Chapter 2.6, Problem 48E

(a)

To determine

The meaning of $f^{\prime }\left(5\right)$ and its units.

Explanation of Solution

Here $f\left(t\right)$ is the number of bacteria after t hours in a controlled laboratory experiment.

Note that, the derivative $f^{\prime }\left(x\right)$ is the instantaneous rate of change of the function $f\left(x\right)$ with respect to x.

The derivative $f^{\prime }\left(t\right)$ is rate of change of number of bacteria with respect to t hours in a controlled laboratory experiment.

Thus, $f^{\prime }\left(5\right)$ is rate of change of number of bacteria with respect to 5 hours in a controlled laboratory experiment.

The instantaneous rate of change is equal to $f^{\prime }\left(t\right)=\underset{\Delta x\to 0}{\lim }\frac{\Delta y}{\Delta t}\text{ here }y=f\left(t\right)$.

Here $\Delta y$ is measured in number of bacteria and $\Delta t$ is measured in hours.

Thus, the units are number of bacteria per hours.

(b)

To determine

To compare: The values $f^{\prime }\left(5\right)\text{ or }{f}^{\prime }\left(10\right)$ when bacteria has unlimited amount of space and nutrients and explain the conclusion when supply of nutrients is limited.

Explanation of Solution

$f\left(t\right)$ is the number of bacteria after t hours in a controlled laboratory experiment.

The growth of population of bacteria is depends on the amount of nutrients, amount of space and the population of bacteria.

If the unlimited amount of space and nutrients,

The growth of bacteria population is depends on the population of bacteria.

The growth of bacteria population increases as the population of bacteria is increases.

The rate of change of bacteria population increases as the population of bacteria is increases.

Therefore, $f^{\prime }\left(5\right)<f^{\prime }\left(10\right)$.

If supply of nutrients is limited.

The growth rate of bacteria is decreasing at some point in time.