Cell theory as alternative to Power formula and Chain rule #38

Cell theory as alternative to Power formula and Chain rule #38 Uni-text 8th ed.: TRUE CALCULUS without the phony limit concept

Alright, I am very glad I stuck to this Chain Rule for the Cell theory can revolutionize not only the Chain Rule with alternative Chain rule but can revolutionize Old Calculus of the integral without using the Power formula as to what the integral is going to be.

I need the University or College student to get out the graph paper, pencil and erasure and mark off in 10 Grid 10 for the y-axis and out to 3.2 for the x-axis. Specifically we are using 1000-Grid with the function y= x^2. Now the integral of y = x^2 by the power formula is int= 1/3x^3. Now the x-axis point in which y ends at the border of infinity is x=3.162. So that marks off specifically a Rectangle of 0 to 3.162 along the x-axis and from 0 to 10 along the y-axis. The total area in this rectangle of 10-Grid is 3.162 x 10 = 31.62.

So plot these points for y = x^2 and for x=0 we have (0,0) for x=1 we have (1,1) for x=2 we have (2,4) for x=3 we have (3,9) for x=3.162 we have (3.162, 9.998).

Now in New Math, in True Calculus we have the Complimentarity function which is that we turn the graph paper upside down and pretend that 10 is 0, that 9 is 1, that 8 is 2 etc etc for the y-axis and pretend 3.162 is 0 for the x-axis, and we have a plot of a different function from that of y = x^2. So what is that function? And how is it related to y = x^2? Is that function y = 1/3x^3? Now that new function has an integral area far larger than y=x^2 because it takes up at least 60% of the area of the rectangle that contains both functions.

If so, that this new function is y = 1/3x^3 then the Cell theory gives us the integral from geometry and without using the Power formula.

For consider the identity function y = x, and if we turn the graph upside down, the Complimentarity function of y= x is also y = x since it is a isosceles right-triangle.


--
Drexel's Math Forum has done an excellent search engine for author posts as seen he
http://mathforum.org/kb/profile.jspa?userID=499986

Now, the only decent search for AP posts on Google Newsgroups, is a search for for it brings up posts that are mostly authored by me and it brings up only about 250 posts. Whereas Drexel brings up nearly 8,000 AP posts. Old Google under Advanced Search
for author, could bring up 20,000 of my authored posts but Google is deteriorating in quality of its searches, likely because AP likes an author search and Google does not want to appear as satisfying to anything that AP likes. If AP likes something, Google is quick to change or alter it.

So the only search engine today doing author searches is Drexel. Spacebanter is starting to do author archive lists. But Google is going in the opposite direction of making author archived posts almost impossible to retrieve.

All the other types of Google searches of AP are just top heavy in hate-spam posts due to search-engine-bombing practices by thousands of hatemongers who have nothing constructive to do in their lives but attack other people.

Now one person claims that Google's deteriorating quality in searches of science newsgroups is all due to "indexing". Well, that is a silly excuse in my opinion, because there is no indexing involved when one simply asks for a author search. No indexing involved if one wants only the pure raw complete list of all posts by a single author. And Google is called the best search engine of our times, yet I have to go to Drexel to see 8,000 of my posts of which I had posted 22,000 to 36,000 posts from 1993 to 2013. It is a shame that Drexel can display 8,000 while Google has a difficult time of displaying 250 of my authored posts. Where the premiere search engine of Google is outclassed by Drexel and even by Spacebanter.

Archimedes Plutonium

What I am trying to do is see if the cell theory can fetch a new route to obtaining the derivative or integral with its complimentarity function. In Old Math, they had no complimentarity function because they had no cell to house such a function.

I was pretty excited this morning in thinking the complimentarity function may offer that new route.

As it turns out, it is not a new route.

If you turn to the Reference page 3 of Stewart's book
Calculus, 5 ed, 2003 there you find a picture of the graphs of y = x^2, y = x^4, y = x^6 which all three are shaped like a U occupying the 1st and 2nd quadrants. Then the next picture shows a graph of y = x^3 and then y = x^5 which looks like this in 1st and 3rd quadrants:

*
* * |
__|
|
|

rather than the U shape

Now all the below calculations are in the 10-Grid where 10 is the infinity border and where math ends at 10.

Now, if you built a x-axis in the 3rd quadrant of the portion of y =x^3 that lies in the 3rd quadrant then you begin to see the compliment of the area under the curve, with y ending at y=10, then we have the complimentarity function, so that if the area under x^3 is 2.154 x 10 = 21.54 and the integral of x^3 is 1/4x^4 in which x= 2.154 would be 5.38 which leaves 21..54 - 5.38 = 16.16 of the area in the complimentarity function.


Now if we rotated the y = x^2 function of the 2nd quadrant and moved it down into the 3rd quadrant via the rotation, so that it looks like the y = x^3 branch. And then compute the area under the x^2 in 1st quadrant as 3.162 x 10 = 31.62 of which the integral of x^2 is 1/3x^3 and plugging in we have 1/3 (3.162^3) = 1/3(31.61) = 10.53 so that of the total area in the rectangle 3.162 x 10 about 1/3 of that area involves the y = x^2 branch but if we transport the other (twisted branch) we get the remaining area of 31.62 - 10.53 = 21.09.

So no new technique to find integrals or derivatives but a lesson in the complimentarity function.

Now maybe something new is gained in this exercise in that the complimentarity function, may allow us to trim away many of the other quadrants and leave only the 1st quadrant for our attention.

Now I am not giving up yet on the Cell theory finding a new technique to fetch the integral or derivative of a function with its structure of a complimentary function. There is the possibility of converting all compounded straightline functions (curves we used to call them curves) into a single straightline such as y = 3x or y = 1/2x and determining whether these full straightline functions can fetch the derivative or integral from the function and its compliment of Cell theory.

For example, in the y = x^2 function, which its integral is 1/3x^3 occupies only 10.53 area of the full rectangle area of 31.62 and the compliment function occupies the remaining 21.09 area. So convert 10.53 into a full straightline that intersects the rectangle of 3.162 x 10 passing from the origin point (0,0) and intersecting the rightward wall of the cell whose x-axis interval is 0 to 3.162. This straightline is going to form a right triangle inside that rectangle and that straightline is going to cut the rectangle into two portions, one having area 10.53 while the other portion of the rectangle has area 21.09. Now one portion is a right triangle but the other portion is a trapezoid (picketfence). So here is an opportunity for the Cell theory to be able to derive either the derivative or integral which Old Math could never do because they had no Cell theory and had only the rain dancing of a limit concept.

AP

mixed results for new technique #39.1 Uni-text 8th ed.: TRUE CALCULUS without the phony limit concept

I am making a valiant effort to find a new technique of Calculus by using the complimentarity function to find either the derivative or integral of a given function. The results are mixed.

For the y = x^2 function, its integral is 1/3x^3 and its derivative y'= 2x and in the 10 Grid it goes beyond infinity at 3.162. So I am working in a rectangle that is 3.162 by 10. Now if I plug into the integral the number 3, I have an area of 9 and 3 plugged into derivative is slope 6. No apparent help there.

For the function y = x^3, its integral is 1/4x^4 and derivative is y' = 3x^2 and in 10 Grid it goes beyond infinity at 2.154, so I am working in a rectangle that is 2.154 by 10 for function and compliment function. Now if I plug into the integral the number 2 as close to 2.154, I have an area of 4 and a derivative of 12.
Now there is a curious aspect to this which was not evident in y=x^2. If we take the total area to be that of x^4 = 16 then the sum of the derivative of 12 with the integral area of 4 we end up with 16. Is that just a special random case or is that a pattern?

So I try it out on the function y=x^5 with integral 1/6x^6 and derivative y' = 5x^4. The number in which the function goes beyond 10 in 10 Grid is somewhere about 1.58 and the total area taking it to be x^6 is about 16 and the derivative plugging in 1.58 to that of 5x^4 is about 32.

So, I may have a pattern here, or, more likely, I have just glimpsed a closer look up front of how the integral is related to derivative in a cell.

And in this tinkering to find a new technique, I ask myself the question of whether beyond y = x^3 whether those higher exponents have any meaning in physics? Do we in physics ever run across the need of x^4 or higher exponents. I cannot remember the use of any higher exponents in physics other than those that daydream in physics. So it maybe the case that in mathematics, the Power formula is over used and that it really belongs to a narrow application of exponents 2 and 3 and occasionally 4.

I cannot think of a single physics application where exponent 5 is required..

AP