91Ó°ÊÓ

Determine whether the statement is true or false. Explain your answer. The normal line to the surface \(z=f(x, y)\) at the point \(P_{0}\left(x_{0}, y_{0}, f\left(x_{0}, y_{0}\right)\right)\) has a direction vector given by \(f_{x}\left(x_{0}, y_{0}\right) \mathbf{i}+f_{y}\left(x_{0}, y_{0}\right) \mathbf{j}-\mathbf{k}\)

Find the directional derivative of \(f\) at \(P\) in the direction of a vector making the counterclockwise angle \(\theta\) with the positive \(x\) -axis. $$ f(x, y)=\frac{x-y}{x+y} ; P(-1,-2) ; \theta=\pi / 2 $$

Sketch the domain of \(f .\) Use solid lines for portions of the boundary included in the domain and dashed lines for portions not included. $$ f(x, y)=\sqrt{x^{2}+y^{2}-4} $$

Locate all relative maxima, relative minima, and saddle points, if any. $$ f(x, y)=x y+\frac{a^{3}}{x}+\frac{b^{3}}{y} \quad(a \neq 0, b \neq 0) $$

Solve using Lagrange multipliers. Find the point on the plane \(4 x+3 y+z=2\) that is closest to \((1,-1,1) .\)

Determine whether the limit exists. If so, find its value. $$ \lim _{(x, y, z) \rightarrow(0,0,0)} \frac{\sin \sqrt{x^{2}+y^{2}+z^{2}}}{x^{2}+y^{2}+z^{2}} $$

Find \(\partial z / \partial x\) and \(\partial z / \partial y\) $$ z=e^{x y} \sin 4 y^{2} $$

Find \(\partial z / \partial x\) and \(\partial z / \partial y\) $$ z=\frac{x y}{x^{2}+y^{2}} $$

Sketch the domain of \(f .\) Use solid lines for portions of the boundary included in the domain and dashed lines for portions not included. $$ f(x, y)=\frac{1}{x-y^{2}} $$

Use a total differential to approximate the change in the values of \(f\) from \(P\) to \(Q\). Compare your estimate with the actual change in \(f .\) $$ f(x, y)=x^{2}+2 x y-4 x ; P(1,2), Q(1.01,2.04) $$