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Calculus BC · FRQ Trainer

Free-response 1 of 24 · Contextual Applications of Differentiation

9 pts official rubric

Released exam question — College Board · AP Calculus BC 2025 · Question 1 (shared with AB). The real Calculus BC Section II is 6 free-response questions in 90 minutes, 50% of your score. The rubric below is that exam’s published scoring guidelines, one row per scoring point. Figures the original printed as graphs are transcribed to tables so the question is answerable here; the values are unchanged.

Prompt

CALCULATOR PERMITTED. An invasive species of plant appears in a fruit grove at time t = 0 and begins to spread. The function C defined by C(t) = 7.6·arctan(0.2t) models the number of acres in the fruit grove affected by the species t weeks after the species appears. It can be shown that C′(t) = 38/(25 + t²). (Your calculator should be in radian mode.)

aA. Find the average number of acres affected by the invasive species from time t = 0 to time t = 4 weeks. Show the setup for your calculations.2 pts

bB. Find the time t when the instantaneous rate of change of C equals the average rate of change of C over the time interval 0 ≤ t ≤ 4. Show the setup for your calculations.2 pts

cC. Assume that the invasive species continues to spread according to the given model for all times t > 0. Write a limit expression that describes the end behavior of the rate of change in the number of acres affected by the species. Evaluate this limit expression.2 pts

dD. At time t = 4 weeks, measures are taken to counter the spread. The function A, defined by A(t) = C(t) − ∫₄ᵗ 0.1·ln(x) dx, models the number of acres affected over 4 ≤ t ≤ 36. At what time t, for 4 ≤ t ≤ 36, does A attain its maximum value? Justify your answer.3 pts

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