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[MUSIC]. 

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Welcome back to Calculus One and welcome 
to week 12 of our time together. 

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We've just got a few more weeks left 
together in the course. 

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Now where are we right now? 
Well two weeks ago we looked at 

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antidifferentiation, we were undoing the 
derivative. 

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And then last week we had a sudden change 
of pace. 

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We looked at integration, computing the 
areas under curve by taking limits of 

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Riemann sums. 
The material from weeks 10 and weeks 11 

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look totally different. 
But this week in week 12. 

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We'll see that they're actually 
practically the same thing, right? 

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Week 12, this week, is about the 
fundamental theorem of calculus. 

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And the content of that theorem is to 
show that antidifferentiation and 

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integration really amount to the same 
thing. 

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This is the key insight, or at least one 
of the key insights of calculus. 

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You know thousands of years ago, people 
knew about areas from this limited 

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perspective, right? 
They used the, the method of exhaustion 

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in Greek mathematics to compute areas of 
things like circles, by taking a limit. 

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And people knew about slopes, you know, 
before Newton and Leibniz. 

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The trick though here is that calculus, 
by thinking very systematically about the 

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change in functions, in particular the 
change in the accumulation function, it's 

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possible to see that finding an area 
really amounts to finding an 

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antiderivative. 
That's the fundamental theorem of 

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calculus. 
And that's what we're going to learn 

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about this week. 
And not just from one perspective, but 

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we're going to show a bunch of different 
perspectives. 

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To really give you a deep, intuitive 
understanding as to why this statement is 

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true. 
Why antidifferentiation and finding areas 

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under curves integrating are so closely 
related. 

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The other good news is that you've got 
more time to take the midterm. 

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You've got until the end of March to turn 
in your midterm. 

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If you've got any questions at all about 
the midterm feel free to post something 

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in the forum. 
We want to make sure that everybody 

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understands those problems. 
Good luck!