Alright, I am just about to start the 3rd edition but let me make special notes of a few things.
In the box function of y = 3 the derivative is 0 and the integral is Int = 3x and in this function we see the picketfences as rectangles only with no triangle atop. Now in the sawtooth function of F(x) = 0 for even numbered x and ?F(x) = 10^603 for odd numbered x. We have no rectangle in the picketfence but just pure triangles and the derivative dy/dx is 10^603/10^-603 = 10^1206 and the integral in the interval 0 to 10^603 is 0.5*10^1206. Now the function y = 3 can be sawtooth also by this manipulation F(x) = 0 for even numbers and F(x) = 3 for odd numbers, so that instead of a flat line across y = 3, we have a sawtooth pattern. And the derivative dy/dx is 3/10^-603 which is 3*10^603 and the integral in the interval 0 to 10^603 is 1/2(3*10^603). So for the function y=3 its area from 0 to 10^603 is 3*10^603 and a sawtooth pattern is 1/2 of that area.
So in the 3rd edition I must realize that sometimes the function has only a rectangle for integration, and sometimes only triangles, but most often it has a combination of rectangle and triangle atop that is a picketfence.
So what the Fundamental Theorem of Calculus concerns itself with is why is Euclidean geometry the only geometry to have the Calculus and that it turns all curves into straight lines for derivative and for integral. So that when we graph the half-circle and find its derivative and integral the circle curve at the 10^-603 is no longer a curve but a tiny straight line segment. That is the essence of what the Calculus does to the function, replaces its curves with straight line segments. And that is why Euclidean geometry is the only one of the three geometries to have a Calculus for we end up replacing the curves of Elliptic and Hyperbolic geometry with straightline segments.
That is why the borderline of infinity cannot be a number smaller than Floor-pi*10^603 because only there is a semicircle function able to be a truncated regular polyhedra.
Alright, I am ready to do the 3rd edition.
-- More than 90 percent of AP's posts are missing in the Google newsgroups author search archive from May 2012 to May 2013. Drexel University's Math Forum has done a far better job and many of those missing Google posts can be seen here: