Time for a summary here of both High-School and Uni texts. The cell is the concept that replaces the phony limit in Old Math. The Cell is what makes True Calculus because the cell gives the integral internal area, whereas the limit makes the integral the adding up of line segments that lack internal area.
The Cell should be the most recognizable figure in all of Calculus. The largest cell is in the smallest Grid System of the 10 Grid. By largest, I mean the x interval is microinfinity of 0.1. The next largest Grid System is the 100 Grid and its microinfinity is 0.01.
The concept of cell was defined and in the 10 Grid the very first cell is this:
10 | | 9.9| | 9.8| |
break in drawing as I leave the reader to fill in the other points from 1.3 to 9.7
The first cell in 10 Grid has a base width of from 0 to 0.1 of .1 metric distance. It is composed of only two finite points of 0 and 0.1 and the rest is empty space in between which comprises infinity number points such as 0.006 or 0.02 or 0.00888 etc etc.
Now, what allows Calculus to exist is because a function always has a left wall of the cell point and then a right wall of the cell point such as in the identity function y = x. , so that for x=0, y = 0 and for x=0.1, y =0.1 so that as we graph that into that cell above we have the 45 degree right triangle:
The hypotenuse is the derivative. No limit is needed to find it because it is part of the function graph itself and is always a straight line segment. The integral is the area of the triangle, of that 45 degree right triangle.
So the integral always has internal area, unless the function is y=0.
Now the integral can be formed of a line segment such as y = 0, or it can be formed of a triangle such as y= x in its first cell or of the sawtooth function has a triangle in all of its cells. The integral can also be a quadrilateral figure or a trapezoid or what I like to call a picketfence figure which has 4 sides, two of the walls of the cell and the floor of the cell and the derivative line segment of the cell.
So, True Calculus is all based upon the fact that finite versus infinity causes there to be a borderline between the two. That means there is a largest finite number and a smallest nonzero finite number, which are called macroinfinity and microinfinity. The microinfinity is the x interval of the cell, and the macroinfinity is the upper reaches of the cell wall. Within the cell there are only points that are increments of the microinfinity so for 10 Grid, we have 0 then 0.1 then 0.2 then 0.3 on up to 9.8 then 9.9 then 10 and those are the only points in existence in 10 Grid. Those are the only points allowed to be used and all graphs connect points with straight line segments. The empty space is all filled with infinity numbers which are not allowed to be used for mathematics is only about finite numbers as precision mathematics.
The Cell is the only means in which the integral has internal area and thus makes the limit concept a phony and irrelevant mechanism.