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 giving 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 Calculus BC FRQs as you prepare for your upcoming AP Calculus BC exam.
Format of the 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 them. 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 10 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:
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Read all the questions thoroughly.
As you begin each section, ensure you read all the questions thoroughly first. This approach helps you identify the topics in which you are strongest. Start by addressing these questions; doing so allows you to get them out of the way and boosts your confidence early. Moreover, this strategy prevents you from spending too much time on the more challenging questions later, ensuring you have ample time to devote to the ones about which you feel confident.
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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 focus on the important aspects of the question. When you hit a vocabulary word, stop and analyze what it means. For instance, if you encounter a question that mentions "relative maximum," pause to recall its meaning in calculus. This includes understanding it as a point where the derivative equals zero and the sign changes from positive to negative or as a situation where the second derivative is negative, among other related concepts.
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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.
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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.
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Save time, draw a line.
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.
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Show your work.
Unless the question explicitly tells you to simplify or give a numerical answer, leave your answers in their original, complete form. The scorers will accept your response in any format as long as you've completed the work required by the question. While preparing for the exam, understand the scoring guidelines and study the sample answers provided by the College Board. This will give you an even better idea of when you should or should not simplify your answers.
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Use your calculator wisely for 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.
You are not expected to show intermediate steps when you need to do any of these four things in a Part A FRQ. Just show the setup correctly, plug it into your calculator, and give the numerical answer.
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Return to Part A.
If you haven't completed your responses in Part A and find extra time after tackling the Part B questions, revisit the Part A questions to complete any unfinished work. Keep in mind, however, that you won't have access to your calculator during this time, which may restrict what you can accomplish.
If you spot a mistake in the way you've set up an integral or equation, correct it, even if you're unable to compute the result without a calculator. Setting up the problem correctly can still earn you points, even without arriving at 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 scenarios. 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
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 take various forms. The example below is from 2021, 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.
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 forms. The example below is from 2021, 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're currently navigating through the AP Calculus BC curriculum, it's advisable to collaborate with your teacher to create a study plan. Ideally, your teacher would incorporate practice with Free-Response Questions (FRQs) throughout the year, drawing from past exams or AP Classroom resources. However, if FRQ practice isn't provided in your coursework, or you're studying independently, make use of past exams by focusing on FRQ sections that are within your current understanding and practicing those throughout the year. Early on, this approach might seem daunting due to your limited knowledge of the material. Still, by selectively working on manageable parts of the FRQs and leaving the rest for later, you can effectively build your skills.
Once you've mastered the course content, begin your test preparation by tackling entire sections of Free-Response Questions (FRQs) in one go. Focus primarily on the most recent exams since the College Board's question formats can evolve 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.
References
- AP Calculus AB and BC Course and Exam Description. (Fall, 2020). apcentral.collegeboard.org. Retrieved on March 18, 2024 from https://apcentral.collegeboard.org/media/pdf/ap-calculus-ab-and-bc-course-and-exam-description.pdf
- AP Calculus BC. (n.d.). apstudents.collegeboard.org. Retrieved on March 18, 2024 from https://apstudents.collegeboard.org/courses/ap-calculus-bc
- AP Calculus BC. (n.d.). apstudents.collegeboard.org. Retrieved on March 18, 2024 from https://apstudents.collegeboard.org/courses/ap-calculus-bc/assessment
- AP Calculus BC Free-Response Questions. (2018). apcentral.collegeboard.org. Retrieved on March 18, 2024 from https://apcentral.collegeboard.org/media/pdf/ap18-frq-calculus-bc.pdf
- AP Calculus BC Free-Response Questions. (2021). apcentral.collegeboard.org. Retrieved on March 18, 2024 from https://apcentral.collegeboard.org/media/pdf/ap21-frq-calculus-bc.pdf
- AP Calculus BC Free-Response Questions. (2022). secure-media.collegeboard.org. Retrieved on March 18, 2024, from https://secure-media.collegeboard.org/apc/ap22-frq-calculus-bc.pdf
Frequently Asked Questions
How are the AP Calculus BC Free-Response questions graded?
High school AP Calculus teachers and college professors who teach calculus grade AP Calculus BC FRQs by hand. 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: 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.