Shaun Ault

Understanding AP Calculus Free Response Questions

What are the AP Calculus free response questions like? Free response means that you will not be given any answer choices. Instead, you will have to work out the problem, showing all steps, and come up with a final answer yourself. In this short article, you’ll discover what to expect on the AP Calculus free response sections.

AP Calculus Free Response Question Format

A typical free response problem states a scenario and then asks 3-4 questions about that scenario.

For example, the following problem was pulled from a practice Calculus BC exam.

Example

Consider the equation x2 – 2xy + 4y2 = 64.

(a) Write an expression for the slope of the curve at any point (x, y).

(b) Find the equations of the tangent lines to the curve at the point x = 2.

(c) Find AP Calculus Free Response at (0, 4).

Put your thinking cap on

Ok, so let’s pick this question apart a little. We won’t solve it here, but we will build a plan for tackling the problem.

  • First of all, the given equation applies to all three parts.
  • Part (a) asks for the slope at any point. That sounds like a derivative. However the given function is not solved explicitly for y. So you’ll have to do implicit differentiation.
  • Part (b) asks for the equations of the tangent lines at a specific point. Notice the plural equations and tangent lines. That’s a clue that you should have more than one answer. But also notice in this case, part (b) depends on part (a). We need the derivative in order to find the slope of any tangent line.
  • Finally, Part (c) asks for a second derivative. Of course, this depends on finding the first derivative (Part (a)).

As you can see from this example, the parts of the question depend on one another. However, Part (c) did not depend on Part (b), so this is not a hard and fast rule.

How Many Free Response Questions are there on the Exam?

Both the AP Calculus AB and BC tests have the same number of free response problems. Section II of the test consists of 6 free response questions, separated into two parts.

  • Part A: 4 problems in 60 minutes. Requires a graphing calculator.
  • Part B: 2 problems in 30 minutes. No calculator.

Budgeting your Time

In order to complete all of the questions, you would have to spend an average of 15 minutes per problem or less. However, I recommend sticking with a free response problem until you can complete all parts.

The graders are looking for significant progress towards the final answer, rather than a few stabs in the dark. Therefore, it is better to completely work out the problems that you know how to answer before trying any that you might have trouble with.

Practice, Practice, Practice!

The best way to understand the AP Calculus free response questions is to do some yourself. Try these ones for starters: AP Calculus AB Free Response Practice Problems and AP Calculus BC Exam Free Response Practice Problems.

Then be sure to practice on an actual AP Calculus exam. In fact, the more practice exams that you do, the better your performance will be on test day!

About Shaun Ault

Shaun earned his Ph. D. in mathematics from The Ohio State University in 2008 (Go Bucks!!). He received his BA in Mathematics with a minor in computer science from Oberlin College in 2002. In addition, Shaun earned a B. Mus. from the Oberlin Conservatory in the same year, with a major in music composition. Shaun still loves music -- almost as much as math! -- and he (thinks he) can play piano, guitar, and bass. Shaun has taught and tutored students in mathematics for about a decade, and hopes his experience can help you to succeed!


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