importance of fixed angles in Grid that creates the Calculus #5 HS-Textbook 7th ed.: TRUE CALCULUS; without the phony limit concept
Alright, I decided that this is the ripe moment to discuss how the Cartesian Coordinate System creates a grid or network of points of numbers and this network creates fixed angles for which the Calculus can then arise and be formed and come into existence. I am going to need a special function to demonstrate this importance of the fixed angle to form Calculus. And that special function is a sawtooth function in 10-Grid, 1st quadrant only. In math we often use F(x) to mean y, so that some prefer to write y and some prefer to write F(x). I prefer to write F(x) whenever the function is broken into parts and assembled as bits and pieces. When a function is able to be written as a whole in one equation such as y = x or y = x^2 or y = cos(x), then I use y, but when the function has different instructions for different values of x, I call it a broken function and use F(x) rather than y. Calculus texts often never make any distinction between a whole function and a cobbled together function called broken function. In True Calculus, anything that requires logic and precision, requires our attention.
So this special sawtooth function is written like this:
F(x) = 0 when x is even number and F(x) = 10 when x is odd number.
So the graph of this Sawtooth function in 10-Grid, 1st quadrant only, looks like this:
(Strange side note in that it is funny how even and odd in mathematics is relative to the Grid system and finite to infinity border, because 1 is thought to be odd, whereas in 10-Grid, it is even. So beware that odd and even relationship is relative to a Grid.)
So, now we connect those x-points forming a steep sawtooth looking function. The line that connects is the derivative of the function at that x value. It alternates up and down looking like this /\/\/\/\. The picketfences for the function involves no rectangle but just the triangles. The function is continuous everywhere and differentiable everywhere and integrable everywhere. In True Calculus, we have no "pathological functions" such as y = sin(1/x) or the Weierstrass function. In True Calculus those would be continuous, differentiable and integrable everywhere, except for discontinuous points at x=0 where division by 0 is undefined. In the Uni-textbook we get rid of all discontinuities of functions so that the function y = 1/x is continuous everywhere, differentiable everywhere and integrable everywhere, but here we keep it simple for the High School text.
Now this sawtooth function tells us the importance of the Coordinate System providing fixed angles to all its points, because when we alter the function a small bit by this function:
F(x) = 0 when x is even and F(x) = 9.9 when x is odd, we have another sawtooth function but with a different y value of 9.9 rather than 10 and thus a different angle. The different angle means the derivative is different and integral is different for these two functions.
In fact, if we were to draw the entire 10-Grid system of 1st quadrant only of all 100 times 100 = 10,000 points and were to select at random any two points of those 10,000 and those two points were points of a function, then those two points form a unique angle that allows for another function to be a derivative and another function to be an integral.
So in mathematics, we have functions that require a unique y value for a given x value, and, we have a Grid system that requires a unique fixed angle given any two points of a function. So, this is important, for the Calculus would not exist if it were not for "fixed angles". Calculus is a mix, a blend of numbers with geometry, so that numbers are fixed points relative to other numbers and this fixed points yields fixed angles.
The coordinate points of (0,0), (1,1), (1.4, 1.4), (5.9,5.9) when plotted always produce a fixed angle of 45 degrees because they are points of the identity function y= x. I have never seen a Calculus textbook ever discuss the importance of coordinate points causing there to be unique fixed angles.
One of the very best ways to learn subject, as I have found out, is by writing a book on the subject. This is a pragmatist saying that "learning is in the doing". We can learn by having teachers lecture to us, or learn by reading a textbook, but better yet is that when writing that textbook you learn so much more than reading it because it involves "doing" which requires more thought and more research. The organizing and order of the book demands us to think about what we are writing and demands some research.