2025 AP Calculus BC Free-Response Questions

1. 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+t^2). (Note: Your calculator should be in radian mode.) A. 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. B. 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. C. 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. D. At time t = 4 weeks after the invasive species appears in the fruit grove, measures are taken to counter the spread of the species. The function A, defined by A(t) = C(t) - ∫_4^t 0.1 ln(x) dx, models the number of acres affected by the species over the time interval 4 ≤ t ≤ 36. At what time t, for 4 ≤ t ≤ 36, does A attain its maximum value? Justify your answer.