91Ó°ÊÓ

\(11-17\) Find the directional derivative of the function at the given point in the direction of the vector v. $$f(x, y, z)=x e^{y}+y e^{z}+z e^{x}, \quad(0,0,0), \quad \mathbf{v}=\langle 5,1,-2\rangle$$

Use Lagrange multipliers to find the maximum and minimum values of the function subject to the given constraint(s). \(f(x, y, z)=3 x-y-3 z\) \(x+y-z=0, \quad x^{2}+2 z^{2}=1\)

Find and sketch the domain of the function. $$f(x, y)=\sqrt{y}+\sqrt{25-x^{2}-y^{2}}$$

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

\(11-16\) Explain why the function is differentiable at the given point. Then find the linearization \(L(x, y)\) of the function at that point. $$ f(x, y)=\sin (2 x+3 y), \quad(-3,2) $$

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 first partial derivatives of the function. $$f(x, y)=x^{4} y^{3}+8 x^{2} y$$

\(11-17\) Find the directional derivative of the function at the given point in the direction of the vector \(\mathbf{v}\) . $$f(x, y, z)=\sqrt{x y z}, \quad(3,2,6), \quad \mathbf{v}=\langle- 1,-2,2\rangle$$

\(11-17\) Find the directional derivative of the function at the given point in the direction of the vector \(\mathbf{v}\) . $$g(x, y, z)=(x+2 y+3 z)^{3 / 2}, \quad(1,1,2), \quad \mathbf{v}=2 \mathbf{j}-\mathbf{k}$$

Find the first partial derivatives of the function. $$f(x, t)=e^{-t} \cos \pi x$$