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

Find the extreme values of \(f\) subject to both constraints. $$ f(x, y, z)=x+2 y ; \quad x+y+z=1, \quad y^{2}+z^{2}=4 $$

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

Sketch the graph of the function. $$f(x, y)=10-4 x-5 y$$

Explain why the function is differentiable at the given point. Then find the linearization \(L(x, y)\) of the function at that point. $$ f(x, y)=e^{-x y} \cos y, \quad(\pi, 0) $$

Find the limit, if it exists, or show that the limit does not exist. $$\lim _{(x, y) \rightarrow(0,0)} \frac{x^{2}+y^{2}}{\sqrt{x^{2}+y^{2}+1}-1}$$

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

Use a tree diagram to write out the Chain Rule for the given case. Assume all functions are differentiable. \(R=f(x, y, z, t), \quad\) where \(x=x(u, v, w), y=y(u, v, w)\) \(z=z(u, v, w), t=t(u, v, w)\)

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

Find the directional derivative of \(f(x, y, z)=x y+y z+z x\) at \(P(1,-1,3)\) in the direction of \(Q(2,4,5)\)