Chapter 7.7, Problem 33E

Psychologists interested in learning theory study learning curves. A learning curve is the graph of a function P(t), the performance of someone learning a skill as a function of the training time t. The derivative dP/dt represents the rate at which performance improves.

(a) When do you think P increases most rapidly? What happens to dP/dt as t increases? Explain.

(b) If M is the maximum level of performance of which the learner is capable, explain why the differential equation

$\frac{dP}{dt}=k(M-P)kapositivecons\tan t$

is a reasonable model for learning.

(c) Make a rough sketch of a possible solution of this differential equation.