2024 AP Calculus BC Free-Response Questions

The temperature of coffee in a cup at time t minutes is modeled by a decreasing differentiable function C, where C(t) is measured in degrees Celsius. For 0 ≤ t ≤ 12, selected values of C(t) are given in the table shown. t (minutes) | 0 | 3 | 7 | 12 C(t) (degrees Celsius) | 100 | 85 | 69 | 55 (a) Approximate C′(5) using the average rate of change of C over the interval 3 ≤ t ≤ 7. Show the work that leads to your answer and include units of measure. (b) Use a left Riemann sum with the three subintervals indicated by the data in the table to approximate the value of ∫₀¹² C(t) dt. Interpret the meaning of (1/12)∫₀¹² C(t) dt in the context of the problem. (c) For 12 ≤ t ≤ 20, the rate of change of the temperature of the coffee is modeled by C′(t) = −(24.55e⁰·⁰¹ᵗ)/t, where C′(t) is measured in degrees Celsius per minute. Find the temperature of the coffee at time t = 20. Show the setup for your calculation. (d) For the model defined in part (c), it can be shown that C″(t) = [0.2455e⁰·⁰¹ᵗ(100 − t)]/t². For 12 < t < 20, determine whether the temperature of the coffee is changing at a decreasing rate or at an increasing rate. Give a reason for your answer. Write your responses to this question only on the designated pages in the separate Free Response booklet. Write your solution to each part in the space provided for that part.