Chapter 1.1, Problem 14E

Exercises 13—15 consider an elementary model of the learning process: Although human learning is an extremely complicated process, it is possible to build models of certain simple types of memorization. For example, consider a person presented with a list to be studied. The subject is given periodic quizzes to determine exactly how much of the list has been memorized. (The lists are usually things like nonsense syllables, randomly generated three-digit numbers, or entries from tables of integrals.) If we let L(t) be the fraction of the list learned at time t, where L = 0 corresponds to knowing nothing and L = 1 corresponds to knowing the entire list, then we can form a simple model of this type of learning based on the assumption:

• The rate dL/dt is proportional to the fraction of the list left to be learned.
Since L = 1 corresponds to knowing the entire list, the model is
   $\frac{dL}{dt}=k\left(1-L\right)$
where k is the constant of proportionality.
14. Suppose two students memorize lists according to the model
   $\frac{dL}{dt}=2\left(1-L\right)$
(a) If one of the students knows one-half of the list at time t = 0 and the other knows none of the list, which student is learning more rapidly at this instant?
(b) Will the student who starts out knowing none of the list ever catch up to the student who starts out knowing one-half of the list?