# The number of bacteria after t hours in a controlled laboratory experiment is n=f(t) ....

## Question:

The number of bacteria after {eq}t {/eq} hours in a controlled laboratory experiment is {eq}n=f(t) {/eq}. Suppose there is an unlimited amount of space and nutrients for the bacteria. Which do you think is larger, {eq}f'(5) \ \mathrm{or} \ f'(10) {/eq}? If the supply of nutrients is limited, would that affect your conclusion? Explain.

## Exponential and Logistic Growth:

Two mathematical models that can express population growth are an exponential and logistic model. The former assumes that a population can grow infinitely large unchecked, and the latter assumes that there is some limit on how large the population can grow. This means that the exponential growth would slow down after a certain point.

## Answer and Explanation:

Bacteria reproduce by binary fission, which means that they split in half. Thus, the more bacteria exist in a population, the more that can split in half to form new bacteria. Thus, the rate of change of the population after ten hours will be larger than the rate of change of the population after five hours. Since the rate of change of a function is the derivative, this means that the value of the derivative at ten hours will be larger than the value of the derivative at five. Mathematically, this means that {eq}f'(10)>f'(5) {/eq}.

However, in real life, habitats are limited in the amount that they can handle. The space that a population can occupy or the nutrients that they consume may limit the growth of this population. Thus, the rate of change of the population may be affected by the limit on that population. If the limit is low compared to the initial value of the population and the initial rate of change, then perhaps the growth in the population would slow down as it approaches this limit. In this case, then {eq}f'(5) < f'(10) {/eq}. If this limit is high compared to the initial value of the population and the initial rate of change, then the growth may not have needed to slow down even by ten hours. In this case, the conclusions drawn previous paragraph would still hold.