exponential growth over time - how do I calculate the increase over a delta-time?

This is probably a si开发者_如何学运维lly / stupid question, but I'm still gonna ask it : if I have an initial start value at Time 0 (which is in my case always 1.0) and a rate of growth, how do I figure out the increase between Time1 and Time2 ?

Tn = T1 + 0.5 (rate of growth)^2

Dose this make sense? The last term one half of the square of the rate of growth.

Thus the difference between the time periods is

diff(Tn - T1) = 0.5 (rate of growth)^2

increase = value at time2 - value at time1. Seems simple, is simple. The value is equal to T0*(rate of growth)^Ti, where Ti is your time.

If i understand correctly:

f(t2) - f(t1) where f(t) = initial * growth_factor^t

If you assume a relative rate of growth r your value as a function of time is given by

f(t) = exp(r*t)

(already incorporated f(0)=1 and f'(0)=r) and thus the absolute difference is

D = f(t2) - f(t1) = exp(r*t2) - exp(r*t1)

while the relative increase is given by

d = f(t2)/f(t1) - 1 = exp(r*t2)/exp(r*t1) - 1 = exp(r*(t2-t1)) - 1