How many points is each AP Calculus BC FRQ worth?

Each of the six free-response questions is worth 9 points, and the section is half of your total score.

Do I need to simplify my answers on free response?

Generally no. Unless a question says otherwise, answers don't need to be simplified, and decimal answers should be correct to three places.

Where can I see real scoring guidelines?

The College Board publishes past free-response questions and their scoring guidelines on AP Central. Reading them is a good way to learn what readers look for.

What if I can't do an earlier part of a question?

Keep going. Later parts can often be done on their own, and if one needs your earlier answer, use your best value and show your work.

How to earn points on AP Calculus BC free response

Updated November 9, 2026

Free response is half of your AP Calculus BC score, and it's the one part where correct math can still lose points. Readers score what's on the page, against a rubric, so an answer needs to show how you got there and why it's true.

This guide covers the habits that show up again and again in published scoring guidelines, with sample wording you can adapt. For the exact rubric on any past question, see the College Board's AP Central site.

How free response is set up

Six questions, 9 points each. Two allow a calculator, and four don't.
You get 15 minutes per question. Parts usually build on each other, but not always, so a stuck part (b) doesn't mean you should skip (c).
You write your answers by hand on paper. Prompts appear in Bluebook.

The rules that apply to every question

Show the setup. On a calculator question, write the integral, derivative, or equation you evaluated, then the value.
Use math notation. Calculator syntax isn't accepted as work.
Name the function. Write " $f'$ changes from positive to negative," not "it changes." Vague pronouns can cost a point.
Include units when asked, and read carefully. An integral of a rate has the units of the amount.
Give three decimal places unless told otherwise, and don't round in the middle of a calculation.
You don't have to simplify. An unsimplified correct answer is fine, and simplifying only adds chances for a slip.
Use your earlier answer. If part (b) needs part (a), use the value you found, even if you're unsure of it.

Justifying a maximum or minimum

"Justify your answer" needs a reason stated in terms of the function. For a relative extremum, name the sign change of the derivative, or use the Second Derivative Test.

Sample wording

The function

g

is differentiable,

g'(3) = 0

g'(x) > 0

for

x

just less than

3

, and

g'(x) < 0

for

x

just greater than

3

. Does

g

have a relative maximum, a relative minimum, or neither at

x = 3

? Justify your answer.

Write: " $g$ has a relative maximum at $x = 3$ because $g'$ changes from positive to negative at $x = 3$ ." That sentence has the conclusion and the reason, and it names the function.

Answer: Relative maximum

Using the Intermediate Value Theorem

State continuity, give the two values, and say that the target value lies between them. Differentiable functions are continuous, so if you're told $f$ is differentiable, say that.

IVT from two values

The function

f

is differentiable, with

f(1) = 2

and

f(4) = 7

. Must there be a value

c

(1, 4)

with

f(c) = 5

? Justify your answer.

Write: "Yes. $f$ is differentiable and therefore continuous on $[1, 4]$ . Since $f(1) = 2 < 5 < 7 = f(4)$ , the Intermediate Value Theorem guarantees a value $c$ in $(1, 4)$ with $f(c) = 5$ ."

Answer: Yes, by the Intermediate Value Theorem

Using the Mean Value Theorem

State continuity on the closed interval and differentiability on the open one, then calculate the average rate of change.

MVT from two values

The velocity

v

of a car is differentiable, with

v(2) = 10

and

v(6) = 18

. Must there be a time

c

(2, 6)

when the acceleration is

2

? Justify your answer.

The average rate of change is $\frac{18 - 10}{6 - 2} = 2$ .

Write: "Yes. $v$ is differentiable, so it's continuous on $[2, 6]$ and differentiable on $(2, 6)$ . Since $\frac{v(6) - v(2)}{6 - 2} = 2$ , the Mean Value Theorem guarantees a value $c$ in $(2, 6)$ with $v'(c) = 2$ ."

Answer: Yes, by the Mean Value Theorem

Working from a table

Table questions often have uneven spacing, so calculate each piece separately and show the arithmetic setup.

A trapezoidal sum with uneven widths

A rate

R(t)

, in liters per hour, has

R(0) = 4

R(2) = 8

, and

R(5) = 6

. Use a trapezoidal sum with the two intervals given to approximate

\int_0^5 R(t)\,dt

. Include units.

The widths are $2$ and $3$ , so the sum is $\frac{1}{2}(2)(4 + 8) + \frac{1}{2}(3)(8 + 6) = 12 + 21 = 33$ . The integral of liters per hour over hours is liters.

Answer: $33$ liters

Series and BC-only questions

BC free response almost always includes a series question, often with a convergence test or an error bound. Name the test and check its conditions in writing. See which series test to use and Lagrange and alternating series error bounds.

Keep going

Practice on exam-style questions

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Questions

How many points is each AP Calculus BC FRQ worth?: Each of the six free-response questions is worth 9 points, and the section is half of your total score.
Do I need to simplify my answers on free response?: Generally no. Unless a question says otherwise, answers don't need to be simplified, and decimal answers should be correct to three places.
Where can I see real scoring guidelines?: The College Board publishes past free-response questions and their scoring guidelines on AP Central. Reading them is a good way to learn what readers look for.
What if I can't do an earlier part of a question?: Keep going. Later parts can often be done on their own, and if one needs your earlier answer, use your best value and show your work.