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Topic: What is a function #1 HS-Textbook 7th ed. : TRUE CALCULUS; without
the phony limit concept

Replies: 12   Last Post: Jul 13, 2013 2:38 AM

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 plutonium.archimedes@gmail.com Posts: 18,572 Registered: 3/31/08
What is the derivative #2 HS-Textbook 7th ed. : TRUE CALCULUS;
without the phony limit concept

Posted: Jul 10, 2013 5:03 PM

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

.    .    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
.     .     .     .     .
.     .     .     .     .     .     .      .      .     .
x
.     .     .      x     .
.     .     .     .     .     .     .      .      .     .

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.

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