Its really quite interesting that you say that, and going off the OPS picture, one of the very foundations of Calculus, the derivatives is based of that idea. The Greek philosopher Zeno, http://en.wikipedia.org/wiki/Zeno_of_Elea, has a paradox about the flight of an arrow. Namely, at each moment of the arrows flight is the arrow moving? That is to say, if you look at the arrow during each moment, its actually frozen in time, its not moving. At each moment its stopped. So how do you know its moving?

Newton and Leibniz come along and say ah well we know its moving because if you look at where it is at the time that you stopped it, then looked at where it is 1 minute later, then we take the distance and divide it by the time to get its speed, so therefore its moving at such and such speed. Zeno comes back and says well all youve proven to me is that at this time it was here and at this time it was here, you still havent proven to me that its moving. So Newton and Leibniz shoot back ok, well Ill take its time and position at the moment, but instead of 1 minute later, Ill go 5 seconds later. So here now I have the time and position at the moment, and then I have the time and position 5 seconds later. I divide the distance by the time, and there you go it was going at this speed. Zeno shoots back again and goes no no, still all youve given me is where its at at this moment and where its at 5 seconds later, it still doesnt prove anything. Newton and Leibniz shoot back and go ok, look what Im gonna do is keep taking smaller and smaller intervals of time, as those intervals approach 0, I keep getting this same speed, this same number, so therefore I conclude that at this moment it was traveling at this speed, therefore it was moving (which is essentially what you do to get the derivative)

My apologies for the long winded response, I totally copied what I learned from this DVD with this Calculus professor, but after going through all the way up to Calculus 3 and after watching that DVD the derivative is starting to make a lot more sense to me and I thought it was appropriate to your comment and the OPs picture.