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Show tha:the function differentiable by finding values of 21 and {2 as designated the definition of differentiability, and verify that both and 22 approach 2s (1x, ...

Question

Show tha:the function differentiable by finding values of 21 and {2 as designated the definition of differentiability, and verify that both and 22 approach 2s (1x, Jy) (0,0),Rx %) = 3x - y2flx + Ax, y - By) flx, y)3* 1x) - y _ Zy(Ay) (Ly) -(A} 2y(Ly) By( Ly)fxx, Y)(Ax) fylx, Y)(Dy)(Ax) 'z(Ay), wnere ? 1As (1x, By) (0, 0), 81and 22

Show tha:the function differentiable by finding values of 21 and {2 as designated the definition of differentiability, and verify that both and 22 approach 2s (1x, Jy) (0,0), Rx %) = 3x - y2 flx + Ax, y - By) flx, y) 3* 1x) - y _ Zy(Ay) (Ly) - (A} 2y(Ly) By( Ly) fxx, Y)(Ax) fylx, Y)(Dy) (Ax) 'z(Ay), wnere ? 1 As (1x, By) (0, 0), 81 and 22



Answers

Show that the function is differentiable by finding values of $ \varepsilon_1 $ and $ \varepsilon_2 $ that satisfy Definition 7.
$ f(x, y) = x^2 + y^2 $

First recall that definition seven says that Delta have of X Y is equal to FX. That's why times Delta X plus F y If x y times Delta Why plus absolute one Delta Eggs plus absolute to Delta Way and first, we'll see that Delta F of X y is f of exports. Stilton X. Why post ilta? Why, minus f of X y and that expands to since F It's x squared plus y squared ex postal toe X squared. Plus why plus Delta y squared minus X squared plus y squared. And if we foil these, we get X squared. Plus two x Delta X Postal two X squared plus y squared plus two. Why Don't know Why was still toe y squared minus X squared minus y squared and we see that will cancel X squared with negative X squared and y squared with negative y squared. So we're left with two x Delta X plus Delta X Squared, which is Delta X Times, Delta X Plus two. Why Teldta. Why? Plus don't y squared, which is Delta White and Delta y. And remember, this was all equal to Delta F E x. Why, so this will be equal to FX of X y. Well, the partial derivative with respect to X is two x times Delta X plus F Y X y and the partial derivative Respect. A wise to why so, too, why don't know why. Plus absolute one Delta eggs put steps onto Delta y and we see that we have to extol two x cancels with two x Delta X and two by Delta Y cancels with you, I don't know why. And so we're left with Delta Eggs Delta X Plus Delta. Why Delta y? Because apps on one dot exe plus absolute to Delta Y, and we see that absolute one equals Delta X and absolute two equals Delta Y.

To begin the problem. We are going to first recall definition seven. So the definition seven seas if it's different. Shipper at X. Com away If I can write the inclement off If Delta Psi s Delta If Delta X types Still the Times X plus then get me it. Is that Delta? If Delta y times Delta off Why plus Delta off X times It's still on one plus dealt off Why time slips along Too less What is my daughter's? The Delta Z is equal to if off X plus still tow X comma y plus still tell why minus if off x comma y So this is the increment off if with respect to X and y so we call it by Delta Z on dhe off course I know the deal. Drive till to exact apart. Shell off if with respect to X on partial off it if with respect to why respectively. And we have the condition on Epsilon one. And if Sloane to what is that? Absalon won an excellent to goes to zero. That's do tow X dealt Why goes to zero. Okay, so we're going to sister finish in Dewberry fight the defense ability off the function. Phyllis. Right down the function if off X comma y physical to exploit minus five White Square So then what is my tell? Tell C. It's a good too. If off exploits Toto X comma y plus tota y minus If off X com away So what is this X plus Delta X. We can put a parenthesis. It really doesn't matter. The book doesn't use. The parent insists. It's fine. So explodes. Kotex times wipe us Why minus Fife by plus tilt away Whole square, minus what is safe or fix Calmer, Wiser Linus extremes White on two minus. Makes it positive. Five. Y squared Phillips Evaluate this so this becomes extents. Lie plus X times dealt off My plus y times still ter fix plus d'eau tex time Still tha Why on then? Negative five. Life's Claire on, then. Negative Fife. Tilt away. Hold square on then. So this will give me plus two a Be like so miners 10 white times don't off why And then I also have expired plus five square six million x y cancels five by square. This fiber is squared cancels as well. So now I have this time X times dough Tough lie plus y times Delta affects Plus still takes time. Still tough. Why minus five times dealt off. Why Hold square on, then. Negative 10. Why? Times dealt off Why? Okay, so next we're going to valve it. Our dill tough del Tex attics come away So what is that? So the delta off till decks off X y minus fight White Square which is going to be why minus zero. So why on similar little toe after toe I at X comma Why, it's going to be so x minus 10 times Why? So this in place dealt if Delta x time six Come away times Delta X It's going to be Whiting's do Tex Thank you Deal Turf to Dubai Time six comma y times tilt off Why is going to be extend? Still tell why minus 10 White dance to tell way sore If I just go back to the expression so I can combine this tone on this term on as Phyllis this term onto this term. Sorry. So we cannot combine this storm so interest rate that down. So this three underlying turn into deal drive till take strengths till two times X then tell if tell why time still don't fly. So let me just write that down. So I end up getting Delta Psi Musical, too, Until life do tow x X comma y time Still today in six plus don't have to tell by times Delta y on Let's go back. Let's see watered expression we still left with So we have this extra tar So the delta X Delta y minus five dental by whole square. So these are the only two Xs term. So just write that down. So don't takes of July minus Fife Delta y Square So what is our choice off Epsilon? We can take insulin. One is the call to our delta. Why? And it's still on to physical Too negative. Five off deal tough. Why? Why? So then I can see that Del takes time Still till I can be bitterness. It's learned one times Dealt off fix so don't tell Fix and negative. Five Delta y whole square can be return. US ifs Ellen two times still tough. Why and absolute one an absolute to goes to zero Yes, till tricks dealt away cost to zero. So Q E. T

If we recall definition. Seven. We know that Delta if the thanks, Why equals FX X y don't X plus F Y x Y times Delta y plus eps on one that's X plus absolute, too. Delta y If we will get Delta f x y, we see that this is equal to F of X postal. Two x Why plus Delta y minus f x y but which, if we plug into, I think given formula for F, we get X plus Delta x Times y plus Delta y minus five times why Post Delta y squared minus X Y minus five. Pi squared and now if we expand this out, we get X Y, plus Delta X y plus x delta y were still two x delta y minus five y squared minus 10. Why Delta y minus five Delta y squared minus X y plus five y squared and we see that we can start to cancel minus five y squared plus five y squared and x y with negative X y. And so we're left with Delta. X Times y plus x times Delta y We're still two x daughter. Why minus 10 y Delta y minus five does a Y squared? So now if we look at the right hand side of the equation, we have been FX and F y that we can plug in. So this will be equal to Yeah, FX, Which is why Times Delta, X plus F y, which is X minus 10 way times. Delta. Why plus absolute one. Don't X plus eps on to Delta y, and that's going to expand to why don't x plus X minus 10 plus x times Delta Why minus 10 y times Delta y plus Absalon one dot exe footsteps on $2 a while And if we look at this part here, receive that will cancel a negative 10 wide daughter. Why with native and my daughter y on both sides and we'll also cancel X Delta y and why don't X and so we're left with don't x Delta y minus five. Delta y squared equals absolute one. Does X Plus Absalon, too? Don't know why. Which is going to show us that Absalon won equals Delta Y and Absalom to equals negative five times Delta y

Were given a function in rest to show this function is differential by finding values of Epsilon one in Epsilon, too. Satisfy the definition for differential function. Function is F X y equals X squared plus y squared. First of all, you have a Delta Z. This is, um, f of a plus. Don't X B plus Delta. Why minus as they be, which is equal to a plus Delta X squared plus B plus Delta y squared, minus a squared plus B squared and multiplying out. This is equal to a squared plus to a Delta X plus Delta X squared plus B squared plus to be Delta Y plus Delta y squared, minus a squared, minus B squared and simplifying. This is to a tennis till two x plus Delta X squared plus two B times Delta y plus still toe Why squared? We have that the partial derivative of F with respect to X and I waited a B. This is one week to a in the partial derivative F with respect, Why evaluated the A B is to be, and therefore we have a Delta Z. This is also equal to the partial derivative of F with respect to Exit A B Times, Delta X Plus partial derivative event with Respect Toe Wyatt A B times, Delta y plus. And then we have Delta X. We have a Delta X term and a Delta y term. But to get this above expression, need to have Delta, X Times, Delta X, and we need to have Delta y times Delta y to get Delta y squared. Therefore, it follows that in the definition definition. Seven. You'll want to take Epsilon one to be Delta X and Epsilon, too to be Delta. Why, by choosing these values of Epsilon One and EPS 12 between them are positive. It then follows the F is differential by definition.


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