# Thread: Integrating a graph problem

1. ## Integrating a graph problem

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.

2. The quantity...

$\displaystyle \displaystyle \int_{0}^{24} v(t)\ dt$ (1)

... is simply the area of the 'trapetium' of the figure and it can be computed with a simple geometric approach...

Kind regards

$\displaystyle \chi$ $\displaystyle \sigma$

3. Thanks for the reply, as I said I figured out the area, but I could use some help on the other questions. Thanks for your time.

4. Sorry for the double post, but could anyone help me with c and d, thanks.