pretend infinity border is 10 grid and derivative #4 Textbook 2nd ed. : TRUE CALCULUS; without the phony limit concept

[Archimedes Plutonium]

Alright, in the next edition I need to use the 10 grid more than the
10^-603 grid so that High School students have an easy time with True
Calculus.

Now I have the four functions:

y = 3 (type of a box function)
y = x (identity function)
y = x^2
y = 1/x (similar to a type of a logarithm function)

their derivates are respectively (y' denotes derivative)

y' = 0
y' = 1
y' = 2x
y' = -x^-2

Now let me focus on the function y= 1/x and its derivative using the
10 grid. The 10 grid is where infinity borderline is 10 and thus the
smallest nonzero number is 1/10 and where all the numbers that exist
for mathematics in the 1st quadrant are these:

0, 1/10, 2/10, 3/10, 4/10, . . on up to 10

Those are 100 individual numbers and since the y-axis has the same
amount, there are 100X100 = 10,000 numbers in all.


Now let me focus in on the first four numbers of this 10 grid and
since I cannot ascii art 100 rows, let me mark the first four
coordinate points:

for x=0, it is undefined for 1/x
for x=.1 then we have (.1, 10)
for x=.2 then we have (.2, 5)
for x=.3 then we have (.3, 3.333..)
for x=.4 then we have (.4, 2.5)
for x=.5 then we have (.5, 2)

Now here is a truncated version to get a sense of the picture but the
student is asked to plot it in pencil on their graph paper to use an
erasure so you can use over again.

. x . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . x . . .

. . . . . .
x
. . . . . .
x
. . . . . x

. . . . . .

. . . . . .
0 .1 .2 .3 .4 .5



Now we can draw a picketfence in the interval of dx for .3, .4, .5.

That interval of dx involves .1 + .1 = .2

That interval of dy involves 3.33.. - 2 = 1.33..

So that dy/dx is 1.33../.2 = 6.65 slope and negative

Whereas the derivative of y = 1/x is -1/x^2 which is
-1/.16 which is - 100/16 = -6.25 slope

Now let me draw the two picketfences
. . . . . .
x
. . . . . .
x
. . . . . x

. . . . . .

. . . . . .

. . . . . .
|\
. . . |. \ . .
| x
. . . |____\

. . . . . .

. . . . . .

Now the interval .3, .4, .5 involves two picketfences
in this picture, but in True Calculus we end up with just one
picketfence that models the derivative and integral. The limit concept
of Old Math is the use of two picketfences as the delta gets smaller
and smaller. However, as the above shows, no matter how small, the two
picketfences do not give the proper correct value. What does give the
proper and correct value is that we use just one picketfence and we
have that empty space in which the angle of the hypotenuse can become
the exact angle for the slope or tangent at the given point.

Now let me do another example in the 10^603 grid instead of the
pretend 10 grid. In this example we can also see how the Weierstrass
function and the sin(1/x) function are differentiable and continuous
everywhere except perhaps 0 since division by zero is undefined.

. x . x . x

. . . . . .

. . . . . .


break in scale


. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

x . x . x .
0 a b c d e

Now a is 1*10^-603 and b is 2*10^-603 and c is 3*10^-603 etc etc

and the x's on top are where y = 10^603 and the x's on bottom are
where y= 0.

So the function here is F(x) = 0 for even x and F(x) = 10^603 for odd
x. Now this graph is like the world's steepest mountain range or it is
a sawblade. It has the steepest angle except for 90 degrees itself, so
the angle involved is (90- 1*10^-603). Now in Old Math, this function
would be a step function and discontinuous everywhere and
differentiable nowhere. In Old Math the Weierstrass function is
continuous everywhere but differentiable nowhere. In New Math this
sawblade function is continuous everywhere and differentiable
everywhere. In New Math the function sin(1/x) is discontinuous at 0
but is continuous everywhere else and differentiable everywhere else.

So let us examine the sawblade function just described. If we draw a
straight line connecting the three points (0, 0), (1*10^-603, 10^603),
(2*10^-603, 0) we get our first tooth, a sharp tooth of the sawblade
for it stretches from 0 to the infinity borderline of 10^603. Now Old
Math with their limit concept has no way of handling or dealing with
such a function, not to mention the Weierstrass or sin(1/x) but New
Math handles every function that exists in Nature because we can list
all the grid points and examine any neighborhood of grid points.

Now the picketfence of this function is rather different from most
picketfences because it is all one long slender triangle and not a
rectangle involved. Most derivatives and integrals involve the
rectangle with triangle sitting atop:

/|
||
||
||

Here there is no rectangle portion but just the triangle, and it has
width 1*10^-603 and it has height of 1*10^603 so it has area and it
has a hypotenuse and thus it has a derivative.

Now we can do the same for Weierstrass function and see that it is
indeed continuous and differentiable everywhere and we can do the same
for y= sin(1/x) although discontinuous at 0, it is continuous
everywhere else and differentiable everywhere else.


--
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

[Archimedes Plutonium]

Alright for integration, I am going to pare down to these three
functions and not use the y= 1/x.
y = 3
y = x
y = x^2
there derivates are respectively

y' = 0
y' = 1
y' = 2x

there integrals (abbreviated Int) are respectively
Int = 3x
Int = 1/2x^2
Int = 1/3x^3

Antiderivative Technique
(1) for the derivative of a function x^n the derivative is
n(x)^(n-1).
(2) for the integral the antiderivative works backward. So for x^n,
the antiderivative is (1/(n+1)(x^(n+1))

Now we know the derivative was the slope or tangent or rate of change,
and we know the integral is the area under the curve with respect to
the x-axis.
Now we have the antiderivative technique that gives us the derivative
and integral for a class of functions of form y = x^n.
Now we have the technique and we know the integral is the area under
the graph of the function and we want to see how that technique gives
us that area. That technique was known by Newton and Leibniz circa
about 1675. And both Newton and Leibniz probably understood the
antiderivative by examining two of our three functions, the identity
function along with what I call a box function y=3. I do not know if
Newton or Leibniz had in their writings on how they came upon the
Antiderivative Technique
but I would hazard to guess that they discovered it from a box like
function of y=3 and the identity function.
If you look at y=3 its intervals for integration are squares or
rectangles and the triangle top of the picketfence has no triangle
for 
the derivative is 0. 
And the area of a rectangle is length by
width. So the area under the 
graph of the function y=3 for interval 0
to 2 would be 2x3 or area 6 
just as the antiderivative as integral
gives us Int = 3x and that is 
also 6. When x is 3 we have a square
box and thus the area is 3x3 =9. 
And then Newton and Leibniz probably
noticed that the identity 
function, y= x is a equilateral triangle
itself with the dy and dx and 
the area of an equilateral triangle is
1/2x^2 or 1/2 of a square box. 
So that the entire identity function
is the magnified tiny triangle 
atop the picketfence for the function
y=x.
So I reckon that both Newton and Leibniz analyzed and saw this box
function and identity function and then discovered the Antiderivative
Technique. I do not know for sure how Newton and Leibniz discovered
the Antiderivative technique but the above is a good guess.
Now let me use the Pretend Infinity borderline 10-grid where the
smallest nonzero number is 1/10 for the Identity function y= x.

.    .    .    .    .    x

.    .    .    .    x    .
.    .    .    x    .    .

.    .    x    .    .    .
.    x    .    .    .    .
x    .    .    .    .    .

0 .1 .2 .3 .4 .5

So the function y= x is a straight line diagonal of 45degrees in 1st
quadrant. It is a fine function to illustrate the Picketfence model.
So for the interval .4 to .5 for x then
y is .4 and .5. And the picketfence looks like this:

/|
||
||
||
||

where its rectangle width is 1/10 and its height is 4/10 and the
triangle atop has width 1/10 and height 1/10. So an area is found for
this picketfence.

Now here is problem for Old Math, in that the area is found for
rectangles alone and the limit is used as the tool to make the
rectangle width as small as possible to gain the most accurate area.
That is the problem, the only width to gain a precise accurate area is
for the limit to go to zero and thus the width of the rectangle is
also zero. So in Old Math, they wanted you to believe, not that the
Moon is made of cheese, but something almost as bad, that rectangles
of 0 width can have area.

In New Math, we escape that contradiction of integration of rectangles
with 0 width, because in New Math, we have holes and gaps of 1/10 in
the 10 grid and of 1*10^-603 in the 10^603 grid. These empty spaces
between successive numbers insures that the integral of myriad
picketfences each have an internal nonzero width and thus a true
internal area.

So let me summarize why Old Math fails to have a Calculus with a limit
concept and why New Math succeeds in having a Calculus. Because
successive numbers have a hole or gap or empty space between them,
they allow enough freedom to form any angle between 0 and 90 degrees
for the hypotenuse of the triangle atop the picketfence, and because
the empty space has a width that gives the picketfence internal area
allows the formation of the integral.

In Old Math, the limit concept was just a yakkity yak time filling
exercise, for in Old Math, they needed something to talk about in
deriving the derivative and integral and the limit became just a
mouthpiece. Sort of like in medicine when we do not know what cancer
truly is, yet we have to talk about something to fill up the time, so
we spend all the lecture time on talking of how various drugs have
effective use. In New Math, we see that it is the empty space between
two successive numbers that allows a derivative and integral to form
and that they are related so that if you change one, you change the
other.

[Archimedes Plutonium]

Alright, we covered derivative and integral. We understand that the
picketfence model is the essence of the derivative and integral, in
that a empty space is required between successive numbers. The metric
of that empty space is the inverse of the borderline of finite with
infinite. That metric is Floor-pi*10^603 with its inverse abbreviated
as 10^-603.
In Old Math, the limit concept destroys the integral because it
demands the summation of rectangles each having 0 width, in other
words, 
rectangles that are not rectangles but rather line segments.
Now if 
you believe a line segment or line can have area, well, then
join the 
millions of other fools who accept Old Math Calculus.
The limit concept destroys the derivative because it allows no room,
no empty space for the dy/dx to form an angle without the
interference 
of neighboring points of the function. Now when the
function is a straight line and not a curve like that of y = 1/x, then
you do get a true derivative. But when you have a curve function, your
dy/dx is never able to match the true value. When you have a empty
space, a hole or gap that is 10^-603 from neighboring points, that
empty space gives the function the freedom to form the correct angle
between 0 degrees and 90 degrees (excluding 90 itself for then it is
no longer a function) and not be constrained with the limit delta
interval.
So, a empty space between successive numbers is essential and
required 
for the Calculus to exist. The empty space gives the
integral a 
nonzero area for its picketfences that are summed over the
graph of 
the function, and the empty space gives the derivative room
enough 
that a correct angle is formed between a point and its
neighboring points.
So that in Old Math with their phony limit concept, they coughed up
silly and fictional functions such as the Weierstrass function. They
did get something correct about the Weierstrass function for they
call 
it a pathological function as seen in the Wikipedia entry of
this 
function. In New Math with holes of 10^-603 there are no
pathological 
functions. In New Math, all continuous functions are
differentiable 
everywhere.
Now even Strang in his textbook CALCULUS, 1991 on page 86 shows us
what can be considered another pathological function when you do not
have a borderline of finite to infinite and instead use a limit
concept. Strang shows us the function y= sin(1/x). 
Strang thinks it
is discontinous because it fluctuates so wildly when 
it comes near 0.
It is discontinuous at zero for division by zero is undefined.
However, it is continuous thereafter.
So that if Weierstrass or Strang had done a Cartesian Coordinate
System Grid of the x-axis and y-axis in the First Quadrant of those
dots with 10^-603 holes, that the Weierstrass function and the
function y= sin(1/x) are simply normal functions.

Now here we have an interesting consequence of the fact that the
Calculus requires a empty space between successive numbers. So that,
by logic we require the existence of Calculus, which then implies,
holes or empty spaces must exist. That then implies a borderline
between finite and infinity must exist and we just have to find it. So
in this textbook, I started out with a borderline between finite and
infinity and then built the Calculus. Be we can start out in the
opposite direction of insisting Calculus exists and then end up
looking for a infinity borderline. Now I wonder if there is any
special function in Calculus of its derivative or integral that would
seek out the number Floor-pi*10^603? When I did the textbook
Correcting Math, the tractrix and pseudosphere were such functions.
But there could well be others that are more simple and easy and
intuitive.

[Archimedes Plutonium]

Looks like I learned a second new thing in this book, other than a
definition of continuous function. Looks like I may have found a new
means of discovery of the infinity number borderline. In my last post
#6, I discussed a possibility of a function that requires the infinity
borderline to be a specific number or else the Calculus does not
exist. I am always on the look-out for ways of arriving at Floor-
pi*10^603 as the infinity borderline.

If we draw a half-circle function in the 1st quadrant we involve
ourselves with pi. So we look for a number to be the borderline and
then take its inverse to be the width of the holes or empty space
between successive numbers in order to have a Calculus at all. So now,
can the infinity borderline be 10^500 say, and which would make the
size of the holes as 10^-500?

Well, if we look at this function of a half circle
y = sqrt(r^2-x^2), centered at origin so we have a quarter circle in
the 1st quadrant. Now we ask the question what should the size of
holes or empty space between successive numbers be in order for the
Calculus to exist at all? It cannot be 10^500 because we need there to
be a place in the digits of pi where it is evenly divisible by 5! =
120 so that the entire quarter circle picketfences of derivatives
(slopes or tangents) become a truncated regular polyhedra. In other
words, a place in the digits of pi so that the derivative of the
circle function is a truncated regular polyhedra that becomes the
circle. This happens to pi at Floor-pi*10^603. So at each successive
number points on the graph the curve of the circle is replaced by a
straight line segment of the truncated regular polyhedra as the circle
itself. That can only happen first when infinity borderline is Floor-
pi*10^603. It could happen further down in pi digits beyond 10^603 but
it happens first in pi at 10^603 and so that is the infinity
borderline.