# Finding a smooth(infinitely differentiable) function

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• Oct 11th 2016, 01:17 PM
shea
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!
• Oct 11th 2016, 01:33 PM
romsek
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

$a(t) = \dfrac{1+\cos(\pi t)}{2}$
• Oct 11th 2016, 03:29 PM
shea
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 :)