SeriesB AP Exam Problems
AP® Calculus BC 2023 Free-Response Questions
| \(t\) (seconds) | 0 | 60 | 90 | 120 | 135 | 150 |
|---|---|---|---|---|---|---|
| \(f(t)\) (gal/second) | 0 | 0.1 | 0.15 | 0.1 | 0.05 | 0 |
- 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 \(\int_{60}^{135} f(t)\, dt\)
\(\int _{60}^{135} f(t)\, dt\) is the total gas (in gallons) that the customer receives between 60 and 135 seconds.
\(\sum_{n=2}^{5} (t_{n} - t_{n-1}) * f(t)\)
a = (90-60) * 0.15
b = (120 - 90) * 0.1
c = (135 - 120) * 0.05
a + b + c8.25 gallons
(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 \(g(t) = (\frac{t}{500}) \cos( ( \frac{t}{120})^2)\) for \(0 < t < 150\). Using this model, find the average rate of flow of gasoline over the time interval \(0 < t < 150\). 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.