91Ó°ÊÓ

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. $$f(x, y)=\left(x^{2}+y^{2}\right) e^{y^{2}-x^{2}}$$

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

\(7-30\) Find the first partial derivatives of the function. $$f(x, y)=\frac{x}{y}$$

Find and sketch the domain of the function. $$f(x, y, z)=\ln \left(16-4 x^{2}-4 y^{2}-z^{2}\right)$$

Use Lagrange multipliers to find the maximum and minimum values of the function subject to the given constraint. $$ \begin{array}{l}{f\left(x_{1}, x_{2}, \ldots, x_{n}\right)=x_{1}+x_{2}+\cdots+x_{n}} \\\ {x_{1}^{2}+x_{2}^{2}+\cdots+x_{n}^{2}=1}\end{array} $$

\(7-30\) Find the first partial derivatives of the function. $$f(x, y)=\frac{x}{(x+y)^{2}}$$

Explain why the function is differentiable at the given point. Then find the linearization \(L(x, y)\) of the function at that point. $$ f(x, y)=y+\sin (x / y), \quad(0,3) $$

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 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. $$f(x, y)=e^{y}\left(y^{2}-x^{2}\right)$$

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. $$f(x, y)=y^{2}-2 y \cos x, \quad-1 \leqslant x \leqslant 7$$