91Ó°ÊÓ

Find all biologically relevant equilibria of the negative binomial host- parasitoid model $$ \begin{array}{l} N_{t+1}=4 N_{t}\left(1+\frac{0.01 P_{t}}{2}\right)^{-2} \\ P_{t+1}=N_{t}\left[1-\left(1+\frac{0.01 P_{t}}{2}\right)^{-2}\right] \end{array} $$ and analvze their stability.

Use Lagrange multipliers to find the maxima and minima of the functions under the given constraints. \(f(x, y)=x^{2}-y^{2} ; 2 x+y=1\)

Find a unit vector that is normal to the level curve of the function $$f(x, y)=x^{2}+\frac{y^{2}}{9}$$ at the point \((1,3)\).

Find a linear approximation to each function \(f(x, y)\) at the indicated point. $$ \mathbf{f}(x, y)=\left[\begin{array}{c} e^{x} \sin y \\ e^{-y} \cos x \end{array}\right] \text { at }(0,0) $$

Find a unit vector that is normal to the level curve of the function $$f(x, y)=x^{2}-y^{3}$$ at the point \((1,3)\).

Find all biologically relevant equilibria of the negative binomial host- parasitoid model $$ \begin{array}{l} N_{t+1}=4 N_{t}\left(1+\frac{0.01 P_{t}}{0.5}\right)^{-0.5} \\ P_{t+1}=N_{t}\left[1-\left(1+\frac{0.01 P_{t}}{0.5}\right)^{-0.5}\right] \end{array} $$ and analyze their stability.

In Problems \(39-48\), find the indicated partial derivatives. $$ f(x, y)=x e^{y} ; \frac{\partial^{2} f}{\partial x \partial y} $$

Use Lagrange multipliers to find the maxima and minima of the functions under the given constraints. \(f(x, y)=x^{2}+y^{2} ; 3 x-2 y=4\)

Find a linear approximation to each function \(f(x, y)\) at the indicated point. $$ \mathbf{f}(x, y)=\left[\begin{array}{c} (x+y)^{2} \\ x y \end{array}\right] \text { at }(-1,1) $$

Find a unit vector that is normal to the level curve of the function $$f(x, y)=x y$$ at the point \((2,3)\).