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[MUSIC].

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

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

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

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

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

Â And the we did repeated linear approximation.

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

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

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

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

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

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

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

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

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

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

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

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

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

Â change. This is what I get by doing another

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

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

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

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

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

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

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

Â So then this is the method of Euler, right?

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

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

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

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

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

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

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

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

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

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

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

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

Â small. So, really, if I wanted to approximate

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

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

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

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

Â functions? Well, I know that the derivative of the

Â accumulation function, right, is the original function.

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

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

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

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

Â antiderivative. it all makes sense.

Â Right? All of these things that appear different

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

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

Â