The problem calculus solves

Change that will not sit still.

Average against instant

Average speed over an hour is easy: distance divided by time. Speed at one instant is harder, because at a single moment no time passes and no distance is covered.

Calculus answers by asking what the average approaches as the interval shrinks.
That approaching value is a limit, and it is the foundation of everything else.

Two directions

Given position, find speed: that is differentiation. Given speed, find distance: that is integration. Nearly all of calculus is those two questions in different clothing.