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Second derivative question

  1. Dec 13, 2006 #1
    1. The problem statement, all variables and given/known data

    Show that lim(h-->0) [f(x+h)-2f(x) + f(x-h)]/h^2
    is equal to f''(x) for any given value of x where the second derivative exists.


    I'm supposed to use L'Hopitals rule for this problem. I did and got
    [f(x+h)-f(x-h)]/2h

    Now I am stuck. I thought about adding and subtracting say f(x) but don't see how that solves to f''(x)
     
  2. jcsd
  3. Dec 13, 2006 #2

    StatusX

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    You're almost there. You can rewrite that as the average of the left and right limits for the derivative of f(x), which will be the same if f is differentiable at x.
     
  4. Dec 13, 2006 #3

    Hurkyl

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    Gold Member

    f(x+h)-f(x-h)

    is not the derivative of

    f(x+h)-2f(x) + f(x-h)


    p.s.: for the record, you have to assume f is differentiable in an entire neighborhood of x in order to invoke L'Hôpital's rule.
     
    Last edited: Dec 13, 2006
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