Chapter 10

Determine the equation of the level curves \(f(x, y)=c\) and sketch the level curves for the specified values of \(c\). \(f(x, y)=y-x^{2} ; c=0,1,2\)

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

Host-Parasitoid Interactions 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 analyze their stability.

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

Determine the equation of the level curves \(f(x, y)=c\) and sketch the level curves for the specified values of \(c\). \(f(x, y)=x y ; c=0,1,2\)

Find a linear approximation to each func\mathrm{tion } \(f(x, y)\) at the indicated point. \(\mathbf{f}(x, y)=\left[\begin{array}{l}\frac{x}{y} \\\ \frac{y}{x}\end{array}\right]\) at \((1,2)\)

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

Host-Parasitoid Interactions 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.

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

Determine the equation of the level curves \(f(x, y)=c\) and sketch the level curves for the specified values of \(c\). \(f(x, y)=x^{2}-y^{2} ; c=0,1,-1\)

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

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

Determine the equation of the level curves \(f(x, y)=c\) and sketch the level curves for the specified values of \(c\). \(f(x, y)=y / 2 ; c=0,1,2\)

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. $$ f(x, y)=x^{2} y ; x^{2}+3 y=1 $$

Find the indicated partial derivatives. \(g(s, t)=\ln \left(s+t^{2}\right) ; \frac{\partial^{2} g}{\partial s^{2}}\)

Determine the equation of the level curves \(f(x, y)=c\) and sketch the level curves for the specified values of \(c\). \(f(x, y)=\frac{x-y}{x+y} ; c=0,1,2\)

Find a linear approximation to $$ \mathbf{f}(x, y)=\left[\begin{array}{l} x / y \\ 2 x y \end{array}\right] $$ at \((-1,1)\). Use your result to find an approximation for \(f(-0.9,1.05)\), and compare the approximation with the value of \(f(-0.9,1.05)\) that you get when you use a calculator.

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

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

Let $$f_{a}(x, y)=a x^{2}+y^{2}$$ for \((x, y) \in \mathbf{R}\), where \(a\) is a positive constant. (a) Assume that \(a=1\) and describe the level curves of \(f_{1}\). The graph of \(f_{1}(x, y)\) intersects both the \(x-z\) and the \(y-z\) planes; show that these two curves of intersection are parabolas. (b) Assume that \(a=4\). Then $$f_{4}(x, y)=4 x^{2}+y^{2}$$ and the level curves satisfy $$4 x^{2}+y^{2}=c$$ Use a graphing calculator to sketch the level curves for \(c=\) \(0,1,2,3\), and \(4 .\) These curves are ellipses. Find the curves of intersection of \(f_{4}(x, y)\) with the \(x-z\) and the \(y-z\) planes. (c) Repeat (b) for \(a=1 / 4\). (d) Explain in words how the surfaces of \(f_{a}(x, y)\) change when \(a\) changes.

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.

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

Find a linear approximation to $$ \mathbf{f}(x, y)=\left[\begin{array}{c} \sqrt{2 x+y} \\ x-y^{2} \end{array}\right] $$ at \((1,2) .\) Use your result to find an approximation for \(f(1.05,2.05)\), and compare the approximation with the value of \(f(1.05,2.05)\) that you get when you use a calculator.

Let $$f(x, y)=x+y \quad(x, y) \in \mathbf{R}^{2}$$ with constraint function \(x y=1\). (a) Use Lagrange multipliers to find all local extrema. (b) Are there global extrema?

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

The Lotka-Volterra equations are often used to model the links between a particular of prey organisms (e.g., sardines) and a population of predatory organisms (e.g., sharks), (see Chapter 11.) In a particular ecosystem we will use \(u\) to represent the number of sharks and \(v\) to represent the number of sardines. Suppose the growth rate of the shark population is $$ f(u, v)=-0.5 u+\frac{u v}{100} $$ and of the sardine population is $$ g(u, v)=3 v-10 u v $$ (a) Show that if \(u=0.3\) and \(v=50\), then \(f(u, v)=0\), and \(g(u, v)=0\). (The populations are said to be in equilibrium.) (b) Find the linear approximation of the vector valued function $$ \mathbf{h}:(u, v) \mapsto\left[\begin{array}{l} f(u, v) \\ g(u, v) \end{array}\right] $$ if \(u\) is close to \(0.3\) and \(v\) is close to 50 .

Let $$f(x, y)=x+y$$ with constraint function $$\frac{1}{x}+\frac{1}{y}=1, x \neq 0, y \neq 0$$ (a) Use Lagrange multipliers to find all local extrema. (b) Are there global extrema?

Find the indicated partial derivatives. \(f(x, y)=\sin (3 x y) ; \frac{\partial^{2} f}{\partial y^{2}}\)

Two different species of organisms may compete for the same limited resource, for example, hyenas and lions may compete for territory and prey. A large lion population will reduce the growth rate of the hyena population, and vice versa. Let the number of lions be \(x\) and the number of hyenas be \(y .\) The growth rates of the lion and hyena populations may be modeled using Lotka- Volterra equations (see Chapter 11). Suppose the growth rates are given by functions: $$ \begin{array}{l} f(x, y)=0.5 x-\frac{x^{2}}{10}-\frac{x y}{20} \\ g(x, y)=2 y-\frac{y^{2}}{20}-\frac{x y}{5} \end{array} $$ (a) Show that \(f(x, y)=0\) and \(g(x, y)=0\) when \(x=0\) and \(y=40\). The two populations are said to be in equilibrium when at this size. (b) Linearize the vector valued function $$ \mathbf{h}:(x, y) \mapsto\left[\begin{array}{l} f(x, y) \\ g(x, y) \end{array}\right] $$ when \(x\) is approximately 0, and \(y\) is approximately 40 .

Let $$f(x, y)=x y, \quad(x, y) \in \mathbf{R}^{2}$$ with constraint function \(y-x^{2}=0\). (a) Use Lagrange multipliers to find candidates for local extrema. (b) Use the constraint \(y-x^{2}=0\) to reduce \(f(x, y)\) to a singlevariable function, and then use this function to show that \(f(x, y)\) has no local extrema on the constraint curve.

The prey-density responses of some predators are sigmoidal: the number of prey attacked has a sigmoidal shape when plotted as a function of prey density. If we denote the number of nearby prey by \(N\), and the handling time of each prey item by \(T_{h}\), then the rate of prey attacks per predator as a function of \(N\) and \(T_{h}\) can be expressed as $$P\left(N, T_{h}\right)=\frac{b^{2} N^{2}}{1+c N+b T_{h} N^{2}}$$ where \(b\) and \(c\) are positive constants. (a) Investigate how an increase in the prey density \(N\) affects the function \(P\left(N, T_{h}\right)\). (b) Investigate how an increase in the handling time \(T_{h}\) affects the function \(P\left(N, T_{h}\right)\). (c) Graph \(P\left(N, T_{h}\right)\) as a function of \(N\) when \(T_{h}=0.2\) hours, \(b=0.8\), and \(c=0.5\)

Parasites live by stealing resources from hosts. When parasites reproduce their offspring must find new hosts. However, if a potential host is already infected by parasites, then new parasites will not be able to infect it. This leads to interference between parasites, and we will build a model for these effects in this Problem. We assume that \(N\) is the number of hosts in a given area, and \(P\) is the number of parasites. A frequently used model for host- parasite interactions is the Nicholson-Bailey model (see Nicholson and Bailey, 1935 ), in which it is assumed that the number of parasitized hosts, denoted by \(N_{a}\), is given by $$N_{a}=N\left[1-e^{-b P}\right]$$ where \(b\) is the searching efficiency. (a) Let's treat \(N\) and \(P\) as independent variables and \(N_{n}\) as a function of \(N\) and \(P .\) By calculating the appropriate partial derivatives investigate how: (i) an increase in the number of hosts \(N\) affects the number of parasitized hosts \(N_{a}(N, P)\) (ii) an increase in the number of parasites affects the number of parasitized hosts \(N_{a}(N, P)\) (b) Show that $$b=\frac{1}{P} \ln \frac{N}{N-N_{a}}$$ by solving \((10.6)\) for \(b\). (c) Consider $$b=f\left(P, N, N_{\alpha}\right)=\frac{1}{P} \ln \frac{N}{N-N_{a}}$$ That is, we regard searching efficiency as a function of \(P, N\), and \(N_{n} .\) How is the searching efficiency \(b\) affected when the, number of parasites increases?

In Section \(10.1\), Problem 3 , we introduced wind chill as a way of calculating the apparent temperature a person would feel as a function of the real air temperature, \(T\), and \(V\) in mph. Then the wind chill (i.e., the apparent temperature) is: $$\begin{array}{c}W(T, V)=35.74+0.6215 T-35.75 V^{0.16} \\\0.4275 T V^{0.16}\end{array}$$ (a) By calculating the appropriate partial derivative, show that increasing \(T\) always increases \(W\). (b) Under what conditions does increasing \(V\) decrease \(W\) ? Your answer will take the form of an inequality involving \(T\). (c) Assuming that \(W\) should always decrease when \(V\) is increased, use your answer from (b) to determine the largest domain in which this formula for \(W\) can be used.

In the introductory example in this subsection, we discussed how egg size depends on maternal age. Assume now that the fitness function is given by $$f\left(x_{1}, x_{2}\right)=\frac{5}{3} \rho\left(x_{1}\right)+\frac{5}{6} \rho\left(x_{2}\right)$$ with $$\rho(x)=\frac{3 x}{4+x}$$ The constraint function is given by $$5 x_{1}+5 x_{2}=7$$ (a) Compare the given functions with the corresponding ones in the text, and identify the parameters \(n, p_{1}, p_{2}\), and \(R\) from the text. (b) Solve the constraint function for \(x_{2}\) and substitute your expression for \(x_{2}\) into the function \(f .\) This then yields a function of one variable. Find the domain of this single-variable function and use single- variable calculus to determine optimal egg sizes for clutch 1 and clutch 2.

The amount of plant biomass (that is, weight of living plant matter), \(y\), produced in a particular soil patch is thought to depend linearly on the amount of nitrogen added to the soil. Let \(x\) represent the amount of added nitrogen. Suppose you measure the following data by adding different amounts of nitrogen $$\begin{array}{ccccccc} \hline \boldsymbol{x}_{\boldsymbol{i}} & 0 & 0.5 & 1 & 1 & 4 & 6 \\ \boldsymbol{y}_{\boldsymbol{i}} & 0.79 & 1.62 & 5.54 & 8.14 & 5.00 & 24.62 \\ \hline \end{array}$$ Assuming that \(y_{i}=m x_{i}+c\), find least squares estimated for the coefficients \(m\) and \(c\).

One of the most fundamental power laws in biology is how the energy (or metabolic) needs of an animal increase as a function of its body mass. Kolokotrones et al. (2010) examined how the metabolic demands (measured in watts) of 636 species of mammals depend on their mass, \(M\), measured in grams. Here are some representative data points from their paper: $$\begin{array}{lccccccc} \hline \boldsymbol{M} & 5.28 & 22.6 & 121 & 180 & 608 & 2.99 \times 10^{3} & 2.68 \times 10^{4} \\ \boldsymbol{B} & 0.109 & 0.215 & 0.455 & 0.9921 & 2.21 & 6.81 & 37.3 \\ \hline \end{array}$$ It has been hypothesized that \(B\) has a power law dependence on \(M\) \(B=c M^{a}\) for some coefficients \(c\) and \(a\). (a) Explain how the data can be transformed so that it may be plotted as a straight line [Hint: what if \(\log B\) is plotted against \(\log M ?]\) (b) Use the method of least squares errors to estimate the coefficient \(a\). (c) A long-standing hypothesis, known as Kleiber's law, states that \(a=3 / 4\). Is that consistent with your estimate from (b)?

A radioactive isotope decays over time, following an exponential decay law. That is, the amount of isotope left at time \(t\) is predicted to be: $$W(t)=W_{0} e^{-\lambda t}$$ where \(W_{0}\) and \(\lambda\) are both coefficients. You measure the following data on the amount of isotope left in a particular sample, \(W\), at different times \(t .\) $$\begin{array}{lcccccc} \hline \boldsymbol{t} & 0 & 0.1 & 0.2 & 0.4 & 0.8 & 1.0 \\ \boldsymbol{W} & 113.2 & 63.7 & 66.0 & 32.1 & 13.1 & 3.89 \\ \hline \end{array}$$ (a) Use a least squares method to estimate the coefficients \(W_{0}\) and \(\lambda\). (b) When the fitted coefficients \(W_{0}\) and \(\lambda\) are input into the model, what is the predicted half-life of the isotope (that is, the time taken for the amount of isotope present to decay from \(W_{0}\) to \(\frac{1}{2} W_{0}\) )?

A particular chemical reaction is predicted to have Michaelis-Menten kinetics, meaning that the rate of reaction, \(r\), is related to the concentration of the reacting chemical, \(C\), by the function: $$r=\frac{k C}{C+a}$$ where \(k\) and \(a\) are constants. (a) Show that the reaction rate equation can be rewritten as: $$\frac{1}{r}=\frac{a}{k} \cdot \frac{1}{C}+\frac{1}{k}$$ (b) Explain using (a) how the constants \(a\) and \(k\) could be fit from a plot of \(\frac{1}{r}\) against \(\frac{1}{C}\). You measure the following data for reaction rates, \(r\), at different chemical concentrations, \(C\). $$\begin{array}{ccccccc} \hline \boldsymbol{C} & 0 & 0.1 & 0.5 & 1 & 2 & 4 \\ \boldsymbol{r} & 0 & 0.28 & 1.01 & 1.21 & 1.65 & 1.42 \\ \hline \end{array}$$ Use the least squares error method to estimate \(k\) and \(a\) from these data. (Hint: Omit the point \((C, r)=(0,0)\) ).

Suppose the size of an insect population, \(N(t)\), grows with time \(t\), according to the function $$N(t)=M t e^{-m t}$$ where \(M\) and \(m\) are coefficients. (a) Show that the model can be rewritten as: $$\ln \left(\frac{N}{t}\right)=\ln M-m t$$ (b) Explain how the coefficients \(m\) and \(M\) can be estimated from a plot of \(\ln (N / t)\) against \(t\). $$\begin{array}{lccccc} \hline \boldsymbol{t} & 0.1 & 0.3 & 0.5 & 0.8 & 1 \\ \boldsymbol{N} & 6.11 & 1.64 & 1.00 & 0.196 & 0.0633 \\ \hline \end{array}$$ (c) Use a least squares error method to fit \(M\) and \(m\) from the following experimental data.