Chapter 7.1, Problem 15E

a.

To determine

To comment: When $P$ increases most rapidly and what happens to $\frac{dP}{dt}$ when $t$ increases.

Answer to Problem 15E

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Explanation of Solution

Given information:

The performance of someone learning a skill is a function of $t$ which is represented by $P\left(t\right)$

I think $P$ increases most rapidly at the beginning. Initially the learning is generally easy and the learner has the most spirit. And my intuition is telling so.

As the time passes learner has to learn more complex things to achieve mastery. For this reason and also for the above reasons, I think the value of $\frac{dP}{dt}$ will decrease as the time increases.

b.

To determine

To explain: Why the given differential equation is a reasonable model for learning.

Explanation of Solution

Given information:

  $\frac{dP}{dt}=K\left(M-P\right)K\text{ is a positive constant}$

  $M$ is the maximum level of performance.

  $\frac{dP}{dt}=K\left(M-P\right)K\text{ is a positive constant}$

This differential equation tells us as $P$ increase the value of $\frac{dP}{dt}$ decreases. It implies that the slope of the learning curve decreases with time. When $\text{P=M}$ , $\frac{dP}{dt}=0$ i.e. when a learner achieve maximum level, he cannot learn any more. So, differential equation combines all requirements. Hence the given differential equation is reasonable model for learning.

c.

To determine

To draw: A rough sketch of the solution of the given differential equation.

Explanation of Solution

Given information:

  $\frac{dP}{dt}=K\left(M-P\right)K\text{ is a positive constant}$

Graph:

  

Interpretation :

When $\text{P=M}$ , the graph will be flat and represents the performance of the learner is not increasing.