
picketfence model of Calculus #3 HSTextbook 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 xaxis and y axis, making a total of 100x100= 10,000 coordinate points, that the dx in 10Grid can be no smaller than 0.1. So the width of all the picketfences in Calculus in the 10Grid 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 xaxis 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 xaxis 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 electrondotcloud are galaxies

