2. As you know, when a course ends, students start to forget the material they h
ID: 2830936 • Letter: 2
Question
2. As you know, when a course ends, students start to forget the material they have learned. The
Ebbinghaus model assumes that the rate at which a student forgets material is proportional to the
difference between the material currently remembered and some positive constant .
(a) Let y = f(t) be the fraction of the original material remembered t weeks after the course has ended.
Set up a differential equation for y. Your equation will contain two constants: the constant is
less than y for all t.
(b) Solve the differential equation.
(c) Describe the practically meaning (in terms of the amount remembered) of the constants in the
solution y = f(t).
Explanation / Answer
dy/dt = -k(y-a)
& y(0) = 1
by solving above differential equation we get,
general solution
y = a + (1 - a)?e^(-k?t)
Since k is a positive constant the exponential function vanishes as t approaches infinity, i.e.
limt?? { y } = a
That means parameter 'a' represents the fraction of the material, which will never be forgotten.
You can consider 'a' as persistent knowledge. The fraction of material remembered above the level 'a'
is some temporary knowledge which fades away as times goes by. Let x be this temporary knowledge, i.e.
x = y - a
SInce
dx/dt = dy/dt
it follows from the differential equation:
dx/dt = - k?x
That's simple exponential decay model, in which k represents the fraction of x which decreases per unit time. That means 'k' represent the fraction of temporary knowledge which vanishes per unit time.
To determine the parameters rearrange the general solution:
y = a + (1 - a)?e^(-k?t)
<=>
ln( (y - a)/(1 - a) ) = ?-k?t
=>
ln( (y - a)/(1 - a) ) / t = ?- k = constant
Right hand side is constant. So you can apply the two data values for y and the respective time t on left hand side and equate the resulting expressions:
ln( (0.75 - a)/(1 - a) ) / 1 = ln( (0.61 - a)/(1 - a) ) / 2
<=>
2?ln( (0.75 - a)/(1 - a) ) = ln( (0.61 - a)/(1 - a) )
Using the logarithmic identity
Since ln(a))= n?ln(a)
you can rewrite
ln( (0.75 - a)
Related Questions
Navigate
Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.