AP Calculus AB Practice Tests and Questions

Practice for AP Calculus AB is abundant and mostly badly used. The most valuable material — released free-response questions with their official scoring guidelines — is free, published every year, and skipped by most students in favour of question banks with answer keys and no rubrics. This guide covers where to find genuine practice, how to use it, and includes original AB-style problems worked in full.

(AI-Math is independent and not affiliated with the College Board. We do not reproduce copyrighted exam questions; the problems below are original.)

Where real AP Calculus AB practice comes from

Released free-response questions. The College Board publishes the free-response questions from each administration together with scoring guidelines and samples of scored student work. This is the single best resource in existence for this exam and it costs nothing. Years of them are available.

Released multiple-choice exams. Complete multiple-choice sections from a handful of past administrations have been released publicly over the years. There are far fewer of these than of FRQs, so ration them: use them as timed diagnostics, not as daily drills.

AP Classroom. If you are enrolled in an AP course, your teacher has access to secure practice exams and a large question bank. Ask for them explicitly — many teachers only assign a fraction of what is available.

Textbook and review-book sets. Useful for drilling a specific technique. Their difficulty calibration is often off, so do not read a review-book score as a predicted AP score.

How to use a practice test

A practice test used as a test tells you a number. A practice test used as a diagnostic tells you what to do next. The difference is what happens afterwards:

1. Sit it under real conditions — correct timing, correct calculator restrictions per part, no interruptions.
2. Before checking answers, mark every question you were unsure about. Guessed-right questions are the ones that will hurt you next time.
3. Sort every miss into concept, algebra, or misread. This is the whole point of the exercise.
4. Rework each miss from a blank page. Reading a solution feels like learning and is not.
5. For free response, grade yourself against the published scoring guidelines line by line.

If your misses are mostly "algebra," your revision plan is algebra, not calculus. That is the most common finding and the most commonly ignored one.

Original practice questions, worked

1. Limits (no calculator)

Evaluate $\displaystyle\lim_{x \to 3} \frac{x^2 - 9}{x^2 - 2x - 3}$.

Solution. Direct substitution gives $0/0$, so factor both parts:

$\frac{x^2 - 9}{x^2 - 2x - 3} = \frac{(x-3)(x+3)}{(x-3)(x+1)} = \frac{x+3}{x+1}, \quad x \neq 3.$

The limit is $6/4 = \frac{3}{2}$. Check work like this on the limit calculator.

2. Related rates (calculator active)

A spherical balloon is inflated so that its volume increases at 40 cubic centimetres per second. How fast is the radius increasing when the radius is 5 cm?

Solution. From $V = \frac{4}{3}\pi r^3$, differentiate with respect to time:

$\frac{dV}{dt} = 4\pi r^2 \frac{dr}{dt}.$

Substituting $dV/dt = 40$ and $r = 5$: $40 = 4\pi(25)\frac{dr}{dt}$, so

$\frac{dr}{dt} = \frac{40}{100\pi} = \frac{0.4}{\pi} \approx 0.127 \text{ cm/s}.$

Note the two things that earn points beyond the number: the differentiated relationship written explicitly, and the units. A bare 0.127 is worth very little.

3. Analytical applications of the derivative (no calculator)

Let $f(x) = x^3 - 6x^2 + 9x + 2$. Find all local extrema and classify them.

Solution. $f'(x) = 3x^2 - 12x + 9 = 3(x-1)(x-3)$, so the critical points are $x = 1$ and $x = 3$.

Sign of $f'$: positive on $(-\infty, 1)$, negative on $(1,3)$, positive on $(3,\infty)$. So $f$ has a local maximum at $x = 1$ with $f(1) = 6$, and a local minimum at $x = 3$ with $f(3) = 2$.

The second derivative confirms it: $f''(x) = 6x - 12$, so $f''(1) = -6 < 0$ and $f''(3) = 6 > 0$. Either justification earns credit, but you must state one — a table of values alone does not.

4. Accumulation and the Fundamental Theorem (calculator active)

Water flows into a tank at a rate of $R(t) = 12 + 8\sin(t/2)$ litres per minute for $0 \leq t \leq 10$. How much water enters during the first 10 minutes?

Solution. Total accumulation is the integral of the rate:

$\int_0^{10} \left(12 + 8\sin\frac{t}{2}\right) dt = \left[12t - 16\cos\frac{t}{2}\right]_0^{10} = (120 - 16\cos 5) - (0 - 16).$

Since $\cos 5 \approx 0.28366$, this is $120 - 4.539 + 16 \approx 131.461$ litres.

On a calculator-active question you may evaluate the integral numerically, but write the integral first. Practise the antiderivatives on the integral calculator.

5. Volume of revolution (no calculator)

The region bounded by $y = \sqrt{x}$, $y = 0$ and $x = 4$ is revolved about the x-axis. Find the volume.

Solution. Disks perpendicular to the axis of revolution give

$V = \pi \int_0^4 (\sqrt{x})^2 dx = \pi \int_0^4 x \, dx = \pi \left[\frac{x^2}{2}\right]_0^4 = 8\pi.$

The most common error here is forgetting to square the radius. The second most common is integrating with respect to the wrong variable — when revolving about the x-axis with disks, integrate in $x$.

6. Separable differential equation (no calculator)

Solve $\frac{dy}{dx} = 2xy$ with $y(0) = 3$.

Solution. Separate: $\frac{dy}{y} = 2x\,dx$. Integrating gives $\ln|y| = x^2 + C$, so $y = Ae^{x^2}$. Applying $y(0) = 3$ gives $A = 3$, hence

$y = 3e^{x^2}.$

Scoring guidelines award separate points for the separation, the antiderivatives, the constant of integration, and the use of the initial condition. Skipping the $+C$ and back-solving at the end usually loses a point. Check your work on the differential equation solver.

A six-week practice schedule

Week	Focus
1	Full timed multiple-choice section as a diagnostic; sort every miss
2	Weakest two units only, from the textbook, with step-by-step checking
3	Six free-response questions, self-graded against scoring guidelines
4	Second timed multiple-choice section; compare error categories with week 1
5	Six more free-response questions, prioritising types you scored lowest on
6	Two complete timed exams; rewrite your three weakest FRQ answers

The one rule that matters: never mark your own free response generously. Read the scoring guidelines and award only what the rubric says, including the units and the justification. The gap between how you would mark yourself and how a reader would mark you is the score you are leaving on the table.

Related: AP Calculus AB and BC complete exam guide