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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
importance of fixed angles in Grid that creates the Calculus #5
HS-Textbook 7th ed.: TRUE CALCULUS; without the phony limit concept

Posted: Jul 11, 2013 1:02 PM

importance of fixed angles in Grid that creates the Calculus #5
HS-Textbook 7th ed.: TRUE CALCULUS; without the phony limit concept

Alright, I decided that this is the ripe moment to discuss how the
Cartesian Coordinate System creates a grid or network of points of
numbers and this network creates fixed angles for which the Calculus
can then arise and be formed and come into existence. I am going to
need a special function
to demonstrate this importance of the fixed angle to form Calculus.
And that special function is a sawtooth function in 10-Grid, 1st
quadrant only. In math we often use F(x) to mean y, so that some
prefer to write y and some prefer to write F(x). I prefer to write
F(x) whenever the function is broken into parts and assembled as bits
and pieces. When a function is able to be written as a whole in one
equation such as y = x or y = x^2 or y = cos(x), then I use y, but
when the function has different instructions for different values of
x, I call it a broken function and use F(x) rather than y. Calculus
texts often never make any distinction between a whole function and a
cobbled together function called broken function. In True Calculus, anything that requires logic and precision, requires our attention.

So this special sawtooth function is written like this:

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

(Strange side note in that it is funny how even and odd in mathematics
is relative to the Grid system and finite to infinity border, because
1 is thought to be odd, whereas in 10-Grid, it is even. So beware that
odd and even relationship is relative to a Grid.)

So, now we connect those x-points forming a steep sawtooth looking
function. The line that connects is the derivative of the function at
that x value. It alternates up and down looking like this /\/\/\/\.
The picketfences for the function involves no rectangle but just the
triangles. The function is continuous everywhere and differentiable
everywhere and integrable everywhere. In True Calculus, we have no
"pathological functions" such as y = sin(1/x) or the Weierstrass
function. In True Calculus those would be continuous, differentiable
and integrable everywhere, except for discontinuous points at x=0 where division by 0 is undefined. In the Uni-textbook we get rid of all discontinuities of functions so that the function y = 1/x is continuous everywhere, differentiable everywhere and integrable everywhere, but here we keep it simple for the High School text.

Now this sawtooth function tells us the importance of the Coordinate
System providing fixed angles to all its points, because when we alter
the function a small bit by this function:

F(x) = 0 when x is even and F(x) = 9.9 when x is odd, we have another
sawtooth function but with a different y value of 9.9 rather than 10
and thus a different angle. The different angle means the derivative
is different and integral is different for these two functions.

In fact, if we were to draw the entire 10-Grid system of 1st quadrant
only of all 100 times 100 = 10,000 points and were to select at random
any two points of those 10,000 and those two points were points of a
function, then those two points form a unique angle that allows for
another function to be a derivative and another function to be an
integral.

So in mathematics, we have functions that require a unique y value for
a given x value, and, we have a Grid system that requires a unique
fixed angle given any two points of a function. So, this is important, for the Calculus would not exist if it were not for "fixed angles". Calculus is a mix, a blend of numbers with geometry, so that numbers are fixed points relative to other numbers and this fixed points yields fixed angles.

The coordinate points of (0,0), (1,1), (1.4, 1.4), (5.9,5.9) when
plotted always produce a fixed angle of 45 degrees because they are
points of the identity function y= x. I have never seen a Calculus
textbook ever discuss the importance of coordinate points causing
there to be unique fixed angles.

One of the very best ways to learn subject, as I have found out, is by writing
a book on the subject. This is a pragmatist saying that "learning is in
the doing". We can learn by having teachers lecture to us, or learn by
reading a textbook, but better yet is that when writing that textbook
you learn so much more than reading it because it involves "doing"
which requires more thought and more research. The organizing and
order of the book demands us to think about what we are writing and
demands some research.

--

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