Fisher & Ziebur, 2nd ed. CALCULUS, 1965, fake proof of Fundamental Theorem of Calculus, starting page 234. Any better than Stewart's fake FTC of 2003??

We may as well focus some detail of fake FTC, Fundamental Theorem of Calculus proofs of Old Math, for we can learn so much about how dreadful lacking in logic are most math professors. And how math classes for the most part are exercises in memorization of fake math.

I must note that this textbook of 1965 was the one I learned calculus from in Freshman College Univ Cincinnati in 1968.

Actually, Fisher & Ziebur give a better discussion of FTC than does Stewart.. Stewart relies totally on a "limit analysis" saving him from actually proving FTC and plunges directly into a fake Non Sequitur limit analysis. Whereas Fisher & Ziebur keep backing-up-- going to the Theorem of the Mean on page 112, and going to the Extended Theorem of the Mean on page 112, and going further back to Rolle's theorem on page 111.

And so what Fisher & Ziebur do is the same as fakery as Stewart in the end with a "limit analysis" saving Fisher & Ziebur with their "So our problem is to find numbers x_1, x_2, ... x_n for which Equation 37-5 is valid." And here, they apply Limit and because they say-- so, it is true and hence their fake proof. Only Fisher & Ziebur go backwards in a tangled mess to Mean Value to Rolle's and even further back.

Note: the Mean Value theorem of Old Math is what AP's pure geometry proof that takes the midpoint of a interval of a specific Grid System. Old Math had to look for theorems talking about midpoints, the Mean Value theorem because it is the midpoint of empty space that we Borrow from the 100 Grid System to apply to the 10 Grid System and flip up the right triangle on a hinge at the midpoint.

So anyone in Old Math, whose mind is not completely destroyed and polluted by Old Math, can recognize why Old Math needed a Mean Value theorem because midpoints in Empty Space of discrete numbers is essential in Calculus.

I have seen so so many people in math questioning where on Earth is this need of a Mean Value Theorem. And they will never get an answer in the error filled Old Math, because the reason they have a Mean Value Theorem but New True Math needs no Mean Value theorem, is because in New Math we have empty space between one number and its next successor number but we need a midpoint for the derivative and integral.

What fake proofs in math go to show is that there are immense variations in the fakery, from Stewart's fakery to Fisher & Ziebur's fakery. A true proof of FTC would be the same thing no matter what Old Math authors wrote it.

And also, it is highly worth repeating over and over again. That when you have True Mathematics, there is never a buildup of rules or chains of theorems. In Old Math they had the trapezoid rule, Simpson's rule, l'Hospital Rule and thousands of rules for derivatives and integrals. In New Math all our functions are polynomials and all derivatives and integrals are easy cool as a breeze in solving with only the Power rule.

AP can teach calculus in High School. Old Math saddled neck deep in error is lucky to teach their garbage in College.

AP
King of Science, King of Logic