Let's explore a logistic function. We are given the formula for the milligrams of bacteria

in a certain population as a function oftin hours:P(t) =300/3+97e^-0.2t mg of bacteria.

(a) What is the initial amount of bacteria?

(b) GraphP,P, andP.

(c) What, if any, are the local maxima, minima, and points of inflection?

(d) What is the asymptotic behavior of this function? That is, what does the function

look like for massivet, such as att= 1000.

(e) At what time, if any, does the population become 20 mg?

(f) At what time, if any, does the population become 200 mg?

(g) When is the population growing the fastest and what is the amount of the population at that time?

(h) If the population has a limiting value, when is it within 1mg of that limit?

(i) List the doubling times of the population relative to the starting value.

(j) (Bonus) It is a fact thatPsatisfies the differential equationP(t) = 0.2P(t)[1P(t)/100]. Give some evidence for this; graphical is fine as is algebraic.

(k) (Bonus) Using a step-size of half an hour, approximate this differential equation with a difference equation. Go out to a time of 50 hours (100 steps). How does the value compare toP(50)?