Which of the following limits are equal to 6?
Select all that apply.
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About this test
This is a free 15-question sampler in the style of the AP Calculus AB multiple-choice section, written by the StudyPDF team. It covers all eight units, from limits and continuity through derivatives, applications of differentiation, integration and differential equations, and mixes multiple choice, select all that apply and fill in the blank. Every question can be done without a calculator. Answer all 15, then check your score and read the worked explanation for every question.
All 15 questions at a glance
Prefer to read before you play? Every question in the sampler is listed below. Answers and explanations stay collapsed until you open them.
1.Which of the following limits are equal to 6?
Select all that apply.
- A. lim x->0 of 6/x
- B. lim x->3 of (x^2 - 9)/(x - 3)
- C. lim x->0 of sin(6x)/x
- D. lim x->6 of (x - 6)/x
- E. lim x->infinity of (6x^2 + 1)/(x^2 + 5)
Answer: B. lim x->3 of (x^2 - 9)/(x - 3), C. lim x->0 of sin(6x)/x, E. lim x->infinity of (6x^2 + 1)/(x^2 + 5)
For the second, factor x^2 - 9 into (x - 3)(x + 3), cancel the common factor and substitute: x + 3 becomes 6. For the third, sin(kx)/x approaches k as x approaches 0, and here k is 6. For the last, divide top and bottom by x^2: the limit is 6/1 = 6 because the leading coefficients decide the end behavior. The first does not exist, since 6/x grows without bound from the right and falls without bound from the left. The fourth is 0/6 = 0 at x = 6.
2.Let f(x) = x^2 + 1 for x < 2 and f(x) = kx - 1 for x >= 2. For what value of k is f continuous at x = 2?
- A. 2
- B. 3
- C. 4
- D. 5
Answer: B. 3
Continuity at x = 2 needs the left limit, the right limit and f(2) to agree. From the left, x^2 + 1 approaches 4 + 1 = 5. At and from the right, kx - 1 gives 2k - 1. Setting 2k - 1 = 5 gives k = 3. Always check all three pieces: a function can have equal one-sided limits and still be undefined or different at the point itself.
3.If f(x) = x * e^x, what is f'(1)?
- A. e
- B. e + 1
- C. 2e + 1
- D. 2e
Answer: D. 2e
Use the product rule: the derivative of x times e^x is (1)(e^x) + (x)(e^x) = e^x (1 + x). At x = 1 that is e * 2 = 2e. The common slip is to differentiate each factor and multiply the results, which gives e^x and loses the x e^x term.
4.What is the value of lim h->0 of ((2 + h)^3 - 8)/h ?
- A. 0
- B. 8
- C. 12
- D. The limit does not exist
Answer: C. 12
This is the limit definition of the derivative: it equals f'(2) for f(x) = x^3, and f'(x) = 3x^2 gives 3(4) = 12. You can also expand directly: (2 + h)^3 = 8 + 12h + 6h^2 + h^3, so the numerator is 12h + 6h^2 + h^3. Dividing by h leaves 12 + 6h + h^2, which approaches 12. Direct substitution gives 0/0, which means the form is indeterminate, not that the limit is 0.
5.If y = (3x + 1)^4, what is dy/dx?
- A. 12(3x + 1)^3
- B. 4(3x + 1)^3
- C. 12(3x + 1)^4
- D. 3(3x + 1)^4
Answer: A. 12(3x + 1)^3
Apply the chain rule. The outside function is something to the 4th power, so its derivative is 4 times the inside raised to the 3rd power. Then multiply by the derivative of the inside, which is 3. That gives 4 * 3 * (3x + 1)^3 = 12(3x + 1)^3. The answer 4(3x + 1)^3 is what you get if you forget the factor from the inside function.
6.The curve x^2 + y^2 = 25 passes through the point (3, 4). What is dy/dx at that point?
- A. 3/4
- B. -3/4
- C. 4/3
- D. -4/3
Answer: B. -3/4
Differentiate both sides with respect to x and remember that y depends on x: 2x + 2y(dy/dx) = 0. Solving gives dy/dx = -x/y. At (3, 4) that is -3/4. It also makes geometric sense: the radius to (3, 4) has slope 4/3, and a tangent to a circle is perpendicular to the radius, so its slope is the negative reciprocal, -3/4.
7.A particle moves along a line with position s(t) = t^3 - 6t^2 + 9t for t >= 0. Which statements are true?
Select all that apply.
- A. The acceleration is positive at t = 1
- B. The particle is at rest at t = 1 and at t = 3
- C. The particle is speeding up at t = 2
- D. The particle is moving to the left at t = 2
- E. The particle is slowing down at t = 0.5
Answer: B. The particle is at rest at t = 1 and at t = 3, D. The particle is moving to the left at t = 2, E. The particle is slowing down at t = 0.5
Velocity is s'(t) = 3t^2 - 12t + 9 = 3(t - 1)(t - 3), which is 0 at t = 1 and t = 3. Acceleration is s''(t) = 6t - 12. At t = 2, velocity is 3(1)(-1) = -3, so the particle moves left, and acceleration is 0, so it is neither speeding up nor slowing down at that instant. At t = 1, acceleration is -6, which is negative. At t = 0.5, velocity is positive (3.75) and acceleration is negative (-9), so the signs differ and the particle is slowing down. A particle speeds up only when velocity and acceleration have the same sign.
8.The radius of a circle is increasing at a constant rate of 2 cm per second. How fast is the area increasing at the instant the radius is 5 cm?
- A. 10 pi cm^2 per second
- B. 25 pi cm^2 per second
- C. 100 pi cm^2 per second
- D. 20 pi cm^2 per second
Answer: D. 20 pi cm^2 per second
Write the relationship, A = pi r^2, then differentiate with respect to time: dA/dt = 2 pi r (dr/dt). Only now substitute the numbers: 2 pi (5)(2) = 20 pi. Substituting r = 5 before differentiating is the classic related rates mistake, because it turns a changing quantity into a constant. The answer 10 pi comes from leaving out dr/dt, and 25 pi is the area itself at that instant.
9.For f(x) = x^2 on the interval [1, 3], the Mean Value Theorem guarantees a number c in (1, 3) where f'(c) equals the average rate of change. What is c?
- A. 1
- B. 1.5
- C. 2
- D. 2.5
Answer: C. 2
The average rate of change is (f(3) - f(1))/(3 - 1) = (9 - 1)/2 = 4. Now solve f'(c) = 2c = 4, which gives c = 2. The theorem needs f to be continuous on [1, 3] and differentiable on (1, 3), and a polynomial meets both. It promises that such a c exists. It does not say how to find it, so you still have to solve for it.
10.Let f(x) = x^3 - 3x. Which statements are true?
Select all that apply.
- A. f is increasing on the interval (-1, 1)
- B. f has a local maximum at x = -1
- C. f is concave up on the interval (-infinity, 0)
- D. f has a point of inflection at x = 0
- E. f has a local minimum at x = -1
Answer: B. f has a local maximum at x = -1, D. f has a point of inflection at x = 0
The first derivative is f'(x) = 3x^2 - 3 = 3(x - 1)(x + 1). It is positive for x < -1, negative between -1 and 1, and positive for x > 1. So f rises, then falls, then rises: x = -1 is a local maximum and x = 1 is a local minimum, and f is decreasing on (-1, 1). The second derivative is f''(x) = 6x. It is negative for x < 0, so f is concave down there, and positive for x > 0. Because f'' changes sign at x = 0, that point is an inflection point.
11.Complete the sentence.
On the closed interval [0, 4], f(x) = x^2 - 4x + 1 has an absolute minimum value of _____ and an absolute maximum value of _____.
Options for blank 1: 1, -3, -4, -2
Options for blank 2: -3, 0, 1, 5
Answer: -3, 1
Use the candidates test. Set f'(x) = 2x - 4 = 0 to get the critical point x = 2, where f(2) = 4 - 8 + 1 = -3. Then check the endpoints: f(0) = 1 and f(4) = 16 - 16 + 1 = 1. The smallest value is -3 and the largest is 1. On a closed interval the absolute extremes happen at a critical point or an endpoint, so skipping the endpoints is the usual way to lose this question.
12.Let g(x) be the integral of sqrt(t^2 + 9) dt from t = 0 to t = x. What is g'(4)?
- A. 5
- B. 4
- C. 12
- D. 25
Answer: A. 5
This is the first part of the Fundamental Theorem of Calculus: when the upper limit is x, the derivative of the accumulated integral is the integrand evaluated at x. So g'(x) = sqrt(x^2 + 9), and g'(4) = sqrt(16 + 9) = sqrt(25) = 5. You never need to find an antiderivative, and trying to would be hard here anyway.
13.A left Riemann sum with three subintervals of equal width approximates the integral of f(x) = x^2 from x = 0 to x = 3. What is its value?
- A. 4
- B. 5
- C. 9
- D. 14
Answer: B. 5
Each subinterval has width 1, and a left sum uses the left endpoint of each: x = 0, 1 and 2. The sum is 1 * (0 + 1 + 4) = 5. A right sum would use x = 1, 2 and 3 and give 1 + 4 + 9 = 14. The exact integral is 9, which lies between the two. Because x^2 is increasing on [0, 3], the left sum underestimates and the right sum overestimates.
14.Complete the sentence.
The definite integral of 3x^2 dx from x = 0 to x = 2 equals _____, so the average value of 3x^2 on that interval equals _____.
Options for blank 1: 4, 6, 8, 12
Options for blank 2: 2, 4, 6, 8
Answer: 8, 4
An antiderivative of 3x^2 is x^3, so the integral is 2^3 - 0^3 = 8. The average value of a function on [a, b] is the integral divided by the length of the interval: 8/(2 - 0) = 4. Notice that the average value is not the average of the endpoint values. At the endpoints 3x^2 is 0 and 12, and their mean would be 6.
15.Complete the sentence.
The solution to the differential equation dy/dx = 2y with y(0) = 3 is y = _____ e^(_____x).
Options for blank 1: 1, 2, 3, 6
Options for blank 2: 1, 2, 3, 6
Answer: 3, 2
This is exponential growth. Separate the variables: dy/y = 2 dx, so ln|y| = 2x + C, which gives y = A e^(2x). The initial condition y(0) = 3 means A e^0 = A = 3, so y = 3e^(2x). The rate constant 2 sits in the exponent and the starting value 3 sits in front. A quick check: differentiating 3e^(2x) gives 6e^(2x), which is 2 times y.
What the AP Calculus AB exam looks like and how it is weighted
The AP Calculus AB exam is scored 1 to 5 and has two sections of equal weight. According to the College Board, Section I is 42 multiple-choice questions in 1 hour 40 minutes: Part A has 29 questions with no graphing calculator, worth 35 percent of the score, and Part B has 13 questions where a graphing calculator is required for some, worth 15 percent. Section II is 6 free-response questions in 1 hour 30 minutes: Part A has 2 problems that need a graphing calculator and Part B has 4 problems that do not. The exam is hybrid digital, so you answer the multiple-choice questions in the Bluebook app and handwrite your free-response answers in a paper booklet. Older review books and blog posts list a different question count and length, so trust the current College Board page for the format and the date.
The course is organized into eight units, and the College Board publishes how much of the exam each one gets. Limits and continuity is roughly 10 to 12 percent, differentiation basics roughly 10 to 12, composite, implicit and inverse functions roughly 9 to 13, contextual applications of differentiation roughly 10 to 15, analytical applications of differentiation roughly 15 to 18, integration and accumulation of change roughly 17 to 20, differential equations roughly 6 to 12, and applications of integration roughly 10 to 15. Units 5 and 6 together are about a third of the exam, so they are where a weak spot costs the most.
A 5 is the top score, but what a score is worth is up to each college. Some give credit for a 3, many ask for a 4 or a 5, and policies differ by course. Look up the schools you care about before you decide how hard to push.
How to use this sampler, and how to take it on paper
The 15 questions are spread across the eight units, with the most on the two heaviest ones, analytical applications of differentiation and integration. They ask for the reasoning behind the move rather than a memorized result: why the chain rule needs the derivative of the inside function, why a left Riemann sum underestimates an increasing function, why a candidate has to be checked against the endpoints, why acceleration alone does not tell you whether a particle speeds up. That is the thinking the free-response section rewards too.
Take it in one sitting with scratch paper and no calculator, then read the explanation for every question, including the ones you got right. Treat a miss as a topic. If you missed the implicit differentiation question and the related rates question, the gap is probably the habit of differentiating with respect to time or to x and carrying the dy/dx along, not two separate facts. Math is written in plain text here, so x^2 means x squared and e^(2x) means e raised to the power 2x.
If you would rather work on paper, use the Download test as PDF button above the questions. The file has all 15 questions with an answer key on the last page, and it needs a free StudyPDF account, while playing online is free. The questions are original and written to the published course outline. They are not released exam questions, and this sampler does not practice the free-response section, so pair it with past free-response questions from the College Board.
How StudyPDF builds AP Calculus AB practice tests from your class notes
Fifteen questions can show you which unit is shaky. They cannot get you ready on their own, and a generic bank rarely matches the problem types and notation your teacher actually uses.
StudyPDF works from your material instead. Upload your textbook chapters, your teacher's slides, a worked problem set or your review packet, press New material, and StudyPDF builds practice tests from those exact pages. Every question is grounded in your material and every explanation cites where the answer comes from, so you can go straight back to the page when a step does not make sense.
You do not need to upload anything to start. Name a topic instead, for example related rates or the fundamental theorem of calculus, and Bo sets it up with its key concepts so you can build quizzes and flashcards on it. Your results track which concepts you keep missing, so the next practice test can target them. Starting is free.
Written by the StudyPDF team. Last updated August 19, 2026.
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Is this AP Calculus AB practice test free?
Yes. All 15 questions, the score screen and every worked explanation are free, and the quiz runs in your browser. A free StudyPDF account is needed to download the test as a PDF with an answer key or to build more practice tests from your own notes.
Who wrote these questions?
The StudyPDF team wrote every question and explanation, following the eight units of the published AP Calculus AB course outline. They are original items, not real or released exam questions, and StudyPDF is not affiliated with the College Board, which does not endorse this page.
How is this different from the real AP Calculus AB exam?
The real multiple-choice section has one correct answer per question and is split into a part without a graphing calculator and a part with one. This sampler also uses select all that apply and fill in the blank, because judging each option separately is a good check of a unit, and every question can be done by hand. It does not practice the six free-response questions.
Can I get practice tests built from my own class materials?
Yes. Upload your textbook chapters, slides, problem sets or review packet to StudyPDF and you can build practice tests from those exact pages, with explanations tied to your sources. If you have nothing to upload, name a topic or a unit instead.