AP® Calculus BC Free-Response Questions (FRQs)
The AP® Calculus BC exam consists of two major sections: multiple-choice questions (MCQs) and free-response questions (FRQs). In this guide, we’ll take a look at the FRQ section of the exam.
We will start by breaking down the format of Section II of the AP Calculus BC exam and give you tips to maximize your points on the FRQs. In the sections that follow, we have also included a few examples of AP Calculus BC free-response questions you might see on the exam. By the end of this article, you will know how to practice AP Calc BC FRQs as you get ready for your upcoming AP Calc BC exam.
Format of AP Calculus BC Free Response Section
Let’s start this section by answering the most frequently asked question: “How many FRQs are on the Calculus BC exam? ” The FRQ section of the AP Calculus BC exam is split into two parts: Part A and Part B. Part A consists of two questions on which you are allowed to use a calculator, and you are given 30 minutes to complete it. Part B consists of four questions for which a calculator is not allowed, and you get one hour to address this part.
The entire Section II accounts for 50% of the total points on the exam, where each question is weighted the same regardless of whether it is in Part A or B. The questions cover all ten units of the AP Calculus BC course. The table below summarizes the crucial information about the format of the MCQ section on the AP Calculus BC exam.
| Section II | Part A | Part B |
|---|---|---|
| No. of Questions | 2 FRQs | 4 FRQs |
| Exam Weight | 16.7% | 33.3% |
| Time Allotted | 30 minutes | 1 hour |
| Calculator Usage | Permitted | Not Permitted |
How to Answer AP Calculus BC’s Free-Response Questions?
Here are some tips for approaching the FRQ section of the AP Calculus BC exam:
- Read all the questions thoroughly.
When each section begins, start by reading all of the questions available in their entirety. As you read through the questions, you will notice concepts that you are more proficient with. Start by answering those questions to get them out of the way and build your confidence. This also keeps you from spending too much time on the questions you struggle with and not having enough time to address the questions you are confident about.
- Underline important information.
When reading through the question stem, underline things like vocabulary, given values, function definitions, and the actual quantity the question is asking for. This will help you key in on the important aspects of the question. When you hit a vocabulary word, stop and analyze what that means. For example, if the question says “relative maximum,” take a moment to think about what that means in calculus, like “derivative equals 0 and changes sign from positive to negative,” “second derivative is negative,” and so on.
- Treat each part of every FRQ independently.
Some parts of the questions are related to each other. For example, in part a, they may ask you to estimate a Riemann sum, and then in part b, they may ask you to interpret whether your answer in part a is an overestimate or underestimate. Even if you are not confident in your answer to part a, answer part b as if your answer to part a was correct. Readers are instructed to grade each answer independently. Therefore, you can still get full points on part b even if your answer to part a is incorrect but you interpreted it correctly.
- Organize your work.
As you prepare for the exam, use the College Board® scoring guidelines to study what kinds of things they instruct the readers to look for. When you answer questions on the exam, only include the information you need to score points. If you need to write things down as you work toward your answers, do so in the question booklet.
- Don’t erase.
If you make an error and catch it, do not erase it. Instead, draw a line or X through the sections you don’t want the readers to consider. Doing this saves time over erasing.
- Don’t simplify.
Unless the question explicitly tells you to simplify or give a numerical answer, leave your answers in their unsimplified form. As long as you have done the work that the question asks for, the readers will accept your answer in any form. Understand the scoring guidelines and study the sample answers provided by the College Board while preparing for the exam. This will give you an even better idea of when you should or should not simplify your answers.
- Don’t overwork on Part A.
The following are four functionalities that the College Board expects your graphing calculator to do:
- Plot the graph of a function within an arbitrary viewing window.
- Find the zeros of functions (solve equations numerically).
- Numerically calculate the derivative of a function.
- Numerically calculate the value of a definite integral.
When you need to do any of these four things in a Part A FRQ, you are not expected to show any intermediate steps. Just show the set up correctly, plug it into your calculator, and give the numerical answer.
- Return to Part A.
If you didn’t finish writing out your answers in Part A and if you have time after finishing the Part B questions, turn back to the questions in Part A and finish up anything you need to. However, you will no longer have access to your calculator, so what you can do may be limited.
If you notice an error in setting up an integral or equation, go ahead and fix the error, even if you can’t calculate the result without a calculator. You may get points for the correct setup even without the correct numerical answer.
AP Calculus BC Free-Response Question Examples
Here are some examples of AP Calc BC FRQs from past exams to illustrate the different kinds of questions you will see on the exam. These questions come directly from the College Board and are an excellent source to practice with.
One of the question types that appear most frequently on the AP Calculus BC exam is a table of data modeling some real-world scenario. The College Board will use the table to test concepts that allow you to calculate approximate values, like average rate of change and Riemann sums. The intermediate value theorem is another concept that is easily tested with tabular data. An example of this type of question (#4 from 2022) is shown below with highlights demonstrating these common elements.
| t (days) |
0 | 3 | 7 | 10 | 12 |
| r ‘ (t ) (centimeters per day) |
−6.1 | −5.0 | −4.4 | −3.8 | −3.5 |
An ice sculpture melts in such a way that it can be modeled as a cone that maintains a conical shape as it decreases in size. The radius of the base of the cone is given by a twice-differentiable function r , where r (t ) is measured in centimeters and t is measured in days. The table above gives selected values of r’ (t ), the rate of change of the radius, over the time interval 0 ≤ t ≤ 12.
- Approximate r” (8.5) using the average rate of change of r’ over the interval 7 ≤ t ≤ 10. Show the computations that lead to your answer, and indicate units of measure.
- Is there a time t, 0 ≤ t ≤ 3, for which r’ ( t ) = −6 ? Justify your answer.
- Use a right Riemann sum with the four subintervals indicated in the table to approximate the value of ∫120 r’ (t ) dt.
- The height of the cone decreases at a rate of 2 centimeters per day. At time t = 3 days, the radius is
100 centimeters and the height is 50 centimeters. Find the rate of change of the volume of the cone with respect
to time, in cubic centimeters per day, at time t = 3 days. (The volume V of a cone with radius r and height h is V = 1⁄3 πr 2h. )
Ref: College Board: Example 1 is on page 8 of
https://secure-media.collegeboard.org/apc/ap22-frq-calculus-bc.pdf
Another commonly asked free-response question in the AP Calculus BC exam is understanding meaning within a context. You will be asked to calculate a quantity and give an interpretation of what that quantity means within the given scenario. You will have to present your inference with the correct units, requiring you to understand how a derivative or integral changes the units of a function. This often occurs within a table question, but not always. The question below (2018 #2) is one such question.
Researchers on a boat are investigating plankton cells in a sea. At a depth of h meters, the density of plankton cells, in millions of cells per cubic meter, is modeled by p(h) = 0.2h2e-0.0025h2 for 0 ≤ h ≤ 30 and is modeled by f(h) for h ≥ 30. The continuous function f is not explicitly given.
- Find p’ (25). Using correct units, interpret the meaning of p’ (25) in the context of the problem.
- Consider a vertical column of water in this sea with horizontal cross sections of constant area 3 square meters. To the nearest million, how many plankton cells are in this column of water between h = 0 and h = 30 meters?
- There is a function u such that 0 ≤ f(h) ≤ u(h) for all h ≥ 30 and ∫∞30 u (h) dh = 105. The column of water in part (b) is K meters deep, where K > 30. Write an expression involving one or more integrals that gives the number of plankton cells, in millions, in the entire column. Explain why the number of plankton cells in the column is less than or equal to 2000 million.
- The boat is moving on the surface of the sea. At time t ≥ 0, the position of the boat is (x(t), y(t)), where x'(t) = 662 sin(5t ) and y'(t) = 880 cos(6t). Time t is measured in hours, and x(t) and y(t) are measured in meters. Find the total distance traveled by the boat over the time interval 0 ≤ t ≤ 1.
Ref: College Board: Example 2 is on page 3 of
https://apcentral.collegeboard.org/pdf/ap18-frq-calculus-bc.pdf
Another thing to note about the FRQ section of the AP Calculus BC exam is that the College Board aims to test your ability to interpret data from a number of different presentations: functions, graphs, and tables. The question below (2022 #5) shows a graph of a function and asks about several concepts that tend to give students a lot of trouble: area and volume from Unit 8 and improper integrals from Unit 6.
Figures 1 and 2, shown above, illustrate regions in the first quadrant associated with the graphs of y = 1⁄x and y = 1⁄x 2, respectively. In Figure 1, let R be the region bounded by the graph of y = 1⁄x, the x-axis, and the vertical lines x = 1 and x = 5. In Figure 2, let W be the unbounded region between the graph of y = 1⁄x 2 and the x-axis that lies to the right of the vertical line x = 3.
- Find the area of region R
- Region R is the base of a solid. For the solid, at each x the cross section perpendicular to the x-axis is a rectangle with area given by xe x/5. Find the volume of the solid.
- Find the volume of the solid generated when the unbounded region W is revolved about the x-axis.
Reference College Board: Example 3 is on page 9 of
https://secure-media.collegeboard.org/apc/ap22-frq-calculus-bc.pdf
On the AP Calculus BC exam, the sixth and final FRQ is always a power series from Unit 10. Most often, it includes a Taylor or Maclaurin polynomial or series, but it can come in various different forms. The example below is from 2022, and it includes a number of different techniques and concepts from throughout Unit 10.
Study power series thoroughly and be prepared for this concept to feature prominently in an FRQ.
The function g has derivatives of all orders for all real numbers. The Maclaurin series for g is given by g(x) = Σ∞n=0 (-1)n xn⁄2e n + 3 on its interval of convergence.
- State the conditions necessary to use the integral test to determine convergence of the series Σ∞n=0 1⁄e n . Use the integral test to show that Σ∞n=0 1⁄e n converges.
- Use the limit comparison test with the series Σ∞n=0 1⁄e n to show that thee series g (1) = Σ∞n=0 (-1)n⁄2e n + 3 converges absolutely.
- Determine the radius of convergence of the Maclaurin series for g
- The first two terms of the series g (1) = Σ∞n=0 (-1)n⁄2e n + 3 are used to approximate g (1). Use the alternating series error bound to determine an upper bound on the error of approximation.
Ref: College Board: Example 4 is on page 10 of
https://apcentral.collegeboard.org/pdf/ap21-frq-calculus-bc.pdf
How Can I Practice AP Calculus BC Free-Response Questions?
The best way to practice for the AP Calculus BC exam is to use the College Board's previous years’ exams.
If you are still learning the AP Calculus BC curriculum, it may be best if you could let your teacher help you with a plan. They should provide you with FRQ practice throughout the year, using the questions from past exams or questions from the AP Classroom. However, if your teacher does not give you FRQ practice during the school year or if you are self-studying, using past exams, look for parts of FRQs that you know how to do and practice throughout the year. It may be challenging early on because you haven’t learned much of the material yet, but pick and choose pieces of FRQs that you can do and save the rest for later.
As you start your test prep after learning the course content, start to take entire sections of FRQs at a time. Prioritize the most recent exams as the College Board changes how they ask questions over time. Use the scoring guidelines to grade yourself and study the rubrics that the College Board provides to readers. After taking a few sections, start timing yourself according to the time limits of the AP Calculus BC exam to prepare yourself for the stress of a time limit.
Frequently Asked Questions
How are the AP Calculus BC Free-Response questions graded?
AP Calculus BC FRQs are graded by hand by high school AP Calculus teachers and college professors who teach calculus. The College Board provides scoring guidelines to tell the readers what to look for, which tend to be similar from year to year. There is a designated head reader to make decisions if there is any confusion about how a question should be graded.
How long is the FRQ section of the AP Calculus BC exam?
The FRQ section of AP Calculus BC is an hour and a half long, split into two parts of 2 questions (30 minutes) and 4 questions (1 hour). You have an average of 15 minutes to answer each FRQ.
Where can I get the AP Calculus BC past exam FRQs?
You can find released questions from past exams on the AP Central website.
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