the derivative #3 Textbook 2nd ed. : TRUE CALCULUS; without the phony limit concept

[Archimedes Plutonium]

the derivative #3 Textbook 2nd ed. : TRUE CALCULUS; without the phony
limit concept

Alright, I will work with these four functions as listed previously:

y = 3
y = x
y = x^2
y = 1/x

The key geometry model for both the derivative and the integral is the
Picket-fence model. It is a slender thin rectangle which atop sits a
triangle. I first learned the picket-fence model from this book in the
middle 1960s.
Life Science Library titled MATHEMATICS by 
David Bergamini, 1963 on
page 109 shows a picture of the picket-fence 
model and calls it
"Picket-Fence Integrals". However, reading that 
caption, the author
thought the triangle portion was bad for calculus 
saying that "..by
making the pickets so thin the tops become 
negligible." Sadly, if
David had read this textbook, the entire 
picket, both the triangle
and rectangle portions are essential for the 
triangle determines the
derivative and the triangle with rectangle 
determine the integral,
all of which happens when the point has a hole 
or gap on both right
and left side.
Then when I went to College at University of Cincinnati in freshman
Calculus I remember the picket-fence model was again used, along with
the textbook by Fisher and Ziebur.
I finally tracked down the history of the picket fence model and it
appears as though Leibniz circa 1675 was the first one to use a
portion of the picket-fence model. And for the first time the text
Calculus by Ellis & Gulick, 1986 comes 
to a rescue in a big way. On
page 115, the authors describe how Newton 
came to the derivative and
more importantly how Leibniz came to the 
derivative in a different
manner. And although picket-fence is not 
mentioned, what is mentioned
is that Leibniz is motivated by small 
triangles and when you think
about it, Leibniz is using the small 
triangle atop the picket fence
but not mentioning the picket fence 
structure. Quoting that passage:
"Motivated by the small triangles that appeared when he attempted to
find tangents to curves (Figure 3.13), he adopted the notation dy/dx
for the derivative. Here dy and dx signified small changes in y and
in 
x, respectively."

Now I am rather sure that Germany has picketfences in the 1600s and
after, but just the triangle portion is enough evidence for me to say
that Leibniz discovered this model.

Now here is a picture of a picketfence

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and here is the reverse angle:

/|
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Now why is this model so key, so important? Because the hypotenuse of
that triangle atop the rectangle is the slope or derivative of a
specific point on the function graph, and since the dx has a minimum
hole or gap of 1*10^-603 on both sides of a x-value that the
hypotenuse serves as the derivative and the hole or gap gives the
rectangle a width that gives it an area and thus creates a integral
about that specific point. Without the hole or gap, the blizzard of
infinite numbers surrounding any point gives no room for the slope,
the dy, to form precise and exact angle, and the infinite points of dx
requires a phony limit which makes the rectangle have no width at all,
just line segments. So in Old Math, they had clutter of infinite
points for dy to suffocate a derivative and they had the limit turn
the rectangles into line segments with no width and thus, no area. And
the picketfence is key in seeing how the derivative is the reverse of
integration or vice versa. Because if you change the hypotenuse angle
you change the area inside the picketfence and if you change the area,
you automatically cause the angle of the hypotenuse to change.

Now to start the derivative, perhaps I should 
discuss the
antiderivative as a technique of Calculus of easily 
knowing what the
derivative and integral are. And it is one of the reasons I picked the
identity function y= x and the box function of y=3 because those two
functions
can determine the derivative and the antiderivative.

So here is the 
Antiderivative technique that many of those 700 page
standard college 
textbooks, such as Strang, such as Ellis & Gulick,
such as Stewart and such as Fisher & Ziebur, cover this technique.

Antiderivative Technique

(1) for the derivative of a function x^n the derivative is
n(x)^(n-1). 
So of our four functions, y=3, and y=x 
and y=x^2 and y=
1/x
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.
for y = 1/x we have the derivative is -x^-2.
Alright, with these functions
y = 3
y = x
y = x^2
y = 1/x
their derivates are respectively (y' denotes derivative)
y' = 0
y' = 1
y' = 2x
y' = -x^-2

Now I need to show how the box function y = 3
and the identity function y = x delivers those rules of derivative and
antiderivative. The identity function easily delivers that technique
for integral in that we note the area is 1/2 of a square.

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 four 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 Integral = 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. And we saw that Ellis & Gulick noted that Leibniz history
of
focusing on small triangles with the dy/dx for derivative.
Now the reason I have the fourth function be y =1/x
is because it is easy to see that the dy/dx values do not match the
derivative value when x=3. A function where the limit value is
incongruent with the actual derivative value as seen by this function
y= 1/x, and I should have realized this before that a log type of
function 1/x would show this incongruity. This function is seen in the
Strang textbook CALCULUS, 1991, page 47. So that in the case of x=2
the slope is valued by the 
limit concept to be -1/3 with a delta as 1
unit of 1 and 3 with a dy/ 
dx as -(2/3)/2 whereas the true slope is
-1/4 at x =2.

So when the limit concept is used we get incongruity with true values.
When the picketfence is used around x=2 where to each side 2 is 2-
(1*10^-603) and 2 +(1*10^-603) that we get the exact value of the
derivative at x=2.

Now I have not covered the integral and integration yet, but the
vision is coming in very clear, that derivative is a hypotenuse of a
triangle that sits atop a rectangle that occupies a hole or gap of
10^-603. The hole provides the derivative plenty of free space and
room to form an angle with its neighbor point to the leftward and
rightward, and the hole provides the rectangle with a width so the
picket fence has internal area. In Old Math, integration involved
limit which was a summing up of line segments of no internal area.


--
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Archimedes Plutonium
http://www.iw.net/~a_plutonium
whole entire Universe is just one big atom
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