I hate the terms Concave and Convex in relation to functions.
Agreed.
the line (which should be considered to be the open side because integration makes the side below the line the solid side)
This is terrible: one pretty basic property you want in your definition describing the shape of functions is that it shouldn’t change if you translate the function around.
Consider logx (for x>0). Pretty much the point of this definition is to be able to say it’s the opposite kind as ex, but your choice wouldn’t have that feature.
The “line at infinity” is a better choice for the imagined boundary. That’s how we can think of parabolae as a kind of ellipse, for example.
if we called them decelerating (concave) or accelerating (convex) functions.
That’d be at least as confusing as the current terms for functions like or e−x or −x2.
Agreed.
This is terrible: one pretty basic property you want in your definition describing the shape of functions is that it shouldn’t change if you translate the function around.
Consider logx (for x>0). Pretty much the point of this definition is to be able to say it’s the opposite kind as ex, but your choice wouldn’t have that feature.
The “line at infinity” is a better choice for the imagined boundary. That’s how we can think of parabolae as a kind of ellipse, for example.
That’d be at least as confusing as the current terms for functions like or e−x or −x2.