AP Calculus AB — Practice Questions flashcards

6 of 60 questions free to study here · 8 topics · the whole deck is built into Willys

Multiple-choice practice questions, one best answer each, on limits, derivatives, related rates, MVT and curve analysis, Riemann sums, the FTC, integration, differential equations, area and volume.

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Practice deck · part of AP Calculus AB · written against: College Board AP Calculus AB Course and Exam Description (Units 1–8). Updated 2026-10-10.

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What is $\lim_{x \to 2} \frac{x^2 + x - 6}{x^2 - 4}$ ⁠?

A) $0$
B) $\frac{3}{4}$
C) $\frac{5}{2}$
D) $\frac{5}{4}$

D) $\frac{5}{4}$

Factor to $\frac{(x + 3)(x - 2)}{(x + 2)(x - 2)}$ and cancel for $x \neq 2$ ⁠; substituting into $\frac{x + 3}{x + 2}$ gives $\frac{5}{4}$ ⁠. The $\frac{0}{0}$ form at $x = 2$ is indeterminate, so it does not make the limit 0.
Limits and Continuity › Evaluating Limits
What is $\lim_{h \to 0} \frac{(2 + h)^5 - 32}{h}$ ⁠?

A) $80$
B) $40$
C) $32$
D) $0$

A) $80$

This is the definition of the derivative of $x^5$ at $x = 2$ ⁠, so it equals $5 \cdot 2^4 = 80$ ⁠. Substituting $h = 0$ directly gives the indeterminate form $\frac{0}{0}$ ⁠, not 0.
Differentiation Definition and Basic Rules › Definition of the Derivative
A particle moves along a line with position $x(t) = t^3 - 6t^2 + 9t$ for $t \ge 0$ ⁠. On which intervals is the particle speeding up?

A) $0 \le t < 1$ and $t > 3$
B) $1 < t < 2$ and $t > 3$
C) $1 < t < 3$ only
D) $t > 2$ only

B) $1 < t < 2$ and $t > 3$

$v(t) = 3(t - 1)(t - 3)$ and $a(t) = 6t - 12$ ⁠. Speed increases where $v$ and $a$ share a sign: both negative on $(1, 2)$ and both positive for $t > 3$ ⁠. On $(2, 3)$ the acceleration is positive but the velocity is still negative, so the particle is slowing down.
Contextual Applications of Differentiation › Rates of Change and Motion
For $f(x) = x^3$ on $[0, 3]$ ⁠, which value of $c$ satisfies the conclusion of the Mean Value Theorem?

A) $3$
B) $\frac{3}{2}$
C) $\sqrt{3}$
D) $1$

C) $\sqrt{3}$

The average rate of change is $\frac{27 - 0}{3} = 9$ ⁠, and $f'(c) = 3c^2 = 9$ gives $c = \sqrt{3}$ ⁠, which lies in $(0, 3)$ ⁠. The midpoint $\frac{3}{2}$ is the Mean Value Theorem point for a quadratic, not for this cubic.
Analytical Applications of Differentiation › MVT and EVT
A slope field has horizontal segments at every point of the line $y = 2x$ and positive slopes at every point above that line. Which differential equation could it represent?

A) $\frac{dy}{dx} = x - 2y$
B) $\frac{dy}{dx} = y - 2x$
C) $\frac{dy}{dx} = 2y - x$
D) $\frac{dy}{dx} = 2x - y$

B) $\frac{dy}{dx} = y - 2x$

$y - 2x = 0$ exactly on $y = 2x$ ⁠, and above the line $y > 2x$ makes the slope positive. $2x - y$ has the right zero line but negative slopes above it.
Differential Equations › Modeling and Slope Fields
What is the average value of $f(x) = 3x^2 - 2x$ on $[1, 3]$ ⁠?

A) $18$
B) $9$
C) $8$
D) $10$

B) $9$

$\frac{1}{3 - 1}\int_1^3 \left(3x^2 - 2x\right) dx = \frac{1}{2}\left((27 - 9) - (1 - 1)\right) = 9$ ⁠. The value 10 is the average rate of change, $\frac{f(3) - f(1)}{2}$ ⁠, and 18 forgets to divide by the interval length.
Applications of Integration › Average Value and Motion

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Review status: Second-reader reviewed, with corrections applied.

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Topic outline follows the College Board AP Calculus AB and BC Course and Exam Description (effective Fall 2020), Units 1–8 (the AP name is used nominatively; no course or exam text reproduced).
OpenStax Calculus Volumes 1 and 2, consulted as references; no text copied.

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