2007-09-25 - More logistics
Continuing our derivation from yesterday. This is related to logistical regressions, that tries to find a function that will fit the data. The final equation we came up with was: P = m / (1 + C_{2}e^{-kmT}). What's the limit as T->infinity? It approaches "m." So on the TI-84 calculators, go to [STAT] and then to [CALC] and then to the "Logistic" regressional analysis entry. So it will take some time ... and on the calculator the form that is produced is = C/(1 + ae^(-bx)). With the specific numbers that it found: (122.89)/(1+1137.113196e^(-.9378989775x)).
When was the growth the fastest? It'd be at your inflection point, which will be at the middle. We can do all of this mathematically, even if you forget these simple answers, well, you can derive it yourself. If we can remember, of course we might do it in physics instead of calculus, is some sample data and do some regressions. So if you're not in physics, sorry- trying to get a rise out of Ketteman but he doesn't seem to be paying attention.
Calculator version of logistic regresional analysis: y = C/(1+ae^{-bx}). So you notice, so then, yeah, Dakota pointed out that the B on the calculator is "m*k" on the version we did, and "C" is "m", so what is k? So "k" in our version is "b/c" in the calculator's version.
So we had some questions. (A) At what value of P is the growth (of the population of the people with the disease) the greatest? The equation for growth is dP/dT = kP(m-P). So when is that the greatest? So what is the P value at the inflection point? It'll be (1/2)m. Well, you're asked to find the maximum and minimum here, so you have to find the zeroes of the derivative, so if you want to know when dP/dT is the greatest, well, that translates to inflection points, but let's take the derivative of dP/dT, and what's the derivative? So (dP/dT)' = k(m-P) + -kP. So how do you find the maximums and minimums? Well, set it to zero and solve for P, what does P have to be to make that derivative of dP/dT to be zero? So: km=2kp, so P=km/2k = m/2. So there you go: the rate is the greatest when P is half of the maximum population. Once half has the disease, you start slowing down again. This is the general logistic assumption.
Study Taylor polynomails which involve factorials.