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Math Help - Revenue Function - Integrals

  1. #1
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    Revenue Function - Integrals

    The marginal-revenue function for a manufacturer's product is of the form
     <br />
\frac{dr}{dq}=\frac{a}{e^q+b}<br />
    for constands a and b, where r is total revenue received when q units are produced and sold.

    Find the demand function, and express it in the form p = f(q)

    I'm not sure where to even begin, any help is appreciated....thank you!
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  2. #2
    Member Black's Avatar
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    To find the demand function p, you must find r, then use p=\frac{r}{q}.

    To solve the integral

    r(q)=a\int\frac{dq}{e^q+b},

    you use the substitution u=e^q+b. Then du=e^q\,dq=(u-b)\,dq,so

    r(q)=a\int\frac{dq}{e^q+b}=a\int\frac{du}{u(u-b)}=\frac{a}{b}\int\left(\frac{1}{u-b}-\frac{1}{u}\right)\, du ...
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  3. #3
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    Evaluate \int \frac{a}{e^q + b} dq to find revenue in terms of quantity sold. You know that if you sell nothing then you make no revenue, so use the fact that q = 0 implies r = 0 to solve for the constant of integration. Since r = pq = f(q)q, once you have r you can find f(q).
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  4. #4
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    \int \frac{a}{e^q + b} \, dq<br />

    a \int \frac{1}{e^q + b} \cdot \frac{e^{-q}}{e^{-q}} \, dq<br />

    a \int \frac{e^{-q}}{1 + be^{-q}} \, dq

    u = 1 + be^{-q}

    du = -be^{-q} \, dq

    -\frac{a}{b} \int \frac{-be^{-q}}{1 + be^{-q}} \, dq

    -\frac{a}{b} \int \frac{du}{u}<br />

    -\frac{a}{b} \ln|u| + C

    -\frac{a}{b} \ln(1+be^{-q}) + C
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