2021 AP Calculus AB FRQ #6

A medication is administered to a patient. The amount, in milligrams, of the medication in the patient at time t hours is modeled by a function y = A(t) that satisfies the differential equation dy/dt = 12 − y / 3 . At time t = 0 hours, there are 0 milligrams of the medication in the patient.
(a) A portion of the slope field for the differential equation dy/dt = 12 − y /3 is given below. Sketch the solution curve through the point (0, 0).
(b) Using correct units, interpret the statement lim A(t) t→∞ = 12 in the context of this problem.
(c) Use separation of variables to find y = A(t), the particular solution to the differential equation dy/dt = 12 − y / 3 with initial condition A(0) = 0
(d) A different procedure is used to administer the medication to a second patient. The amount, in milligrams, of the medication in the second patient at time t hours is modeled by a function y = B(t) that satisfies the differential equation dy / dt = 3 - y / t + 2 . At time t = 1 hour, there are 2.5 milligrams of the medication in the second patient. Is the rate of change of the amount of medication in the second patient increasing or decreasing at time t = 1 ? Give a reason for your answer.
Timestamps
Intro: 00:00
Part a: 00:26
Part b: 01:35
Part c: 02:46
Part d: 07:22

[@tarekdajani4301]

hey, thanks for the videos they are really helping. i just have a question about the final part (d) and why dy/dx wouldn't be considered the rate of change function. isn't f'(x) always considered the rate of change, or would it be f"(x) in this instance because we started off with f'(x)? thanks once again

[@OlgaAndreevaVideos]

Yep, you're exactly right! Since we need to consider the rate of change of the rate of change, we would use f''(x). If we just used f'(x), we would be getting just the rate of change of the amount of medication at t = 1, not necessarily whether it's increasing or decreasing.

[@zirakguzder3764]

W video

[@OlgaAndreevaVideos]

W comment