You can get an idea when you try to understand why the function
y=0 for x<0 y=x^2 for x>=0
has two derivativea but not three.
But the issue is infinitesimal, so very hard to tell.
Jerk you can “linearise” if you think of a car (with no air friction) and its accelerator. Somehow…
Why not, though? Why does third order "look like" second order but second order is starkly different to first order?
Secondly, presumably the distinction of "straight" vs. "curved" is quite deeply programmed into the brain's pattern recognition machinery. The degree of curvature is a quantitative parameter on top of the qualitative categorization. This may or may not have something to do with the fact that a modern human sees straight lines everywhere (something that very much was not the case in the ancestral environment).
These are mathematical derivatives, I think of them as the slope of the thing it's derived of, aka the change in the thing that it's a derivation of.
I think I don't have a sophisticated mathematical understanding, but my basic mechanic understanding makes it feel simpler than your question is acting.