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Read my recent posts in peace and quiet.
https://groups.google.com/forum/?hl=en#!forum/plutonium-atom-universe Archimedes Plutonium
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Archimedes Plutonium
2:12 AM (15 hours ago)
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Alright I come to realize I have no graphic explanation for the proof of the Fundamental Theorem of Calculus for a downward slope function graph. I gave a proof for the upward slope function.
We start with the integral rectangle in the Cell, a specific cell of the function graph. In 10 Decimal Grid there are exactly 100 cells for each number interval, say from 0 to 0.1, then the next cell is 0.1 to 0.2. The midpoint in each cell belongs to a
number in the next higher Grid System, the 100 Grid. So the midpoint of cell 1.1 to 1.2 is 1.15 as midpoint.
Now the integral in that cell of 1.1 to 1.2 is a rectangle and say our function is x^2 --> Y. So the function graph is (1.1, 1.21) and (1.2, 1.44). Now we are strictly in 10 Grid borrowing from 100 Grid.
So say this is our Integral rectangle in cell 1.1 to 1.2.
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1.1 1.2
More later,...
What I am getting at is that in a upward slope the right triangle whose tip is 1.44 hinged at the midpoint 1.15 predicts that future point in the derivative as the right triangle hypotenuse.
But the geometry is different for a downward slope function such as 10 -x --> Y. In this case we have the rectangle integral, but instead of hinging up the right triangle to predict the next point of the function graph, we totally remove the right
triangle from the graph and the missing right-triangle is the successor point.
Teaches that derivative predicts next point of function graph--silly Old Math has derivative as tangent to function graph unable to predict. The great power of Calculus is integral is area under function graph thus physics energy, and its prediction
power of the derivative to predict the next future point of function graph thus making the derivative a "law of physics as predictor". Stupid Old Math makes the derivative a tangent line, while New Math makes the derivative the predictor of next point of
function graph. No wonder no-one in Old Math could do a geometry, let alone a valid proof of Fundamental Theorem of Calculus, for no-one in Old Math even had the mind to realize Calculus predicts the future point in the derivative.
TEACHING TRUE MATHEMATICS-- only math textbooks with a valid proof of Fundamental Theorem of Calculus--teaches that derivative predicts next point of function graph--silly Old Math has derivative as tangent to function graph unable to predict. This is
why calculus is so important for physics, like a law of physics-- predicts the future given nearby point, predicts the next point. And of course the integral tells us the energy. Silly stupid Old Math understood the integral as area under the function
graph curve, but were stupid silly as to the understanding of derivative-- predict the next point as seen in this illustration:
From this rectangle of the integral with points A, midpoint then B
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To this trapezoid with points A, m, B
B
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m /----|
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The trapezoid roof has to be a straight-line segment (the derivative)
so that it can be hinged at m, and swiveled down to form rectangle for integral.
Or going in reverse. From rectangle, the right triangle predicts the next successor point of function graph curve of B, from that of midpoint m and initial point of function graph A.
AP
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Archimedes Plutonium
1:04 PM (4 hours ago)
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In the case of a upward slope function, the derivative requires a midpoint in the integral rectangle for which the right triangle is hinged at the midpoint and raised to rest upon the 4 sided trapezoid that the rectangle becomes. Thus the vertex tip of
right triangle predicts the next future point of the function graph by this vertex tip.
However, a different situation arises as the function graph has a downward slope. There is no raising of a right triangle cut-out of the integral rectangle. And there is no need for a midpoint on top wall of the integral rectangle. For a downward slope
Function Graph, we cut-away a right triangle and discard it. Here the vertex tip is below the level of the entering function graph and is predicted by the derivative.
So there are two geometry accounting for the Fundamental Theorem of Calculus proof. There is the accounting of a function graph if the function has a upward slope and there is the accounting if the function graph is a downward slope. Both involve the
Integral as a rectangle in a cell of whatever Grid System one is in. In 10 Grid there are 100 cells along the x-axis, in 100 Grid there are 100^2 cells. If the function is upward slope we need the midpoint of cell and the right triangle is hinged at that
midpoint. If the function is downward slope, the right triangle is shaved off and discarded-- no midpoint needed and the resultant figure could end up being a rectangle becoming a triangle. In the upward slope function graph, the rectangle becomes a
trapezoid, possibly even a triangle.
AP
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Archimedes Plutonium
3:32 PM (2 hours ago)
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So for an upward slope function, the Proof of Fundamental Theorem of Calculus would have the integral rectangle turned into this.
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To this trapezoid with points A, m, B
B
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m /----|
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While for a downward slope function, the Proof of Fundamental Theorem of Calculus would have the integral rectangle turned into this.
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Where the right-triangle is now swiveled at midpoint but rather where a right triangle is cut-away from the Integral that is a rectangle and that right triangle is then discarded.
AP
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Archimedes Plutonium
11:18 PM (1 hour ago)
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Now two of the most interesting and fascinating downward slope functions in 10 Grid of 1st Quadrant Only would be the quarter circle and the tractrix.
Many of us forget that functions are Sequence progressions, starting at 0 and moving through all 100 cells of the 10 Decimal Grid System.
Here, I have in mind for the quarter circle a radius of 10 to be all inclusive of the 10 Grid.
AP
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Archimedes Plutonium
11:27 AM (4 hours ago)
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By insisting that the only valid function in the world is a polynomial function, we thus reduce Calculus to the ultra simple task of the Power Rule.
So we have a function of x^3, the derivative by Power Rule is (3)x^2. The integral by Power Rule is (1/4)x^4, and to check to see if integral is correct, we take the derivative of (1/4)x^4 to see if it becomes x^3, and surely it does so.
So what AP teaches math to the world, is that Calculus can be mastered by 13 and 14 year olds. Students just beginning High School.
Impossible in Old Math because Old Math is filled with mistakes and errors and crazy idiotic and stupid math.
In New Math, we clean house. We do not let creeps and kooks fill up math that causes students to have nightmares and nervous breakdowns and vomit before tests.
In New Math, we think only of our young students, we do not think of kooks like Dr.Hales, Dr.Tao, Dr. Wiles trying to achieve fame and fortune at the expense of our young students-- who, all they wanted was to learn the truth of mathematics.
If you run to a teacher of New Math with a function, and that function is not a polynomial, then the teacher is going to tell you "that is not a valid function, and you simply convert it to a polynomial".
In AP math class in 9th grade USA, AP makes students of 13 and 14 year old master Calculus. Master calculus better, far better than 1st year college students in Old Math at any college or university across the globe.
14 year old students in AP math class master calculus and "have fun and joy" in math class.
19 or 20 year olds in colleges and universities go through nightmares, vomiting, and even nervous breakdowns in their learning calculus.
I am not exaggerating here, but obvious observations of education of mathematics.
No-one in math education cares about students in Old Math. No-one has ever Cleaned House of Old Math, but let the rotten fetid Old Math stench increase.
AP, King of Science
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Archimedes Plutonium
3:56 AM (10 hours ago)
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Now I need to add more to the Power Rules of Calculus as we make Polynomials be the only valid functions of mathematics. If you come to math with a function not a polynomial, you are sent home to convert your silly contraption into a polynomial over a
interval in 1st Quadrant Only, a interval of concern.
But in all the years I did calculus, I seem to not have registered in my mind the geometrical significance of the Power Rules. What is the geometry of taking x^2 to the power rule of n(x^n-1) for derivative. Then what is the geometry significance of
taking the integral power rule-- (1/(n+1)) (x^(n+1)).
It seems to me that at one moment in time, that geometry stuck to my mind, but is now elusive, I cannot recall the geometry significance of either Power Rule when played out on x^n.
Cavalieri 1598-1647
So that if we start with a polynomial function such as x^2 -> Y, we instantly know from the power rules that the derivative is 2x and the integral is 1/3x^3.
Derivative Power Rule of a polynomial x^n that the derivative is n(x^n-1).
The Integral Power Rule is sort of the opposite of the derivative rule so for polynomial x^n that the integral is (1/(n+1)) (x^(n+1)).
On Tuesday, September 5, 2023 at 3:00:37 AM UTC-5, Archimedes Plutonium wrote:
Now I need to add more to the Power Rules of Calculus as we make Polynomials be the only valid functions of mathematics. If you come to math with a function not a polynomial, you are sent home to convert your silly contraption into a polynomial over a
interval in 1st Quadrant Only, a interval of concern.
But in all the years I did calculus, I seem to not have registered in my mind the geometrical significance of the Power Rules. What is the geometry of taking x^2 to the power rule of n(x^n-1) for derivative. Then what is the geometry significance of
taking the integral power rule-- (1/(n+1)) (x^(n+1)).
It seems to me that at one moment in time, that geometry stuck to my mind, but is now elusive, I cannot recall the geometry significance of either Power Rule when played out on x^n.
Cavalieri 1598-1647
So that if we start with a polynomial function such as x^2 -> Y, we instantly know from the power rules that the derivative is 2x and the integral is 1/3x^3.
Derivative Power Rule of a polynomial x^n that the derivative is n(x^n-1).
The Integral Power Rule is sort of the opposite of the derivative rule so for polynomial x^n that the integral is (1/(n+1)) (x^(n+1)).
Now I need to include the Cavalieri proof, a geometry proof that rectangles under a function graph such as Y--> x^2 yields the power rule formula (1/(n+1))(x^(n+1)) so for x^2 the integral is (1/3)x^3.
I would think that showing Cavalieri's proof would be standard fare in all 1st year college calculus textbooks. To my surprise, not Stewart, not Apostol, not Fisher& Zieber, not Ellis & Gulick, not Strang, no-one is up to the task of showing how
Cavalieri got that formula from summing rectangles.
Morris Kline in volume 1 "Mathematical Thought" shows a picture.
Stillwell in "Mathematics and its History" shows a picture.
But it must be too difficult for college authors to replicate Cavalieri's proof of approximating rectangles for x^2.
Now if I were back in the days of Cavalieri and tasked to find a formula, I would do rectangles and trial and error. First finding a formula for easy ones such as Y--> x, then Y-->x^2, then a third trial, Y--> 2x to see if the formula is good, sort of a
math induction settling upon (1/(n+1))(x^(n+1)).
But I am very disappointed that none of my college calculus books derives the formula (1/(n+1))(x^(n+1)) via approximation.
There were no standards for math proof in the days of Cavalieri for his genius of deriving the Integral Power rule. Y--> x^n is integral (1/(n+1))(x^(n+1))
So what I am going to do is prove (1/(n+1))(x^(n+1)) in New Math.
I looked through the literature and there was no actual Old Math proof of (1/(n+1))(x^(n+1))
This is worthy of a whole entire new book of itself.
And the beauty is that it is a Mathematical Induction proof.
And the beauty also is that functions are chains of straightline connections from one point to the next in Discrete Geometry.
That means we no longer approximate the integral but actually derive the Integral from a Right Trapezoid whose area is 1/2(base_1 + base_2)(height).
We see that in a function such as 3x becomes integral (1/2)(3)x^2 due to that right-trapezoid area.
The right-trapezoid is such that its base_1 and base_2 are the Y points for cells of calculus in Decimal Grid Systems.
Trouble in Old Math is when the "so called historian" reads a passage in old works, they become overgenerous in crediting a proof when none really existed -- Fermat, Cavalieri. And this is the reason that no-one in modern times who wrote a Calculus
textbook features the Cavalieri Integral Power Rule, because there never was a proof, .... until now... a Mathematical Induction proof.
AP, King of Science
None of this is a proof of Cavalieri's integral power rule formula. Because Geometry is discrete and all curves in geometry are chains of straightline segments. The Internet boasts of some modern recent proofs of Cavalieri, but I suspect all those are
bogus claims, being victims of computer graphics and no honest down to earth proof at all. I myself was a victim of computer graphics, for a computer can really spit out any image you ask it to spit out, such as hexagon tiling of sphere surface.
--- quoting Wikipedia ---
The modern proof is to use an antiderivative: the derivative of xn is shown to be nxn−1 – for non-negative integers. This is shown from the binomial formula and the definition of the derivative – and thus by the fundamental theorem of calculus the
antiderivative is the integral. This method fails for
∫1/x dx
which is undefined due to division by zero. The logarithm function, which is the actual antiderivative of 1/x, must be introduced and examined separately.
The derivative
(x^n)'=nx^{n-1} can be geometrized as the infinitesimal change in volume of the n-cube, which is the area of n faces, each of dimension n − 1.
Integrating this picture – stacking the faces – geometrizes the fundamental theorem of calculus, yielding a decomposition of the n-cube into n pyramids, which is a geometric proof of Cavalieri's quadrature formula.
For positive integers, this proof can be geometrized: if one considers the quantity xn as the volume of the n-cube (the hyperc