Groups | Search | Server Info | Keyboard shortcuts | Login | Register [http] [https] [nntp] [nntps]
Groups > sci.math > #434735 > unrolled thread
| Started by | Archimedes Plutonium <plutonium.archimedes@gmail.com> |
|---|---|
| First post | 2019-02-18 16:15 -0800 |
| Last post | 2019-03-02 16:11 -0800 |
| Articles | 17 on this page of 37 — 6 participants |
Back to article view | Back to sci.math
1meaning of dy+dx and dy-dx(or dx-dy depending on which is larger) //AP's Geometry Proof of Fundamental Theorem of Calculus, explaining the dy/dx =derivative=angle, dy*dx=integral=area Archimedes Plutonium <plutonium.archimedes@gmail.com> - 2019-02-18 16:15 -0800
replace derivative with angle and angulation Re: 1meaning of dy+dx and dy-dx(or dx-dy depending on which is larger) //AP's Geometry Proof of Fundamental Theorem of Calculus, explaining the dy/dx =derivative=angle, dy*dx=integral=area Archimedes Plutonium <plutonium.archimedes@gmail.com> - 2019-02-18 21:33 -0800
Re: replace derivative with angle and angulation Re: 1meaning of dy+dx and dy-dx(or dx-dy depending on which is larger) //AP's Geometry Proof of Fundamental Theorem of Calculus, explaining the dy/dx =derivative=angle, dy*dx=integral=area Archimedes Plutonium <plutonium.archimedes@gmail.com> - 2019-02-19 06:25 -0800
inspecting dx+dy and dx-dy for a straightline arc function Re: 1meaning of dy+dx and dy-dx(or dx-dy depending on which is larger) //AP's Geometry Proof of Fundamental Theorem of Calculus, explaining the dy/dx =derivative=angle, dy*dx=integral=area Archimedes Plutonium <plutonium.archimedes@gmail.com> - 2019-02-19 19:09 -0800
inspecting dx+dy and dx-dy for a straightline arc function Re: 1meaning of dy+dx and dy-dx(or dx-dy depending on which is larger) //AP's Geometry Proof of Fundamental Theorem of Calculus, explaining the dy/dx =derivative=angle, dy*dx=integral=area Archimedes Plutonium <plutonium.archimedes@gmail.com> - 2019-02-20 10:10 -0800
Y=x^2 and Y =x^3 Re: 1meaning of dy+dx and dy-dx(or dx-dy depending on which is larger) //AP's Geometry Proof of Fundamental Theorem of Calculus, explaining the dy/dx =derivative=angle, dy*dx=integral=area Archimedes Plutonium <plutonium.archimedes@gmail.com> - 2019-02-20 18:20 -0800
Y=x^2 and Y =x^3 Re: 1meaning of dy+dx and dy-dx(or dx-dy depending on which is larger) //AP's Geometry Proof of Fundamental Theorem of Calculus, explaining the dy/dx =derivative=angle, dy*dx=integral=area Archimedes Plutonium <plutonium.archimedes@gmail.com> - 2019-02-20 21:38 -0800
Y=x^2 and Y =x^3 Re: 1meaning of dy+dx and dy-dx(or dx-dy depending on which is larger) //AP's Geometry Proof of Fundamental Theorem of Calculus, explaining the dy/dx =derivative=angle, dy*dx=integral=area Archimedes Plutonium <plutonium.archimedes@gmail.com> - 2019-02-21 10:01 -0800
Y=x^2 and Y =x^3 Re: 1meaning of dy+dx and dy-dx(or dx-dy depending on which is larger) //AP's Geometry Proof of Fundamental Theorem of Calculus, explaining the dy/dx =derivative=angle, dy*dx=integral=area Archimedes Plutonium <plutonium.archimedes@gmail.com> - 2019-02-25 22:31 -0800
Y=x^2 and Y =x^3 Re: 1meaning of dy+dx and dy-dx(or dx-dy depending on which is larger) //AP's Geometry Proof of Fundamental Theorem of Calculus, explaining the dy/dx =derivative=angle, dy*dx=integral=area Archimedes Plutonium <plutonium.archimedes@gmail.com> - 2019-02-26 22:02 -0800
Re: Y=x^2 and Y =x^3 Re: 1meaning of dy+dx and dy-dx(or dx-dy depending on which is larger) //AP's Geometry Proof of Fundamental Theorem of Calculus, explaining the dy/dx =derivative=angle, dy*dx=integral=area Zelos Malum <zelos.malum@gmail.com> - 2019-02-26 22:37 -0800
Re: Y=x^2 and Y =x^3 Re: 1meaning of dy+dx and dy-dx(or dx-dy depending on which is larger) //AP's Geometry Proof of Fundamental Theorem of Calculus, explaining the dy/dx =derivative=angle, dy*dx=integral=area Archimedes Plutonium <plutonium.archimedes@gmail.com> - 2019-02-27 17:21 -0800
Re: Y=x^2 and Y =x^3 Re: 1meaning of dy+dx and dy-dx(or dx-dy depending on which is larger) //AP's Geometry Proof of Fundamental Theorem of Calculus, explaining the dy/dx =derivative=angle, dy*dx=integral=area Archimedes Plutonium <plutonium.archimedes@gmail.com> - 2019-02-27 17:45 -0800
Geometry of dy-dx, and power rules //AP's Geometry Proof of Fundamental Theorem of Calculus, explaining the dy/dx =derivative=angle, dy*dx=integral=area Archimedes Plutonium <plutonium.archimedes@gmail.com> - 2019-02-27 19:34 -0800
Re: Geometry of dy-dx, and power rules //AP's Geometry Proof of Fundamental Theorem of Calculus, explaining the dy/dx =derivative=angle, dy*dx=integral=area eastside.erik@gmail.com - 2019-02-27 20:46 -0800
Re: Geometry of dy-dx, and power rules //AP's Geometry power rules and dy-dx //Proof of Fundamental Theorem of Calculus, explaining the dy/dx =derivative=angle, dy*dx=integral=area Archimedes Plutonium <plutonium.archimedes@gmail.com> - 2019-02-27 21:27 -0800
need about a 27degree angle //AP's Geometry power rules and dy-dx //Proof of Fundamental Theorem of Calculus, explaining the dy/dx =derivative=angle, dy*dx=integral=area Archimedes Plutonium <plutonium.archimedes@gmail.com> - 2019-02-27 22:19 -0800
too much sigma error //AP's Geometry power rules and dy-dx //Proof of Fundamental Theorem of Calculus, explaining the dy/dx =derivative=angle, dy*dx=integral=area Archimedes Plutonium <plutonium.archimedes@gmail.com> - 2019-02-27 22:40 -0800
proving the Power Rules and no longer they are rules but actually a full part of Calculus //AP's Geometry power rules and dy-dx Archimedes Plutonium <plutonium.archimedes@gmail.com> - 2019-02-28 14:52 -0800
trying another point on Y=x^2 Re: proving the Power Rules and no longer they are rules but actually a full part of Calculus //AP's Geometry power rules and dy-dx Archimedes Plutonium <plutonium.archimedes@gmail.com> - 2019-02-28 17:28 -0800
one of the beauties of true math— chart of angles Re: proving the Power Rules and no longer they are rules but actually a full part of Calculus //AP's Geometry power rules and dy-dx Archimedes Plutonium <plutonium.archimedes@gmail.com> - 2019-02-28 21:28 -0800
still too big sigma error on dy-dx Re: proving the Power Rules and no longer they are rules but actually a full part of Calculus //AP's Geometry power rules and dy-dx Archimedes Plutonium <plutonium.archimedes@gmail.com> - 2019-02-28 22:02 -0800
math induction proof of Power rules Re: proving the Power Rules and no longer they are rules but actually a full part of Calculus //AP's Geometry power rules and dy-dx Archimedes Plutonium <plutonium.archimedes@gmail.com> - 2019-03-01 08:45 -0800
Re: math induction proof of Power rules Re: proving the Power Rules and no longer they are rules but actually a full part of Calculus //AP's Geometry power rules and dy-dx eastside.erik@gmail.com - 2019-03-01 09:46 -0800
Re: math induction proof of Power rules Re: proving the Power Rules and no longer they are rules but actually a full part of Calculus //AP's Geometry power rules and dy-dx Archimedes Plutonium <plutonium.archimedes@gmail.com> - 2019-03-01 15:07 -0800
Re: Archimedes "just not bolted down too tight in the first place" Plutonium flunked the physics lifetime-generation test moroney@world.std.spaamtrap.com (Michael Moroney) - 2019-03-01 23:14 +0000
Re: math induction proof of Power rules Re: proving the Power Rules and no longer they are rules but actually a full part of Calculus //AP's Geometry power rules and dy-dx eastside.erik@gmail.com - 2019-03-01 15:26 -0800
review of Cavalieri's integral power rule //AP's Geometry power rules Archimedes Plutonium <plutonium.archimedes@gmail.com> - 2019-03-01 22:02 -0800
review of Cavalieri's integral power rule //AP's Geometry power rules Archimedes Plutonium <plutonium.archimedes@gmail.com> - 2019-03-01 23:40 -0800
only functions that exist are polynomials Re: review of Cavalieri's integral power rule //AP's Geometry power rules Archimedes Plutonium <plutonium.archimedes@gmail.com> - 2019-03-02 15:08 -0800
Re: only functions that exist are polynomials Re: review of Cavalieri's integral power rule //AP's Geometry power rules eastside.erik@gmail.com - 2019-03-02 15:40 -0800
Re: only functions that exist are polynomials Re: review of Cavalieri's integral power rule //AP's Geometry power rules Earle Jones <earledjones@icloud.com> - 2019-03-02 17:14 -0800
Need to emphasize the derivative of True Math is a Angle Re: AP's Geometry power rules Archimedes Plutonium <plutonium.archimedes@gmail.com> - 2019-03-03 10:22 -0800
Re: Archimedes "just not bolted down too tight in the first place" Plutonium flunked the physics lifetime-generation test moroney@world.std.spaamtrap.com (Michael Moroney) - 2019-03-05 17:27 +0000
dy-dx cleared up Re: proving the Power Rules and no longer they are rules but actually a full part of Calculus //AP's Geometry power rules and dy-dx Archimedes Plutonium <plutonium.archimedes@gmail.com> - 2019-03-01 19:56 -0800
Re: Archimedes "just not bolted down too tight in the first place" Plutonium flunked the physics lifetime-generation test moroney@world.std.spaamtrap.com (Michael Moroney) - 2019-03-01 19:51 +0000
Re: 1meaning of dy+dx and dy-dx(or dx-dy depending on which is larger) //AP's Geometry Proof of Fundamental Theorem of Calculus, explaining the dy/dx =derivative=angle, dy*dx=integral=area Jan <filmart@gmail.com> - 2019-03-02 16:11 -0800
Page 2 of 2 — ← Prev page 1 [2]
| From | Archimedes Plutonium <plutonium.archimedes@gmail.com> |
|---|---|
| Date | 2019-02-28 21:28 -0800 |
| Subject | one of the beauties of true math— chart of angles Re: proving the Power Rules and no longer they are rules but actually a full part of Calculus //AP's Geometry power rules and dy-dx |
| Message-ID | <7a6a09ef-940a-4629-94d5-e7c057847c7f@googlegroups.com> |
| In reply to | #435793 |
Unlike Old Math with their juju-totem oijee board of fake trig tables that costs money for fakery, New Math gets by totally with Chart of Angles that 10 year old can assemble with graph paper and protractor.
Chart of Angles
1by10 2by10 3by10 4by10...
0 10/1 10/2 10/3 10/4 10/5 10/6 10/7 10/8 10/9 10/10 dx/dy, the inverse or complement
90 84 78.7 73 68 63 59 55 51 48 45, slope 1, rt-triangle 10by10
42, slope .9, rt-triangle 10by9
39, slope .8, rt-triangle 10by8
35, slope .7, rt-triangle 10by7
31, slope .6, rt-triangle 10by6
27, slope .5, rt-triangle 10by5
22, slope .4, rt-triangle 10by4
17, slope .3, rt.triangle 10by3
11.3 , slope .2, rt-triangle 10by2
6, slope .1, rt-triangle 10by1
0 .1 .2 .3 .4 .5 .6 .7 .8 .9 1.0
And we see our derivative of 4.2 is near 78degrees indicating our dy-dx is near 12degrees.
AP
[toc] | [prev] | [next] | [standalone]
| From | Archimedes Plutonium <plutonium.archimedes@gmail.com> |
|---|---|
| Date | 2019-02-28 22:02 -0800 |
| Subject | still too big sigma error on dy-dx Re: proving the Power Rules and no longer they are rules but actually a full part of Calculus //AP's Geometry power rules and dy-dx |
| Message-ID | <cbd27a09-762a-486b-9c3a-8c9f0373cab3@googlegroups.com> |
| In reply to | #435802 |
Alright everything works out great except dy-dx as i get too much sigma error of angle. Hinge at (2.05,4.2) Rectangle (2,4) (2.1,4) (2.1,4.2) (2,4.2) Derivates match .21/.05 = 4.2 Integral matches 3.087- 2.666 = .42 with 4.2x.1= .42 What does not match perfectly is dy-dx 2.1 -.5= 1.6 which is an angle of 9degrees not 12, large sigma error. Maybe if i took this to 100grid the sigma error would disappear. AP
[toc] | [prev] | [next] | [standalone]
| From | Archimedes Plutonium <plutonium.archimedes@gmail.com> |
|---|---|
| Date | 2019-03-01 08:45 -0800 |
| Subject | math induction proof of Power rules Re: proving the Power Rules and no longer they are rules but actually a full part of Calculus //AP's Geometry power rules and dy-dx |
| Message-ID | <1ed2f4be-b29f-42e0-8a3d-c8acf571edc0@googlegroups.com> |
| In reply to | #435803 |
Now i suspect, not sure though, that the power rules of calculus can be proven true universally, thus no longer rules of math but the internal heart of calculus. This matches my quest to make all of function theory be Polynomials— every function a polynomial and only a polynomial. AP
[toc] | [prev] | [next] | [standalone]
| From | eastside.erik@gmail.com |
|---|---|
| Date | 2019-03-01 09:46 -0800 |
| Subject | Re: math induction proof of Power rules Re: proving the Power Rules and no longer they are rules but actually a full part of Calculus //AP's Geometry power rules and dy-dx |
| Message-ID | <b7717cbc-593f-4870-a297-571fdb9b9662@googlegroups.com> |
| In reply to | #435879 |
On Friday, March 1, 2019 at 8:46:03 AM UTC-8, Archimedes Plutonium wrote: > Now i suspect, not sure though, that the power rules of calculus can be proven true universally, thus no longer rules of math but the internal heart of calculus. This matches my quest to make all of function theory be Polynomials— every function a polynomial and only a polynomial. > > AP The weeds are getting pretty deep, Archie. Anybody who took first-semester calculus did the calculation in a few seconds.
[toc] | [prev] | [next] | [standalone]
| From | Archimedes Plutonium <plutonium.archimedes@gmail.com> |
|---|---|
| Date | 2019-03-01 15:07 -0800 |
| Subject | Re: math induction proof of Power rules Re: proving the Power Rules and no longer they are rules but actually a full part of Calculus //AP's Geometry power rules and dy-dx |
| Message-ID | <90577c55-5dd9-4dd1-b03f-a7e9ece0e4fe@googlegroups.com> |
| In reply to | #435879 |
On Friday, March 1, 2019 at 10:46:03 AM UTC-6, Archimedes Plutonium wrote: > Now i suspect, not sure though, that the power rules of calculus can be proven true universally, thus no longer rules of math but the internal heart of calculus. This matches my quest to make all of function theory be Polynomials— every function a polynomial and only a polynomial. > Perhaps as a first induction step I use Y = x
[toc] | [prev] | [next] | [standalone]
| From | moroney@world.std.spaamtrap.com (Michael Moroney) |
|---|---|
| Date | 2019-03-01 23:14 +0000 |
| Subject | Re: Archimedes "just not bolted down too tight in the first place" Plutonium flunked the physics lifetime-generation test |
| Message-ID | <q5cec8$qs3$4@pcls7.std.com> |
| In reply to | #435939 |
AlzheimerPlutonium <plutonium.archimedes@gmail.com> fails at math and science: >>Autistic Archimedes Plutonium <plutonium.archimedes@gmail.com> fails at math: MM writes: never any math, never any physics from Plutonium, only stalk stalk hatred for 26 years x-no-archive: yes
[toc] | [prev] | [next] | [standalone]
| From | eastside.erik@gmail.com |
|---|---|
| Date | 2019-03-01 15:26 -0800 |
| Subject | Re: math induction proof of Power rules Re: proving the Power Rules and no longer they are rules but actually a full part of Calculus //AP's Geometry power rules and dy-dx |
| Message-ID | <ac8e6e76-5d00-415d-8169-a540cba2ae94@googlegroups.com> |
| In reply to | #435939 |
On Friday, March 1, 2019 at 3:07:53 PM UTC-8, Archimedes Plutonium wrote: > On Friday, March 1, 2019 at 10:46:03 AM UTC-6, Archimedes Plutonium wrote: > > Now i suspect, not sure though, that the power rules of calculus can be proven true universally, thus no longer rules of math but the internal heart of calculus. This matches my quest to make all of function theory be Polynomials— every function a polynomial and only a polynomial. > > > > Perhaps as a first induction step I use Y = x Well, what are you going to do about exponential functions, logarithmic functions, periodic functions, or (horrors!) factorial functions? Pretty tough with just polynomials. You're gonna have lots of "sigma error".
[toc] | [prev] | [next] | [standalone]
| From | Archimedes Plutonium <plutonium.archimedes@gmail.com> |
|---|---|
| Date | 2019-03-01 22:02 -0800 |
| Subject | review of Cavalieri's integral power rule //AP's Geometry power rules |
| Message-ID | <6ced6193-cfd1-432e-bd13-389c2c49fa44@googlegroups.com> |
| In reply to | #435939 |
On Friday, March 1, 2019 at 5:07:53 PM UTC-6, Archimedes Plutonium wrote:
> On Friday, March 1, 2019 at 10:46:03 AM UTC-6, Archimedes Plutonium wrote:
> > Now i suspect, not sure though, that the power rules of calculus can be proven true universally, thus no longer rules of math but the internal heart of calculus. This matches my quest to make all of function theory be Polynomials— every function a polynomial and only a polynomial.
> >
>
> Perhaps as a first induction step I use Y = x
I remember a picture of Cavalieri's integral rule but seem not to spot it at this moment. What I am after here is a proof that makes the Power Rules a theorem and universally true, no longer just a mere Rule, but a fact. Of course we turn all functions into being all polynomials, and skip as trash, any function not turned into a polynomial.
Historic math thread: Integral Power Rule & Derivative Power Rule-- Purely GEOMETRICAL PROOF
me (Archimedes Plutonium change)
11/15/16
Alright, this is a historic thread in math history because it is the first time that both Integral Power Rule plus the Derivative Power Rule are proven in terms of pure Geometry.
Now let me define Integral Power Rule (which some call the antiderivative), which is stated as this:
Integral x^n = (1/(n+1)) x^(n+1)
And the Derivative Power Rule is this:
(x^n)' = n x^(n-1)
Now I need to correct myself on the fact that the Integral Power Rule with geometry proof was first given by Cavalieri circa 1639 and what he called quadrature formula.
So the above Integral Power Rule was discovered in 1639, and all I am doing is clarifying and sharpening it up (see below).
But my own original contribution is a pure geometry proof of the Derivative Power Rule which has never been given before, until now.
Let me hit this proof right now:
Why is derivative of x^n that of n x^(n-1) ?
Proof: definition of derivative is dy/dx. Definition of Integral is dy*dx. As Cavalieri showed the dy for Y=x^2 is x^2 and the dx as Cavalieri showed was x so we have for integral of x^2 that of x^3 and the constant for Integral Y=x^2 is that of 1/n+1 because the area occupied by x^2 in the 1 by 1 square is 1/3 of the area which by Mathematical Induction proves the Integral Power Rule. Back now to the Derivative Power Rule as the derivative is defined by dy/dx and the case of Y=x^2 the dy is x^2, the dx is x, and by the definition of derivative we have dy/dx = dy*dx^-1 and since the dy is a x^n and by arithmetic we have x^(n+(-1)). We have x^(n-1) and lastly we need the constant of n in nx^n-1. We have the x^n-1 but not the prefix n. How do we get that purely geometrically? We get it purely geometrically by noting that the most simple polynomial is Y=x and the n is 1, we also note that all straightline function of Y = mx + b that the derivative of mx is m since the exponent of mx is 1. Thus, the Derivative Power Rule. QED
Now, leading up to this proof were several posts from yesterday and today and I feel like repeating them, since this is a historic day in Calculus history. The bulk of Calculus is the Power Rule for integral and derivative.
Newsgroups: sci.math
Date: Tue, 15 Nov 2016 03:36:10 -0800 (PST)
Subject: History of the Power Rule Re: Do we build the Power Rule from just
one example Y=x??
From: Archimedes Plutonium <plutonium....@gmail.com>
Injection-Date: Tue, 15 Nov 2016 11:36:10 +0000
History of the Power Rule Re: Do we build the Power Rule from just one example Y=x??
Alright, learned a lot tonight that it kept me from sleep.
Looks as though the integral power rule-- often called antiderivative was discovered by Cavalieri, apparently before the derivative power rule.
And my search for a geometry meaning for why integral of x^2 is 1/3x^3 appears to involve the summation of partition rectangles.
And according to one author that involves the summation of series 1^p + 2^p + ..+ n^p
But so far I fail to see how the partitions gets 1/3 and x^3 for
Y=x^2.
Do you, see how the partitions retrieve a 1/3x^3 from out of y=x^2???
iPhone post
AP
Newsgroups: sci.math
Date: Tue, 15 Nov 2016 05:23:32 -0800 (PST)
Subject: History of the Power Rule Re: Do we build the Power Rule from just
one example Y=x??
From: Archimedes Plutonium <plutonium....@gmail.com>
Injection-Date: Tue, 15 Nov 2016 13:23:32 +0000
History of the Power Rule Re: Do we build the Power Rule from just one example Y=x??
Alright, good, great and well worth the all nighter.
I figured out how Cavalieri got the integral Power Rule of x^n is x^n+1/n+1
But before I explain Cavalieri, let me suggest the math community recognize the fact that the Antiderivative is the same as the integral.
In the past weeks, I was hungering for a geometry explanation of why the area for x^2 function involved 1/3 and x^3
It is simple to see that the integral of y=x is 1/2x^2 because the function y=x forms a right triangle. But no geometry explanation for function y=x^2 until I came across Cavalieri.
What Cavalieri method was to make rectangles have 0 width and so the height of his rectangles for x^2 would be the y value which in the case of Y=x^2 is x^2, and Cavalieri ' width value is x itself, thus x^2 times x is x^3. But it does not explain the 1/3 from 1/(2+1)
So I still am short of 1/(n+1) explanation.
iPhone post
AP
Newsgroups: sci.math
Date: Tue, 15 Nov 2016 07:13:28 -0800 (PST)
Subject: Re: History of the Power Rule Re: Do we build the Power Rule from
just one example Y=x??
From: Archimedes Plutonium <plutonium....@gmail.com>
Injection-Date: Tue, 15 Nov 2016 15:13:31 +0000
Re: History of the Power Rule Re: Do we build the Power Rule from just one example Y=x??
Alright, I know where the 1/3 in integral of x^2 comes from, in the fact that the square of 1 by 1 from 0 to 1 on x axis and from 0 to 1 on y axis that the function x^2 occupies 1/3 of the area. For function x^3, the occupation of area of 1by1 is 1/4, for function x^4, the occupation of area is 1/5 of 1 by 1 square.
AP
Newsgroups: sci.math
Date: Tue, 15 Nov 2016 14:04:02 -0800 (PST)
Subject: x^0 is crucial in calculus Re: History of the Power Rule Re: Do we
build the Power Rule from just one example Y=x??
From: Archimedes Plutonium <plutonium....@gmail.com>
Injection-Date: Tue, 15 Nov 2016 22:04:02 +0000
x^0 is crucial in calculus Re: History of the Power Rule Re: Do we build the Power Rule from just one example Y=x??
On Tuesday, November 15, 2016 at 9:13:38 AM UTC-6, Archimedes Plutonium wrote:
> Alright, I know where the 1/3 in integral of x^2 comes from, in the fact that the square of 1 by 1 from 0 to 1 on x axis and from 0 to 1 on y axis that the function x^2 occupies 1/3 of the area. For function x^3, the occupation of area of 1by1 is 1/4, for function x^4, the occupation of area is 1/5 of 1 by 1 square.
>
> AP
Alright, beautiful, and none of this turned out the way I expected, because here is where Algebra and Geometry come together to fulfill the Calculus.
I need a whole brand new thread for this, what I call the Normalization of Calculus, just as in Physics we Renormalize infinity.
So the beauty of Cavalieri was that he took the height of a polynomial function such as x^2 as y= x^2 and took that height as being a rectangle with height x^2 and then his rectangle base would be x so we have a rectangle area of x^2 times x = x^3
And to achieve the Integral Power Rule of (1/n+1)x^(n+1) so we see the x^3 area for function Y=x^2 but we do not see the reason for 1/3 for y=x^2 and the reason for that 1/3 or in general for (1/(n+1) is because the polynomial x^n divides the unit square area of 1 by 1 square into 1/3 for x^2, into 1/4 for x^3, into 1/5 for x^4 etc etc.
Now, all I need and then I am done is geometrically explain why dy/dx, the derivative has Power Rule of nx^(n-1).
All I need to finish, is geometrically explain that.
And it comes as a shocker to me, because it involves 0, my hated enemy turns out to be essential here for it is the interloper between fixing the derivative with the integral.
Earlier, some weeks or days ago, I posted a THEOREM that is crucial to Calculus, essential, and is a huge theorem, probably on par with the Fundamental Theorem of Calculus. Let me call it the LINE POINT DUALITY THEOREM. For the essential idea is that if we mark any point that we can see, if you put a microscope to that point, it is no longer a point but is a line or rectangle. Now, a atom a single atom is a point, but we cannot see it and a atom has no visible length, width or depth because it is so small, but, when we get a Grid System small enough, say 10^30 our atom point becomes a line segment or a rectangle or circle, because we matched the smallness with a Grid.
So, here in Calculus we have Cavalieri with the Integral Power Rule for x^2 being 1/3 x^3 and he gets x^3 because his rectangles are x^2 by x in area. Now we would look upon that as line segments that have no area. And here comes the LIne Point Duality theorem to step in and say the 10^2 Grid is too larger and if you went down to a 10^3 or 10^4 Grid you would recognize your Cavalieri rectangle of x^2 by x = x^3 area very easily recognize it. Same as recognizing a Cavalieri rectangle for x^3 as that of x^3 by x = x^4 if we went to a 10^5 Grid. So what appears to us in 10 Grid of a line segment if we go to a larger Grid of 10^2 or 10^3, our line segment is now a rectangle.
So, now, I mentioned 0 as the Renormalization factor. And here is how it works. For the function Y=x the derivative is 1 and integral is 1/2x^2 due to Cavalieri integral, but how do we explain the Derivative Power Rule of nx^{n-1). For function x we have 1*x^(1-1) and that is x^0.
Math defines x^0, whether Old Math or New Math, both are the same when it comes to x^0=1.
And that is very important for Calculus, because what it is saying is that our Cavalieri rectangle, which looks like a line segment but is really a thin rectangle if we go to the correct Grid, and so the Integral is really and truly that of length times width or x^2 times x for function x^2. But that the derivative of x^2 is 2x which is not a flat line segment of Cavalieri 1/3x^3 but rather is a slanted line segment of 2x.
So, here, we come to a pretty and beautiful conundrum of how does a dy/dx of derivative be slanted whereas the dy*dx integral is Cavalieri's thin rectangle? How do we fix that paradox that integral is a flat rectangle dy/dx yet the real dy/dx is slanted. How do we fix that?
We fix it because we realize that x^0 is 1.
The x^0 allows us to see the Cavalieri rectangle width is both a flat width of x and also that x is x^0 for a dy/dx = 1.
Now, we can derive Geometrically the Derivative Power Rule of x^n = nx^(n-1). For instance Y=x means the Cavalieri integral is 1/2x^2 where the rectangles are x*x, occupying 1/2 of the 1 by 1 square of 10 Grid. And the slope of the function graph Y=x is a 45 by 45 by 90 right triangle and the Derivative Power Rule is 1*x^(1-1) = x^0 =1
So, the Cavalieri rectangle width is a flat line but the Derivative is a slanted line and all allowed because of x^0 = 1.
Let me start a new thread on this. And let me derive the Derivative Power Rule formula nx^(n-1).
AP
[toc] | [prev] | [next] | [standalone]
| From | Archimedes Plutonium <plutonium.archimedes@gmail.com> |
|---|---|
| Date | 2019-03-01 23:40 -0800 |
| Subject | review of Cavalieri's integral power rule //AP's Geometry power rules |
| Message-ID | <dd7aed5c-a4f1-478b-bfbe-161669fc5c23@googlegroups.com> |
| In reply to | #435959 |
Alright i am going to offer this up as a Mathematical Induction proof the the Power Rules are no longer Rules of mathematics but universal facts as integral and derivative of all functions of mathematics, where all functions are polynomials, and if not a polynomial it is not a function of mathematics. Why is derivative of x^n that of n x^(n-1) ? Proof: definition of derivative is dy/dx. Definition of Integral is dy*dx. As Cavalieri showed the dy for Y=x^2 is x^2 and the dx as Cavalieri showed was x so we have for integral of x^2 that of x^3 and the constant for Integral Y=x^2 is that of 1/n+1 because the area occupied by x^2 in the 1 by 1 square is 1/3 of the area which by Mathematical Induction proves the Integral Power Rule. Back now to the Derivative Power Rule as the derivative is defined by dy/dx and the case of Y=x^2 the dy is x^2, the dx is x, and by the definition of derivative we have dy/dx = dy*dx^-1 and since the dy is a x^n and by arithmetic we have x^(n+(-1)). We have x^(n-1) and lastly we need the constant of n in nx^n-1. We have the x^n-1 but not the prefix n. How do we get that purely geometrically? We get it purely geometrically by noting that the most simple polynomial is Y=x and the n is 1, we also note that all straightline function of Y = mx + b that the derivative of mx is m since the exponent of mx is 1. Thus, the Derivative Power Rule. QED AP
[toc] | [prev] | [next] | [standalone]
| From | Archimedes Plutonium <plutonium.archimedes@gmail.com> |
|---|---|
| Date | 2019-03-02 15:08 -0800 |
| Subject | only functions that exist are polynomials Re: review of Cavalieri's integral power rule //AP's Geometry power rules |
| Message-ID | <a79e4d7b-c52d-496d-aab9-df6a037a7819@googlegroups.com> |
| In reply to | #435963 |
On Saturday, March 2, 2019 at 1:41:03 AM UTC-6, Archimedes Plutonium wrote: > Alright i am going to offer this up as a Mathematical Induction proof that the Power Rules are no longer Rules of mathematics but universal facts as integral and derivative of all functions of mathematics, where all functions are polynomials, and if not a polynomial it is not a function of mathematics. > When a math professor has little to no logical mind, they make errors, left and right. But also, their mental hygiene is out of balance, such as allowing any any crazy type of function. > > Why is derivative of x^n that of n x^(n-1) ? > Proof: definition of derivative is dy/dx. Definition of Integral is dy*dx. As Cavalieri showed the dy for Y=x^2 is x^2 and the dx as Cavalieri showed was x so we have for integral of x^2 that of x^3 and the constant for Integral Y=x^2 is that of 1/n+1 because the area occupied by x^2 in the 1 by 1 square is 1/3 of the area which by Mathematical Induction proves the Integral Power Rule. Back now to the Derivative Power Rule as the derivative is defined by dy/dx and the case of Y=x^2 the dy is x^2, the dx is x, and by the definition of derivative we have dy/dx = dy*dx^-1 and since the dy is a x^n and by arithmetic we have x^(n+(-1)). We have x^(n-1) and lastly we need the constant of n in nx^n-1. We have the x^n-1 but not the prefix n. How do we get that purely geometrically? We get it purely geometrically by noting that the most simple polynomial is Y=x and the n is 1, we also note that all straightline function of Y = mx + b that the derivative of mx is m since the exponent of mx is 1. Thus, the Derivative Power Rule. QED > You ever ask yourself, why is there kilograms and kilograms of all types of functions-- exp, trig, hyperbolic, log, inverse trig, on and on and on. Is math truly like this-- a huge kooky bag of assortment of functions? No, math is not like that, for the ugly bag of function types is because of math professors with only a microgram brain of logic. When you define Function to be only POLYNOMIALS, you get rid of the kook math of functions. All functions that exist must be a Polynomial. So you can take a log or trig or exp function and convert it to a polynomial and only when it is converted, is it a function to work with. AP
[toc] | [prev] | [next] | [standalone]
| From | eastside.erik@gmail.com |
|---|---|
| Date | 2019-03-02 15:40 -0800 |
| Subject | Re: only functions that exist are polynomials Re: review of Cavalieri's integral power rule //AP's Geometry power rules |
| Message-ID | <7d34bb59-6847-4e68-bf2f-168b336e8f1d@googlegroups.com> |
| In reply to | #436042 |
On Saturday, March 2, 2019 at 3:08:53 PM UTC-8, Archimedes Plutonium wrote: > On Saturday, March 2, 2019 at 1:41:03 AM UTC-6, Archimedes Plutonium wrote: > > Alright i am going to offer this up as a Mathematical Induction proof that the Power Rules are no longer Rules of mathematics but universal facts as integral and derivative of all functions of mathematics, where all functions are polynomials, and if not a polynomial it is not a function of mathematics. > > > > When a math professor has little to no logical mind, they make errors, left and right. But also, their mental hygiene is out of balance, such as allowing any any crazy type of function. > > > > > Why is derivative of x^n that of n x^(n-1) ? > > Proof: definition of derivative is dy/dx. Definition of Integral is dy*dx. As Cavalieri showed the dy for Y=x^2 is x^2 and the dx as Cavalieri showed was x so we have for integral of x^2 that of x^3 and the constant for Integral Y=x^2 is that of 1/n+1 because the area occupied by x^2 in the 1 by 1 square is 1/3 of the area which by Mathematical Induction proves the Integral Power Rule. Back now to the Derivative Power Rule as the derivative is defined by dy/dx and the case of Y=x^2 the dy is x^2, the dx is x, and by the definition of derivative we have dy/dx = dy*dx^-1 and since the dy is a x^n and by arithmetic we have x^(n+(-1)). We have x^(n-1) and lastly we need the constant of n in nx^n-1. We have the x^n-1 but not the prefix n. How do we get that purely geometrically? We get it purely geometrically by noting that the most simple polynomial is Y=x and the n is 1, we also note that all straightline function of Y = mx + b that the derivative of mx is m since the exponent of mx is 1. Thus, the Derivative Power Rule. QED > > > > You ever ask yourself, why is there kilograms and kilograms of all types of functions-- exp, trig, hyperbolic, log, inverse trig, on and on and on. > > Is math truly like this-- a huge kooky bag of assortment of functions? No, math is not like that, for the ugly bag of function types is because of math professors with only a microgram brain of logic. When you define Function to be only POLYNOMIALS, you get rid of the kook math of functions. > > All functions that exist must be a Polynomial. So you can take a log or trig or exp function and convert it to a polynomial and only when it is converted, is it a function to work with. > > AP Let's see a polynomial that works for sine.
[toc] | [prev] | [next] | [standalone]
| From | Earle Jones <earledjones@icloud.com> |
|---|---|
| Date | 2019-03-02 17:14 -0800 |
| Subject | Re: only functions that exist are polynomials Re: review of Cavalieri's integral power rule //AP's Geometry power rules |
| Message-ID | <2019030217145636869-earledjones@icloudcom> |
| In reply to | #436042 |
On 2019-03-02 23:08:43 +0000, Archimedes Plutonium said: > On Saturday, March 2, 2019 at 1:41:03 AM UTC-6, Archimedes Plutonium wrote: >> Alright i am going to offer this up as a Mathematical Induction proof >> that the Power Rules are no longer Rules of mathematics but universal >> facts as integral and derivative of all functions of mathematics, where >> all functions are polynomials, and if not a polynomial it is not a >> function of mathematics. >> > > When a math professor has little to no logical mind, they make errors, > left and right. But also, their mental hygiene is out of balance, such > as allowing any any crazy type of function. > >> >> Why is derivative of x^n that of n x^(n-1) ?> Proof: definition of >> derivative is dy/dx. Definition of Integral is dy*dx. As Cavalieri >> showed the dy for Y=x^2 is x^2 and the dx as Cavalieri showed was x so >> we have for integral of x^2 that of x^3 and the constant for Integral >> Y=x^2 is that of 1/n+1 because the area occupied by x^2 in the 1 by 1 >> square is 1/3 of the area which by Mathematical Induction proves the >> Integral Power Rule. Back now to the Derivative Power Rule as the >> derivative is defined by dy/dx and the case of Y=x^2 the dy is x^2, the >> dx is x, and by the definition of derivative we have dy/dx = dy*dx^-1 >> and since the dy is a x^n and by arithmetic we have x^(n+(-1)). We have >> x^(n-1) and lastly we need the constant of n in nx^n-1. We have the >> x^n-1 but not the prefix n. How do we get that purely geometrically? We >> get it purely geometrically by noting that the most simple polynomial >> is Y=x and the n is 1, we also note that all straightline function of >> Y = mx + b that the derivative of mx is m since the exponent of mx is >> 1. Thus, the Derivative Power Rule. QED> > You ever ask yourself, why is there kilograms and kilograms of all > types of functions-- exp, trig, hyperbolic, log, inverse trig, on and > on and on. > > Is math truly like this-- a huge kooky bag of assortment of functions? > No, math is not like that, for the ugly bag of function types is > because of math professors with only a microgram brain of logic. When > you define Function to be only POLYNOMIALS, you get rid of the kook > math of functions. > > All functions that exist must be a Polynomial. So you can take a log or > trig or exp function and convert it to a polynomial and only when it is > converted, is it a function to work with. > > AP AP: What is the sine of 45 degrees? (In polynomials.) earle *
[toc] | [prev] | [next] | [standalone]
| From | Archimedes Plutonium <plutonium.archimedes@gmail.com> |
|---|---|
| Date | 2019-03-03 10:22 -0800 |
| Subject | Need to emphasize the derivative of True Math is a Angle Re: AP's Geometry power rules |
| Message-ID | <dfcb4f7e-1755-42b8-8451-dde6d00b110e@googlegroups.com> |
| In reply to | #436042 |
This is something Old Math failed to recognize, that dy/dx is in fact a angle so that Y=x is derivative 1 is angle 45degrees, and Y=x^2 at x=1 is derivative 2 is angle 63degrees. Oh yes, Old Math comprehended that integral is “area under the function graph”, but failed to recognize that derivative was Angle. Why did they fail? Because they focused all their screwball attention on Limits and on tangents to a graph. When they should have focused on the fact that dy*dx for integral was a area, placing dy/dx as a division ratio— just what an angle is all about. But there is more good news to the story, because dy/dx is an angle the derivative gives out at 90degrees, for the larger the number dy/dx becomes 2 say 10 say 150 etc etc the closer the angle gets to 90degrees but never reaching it. This is proof by Calculus that all of mathematics takes place in 1st Quadrant Only. AP
[toc] | [prev] | [next] | [standalone]
| From | moroney@world.std.spaamtrap.com (Michael Moroney) |
|---|---|
| Date | 2019-03-05 17:27 +0000 |
| Subject | Re: Archimedes "just not bolted down too tight in the first place" Plutonium flunked the physics lifetime-generation test |
| Message-ID | <q5mbh4$9o9$2@pcls7.std.com> |
| In reply to | #436116 |
AlzheimerPlutonium <plutonium.archimedes@gmail.com> fails at math and science: >AP writes: Moroney round the clock stalking So that's what you are doing by spamming that "song"? You are stalking me around the clock? x-no-archive: yes
[toc] | [prev] | [next] | [standalone]
| From | Archimedes Plutonium <plutonium.archimedes@gmail.com> |
|---|---|
| Date | 2019-03-01 19:56 -0800 |
| Subject | dy-dx cleared up Re: proving the Power Rules and no longer they are rules but actually a full part of Calculus //AP's Geometry power rules and dy-dx |
| Message-ID | <eb5f6f05-ddd6-4f92-911c-e8c88c5de82c@googlegroups.com> |
| In reply to | #435803 |
On Friday, March 1, 2019 at 12:02:53 AM UTC-6, Archimedes Plutonium wrote: > Alright everything works out great except dy-dx as i get too much sigma error of angle. > > Hinge at (2.05,4.2) > > Rectangle (2,4) (2.1,4) (2.1,4.2) (2,4.2) > > Derivates match .21/.05 = 4.2 > > Integral matches 3.087- 2.666 = .42 with 4.2x.1= .42 > > What does not match perfectly is dy-dx 2.1 -.5= 1.6 which is an angle of 9degrees not 12, large sigma error. > > Maybe if i took this to 100grid the sigma error would disappear. > Alright, I have solved this sigma error issue with dy-dx. It is not a problem at all and gives real good results not even worth bothering about the sigma error. What I had forgotten was the dy-dx is the distance along the diagonal of the picketfence right-triangle once it is hinged down forming a rectangle for integral. Whereas I was just using the right-triangle, no, it is the right-triangle hinged down into the rectangle and the Diagonal of this rectangle. I had used two sample exercises (1,1) to (1.05,1.1025) to (1.1, 1.21) that gave 65degree with 25degree (2,4) to (2.05,4.20) to (2.1, 4.41) that gave 75degree with 15degree The dy-dx is the distance along the diagonal of the rectangle formed from folding down the right triangle to form the rectangle. We instantly know the angle once we have the derivative of 2.2 or of 4.2 so we know both angles, and the dy-dx was not the other angle, no, it was the distance along the diagonal which is the perpendicular altitude. So that clears up the dy-dx. AP
[toc] | [prev] | [next] | [standalone]
| From | moroney@world.std.spaamtrap.com (Michael Moroney) |
|---|---|
| Date | 2019-03-01 19:51 +0000 |
| Subject | Re: Archimedes "just not bolted down too tight in the first place" Plutonium flunked the physics lifetime-generation test |
| Message-ID | <q5c2fn$cml$1@pcls7.std.com> |
| In reply to | #434735 |
AlzheimerPlutonium <plutonium.archimedes@gmail.com> fails at math and science: >Subject: Alan H. Guth,Michael E. Brown,Konstantin Batygin,Ben Bullock,Edward > Witten is Moroney giving away supersonic missile tech of MIT to Russia?? Too Oh man! AlzheimerPlutonium is failing all over again, and now he is throwing his failures at everyone, like the angry zoo ape! Can't anyone stop it? At least quit letting him go to Taco Bell??? x-no-archive: yes
[toc] | [prev] | [next] | [standalone]
| From | Jan <filmart@gmail.com> |
|---|---|
| Date | 2019-03-02 16:11 -0800 |
| Message-ID | <0781b92e-5b17-431b-b182-dc10e2633bab@googlegroups.com> |
| In reply to | #434735 |
On Monday, February 18, 2019 at 4:15:21 PM UTC-8, Archimedes Plutonium wrote: > 1meaning of dy+dx and dy-dx(or dx-dy depending on which is larger) //AP's Geometry Proof of Fundamental Theorem of Calculus, explaining the dy/dx =derivative=angle, dy*dx=integral=area This is all standard real analysis, very old. Just pick any decent textbook on integration on manifolds, like Lang, Marsden, Spivak, etc. etc. There is nothing mysterious left to discuss here. You are almost a century behind. -- Jan
[toc] | [prev] | [standalone]
Page 2 of 2 — ← Prev page 1 [2]
Back to top | Article view | sci.math
csiph-web