Find the inflection point

Question

For some tasks, a sigmoid learning curve is used to model proficiency. Suppose the percent proficiency is given by the model P(t)=(100)/(1+100e^-.6t), where t is days of practice.

 

Find the inflection point, the point where additional practice starts producing diminishing returns.

days=?

Answer 1

If you really want the inflection point (not the maximum), you can plot the function over a domain of -∞ to ∞ and see that there is an inflection point where P(t) = P_max/2 = 100/2 = 50.

 

Set P(t) = 50 and solve for t.

50 = 100 / (1 + 100e^-0.6t)

1 + 100e^-0.6t = 100/50 = 2

e^-0.6t = (2 - 1)/100 = 0.01

 

t = ln(0.01) / (-0.6) ≅ 7.675 days