A biologist is studying the growth of a bacterial culture in a lab. The graph shows that bacteria's population $P$ varies with time $t$. Determine the behavior of $P^{\prime }$.

Given the graph of a function $g\left(x\right)$, draw the graph of $g^{\prime }\left(x\right)$.

For a linear function $f\left(x\right)$ defined on the interval $[-3,7]$, sketch the function given the following conditions:

i) The graph of $f\left(x\right)$ is composed of connected line segments.

ii) The initial point of the graph is $(-3,0)$.

iii) The derivative of $f\left(x\right)$ is represented by a step function.