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 Special Calculus

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Romp
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PostSubject: Special Calculus   Sat Jul 20, 2013 9:46 am

Introduction
 
This page is for Dewey, but all brash Fury pirates are welcome here.  If you read through this and make it all the way to the end, announce to me brashly in guild chat that you read the whole thing and punctuate with an exclamation mark.  No, it doesn’t matter if you didn’t understand it entirely on the first reading.  
 
Do words like Calculus and Relativity carry with them the fear and the dread we first felt when facing Jolly Roger or Foulberto Smasho?  It is okay to admit to the existence of the fear.  Accept  and embrace it, count five seconds, and then let it go away.  Let it go away.
 
Special Calculus
 
If you have heard the words Special Relativity and General Relativity you may have assumed that “special” meant “better”.  It does not.  It means “limited” and after a few drinks of rum we could also translate it as meaning  “beginner”.
 
Special Relativity = Beginner’s Relativity
 
We could do the same thing for calculus—if we are drinking coconut rum.
 
Calculus is the collection of magical tricks used to calculate things found both on curves and under curves.  
 
What if we taught the introduction to calculus using only straight lines?  You could acquire the principles and the examples would be incredibly easy.  Let’s do it.
Come with me to my new magic ship that travels 30 knots per hour.  
 
30 k/hr
 
At this maximum speed of thirty, after four hours we have traveled 120 knots.
 

 
Can you see that we have built a box with an area of 120? (4*30=120)
 
Look at the graph with your imagination and see the box corresponding to a trip of three hours.  Do the same for a trip of five hours.  90 knots and 150 knots, right?
Integration is a trick to calculate the area under a curve (for us it was a straight horizontal line).
 
Let’s do an example that is a little more complicated.  Assume we travel for an hour at 30 k/hr.  Assume for the next two hours, repairs are in progress and we travel at 10 k/hr.  Assume that for the fourth hour we resume our velocity of 30 k/hr.
 
In your head you can see that we will travel 80 knots.  You can look at the graph and get the same thing from counting and adding up the squares (and thus computing the area under the function.
 

 
I ask you to look at the same graph again, this time seeing four rectangles.
 
 
 
We are now ready to take a brief look at General Calculus where the curves can be curvy—we won’t do any actual math, we’re just window shopping.
 

 
This last graph shows velocity as a curve.  We can’t do the easy squares as we could for the first two graphs.  The third graph is NOT a part of what we call Special Calculus.
Calculus puts rectangles under the curve to add up “rectangle areas” just as we did for the first two examples. The next graphic is intended to give you the idea that with more rectangles, the area of the rectangles is closer to the area under the curve.




The BIG THING is that as the number of rectangles increases, the error goes down (kind of like adding 9's to 3.99999 gets it closer to 4).

If the number of rectangles is infinite, the error is zero. Calculus gives us an infinite number of rectangles under the curve. And it is magic. Smile 

 

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PostSubject: Re: Special Calculus   Sat Jul 20, 2013 10:23 am

I am...gonna go...cry...in the corner...

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