n 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. Determine a differential equation for the amount A(t). (SET
UP ONLY. DO NOT SOLVE.)
(b) (2 pts) Now assume that the rate at which material is forgotten is proportional to the amount memorized
in time t. Determine a differential equation for the amount A(t) when forgetfulness is taken into account.
(SET UP ONLY. DO NOT SOLVE.)
3. (4 pts) Find the critical points and phase portrait of the autonomous first-order differential equation
dy
dx = y2(y + 4)(y − 2). Classify each critical point as an attractor (asymptotically stable), repeller (unstable), or
semi-stable.