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

    To do this, do I have to show the proof? If so what is the proof?
    Write out the Taylor expansion of $\displaystyle f(x)$ about $\displaystyle x_1$ evaluated at $\displaystyle x_2$, 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 $\displaystyle [x_1,x_2]$ if $\displaystyle x_2>x_1$ and $\displaystyle [x_2,x_1]$ if $\displaystyle x_2<x_1$.

    Now rearrange into the form:

    $\displaystyle f'(x_1)= ...$

    CB
    Last edited by CaptainBlack; Apr 10th 2010 at 11:43 PM.
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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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