91Ó°ÊÓ

Find the linear approximation of $$f(x, y)=\ln \left(x^{2}-3 y\right)$$ at \((1,0)\), and use it to approximate \(f(1.1,0.1)\). Using a calculator, compare the approximation with the exact value of \(f(1.1,0.1)\).

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

Show that the equilibrium \(\left[\begin{array}{l}0 \\ 0\end{array}\right]\) of $$ \begin{array}{l} x_{1}(t+1)=\frac{x_{2}(t)}{4\left(1+x_{1}^{2}(t)\right)} \\ x_{2}(t+1)=\frac{2 x_{1}(t)}{1+x_{2}^{2}(t)} \end{array} $$ is locally stable.

Suppose crop yield \(Y\) depends on nitrogen \((\mathrm{N})\) and phosphorus (P) concentrations as $$ Y(N, P)=N P e^{-(N+P)} $$ Find the value of \((N, P)\) that maximizes crop yield.

(a) Write $$ h(x, y)=\sin \left(x^{2}+y^{2}\right) $$ as a composition of two functions. (b) For which values of \((x, y)\) is \(h(x, y)\) continuous?

Choose three numbers \(x, y\), and \(z\) so that their sum is equal to 60 and their product is maximal.

Show that the equilibrium \(\left[\begin{array}{l}0 \\ 0\end{array}\right]\) of $$ \begin{array}{l} x_{1}(t+1)=\frac{3 x_{2}(t)}{1+x_{1}^{2}(t)} \\ x_{2}(t+1)=\frac{2 x_{1}(t)}{1+x_{2}^{2}(t)} \end{array} $$ is unstable.

(a) Write $$ h(x, y)=\sqrt{x+y} $$ as a composition of two functions.

Find the linear approximation of $$f(x, y)=\tan \left(2 x-3 y^{2}\right)$$ at \((0,0)\), and use it to approximate \(f(0.03,0.05) .\) Using a calculator, compare the approximation with the exact value of \(f(0.03,0.05).\)

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