91Ó°ÊÓ

Show that the equilibrium \(\left[\begin{array}{l}0 \\ 0\end{array}\right]\) of $$ \left[\begin{array}{l} x_{1}(t+1) \\ x_{2}(t+1) \end{array}\right]=\left[\begin{array}{rr} -0.2 & -0.4 \\ 0.6 & 0.1 \end{array}\right]\left[\begin{array}{l} x_{1}(t) \\ x_{2}(t) \end{array}\right] $$ is stable.

Find a unit vector that is normal to the level curve of the function $$ f(x, y)=3 x+4 y $$ at the point \((-1,1)\).

Find the indicated partial derivatives. \(f(x, y)=\frac{x y}{x^{2}+2} ; f_{x}(-1,2)\)

Find the linearization of \(f(x, y)\) at the indicated point \(\left(x_{0}, y_{0}\right) .\) \(f(x, y)=x^{2} e^{y} ;(1,0)\)

Find the global maxima and minima of $$f(x, y)=x^{2}+y^{2}+x y-2 y$$ on the disk $$ D=\left\\{(x, y): x^{2}+y^{2} \leq 4\right\\} $$

Find a unit vector that is normal to the level curve of the function $$ f(x, y)=x^{2}+\frac{y^{2}}{9} $$ at the point \((1,3)\).

Find the indicated partial derivatives. \(f(u, v)=e^{\mu+3 v^{2}} ; f_{u}(2,1)\)

Show that $$f(x, y)=\left\\{\begin{array}{cl}\frac{3 r(y+x)}{x^{2}+y^{3}} & \text { for }(x, y) \neq(0,0) \\\0 & \text { for }(x, y)=(0,0)\end{array}\right.$$ is discontinuous at \((0,0)\).

Show that the equilibrium \(\begin{array}{ll}0 & \text { of } \\ 0\end{array}\) $$ \left[\begin{array}{l} x_{1}(t+1) \\ x_{2}(t+1) \end{array}\right]=\left[\begin{array}{rr} 0.2 & 0.3 \\ -0.5 & -0.4 \end{array}\right]\left[\begin{array}{l} x_{1}(t) \\ x_{2}(t) \end{array}\right] $$ is stable.

Find the range of each function \(f(x, y)\), when defined on the specified domain \(D\). \(\left.f(x, y)=x^{2} y ; D=(x, y):-2 \leq x \leq 1,0 \leq y \leq 1, y