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)=\sin x \sin y, \quad-\pi

Find the extreme values of \(f\) subject to both constraints. $$ \begin{array}{l}{f(x, y, z)=3 x-y-3 z} \\ {x+y-z=0, \quad x^{2}+2 z^{2}=1}\end{array} $$

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

\(7-30\) Find the first partial derivatives of the function. $$w=\frac{e^{v}}{u+v^{2}}$$

Explain why the function is differentiable at the given point. Then find the linearization \(L(x, y)\) of the function at that point. $$ f(x, y)=\sqrt{x+e^{4 y}}, \quad(3,0) $$

Use a graph or level curves or both to estimate the local maximum and minimum values and saddle point(s) of the function. Then use calculus to find these values precisely. $$f(x, y)=3 x^{2} y+y^{3}-3 x^{2}-3 y^{2}+2$$

Find the maximum rate of change of \(f\) at the given point and the direction in which it occurs. $$f(x, y)=\sin (x y), \quad(1,0)$$

\(7-30\) Find the first partial derivatives of the function. $$g(u, v)=\left(u^{2} v-v^{3}\right)^{5}$$

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

Verify the linear approximation at \((0,0)\) $$ \frac{2 x+3}{4 y+1} \approx 3+2 x-12 y $$