2023 AP Calculus AB FRQ #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 ∫ f(t) dt from 60 to 135 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 ∫ f(t) dt from 60 to 135.
(b) Must there exist a value of c, for c is between 60 and 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) = (t/500) * cos ((t/120)^2)) for t is between 0 and 150. Using this model, find the average rate of flow of gasoline over the time interval t is between 0 and 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.
Intro: 00:00
Problem a: 00:30
Problem b: 07:49
Problem c: 10:59
Problem d: 12:41