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0:00

Lowie T.

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

01:59

Nutan C.

Felicia S.

00:51

Heather Z.

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Okay, so the first example we're just going to get our hands dirty and do a couple of premature is ations of a few pretty common surfaces that you may see in space. So the first one is the sphere, and now I'm going to cheat a little bit and say that well, we really actually have already parameter rised this fear when we talked about spherical coordinates. It's a spherical coordinates, more or less give a parameter ization of this sphere in terms of Thea Angel's. So the angle fee and the angles data. So for the sphere of what we can do is we can actually just use spherical coordinates. And so now this is a sphere of radius one. So that's just gonna make row equal one. But what is exe in spherical coordinates? What's row? Which is one times sign of feet and then Cassina theta. Okay, so I guess really here. If I want my parameter station to be in terms of S and T, I need to call one of these variables s and one of them T. So that was just a reminder. We're really thinking about feeding data, so let's just say that we have signed s and a sign of tea And then for why we have sign of s or sign of fee, really and then sign of tea and then for Z, we just have co sign of Yes, like that. So that's the premature ization. Now we need to give the range. So where did these esti points live? This is what's really important. So they live in a rectangle. So for s this is really like the fee variable. So that's going from zero to pi and then for t that's like data that's going from 0 to 2 by and so you see, for as S and T vary in this rectangle, we're really just getting points on this fear. Okay, so the next one is the side of a cylinder, A radius one and height one. So this looks something like this. We're gonna kind of cheat again. I mean, it's not really cheating. We're just using what we know and think about. Well, what is a cylinder and two dimensions? It's really just a circle. So in the X and Y direction, I can just think, Well, I'm living on a circle of radius one. And so if I put that in, what does that mean? Well, the X component I could just say is co sign of tea and notice that it doesn't matter which parameter I put where you know, I could have used s for the angle, Uh, kind of along the circle, But I'm just using t It doesn't matter. And so why? I want to be sign t. Okay, so x and y, that's just, you know, the circle dramatized. And then what about for the Z component? Well, the Z component is just increasing constantly from zero toe one because it has hight one. So I'm just gonna put us here, come back, and here s t so s go say from 0 to 1 because I'm just thinking about s Is the height going up along the cylinder? So this will be zero the one and then crossed with Well, im going all the way around the circle of radius one so and 0 to 2 pi. Okay, cool. So now the last one, we have a cone, and this is a little bit trickier, I'll say, but a cone is sort of like a cylinder and I want to think about. You know, if this is the Z direction increasing like this, the cross sections of the cone really are just circles. But now the radius of the circle is going to depend on sort of the Z coordinate or the height. So if I Let s be the height. So you see, like if this is s equals one at the top and s equals zero at the bottom, When s zero? The radius is zero, and when s is one The radius is one. And then I'm just increasing kind of in 1 to 1 correspondence between. So my premature ization, instead of just being ko 70 70 is going to be s times because anti and s time 70 and then just s for the Z component. So you see that what's gonna happen is s goes to zero. The radius of the circle is going to get smaller. But in a 1 to 1 correspondence, I mean, if the radius in hyper different, I would kind of have toe account for how the radius changed is a function of the height of the cylinder. But this is what I have and this is again for S and T in the same domain s is just going from 0 to 1 and then T is going from 0 to 2 pi. Okay, so that's just three examples of commonly parameter rise surfaces in space.

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