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!

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

$a(t) = \dfrac{1+\cos(\pi t)}{2}$

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 :)