91Ó°ÊÓ

Compute $$ \lim _{(x, y) \rightarrow(0,0)} \frac{2 x y}{x^{3}+y x} $$ along lines of the form \(y=m x\), for \(m \neq 0\), and along the parabola \(y=x^{2} .\) What can you conclude?

Show that \(\begin{array}{ll}0 & \text { is an equilibrium of }\end{array}\) $$ \left[\begin{array}{l} x_{1}(t+1) \\ x_{2}(t+1) \end{array}\right]=\left[\begin{array}{ll} -1.4 & 0 \\ -0.5 & 0.1 \end{array}\right]\left[\begin{array}{l} x_{1}(t) \\ x_{2}(t) \end{array}\right] $$ and determine its stability.

Find the gradient of each function. $$ f(x, y)=\sqrt{x^{3}-3 x y} $$

Maximize the function $$ f(x, y)=2 x y-x^{2} y-x y^{2} $$ on the triangle bounded by the line \(x+y=2\), the \(x\) -axis, and the \(y\) -axis.

Maximize the function $$ f(x, y)=x y(15-5 y-3 x) $$ on the triangle bounded by the line \(5 y+3 x=15\), the \(x\) -axis, and the \(y\) -axis.

In Problems 17-24, find the indicated partial derivatives. $$ h(u, v)=e^{l \prime} \sin (u+v) ; h_{u}(1,-1) $$

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

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

Find the gradient of each function. $$ f(x, y)=x\left(x^{2}-y^{2}\right)^{2 / 3} $$

Compute $$ \lim _{(x, y) \rightarrow(0,0)} \frac{3 x^{2} y^{2}}{x^{3}+y^{6}} $$ along lines of the form \(y=m x\), for \(m \neq 0\), and along the parabola \(x=y^{2} .\) What can you conclude?