AP Calculus AB · Unit 6 · Integration and Accumulation of Change

The rate at which visitors enter a museum through its main entrance, $R(t)$ people per hour, is differentiable, with selected values shown in the table for $0 \le t \le 8$ hours after opening. At $t = 0$, 40 people are already inside. Let $N(t) = 40 + \int_0^t R(s)\,ds$ be the number of people who have entered the museum by time $t$.

(a) Approximate $R'(7)$ using the data in the table. Show the computation that leads to your answer, and indicate units.
(b) Use a trapezoidal sum with the four subintervals indicated by the table to approximate $\int_0^8 R(t)\,dt$. Then use this approximation to approximate $N(8)$, the total number of people who have entered the museum by time $t=8$ hours. Show the work that leads to your answer, and indicate units.
(c) Use your approximation from part (b) to approximate the average rate at which visitors enter through the main entrance for $0 \le t \le 8$. Must there be a time $t$, for $0 < t < 8$, at which $R(t)$ equals this approximation? Justify your answer.
(d) For $8 \le t \le 10$, the rate at which visitors enter through the main entrance remains constant at $R(8) = 40$ people per hour. Approximate $N(10)$.