Answer
(a) P increases most quickly at the start when a person first begins to learn a new skill. As time goes by, the rate of improvement decreases as a person's ability gets closer to the maximum performance level.
(b) This differential equation seems appropriate since the rate of improvement decreases as the value of $M-P$ decreases. That is, the rate of improvement decreases as the person's ability gets closer to the maximum performance level.
(c) Please see the graph below.
Work Step by Step
(a) P increases most quickly at the start when a person first begins to learn a new skill. It is normally quite easy to learn the most basic skills quite quickly. As time goes by, the rate of improvement decreases as a person's ability gets closer to the maximum performance level.
(b) Differential equation: $\frac{dP}{dt} = k(M-P)$
This differential equation seems appropriate since the rate of improvement decreases as the value of $M-P$ decreases. That is, the rate of improvement decreases as the person's ability gets closer to the maximum performance level.
(c) Please see the graph below.
