The AP Calculus BC Free Response section can be intimidating if you aren't prepared. You'll need to answer six multi-part questions to show your skills in numerous calculus topics. However, when you know what to expect from this section, you'll save yourself a lot of anxiety and can create a study plan to help you get a top AP score. In this guide, we explain everything you need to know about the AP Calc BC free response section, including what the format is, when you can and can't use a calculator, what sample questions look like and how to get a top score on them, and what study tips you need to keep in mind.
Overview of the AP Calculus BC Exam
AP Calculus BC is one of two AP Calculus tests, and it covers slightly more advanced material than AP Calculus AB. The exam tests your knowledge of various calculus topics such as limits, integers, and differential equations. You can learn more about exactly what you'll be tested on by looking at the official AP Calc AB and BC Course and Exam Description. The AP Calculus BC exam is divided into two major sections: multiple choice and free response.
Each section is worth half of your total exam score, and the total exam will take three hours and 15 minutes. Both the multiple-choice and free-response sections are divided into a Part A and a Part B. For some sections you'll be able to use a graphing calculator, while for others you won't. Below is an overview of the exam's format.
| AP Calc BC Section | % of Score | Time | # of Questions | Calculator? |
| Multiple Choice | Part A: 33.3% | 60 mins | 30 | No |
| Part B: 16.7% | 45 mins | 15 | Yes | |
| Total: 50% | Total: 1 hr 45 mins | Total: 45 | — | |
| Free Response | Part A: 16.7% | 30 mins | 2 | Yes |
| Part B: 33.3% | 60 mins | 4 | No | |
| Total: 50% | Total: 1 hr 30 mins | Total: 6 | — | |
| TOTAL | 100% | 3 hrs 15 mins | 51 | — |
What Is the AP Calculus BC Free Response Section Like?
You'll take the AP Calculus BC free response section after you've completed the multiple-choice section. Free response is the second and final section of the test.
You'll have a total of 90 minutes to complete this section, which consists of six questions divided into two parts: Part A and Part B. According to the College Board, the questions will "include various types of functions and function representations and a roughly equal mix of procedural and conceptual tasks." There will also be at least two questions that incorporate "a real-world context or scenario into the question."
Here's the format of the free-response section:
| AP Calculus BC Free Response | Time | # of Questions | % of Total Score | Calculator Policy |
| Part A | 30 mins | 2 | 16.7% | Required |
| Part B | 60 mins | 4 | 33.3% | Not permitted |
For the AP Calculus BC FRQ section, you'll be tested on four main skill areas, according to the College Board:
| Skill Area | Description | % of AP Calculus BC Free-Response Section |
| Implementing Mathematical Processes | Determine expressions and values using mathematical procedures and rules | 37-59% |
| Connecting Representations | Translate mathematical information from a single representation or across multiple representations | 9-16% |
| Justification | Justify reasoning and solutions | 37-59% |
| Communication and Notation | Use correct notation, language, and mathematical conventions to communicate results or solutions | 9-20% |
For each AP Calc BC FRQ, you'll receive a score from 0-9. Free-response questions contain multiple parts, and you'll receive a certain number of points (typically 1-3) for each part of the question. The points you earn from the free-response section will be combined with your multiple-choice section score, then converted to your final AP score on a scale of 1-5.
AP Calculus BC Free Response Sample Questions
What do AP Calc BC FRQ look like, and how do you go about solving them? In this section, we'll show you two official AP Calculus BC FRQ from the 2019 AP Calc BC exam. For each question, we'll explain how it is scored and what your answer needs to include for you to receive a full score.
Sample Question 1: Part A AP Calculus BC FRQ (Calculator Required)
Part A
To solve the first part of the question, you need to take the integral, which will get you the answer of 153.46. Round to the nearest whole number (153) to get your final answer. Part A is worth two points: one for showing the correct integral, and one for having the final correct answer.
Part B
For Part B, you again need to take an integral, this time for the average number of fish leaving the lake each hour from midnight to 5AM. This gets you a value of 6.059, so the final answer is 6.059 fish per hour. Part B is worth two points, one for the correct integral, and one for the correct final answer.
Part C
Part C is a little more complicated than the first two parts of the question, and the solution above shows two ways to solve the problem. In either case, first you need to know that E(t) - L(t) is equal to zero. That then allows you to solve for t which is 6.20356. You then know that the greatest change in the number of fish is 6.204 hours after midnight. This part is worth three points: one for knowing E(t)-L(t)=0, one point for correctly solving for t (both 6.204 and 6.203 are accepted answers), and one point for the justification.
Part D
For Part D, you need to solve for E'(5) - L'(5) and understand what the value means. Part D is worth two points. You can earn one point for solving for E'(5) - L'(5) (the correct answer is -10.7228) and one point for the correct explanation (because the value is negative, the rate in change in number of fish is decreasing at t = 5.)
Sample Question 2: Part B AP Calculus BC FRQ (Calculator Not Permitted)
Part A
Part A is worth three points. You'll earn one point for knowing the denominator of f'(x), which is (x2 -2x + k)2. You'll earn one point for the correct complete f'(x) equation, and one point for finding the correct value of k.
Part B
Part B is also worth three points. The first point is for showing the partial fraction decomposition. Partial fraction decomposition is when you start with the simplified answer and change it to polynomial fractions. In this case, to earn the point you'd need to take x2 -2x + 8 and convert it to (x-4)(x+2), then be able to find that A= ⅙ and B= -⅙. The second point is for writing out the correct antiderivatives, and the third point is for working them out and getting the answer of -⅙ ln2.
Part C
Part C is, as you likely guessed, also worth three points. You earn one point for writing out the improper integral, one point for the antiderivative, and one point for finding both that the limit does not exist and explaining that, because the limit doesn't exist, the integral diverges.
Tips for Answering AP Calculus BC FRQ
As you're preparing for the AP Calc BC exam, keep these three tips in mind when studying and working through practice FRQ.
Tip 1: Know How to Use Your Calculator
For Part A of AP Calculus BC free response (the first two questions of the section), you're required to use a calculator. It's not just allowed; you absolutely need a calculator to solve those questions.
While it might seem obvious that you should know how to work your calculator, knowing exactly how and when to use its different functions can save you a lot of time on the exam and increase your chances of getting the correct answer.
According to the College Board, the four calculator capabilities you'll use the most during the AP Calculus AB exam and should easily be able to do with your calculator are:
- 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're preparing for the exam, be sure that you know how to do each of these (completing practice problems can help!).
Tip 2: Study With High-Quality Practice Questions
Your studying will only be as effective as your study materials, so you want to use the best. The best way to prepare for the AP Calculus BC exam is to use official practice questions and tests. Official practice materials are those produced by the College Board, and you can be sure they're high-quality because they're made by the same people making the actual AP exam.
We recommend you begin your AP exam prep by taking a full-length practice test (with proper timing). That will let you get used to the difficulty, content, and pacing of the test and the free-response section specifically.
We have a guide to the best AP Calculus BC practice tests to help you get started. For free-response questions specifically, the College Board offers 20 years worth of FRQ used in old AP exams. These are a great resource, and you should absolutely take advantage of them.
Tip 3: Show Your Work
For most free-response questions on the test, the final answer to a problem is only worth 1-2 points out of a possible 9. This means that the majority of points are earned through intermediate steps of the problem—and if you don't show how you reached those intermediate steps, you'll get a very low score on AP Calc BC FRQ.
Even if you solve a problem by using your calculator, you have to write the setup (such as the equation being solved or the derivative being evaluated) as well as the answer in order to get credit for your work.
You might be used to not writing down certain work that seems particularly obvious on homework and class tests. However, even if your teacher doesn't mind, AP graders will.
Remember that AP graders are more interested in how you reached your final answer than they are in what that final answer is, so get yourself in the habit of showing each step of your work well before exam time.
What's Next?
AP Calculus BC is known for being a tough class, but how does it compare to other APs? Learn what the hardest AP classes and tests are by reading our guide.
How many students actually pass the AP Calculus BC test? Our guide has average scores, passing rates, and 5 rates for every AP exam.
When's the ideal time in your high school career to take the ACT/SAT? Find out here.
