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

Replies: 1   Last Post: May 23, 2013 7:40 AM

 Messages: [ Previous | Next ]
 Inverse 18 Mathematics Posts: 175 Registered: 7/23/10
Re: the derivative #3 Textbook 2nd ed. : TRUE CALCULUS; without the
phony limit concept

Posted: May 23, 2013 7:40 AM

On May 23,2013, Archimedes Plutonium wrote:
> 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
>
>
>
> |\
>
> ||
>
> ||
>
> ||
>
>
>
> and here is the reverse angle:
>
>
>
> /|
>
> ||
>
> ||
>
> ||
>
>
>
> 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.
>
>
>
>
>
> --
>
> 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

The only phony concept is your so-called "true calculus"

Date Subject Author
5/23/13 plutonium.archimedes@gmail.com
5/23/13 Inverse 18 Mathematics