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"