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Thread: Finding a smooth(infinitely differentiable) function

  1. #1
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    Finding a smooth(infinitely differentiable) function

    So I have to find a smooth (infinitely differentiable) function where a(t) = 1 if t<=0 and a(t) = 0 if t>=1, so the major thing is finding what a(t) is when 0<t<1, and I know it'll have something to do with e^(something) but I just can't seem to find it. Any help would be much appreciated!
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  2. #2
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    Re: Finding a smooth(infinitely differentiable) function

    you have 4 restrictions on $a(t)$

    $a(0)=1$

    $a^\prime(0)=0$

    $a(1)=0$

    $a^\prime(1)=0$

    I suggest using half a period of a cosine to join these segments.

    you automatically get the derivatives at the top and bottom being 0.

    to match the endpoints you'd want

    $a(t) = \dfrac{1+\cos(\pi t)}{2}$
    Thanks from topsquark
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  3. #3
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    Re: Finding a smooth(infinitely differentiable) function

    that's great! didn't think about that and it totally works, thanks so much for your help
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