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4-2 Freshman High School Geometry for age 15// Fundamental Theorem of Calculus

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  4-2  Freshman High School Geometry for age 15// Fundamental Theorem of Calculus Archimedes Plutonium <plutonium.archimedes@gmail.com> - 2018-07-19 23:10 -0700
    Calculus taught to 15 year olds, then to college Freshmen Archimedes Plutonium <plutonium.archimedes@gmail.com> - 2018-07-20 10:10 -0700
      comments on true Calculus Re: Calculus taught to 15 year olds, then to college Freshmen Archimedes Plutonium <plutonium.archimedes@gmail.com> - 2018-07-20 14:25 -0700
        comments on true Calculus Re: Calculus taught to 15 year olds, then to college Freshmen Archimedes Plutonium <plutonium.archimedes@gmail.com> - 2018-07-20 19:18 -0700
          tying together the Calculus theorem and Pythagorean theorem Re: comments on true Calculus Archimedes Plutonium <plutonium.archimedes@gmail.com> - 2018-07-21 10:58 -0700
            Re: tying together the Calculus theorem and Pythagorean theorem Re: comments on true Calculus Archimedes Plutonium <plutonium.archimedes@gmail.com> - 2018-07-21 15:36 -0700
              Re: Archimedes "just not bolted down too tight in the first place" Plutonium flunked the physics lifetime-generation test moroney@world.std.spaamtrap.com (Michael Moroney) - 2018-07-21 23:58 +0000
                Re: Archimedes "just not bolted down too tight in the first place" Plutonium flunked the physics lifetime-generation test "Chris M. Thomasson" <invalid_chris_thomasson@invalid.invalid> - 2018-07-21 21:06 -0700

#418588 — 4-2 Freshman High School Geometry for age 15// Fundamental Theorem of Calculus

4-2  Freshman High School Geometry for age 15// Fundamental Theorem of Calculus

Against my better judgement, I am going to see if I can teach Calculus, back to back with Pythagorean theorem.

PICTURE DIAGRAM OF FUNDAMENTAL THEOREM OF CALCULUS 

The fundamental picture of all of Calculus are these two of a 
trapezoid and rectangle. In fact, call the picture, the 

FUNDAMENTAL THEOREM OF CALCULUS, Picture 

Remember the trapezoid you drew in Lesson 1-8

Trapezoid: (13,15) (17,15) (17,20) (13,16) well let us redraw that. Just to remind us of what a trapezoid looks like.

Now draw a new trapezoid so it is easier to explain this Calculus. This theorem is the essence of Calculus, so we want to understand it easily.

Draw the trapezoid  (0,0) (10,0) (10, 20) (0, 10)

Now look at the coordinate point (5,10) and mark it with a "m" meaning midpoint.

For it is the midpoint of the line segment that goes from (0,10) to (10,20) a slanted line segment.

Draw it in and draw the whole trapezoid.

Now the derivative in Calculus is this rooftop of a slanted line segment that goes from (0,10) to (10,20).

Here is a picture of what you have 

From this: 
        B 
        /| 
      /  | 
 m /----| 
  /      | 
|A      | 
|____| 


The trapezoid roof has to be a straight-line segment (the derivative) 
so that it can be hinged at m, and swiveled down to form rectangle for 
integral. The area of the rectangle is the integral.

To this: 

______ 
|         | 
|         | 
|         | 
--------- 

So, we have two items in Calculus for this theorem, we have a derivative, the straight line segment A to B with m in the middle. And we have the rectangle area which we call the integral.

We draw in m, the midpoint because that is where we have a hinge, we imagine a hinge there. In fact, some teachers will build this model in wood working class just to use in math class.

So, Calculus has two items-- the derivative which is the rooftop, the straightline. And the other item, the integral which is the rectangle area.

So, what is this theorem all about? 

Well, it says that-- if you have a rectangle with a midpoint on its top side.

__m__ 
|         | 
|         | 
|         | 
--------- 

That you can cut a right triangle from the midpoint

__m__ 
|  /      | 
|/        | 
|         | 
--------- 

Cut that right triangle and swivel it up to make the trapezoid

        B 
        /| 
      /  | 
 m /----| 
  /      | 
|A      | 
|____| 

Or, you can start with that trapezoid and swivel the right triangle downwards to make the rectangle


__m__ 
|  /      | 
|/        | 
|         | 
--------- 


And, basically that is the Calculus at its most simple form. Where the slanted line is the derivative and the rectangle area is the integral. So, there, 15 year olds, you have just learned the fundamental basics of Calculus. Take a rectangle, swivel the right triangle and you have a derivative. Take the trapezoid, swivel the right triangle to form a rectangle area and you have the integral.

Basically, that is all what Calculus is.

AP

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#418636 — Calculus taught to 15 year olds, then to college Freshmen

On Friday, July 20, 2018 at 1:10:46 AM UTC-5, Archimedes Plutonium wrote:
> 4-2  Freshman High School Geometry for age 15// Fundamental Theorem of Calculus
> 
> Against my better judgement, I am going to see if I can teach Calculus, back to back with Pythagorean theorem.
> 

Well, just to the opposite, for here is the opportunity to tie together both Pythagoreaan Theorem and Calculus, all in one stroke. In College when these two pictures are retaught.

Pythagorean theorem is this picture in letters since I cannot ascii art them


      n    m
         =
        H

H = hypotenuse square, = is the right triangle, n is the smallest square, m is the second smaller square


The Calculus in ascii art is this:


  ______ 
 |         | 
 |         | 
 |         | 
 -------

goes to 

          
         /| 
       /  | 
     /---| 
   /      | 
 |        | 
 |_____| 

and that goes back to a rectangle


What Calculus is basically is the rectangle is angular momentum, the derivative is linear momentum

The Pythagorean theorem is actually far more complicated and complex than is Calculus for its basic understanding is conservation of laws of physics and being quantized energy-- all of this saved for College Calculus.  The energy in physics is mc^2 dealing with squares, not rectangles, so we have a square as H rather than in Calculus we have the rectangle, and we go from one large square, to two smaller squares.

Calculus is Momentum, Pythagorean theorem is energy conservation.

Now, none of this is going to be taught 15 year olds. They will be taught the basic two pictures. But the aim is, teach 15 year olds the picture so it sticks in their minds so that when they reach college and taught calculus over again, all of that can be easily done, for they have all the preliminaries.


> PICTURE DIAGRAM OF FUNDAMENTAL THEOREM OF CALCULUS 
> 
> The fundamental picture of all of Calculus are these two of a 
> trapezoid and rectangle. In fact, call the picture, the 
> 
> FUNDAMENTAL THEOREM OF CALCULUS, Picture 
> 
> Remember the trapezoid you drew in Lesson 1-8
> 
> Trapezoid: (13,15) (17,15) (17,20) (13,16) well let us redraw that. Just to remind us of what a trapezoid looks like.
> 
> Now draw a new trapezoid so it is easier to explain this Calculus. This theorem is the essence of Calculus, so we want to understand it easily.
> 
> Draw the trapezoid  (0,0) (10,0) (10, 20) (0, 10)
> 
> Now look at the coordinate point (5,10) and mark it with a "m" meaning midpoint.
> 
> For it is the midpoint of the line segment that goes from (0,10) to (10,20) a slanted line segment.
> 
> Draw it in and draw the whole trapezoid.
> 
> Now the derivative in Calculus is this rooftop of a slanted line segment that goes from (0,10) to (10,20).
> 
> Here is a picture of what you have 
> 
> From this: 
>         B 
>         /| 
>       /  | 
>  m /----| 
>   /      | 
> |A      | 
> |____| 
> 
> 
> The trapezoid roof has to be a straight-line segment (the derivative) 
> so that it can be hinged at m, and swiveled down to form rectangle for 
> integral. The area of the rectangle is the integral.
> 
> To this: 
> 
> ______ 
> |         | 
> |         | 
> |         | 
> --------- 
> 
> So, we have two items in Calculus for this theorem, we have a derivative, the straight line segment A to B with m in the middle. And we have the rectangle area which we call the integral.
> 
> We draw in m, the midpoint because that is where we have a hinge, we imagine a hinge there. In fact, some teachers will build this model in wood working class just to use in math class.
> 
> So, Calculus has two items-- the derivative which is the rooftop, the straightline. And the other item, the integral which is the rectangle area.
> 
> So, what is this theorem all about? 
> 
> Well, it says that-- if you have a rectangle with a midpoint on its top side.
> 
> __m__ 
> |         | 
> |         | 
> |         | 
> --------- 
> 
> That you can cut a right triangle from the midpoint
> 
> __m__ 
> |  /      | 
> |/        | 
> |         | 
> --------- 
> 
> Cut that right triangle and swivel it up to make the trapezoid
> 
>         B 
>         /| 
>       /  | 
>  m /----| 
>   /      | 
> |A      | 
> |____| 
> 
> Or, you can start with that trapezoid and swivel the right triangle downwards to make the rectangle
> 
> 
> __m__ 
> |  /      | 
> |/        | 
> |         | 
> --------- 
> 
> 
> And, basically that is the Calculus at its most simple form. Where the slanted line is the derivative and the rectangle area is the integral. So, there, 15 year olds, you have just learned the fundamental basics of Calculus. Take a rectangle, swivel the right triangle and you have a derivative. Take the trapezoid, swivel the right triangle to form a rectangle area and you have the integral.
> 
> Basically, that is all what Calculus is.
> 
> AP

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#418675 — comments on true Calculus Re: Calculus taught to 15 year olds, then to college Freshmen

On Friday, July 20, 2018 at 12:10:22 PM UTC-5, Archimedes Plutonium wrote:
> On Friday, July 20, 2018 at 1:10:46 AM UTC-5, Archimedes Plutonium wrote:
> > 4-2  Freshman High School Geometry for age 15// Fundamental Theorem of Calculus
> > 
> > Against my better judgement, I am going to see if I can teach Calculus, back to back with Pythagorean theorem.
> > 
> 
> Well, just to the opposite, for here is the opportunity to tie together both Pythagoreaan Theorem and Calculus, all in one stroke. In College when these two pictures are retaught.
> 


Now, take a right triangle, any right triangle you want and place the hypotenuse parallel to the x-axis


     _____________> x axis


    A  /_______\B
           hypot.


Now, draw on the square on side A and the square on side B

Draw in the square on the hypotenuse


It is going to look like this when finished
  A    B
    =
    H


Now, does that not look like the trapezoid in Calculus, to some extent?



> Pythagorean theorem is this picture in letters since I cannot ascii art them
> 
> 
>       n    m
>          =
>         H
> 
> H = hypotenuse square, = is the right triangle, n is the smallest square, m is the second smaller square
> 
> 
> The Calculus in ascii art is this:
> 
> 
>   ______ 
>  |         | 
>  |         | 
>  |         | 
>  -------
> 
> goes to 
> 
>           
>          /| 
>        /  | 
>      /---| 
>    /      | 
>  |        | 
>  |_____| 
> 
> and that goes back to a rectangle
> 
> 
> What Calculus is basically is the rectangle is angular momentum, the derivative is linear momentum
> 
> The Pythagorean theorem is actually far more complicated and complex than is Calculus for its basic understanding is conservation of laws of physics and being quantized energy-- all of this saved for College Calculus.  The energy in physics is mc^2 dealing with squares, not rectangles, so we have a square as H rather than in Calculus we have the rectangle, and we go from one large square, to two smaller squares.
> 
> Calculus is Momentum, Pythagorean theorem is energy conservation.
> 
> Now, none of this is going to be taught 15 year olds. They will be taught the basic two pictures. But the aim is, teach 15 year olds the picture so it sticks in their minds so that when they reach college and taught calculus over again, all of that can be easily done, for they have all the preliminaries.
> 
> 
> > PICTURE DIAGRAM OF FUNDAMENTAL THEOREM OF CALCULUS 
> > 
> > The fundamental picture of all of Calculus are these two of a 
> > trapezoid and rectangle. In fact, call the picture, the 
> > 
> > FUNDAMENTAL THEOREM OF CALCULUS, Picture 
> > 
> > Remember the trapezoid you drew in Lesson 1-8
> > 
> > Trapezoid: (13,15) (17,15) (17,20) (13,16) well let us redraw that. Just to remind us of what a trapezoid looks like.
> > 
> > Now draw a new trapezoid so it is easier to explain this Calculus. This theorem is the essence of Calculus, so we want to understand it easily.
> > 
> > Draw the trapezoid  (0,0) (10,0) (10, 20) (0, 10)
> > 
> > Now look at the coordinate point (5,10) and mark it with a "m" meaning midpoint.
> > 
> > For it is the midpoint of the line segment that goes from (0,10) to (10,20) a slanted line segment.
> > 
> > Draw it in and draw the whole trapezoid.
> > 
> > Now the derivative in Calculus is this rooftop of a slanted line segment that goes from (0,10) to (10,20).
> > 
> > Here is a picture of what you have 
> > 
> > From this: 
> >         B 
> >         /| 
> >       /  | 
> >  m /----| 
> >   /      | 
> > |A      | 
> > |____| 
> > 
> > 
> > The trapezoid roof has to be a straight-line segment (the derivative) 
> > so that it can be hinged at m, and swiveled down to form rectangle for 
> > integral. The area of the rectangle is the integral.
> > 
> > To this: 
> > 
> > ______ 
> > |         | 
> > |         | 
> > |         | 
> > --------- 
> > 
> > So, we have two items in Calculus for this theorem, we have a derivative, the straight line segment A to B with m in the middle. And we have the rectangle area which we call the integral.
> > 
> > We draw in m, the midpoint because that is where we have a hinge, we imagine a hinge there. In fact, some teachers will build this model in wood working class just to use in math class.
> > 
> > So, Calculus has two items-- the derivative which is the rooftop, the straightline. And the other item, the integral which is the rectangle area.
> > 
> > So, what is this theorem all about? 
> > 
> > Well, it says that-- if you have a rectangle with a midpoint on its top side.
> > 
> > __m__ 
> > |         | 
> > |         | 
> > |         | 
> > --------- 
> > 
> > That you can cut a right triangle from the midpoint
> > 

For some reason or other, the midpoint plays a key role in the true definition of Calculus. In Old Math they played around much with midpoints, but in true New Math, midpoints are seen as entries from a given Grid system like the 10 Grid into the 100 Grid, so that no matter how small the infinitesimal is in one Grid, the midpoint requires us to go to a higher Grid to insure the midpoint.





> > __m__ 
> > |  /      | 
> > |/        | 
> > |         | 
> > --------- 
> > 
> > Cut that right triangle and swivel it up to make the trapezoid
> > 
> >         B 
> >         /| 
> >       /  | 
> >  m /----| 
> >   /      | 
> > |A      | 
> > |____| 
> > 
> > Or, you can start with that trapezoid and swivel the right triangle downwards to make the rectangle
> > 
> > 
> > __m__ 
> > |  /      | 
> > |/        | 
> > |         | 
> > --------- 
> > 
> > 
> > And, basically that is the Calculus at its most simple form. Where the slanted line is the derivative and the rectangle area is the integral. So, there, 15 year olds, you have just learned the fundamental basics of Calculus. Take a rectangle, swivel the right triangle and you have a derivative. Take the trapezoid, swivel the right triangle to form a rectangle area and you have the integral.
> > 
> > Basically, that is all what Calculus is.
> > 
> > AP

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#418698 — comments on true Calculus Re: Calculus taught to 15 year olds, then to college Freshmen

So what can be made of this that the calculus is a trapezoid rectangle while pythagorean theorem is a Y shaped figure. It is not a inverse relationship, but rather, where one wants direction— calculus derivative the other wants a balanced equality from area.

AP

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#418751 — tying together the Calculus theorem and Pythagorean theorem Re: comments on true Calculus

On Friday, July 20, 2018 at 9:18:34 PM UTC-5, Archimedes Plutonium wrote:
> So what can be made of this that the calculus is a trapezoid rectangle while pythagorean theorem is a Y shaped figure. It is not a inverse relationship, but rather, where one wants direction— calculus derivative the other wants a balanced equality from area.
> 
> 

Alright now, well, the Pythagorean theorem sort of looks like that radioactive decay warning logo



       H       H
            K

            H




But now, let us orient the Pythagorean theorem in such a way that we can relate to the Calculus Trapezoid in the easiest manner possible.


We orient the Pythagorean theorem as such

       S       H
            K

            M

Where the largest square is H, middle large square is M and smallest square S, and the right-triangle is K

Now the Calculus is all about this singular trapezoid that can be fold up or folded down into a rectangle.


 __m__ 
 |  /      | 
 |/        | 
 |         | 
 -------


         B 
         /| 
       /  | 
 m /---| 
   /      | 
 |A      | 
 |_____| 

What I am trying to do here is get a direct link, a connection between the right triangle of Calculus being lifted up onto the rectangle to form the derivative. And then the large hypotenuse square in Pythagorean theorem.

As I said earlier, Calculus is linear momentum as derivative and angular momentum as integral, but the Pythagorean theorem is all about the Conservation of the momentum, whether it be linear or angular momentum. Pythagorean theorem is the conservation of energy.

AP

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#418775 — Re: tying together the Calculus theorem and Pythagorean theorem Re: comments on true Calculus

Alright, a link up, a tie between Pythagorean theorem and FTC, Fundamental theorem of Calculus.

Take out a sheet of graph paper and plot the function Y= x and Y=x^2 from interval 0 to 4.

Now, the right triangle formed in FTC with a midpoint of the rectangle where we hinge the right triangle upwards, for Y = x we have sqrt32= 5.65 but this is a midpoint square which would be 2.82 by 2.82 square with area 8, and the area by integral is also 1/2x^2 for 4 is 8. So we have a direct relationship of area of hypotenuse square to that of the area of integral under the graph-function.

But let me check that out on Y=x^2, just to make sure.

So I will check for integral 1/3x^3 interval 3 to 4
The integral gives me 21.3 - 9 = 12.3
The Pythagorean theorem on the hypotenuse right triangle   7^2 + 1^2 = 50 and sqrt50 = 7.07 and midpoint of that is 3.53 and the square of 3.53 by 3.53 is area 12.5

So that is a sigma error of 12.5/12.3 = 1.01 and thus acceptable as equal.

So, what I think I discovered is that the midpoint of the right triangle forms a square on that hypotenuse which is equal to the area under the function graph in that specific interval.

So this would be a direct tie up, or link between the Pythagorean theorem and the integral of Calculus.

AP

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#418787 — Re: Archimedes "just not bolted down too tight in the first place" Plutonium flunked the physics lifetime-generation test

> AP is on your side-- dear students Re: 6-Trigonometry: Why AP gutted and 
>disemboweled Old Math's evil monster Trigonometry 

Beware, students and teachers of South Dakota! Mr. Plutonium is not 
content to be a Failure of Math and Physics all by himself. He is trying 
to get friendly with you because he wants you to fail as well!

Is he being paid to destroy math by Putin/Russia, or by China?

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#418811 — Re: Archimedes "just not bolted down too tight in the first place" Plutonium flunked the physics lifetime-generation test

On 7/21/2018 4:58 PM, Michael Moroney wrote:
>> AP is on your side-- dear students Re: 6-Trigonometry: Why AP gutted and
>> disemboweled Old Math's evil monster Trigonometry
> 
> Beware, students and teachers of South Dakota! Mr. Plutonium is not
> content to be a Failure of Math and Physics all by himself. He is trying
> to get friendly with you because he wants you to fail as well!

Fwiw, AP has inspired me to create several interesting fractals. Mostly 
with his semi-circle ideas. Another is brewing wrt right triangles and 
the progression of the unsigned integer based variety. The interpolated 
"journey" from a (3, 4, 5) to a (3 * 2, 4 * 2, 5 * 2) triangle can 
contain an interesting infinite fractal in and of itself.

Take all of the triangle within the "journey". We can get as fine 
grained as we want here. Precision? Zoom level? Ahhh. Very fun to work with.

> Is he being paid to destroy math by Putin/Russia, or by China?

Yea right. lol.

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