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[MUSIC]. 

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Remember back to those good old days when 
we were approximating antiderivatives by 

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using Euler's Method? 
I'm going to be started with some 

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function little f and I wanted to 
numerically approximate a function big F, 

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whose derivative was little f. 
And maybe I know big F's value at 0 is 

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exactly 0, and I want to numerically 
approximate big F for values away from 0. 

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And the we did repeated linear 
approximation. 

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I pick some tiny h, and then I 
approximated big F of h. 

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Well, what do I know about big F Big F's 
derivative is little f. 

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So my approximation for big F at h is the 
value of big F at 0 plus h times the 

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derivative of big F at 0. 
Well, what is this? 

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The value of big F at 0 is exactly equal 
to 0. 

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And, I got plus h times, and the 
derivative of big F is little f. 

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So times little f at 0. 
So I can use this as my approximation for 

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big F at h. 
And then I did it again. 

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So, I want to approximate big F at 2h. 
Well that's big F at h approximately. 

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I mean I'm writing equals but I really 
mean approximately. 

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Big F of h plus h times the derivative of 
big F at h, right? 

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I start at F of h and I'm going to wiggle 
over by h and the derivative is encoding 

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at least approximately, how much the 
output should change for a given input 

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change. 
This is what I get by doing another 

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linear approximation. 
But now I've already got an approximation 

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for big f at h. 
It's h times f of zero. 

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So I'll use that for my value of big F at 
h. 

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H times f of 0 plus h times. 
And I know F prime of h. 

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I know big F's derivative is little f. 
So I can use that here. 

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So this is just little f. 
At h, so this is an approximation for big 

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F at 2 h. 
And then I did it a third time. 

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So then this is the method of Euler, 
right? 

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I want to approximate big F at 3 h. 
Well that'll be big F at 2 h, plus how 

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much I wiggled by, which is h times the 
derivative ff big F at 2 h. 

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And what do I know? 
Well, I've already got an approximation 

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for big F at 2 h, it's right here. 
So, it's h times little f of 0 plus h 

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times little f of h plus h times, and now 
what's my derivative of big F at 2 h? 

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Well, big F's derivative is little f. 
So I can use that here. 

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This will be little f at 2 h. 
And I just keep on going. 

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I want to approximate big F at 10 h, 
right. 

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I just be repeating this process. 
It'll be h times f of 0 plus h times f of 

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h. 
Plus h times f of 2h, and it will keep on 

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going until I get to h times f of 9h. 
Now, what does that look like? 

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This looks like a Riemann Sum, right, and 
I would want to choose h to be very 

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small. 
So, really, if I wanted to approximate 

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big F of x using the method of Euler I'd 
be using smaller and smaller values of h 

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and calculating it like this, and what 
would I be calculating? 

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I'd just be calculating the integral from 
0 to x of my function, right, of little f 

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of td, dt. 
And what do I know about accumulation 

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functions? 
Well, I know that the derivative of the 

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accumulation function, right, is the 
original function. 

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And that's exactly what I want, right? 
I mean, this is saying that the 

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derivative of big F is little f. 
So Euler's method amounts to calculating 

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a Riemann's sum. 
And Riemann's sum approximates an 

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integral, the accumulation function, and 
the accumulation function is an 

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antiderivative. 
it all makes sense. 

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Right? 
All of these things that appear different 

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are really the same thing. 
Euler's method then gives another 

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perspective on why the fundamental 
theorem of calculus should be true.