91Ó°ÊÓ

In what direction does \(f(x, y)=e^{x} \cos y\) increase most rapidly at \((0, \pi / 2) ?\)

Show that \(\left[\begin{array}{l}0 \\ 0\end{array}\right]\) is an equilibrium of $$ \left[\begin{array}{l} x_{1}(t+1) \\ x_{2}(t+1) \end{array}\right]=\left[\begin{array}{rr} 0.1 & 0.3 \\ 0.1 & -1.8 \end{array}\right]\left[\begin{array}{l} x_{1}(t) \\ x_{2}(t) \end{array}\right] $$ and determine its stability.

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.

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

Use the definition of continuity to show that $$f(x, y)=2 x^{2}+y^{2}+1$$ is continuous at \((0,0)\).

In what direction does \(f(x, y)=\sqrt{x^{2}-y^{2}}\) increase most rapidly at \((5,3)\) ?

Find the indicated partial derivatives. \(f(x, z)=\ln (x z) ; f_{z}(e, 1)\)

Find the global maxima and minima of $$f(x, y)=x^{2}+y^{2}+4 x-1$$ on the disk $$ D=\left\\{(x, y): x^{2}+y^{2} \leq 9\right\\} $$

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

Show that \(\left[\begin{array}{l}0 \\ 0\end{array}\right]\) is an equilibrium of $$ \left[\begin{array}{l} x_{1}(t+1) \\ x_{2}(t+1) \end{array}\right]=\left[\begin{array}{ll} 1.5 & 0.2 \\ 0.08 & 0 \end{array}\right]\left[\begin{array}{l} x_{1}(t) \\ x_{2}(t) \end{array}\right] $$ and determine its stability.