91Ó°ÊÓ

\(\begin{array}{l}{9-10 \text { Draw the graph of } f \text { and its tangent plane at the given }} \\ {\text { point. (Use your computer algebra system both to compute the }} \\ {\text { partial derivatives and to graph the surface and its tangent plane. }} \\ {\text { Then zoom in until the surface and the tangent plane become }} \\ {\text { indistinguishable. }}\end{array}\) $$ f(x, y)=e^{-x y / 10}(\sqrt{x}+\sqrt{y}+\sqrt{x y}), \quad\left(1,1,3 e^{-01}\right) $$

Use Lagrange multipliers to find the maximum and minimum values of the function subject to the given constraint(s). \(f(x, y, z)=x^{2}+y^{2}+z^{2} ; \quad x^{4}+y^{4}+z^{4}=1\)

Find and sketch the domain of the function. $$f(x, y)=\sqrt{x+y}$$

\(5-22\) Find the limit, if it exists, or show that the limit does not exist. $$\lim _{(x, y) \rightarrow(0,0)} \frac{x y \cos y}{3 x^{2}+y^{2}}$$

\(7-12\) Use the Chain Rule to find \(\partial z / \partial s / \partial s\) and \(\partial z / \partial t\) $$z=e^{r} \cos \theta, \quad r=s t, \quad \theta=\sqrt{s^{2}+t^{2}}$$

\(11-17\) Find the directional derivative of the function at the given point in the direction of the vector \(\mathbf{v}\) . $$f(x, y)=1+2 x \sqrt{y}, \quad(3,4), \quad \mathbf{v}=\langle 4,-3\rangle$$

If \(f(x, y)=16-4 x^{2}-y^{2},\) find \(f_{x}(1,2)\) and \(f_{y}(1,2)\) and interpret these numbers as slopes. Illustrate with either hand-drawn sketches or computer plots.

\(11-16\) Explain why the function is differentiable at the given point. Then find the linearization \(L(x, y)\) of the function at that point. $$ f(x, y)=x \sqrt{y}, \quad(1,4) $$

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. $$f(x, y)=x^{3}-12 x y+8 y^{3}$$

\(5-22\) Find the limit, if it exists, or show that the limit does not exist. $$\lim _{(x, y \rightarrow(0,0)} \frac{6 x^{3} y}{2 x^{4}+y^{4}}$$