91Ó°ÊÓ

Let $$ f(x, y)=2 x^{3}-3 y x $$ Compute \(f_{x}(1,2)\) and \(f_{y}(1,2)\), and interpret these partial derivatives geometrically.

Compute the directional derivative of \(f(x, y)\) at the given point in the indicated direction. $$ f(x, y)=2 x y^{3}-3 x^{2} y \text { at }(1,-1) \text { in the direction }\left[\begin{array}{l} 3 \\ 1 \end{array}\right] $$

Show that the equilibrium \(\left[\begin{array}{l}0 \\ 0\end{array}\right]\) of $$ \begin{array}{l} x_{1}(t+1)=x_{2}(t) \\ x_{2}(t+1)=\frac{2 x_{2}(t)-x_{1}(t)}{2+x_{1}(t)} \end{array} $$ is locally stable.

Find the Jacobi matrix for each given function. $$ \mathbf{f}(x, y)=\left[\begin{array}{c} x+y \\ x^{2}-y^{2} \end{array}\right] $$

(a) Write $$ h(x, y)=e^{x y} $$ as a composition of two functions. (b) For which values of \((x, y)\) is \(h(x, y)\) continuous?

Find the maximum volume of a rectangular closed (top, bottom, and four sides) box with surface area \(48 \mathrm{~m}^{2}\).

Find the maximum volume of a rectangular open (bottom and four sides, no top) box with surface area \(75 \mathrm{~m}^{2}\).

a) Write $$ h(x, y)=\cos (y-x) $$ as a composition of two functions. (b) For which values of \((x, y)\) is \(h(x, y)\) continuous?

Find the Jacobi matrix for each given function. $$ \mathbf{f}(x, y)=\left[\begin{array}{c} 2 x-3 y \\ 4 x^{2} \end{array}\right] $$

Show that, for any \(a>1\), the equilibrium \(\left[\begin{array}{l}0 \\\ 0\end{array}\right]\) of $$ \begin{aligned} x_{1}(t+1) &=x_{2}(t) \\ x_{2}(t+1) &=\frac{a x_{2}(t)-(a-1) x_{1}(t)}{a+x_{1}(t)} \end{aligned} $$ is locally stable.