91Ó°ÊÓ

Find the limit, if it exists, or show that the limit does not exist. $$ \lim _{(x, y) \rightarrow(0,0)} \frac{x y^{4}}{x^{2}+y^{8}} $$

Find the extreme values of \(f\) subject to both constraints. $$ f(x, y, z)=z ; \quad x^{2}+y^{2}=z^{2}, \quad x+y+z=24 $$

Find the local maximum and minimum values and saddle point(s) of the function. If you have three-dimensional graphing software, graph the function with a domain and viewpoint that reveal all the important aspects of the function. $$ f(x, y)=\left(x^{2}+y^{2}\right) e^{-x} $$

Find and sketch the domain of the function. $$ f(x, y)=\frac{\sqrt{y-x^{2}}}{1-x^{2}} $$

Find the directional derivative of \(f(x, y)=\sqrt{x y}\) at \(P(2,8)\) in the direction of \(Q(5,4) .\)

Use a tree diagram to write out the Chain Rule for the given case. Assume all functions are differentiable. $$ \begin{array}{l}{T=F(p, q, r), \quad \text { where } p=p(x, y, z), q=q(x, y, z)} \\ {r=r(x, y, z)}\end{array} $$

Find the extreme values of \(f\) subject to both constraints. $$ f(x, y, z)=y z+x y ; \quad x y=1, \quad y^{2}+z^{2}=1 $$

Find the local maximum and minimum values and saddle point(s) of the function. If you have three-dimensional graphing software, graph the function with a domain and viewpoint that reveal all the important aspects of the function. $$ f(x, y)=y^{2}-2 y \cos x, \quad-1 \leqslant x \leqslant 7 $$

Find the first partial derivatives of the function. $$ z=\ln \left(x+t^{2}\right) $$

Find the limit, if it exists, or show that the limit does not exist. $$ \lim _{(x, y, z) \rightarrow(\pi, 0,1 / 3)} e^{y^{z} \tan (x z)} $$