Question

Learning Theory In the theory of learning, the rate at which

a subject is memorized is assumed to be proportional to the amount that is left to be memorized. Suppose M denotes the total amount of a subject to be memorized and A(t) is the amount memorized in time t > 0. Determine a differential

equation for the amount A(t).

Short answer

A solution to this DE should be a monotonically increasing function with inflection point at a point with y-coordinate equal to 0.

Step-by-step solution

Definition of Newton’s law of cooling with mathematical statement.

Newton’s law ofcooling/warming translates into the mathematical statement

$\frac{dT}{dt}\propto T-T_m\text{or}\frac{dT}{dt}=k\left(T-T_m\right)$

where k is a constant of proportionality. In either case, cooling or warming, if $T_m$is a

constant, it stands to reason that $k<0$.

Compute Differentiation.

Given M- Total amount of a subject, A(t) is the amount of memorized time.

Now, to find the differential equation

$\frac{dA}{dt}=kL\left(t\right)$

$M=A\left(t\right)+L\left(t\right)$

M (The total amount of study material) = A (the amount memorized)\[ + L\](The

amount left to memorize)

$M-A\left(t\right)=L\left(t\right)$

Solve for \[L\] and place it into given equation, we get

$\frac{dA}{dt}=k\left(M-A\left(t\right)\right)$.