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Thread: Using limits to say where a derivative is defined

  1. #1
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    Using limits to say where a derivative is defined

    I have been asked to say where the following derivative is defined;

    f'(x)=cosx-sinx for 2kPi-Pi/2<x<2kPi+Pi/2
    =cosx+sinx for 2kPi+Pi/2<x<2kPi+3Pi/2

    Im pretty sure i need to be using limits as I hav seen from a similar question but I am unsure how?

    Thanks for any help
    Last edited by CaptainBlack; Dec 30th 2010 at 02:17 AM.
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  2. #2
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    Hello, leigh!

    Where is the following derivative defined?

    . . $\displaystyle f'(x)\;=\; \begin{Bmatrix}\cos x-\sin x & \text{for }2k\pi-\frac{\pi}{2} < x < 2k\pi+\frac{\pi}{2} \\ \\[-3mm]
    \cos x+ \sin x & \text{for }2k\pi+\frac{\pi}{2} <x< 2k\pi+\frac{3\pi}{2} \end{array}$

    For clarity, I wrote the function like this:

    $\displaystyle f'(x) \;=\;\begin{Bmatrix}\cos x - \sin x & x \in (\text{-}\frac{\pi}{2}.\,\frac{\pi}{2}) & (\frac{3\pi}{2},\,\frac{5\pi}{2}) & (\frac{7\pi}{2},\,\frac{9\pi}{2}) & \hdots \\ \\[-3mm] \cos x + \sin x & x \in (\frac{\pi}{2},\,\frac{3\pi}{2}) & (\frac{5\pi}{2},\,\frac{7\pi}{2}) & (\frac{9\pi}{2},\,\frac{11\pi}{2}) & \hdots \end{array}$


    Clearly the derivative is defined within those intervals.
    The only doubts occur at the endpoints.


    Test an endpoint using one-sided limits.

    . . $\displaystyle \displaystyle \lim_{x\to\frac{\pi}{2}^-} f'(x) \;=\;\lim_{x\to\frac{\pi}{2}^-}(\cos x - \sin x) \;=\;\cos\tfrac{\pi}{2} - \sin\tfrac{\pi}{2} \;=\;0 - 1 \;=\;-1$

    . . $\displaystyle \displaystyle \lim_{x\to\frac{\pi}{2}^+}f'(x) \;=\;\lim_{x\to\frac{\pi}{2}^+}(\cos x + \sin x) \;=\;\cos\tfrac{\pi}{2} + \sin\tfrac{\pi}{2} \;=\;0 + 1 \;=\;+1$

    The limit does not exist at $\displaystyle x = \frac{\pi}{2}.$

    Hence, the derivative is not defined at $\displaystyle x = \frac{\pi}{2}.$


    It can be shown that the derivative is undefined for any odd multiple of $\displaystyle \frac{\pi}{2}.$


    $\displaystyle f'(x)$ is defined for all $\displaystyle \,x$ except at $\displaystyle x \:=\:\dfrac{(2k+1)\pi}{2}$ for any integer $\displaystyle \,k.$
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  3. #3
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    The limit does not exist at Pi/2
    How do you know this from finding the limit?

    Thank you very much for your help
    Last edited by CaptainBlack; Dec 30th 2010 at 02:25 AM.
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