Paper Homework 4

Problem 3: The level curve of a function with multiple variables, say f(x,y,z), is given by f(x,y,z)=k. Notice that to figure out the shape of each level curve (or to be more precise, level surface), we have to discuss three cases k<0, k=0, k>0. Some students missed one or two of those cases.

Problem 6: To show \lim_{(x,y)\to(0,0)}f(x,y)=0, it is not enough to just check the limit goes to 0 along any linear approach of the point (x,y) to the origin. We need to either use the definition to show the limit exists and equals 0 or use squeeze theorem.

Problem 7: Some students didn’t explain the limit of r^2\ln(r^2) is 0 as r approaches 0.

Problem 10: For (a), I took points off if one didn’t simplify his or her answer for f_x, f_y. For (d), the reason that Clairaut’s theorem fails is that f_{xy}, f_{yx} are not continuous at 0. Few people got this correct.

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