[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.