[fusiongyro]

Very neat article. This is essentially calculus with infinitesimals (also called "nonstandard analysis") implemented on the machine. If you like the approach, a more general and rigorous investigation can be had by reading H. Jerome Keisler's book Elementary Calculus, which is freely available online in the 2nd edition here:

  http://www.math.wisc.edu/~keisler/calc.html

The third edition is now in print. I've been studying calculus with it off-and-on for a while and I find the approach very intuitive, though Spivak's Calculus is probably a better book, the "standard analysis" is a little less intuitive (and now, evidently, harder to teach a machine).