[eiejktcd] S shaped curves

Sigmoid curves
Logistic function 1/(1+exp(-x))
Arctangent atan(x)
Error function erf(x)

A clever approximation from the Wikipedia talk page of Sigmoid function https://en.wikipedia.org/wiki/Talk:Sigmoid_function#Another_sigmoid.3F which evaluates quickly on a computer, but whose domain is bounded between -1 and 1.
a*x^b+(a-1)*x^(ab/a-1) (try a=3, b=2)

The derivatives are hat or bell shaped curves:
f(x)*(1-f(x))
1/(1+x^2)
2/(sqrt pi)*exp(-x^2) (scaled Normal PDF)