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[MUSIC].

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

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

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

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

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

And the we did repeated linear approximation.

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

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

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

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

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

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

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

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

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

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

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

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

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

change. This is what I get by doing another

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

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

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

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

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

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

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

So then this is the method of Euler, right?

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

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

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

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

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

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

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

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

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

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

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

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

small. So, really, if I wanted to approximate

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

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

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

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

functions? Well, I know that the derivative of the

accumulation function, right, is the original function.

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

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

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

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

antiderivative. it all makes sense.

Right? All of these things that appear different

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

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