MTH 264
DELTA COLLEGE
SECTION 1.1 14
Suppose two students memorize lists according to the same model
dL
= 2 (1 − L)
dt
(a) If one student knows one-half of the list at time t = 0 and the other knows none of the list, which student
is learning more rapidly at this instant?
(b) Will the student who starts out knowing none of the list ever catch up to the student who starts out knowing
one-half of the list?
Solution:
Assume that student 1 knows one-half of the list and student 2 knows none of the list at time t = 0.
(a) Rate of learning
1
dL
=2 1−
=1
dt
2
dL
= 2 (1 − 0) = 2
dt
(L = 1/2 when t = 0)
student 1
(L = 0 when t = 0)
student 2
Student 2 is learning more rapidly when t = 0.
(b) No, student 2 will never catch up to student 1.
The two students’ rates of learning are described by the same function “2(1 − L)” and this function only
depends on the amount of the list memorized. This turns out to be a strong constraint on the amount of the list
learned as a function of time “L(t)”.
For example, student 2 cannot catch student 1 before student 2 has learned 1/2 of the list (L = 1/2), so
assume that student 2 reaches L = 1/2 at time t = t0 . This means that at time t = t0 student 2 s learning at the
same rate that student 1 was learning at t0 time units ago. It then follows that student 2 begins tracing the same
“learning curve” as student 1, but shifted t0 time units to the right. Student 2 will thus never catch student 1.