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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). (Assume the constant of proportionality is k > 0. Use A for A(t).)

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Step-by-step explanation:

From the statement:

M: is total to be memorized

A(t): the amount memorized.

The key issue is translate this statement as equation "rate at which a subject is memorized is assumed to be proportional to the amount that is left to be memorized"

memorizing rate is [tex]\frac{dA(t)}{dt}[/tex].

the amount that is left to be memorized can be expressed as the total minus the amount memorized, that is [tex]M-A(t)[/tex].

So we can write

[tex]\frac{dA(t)}{dt}=k(M-A(t))[/tex]

And that would be the differential equation for A(t).

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