Suppose two students are memorizing a list according to the rate of change equat

Question

Suppose two students are memorizing a list according to the rate of change equation dL/dt = 0.5(1 - L), where L represents the fraction of the list that is memorized at any time t. If one of the students knows one-third of the list at time t = 0 and the other student knows none of the list, which student is learning most rapidly at this instant? Why? What does the rate of change equation predict for someone who starts off with the list completely memorized? Explain. Suppose now that the list is infinitely long, like the decimal representation for pi. In reality no one can memorize all the digits to pi. but what does the rate of change equation predict will happen for a person who starts out not knowing any of the digits? That is, according to the rate of change equation, if L = 0 at time t = 0, is there every a value of/for which L = 1? Explain.

Explanation / Answer

Let L1(t) represent the portion of the list the first student has memorized by time t,and let
L2(t) represent the portion of the list the second student has memorized by time t. Then L1(0) = 1/2,
and L2(0) = 0. Substituting in to the differential equation for the model,we have

dL1/dt=
t=0= 0.5(1 L1(0)) = 0.5(1 1/2) = 1/4,
and
dL2/dt
t=0= 0.5(1 L2(0)) = 0.5(1 0) = 0.5,
so the second student is learning faster initially.

b) : No,not according to the model. Although both students will tend toward knowing the entire list,neither will actually reach that goal,and the second student will always trail the first in
how much (s)he knows.

The student who knows none of the list begins with a greater dL/dt than the student of the list, but since the rate of change decreases
as each student learns more, mathematically, the student who started knowing nothing will never catch up to the student
. How-ever, as time increases, the difference between the knowledges of the two students will approach 0.

If a student starts off with the list completely memorized, dL/dt = 0.5(1 1) = 0. Their rate of change is 0 because, having already memorized the
whole list, there is nothing left for them to memorize.

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