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Sketch the indicated curves and surfaces.Curves that represent a constant temperature are called isotherms. The temperature at a point $(x, y)$ of a flat plate is $t\left(^{\circ} \mathrm{C}\right),$ where $t=4 x-y^{2} .$ In two dimensions, draw the isotherms for $t=-4,0,8$

Calculus 3

Chapter 29

Partial Derivatives and Double Integrals

Section 2

Curves and Surfaces in Three Dimensions

Partial Derivatives

Johns Hopkins University

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University of Michigan - Ann Arbor

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Okay, so we're giving a function t of X y, and it's equal to 100 over one plus x squared plus two y squared. And so we want to know the level curves of dysfunction. So we're going to send it equal to zero right now, it's pretty hard to tell, but why don't me rearrange this equation into something we're more familiar with? So let's raise both sides to the negative first power, and then, oops, it should be raised to the negative first. Yeah, right. And so now let's multiply through by 100 on each side and rewrite, All right, And then let's just get are very opposable. So I'm going to subtract one to the side and this entire fractions over Aziz. So that's the same thing as 100 minus Z Rizzi. All right. And so the reason we're able, by the way to rearrange this without any worry of changing the graph is there are no discontinuities anywhere on this crash and our variables or strictly contributing positive values to the denominator and the denominators, never zero. So you can be pretty confident that this won't change our graph, right? And so it's just me. Corrects white clean. All right. And so what is the largest? What is this? Actually, why did we rearrange like this? I'll tell you what this is a radius this on this site you should be familiar with as some type of parable. They're not problems in Ellipse where an ellipse has the equation. A X squared once me Why squared is equal. Do some constancy square. Well, we already send that Z is fixed, So this is a constant and the square root of it is going to be not necessarily our radius, but it's just a constant all right. And so another thing to notice is our X is going to extend How much further than our way. Let's assume that why is your oh so X squared equals? Let's just call this some constancy. See, let's leave one to distinguish from that formula. So X is going to extend between negative and positive. See, you want why, on the other again, we'll extend between negative and positive. She 1/2 squared of C 1/2 because we have to divide through by two before we can take the square root. All right. And so When is this going to look like? Well, what's the biggest that this radius can get? You can actually get fairly large if we take ze to be some extremely small number for practical for practical purposes. Let's just see that Z is equal to want. Come on. So, what do we get that? So what mean get is 100 minus one? So that's 99 over one. All right. And so we already established that are X is going to run between the square root of this positive and negative. So X is going to go from here. The Here, This is about 10 just a little longer. 10. So let's call this negative 10. It's called this positive 10. And our why is going the running between, uh, 99 over to they're sorry, the square root of 99 over to which is going to look something like the following spirit, too. So what is this? When the inside this is equal to, we're plus or minus off 49 point site. So you're going to get something very, very close to seven, but just a little bit about negative. And so our curve here looks like in a lip that extends beyond seven and and just before negative, the poor drawing. All right, what else? Them. Let's take another value of Scene Z is equal to, Ah four. So you'll have 96 divided by four. That's too or 24. And so we know that are X is going to run. Our X is going to run between square 24. Plus, they've been negative. Marwan is going to run between the square root 12 positive and negative. This is about five. That is, adult somewhere between three and four. Like we around 3.3. Maybe. Would you see? Actually, likely around a little over. So let's call it three points were just guests to meeting because we want to get a an idea of the group. Three of three years. Negative three. Let me out. Well, you five. So we're going to end a little bit or in a little bit over little bit for a little bit over. All right, so we've seen that all of them were Hey, so thermic, um, regions are forever. The lips used where the ex extends approximately square root to greater than the wine. And so we're done

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