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Math Help - Coming up with an exponential equation

  1. #1
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    Definite Integrals

    I need to come up with an exponential equation which fulfils the following criteria;
    -is an increasing function
    -is concave up
    -pass through the points (0,3) and (4,11)
    -has a definite integral of 22 between x=0 and x=4.
    Last edited by asaver; July 26th 2011 at 01:42 AM.
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  2. #2
    MHF Contributor Siron's Avatar
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    Re: Coming up with an exponential equation

    Why are you so sure the exponential function has base e? We know the exponential function is increasing so the base >1.
    Last edited by Siron; July 25th 2011 at 04:15 AM.
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  3. #3
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    Re: Coming up with an exponential equation

    Hello, asaver!

    I need an exponential equation which fulfils the following criteria:
    . . [1] is an increasing function
    . . [2] is concave up
    . . [3] pass through the points (0,3) and (4,11)
    . . [4] has a definite integral of 22 between x=0 and x=4.

    Because of the y-intercept (0,3)
    . . you believe the function has the form: f(x) \,=\,3e^x

    But it could have the form: . f(x) \,=\,e^x + 2


    You had best begin with the general exponential function: . f(x) \:=\:ae^{bx} + c


    [1] and [2] tells us that a > 0


    [3] says f(0) = 3 and f(4) = 1.

    . . We have: . \begin{Bmatrix} a + c &=& 3 \\ ae^{4b} + c &=& 11 \end{Bmatrix}

    [4] gives us: . \int^4_0\left(ae^{bx} + c\right)\,dx \:=\:22

    We have three equations in three variables \{a,b,c\}.
    . . Solve the system.

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