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

Posted: Jul 10, 2013 3:08 PM

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

The first ten pages of this textbook is devoted to High School (HS)
students learning Calculus for the first time. The first ten pages is
so well written that it requires no teacher, simply the interest and
desire of a student to learn true Calculus, not the false and phony
calculus with its limit concept.

Let us begin, and we define a function.

Definition : A function is a graph where
every x value has a unique, one and only one y value.

A line segment or line can be a function because for every x-value,
there is just one and only one y-value.
But a square or rectangle or circle or ellipse drawn on a graph cannot
be a function because for some x-values, they have more than one y-
value.

Now the circle, if cut in half could be a function because for every x-
value, it can have one and only one y-value.

Now we plot three functions in the 10-Grid Coordinate System and using
the 1st quadrant only. Throughout this textbook we plot only in the 1st quadrant with positive values for x and for y.

Definition: 10 Grid system is where we use all the counting numbers from 0 to 10 and where we use only 1/10 decimal fractions. And we thus have 0, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1.0, 1.1, 1.2, . . on and on till we come to 10.0 and the Grid are points of these numbers looking like this:

. . . . .
. . . . .
. . . . .
. . . . .
. . . . .
0 .1 .2 .3 .4 x-axis

We are going to use three functions:

y = 3

y = x

y = 1/x

And the 10 Grid system is where we pretend 10 is infinity borderline
and so all the numbers that exist in 10-Grid are

0, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1.0, 1.1, 1.2, .. on
up to the number 10

Not counting 0, there are 100 numbers along the x-axis, and likewise
there are 100 numbers along the y-axis.

Now get a sheet of graph paper and plot these three functions with me:

y= 3 looks like this:

|
..........................> y=3
|
|
...............................> x-axis

The function y=x looks like this where the diagonal splits the 1st
|
|    /
|  /___

The function y = 1/x looks like this:

.    x   .    .    .    .
.    .    .    .    .    .
.    .    .    .    .    .
.    .    .    .    .    .
.    .    .    .    .    .
.    .   x    .    .    .
.    .    .    .    .    .
???           x
.    .    .    .    .    .
? ?                x
.    .    .    .    .    x
.    .    .    .    .    .
.    .    .    .    .    .
0  .1  .2  .3   .4  .5

The student is going to need graph paper and pencil and erasure so as
to keep on re-using the graph paper. Since ascii art on the sci.math
newsgroup is difficult, the student must follow along by doing the
graphs.

There is a famous pragmatist school of thought that "learning is in
the doing" and this text requires the student to do, along with read.

--
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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