91Ó°ÊÓ

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

In Problems 17-24, 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 \(\left(x_{0}, y_{0}\right).\) $$ f(x, y)=\ln \left(x^{2}+y\right) ;(1,1) $$

Find the absolute maxima and minima of $$ f(x, y)=x^{2}+y^{2}+x-y $$ on the disk $$ D=\left\\{(x, y): x^{2}+y^{2} \leq 1\right\\} $$

Find the gradient of each function. $$ f(x, y)=\ln \left(\frac{x}{y}+\frac{y}{x}\right) $$

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

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

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

In Problems 17-24, find the indicated partial derivatives. $$ f(u, v)=e^{u^{2} / 2} \ln (u+v) ; f_{u}(2,1) $$

Find the linear approximation of $$f(x, y)=e^{x+y}$$ at \((0,0)\), and use it to approximate \(f(0.1,0.05)\). Using a calculator, compare the approximation with the exact value of \(f(0.1,0.05)\).