This is also what's happening in an elevator. You not only want the speed to increase slowly, you also want the acceleration to increase slowly, cause that's what actually makes your guts go down. And the best way to do this is to have the acceleration of the acceleration continuous.
In the end the position of the elevator is 3-continuous (why is it called G3? in France we call this C3). And the apple corner is just a graph of the position of an elevator wrt time. Mind blowing
G^n curvature solely depends on the geometry of the curve, while C^n continuity also depends on how you parameterize the curve. So, G^n is what you want if you're talking about a purely geometric shape rather than an (x(t), y(t)) trajectory.
* reference: section 2.1 of https://graphics.stanford.edu/courses/cs348a-21-winter/Reade...
G and C continuity have slightly different meanings. You can have curves that are G^n but not C^n and vice-versa. I'll leave it to you to find a maths textbook that gives a better explanation than I would if I attempted to here.
I'm always reminded of snap, crackle and pop (https://en.wikipedia.org/wiki/Fourth,_fifth,_and_sixth_deriv...) for this topic. Essentially it's not enough to just have continuous acceleration, you have to ease into it (low snap), you can probably go into further derivatives for ultra smoothness but maybe not worth it?
As your link says, acceleration is 2nd derivative of position, so rate of change of acceleration is 3rd derivative, often called jolt. As you say, you want acceleration to vary slowly, so it's low jolt that you want.
Snap (or jounce), crackle and pop are 4th/5th/6th derivative. They're probably less of a problem.
It can help because the higher derivatives also tend to promote vibrations in the system, but I doubt it'd be perceptible by people. I have heard of 5th-order smooth curves being used for very sensitive structures, like the movement of big observatory telescopes.
I’ve also heard it called “jerk”.
Oops, yeah you’re right!