I often deal with some functions that are increasing, but where some are increasing less than linearly as a function of x, and others are increasing more than linearly. I would like to know the technical terms to describe such functions. Intuitively, the terms 'sub-linear' and 'super-linear' make sense to me to describe such functions, but evidently that is not correct terminology. It seems the term 'exponential' is often used loosely to refer to anything that is greater than linear, but I'm not sure about functions that are less than linear.
Heavens! "Exponential" means exponential, and nothing else.
If the derivative $f'$ of a function $f$ is increasing with $x$ then this function is called convex, and if $f'$ is decreasing then $f$ is called concave.
A convex function can be decreasing or increasing. In any case it has at most one interior minimum, and is maximal at one of the endpoints of the domain interval.
A concave function can be increasing or decreasing. In any case it has at most one interior maximum, and is minimal at one of the endpoints of the domain interval.