Question

Psychologists interested in learning theory study LEARNING CURVES. A learning curve is the graph of a function \(P\left( t \right)\), the performance of someone learning a skill as a function of the training time \(t\). The derivative \(\frac{{dP}}{{dt}}\) represents the rate at which performance improves.

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

\(\frac{{dP}}{{dt}} = k\left( {M - P} \right)\) \(k\)a positive constant

is a reasonable model for learning.

(b) Solve the differential equation in part (a) to find an expression for \(P\left( t \right)\). What is the limit of this expression?

Short answer

(a) At starting learner’s process increase most rapidly.

(b) Rate of learning approaches zero.

Step-by-step solution

Step 1

(a)

At the beginning\(P\)increases most rapidly, it is reasonable to assume that on beginner learns more at starting time, Initially\(\frac{{dP}}{{dt}}\)is very large.

As time increases rate of learning process decreases that is\(\frac{{dP}}{{dt}} \to 0\)as\(t \to \infty \).

Thus, at starting learner’s process increase most rapidly.

Step 2

(b)

The differential equation\(\frac{{dP}}{{dt}} = k\left( {M - P} \right)\).

Here,\(M\)is maximum level of learning.

Initially\(P\)is small, assume that\(P \prec M\).

So\(\left( {M - P} \right)\)is positive

Thus,\(\frac{{dP}}{{dt}}\)is positive.

That is, the level of performance\(P\)is increasing.

The level of performance\(P\)is increasing means\(P\)gets close to\(M\).

Thus,\(\frac{{dP}}{{dt}}\)close to\(0\).

That is rate of learning approaches zero.

Therefore, Rate of learning approaches zero.

Step 3

(c)

From the graph, it is observed that the function \(P\left( t \right)\) is increasing and it is approach to the maximum level of the performs \(M\), that is this \(P\left( t \right)\) is the solution of the given differential equation