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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: 18,263 Registered: 3/31/08
#13.5 Re: Number "e" and Straightlinecurves; derivative & integral of
y=1/x Uni-textbook 6th ed.:TRUE CALCULUS

Posted: Jul 1, 2013 12:51 PM

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

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.

Note: negative sign in y'= -x^-2 is not needed for it only tells us the orientation of the derivative.

10 GRID graphing

(.1,10) and (.2,5) and (.3, 3.3) = dy/dx = 6.7/.2 = 33.5 whereas x^-2 delivers 25

(.3,3.3) and (.4,2.5) and (.5, 2) = dy/dx = 1.3/.2 = 6.5 whereas x^-2 delivers 6.2

(.9,1.1) and (1.0,1.0) and (1.1, .9) = dy/dx = .2/.2 = 1 and x^-2 delivers 1

(1.1,.9) and (1.2, .8) and (1.3, .7) = dy/dx = .2/.2 = 1  whereas x^-2 delivers 0.6

(2.9,.3) and (3.0, .3) and (3.1, .3) = dy/dx = 0/.2 = 0  whereas x^-2 delivers 0.1

100 GRID graphing

(.01,100) and (.02,50) and (.03, 33.33) = dy/dx = 66.67/.02 = 3333.5 whereas x^-2 delivers 2500

(.03,33.33) and (.04,25) and (.05, 20) = dy/dx = 13.3/.02 = 665 whereas x^-2 delivers 625

(.99,1.01) and (1.0,1.0) and (1.01, .99) = dy/dx = .02/.02 = 1 and x^-2 delivers 1

(1.01,.99) and (1.02, .98) and (1.03, .97) = dy/dx = .02/.02 = 1  whereas x^-2 delivers 0.96

(2.99,.33) and (3.00, .33) and (3.01, .33) = dy/dx = 0/.02 = 0  whereas x^-2 delivers 0.11

1000 GRID graphing

(.001,1000) and (.002,500) and (.003, 333.333) = dy/dx = 666.667/.002 = 333333.5 whereas x^-2 delivers 250000

(.003,333.333) and (.004,250) and (.005, 200) = dy/dx = 133.333/.002 = 66666.5 whereas x^-2 delivers 62500

(.999,1.001) and (1.0,1.0) and (1.001, .999) = dy/dx = .002/.002 = 1 and x^-2 delivers 1

(1.001,.999) and (1.002, .998) and (1.003, .997) = dy/dx = .002/.002 = 1 whereas x^-2 delivers 0.996

(2.999,.333) and (3.000, .333) and (3.001, .333) = dy/dx = 0/.002 = 0 whereas x^-2 delivers 0.111

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.

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.

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.

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.

--

More than 90 percent of AP's posts are missing in the Google ?newsgroups author search archive from May 2012 to May 2013. Drexel ?University's Math Forum has done a far better job and many of those ?missing Google posts can be seen here:

http://mathforum.org/kb/profile.jspa?userID=499986

Archimedes Plutonium ?
http://www.iw.net/~a_plutonium ?
whole entire Universe is just one big atom ?
where dots of the electron-dot-cloud are galaxies