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Math Help - Find if function is increasing or decreasing (Find the derivative)

  1. #1
    Junior Member moonman's Avatar
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    Find if function is increasing or decreasing (Find the derivative)

    ne^{-\frac{n^2}{8}}

    ne^{-\frac{n^2}{8}} = \frac{n}{e^\frac{n^2}{8}}

    with quotient rule

    \frac{n}{e^\frac{n^2}{8}}= \frac{(n)'(e^{\frac{n^2}{8}})-(n)(e^{\frac{n^2}{8}})'}{(e^{\frac{n^2}{8}})^2}

    = \frac{(e^{\frac{n^2}{8}})-(n)(\frac{n}{4})}{(e^{\frac{n^2}{8}})^2}

    = \frac{1 - \frac{n^2}{4}}{e^\frac{n^2}{8}}

    What can I do next? Or can I assume the function is decreasing now?
    Last edited by moonman; November 26th 2012 at 10:36 AM.
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  2. #2
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    Re: Find if function is increasing or decreasing (Find the derivative)

    Quote Originally Posted by moonman View Post
    ne^{-\frac{n^2}{8}}

    ne^{-\frac{n^2}{8}} = \frac{n}{e^\frac{n^2}{8}}

    with quotient rule

    \frac{n}{e^\frac{n^2}{8}}= \frac{(n)'(e^{\frac{n^2}{8}})-(n)(e^{\frac{n^2}{8}})'}{(e^{\frac{n^2}{8}})^2}

    = \frac{(e^{\frac{n^2}{8}})-(n)(\frac{n}{4})}{(e^{\frac{n^2}{8}})^2}

    = \frac{1 - \frac{n^2}{4}}{e^\frac{n^2}{8}}

    What can I do next? Or can I assume the function is decreasing now?
    ... what if |n| < 2 ?
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  3. #3
    Junior Member moonman's Avatar
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    Re: Find if function is increasing or decreasing (Find the derivative)

    Quote Originally Posted by skeeter View Post
    ... what if |n| < 2 ?
    If |n| < 2 then the function would be positive...so then then the function is positive?
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  4. #4
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    Re: Find if function is increasing or decreasing (Find the derivative)

    no ... what does f' > 0 tell you about f ?
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  5. #5
    Junior Member moonman's Avatar
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    Re: Find if function is increasing or decreasing (Find the derivative)

    Quote Originally Posted by skeeter View Post
    no ... what does f' > 0 tell you about f ?
    That tells me that f is increasing?
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  6. #6
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    Re: Find if function is increasing or decreasing (Find the derivative)

    The derivative of the function vanishes at n = 2, -2? This function cannot be monotonic. Looks to me that it has points of extrema. It can only be locally increasing or decreasing.
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  7. #7
    Junior Member moonman's Avatar
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    Re: Find if function is increasing or decreasing (Find the derivative)

    So can the Integral Test be performed on this if it were a series?
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  8. #8
    Junior Member moonman's Avatar
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    Re: Find if function is increasing or decreasing (Find the derivative)

    Ok, I found an error in differentiation.
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  9. #9
    Junior Member moonman's Avatar
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    Re: Find if function is increasing or decreasing (Find the derivative)

    ne^{-\frac{n^2}{8}}

    ne^{-\frac{n^2}{8}} = \frac{n}{e^\frac{n^2}{8}}

    with quotient rule

    \frac{n}{e^\frac{n^2}{8}}= \frac{(n)'(e^{\frac{n^2}{8}})-(n)(e^{\frac{n^2}{8}})'}{(e^{\frac{n^2}{8}})^2}

    = \frac{(e^{\frac{n^2}{8}})-(n)(e^\frac{n^2}{8})(\frac{n}{4})}{(e^{\frac{n^2}{  8}})^2}

    =\frac{1-\frac{n^2}{4}}{1} always negative
    Correct? Can someone please check the derivative?
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  10. #10
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    Re: Find if function is increasing or decreasing (Find the derivative)

    you had it correct the first time ...

    \frac{d}{dx} \left[x \cdot e^{-\frac{x^2}{8}}\right]

    x\left(-\frac{x}{4}\right)e^{-\frac{x^2}{8}} + e^{-\frac{x^2}{8}}

    e^{-\frac{x^2}{8}}\left[1 - \frac{x^2}{4}\right]

    the derivative is negative for |x| > 2 and positive for |x| < 2
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