#2 TRUE CALCULUS; without the phony limit concept; Textbook for Middle-School on up (1st edition)

[Archimedes Plutonium]

Chapter 2 What the Derivative truly is

Alright, let me change the title with the word "phony" instead of
"fiction".

Now let us pursue the true meaning of derivative. Before I do that,
let me tell you the gross mistakes of Old Math about the derivative so
that you can keep that in mind as you read about the true derivative.

Mistakes of Old Math:
(1) never had a borderline between finite and infinite
(2) never had any gaps between one number and the next successive
number

To the credit of Old Math, they did figure out, or see in their mind's
eye that the derivative and the integral had something to do with
"picketfence construction".

Here is a picketfence construction:

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or switch around:

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Most mathematicians had a mind's eye and realized that calculus had
much in common with picketfence constructions, both for the derivative
and integral.
But they failed to seize on the most important aspect of that
construction analogy-- that you need an empty space gap from one
number to its immediate successor number. In other words, you need
holes in geometry in order to even have a Calculus. Without holes in
geometry, there is no Calculus at all.

And the sad thing about Old Math, is that they went entirely in the
opposite and fakery direction by coughing up a concept called the
"limit concept". The limit concept is their rug sweeping away of their
mistakes.

Now in the first chapter we have the three simple functions of

y = 3

y = x

y = x^2

Now their derivates (denoted by y'), respectively, are these:

y' = 0

y' = 1

y' = 2x

Now the derivative in Old Math was called the slope of the function at
a given point, or the tangent. In New Math the derivative is the angle
measure at that given point of the function. It is the angle at the
top of that picketfence that we place at the point of the function.

So here we begin to see or dawn upon our minds or mind's eye that the
derivative is that angle at the top of the picketfence and that angle
has to be able to pivot between 0 degrees and 90 degrees. And to be
able to pivot without any numbers in between the sides of that
picketfence, in other words a gap or hole where no numbers can alter
the picketfence angle.

So let us take that picketfence and place it onto those three
functions and see what happens at some specific points.

y = 3 has y' = 0
and that function is a straight line whose y value is always 3. So we
draw a picketfence for the point x =2, and F(x)=3 and y is thus 3. Now
the picketfence has no angle slanted at the top but is flat with the x-
axis. It means that the slope or tangent is zero. The derivative is
zero.

y= x has y' = 1
and that function is a 45 degree slanted line in the first quadrant.
It is normally called the identity function. Now a picket fence on a
point such as x=3 would have a y value of 3. Now no matter what x
value I have, the y value is the same so when I divide y/x it ends up
being a slope or tangent of 1.

y= x^2 has a y' = 2x
In this textbook I need at least one function that is a curve and not
a straight line. If we pick the x value of 3, x=3, the y value is 9,
and the slope or tangent is 6.
Now if I look at x= 2 and x=4, I have y values of 4 and 16
respectively. So the slope of a picketfence drawn from interval x =2
to x=4 has y values from the interval 16 to 4, so the slope or tangent
in that interval is (16- 4)/(4-2) which is 12/2 = 6

Now with derivatives as picketfences angle at the top allows us to get
an angle because there are no interfering numbers between 2 and 3 or 3
and 4 or 4 and 5 etc etc. There are just holes and gaps.

What if there were no holes and gaps and that there were a mess of
numbers between any two numbers. Then you would not have any assurance
or guarantee that the picketfence has a top angle.

If we are to believe in Old Math of the Weierstrass function, known as
a pathological function, we are never able to construct a picketfence
in any interval no matter how tiny the interval. We cannot do it
because there are no gaps or holes to allow a picketfence top to form
an angle.

Now in New Math, it is discovered that the infinity borderline is root-
pi 10^603 where pi has its first three zero digits in a row and evenly
divisible by 2, 3, 4, 5. Let me call this number for short 10^603.
Since that is the borderline of finite to infinite, then 10^-603 is
the smallest possible number and that there are no numbers between 0
and 10^-603. The number 10^-603 is the measure of the holes and gaps
between every two successive numbers in geometry and number theory.

So that the functions, y =3, and y=x and y =x^2 have no numbers
between 0 and 1*10^-603 and between
1*10^-603 and 2*10^-603.

Because all the numbers have at least that size of hole or gap between
them and the neighboring numbers, allows there to be a derivative of
all functions in mathematics, even a Weierstrass function.

Now when I get to the integration chapter, those gaps and holes allow
us to see how the derivative is the inverse of the integral.

But before I get to that chapter I have to elaborate more of what I
said in this post.


--
More than 90 percent of AP's posts are missing in the Google
newsgroups author search from May 2012 to May 2013. 
Drexel
University's Math Forum has done a better job and many of those
missing 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

[1treePetrifiedForestLane]

when are you going to show an example problem & solution,
or at least a problem?

the Liebnizian method is N times better than Newton's!

> More than 90 percent of AP's posts are missing in the Google

> newsgroups author search from May 2012 to May 2013.; just cannot be avoided!