91Ó°ÊÓ

Compute the directional derivative of \(f(x, y)\) at the given point in the indicated direction. $$ f(x, y)=y e^{x^{2}} \text { at }(0,2) \text { in the direction }\left[\begin{array}{r} 4 \\ -1 \end{array}\right] $$

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

In Problems \(31-38\), find \(\partial f / \partial x, \partial f / \partial y\), and \(\partial f / \partial z\) for the given functions. $$ f(x, y, z)=x^{2} z+y z^{2}-x y $$

Compute the directional derivative of \(f(x, y)\) at the point \(P\) in the direction of the point \(Q\). $$ f(x, y)=2 x^{2} y-3 x, P=(2,1), Q=(3,2) $$

Find the minimum surface area of a rectangular closed (top, bottom, and four sides) box with volume \(216 \mathrm{~m}^{3}\).

Show that \(\left[\begin{array}{l}0 \\ 0\end{array}\right]\) is an equilibrium point of $$ \begin{array}{l} x_{1}(t+1)=a x_{2}(t) \\ x_{2}(t+1)=2 x_{1}(t)-\cos \left(x_{2}(t)\right)+1 \end{array} $$ Assume that \(a>0\). For which values of \(a\) is \(\left[\begin{array}{c}0 \\\ 0\end{array}\right]\) locally stable?

Draw an open disk with radius 2 centered at \((1,-1)\) in the \(x-y\) plane, and give a mathematical description of this set.

Draw a closed disk with radius 3 centered at \((2,0)\) in the \(x-y\) plane, and give a mathematical description of this set.

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

Find the minimum surface area of a rectangular open (bottom and four sides, no top) box with volume \(256 \mathrm{~m}^{3}\).