"I like the approach, as I explained before, because, in past times, I simply felt for my own satisfaction, that I should be able to easily see the relation between a rate of flow (water streaming into a container,) a rate of accumulation (total water in container,) and the relation of those things to the symbols on paper (equations an functions) and the numbers they might represent for a given instant of time. I felt it *should* be comfortable intuition, and not a mysterious wonder."
I am kind of surprised that you weren't looking at it that way early on. One of the early connections that helped me see the relationship was comparing the geometric area of things, starting with simple things like the area under a linear graph, and seeing how the process of integration was coming to the same result and how it was doing it. For example, the graph of y = x is a line at 45 degrees, and using the area of a triangle as we know from elementary school, 1/2*ab, so in that case, 1/2*x^2, which is the same result gotten in calculus. And if we plot that, we get a parabola and see that dy/dx giving us the change in that area per dx, and we get back to the equation of our line. Certainly, my experience algebraically with sequences helped, because I was well aware of the ways we can get x^2 through summation of arithmetic sequences, and we did do the whole epsilon-delta bit as well. Oh, and I took two years of physics as well, where we were dealing with the same relatio! nship in many problems. In any event, the relationship between the derivative and integration was clicking with me on multiple fronts.
Did you know the math, but were unable to connect it to an analogy, or were there simply no suitable analogies to connect it to?