Question 1 (4 points) Skyler and Denver are two students in Calculus IV. Studying for the Midterm, they are having an argument. - Skyler: I have a function f(x,y) and I know that fx(0,0) and fy(0,0) both exist. Then the problem is asking me to check if the function f(x,y) is continuous at (0,0). Isn't this obvious? Denver: Why do you think so? - Skyler: Because if f has partial derivatives at (0,0), that means it's differentiable at (0,0), so it has to be continuous at (0,0). I don't think there's anything to check here! - Denver: Hold on a second! Are you sure about what you just said? I think there's a hole in your logic, let me check my notes... (a). (2 points) If you think that Skyler is right, justify your answhir using the theorems and theory that we discussed in class. Otherwise, write down and explain all the details of an example that Denver can use to show that Skyler's argument is mistaken (i.e. an exam- ple to show that the claim "if the partial derivatives exist at a point, then the function is continuous at that point" is false). Continuing with their study session, a few hours later, they disagree again. - Skyler: Since this function is smooth (C), that means it has derivatives of any order and they are all continuous, right? - Denver: That's correct. - Skyler: So then, when I'm classifying it's critical points, I don't need to compute fry and fyx separately, right? They will be equal to each other, correct? - Denver: I don't think so... I remember that we worked out an example in class where fxy + fyx, so I'm pretty sure that what you are saying is not always true. Skyler: Are you sure? Denver: Yes. The theorem tells you that if a function is C then fry = fyx, but it doesn't say anything about smooth functions... (b). (2 points) If you think that Skyler is right, justify your answer using the theorems and the ory that we discussed in class, and explain Denver's confusion. Otherwise, write down and explain all the details of an example that Denver can use to show that Skyler's argu- ment is mistaken.

3 But Take f(x,y). x3y -zy gry? 2 Iz G, y) = 32 - 2ably sy => Ixcoy = -2 x²x4² Ixy (0,0) = -1 x²-3xy fy(x,y) zyliyyat => Fy (%,0). a 52 Fayx (0,0)=1. 7 Jay & fyn a) Take 7(x, y) = (X17) 0,0) otherwise Check Partial derivatives al co,o)- 72(0,0) = Lim JCh,o)-100) if the limit exists h ho zlim h Tycoo) lim 300, le) - 110,0) it.The limeto exists Tim the But lin Fx, y) - lim [Taking y along the x²mge? line ma does not existe ] Taman Hence both partial Derivative, existe but the Jun is not continuous at coo) Hence whatever. Skylese is is false, is saying 0 b) By Clairot's Theoremo It all the partial derivacitives of f(x, you exists and be Jay, Tyne are both continuous then Fay - Tyx, For a smooth Coo) fue" all portial dosuratives of feed exists and continuary hence Ixy 2 lyx .

Question 1 (4 points) Skyler and Denver are two stude…

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3 But Take f(x,y). x3y -zy gry? 2 Iz G, y) = 32 - 2ably sy => Ixcoy = -2 x²x4² Ixy (0,0) = -1 x²-3xy fy(x,y) zyliyyat => Fy (

a) Take 7(x, y) = (X17) 0,0) otherwise Check Partial derivatives al co,o)- 72(0,0) = Lim JCh,o)-100) if the limit exists h ho

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Question 1 (4 points) Skyler and Denver are two students in Calculus IV. Studying for the...

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