2023 AP Calculus BC Free-Response Questions

t (seconds) 0 60 90 120 135 150 f(t) (gallons per second) 0 0.1 0.15 0.1 0.05 0 1. A customer at a gas station is pumping gasoline into a gas tank. The rate of flow of gasoline is modeled by a differentiable function f, where f(t) is measured in gallons per second and t is measured in seconds since pumping began. Selected values of f(t) are given in the table. (a) Using correct units, interpret the meaning of $\int_{60}^{135} f(t)\,dt$ in the context of the problem. Use a right Riemann sum with the three subintervals [60, 90], [90, 120], and [120, 135] to approximate the value of $\int_{60}^{135} f(t)\,dt$. (b) Must there exist a value of c, for $60<c<120$, such that $f'(c)=0$? Justify your answer. (c) The rate of flow of gasoline, in gallons per second, can also be modeled by $g(t)=\left(\frac{t}{500}\right)\cos\left(\left(\frac{t}{120}\right)^2\right)$ for $0\le t\le 150$. Using this model, find the average rate of flow of gasoline over the time interval $0\le t\le 150$. Show the setup for your calculations. (d) Using the model $g$ defined in part (c), find the value of $g'(140)$. Interpret the meaning of your answer in the context of the problem. 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.