A population of bacteria in a laboratory culture is modeled by the differentiable function P, measured in hundreds of bacteria, where P(t) is recorded t hours after the beginning of the experiment. For t near 6, values of P are given by the table below.
t (hours)
5.97
5.99
6.01
6.03
P(t) (hundreds)
12.31
12.36
12.41
12.46
Answer the following.
(1) Part 1: (a) Calculate the average rate of change of the population, in hundreds of bacteria per hour, over the interval [5.97,6.03].
(2) Part 2: (b) Using two non-overlapping subintervals of length 0.02 centered at t=6, compute two additional average rates of change and use them to produce a numerical estimate for P′(6).
(3) Part 3: (c) Interpret the value obtained in part (b) in the context of the laboratory culture, including appropriate units.