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This is great. I remember studying Calculus in high school in the halcyon days of the early 90's. We spent far too much time working through proofs, and not nearly enough time discussing the actual concepts.
Fundamental Theorem of Calculus is really incredible. Most people don't understand it's consequences.
Think of numbers as resolution as well as quantity. Zooming in and out of a screen. What happens if you zoom in all the way? Well that's a lot like derivatives.
So at infinite resolution, if you sum up everything in the past, you can predict one step in the future. And the inverse holds. If you are perfect, and zoom in all the way at one point, you can know the entirely of the past. That function line is not just a bunch of dumb dots. Zoom in all the way, and it's a miniature function line.
With integrals, I remember wondering why a first power function integrates to an ellipse instead of a circle on my graphing calculator. r^2 is the area of a circle right? What solved it for me was seeing the output value of the function as the area. And looking back, what would have worked is if my calculator went into polar coordinate mode, with input as the radius and output the area of a sweep. From there I went on to seeing squares as the output of controlling two legs of input, and pressure and velocity in fluids really made sense. To go forward, you need to get the pressure to hit all sealing surfaces.