You can build an intuition by drawing a curve on a sheet of very thin, high quality paper, mark an interval and cut out a slice that contains the area under the curve, roll some weed into the slice and smoke it.
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It becomes obvious that the Fundamental Theorem defines the nature of movements, as the relationship between position and velocity, velocity and acceleration, acceleration and impulse and so on. From there, you see that it generalizes to all (approximately) continuous changes.
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It’s important to realize that the Fundamental Theorem does not describe some arkane mathematical mystery, but merely points at something that your brain learned to do when you were a toddler, and allows you to write it down with a notation that can be plugged into a calculator.
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