See comments below, for I need an extra chapter. I thought last night I was done with revision for a good while. But no, I quickly realized the proof needs a Mathematical Induction Proof on top of the "arbitrary function yields general proof". And in this light, can we ask whether Mathematical Induction proofs are the same as "arbitrary selection proof gives general proof".. So, well that would be AP's 177th book of science that Mathematical Induction = proof in general via arbitrary selection. Please see comments below.

11th published book

World's First Geometry Proof of Fundamental Theorem of Calculus// Math proof series, book 2 Kindle Edition
by Archimedes Plutonium (Author)

Last revision was 16May2021. This is AP's 11th published book of science.
Preface:
Actually my title is too modest, for the proof that lies within this book makes it the World's First Valid Proof of Fundamental Theorem of Calculus, for in my modesty, I just wanted to emphasis that calculus was geometry and needed a geometry proof. Not being modest, there has never been a valid proof of FTC until AP's 2015 proof. This also implies that only a geometry proof of FTC constitutes a valid proof of FTC.

Calculus needs a geometry proof of Fundamental Theorem of Calculus. But none could ever be obtained in Old Math so long as they had a huge mass of mistakes, errors, fakes and con-artist trickery such as the "limit analysis". To give a Geometry Proof of Fundamental Theorem of Calculus requires math be cleaned-up and cleaned-out of most of math's mistakes and errors. So in a sense, a Geometry FTC proof is a exercise in Consistency of all of Mathematics. In order to prove a FTC geometry proof, requires throwing out the error filled mess of Old Math. Can the Reals be the true numbers of mathematics if the Reals cannot deliver a Geometry proof of FTC? Can the functions that are not polynomial functions allow us to give a Geometry proof of FTC? Can a Coordinate System in 2D have 4 quadrants and still give a Geometry proof of FTC? Can a equation of mathematics with a number that is _not a positive decimal Grid Number_ all alone on the right side of the equation, at all times, allow us to give a Geometry proof of the FTC?

Cover Picture: Is my hand written, one page geometry proof of the Fundamental Theorem of Calculus, the world's first geometry proof of FTC, 2013-2015, by AP.

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Table of Contents
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1) My history of Fundamental Theorem of Calculus from 2013-2015.

2) Preliminary mathematics needed to do the Geometry Proof of Fundamental Theorem of Calculus.

3) How the dumb limit concept was borne, that is a con-artist job.

4) When you know the true numbers of mathematics, Decimal Grid Numbers, you need no limit concept.

5) Mathematics has two houses, one is numbers, one is geometry.

6) All numbers come from physics because the Universe is just one big atom of 231Pu.

7) History of my discovery of Decimal Grid Numbers.

8) The error of having a proper Coordinate System to do the Calculus as 1st Quadrant Only with all positive Decimal Grid Numbers.

9) Concept of Infinity versus Finite for Calculus.

10) Brief proofs of the Infinity borderline, especially Huygens tractrix.

11) World's first picture diagram proof of the Fundamental Theorem of Calculus by 2015.

12) Calculus the very most important math to date.

13) Everyone in mathematics knows that Calculus is geometry.

14) As of 2021, the leading mathematicians in the world are failures of calculus by never providing a geometry proof of Fundamental Theorem of Calculus, Tao, Wiles, Hales, Stillwell to list a few.

15) In the opinion of AP, cancel, or cast into oblivion, all medals and rewards of honor in mathematics such as Fields, due to the fact, only AP can do a geometry proof of Fundamental Theorem of Calculus.

16) Set up a world wide system of mathematics oversight, wherein professors of mathematics can be reprimanded or thrown out of mathematics simply because they are not logical.

17) The Old Hilbert Math Problems-- Standard Bearer becomes the AP Standard Bearer.

18) Examples of fake Old Math FTC proofs by Stewart, Fisher & Ziebur, Strang, Apostol, Ellis & Gulick.

19) Perhaps a new idea and theorem of Calculus, call it the Restriction Calculus Theorem if true.

20) Review and proof on how the Power Rules of Derivative and Integral are derived.

21) Explanation for why integral power rule is add 1 and derivative power rule is subtract 1.

22) Mystery of the derivative as it projects forward into the future of the next successor point of function graph.

23) Why the integral comes before the derivative.

24) History summary of Fundamental Theorem of Calculus and its proof.

25) The Fundamental Theorem of Calculus as the pinnacle peak of all mathematics.

Length: 100 pages

Product details
ASIN : B07PQTNHMY
Publication date : March 14, 2019
Language : English
File size : 1281 KB
Text-to-Speech : Enabled
Screen Reader : Supported
Enhanced typesetting : Enabled
X-Ray : Not Enabled
Word Wise : Not Enabled
Print length : 100 pages
Lending : Enabled
Amazon Best Sellers Rank: #128,729 Paid in Kindle Store (See Top 100 Paid in Kindle Store)
#2 in 45-Minute Science & Math Short Reads
#134 in Calculus (Books)
#20 in Calculus (Kindle Store)

Comments as of 17May2021. I thought for sure I was finished for awhile on this book. Having reached 100 pages. But when taking a bathe last night sitting in the tub, I realized that all functions of Y=mx+b, cut across the rectangle integral at midpoint. In a picture like this.

__/_
| /    |
|    |
|      |
------
2.0 2.1
So all functions of Y=mx+b, the first order polynomial crisscross through the integral rectangle and all you need do is lift the excess right triangle from left wall and lift it up on its hinge at midpoint to form the new coordinate point of x=2.1.

Now the next order polynomial of order 2 or 3 or 4, etc etc. our right triangle has to be manufactured from the left wall of rectangle integral and is not ready made as in order 1 polynomial.

And here I should have defined early on first beginning chapters that there exists no curved line. I should have made these two definitions plain and clear.

Straightline: a straightline is a line whose algebra form if Y = mx+b for its duration.

Curvedstraightline: this is one word, and what it means is that a curved line is a string of straightlines connected up. A circle is a curvedstraightline. A circle in New Math is a regular polygon of many sides, so many that we are fooled and tricked into thinking it is a smooth circle. In Old Math, they believed a smooth curve actually exists. In New Math, there is _no smooth curve_ but only Curvedstraightlines.

So the reason in my Arbitrary to General Proof, that I had a large sigma error in the cell 2.0 to 2.1 of 10 Grid and then switched to the 100 Grid for cell 2.00 to 2.01 and reduced that sigma error to acceptable measure, is because we straighten out the "curvedstraightline" more in the 100 Grid and even more so in the 1000 Grid.

So I need to add these to the 100 page book, and probably end up writing how the Proof Method of Mathematical Induction is the same as the proof method of going from Arbitrary to General constitutes a proof of Logic.

AP
King of Science

Re: World's first and only valid proof of math's Fundamental Theorem of Calculus.

Now I was looking to see if any of the surveyed authors of Old Math with their invalid FTC proofs had a similar picture of where their integral crisscrosses the integral rectangle and where you simply lift up the right triangle in half of the cell to intersect with the next successor point of the cell.

Of course, well, the Old Math author would need to show a function of Y = mx+b, a first order polynomial, not a curvedstraightline function of higher order than 1.

Looking at Stewart's 5th ed. on page 368 he shows cells of a curvedlinefunction. On pages 372 to 373, Stewart shows pictures of the cell rectangle sticking out above a curvedstraightline function then below the function, not sticking out. So in this scheme of things, Stewart is going to argue that the area under the function graph is a summation of the sticking out and the not sticking out.

Looking at Apostol on page 5, he also is going to make a "average summation" of sticking out cells with not sticking out cells to come up with a area of integral.

What AP does instead of a average of sticking out and not sticking out. Is that AP has no curves in geometry, and that once you form a cell, inside that cell every line is a Straightline, never a curve. And so there is no averaging in AP, there is just a carving of a right triangle to coincide with the next point of the function graph. The going of from x=2.0 to x=2.1 and its y coordinate point coinciding with the right triangle carved out of the x=2.0 wall.

And the reason I need to write an additional chapter is that FTC has not only a proof by Arbitrary going to General, but a proof by good old natural Mathematical Induction.

A shame that no Old Math author used a first order polynomial to explain integral, for if they did, they could not help but notice the right triangle you flip up at hinge of midpoint of Cell.

AP
King of Science

Re: World's first and only valid proof of math's Fundamental Theorem of Calculus.

Have to check and be sure that Fisher & Ziebur, Ellis & Gulick, and Strang in their Calculus books never show the Integral with a 1st order polynomial..

Checking on Strang of his 1st ed. Calculus page 191, is worse than Stewart or Apostol where Strang only shows "sticking out integral rectangles" not even a averaging procedure.

Looking at Ellis & Gulick 3rd ed. page 235, they do a Y= x^2 and shows them both sticking out rectangles and not sticking out for a average, much like Apostol and Stewart.

Looking at Fisher & Ziebur, 2nd ed. page 200, they come awfully close to a 1st degree polynomial where part sticks out and part does not stick out for integral rectangle of function Y = 1/(1+x^2). A shame that all Fisher & Ziebur needed to do was show as first examples a function like Y=2x and Y = 10- 2x over interval 0 to 5. Displaying that all you need do for calculus FTC proof is flip on a hinge at midpoint to obtain exact area under function graph.

There is one interesting feature of Fisher & Ziebur as they spent too much time, way too much time on what seems to be their favorite function of the step function on page 204. I have no idea where Fisher & Ziebur got their over emphasis on step functions, perhaps as a me-too-text of Apostol page 50 step functions. And Apostol uses the step function to attain quite a few theorems.

So maybe in AP's Mathematical Induction proof of FTC, maybe AP should not start with Y=mx+b but even a more primal and crude function as that of step function. And here we have to debate whether Y=mx+b is the most basic of 1st order polynomial or whether Y= b is the starting function for all polynomial functions. But what the authors of Apostol, Stewart, Fisher & Ziebur, Ellis & Gulick, Strang all missed in their Calculus was that you start Integral not with a advanced curvedstraightline but with the most basic of lines the Y= mx+b.

It is AP's suspicion that no Old Math mathematician could ever do a valid FTC proof, because it takes more than just logical mind, it takes a logical personality such as AP, to say-- what is wrong in your mindset, is that curves of Old Math do not exist and that the Calculus is totally built from Discrete Space with only tiny fine straightlines crisscrossing about that Space.

The best that Old Math could ever do, is show a sequence of sticking out rectangles then a sequence of not sticking out rectangles and average the two sets together.

In New Math, we construct the right triangle that hinges at the midpoint of the cell and bridges over from x_1 point to its successor point x_2.

So in Old Math, they walk away with the silly crazy notion that the derivative is a tangent to a mythical non existing curve. In New Math, the derivative is the straightline segment that joins the first point (x_1, y_1) to its next successor point (x_2, y_2) and of course the derivative is a tangent line because all lines are straightline segments in New Math.

AP
King of Science