2023 AP Calculus AB FRQ #3

A bottle of milk is taken out of a refrigerator and placed in a pan of hot water to be warmed. The increasing function M models the temperature of the milk at time t, where M(t) is measured in degrees Celsius (°C) and t is the number of minutes since the bottle was placed in the pan. M satisfies the differential equation dM/dt = 1/4 * (40 − M). At time t = 0, the temperature of the milk is 5°C. It can be shown that M(t) is less than 40 for all values of t.

(a) A slope field for the differential equation dM/dt = 1/4 *(40 − M) is shown. Sketch the solution curve through the point (0, 5).

(b) Use the line tangent to the graph of M at t = 0 to approximate M(2), the temperature of the milk at time t = 2 minutes.

(c) Write an expression for d^2M/dt^2 in terms of M. Use d^2M/dt^2 to determine whether the approximation from part (b) is an underestimate or an overestimate for the actual value of M(2). Give a reason for your answer.

(d) Use separation of variables to find an expression for M(t), the particular solution to the differential equation dM/dt = 1/4 *(40 − M) with initial condition M(0) = 5.

Intro: 00:00
Problem a: 00:33
Problem b: 01:40
Problem c: 05:12
Problem d: 08:55