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Topic: Number "e" and Straightlinecurves; derivative & integral of y=1/x #13
Uni-textbook 6th ed.:TRUE CALCULUS

Replies: 19   Last Post: Jul 2, 2013 1:43 AM

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 plutonium.archimedes@gmail.com Posts: 18,572 Registered: 3/31/08
#13.6 all functions are now continuous in New Math; Uni-textbook 6th
ed.:TRUE CALCULUS

Posted: Jul 1, 2013 1:57 PM

On Monday, July 1, 2013 11:51:16 AM UTC-5, Archimedes Plutonium wrote:
> #13.5 Re: Number "e" and Straightlinecurves; derivative & integral of y=1/x Uni-textbook 6th ed.:TRUE CALCULUS
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> Sorry I made a few mistakes in this graphing that is corrected now. And this is the 10 Grid, but now I need to expand to the 100 and 1000 Grid to show that Old Math with their limit concept is never able to provide a true correct answer. Sure, Old Math can provide for a precise answer when the function is a overall straightline, but when the function is an overall straightlinecurve like that of y = 1/x, Old Math has to make the contradiction and hypocrisy that integrals are summations of areas with no width and tangents of no length. In New Math, the integral has width of at minimum 1*10^-603 and the derivative has length for it is the function graph itself.
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> Note: negative sign in y'= -x^-2 is not needed for it only tells us the orientation of the derivative.
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> 10 GRID graphing
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> (.1,10) and (.2,5) and (.3, 3.3) = dy/dx = 6.7/.2 = 33.5 whereas x^-2 delivers 25
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> (.3,3.3) and (.4,2.5) and (.5, 2) = dy/dx = 1.3/.2 = 6.5 whereas x^-2 delivers 6.2
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> (.9,1.1) and (1.0,1.0) and (1.1, .9) = dy/dx = .2/.2 = 1 and x^-2 delivers 1
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> (1.1,.9) and (1.2, .8) and (1.3, .7) = dy/dx = .2/.2 = 1  whereas x^-2 delivers 0.6
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> (2.9,.3) and (3.0, .3) and (3.1, .3) = dy/dx = 0/.2 = 0  whereas x^-2 delivers 0.1
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> 100 GRID graphing
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> (.01,100) and (.02,50) and (.03, 33.33) = dy/dx = 66.67/.02 = 3333.5 whereas x^-2 delivers 2500
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> (.03,33.33) and (.04,25) and (.05, 20) = dy/dx = 13.3/.02 = 665 whereas x^-2 delivers 625
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> (.99,1.01) and (1.0,1.0) and (1.01, .99) = dy/dx = .02/.02 = 1 and x^-2 delivers 1
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> (1.01,.99) and (1.02, .98) and (1.03, .97) = dy/dx = .02/.02 = 1  whereas x^-2 delivers 0.96
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> (2.99,.33) and (3.00, .33) and (3.01, .33) = dy/dx = 0/.02 = 0  whereas x^-2 delivers 0.11
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> 1000 GRID graphing
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> (.001,1000) and (.002,500) and (.003, 333.333) = dy/dx = 666.667/.002 = 333333.5 whereas x^-2 delivers 250000
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> (.003,333.333) and (.004,250) and (.005, 200) = dy/dx = 133.333/.002 = 66666.5 whereas x^-2 delivers 62500
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> (.999,1.001) and (1.0,1.0) and (1.001, .999) = dy/dx = .002/.002 = 1 and x^-2 delivers 1
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> (1.001,.999) and (1.002, .998) and (1.003, .997) = dy/dx = .002/.002 = 1 whereas x^-2 delivers 0.996
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> (2.999,.333) and (3.000, .333) and (3.001, .333) = dy/dx = 0/.002 = 0 whereas x^-2 delivers 0.111
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> Now it is easy to see that some of those will never be precise agreement, such as the (3.000,.333) for the 1000 Grid or any grid thereafter.
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> And that is why the phony limit concept is brought into mathematics. Because mathematicians up to now were so lazy and stupid as to not giving a precise borderline between finite and infinity.
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> When we define precisely the borderline to be 1*10^603, we make the derivative precise in every cell of 1*10^-603 cell width, because the derivative is the connection of the two successor points of the function graph. In New Math, the function y = (-1)x^-2 is not really the derivative of y = 1/x, but a rather good approximation. In New Math, we have to venture into each cell of the function y = 1/x to pull out its precise derivative.
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> In Old Math, they had functions like the Weierstrass function or the y = sin(1/x) which behaved pathological and could be continuous everywhere but differentiable nowhere. In New Math, it is rare that a function is discontinous, and it is impossible for a function to be continuous everywhere yet differentiable nowhere. In New Math, if you are continuous, you are differentiable, because the function graph is the derivative.
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On second thought, here, when we define the borderline of finite to infinite as 1*10^603, it causes an immediate chain reaction that the smallest nonzero positive number is 1*10^-603 which then causes the first infinite number to be 1*10^603 + 1*10^-603. This first infinite number is then the inverse of
0 so that 1/0 is equal to this first infinite number. So division by 0 is now defined. However, any operation using the first infinite number is undefined.

So in New Math, we define division by zero, but we leave undefined any operations with this "first infinite number". It seems like not much of a change for we just shift the "undefined division by zero" to that of "undefined operations with first infinity number". But when we realize that with 0 now in play, we no longer have any discontinuous functions in mathematics.

That our shift of undefined from division by zero to operations by first infinity numbers makes for a Calculus free of discontinuity. Every function is now continuous.

In the function y = 1/x, when x=0, then y = 1*10^603 + 1*10^-603.

In the function y = sin(1/x) when x=0, they y = 1*10^603 + 1*10^-603 and we picture that in a graph with a huge spike, a needle shaped spike. Now that spike contributes some area to the integral and a derivative in the first cell.

So in New Math, all functions are continuous, for there is nothing undefined to cause discontinuity, and step functions are defined by they are continuous because we connect successor points which have empty space in between.

So in Old Math, discontinuity arose because of the laziness and stupidity of never a finite to infinity borderline. Once you define that borderline, functions are all continuous. It sort of reminds me of Algebra, when you are too lazy and stupid to define 1 as the inverse identity function with respect to multiplication so that 1/z (z) = 1, then you fail to have Field theory Algebra.

When you fail to define borderline of finite to infinite, you leave division by 0 as undefined which then causes massive and permanent damage to the concept of functions, leaving many functions as discontinuous.

Science and math are no different than ordinary life, when you are lazy and stupid, then you live a life full of shoddiness and discontinuity. When you are energetic and logical, you strive to gain continuity and precision. And you realize the discontinuity is there, not because Nature has it, but because of your inability to apply logic.

AP