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Math Help - numerical analysis? show that if f is in c^2[a,b]...

  1. #1
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    numerical analysis? show that if f is in c^2[a,b]...

    Show that if f \in C^2[a,b], then for x_1, x_2 \in[a,b], there exists \xi \in[a,b], so that f '( x_1)=(f( x_2)-f( x_1)/( x_2- x_1) )-(( x_2- x_1)/2) f''( eta)

    To do this, do I have to show the proof? If so what is the proof?
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by alice8675309 View Post
    Show that if f \in C^2[a,b], then for x_1, x_2 \in[a,b], there exists \xi \in[a,b], so that f '( x_1)=(f( x_2)-f( x_1)/( x_2- x_1) )-(( x_2- x_1)/2) f''( eta)

    To do this, do I have to show the proof? If so what is the proof?
    Write out the Taylor expansion of f(x) about x_1 evaluated at x_2, with remainder term for the series truncated after the first linear term (this will be in terms of the second derivative at some point in the interval [x_1,x_2] if x_2>x_1 and [x_2,x_1] if x_2<x_1.

    Now rearrange into the form:

    f'(x_1)= ...

    CB
    Last edited by CaptainBlack; April 11th 2010 at 12:43 AM.
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  3. #3
    Grand Panjandrum
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    Quote Originally Posted by nike22 View Post
    I also want to know the answer to this question.
    And what is it about the suggested approach that you don't understand?

    CB
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