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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
picketfence model of Calculus #3 HS-Textbook 7th ed.: TRUE CALCULUS;
without the phony limit concept

Posted: Jul 10, 2013 11:11 PM

Math calls it a trapezoid, but to most people, when seeing it, it looks
like a picketfence. The picketfence model is the best model to
understand the derivative and integral of Calculus.

Here is what a picketfence looks like:

|\
||
||
||

and here is the reverse angle:

/|
||
||
||

The concept or model of picketfence is that we have a slender
rectangle with a triangle sitting atop the rectangle. Now in our 10-
Grid system where we have only 100 number points in the x-axis and y-
axis, making a total of 100x100= 10,000 coordinate points, that the dx
in 10-Grid can be no smaller than 0.1. So the width of all the
picketfences in Calculus in the 10-Grid is 0.1 metric width. That
nonzero width insures every picketfence will have a nonzero area for
the integral.

Now the hypotenuse of that triangle atop the slender rectangle is the
derivative. The hypotenuse is the slope or tangent or rate of change
and connects two successive graph points of the function.

Now sometimes the rectangle has no triangle atop, and when that
happens is when the function is a flat straight line such as y=3. So
as we review the function graph of y=3 we see it is flat or 0 slope.
The dy/dx of y=3 has no change in the y values and thus is 0. In the
case of y=x, the dy always equals the dx and thus we have a slope or
derivative of 1. In the case of y=x, the triangle that sits atop the
rectangle is always a isosceles right triangle of two successive
graph points.
Sometimes the picketfence model has only the triangle and no rectangle
as in the case of step functions for some successive points of the
graph. Step functions are covered in the advanced portion of this
textbook.

The importance of the picketfence model is that it shows us the
derivative and how it relates to the integral. The derivative is the
hypotenuse of the right triangle and it is the actual graph of the
function for those successive points, and the integral is the area of
the picketfence. So in the integral, as we sum all the picketfences
over a interval of the function, we are summing the area of all the individual picketfences in that x-axis interval of the graph of the function. It is an exact area due to the fact that the derivative is connecting two successive points of the graphed function.

So let us break down the picketfence of the function y=x^2 from  0.3
to 0.5 along the x-axis as shown here:

.    .     .     .     .      .      .     x

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

x    x    x     .     .     .     .     .
0   .1   .2    .3   .4    .5   .6   .7

From .3 to .4 we have this picketfence:

;
/ |
|__|

From .4 to .5 we have this picketfence:

/ |
|   |
|__|

So we immediately see that the derivative is the connecting of one
point of the graph to its successor point by a straightline segment
and forms the hypotenuse of the right triangle atop the picketfence
rectangle. And the integral is the area inside each of those
picketfences.

We already begin to see and sense a relationship of derivative to
integral, in that you alter one, you alter the other proportionally.

This altering in proportion becomes the Antiderivative Rule in
Calculus.

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