Chapter 3.1, Problem 92E

Ebbinghaus learning model. Suppose that you are given the task of learning 100% of knowledge. Human nature is such that we retain only a percentage P of knowledge t weeks after we have learned it. The Ebbinghaus learning model asserts that P is given by

$P\left(t\right)=Q+\left(100-Q\right)e^{-kt},$ where Q is the percentage that we would never forget and k is a constant that depends on the knowledge learned. Suppose that $Q=40$ and $k=0.7$.

a. Find the percentage retained after 0 weeks, 1 week, 2 weeks, 6 weeks, and 10 weeks.

b. Find $\underset{t\to \infty }{\lim }P\left(t\right).40\%$

c. Sketch a graph of P.

d. Find the rate of change of $P\left(t\right)$ with respect to time t.

e. Interpret the meaning of the derivative.