Free-response questions

The purpose of this article is to learn more about the free-response section of the AP Calculus BC exam and how to write high-quality responses.

There are two parts to the free-response section of the exam:

  • Part A (calculator) comprised of $2$ multi-part questions for $30$ minutes
  • Part B (no calculator) comprised of $4$ multi-part questions for $60$ minutes

There may be parts of questions in Part A (calculator) that do not require a calculator, and most topics in the course can occur in either section.

These two free-response parts account for half the total exam grade. Each question is worth $9$ points, totaling $54$ possible points.

Characteristics of a good written response

Correct setup

Before writing or evaluating with the particulars of the question, always start by writing mathematically valid formulas, expressions, or models that match the situation.

This demonstrates conceptual understanding before diving into any computations.

Application of calculus principles

Written work must go beyond memorized formulas—it must show the calculus operations and concepts correctly applied.

Justification and reasoning

Explain or justify key steps whenever the question involves interpretation, such as why a derivative models a rate of change.

Proper substitution, evaluation, and precision

After setup, substitute values accurately and consistently with observable notation, such as parentheses. Be especially clear with time or rate values, using units whenever possible.

Make sure to compute the result to the required precision, and do not round until your final step to avoid a cascading rounding issue. Requirements are often $3$ decimal places or reasonable exact form.

Consistency with context

Final answers and interpretations must align with the problem’s situational meaning: no negative counts of people, for example.

Clear communication

Present work with correct notation and structure. Responses must be readable and logically organized.

Strategies

Free-response questions do not have to be completed in the order they appear on the exam.

  • Start by reading the whole problem. It's important to read all parts of the question to understand the overall task and how parts may connect.

  • Select the parts of questions that are easier, and do them first. Only work on the more difficult parts of questions after answering all the easier parts of the questions.

  • The parts in each question are not generally dependent on each other. For example, an answer from one part of a question is usually not needed to answer another part.

  • When parts are dependent, if an incorrect answer from one part is used correctly in another part of the question, then partial points are possible.

  • When using a theorem that has conditions, we must write how those conditions have been met before applying the theorem.

  • Label all graphs and functions clearly.

Scoring

How many points?

Each free-response question on the AP Calculus BC exam is worth $9$ points. In general there are $3$ to $4$ parts to each question. Most parts to a question are worth $2$ to $3$ points, but occasionally a part of a question is worth only $1$ point.

Rubrics

Graders score free-response questions using a scoring rubric for each question, which describes how each point can be earned, including partial credit. There are examples of scoring rubrics in the articles at the end of each unit.

We'll use these rubrics to help illustrate the key elements of a high-quality response. Here's is an example of a scoring rubric.

This part of the question is worth $2$ of the $9$ points.

  • $1$ point for setting $f'(x)$ equal to $g'(x)$.
  • $1$ point for the correct answer.

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Notice that just writing the correct answer will not achieve full credit, nor will just setting up the problem. We must clearly complete both steps to earn both points.