Chapter 10

Find all nonnegative equilibria of $$ \begin{array}{l} x_{1}(t+1)=x_{2}(t) \\ x_{2}(t+1)=\frac{1}{2} x_{1}(t)+\frac{1}{3} x_{2}(t)-x_{2}^{2}(t) \end{array} $$ and analyze their stability.

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

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

Given the symmetric matrix $$ A=\left[\begin{array}{ll} a & c \\ c & b \end{array}\right] $$ where \(a, b\), and \(c\) are real numbers, show that the eigenvalues of \(A\) are real. (Hint: Compute the eigenvalues.)

Give a geometric interpretation of the set $$ A=\left\\{(x, y) \in \mathbf{R}^{2}: \sqrt{x^{2}+6 x+y^{2}-2 y+10}<2\right\\} $$

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

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

Understanding species richness and diversity is a major concern of ecological studies. A frequently used measure of diversity is the Shannon and Weaver index $$ H=-\sum_{i=1}^{n} p_{i} \ln p_{i} $$ where \(p_{i}\) is equal to the proportion of species \(i, i=1,2, \ldots, n\), and \(n\) is the total number of species in the study area. Assume that a community consists of three species with relative proportions \(p_{1}, p_{2}\), and \(p_{3}\) (a) Use the fact that \(p_{1}+p_{2}+p_{3}=1\) to show that \(H\) is of the form $$ \begin{aligned} H\left(p_{1}, p_{2}\right)=&-p_{1} \ln p_{1}-p_{2} \ln p_{2} \\ &-\left(1-p_{1}-p_{2}\right) \ln \left(1-p_{1}-p_{2}\right) \end{aligned} $$ and that the domain of \(H\left(p_{1}, p_{2}\right)\) is the triangular set in the \(p_{1}-p_{2}\) plane bounded by the lines \(p_{1}=0, p_{2}=0\), and \(p_{1}+p_{2}=1\). (b) Show that \(H\) attains its absolute maximum when \(p_{1}=p_{2}=\) \(p_{3}=1 / 3\)

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

For which values of \(a\) is the equilibrium \(\left[\begin{array}{l}0 \\\ 0\end{array}\right]\) of $$ \begin{array}{l} x_{1}(t+1)=x_{2}(t) \\ x_{2}(t+1)=\frac{1}{2} x_{1}(t)+a x_{2}(t)-x_{2}^{2}(t) \end{array} $$ locally stable?

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

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

In Problems 36-45, use Lagrange multipliers to find the maxima and minima of the functions under the given constraints. \(f(x, y)=2 x-y ; x^{2}+y^{2}=5\)

Let $$ f(x, y)=x^{2}+3 y^{2} $$ Use the \(\epsilon-\delta\) definition of limits to show that $$ \lim _{(x, y) \rightarrow(0,0)} f(x, y)=0 $$

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

Denote by \(x_{1}(t)\) the number of juveniles, and by \(x_{2}(t)\) the number of adults, at time \(t\). Assume that \(x_{1}(t)\) and \(x_{2}(t)\) evolve according to $$ \begin{array}{l} x_{1}(t+1)=x_{2}(t) \\ x_{2}(t+1)=\frac{1}{2} x_{1}(t)+r x_{2}(t)-x_{2}^{2}(t) \end{array} $$ (a) Show that if \(r>1 / 2\), there exists an equilibrium \(\left[\begin{array}{c}x_{1}^{*} \\ x_{2}^{*}\end{array}\right]\) with \(x_{1}^{*}>0\) and \(x_{2}^{*}>0\). Find \(x_{1}^{*}\) and \(x_{2}^{*}\). (b) Determine the stability of the equilibrium found in (a) when \(r>1 / 2 .\)

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

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

Use Lagrange multipliers to find the maxima and minima of the functions under the given constraints. \(f(x, y)=3 x^{2}+y ; x^{2}+y^{2}=1\)

In what direction does \(f(x, y)=\ln \left(x^{2}+y^{2}\right)\) increase most rapidly at \((1,1) ?\)

Find all biologically relevant equilibria of the NicholsonBailey model $$ \begin{aligned} N_{t+1} &=2 N_{t} e^{-0.2 P_{t}} \\ P_{t+1} &=N_{t}\left[1-e^{-0.2 P_{t}}\right] \end{aligned} $$ and analyze their stability.

Find a linear approximation to each function \(f(x, y)\) at the indicated point. $$ \mathbf{f}(x, y)=\left[\begin{array}{c} 3 x-y^{2} \\ 4 y \end{array}\right] \text { at }(-1,-2) $$

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

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

Find a unit vector that is normal to the level curve of the function $$f(x, y)=3 x+4 y$$ at the point \((-1,1)\).

Find all biologically relevant equilibria of the NicholsonBailey model $$ \begin{aligned} N_{t+1} &=4 N_{t} e^{-0.1 P_{t}} \\ P_{t+1} &=N_{t}\left[1-e^{-0.1 P_{t}}\right] \end{aligned} $$ and analyze their stability.

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

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

Use Lagrange multipliers to find the maxima and minima of the functions under the given constraints. \(f(x, y)=x y ; 2 x-4 y=1\)

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 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.

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

In Problems \(39-48\), find the indicated partial derivatives. $$ f(x, y)=y^{2}(x-3 y) ; \frac{\partial^{2} f}{\partial y^{2}} $$

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

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

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

Use Lagrange multipliers to find the maxima and minima of the functions under the given constraints. \(f(x, y)=x y^{2} ; x^{2}-y=0\)

Chemotaxis is the chemically directed movement of organisms up a concentration gradient-that is, in the direction in which the concentration increases most rapidly. The slime mold Dictyostelium discoideum exhibits this phenomenon. Single-celled amoebas of this species move up the concentration gradient of a chemical called cyclic adenosine monophosphate (AMP). Suppose the concentration of cyclic AMP at the point \((x, y)\) in the \(x-y\) plane is given by $$f(x, y)=\frac{4}{|x|+|y|+1}$$ If you place an amoeba at the point \((3,1)\) in the \(x-y\) plane, determine in which direction the amoeba will move if its movement is directed by chemotaxis.

Find a linear approximation to $$\mathbf{f}(x, y)=\left[\begin{array}{l} x^{2}-x y \\ 3 y^{2}-1 \end{array}\right]$$ at \((1,2)\). Use your result to find an approximation for \(f(1.1,1.9)\), and compare the approximation with the value of \(f(1.1,1.9)\) that you get when you use a calculator.

Use Lagrange multipliers to find the maxima and minima of the functions under the given constraints. \(f(x, y)=x^{2} y ; x^{2}+3 y=1\)

Suppose an organism moves down a sloped surface along the steepest line of descent. If the surface is given by $$f(x, y)=x^{2}-y^{2}$$ find the direction in which the organism will move at the point \((2,3)\)

In Problems \(39-48\), find the indicated partial derivatives. $$ g(s, t)=\ln \left(s^{2}+3 s t\right) ; \frac{\partial^{2} g}{\partial t^{2}} $$

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

Find a linear approximation to $$\mathbf{f}(x, y)=\left[\begin{array}{c} (x-y)^{2} \\ 2 x^{2} y \end{array}\right]$$ at \((2,-3)\). Use your result to find an approximation for \(f(1.9,-3.1)\), and compare the approximation with the value of \(f(1.9,-3.1)\) that you get when you use a calculator.

In Problems \(39-48\), find the indicated partial derivatives. $$ f(x, y)=x^{3} \cos y ; \frac{\partial^{3} f}{\partial x^{2} \partial y} $$