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00:56

Greninjack Dan

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

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Simon Exley

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Felicia Sanders

00:38

Amy Jiang

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So we have spent the last couple of topics mostly talking about differentiation. So first of all, we talked about how we can talk about partial derivatives, generalizing the notion of the derivative of a function of one variable. We talked about applications of differentiation, how to find local extreme, a absolute extreme A using the Grady int which is basically the derivative finding those critical points, etcetera. Now we want to switch gears and generalize something else that we talked about in Calico one and I'll remind you of. And that's finding the area under the curve of a function of one variable. So if we look at our function of one variable, what we can dio is we can say okay, from a point A to a point B, what is the area under the graph of s. And now this has a lot of really nice applications. Recall. There's the net change serum that allows us to kind of if we know the velocity. This is just a physical application of, you know, the velocity. We can integrate the velocity to get the net change in position, things like that. But then, of course, it also just has this nice geometric application of, you know, we're just finding the area underneath this this curve so we can find the area of some interesting regions. So we wanted again. Cal three really is an extension of Cal Quantum. More than one variable. So we wanna look at things like functions of two variables in the natural question would be Okay, Well, we've seen that the graph of a function of two variables is actually a surface. And so if we look it a surface like this, well, it doesn't make sense to look at the area underneath the surface. It makes sense to look it the volume underneath the surface. So how does this relate to for a function of one variable doing something like this? We know that we can look for the definite integral, which is the area, and that's related to finding an anti derivative of s. And so we'll explore that. So what does this really mean here? Well, we're going to start talking about multiple, integral And so, for a function of two variables, we can look at what's called a double integral of the function, maybe with respect to X and y and will first explore what this means. But the idea is just okay. Instead of integrating over one dimension just over a segment, finding the area, I can integrate over something like a square so I can look in the X direction and in the Y direction and add up bunch of rectangles. Do something similar to a definite integral, like we didn't count wine. Get the volume underneath the surface of a function of two variables, and this idea is gonna have widespread applications. So first of all, of course, we can just find volume. But we can also look at things like mass and moments of inertia, applications to physics, etcetera, etcetera, etcetera. So this idea of multiple integration are basically integrating functions of more than one variable. It's gonna be very analogous to taking partial derivatives, but it's going to have a lot of applications, just like we saw with multi variable optimization. It was a lot of the same ideas, but sort of applied to more general settings

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