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Math Help - when is nth differential non zero

  1. #1
    Junior Member cupid's Avatar
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    when is nth differential non zero

    >>>>>>> Question <<<<<<<<<


    it is easy to find it generally but i was thinking that what if in some question ... say 100th differential is 0 ... there has to be a shortcut ... some help please

    i even found out the general term of what differential would be after the third differential (when 2/3 x^{3} goes away)...

    its f_{n}(x) = e^{x} + (-1)^{n-1}e^{-x} -2 sin[n\pi/2  + (-1)^{n}x]

    But i dont know how to find when f_{n}(x) will be non-zero
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  2. #2
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    Use Taylor series. In my calculations, the series for f(x) starts with 4x^7/7!, so the answer is 7.
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  3. #3
    Forum Admin topsquark's Avatar
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    Quote Originally Posted by cupid View Post
    >>>>>>> Question <<<<<<<<<


    it is easy to find it generally but i was thinking that what if in some question ... say 100th differential is 0 ... there has to be a shortcut ... some help please

    i even found out the general term of what differential would be after the third differential (when 2/3 x^{3} goes away)...

    its f_{n}(x) = e^{x} + (-1)^{n-1}e^{-x} -2 sin[n\pi/2  + (-1)^{n}x]

    But i dont know how to find when f_{n}(x) will be non-zero
    Focus on that last term.
    \displaystyle \left ( \frac{2}{3}x^3 \right ) ^{\prime} = 2x^2

    ( 2x ^2 ) ^{\prime} = 4x

    ( 4x ) ^{\prime} = 4
    definitely non-zero.

    -Dan
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  4. #4
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    Quote Originally Posted by topsquark View Post
    Focus on that last term.
    \displaystyle \left ( \frac{2}{3}x^3 \right ) ^{\prime} = 2x^2

    ( 2x ^2 ) ^{\prime} = 4x

    ( 4x ) ^{\prime} = 4
    definitely non-zero.
    You can't focus on the last term only because it may be canceled by other terms.
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  5. #5
    Junior Member cupid's Avatar
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    And that is what happens .. 1,2,3 differentials are not 0
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  6. #6
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    I have no idea what you are talking about. You can check in WolframAlpha that \displaystyle\frac{d^7f(x)}{dx^7}(0)=4 and \displaystyle\frac{d^nf(x)}{dx^n}(0)=0 for n=1,\dots,6.
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  7. #7
    Junior Member cupid's Avatar
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    sorry i meant that they are 0

    typing mistake

    but isnt there is any general method to do these kind of problems
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  8. #8
    Forum Admin topsquark's Avatar
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    Quote Originally Posted by emakarov View Post
    You can't focus on the last term only because it may be canceled by other terms.
    Good point.

    -Dan
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  9. #9
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    but isnt there is any general method to do these kind of problems
    I don't know the answer because I don't know the exact class of problems you are referring to. For an arbitrary function f(x) given by its formula, I think the easiest way is to compute the nth derivative and evaluate it at 0. This particular problem seems to me to be designed to be solved using Taylor series. However, concerning problems that you may encounter during tests, etc., I don't know how similar to this one they will be.
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