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Math Help - check d differentiability

  1. #1
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    check d differentiability

    will d following function be differentiable at x = 0 ???
    1. cos(|x|)+|x|
    2. sin(|x|)-|x|
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  2. #2
    Senior Member Twig's Avatar
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    Gothenburg
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    Hi

    First one:

    If  h approaches zero from the right, we have:

    \frac{f(x+h)-f(x)}{h} = \frac{cos(h)+h-1}{h}=\frac{cos(h)-1}{h}+\frac{h}{h} \to 1 \mbox{ as } h \to 0

    Now we approach from the left. Note that cos(h)=cos(-h) , cos(x) is an even function.

    \frac{cos(h)-h-1}{h}=\frac{cos(h)-1}{h}-\frac{h}{h} \to -1 \mbox{ as } h \to 0

    So the function is not differentiable at  x = 0 .

    The second one you basically do the same thing.

    \frac{sin(h)-h}{h} \to 1-1 = 0 and

     \frac{-sin(h)+h}{h} \to -1 +1 = 0 .

    So the second one is differentiable.
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