Chapter 10

Given a function \(f(x, y)\) that is defined and differentiable on an open ball containing the point \(\left(x_{0}, y_{0}\right)\), show that the function \(f\) decreases most rapidly in the direction of \(-\nabla f\left(x_{0}, y_{0}\right)\).

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

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

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

Suppose an organism moves down a sloped surface along the steepest line of descent, i.e., the direction in which the surface decreases most rapidly. 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 \((2,3)\)

The oxygen content of blood depends on the partial pressure of oxygen in surrounding tissues \((P)\) and on a reaction rate constant \((K) .\) Blood oxygenation is often modcled using Hill's equation, which predicts that the fraction of hemoglobin molecules in blood that are bound to oxygen will be given by a function of \(P\) and \(K\) : $$f(P, K)=\frac{P^{3}}{K^{3}+P^{3}}$$ (a) Explain why, if \(K>0\) and \(P \geq 0, f(P, K)<1\) and \(f(P, K) \geq 0\) (b) Use partial differentiation to determine the effect of increasing \(P\) on \(f\). (c) Use partial differentiation to determine the effect of increasing \(K\) on \(f\).

(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. \(\mathbf{f}(x, y)=\left[\begin{array}{c}2 x-3 y \\ 4 x^{2}\end{array}\right]\)

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

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

Find \(\partial f / 2 x, \partial f / \partial y\), and \(\partial f / \partial z\) for the given functions. \(f(x, y, z)=x^{2} z+y z^{2}-x y\)

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

Use nine evenly spaced points and five colors to draw heat maps of the following functions, defined on their specified domains. \(f(x, y)=x^{2}+y^{2}\) on \(D=\\{(x, y) ; 0 \leq x \leq 1,0 \leq y \leq 1\\}\)

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

Find the minimum surface area of a rectangular closed (top, bottom, and four sides) box with volume \(64 \mathrm{~m}^{3}\).

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

Find \(\partial f / 2 x, \partial f / \partial y\), and \(\partial f / \partial z\) for the given functions. \(f(x, y, z)=x y(z+x)\)

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

Use nine evenly spaced points and five colors to draw heat maps of the following functions, defined on their specified domains. \(f(x, y)=x-y\) on \(D=\\{(x, y): 0 \leq x \leq 1,0 \leq y \leq 1\\}\)

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 \(256 \mathrm{~m}^{3}\).

Show that \(\left[\begin{array}{l}0 \\ 0\end{array}\right]\) and \(\left[\begin{array}{l}\pi \\ \pi\end{array}\right]\) are equilibria of $$ \begin{array}{l} x_{1}(t+1)=x_{2}(t) \\ x_{2}(t+1)=\sin \left(x_{2}(t)\right)+x_{1}(t) \end{array} $$ and analyze their stability.

Give a geometric interpretation of the set $$A=\left\\{(x, y) \in \mathbf{R}^{2}: \sqrt{x^{2}+y^{2}-4 y+4}<3\right\\}$$

Use nine evenly spaced points and five colors to draw heat maps of the following functions, defined on their specified domains. \(f(x, y)=x^{2}-y^{2}\) on \(D=\\{(x, y):-1 \leq x \leq 1,-1 \leq y \leq 1\\}\)

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

The distance between the origin \((0,0,0)\) and the point \((x, y, z)\) is $$\sqrt{x^{2}+y^{2}+z^{2}}$$ Find the minimum distance between the origin and the plane \(x+y+z=1\). (Hint: Minimize the squared distance between the origin and the plane.)

Find all nonnegative equilibria of $$ \begin{array}{l} x_{1}(t+1)=x_{2}(t) \\ x_{2}(t+1)=\frac{x_{1}(t)}{4}-\frac{x_{2}(t)}{4}+\left(x_{2}(t)\right)^{2} \end{array} $$ and analyze their stability.

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. \(\mathbf{f}(x, y)=\left[\begin{array}{c}\ln (x+y) \\\ e^{x+y}\end{array}\right]\)

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.

Find \(\partial f / 2 x, \partial f / \partial y\), and \(\partial f / \partial z\) for the given functions. \(f(x, y, z)=e^{x+y+z}\)

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

At the beginning of this chapter we introduced the heat index as a way of calculating how temperature and humidity affect the apparent temperature. The equation for the heat index is: \(\begin{aligned} H(T, R)=&-42.38+2.049 T+10.14 R-6.838 \times 10^{-3} T^{2} \\\ &-0.2248 T R-5.482 \times 10^{-2} R^{2}+1.229 \times 10^{-3} T^{2} R \\\ &+8.528 \times 10^{-4} T R^{2}-1.99 \times 10^{-6} T^{2} R^{2} \end{aligned}\) where \(T\) is the actual air temperature (in \({ }^{\circ} \mathrm{F}\) ) and \(R\) is the relative humidity (in \%). Using nine evenly spaced points and five colors, make a heat map for the heat index for the domain \(D=\\{(T, R):\) \(80 \leq T \leq 100,40 \leq R \leq 60]\). (You will find it easiest to calculate the heat index, \(H\), if you program the formula for the heat index into a graphing calculator.)

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

Species Diversity A frequently used measure of the diversity of an habitat is the Shannon index: $$H=-\sum_{i=1}^{n} p_{i} \ln p_{i}$$ where \(p_{i}\) is equal to the proportion organisms in the area that are species \(i, i=1,2, \ldots, n\), and \(n\) is the total number of species in the study area. Assume that a habitat harbors 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 global maximum when \(p_{1}=p_{2}=\) \(p_{3}=1 / 3 .\) (Hint : You may assume \(0 \ln 0=0\) ).

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

Find \(\partial f / 2 x, \partial f / \partial y\), and \(\partial f / \partial z\) for the given functions. . \(f(x, y, z)=\sin (x+y-z)\)

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

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

Use Lagrange multipliers to find the maxima and minima of the functions under the given constraints. $$ f(x, y)=2 x-y ; x^{2}+y^{2}=5 $$

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

Use Lagrange multipliers to find the maxima and minima of the functions under the given constraints. $$ f(x, y)=4 x^{2}+y ; x^{2}+y^{2}=1 $$

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 $$ \begin{array}{l} x_{1}(t+1)=x_{2}(t) \\ x_{2}(t+1)=\frac{1}{2} x_{1}(t)+r x_{2}(t)-\left(x_{2}(t)\right)^{2} \end{array} $$ (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 func\mathrm{tion } \(f(x, y)\) at the indicated point. \(\mathbf{f}(x, y)=\left[\begin{array}{c}3 x-y^{2} \\ 4 y\end{array}\right]\) at \((-1,-2)\)

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

Find the indicated partial derivatives. \(f(x, y)=x^{2} y^{2}+2 x y^{2} ; \frac{a^{2} f}{\partial x^{2}}\)

Determine the equation of the level curves \(f(x, y)=c\) and sketch the level curves for the specified values of \(c\). \(f(x, y)=x+y ; c=0,-1,1\)

Use Lagrange multipliers to find the maxima and minima of the functions under the given constraints. $$ f(x, y)=x y ; x^{2}+4 y^{2}=1 $$

Host-Parasitoid Interactions Find all biologically relevant equilibria of the Nicholson-Bailey model $$ \begin{array}{l} N_{t+1}=4 N_{t} e^{-0.1 P_{i}} \\ P_{t+1}=N_{t}\left[1-e^{-0.1 P_{t}}\right] \end{array} $$ and analyze their stability.