91Ó°ÊÓ

\(11-16\) Explain why the function is differentiable at the given point. Then find the linearization \(L(x, y)\) of the function at that point. $$ f(x, y)=x^{3} y^{4}, \quad(1,1) $$

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

\(7-12\) Use the Chain Rule to find \(\partial z / \partial s / \partial s\) and \(\partial z / \partial t\) $$z=\tan (u / v), \quad u=2 s+3 t, \quad v=3 s-2 t$$

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

If \(f(x, y)=\sqrt{4-x^{2}-4 y^{2}},\) find \(f_{x}(1,0)\) and \(f_{y}(1,0)\) and interpret these numbers as slopes. Illustrate with either hand drawn sketches or computer plots.

II-17 Find the directional derivative of the function at the given point in the direction of the vector \(\mathbf{v}\) . $$f(x, y)=\ln \left(x^{2}+y^{2}\right), \quad(2,1), \quad \mathbf{v}=\langle- 1,2\rangle$$

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

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

\(11-16\) Explain why the function is differentiable at the given point. Then find the linearization \(L(x, y)\) of the function at that point. $$ f(x, y)=\frac{x}{x+y}, \quad(2,1) $$

Find and sketch the domain of the function. $$f(x, y)=\ln \left(9-x^{2}-9 y^{2}\right)$$