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Thread: Integrating a graph problem

  1. #1
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    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.
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  2. #2
    MHF Contributor chisigma's Avatar
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    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

    $\displaystyle \chi$ $\displaystyle \sigma$
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  3. #3
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    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.
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  4. #4
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    Sorry for the double post, but could anyone help me with c and d, thanks.
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