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 $$ 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 $$. Few people got this correct.