What is the derivative #2 HS-Textbook 7th ed. : TRUE CALCULUS; without the phony limit concept
Now we do not waste any time here. In one page we learn what a function is, in the second we learn the derivative of calculus. Now I need the student to draw more graphs of functions.
In the previous page we learned what a function is. It is where every x-value has a unique y-value and for that reason a circle cannot be a function but a half circle or quarter circle can be a function. Now notice something peculiar about the function y = 1/x, in that it is a function because every x-value has a unique y-value but strange in that for x=0, y is undefined since division by zero is undefined. So another feature of functions is that every x-value is taken into account, but sometimes the y-value of a function is undefined or does not exist. Now in the university text, we do something special to make division by zero possible and thus all functions can be rendered continuous, but in this High School text, we leave it undefined.
We find the derivative of these three functions:
y = 3
y = x
y = x^2
The derivative for the three points of the x-axis interval of 0 to 0.1 to 0.2 for y= 3 looks like this:
| . . . > y=3 | | . . . > x-axis 0 .1 .2
For the derivative we learn something called dy/dx and let me explain what dy divided by dx means. The dy means the change of y as per the change in x of an interval of x. So in the function of y = 3, the change in x from the interval [0,0.1] is a change of 0.1 and the change in x from the interval [0.1,0.2] is another change of 0.1; and the change in x from the interval [0, 0.2] is a change of 0.2. The change in y, or dy as it is called has remained the same as always being 0 change since all the y values are just 3. So that 3-3 is 0. So the dy/dx for the function y=3 will always be a 0 for the dy and no matter what the dx value is, when dividing into 0 ends up being 0. So the derivative of y=3 is 0 derivative.
Now the derivative has physical names in science and is called "rate of change" or called the "slope" or called the "speed". So what is the rate of change, well it is 0.
Now we connect those y value points with straight line segments as the derivative is the graph of the function itself and looks like this:
| ______ > y=3 | | . . . > x-axis
The function y=x looks like this where the diagonal splits the 1st quadrant in half:
. . d
. d .
d . . > x-axis 0 .1 .2
The derivative, dy/dx is easy and simple for y = x, for as the first interval of x value is [0,0.1] is a change of 0.1, the y value change in that same x value interval is 0.1 so that the dy/dx is identical for dx as with dy and you have the derivative equal to 1. But the derivative does not just give us a value of the rate of change as dy/dx, but the derivative is part of the graph of the function itself so that it connects the points and their empty space between points to produce the graph of the function y=x as this picture shows:
| | / | /___
The function y = x^2 looks like this, since it rises so much faster than y = x :
| / | / |/ ____
And plotting the function y= x^2 looks like this: . . . . . . . . . . . . . . .
x x x . . . . . . . . . . . . 0 .1 .2 .3 .4 .5 .6 .7 .8 .9 1.0 1.1 1.2
The entire graph of the function is not a straight line whereas the entire graph of the functions y=3 and y=x are overall straight lines.
Now as we connect successive points of the graph of y = x^2 those successive points are the slope of the graph, or the dy/dx. So the overall graph is not a straight line but composed of successive straight line segments and each of these segments is the derivative. In Calculus, True Calculus, there is no curved line in Euclidean Geometry. The half circle or quarter circle does not exist as a curved arc but as a compilation of small straight lines. For as we go from a 10 Grid to a 100 Grid to a 1000 Grid and higher systems of plotted points those connecting line segments are smaller and smaller and gives us the appearance of being a smooth curve, when in reality there are no curves in Euclidean Geometry or in Calculus.
Now let us focus on y=x^2 of the point x=0.3 and so y =.3*.3 = 0.09 and then x=.4 we have y=0.16 and x= .2 we have y=0.04. So what is the dy/dx in the interval x goes from .2 to .3 to .4? The dx is 0.2 and the dy goes from 0.16 to 0.04 and subtracting we have 0.16-0.04 = 0.12 and now dividing by 0.2 leaves us a dy/dx as 0.12/0.2 = 0.6.
So what have we learned of the derivative? The derivative is the connecting of one coordinate point of the graph of the function to its successive coordinate point. In y=x^2 we connect the point (.2, .04) with (.3, .09), and with (.3,.09) the next successive point (.4, .16) we connect those three coordinate points with two straight line segments, each being the derivative in that interval between two successive point coordinates. Call the derivative the slope or rate of change from one coordinate point to the successive coordinate point.
In the next page, I shall explain this straight line segment as the derivative as the hypotenuse of a triangle that sits atop a picket fence.