Asking the Right Questions: An Approach to AP Calculus FRQs

AP Calculus BC is a challenging course, even from the perspective of sheer content memorization. Students are expected to memorize a host of basic derivatives and integrals; differentiation rules; integration by parts, partial fractions, and trig substitution; Taylor polynomial formulas and convergence tests; and a host more. With all that to memorize, it's easy to end up neglecting an actual understanding that makes problem-solving straightforward. Let's begin by considering problem 2 on the 2025 AP exam, below:

Out of a possible 9 points, the mean score on this problem was a 3.09 [1]. There's a lot to say about this problem, but in this discussion I'll focus specifically on part D, where students scored an average of .45 points out of 2.

After seeing those scoring statistics, I chose this as an early problem to work through with my Calculus BC students. Unsurprisingly, many of them did not immediately see an approach when they first read through the question.

Working through the problem on the board would give a student little new insight - each of those students was familiar with related rates, and knew how to compute polar derivatives. The challenge was applying familiar tools in an unfamiliar context. Instead, I asked a simple question:

Can you identify a problem type here? For most students, that was sufficient. We'd discussed related rates as a class of problems, and when thinking through classes of problems, it quickly became obvious that this was a related rates problem. Once they'd determined that, they could follow through with identifying known and unknown rates, and apply implicit differentiation to solve the problem. In the one case where this still didn't clarify the problem, the student caught on as soon as I underlined

and "find the rate at which the particle's distance from the origin changes with respect to time". This was not a new, confusing polar problem: it was a related rates problem, in a polar context. That is often the key move in AP Calculus FRQs. Before asking, “How do I solve this?” students should ask, “What kind of problem is this?”

We'll elaborate on those questions in a later blog post, but some useful examples:

• Am I being asked to maximize/minimize a function? If so,

  • Is this an optimization problem?

  • Can I use first derivative test, second derivative test, or either?

• Is this a related rates problem?

• I need to write out a Taylor series: Do I use the base formula, or can I use substitution into a familiar series instead?

  
  

Those questions do not solve problems automatically, but they give students a way in. And for many (if not most) AP Calculus students, finding the way in is the hardest part.

[1] College Board, AP Calculus AB and BC Chief Reader Report, 2025. https://apcentral.collegeboard.org/media/pdf/ap25-cr-report-calculus-ab-bc.pdf