Now earlier today I wrote the below 14.1 and promised to followup with y = 1/x for x values beyond x = 1.
So let me see if y = 1/x obeys the idea that the derivative is two angles of dy/dx.
x=1.9, y = .5 remember there is truncation in 10 Grid x = 2.0, y = .5 x = 2.1, y = .4
Now one can immediately see there are a lot of flat lines joined from two or more cells where the derivative in that cell would be 0.
But the derivative of dy/dx in interval 1.9 to 2.1 would be a dy of .1 and a dx of .2 so we have .1/.2 = .5 as our derivative using length, but now let us use the angles formed by the triangle atop the picketfence in the cell of 2.0 to 2.1. In the cell of 1.9 to 2.0 the derivative is 0 for there is no triangle. But in the cell 2.0 to 2.1 and applying the protractor I get an angle of approximately 30 degrees so the other angle is 60 degrees and we have 30/60 = .5 for derivative matching the length method.
Now following the function y = 1/x for larger values of x, we end up with pretty much the same answer as either 0 derivative or 0.5 derivative.
Now, if we were to superimpose the 100 Grid upon the 10 Grid by accepting that 1/1.9 = .52 and 1/2 = .50 and 1/2.1 = .47, then we sharpen up the derivative answer and get away from having 0 derivatives.
function y =1/x, the dy and dx angles #14.1 Uni-textbook 7th ed.: TRUE CALCULUS; without the phony limit concept
Alright, let us move on to a compilation of straightline segment function such as y = 1/x rather than the overall straightline functions of y = x, y=2x, y= 3x and y = 4x. We want to see if the dy/dx is able to use pure angles rather than length of line segments. I am sure it is true, but we have to verify it such.
Now my graph paper is not ideal of 100 tiny squares by 100 tiny squares for the 10 Grid. Mine is 50 by 35 tiny squares. So I have to make due with what I have and so I mechanically split each tiny square in half on the y-axis and also the x-axis so that I get a cell that is 100 tiny squares. So I have my 100 tiny squares for each cell on the y-axis, but the trouble with this, is I likely lose on accuracy.
So I plot the function y = 1/x and for x =0, remember we impose the infinity number of 10.01, so that x =0 and y = 10.01. But I will use x=.1, x=.2, and x=.3 and forget about the first cell of 0 to .1.
So, I have plotted y = 1/x
x=.1, y = 1/.1 = 10 x=.2, y = 1/.2 = 10/2 = 5 x=.3, y = 1/.3 = 10/3 = 3.3 (remember we are in 10 Grid and the only numbers that exist are 3.3 and not 3.33 so we truncate.)
So now plotting those points on the graph which we shortly will apply the protractor for angles. But first, let us do the derivative by length.
From x interval .1 to .3 is of length .3-.1 = .2 for a dx. Now for a dy we have 10 - 3.3 = 6.7 for a dy and we have thus dy/dx = 6.7/.2 = 33.5 for a derivative. Let us see what the angles of dy/dx yields.
So now, plot the points (.1,10) and (.2,5) and (.3,3.3) and then with ruler connect them. Now remember, in True Calculus there are no curved lines, but only a collection of straightline segments. In the case of y =1/x every cell has its straightline segment and the overall function is this compilation or collection of straightline segments that may begin to look like a "curve" but is not a curve, for curves do not exist in mathematics. When you have a borderline between finite and infinite, this borderline causes there to be no curves in mathematics.
So now, let us slide the protractor up to the point of the graph of (.1,5) and measure that angle and I get between 87 and 88 degrees which corresponds to the other angle as 3 and 2 degrees and we have 87/3 =29 and 88/2 = 44 for a derivative and we had a derivative of 33.5 using the length means of dy/dx. Well, the length means is more accurate than the angle means, for we never had to measure the length but found it from algebra. And I warned the reader that we lose in accuracy with angle measure.
But that is one side of the function y = 1/x, before the function reaches x = 1 and the function behaves better past x =1, and behaves so well that it is the logarithmic defined function once it reaches x = 1. So in my next post, let me check out three cells past x = 1 to see if the dy/dx angles matches the length measure to determine the derivative.