A car is traveling on a straight road. For $\displaystyle 0 =< t =< 24$ seconds, the cars velocity $\displaystyle v(t)$, in meters per second, is modeled by the piecewise-linear function defined by the graph below.

a.) Find $\displaystyle \int_0^{24} v(t) dt$. Using correct units, explain the meaning of $\displaystyle \int_0^{24} v(t) dt$.

b.) For each of $\displaystyle v'(4)$ and $\displaystyle v'(20)$, find the value or explain why it does not exist. Indicate units of measure.

c.) Let $\displaystyle a(t)$ be the cars acceleration at time $\displaystyle t$, in meters per second per second. For $\displaystyle 0 < t < 24$, write a piecewise-defined function for $\displaystyle a(t)$.

d.) Find the average rate of change of $\displaystyle v$ over the interval $\displaystyle 8 =< t =< 20$. Does the mean value theorem guarentee a value of c, for $\displaystyle 8 < c < 20$, such that $\displaystyle v'(c)$ is equal to the average rate of change. Why or why not?

For a.) I came up with the answer 360 meters, I'm pretty sure that is correct.

For b.), I think the acceleration would be zero, correct me if I'm wrong.

I don't know what to do for c.) and d.) any help is appreciated, thanks.