Chapter 1.1, Problem 15E

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.
15. Consider the following two differential equations that model two students' rates of memorizing a poem. Aly's rate is proportional to the amount to be learned with proportionality constant k = 2. Beth's rate is proportional to the square of the amount to be learned with proportionality constant 3. The corresponding differential equations
   $\frac{d{L}_A}{dt}=2\left(1-L_A\right)$ and $\frac{d{L}_B}{dt}=3{\left(1-L_B\right)}^2$
where L_A(t) and L_B(t) are the fractions of the poem learned at time t by Aly and Beth, respectively.
(a) Which student has a faster rate of learning at t = 0 if they both stan memorizing together having never seen the poem before?
(b) Which student has a faster rate of learning at t = 0 if they both stan memorizing together having already learned one-half of the poem?
(c) Which student has a faster rate of learning at t = 0 if they both stan memorizing together having already learned one-third of the poem?