'(c) = \frac{7}{3}$ for some $c$ in $(1, 4)

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   • D.
     f$ has exactly one zero in $(1, 4)

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4. Question 42 marks · Chain rule
   If $y = \sin(x^2)$, then $\dfrac{dy}{dx} =$
   • A.
     $\cos(x^2)$
   • B.
     $2x\cos(x^2)$
   • C.
     $2x\cos(2x)$
   • D.
     $-2x\cos(x^2)$
5. Question 52 marks · Product rule
   If $f(x) = x^2 e^x$, then $f'(x) =$
   • A.
     $2x e^x$
   • B.
     $x^2 e^x$
   • C.
     $e^x(x^2 + 2x)$
   • D.
     $e^x(x^2 - 2x)$
6. Question 62 marks · Implicit differentiation
   The curve $x^2 + y^2 = 25$ passes through the point $(3, 4)$. The slope of the curve at that point is:
   • A.
     $\frac{3}{4}$
   • B.
     $-\frac{3}{4}$
   • C.
     $\frac{4}{3}$
   • D.
     $-\frac{4}{3}$
7. Question 72 marks · Related rates
   The radius of a sphere is increasing at 2 centimeters per second. In terms of $\pi$, how fast is the volume $V = \frac{4}{3}\pi r^3$ increasing when the radius is 3 centimeters?
   • A.
     $24\pi$ cubic centimeters per second
   • B.
     $36\pi$ cubic centimeters per second
   • C.
     $72\pi$ cubic centimeters per second
   • D.
     $144\pi$ cubic centimeters per second
8. Question 82 marks · Mean Value Theorem
   For $f(x) = x^2$ on the interval $[1, 3]$, the value of $c$ guaranteed by the Mean Value Theorem satisfies $f'(c) =$ the average rate of change. Then $c =$
   • A.
     $\frac{3}{2}$
   • B.
     $2$
   • C.
     $\sqrt{5}$
   • D.
     $\frac{5}{2}$
9. Question 92 marks · u-substitution
   $\displaystyle\int 2x\,(x^2 + 1)^3\,dx =$
   • A.
     $(x^2 + 1)^4 + C$
   • B.
     $\frac{(x^2 + 1)^4}{4} + C$
   • C.
     $2(x^2 + 1)^4 + C$
   • D.
     $x^2 (x^2 + 1)^3 + C$
10. Question 102 marks · Fundamental Theorem of Calculus
    If $g(x) = \displaystyle\int_0^x \sqrt{1 + t^3}\,dt$, then $g'(2) =$
    • A.
      $3$
    • B.
      $9$
    • C.
      $\sqrt{5}$
    • D.
      $0$
11. Question 112 marks · Properties of definite integrals
    Given $\displaystyle\int_0^5 f(x)\,dx = 10$ and $\displaystyle\int_0^2 f(x)\,dx = 4$, find $\displaystyle\int_2^5 f(x)\,dx$.
    • A.
      $6$
    • B.
      $14$
    • C.
      $-6$
    • D.
      $40$
12. Question 122 marks · Average value
    The average value of $f(x) = 3x^2$ on the interval $[0, 2]$ is:
    • A.
      $4$
    • B.
      $8$
    • C.
      $12$
    • D.
      $6$
13. Question 139 marks · Particle motion
    Particle motion. A particle moves along the x-axis so that its position at time $t$ is $x(t) = t^3 - 6t^2 + 9t$ for $0 \leq t \leq 5$, where $x$ is measured in meters and $t$ in seconds.
    $x(t) = t^3 - 6t^2 + 9t, \qquad 0 \leq t \leq 5$
    (a)

    Find the velocity $v(t)$ and acceleration $a(t)$ of the particle.

    [2]
    (b)

    On what interval(s) is the particle moving to the left? Justify your answer.

    [3]
    (c)

    At time $t = 2.5$, is the speed of the particle increasing or decreasing? Justify your answer.

    [2]
    (d)

    Find the total distance traveled by the particle from $t = 0$ to $t = 2$.

    [2]
14. Question 149 marks · Area & volume
    Area and volume. Let $R$ be the region in the first quadrant bounded by the graphs of $y = 4x - x^2$ and $y = x$.
    $y = 4x - x^2, \qquad y = x$
    (a)

    Find the points of intersection and the area of $R$.

    [3]
    (b)

    The region $R$ is the base of a solid whose cross-sections perpendicular to the x-axis are squares. Find the volume of the solid.

    [4]
    (c)

    Write, but do not evaluate, an integral expression for the volume of the solid generated when $R$ is rotated about the x-axis.

    [2]
15. Question 159 marks · Accumulation (rate in / rate out)
    Accumulation. Water flows into a tank at the rate $R(t) = 20 - t^2$ liters per minute and drains out at a constant 11 liters per minute, for $0 \leq t \leq 4$. At time $t = 0$ the tank contains 30 liters.
    $R(t) = 20 - t^2, \qquad D(t) = 11, \qquad 0 \leq t \leq 4$
    (a)

    How much water flows INTO the tank during the 4 minutes?

    [2]
    (b)

    How much water is in the tank at time $t = 4$?

    [3]
    (c)

    At time $t = 2$, is the amount of water in the tank increasing or decreasing? Justify.

    [2]
    (d)

    At what time $t$ in $[0, 4]$ is the amount of water in the tank greatest? Justify, and find that maximum amount.

    [2]