Integrating a graph problem

**lancelot854** · January 22nd 2011, 10:00 AM

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

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

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

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

d.) Find the average rate of change of $v$ over the interval $8 =< t =< 20$ . Does the mean value theorem guarentee a value of c, for $8 < c < 20$ , such that $v'(c)$ 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.

**chisigma** · January 22nd 2011, 11:34 AM

The quantity...

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

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

Kind regards

$\chi$ $\sigma$

**lancelot854** · January 22nd 2011, 11:42 AM

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.

**lancelot854** · January 23rd 2011, 11:24 AM

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

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