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Gives a function $f(x, y)$ and a positive number $\epsilon$ In each exercise, show that there exists a $\delta > 0$ such that for all $(x, y)$$$\sqrt{x^{2}+y^{2}} < \delta \Rightarrow|f(x, y)-f(0,0)| < \epsilon$$$$f(x, y)=\frac{x^{3}+y^{4}}{x^{2}+y^{2}} \text { and } f(0,0)=0, \quad \epsilon=0.02$$

$\delta=0.02$

Calculus 3

Chapter 14

Partial Derivatives

Section 2

Limits and Continuity in Higher Dimensions

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University of Nottingham

Idaho State University

Lectures

12:15

In calculus, partial derivatives are derivatives of a function with respect to one or more of its arguments, where the other arguments are treated as constants. Partial derivatives contrast with total derivatives, which are derivatives of the total function with respect to all of its arguments.

04:14

06:17

Gives a function $f(x, y)$…

Oh Marie in the act of X Y is equal to X Q plus wide, with fourth or Ex crew plus y squared. The EPA, 0000 maps on is equal to 0.2 and we want to show that there exists it also greater than zero such that for all x y we have that the square root of X squared plus y squared lesson Delta implies the F of X y minus A 00 is less than Absalon. So arm angle here is to find the value of our delta here when we're given this right handed quick equation. So my goal is to just simplify this part out and get to relate to this X Square plus y square so I can figure out what my delta needs to be. So first was playing into the right hand side. I get absolute value of X cubed plus y to the fourth over X squared plus y squared, and soon you'll find this out. This bottom is always every positive, since these are both square. So I just have execute Plus why the forest We're X squared plus y squared and then separating out this into two different factions would have x cubed over X squared plus y squared. And again, this is triangle inequality since me less than equal to go. So I was really just going to be x squared over X squared plus y squared times absolutely of x plus for the same thing on the other side. I'm gonna go ahead and separate my wife forthe two to y square. Someone have y squared over expert plus y squared times y squared. And my goal is I need to have this lesson up song, which is their 0.2 So first, I'm just gonna look at these two portions right here and try to relate that to this x squared plus y squared square root. And then my second stuff would just be to show that both of these coefficients are less than equal to one. So first looking at my wife squared. If absolute value of y is listening, good one, then that means that why squared? It's going to be listening to the absolute value of why which is equal to the square root of y squared, which, as we know, is less than equal to X squared plus y squared and then looking at that's a value of X. I again know that that is equal to up to the square root of X squared. She getting this less than equal to square root of X squared plus y squared. So therefore, that means that my ex I'm survive X plus Why squared again? Just dealing with that portion first is going to be less than equal to both of these are together. So it's you times X grade plus y squared, which again I want this Are you back? So now the next part is I want to show that these are both lessening Goto one so that this expression will be less than this because this is what's relating to that X squared plus y squared square root. So looking at that 1st 1 sense X squared is lesson or equal. Tio X squared plus y squared began looking this 1st 1 Since this is obviously less than that, in that fraction, X squared over X squared plus y squared, it's going to be less than a good one and similarly well, I squared over X squared plus y squared is going less than equal to one. So then we get this entire saying So we have again this X squared over expert plus y squared X plus y squared over X squared plus y screwed. Why? Why scared? Since both of these are real estate, including one that's just going to be less than or equal Tio X plus y squared, which we just shown is less than or equal to two times Expert Plus why Screen Square Room and we ready know from the beginning that this square of expert Plus y squared is less than Delta. So therefore, X plus y squared. It's going to be less than two Delta, which we want equal 0.2 So therefore dividing two on both sides. I'm going, Tio said Delta, equal to 0.1 and we do that. We have that spirit of X squared plus y squared a less than Delta's air points there. One implies that my function next cube plus why the forest over X squared plus y squared minus zero. It's going to be less than two Delta, which is the 0.1 she called a zero point there, too, which is equal. Kaplan

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