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Math Help - Help with FTC1 problem

  1. #1
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    Help with FTC1 problem

    The question is:

    The function f is given by {f(x)}=\frac{e^\frac{1}{x}}{x^2} where x ≠ 0. Determine a number a<0 such that:

    \int_a^0 {f(x)}{dx} = {f(a)}

    What I've done so far:

    We know that {f'(x)}=\frac{-e^{-x}{x^2}-{2xe^{-x}}}{x^4}=\frac{{-e^{-x}}{(x+2)}}{x^3}

    And also that -\int_0^a{f(x)}{dx}={f(a)} So using FTC1 =>

    \frac{d}{dx}-\int_0^a{f(x)}{dx}=\frac{d}{dx}{f(a)} =>

    -{f(a)}={-e^{-a}}\frac{a+2}{a^3} =>

    \frac{-e^{-a}}{a^2}={-e^{-a}}\frac{a+2}{a^3}

    What do I do now? Assuming I haven't made a catastrophic mistake in my working? Any tips/suggestions very much welcome.

    Best,
    CP


    Last edited by CrispyPlanet; October 16th 2013 at 10:55 AM.
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  2. #2
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    Re: Help with FTC1 problem

    Just in case a picture helps...



    ... where (key in spoiler) ...

    Spoiler:


    ... is the chain rule. Straight continuous lines differentiate downwards (integrate up) with respect to the main variable (in this case a), and the straight dashed line similarly but with respect to the dashed balloon expression (the inner function of the composite which is subject to the chain rule).

    But this is wrapped inside the legs-uncrossed version of...



    ... the product rule, where, again, straight continuous lines are differentiating downwards with respect to a.



    ... is the FTC


    Full size


    __________________________________________________ __________

    Don't integrate - balloontegrate!

    Balloon Calculus; standard integrals, derivatives and methods

    Balloon Calculus Drawing with LaTeX and Asymptote!
    Last edited by tom@ballooncalculus; October 16th 2013 at 02:10 PM.
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  3. #3
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    Re: Help with FTC1 problem

    Very helpful indeed. Thank you!
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  4. #4
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    Re: Help with FTC1 problem

    Quote Originally Posted by CrispyPlanet View Post
    The question is:
    The function f is given by {f(x)}=\frac{e^\frac{1}{x}}{x^2} where x ≠ 0. Determine a number a<0 such that:
    \int_a^0 {f(x)}{dx} = {f(a)}
    I find this question completely faulted.
    The integral \int_a^0 {{x^{ - 2}}{e^{1/x}}dx} is non-convergence for any a\ne 0.

    You can explore this at this webpage.

    Not even "cute" balloons change that fact.
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