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A Different Definition of Derivative

  1. Jan 28, 2009 #1
    1. The problem statement, all variables and given/known data

    part 1. )

    If f:(a,b)--> R is differentiable at p in (a,b), prove that

    f ' (p) = lim (n -> oo) n [ f(p + 1/n) - f(p) ].

    part 2. )

    Show by example that the existence of the limit of the sequence,

    {n [ f(p + 1/n) - f(p) ] } does not imply the existence of f ' (p).

    2. Relevant equations



    3. The attempt at a solution

    One failed attempt involved comparing this limit to the definition below,

    [tex] \lim_{n \rightarrow \infty} \frac{f(p_n)-f(p)}{p_n-p} [/tex].

    I attempted to break the new definition (up top) into the sum of the limits and proceed down that path but it did not lead to anything.

    I have also tried this new definition with a functions and a point and compared its result to f'(p) to gain some understanding. I still cannot seem to get anywhere though.

    Any help would be appreciated. Thank you.
     
  2. jcsd
  3. Jan 28, 2009 #2

    Dick

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    That sequence looks EXACTLY like your definition where p_n=p+1/n. What's the problem?
     
  4. Jan 28, 2009 #3
    I see how this works for the sequence p_n= p + 1/n. Do we not have to show that the two are equivalent for all sequences that converge to p, and/or is that what we are doing?
     
  5. Jan 28, 2009 #4

    Dick

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    If the function is differentiable at p then all such sequences converge to the derivative. For the second part they want to find a function that is NOT differentiable at p, but where the limit of that particular sequence does exist.
     
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