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Math Help - on what point these function are differentiable..

  1. #1
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    on what point these function are differentiable..

    i know that if a function is differentiable on X_0
    then lim [f(x+x_0)-f(x_0)]/[x-x_0] exist

    [IMG]file:///C:/DOCUME%7E1/lun/LOCALS%7E1/Temp/moz-screenshot.jpg[/IMG]
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  2. #2
    MHF Contributor Mathstud28's Avatar
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    Quote Originally Posted by transgalactic View Post
    i know that if a function is differentiable on X_0
    then lim [f(x+x_0)-f(x_0)]/[x-x_0] exist

    [IMG]file:///C:/DOCUME%7E1/lun/LOCALS%7E1/Temp/moz-screenshot.jpg[/IMG]
    A) What exactly is the first one? I believe it is \lfloor x\rfloor \sin^2(\pi x) with \lfloor x\rfloor being the floor function.


    B) f:x\longmapsto\left\{\begin{array}{rcl} x^2\cos\left(\frac{\pi}{x}\right) & \mbox{if} & x\ne 0\\ 0 & \mbox{if} & x=0\end{array}\right.

    For the second one we see that the only problem is f'(0), to ascertain its value/existence we revert back to the definitons.


    \begin{aligned}f'(0)&=\lim_{x\to 0}\frac{f(x)-f(0)}{x-0}\\<br />
&=\lim_{x\to 0}\frac{x^2\cos\left(\frac{\pi}{x}\right)-0}{x}\\<br />
&=\lim_{x\to 0}x\cos\left(\frac{\pi}{x}\right)\\<br />
&=0\end{aligned}
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  3. #3
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    regarding A:
    you descibed the function correctly
    how do you choose what numbers to pick
    after that i just put them into the formula
    and if i get a final limit then its deferentiable on that point.
    how do you choose what numbers to pick?

    regarding b:
    how did you know to check for 0
    and only for 0
    ??
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  4. #4
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    how to choose what points to test
    ??
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