[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!