I was hoping I could replace the picketfence model by two pure triangle model.
Remember the sawtooth function of F(x) = 0 for even numbered x and F(x) = 10^603 for odd numbered x. Here we can do the integration, not by summation of thin picketfences but rather by summation of two thin pure triangles along the leftside and rightside of each point of the function graph. From that sawtooth function I was hoping to eliminate the rectangle portion of the picketfence model, but I cannot. So that leaves me with the-- having to prove that Calculus best model is the picketfence and it is irreplaceable for the calculus. Proof: I would use the identity function and obviously the picketfence model gives the derivative of 1 and the integral of 1/2x^2, but, can I get a derivative of 1 via 2 pure triangles? So far, no. If the only way to get the derivative and integral of the identity function via picketfence model, then it is irreplaceable.
-- 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: