AP Calculus AB and BC · Exam guide
AP Calculus AB and BC FRQ Guide: How the Free-Response Questions Are Scored
The free-response section of the AP Calculus AB and BC exams has six questions in 90 minutes and counts for 50 percent of your exam score. Each question is worth 9 points. The questions are less about getting a number and more about showing that you know why the number is right. This guide explains the format and the writing habits that earn points.
The format
| Part | Questions | Time | Calculator |
|---|---|---|---|
| A | 2 | 30 minutes | Graphing calculator required |
| B | 4 | 60 minutes | No calculator |
Each question has several parts, often labeled (a) to (d), and the parts build on each other. If you cannot do part (a), you can often still do part (b), because later parts usually give you what you need.
What the questions test
- Rates and accumulation in context, often with a table or a function for a rate: total change, average value, and the meaning of an integral.
- Graphical analysis, where you are given a graph of a function or its derivative and asked about extrema, concavity, and area.
- Differential equations, including slope fields, separation of variables, and particular solutions.
- Analytical applications, such as tangent lines, the Mean Value Theorem, related rates, and optimization.
- Area and volume with integrals.
- BC only: parametric, polar, and vector-valued motion, and series, which is typically the last question.
Writing habits that earn points
| When the question says | What to write |
|---|---|
| Justify, or give a reason | Cite the fact you are using, such as "f'(x) changes from positive to negative at x = 3" or the hypotheses of a theorem. A bare statement of the answer earns no justification point. |
| Explain the meaning in context | Write a full sentence using the context's terms and include units, such as "the amount of water in the tank is increasing at 4 gallons per minute". |
| Show the work that leads to your answer | Write the setup, such as the integral or the equation, before any numerical answer. On Part A, write the integral you are evaluating, not just the calculator's result. |
| Find, to three decimal places | Round or truncate to three decimals, and do not round in the middle of a calculation. |
A worked example, with a weak and a strong answer
Water flows into and out of a tank. For \(0 \le t \le 10\) minutes, the rate at which the amount of water changes is \(R(t) = 12 - 2t\) gallons per minute. The tank holds 30 gallons at \(t = 0\). (a) How many gallons of water are in the tank at \(t = 6\)? (b) At what time is the amount of water in the tank greatest? Justify your answer.
For (a), set up and evaluate the integral: \(30 + \int_0^6 (12 - 2t)\,dt = 30 + \left[12t - t^2\right]_0^6 = 30 + (72 - 36) = 66\) gallons.
It finds a critical point but does not show it is a maximum, and does not check the endpoints.
It names the derivative relationship, shows the sign change, and checks the endpoints.
Common mistakes
- Stating an answer with no justification, which earns the answer point at best.
- Forgetting units when the question asks for meaning in context.
- Using the calculator to evaluate an integral without writing the integral first.
- Rounding too early, which can change the last digit of the answer.
- Skipping a part because you cannot finish part (a). Later parts often give you what you need.
Frequently asked questions
How many points is each calculus FRQ worth?
Each of the six questions is worth 9 points. Together they are 50 percent of your exam score.
When can I use my calculator?
On Part A, the first two questions, a graphing calculator is required. On Part B, the last four, no calculator is allowed.
Do I need to simplify my answer?
Usually not. An unsimplified correct expression is generally accepted, but an incorrect simplification can lose a point, so simplify only if it is easy and safe.
Do I need to include units?
Yes, when you explain a meaning in context or give a quantity with units. A missing unit can cost a point.
What if I do not know how to start?
Write down the definition or rule that applies and set up what you can. Many parts give partial credit for correct setup.
Keep practicing
Every lesson in the AP Calculus courses ends with practice problems and an Exam focus box that shows how the topic is tested, and every unit has a 10-problem practice set with worked answers. These free lessons also build the skills the free-response section tests:
See the AP Calculus AB and BC course Free lessons and lesson list See the AP Calculus BC course
Other free-response guides:
AP is a registered trademark of the College Board, which is not affiliated with and does not endorse this site. This guide is our own explanation of the question types and scoring. For the official scoring guidelines, see the College Board.