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Topic: What is a function #1 HS-Textbook 7th ed. : TRUE CALCULUS; without
the phony limit concept

Replies: 12   Last Post: Jul 13, 2013 2:38 AM

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 plutonium.archimedes@gmail.com Posts: 18,572 Registered: 3/31/08
Cell and integral in Calculus #6 HS-Textbook 7th ed.: TRUE CALCULUS;
without the phony limit concept

Posted: Jul 12, 2013 12:52 AM

Cell and integral in Calculus #6 HS-Textbook 7th ed.: TRUE CALCULUS; without the phony limit concept

Now I need to define "cell" in Calculus. It is a hugely important concept.
For it is the easiest way of teaching the integral and integration. The Cell is where the picketfence is found, and the picketfence area is the integral.

The integration is the summation of picketfences over an
interval of the x-axis.

So let me include a term, a new term in Calculus-- "the cell", for the width of the unit picketfence which in
10-Grid is 0.1 width.

Definition of Cell: a cell in 10 Grid 1st quadrant only, is a rectangle whose width on the x-axis is 0.1 and whose height or length is 10, and the 10 Grid has exactly 100 cells and has 10,000 grid points.

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.    .    .    .    .    .    .    .    .    .    .
0  .1   .2  .3  .4  .5   .6  .7  .8  .9  1.0  -->

Then there are 10 cells there with the first cell being 0 to .1 wide
and then going up along the y axis and reaching y = 10. The second cell would be from 0.1 to 0.2 and then up along the y axis to y = 10. Inside each cell is a picketfence or a rectangle or a triangle which will be the source for the derivative or the integral. So the cell contains the geometry figure of a picketfence, or rectangle or triangle from which the derivative and integral are obtained.

There is one key idea in integrals and integration that involves Set-theory. All intervals in True Calculus are closed intervals. For example in
the 10 Grid system we cannot have open intervals of say (0, 0.2)
because we have those gaps of empty space between successive number-
points. So what does (0,0.2) mean in set theory of an open interval?
It would mean the set {0.1} containing a single member of 0.1. The
closed interval [0, 0.2] has three members of the set {0, 0.1, 0.2}.
So in True Calculus, set theory has no open intervals, no full open
intervals and no half open intervals because we cannot put a gap of
empty space into set theory. Some of the gaps of empty space are more
than 10 metric distance long, for example the derivative of the
sawtooth function is the hypotenuse of the right triangle:

F(x) = 0 when x is even number and F(x) = 10 when x is odd number. So
the graph of this Sawtooth function in 10-Grid, 1st quadrant only,
looks like this:

.    x    .   x    .   x    .   x    .    x   . 10.0
.    .    .    .    .    .    .    .    .    .    .  9.9
.    .    .    .    .    .    .    .    .    .    .  9.8
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out of scale for there should be 10 blocks of 10
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.    .    .    .    .    .    .    .    .    .    .
x    .   x    .    x   .    x    .   x    .   x
0  .1   .2  .3  .4  .5   .6  .7  .8  .9  1.0  -->

So in True Calculus, all the intervals are closed intervals.

And for the sawtooth function above, the integral is the summation of
those steep triangles in every one of those successive number points
of width 0.1 metric, for every cell has one steep triangle. So the
integral is the area of the triangle in the 0 to 0.1 interval added
with the triangle in 0.1 to 0.2 interval, and so on. In fact, the
integral of this sawtooth function is 1/2 of (10 by 10) or 50 square
units. For it is apparent that if we stack the triangles of one cell
with its successive cell we have a rectangle that is half the area of
10x10.

Many integrals will be the summation of triangles only, while many
others will be the summation of picketfences such as the y = x
function. Some functions will be the summation of pure rectangles only
for the integral, such as the function y= 3.

The functions, y= 1/x or y = x^2 will be picketfence integrals, since
each cell of 0.1 width will be partly a rectangle with a triangle
atop that rectangle.

--

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