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Math Help - Showing Differentiation

  1. #1
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    Showing Differentiation

    Let I\subset \mathbb{R} be an open interval, let f:I \rightarrow \mathbb{R} be differentiable on I, and suppose f''(a) exists at a\in I. Show that
    f''(a) = \lim_{h\to 0}\frac{f(a+h)-2f(a)+f(a-h)}{h^2}
    Give an example where this limit exists, but the function does not have a second derivative at a .
    Last edited by CrazyCat87; April 18th 2010 at 04:43 PM.
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  2. #2
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    Quote Originally Posted by CrazyCat87 View Post
    Let I\subset \mathbb{R} be an open interval, let [tex]f:I \rightarrow \mathbb{R} be differentiable on I, and suppose f''(a) exists at a\in I. Show that
    f''(a) = \lim_{h\to 0}\frac{f(a+h)-2f(a)+f(a-h)}{h^2}
    Give an example where this limit exists, but the function does not have a second derivative at a .

    Use Taylor polynomials of order 2 around a\,\,\,for\,\,\,f(a+h)\,\,\,and\,\,\,f(a-h) :

    f(a+h)=f(a)+f'(a)h+\frac{f''(a)h^2}{2!}+O(h^3)

    f(a-h)=f(a)+f'(a)(-h)+\frac{f''(a)h^2}{2!}+O(h^3)

    Now add both eq's above and solve for f''(a) and let h\rightarrow 0

    Tonio
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    Does O(h^3) represent the remainder term of the function? I'm confused as to why it disappears..
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    Quote Originally Posted by CrazyCat87 View Post
    Does O(h^3) represent the remainder term of the function? I'm confused as to why it disappears..


    It doesn't disappear: when we add the corresponding eq's we get 2O(h^3) , and when h\rightarrow 0 this, of course, vanishes in the limit...

    Tonio
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  5. #5
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    ah yes
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