Apostol CALCULUS, 1961, page 128-130, his fake proof of Fundamental Theorem of Calculus.

In my last discussion segment of where Strang wrote a me-too text cobbing mostly from Stewart, we have to wonder if this copy paste from old authors to crank out more me-too books, where you change a few problems and add a few more words but essentially cut and paste for vast swathes of the book. One has to wonder if there is a "granddaddy book source for me-too calculus" and whether Apostol was that original source for copy paste me too authors.

One thing unique about Apostol was he starts with integral while the others start with derivative.

Does New Math weigh in on which is the proper starting point in calculus? Whether it is better to start with derivative or integral?

I think New Math does weigh in on this question. For consider starting with a integral as area under the function graph.

Start with this rectangle as the area or integral of a function from E to D interval: 

__A__C 
|         | 
|B       | 
|         | 
--------- 
E...... D 

And consider 10 Grid System and consider the functions y= x, y= x^2, y = x^60. Of course our original function is actually the derivative as we hinge up the right triangle at midpoint A. And we can extend the rectangle below the x-axis for a function of x^60.

So there is nothing wrong with starting with the integral in calculus and doing derivative later.

Now examine if doing the derivative first in calculus, if it brings any hardship.

And the derivative at x= A where A is the midpoint of top side of the rectangle and where we have to borrow from the next higher Grid System to give us midpoint A for it is empty space between D and E, above is merely the dy/dx involving points A and B. Thus, it can never be a curve in Calculus. And the AB is part of the function graph itself. No curves exist in mathematics and no continuum exists in mathematics. 

So, which is logically superior, to start with Integral or to start calculus with Derivative. Here I believe Apostol wins, for it is more intuitive that we have a rectangle to start with and to hinge up a right triangle. Otherwise, it is more difficult to explain that you want a midpoint of a right triangle hypotenuse in order to produce a rectangle by hinging it down.

So the derivative is a hinge at point A and we lift up the right triangle with hypotenuse AB and set it upon the other line segment AC. 

By starting with integral rectangle, the midpoint of a side of rectangle is more intuitive than the trapezoid figure should we start calculus with derivative.

To this: 
        B 
        /| 
      /  | 
 A /----| C 
  /      | 
|        | 
|____| 
E......D 

On page 128, Apostol says a few interesting key points. He relates the FTC as if we are proving that Squaring is the inverse of Square Root. So this is logically sound, in recognition of what we are proving in the first place-- inverses.

But also, Apostol discusses the fact that in FTC proof, we actually have three entities we must have in the proving argument (1) original function (2) derivative of original function (3) integral of original function. This is very important because in most fake proofs of FTC, no attention is given to original function.

And perhaps one of the largest reasons no-one in Old Math pursued a pure geometry proof of FTC, is that no-one in Old Math ever had the idea that the derivative as we lift the right triangle up to form the derivative that its vertex point is the very next point of the original function graph. That idea never entered the obtuse minds of Old Math, that the derivative was not a stupid silly tangent to a function graph at a point, but that the derivative actually marks out what the next function graph point is going to be. So in New Math, the derivative is part and parcel of the original function graph. That is a difficult concept for everyone in Old Math to understand. Derivative was not a tangent to a curve, for no curves exist in math. Only tiny straightline segments exist and the derivative as the right triangle is hinged up from the midpoint lands on the next point of the original function graph.

And Apostol starts the page 128 with a picture diagram called the "Geometric motivation for the first fundamental theorem of calculus."

But then Apostol devolves into a limit analysis, which is a Non Sequitur of proving derivative inverse to integral and vice versa.

So Apostol starts out with some good ideas, but ends up with a fake math contraption.

AP
King of Science

On Thursday, May 13, 2021 at 12:53:07 AM UTC-4, Archimedes Plutonium wrote:
> Apostol CALCULUS, 1961, page 128-130, his fake proof of Fundamental Theorem of Calculus.
>

That from AP of the master fake math and science!

WARNING TO STUDENTS: Don't be a victim of AP's fake math and science

AP is a malicious internet troll who wants only to mislead and confuse you. He may not be all there, but his fake math and science can only be meant to promote failure in schools. One can only guess at his motives.

Note that AP will often delete his bizarre and hateful postings when his lies are called out, only to repost identical ones moments later in a NEW thread.

Readers should, of course, judge for themselves. In AP's OWN WORDS here:

“Primes do not exist, because the set they were borne from has no division.”
--June 29, 2020

“The last and largest finite number is 10^604.”
--June 3, 2015

“0 appears to be the last and largest finite number”
--June 9, 2015

“0/0 must be equal to 1.”
-- June 9, 2015

“0 is an infinite irrational number.”
--June 28, 2015

“No negative numbers exist.”
--December 22, 2018

“Rationals are not numbers.”
--May 18, 2019

According to AP's “chess board math,” an equilateral triangle is a right-triangle.
--December 11, 2019

Which could explain...

“The value of sin(45 degrees) = 1.”
--May 31, 2019

AP deliberately and repeatedly presented the truth table for OR as the truth table for AND:

“New Logic
AND
T & T = T
T & F = T
F & T = T
F & F = F”
--November 9, 2019

AP seeks aid of Russian agents to promote failure in schools:

"Please--Asking for help from Russia-- russian robots-- to create a new, true mathematics [sic]"
--November 9, 2017

And if that wasn't weird enough...

“The totality, everything that there is [the universe], is only 1 atom of plutonium [Pu]. There is nothing outside or beyond this one atom of plutonium.”
--April 4, 1994

“The Universe itself is one gigantic big atom.”
--November 14, 2019

AP's sinister Atom God Cult of Failure???

“Since God-Pu is marching on.
Glory! Glory! Atom Plutonium!
Its truth is marching on.
It has sounded forth the trumpet that shall never call retreat;
It is sifting out the hearts of people before its judgment seat;
Oh, be swift, my soul, to answer it; be jubilant, my feet!
Our God-Pu is marching on.”
--December 15, 2018 (Note: Pu is the atomic symbol for plutonium)

Dan

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