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01:59

Nutan C.

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

0:00

Kim H.

Dungarsinh P.

01:02

Anshu R.

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All right, So we're going to do a circulation integral and recall. Did this really is just a lining? The girl in the closed, simple, positively oriented curve that we're going to be integrating of her is this triangle. So let's plot this triangle. So we have the point 20 and to to and 00 And now notice we have a positive orientations that's going to be counter clockwise. So that will be this way. Another thing to notice. Is that really here? We have four pieces. I'm sorry. Not four pieces. We have three pieces. We have this first kind of segment that will do a line integral over. Then the second piece, then this third piece. And so the total line integral or circulation, Integral is going to be the contribution from this piece. Plus the contribution for this piece, plus the contribution for this piece. Okay, so you recall we know how to do mine and the girls. So let's start with the first piece we need Thio, do the line integral over peace one of Okay, so we take the vector field and do the dot product with the tangent direction? Yes, and so to really actually evaluate this. We want to give a parameter ization of the curve, so we might as well just go ahead and parameter rise these three line segments. And it's actually pretty easy to parameter rise line segments. So here, for one, we're just going from zero to to in the X. So I'm just gonna put two t in the X in zero and the why and let t go from, well, 01 I'm going to do something similar up here. Here. Now the X coordinate is always to and why is going from 0 to 2? So I'll just put two t in the Y direction again. That's for t end 01 And I like toe always pick the interval to be 01 because it's the easiest to work with. Then for three. Now we really got to think about the orientation here. We're starting at the 0.22 so we're gonna take two, and then we're standing at the 20.0 So we're going to subtract to T from each coordinate and then again, let t be in Sierra Toe one. And so I'll leave it to you to verify that this these three victor value functions really do give the parameters ations of these segments. So, you know, his tea goes from 0 to 1. This is tracing out this curve. This is tracing out this curve and this is tracing out description. All right, so now that we have a premature ization, we're ready to actually say what our line and the girl is. It's just the integral. So along this first part, 0 to 1 and then we have F But now we actually want Thio Plug in. What is so f is going to be zero comma to t So we have to t comma. Yes, So the X coordinate is minus. Why? So that's zero. And then the y coordinate is X, which is to t and then got with actually the derivative of that this parameter ization which is just too common Zero duty And now notice. What is this? Well, this is zero plus zero and the in a grand, So this entire integral is actually zero. So that's awesome. That's one piece out of the way. So the second piece is again Just gonna be yes, dotted with the tangential direction. Yes, okay. And so remember that that's actually just f evaluated at the privatization started with the derivative of the parameter ization d t. So again it's you're the one f evaluated at the premature ization here is going to be well minus Why so negative to t and then x which is to and then dot with the derivative of the parameter ization, which is zero common to U T. And so what did we get here? We get zero plus four. So it's just the integral from 0 to 1 of four DT, which is for great. So then the third piece of the curve is going to be f dot t Yes, which is maybe I should write this down in case you're not following. So f dot t Yes is really if I have a parameter ization, it's just f of the parameters ation started with the derivative. So that's what I'm using here and all these examples. Okay, so this is zero the one. I have my premature ization, and I just want to plug it in now both of the x and Y coordinates or the same. So I just want to make one of them negative. So I'll have tu minus two tee times. Negative one is to t minus two and then to minus two t. So that's f evaluated at the privatization and then dot with derivative, which is negative. Two common negative, too. The derivative of the premature ization of this curve duty. But notice what's gonna happen. These have opposite signs. Do you have the same sign? So I'm going to get this. Plus it's negative. So I'm actually just going to get zero here is well, but now I'm done. So the circulation in her girl, remember, I'm gonna put this circle around the integral sign Thio really signify that this curve, if I call it see given by one and two and three, is it closed? Simple, positively oriented curve and the integral is just the some of this integral. What's this? Integral Plus in this integral zero plus four sirrah, which equals four

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