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00:59

Andy S.

If $f(x)=x+\sqrt{2-x}$ and $g(u)=u+\sqrt{2-u},$ is it true that $f=g ?$

00:56

Felicia S.

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So then we spend a little bit of time learning about functions of more than one variable. Things like continuity limits partial derivatives. Now we want to start understanding absolute extreme in local extreme, just like we did with functions of one variable. So recalled that if we had a function of one variable, we were typically interested in things like this. So we had the graph of our function and maybe it represented a cost function. Maybe it represented the position function of a particle, something else. But it looks something like this and what we identified is points where the function had local extreme, and that was related to the function having horizontal tensions. We were also able to find absolute extreme. And that was one of the central questions of optimization of functions of one variable. So the same thing is going to be true for functions of more than one variable. We're gonna develop methods using what we've developed here, partial derivatives. We're gonna define something related to the tangent line here, having a horizontal tangent. It's gonna be the tangent plane for functions of two variables and then higher generalizations of that. But the idea is that even though these functions are somehow a little bit more complicated, we can still ask the same sorts of questions. What's the highest point on the function? What's the lowest point on the function on the graph of the function where their local extreme on the function and etcetera Yeah. So what we're going to dio is we're going to look at specific functions, try to find local extreme things that look like hills or valleys in the function, and then look for things like absolute extreme. What are the highest valleys or what are the highest mountains? What are the lowest valleys in our function? And so these they're gonna have a lot of really nice applications. So we're gonna look at things like, How do you build boxes that have kind of an ideal or optimal shape to maximize volume or minimize surface area things like that, Andi, they're gonna be a lot of other applications as well. There are a lot of applications just like you saw, with functions of one variable optimizing functions and more than one variable to say economics to engineering to physics. Just a lot of widespread applications of multi variable optimization, and we can really think of multi variable optimization as just being kind of a blanket generalization of some of the baby cases we saw with optimizing functions of one variable.

Multiple Integrals

Vector Calculus

29:31

04:58

04:59

03:29

05:42

33:18

01:38

06:52

09:20

06:47

07:55

04:57

06:55

08:13