Fundamental Theorem of Calculus is superfluous and refocused in New Math Textbook 2nd ed. : #8 TRUE CALCULUS; without the phony limit concept

[Archimedes Plutonium]

I have not yet talked about how Calculus as derivative integral exists
as an operator such as add subtract and multiply divide because in a
Cartesian Coordinate System the number-points are so spaced and
arranged in order that this spatial arrangement yields an angle that
is fixed. So that if you have an identity function y = x, the position
of points (1,1) from (2,2) is always the same and is always a 45
degree angle. So Calculus of derivative and integral is based on this
fact of Euclidean Geometry that the coordinates are so spatially
arranged so as to yield a fixed angle.
Calculus is that Angles stay the same regardless of size of Grid
because numbers remain in the same positions relative to one another.
So that addition is the inverse of subtraction and vice versa and
multiplication is the inverse of division. Area is the inverse of
angle slope and angle slope the inverse of area.

Now one indication that angle slope is inverse of area is seen in the
picketfence construction that the area of the picketfence depends on
that angle of the hypotenuse atop the picketfence triangle and if the
area of the picketfence were to change then the angle has to change to
meet the new area.

But there is another way of showing that the area is the inverse of
derivative angle slope.

Picture our function y= x^2 and starting to plot at x = .4 in the 10
grid:

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

at x=.4 y =.16
at x=.5 y =.25

plotting that gives us this:

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

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

Now notice a fascinating aspect of this, for we could build a picket
fence of width .1 and height .16 where the rectangle portion is .1 by .
1 and the triangle portion is .06 by .1. Or, we can just replace the
one picketfence with two triangles each of which has a base .1 and a
height of .16. So the area of these two triangles would be the area of
one rectangle of .1 width and .16 height.

Remember the sawtooth function of F(x) = 0 for even numbered x and
F(x) = 10^603 for odd numbered x. Here we can do the integration, not
by summation of thin rectangles but rather by summation of two thin
triangles at each point.

Remember the simple function y=3 where a summation of rectangles .1
width by 3 height would yield integration? Well here again we can
switch over to summation of pure triangles whose width is .1 and
height is 3, but we have twice as many triangles as rectangles.

Now what is my gain in switching from rectangle summing to triangle
summing? Well, in the Fundamental Theorem of Calculus we want to prove
that the derivative is inverse of integral and vice versa. Well, if
the derivative is that angle of the hypotenuse of triangle and
integral the area, that we can switch from one to the other, this was
shown by the fact that area is 2 triangles and 2 triangles is the area
of 1 rectangle.

Now I probably do not have this perfectly clear in this edition and
need to make more clear in every future edition.

But because Calculus has holes in between successive numbers, the
limit concept is phony and that raises a problem with the Fundamental
Theorem of Calculus is not what it ought to be.

True Calculus has no limit concept, that most of modern day
calculus of those 700 page texts, much of that gobbleygook phony
baloney or gibberish nattering nutter speak is about the limit. When
you have true math, you need just 10 pages to explain it. When you
have fake math, you need 700 pages of symbolism and abstractions to
hide and cover up.

So these holes and gaps make the limit concept as fictional, for
there 
is no need of a limit. The holes themselves serve as a limit.
The 
holes prevent pathological functions from forming such as the
Weierstrass function or the function y = sin(1/x).
The holes allow the derivative to form or come into existence because
the hole gives the derivative room to form a angle, an angle between
0 
and 90 degrees. Without the hole or empty space, the neighboring
infinite points would obstruct as in the Weierstrass function,
obstruct the formation of the derivative. The holes also give the
picketfences or rectangles or triangles in integration give them
internal area, so the integral is not the summation of line segments
without internal area as in Old Math.
Now in most Old Math calculus texts of those 700 page gibbering
nattering nutter symbolism of limits, once they cover derivative and
integral, they usually want to tie the two together in what is called
the "Fundamental Theorem of Calculus". And they make a big splash and
stir and 
fuss about this. But in New Math, we not only throw out the
limit as 
fakery, but we have no need to show that the derivative is
the inverse 
of integral and vice versa. In mathematics, do we need to
have a 
Fundamental theorem of add subtract or a Fundamental theorem
of 
multiply divide and prove they are inverses? No, we need not go
through that silliness. In fact in mathematics, the addition inverse
to subtraction or multiplication inverse to division is provided for
by the axiom set such as the group theory axioms or the Hilbert
axioms. But in Calculus of Old Math, they must have felt that the
inverses of derivative versus integral needed special attention.
In New Math, the inverses need no special attention because they were
constructed to be inverses. Just as shown above that we can sum over 2
times the triangles of their hypotenuse as derivative, or we can sum
over rectangles.
In New Math, in True Calculus we merely note that the derivative is
the angle of the hypotenuse atop the picketfence which determines a
unique area of the picketfence, so that the derivative is the inverse
of the integral. If I change the area of the picketfence, I change
the 
derivative proportional to the area. If I change the angle of
the 
hypotenuse, I proportionally change the area inside the
picketfence.
So in True Calculus we throw out the phony baloney limit concept and
along with it we have no need for a hyped up exaggerated Fundamental
theorem. Instead what we do in New Math is realize that Calculus does
have a Fundamental Theorem but it is not about inverse relationship,
rather it is about why Calculus is stuck with Euclidean geometry.
Now let me speak more about geometry. It is important to know the
relationship of geometry to numbers and angles but there is also the 3
types of geometry of Euclidean, Elliptic and Hyperbolic and that
should have been what the Fundamental Theorem of Calculus addressed.
The 
fundamental theorem should have embodied the idea of why the
Calculus seems to exist only in Euclidean geometry since slope and
area involve straight line segments not curved line segments. So the
Fundamental Theorem 
of Calculus should have been a theorem that
explores and proves why 
Euclidean Geometry can yield a calculus but
that Elliptic geometry or 
Hyperbolic geometry cannot yield a
calculus.

--
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