Fundamental Theorem of Calculus #8 HS-Textbook 7th ed.: TRUE CALCULUS; without the phony limit concept
Alright, these ten pages of this Calculus textbook is designed for the High School student to learn True Calculus, and the second textbook, the Uni-Textbook, included with the HS textbook is for the college and university student. So if in High School you need to learn just 10 pages, and if in college, you need to learn the 90 pages, but the student can refer to both.
Now the last 3 pages of the 10 pages will be a breeze to the High School student because it is more of a lecture on what the student has worked with on the derivative and integral. There are a few pragmatic exercises in these last 3 pages, but the bulk of exercises has already been done by the student. Remember, learning is in the doing, and if you only read and not graph, well, you do not learn as much.
In these last 3 pages of the 10 pages, we summarize and put it all together into a coherent and consistent mathematics. Page 8 is the Fundamental Theorem of Calculus; page 9 is microinfinity versus macroinfinity and page 10 is the idea that the Maxwell Equations of Physics produces all of mathematics.
This is page 8 and is the Fundamental Theorem of Calculus. This is New Math and not the old phony math with their phony concept of limit. So in New Math, the Fundamental Theorem of Calculus is far different. In Old Math, they had a Fundamental Theorem that spoke of the idea that the derivative was an inverse to an integral. But they never specified what it means to be an "inverse". Does it mean something like 1/2 is the multiplicative inverse of 2 giving 1/2*2 = 1 or is it a additive inverse such as 2 -2 = 0 or some other inverse notion? In the Uni-textbook, we dispel of the inverse and find out it is a concept of "dual".
Now in this textbook, we can easily see the derivative is related to the integral because whatever angle the dy/dx of the derivative creates, affects the area under the picketfence. If you change or alter the angle, the dy/dx, you change or alter the area. So we learned they are related. But how are they related?
Now here we have the picture of the picketfence again:
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And the hypotenuse of the triangle atop the rectangle is the derivative and we can easily see if we change that hypotenuse to a different angle, it makes the area inside the picketfence change also. And likewise if we add more area or take away area inside the picketfence, it affects the hypotenuse with a different angle. So the Fundamental Theorem of Calculus has to address the idea that the derivative is related to integral.
In New Math, the Fundamental Theorem is stated as such:
Fundamental Theorem of Calculus: derivative is the 4th side of a picketfence or the 4th side of a rectangle or the 3rd side of a triangle in a cell, and this picketfence or rectangle or triangle area is the integral.
The proof is almost easier than the statement. For the proof involves the Grid System and the connecting of successive points with a straightline segment. So we have in 10 Grid, exactly 100 cells. We have successive points of the function so that 0 goes to 0.1 and 0.1 goes to 0.2 etc. So in a cell, say the 0.1 to 0.2 cell the width of the cell is fixed from 0.1 to 0.2. The two columns or the height of the cell is any number from 0 to 10 or 100 points in all in each column. The function specifies what point to select on the 0.1 column and what point to select on the 0.2 column. Thence we connect those two points with a straightline segment and that is our derivative, and it is a vector with an arrow of direction. Immediately after the derivative is created, it closes in the cell of its width and two columns forming either a picketfence, a rectangle or a triangle. And that is the proof! The function creates a connecting line segment as the derivative and it encloses a width and columns to be a picketfence, or rectangle or triangle which contain area and is the integral.