Chapt4 Integral and Integration #6 TRUE CALCULUS; without the phony limit concept (textbook 1st ed.)

[Archimedes Plutonium]

Now it is almost a relief to me to do a textbook that is no more than
25 pages and a difficult subject, the true Calculus. A relief because
I am in the stages of completing the Atom Totality textbook which is
going to be around 2,000 pages. Now I can write a 25 page textbook in
one day, but it takes me months and years to write a 2,000 page text
on physics.

Now before I dive into the integral, let me review what was covered so
far and the aim of this text. So far we learned that the Calculus is
two items of derivative and integral and the derivative is rate of
change or slope or tangent whereas integral is area.
And those two items can all be placed into the geometry of
picketfences of long thin rectangles with a triangle atop the
rectangle and the hypotenuse of that triangle is the derivative and
the area of the picketfence becomes the integral.

Here is a picture of a picketfence:

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and here is a picketfence with hypotenuse reversed
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Now that hypotenuse and have any angle from 0 up to 90 degrees but not
90 degrees itself for then it would not be a function of mathematics
because a function has to have a unique y value for a given x value
and at 90 degrees we lose that definition of function.

So the picketfence nicely wraps up the essential content of what the
derivative and integral are. The derivative is that angle atop the
picketfence triangle and the integral is the area of all the
picketfences under the function graph.

Now I did not expect to learn anything new while writing this 25 page
textbook but am happy I did learn something new. A meaningful
definition of continuity when all points have that minimum hole or gap
between them of at least 1*10^-603. So a definition of continuity in
Calculus is that in the picketfence the width of the picketfence the
dx of the dy/dx cannot be larger than 1*10^-603. So that if a function
has no y value when the x = 0, that would constitute a dx that is
larger than 1*10^-603 and be, in fact 2*10^-603 gap or hole, and thus
discontinuous. So I did learn something new.

Chapter 4 Integral and Integration
Now to start chapter 4 of Integral and Integration, perhaps I should
discuss the antiderivative as a technique of Calculus of easily
knowing what the derivative and integral are. So here is the
Antiderivative technique that all those 700 page standard college
textbooks cover this technique.

(1) for the derivative of a function x^n the derivative is n(x)^(n-1).
So of our three functions, y=3, and y=x
and y=x^2

for y=x we have 1(x)^(1-1) which is x^0 which is 1
for y=x^2 we have 2(x)^(2-1) which is 2x

(2) for the integral the antiderivative works backward. So for x^n,
the antiderivative is

(1/(n+1)(x^(n+1))

for y=x we have (1/(1+1))(x^(1+1) which is 1/2x^2. Now to see if that
is correct we take the derivative of that to see if it lands us back
to x. So we have 2(1/2)x^(2-1) which surely is x.

for y=x^2 we have (1/(2+1))(x^(2+1)) which is 1/3x^3. Here again to
see if we have the correct integral we take the derivative and see if
it lands us back to x^2. So we have (1/3)(3)x^(3-1) and sure enough we
end up with x^2.

Alright, this textbook uses just these three functions to explain all
of Calculus.

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

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.

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, and
especially good if both Newton and Leibniz ever brought up the idea of
a "picketfence" in their writings?


--

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