[MUSIC]. 
Welcome back to Calculus One and welcome 
to week 12 of our time together. 
We've just got a few more weeks left 
together in the course. 
Now where are we right now? 
Well two weeks ago we looked at 
antidifferentiation, we were undoing the 
derivative. 
And then last week we had a sudden change 
of pace. 
We looked at integration, computing the 
areas under curve by taking limits of 
Riemann sums. 
The material from weeks 10 and weeks 11 
look totally different. 
But this week in week 12. 
We'll see that they're actually 
practically the same thing, right? 
Week 12, this week, is about the 
fundamental theorem of calculus. 
And the content of that theorem is to 
show that antidifferentiation and 
integration really amount to the same 
thing. 
This is the key insight, or at least one 
of the key insights of calculus. 
You know thousands of years ago, people 
knew about areas from this limited 
perspective, right? 
They used the, the method of exhaustion 
in Greek mathematics to compute areas of 
things like circles, by taking a limit. 
And people knew about slopes, you know, 
before Newton and Leibniz. 
The trick though here is that calculus, 
by thinking very systematically about the 
change in functions, in particular the 
change in the accumulation function, it's 
possible to see that finding an area 
really amounts to finding an 
antiderivative. 
That's the fundamental theorem of 
calculus. 
And that's what we're going to learn 
about this week. 
And not just from one perspective, but 
we're going to show a bunch of different 
perspectives. 
To really give you a deep, intuitive 
understanding as to why this statement is 
true. 
Why antidifferentiation and finding areas 
under curves integrating are so closely 
related. 
The other good news is that you've got 
more time to take the midterm. 
You've got until the end of March to turn 
in your midterm. 
If you've got any questions at all about 
the midterm feel free to post something 
in the forum. 
We want to make sure that everybody 
understands those problems. 
Good luck!