Chapter 10

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.

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

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

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

The functional responses of some predators are sigmoidal; that is, the number of prey attacked per predator as a function of prey density has a sigmoidal shape. If we denote the prey density by \(N\), the predator density by \(P\), the time available for searching for prey by \(T\), and the handling time of each prey item per predator by \(T_{h}\), then the number of prey encounters per predator as a function of \(N, T\), and \(T_{h}\) can be expressed as $$ f\left(N, T, T_{h}\right)=\frac{b^{2} N^{2} T}{1+c N+b T_{h} N^{2}} $$ where \(b\) and \(c\) are positive constants. (a) Investigate how an increase in the prey density \(N\) affects the function \(f\left(N, T, T_{h}\right)\) (b) Investigate how an increase in the time \(T\) available for search affects the function \(f\left(N, T, T_{h}\right)\). (c) Investigate how an increase in the handling time \(T_{h}\) affects the function \(f\left(N, T, T_{h}\right)\) (d) Graph \(f\left(N, T, T_{h}\right)\) as a function of \(N\) when \(T=2.4\) hours, \(T_{h}=0.2\) hours, \(b=0.8\), and \(c=0.5\)

In this problem, we will investigate how mutual interference of parasitoids affects their searching efficiency for a host. We assume that \(N\) is the host density and \(P\) is the parasitoid density. A frequently used model for host- parasitoid interactions is the Nicholson-Bailey model (Nicholson, 1933; 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) Show that $$ b=\frac{1}{P} \ln \frac{N}{N-N_{a}} $$ by solving \((10.3)\) for \(b\). (b) Consider $$ b=f\left(P, N, N_{a}\right)=\frac{1}{P} \ln \frac{N}{N-N_{a}} $$ as a function of \(P, N\), and \(N_{a} .\) How is the searching efficiency \(b\) affected when the parasitoid density increases? (c) Assume now that the fraction of parasitized host depends on the host density; that is, assume that $$ N_{a}=g(N) $$ where \(g(N)\) is a nonnegative, differentiable function. The searching efficiency \(b\) can then be written as follows as a function of \(P\) and \(N\) : $$ b=h(P, N)=\frac{1}{P} \ln \frac{N}{N-g(N)} $$ How does the searching efficiency depend on host density when \(g(N)\) is a decreasing function of \(N ?\) (Use the fact that \(g(N)

Leopold and Kriedemann (1975) measured the crop growth rate of sunflowers as a function of leaf area index and percent of full sunlight. (Leaf area index is the ratio of leaf surface area to the ground area the plant covers.) They found that, for a fixed level of sunlight, crop growth rate first increases and then decreases as a function of leaf area index. For a given leaf area index, the crop growth rate increases with the level of sunlight. The leaf area index that maximizes the crop growth rate is an increasing function of sunlight. Sketch the crop growth rate as a function of leaf area index for different values of percent of full sunlight.

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?

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?

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.

In the introductory example, we discussed how egg size depends on maternal age. Assume now that the total amount of resources available is 10 (in appropriate units), the number of eggs per clutch is 3 , the number of clutches is 2 , and the egg size in clutch number \(i\) is denoted by \(x_{i} .\) (a) Find the constraint function. (b) Suppose the fitness function is given by $$ f\left(x_{1}, x_{2}\right)=\frac{3}{2} \rho\left(x_{1}\right)+\frac{3}{4} \rho\left(x_{2}\right) $$ where \(\rho(x)=\frac{2 x}{5+x}\). Find the optimal egg sizes for clutch 1 and clutch 2 under the constraint in (a).

Show that $$ c(x, t)=\frac{1}{\sqrt{8 \pi t}} \exp \left[-\frac{x^{2}}{8 t}\right] $$ solves $$ \frac{\partial c(x, t)}{\partial t}=2 \frac{\partial^{2} c(x, t)}{\partial x^{2}} $$

Show that $$ c(x, t)=\frac{1}{\sqrt{2 \pi t}} \exp \left[-\frac{x^{2}}{2 t}\right] $$ solves $$ \frac{\partial c(x, t)}{\partial t}=\frac{1}{2} \frac{\partial^{2} c(x, t)}{\partial x^{2}} $$

A solution of $$ \frac{\partial c(x, t)}{\partial t}=D \frac{\partial^{2} c(x, t)}{\partial x^{2}} $$ is the function $$ c(x, t)=\frac{1}{\sqrt{4 \pi D t}} \exp \left[-\frac{x^{2}}{4 D t}\right] $$ for \(x \in \mathbf{R}\) and \(t>0\). (a) Show that, as a function of \(x\) for fixed values of \(t>0, c(x, t)\) is (i) positive for all \(x \in \mathbf{R}\), (ii) is increasing for \(x<0\) and decreasing for \(x>0\), (iii) has a local maximum at \(x=0\), and (iv) has inflection points at \(x=\pm \sqrt{2 D t}\). (b) Graph \(c(x, t)\) as a function of \(x\) when \(D=1\) for \(t=0.01\), \(t=0.1\), and \(t=1\).

A solution of $$ \frac{\partial c(x, t)}{\partial t}=D \frac{\partial^{2} c(x, t)}{\partial x^{2}} $$ is the function $$ c(x, t)=\frac{1}{\sqrt{4 \pi D t}} \exp \left[-\frac{x^{2}}{4 D t}\right] $$ for \(x \in \mathbf{R}\) and \(t>0\) (a) Show that a local maximum of \(c(x, t)\) occurs at \(x=0\) for fixed \(t\) (b) Show that \(c(0, t), t>0\), is a decreasing function of \(t\). (c) Find $$ \lim _{t \rightarrow 0^{+}} c(x, t) $$ when \(x=0\) and when \(x \neq 0\) (d) Use the fact that $$ \int_{-\infty}^{\infty} e^{-u^{2} / 2} d u=\sqrt{2 \pi} $$ to show that, for \(t>0\), $$ \int_{-\infty}^{\infty} c(x, t) d x=1 $$ (e) The function \(c(x, t)\) can be interpreted as the concentration of a substance diffusing in space. Explain the meaning of $$ \int_{-\infty}^{\infty} c(x, t) d x=1 $$ and use your results in (c) and (d) to explain why this means that initially (i.e., at \(t=0\) ) the entire amount of the substance was released at the origin. Mathematically, we can specify such an initial condition (in which the substance is concentrated at the origin at time 0 ) by the \(\delta\) -function \(\delta(x)\), with the property that $$ \delta(x)=0, \quad \text { for } x \neq 0 $$ and $$ \int_{-\infty}^{\infty} \delta(x) d x=1 $$

The two-dimensional diffusion equation $$ \frac{\partial n(\mathbf{r}, t)}{\partial t}=D\left(\frac{\partial^{2} n(\mathbf{r}, t)}{\partial x^{2}}+\frac{\partial^{2} n(\mathbf{r}, t)}{\partial y^{2}}\right) $$ where \(n(\mathbf{r}, t), \mathbf{r}=(x, y)\), denotes the population density at the point \(\mathbf{r}=(x, y)\) in the plane at time \(t\), can be used to describe the spread of organisms. Assume that a large number of insects are released at time 0 at the point \((0,0)\). Furthermore, assume that, at later times, the density of these insects can be described by the diffusion equation (10.41). Show that $$ n(x, y, t)=\frac{n_{0}}{4 \pi D t} \exp \left[-\frac{x^{2}+y^{2}}{4 D t}\right] $$ satisfies (10.41).