[MUSIC]. 
Welcome back to Calculus One, and Welcome 
to Week 13 of our time together. 
Well, we're almost to the end of the 
course. 
Soon we will be posting the final exam, 
which will give you a few weeks to work 
on that. 
But I wanted to say a little bit about 
where we are. 
Last week we looked at how 
anti-differentiation gives us the 
fundamental theorem of calculus, which 
lets us evaluate integrals way more 
easily than by going back to the 
definition in terms of Riemann sums. 
This week we continue on that quest, 
we're going to look at u-substitution 
which is one really great technique for 
actually finding some complicated 
antiderivatives. 
Now, in light of this, I also want to say 
a little about how the course as a whole 
has been organized. 
Remember way back at the beginning of the 
course, we gave the definition of limit 
in terms of epsilons and deltas, but very 
quickly, we abandoned those epsilons and 
deltas, and we gave a bunch of techniques 
for evaluating limits. 
Things like the the limit of a sum is the 
sum of the limits that the limits exist. 
And then, once we had easier techniques 
for playing around with limits, we get 
some applications of limits. 
A big one, being the limit of a 
difference quotient, which was the 
definition of the derivative. 
But we didn't usually calculate the 
derivative by going back to that 
definition. 
Usually we used our techniques of 
differentiation like, the derivative is 
of a sum is the sum of derivatives to 
make calculating derivatives way easier. 
And then we did some applications of 
differentiations, right, things like 
optimization problems, related rates 
problems, and we then did definition of 
integration, right, in terms of Riemann 
sums. 
But those Riemann sums were way to 
complicated, so we very quickly abandoned 
Riemann sums for a, a, better technique 
Involving the fundamental theorem of 
calculus. 
In this week, with use substitution, we 
learned another technique for finding 
anti-derivatives, that is another 
technique for evaluating definite 
integrals. 
And in the last couple weeks, we're 
going to look at some more techniques and 
some applications of integration. 
There's not much time left, hang in 
there. 
We're all going to make it to the end 
together.