Chapter 10

The functions are defined for all \((x, y) \in \boldsymbol{R}^{2} .\) Find all candidates for local extrema, and use the Hessian matrix to determine the type (maximum, minimum, or saddle point). \(f(x, y)=y x e^{-y}\)

In Problems 1-14, use the properties of limits to calculate the following limits: $$ \lim _{(x, y) \rightarrow(1,1)} \frac{x y}{x^{2}+y^{2}} $$

Find \(\frac{d y}{d x}\) if \(\left(x^{2}+y^{2}\right) e^{y}=0\)

Refer to the negative binomial host-parasitoid model. Problems \(7,8,11\), and 12 are best done with the help of a spreadsheet, but can also be done with a calculator. The negative binomial model is a discrete-generation host- parasitoid model of the form $$ \begin{array}{l} N_{t+1}=b N_{t}\left(1+\frac{a P_{t}}{k}\right)^{-k} \\ P_{t+1}=c N_{t}\left[1-\left(1+\frac{a P_{t}}{k}\right)^{ k}\right] \end{array} $$ for \(t=0,1,2, \ldots\) Show that when the initial parasitoid density is \(P_{0}=0\), the negative binomial model reduces to $$ N_{t+1}=b N_{t} $$ With \(N_{0}\) denoting the initial host density, find an expression for \(N_{t}\) in terms of \(N_{0}\) and the parameter \(b\).

The tangent plane at the indicated point \(\left(x_{0}, y_{0}, z_{0}\right)\) exists. Find its equation. $$ f(x, y)=\ln (x+y) ;(2,-1,0) $$

Evaluate each function at the given point. \(h(x, t)=\exp \left[-\frac{(x-2)^{2}}{2 t}\right]\) at \((1,5)\)

In Problems 1-16, find \(\partial f / \partial x\) and \(\partial f / \partial y\) for the given functions. $$ f(x, y)=e^{\sqrt{x+y}} $$

The functions are defined for all \((x, y) \in \boldsymbol{R}^{2} .\) Find all candidates for local extrema, and use the Hessian matrix to determine the type (maximum, minimum, or saddle point). \(f(x, y)=x \cos y\)

In Problems 1-14, use the properties of limits to calculate the following limits: $$ \lim _{(x, y) \rightarrow(1,0)} \frac{x^{2}+y^{2}}{x^{2}-y^{2}} $$

Find \(\frac{d y}{d x}\) if \((\sin x+\cos y) x^{2}=0\)

Refer to the negative binomial host-parasitoid model. Problems \(7,8,11\), and 12 are best done with the help of a spreadsheet, but can also be done with a calculator. The negative binomial model is a discrete-generation host- parasitoid model of the form $$ \begin{array}{l} N_{t+1}=b N_{t}\left(1+\frac{a P_{t}}{k}\right)^{-k} \\ P_{t+1}=c N_{t}\left[1-\left(1+\frac{a P_{t}}{k}\right)^{ k}\right] \end{array} $$ for \(t=0,1,2, \ldots\) When the initial parasitoid density is \(P_{0}=0\), the negative binomial model reduces to $$ N_{t+1}=b N_{t} $$ as shown in the previous problem. For which values of \(b\) is the host density increasing if \(N_{0}>0 ?\) For which values of \(b\) is it decreasing? (Assume that \(b>0 .\) )

The tangent plane at the indicated point \(\left(x_{0}, y_{0}, z_{0}\right)\) exists. Find its equation. $$ f(x, y)=\ln \left(x^{2}+y^{2}\right) ;(1,1, \ln 2) $$

Evaluate each function at the given point. \(g(n, p)-n p(1-p)^{n-1}\) at \((5,0.1)\)

In Problems 1-16, find \(\partial f / \partial x\) and \(\partial f / \partial y\) for the given functions. $$ f(x, y)=x^{2} e^{-x y /} $$

The functions are defined for all \((x, y) \in \boldsymbol{R}^{2} .\) Find all candidates for local extrema, and use the Hessian matrix to determine the type (maximum, minimum, or saddle point). \(f(x, y)=y \sin x\)

In Problems 1-14, use the properties of limits to calculate the following limits: $$ \lim _{(x, y) \rightarrow(-1,3)} \frac{x^{2}-x y}{2 x+y} $$

Find \(\frac{d y}{d x}\) if \(\ln \left(x^{2}+y^{2}\right)=3 x y\)

Refer to the negative binomial host-parasitoid model. Problems \(7,8,11\), and 12 are best done with the help of a spreadsheet, but can also be done with a calculator. The negative binomial model is a discrete-generation host- parasitoid model of the form $$ \begin{array}{l} N_{t+1}=b N_{t}\left(1+\frac{a P_{t}}{k}\right)^{-k} \\ P_{t+1}=c N_{t}\left[1-\left(1+\frac{a P_{t}}{k}\right)^{ k}\right] \end{array} $$ for \(t=0,1,2, \ldots\) Evaluate the negative binomial model for the first 25 generations when \(a=0.02, c=3, k=0.75\), and \(b=1.5 .\) For the initial host density, choose \(N_{0}=100\), and for the initial parasitoid density, choose \(P_{0}=50\).

Evaluate each function at the given point. \(h\left(x_{1}, x_{2}\right)=x_{2} e^{-x_{1} / x_{2}}\) at \((2,-1)\)

In Problems 1-16, find \(\partial f / \partial x\) and \(\partial f / \partial y\) for the given functions. $$ f(x, y)=e^{x} \sin (x y) $$

In Problems 1-14, use the properties of limits to calculate the following limits: $$ \lim _{(x, y) \rightarrow(0,1)} \frac{2 x y-3}{x^{2}+y^{2}+1} $$

Find \(\frac{d y}{d x}\) if \(\cos \left(x^{2}+y^{2}\right)=\sin \left(x^{2}-y^{2}\right)\).

Refer to the negative binomial host-parasitoid model. Problems \(7,8,11\), and 12 are best done with the help of a spreadsheet, but can also be done with a calculator. The negative binomial model is a discrete-generation host- parasitoid model of the form $$ \begin{array}{l} N_{t+1}=b N_{t}\left(1+\frac{a P_{t}}{k}\right)^{-k} \\ P_{t+1}=c N_{t}\left[1-\left(1+\frac{a P_{t}}{k}\right)^{ k}\right] \end{array} $$ for \(t=0,1,2, \ldots\) Evaluate the negative binomial model for the first 25 generations when \(a=0.02, c=3, k=0.75\), and \(b=0.5 .\) For the initial host density, choose \(N_{0}=100\), and for the initial parasitoid density, choose \(P_{0}=50\).

Show that \(f(x, y)\) is differentiable at the indicated point. $$ f(x, y)=x y-3 x^{2} ;(1,1) $$

Evaluate each function at the given point. \(g\left(x_{1}, x_{2}, x_{3}, x_{4}\right)=x_{1} x_{4} \sqrt{x_{2} x_{3}}\) at \((1,8,2,-1)\)

In Problems 1-16, find \(\partial f / \partial x\) and \(\partial f / \partial y\) for the given functions. $$ f(x, y)=e^{-y^{2}} \cos \left(x^{2}-y^{2}\right) $$

Consider the function $$ f(x, y)=a x^{2}+b y^{2} $$ (a) Show that $$ \nabla f(0,0)=\left[\begin{array}{l} 0 \\ 0 \end{array}\right] $$ (b) Find values for \(a\) and \(b\) such that (i) \((0,0)\) is a local minimum, (ii) \((0,0)\) is a local maximum, and (iii) \((0,0)\) is a saddle point.

In Problems 1-14, use the properties of limits to calculate the following limits: $$ \lim _{(x, y) \rightarrow(-1,-2)} \frac{x^{2}-y^{2}}{2 x y+2} $$

Find \(\frac{d y}{d x}\) if \(y=\arccos x\).

In the Nicholson-Bailey model, the fraction of hosts escaping parasitism is given by $$ f(P)=e^{-a P} $$ (a) Graph \(f(P)\) as a function of \(P\) for \(a=0.1\) and \(a=0.01\). (b) For a given value of \(P\), how are the chances of escaping parasitism affected by increasing \(a\) ?

Show that \(f(x, y)\) is differentiable at the indicated point. $$ f(x, y)=\cos (x+y) ;(0,0) $$

Find the largest possible domain and the corresponding range of each function. Determine the equation of the level curves \(f(x, y)=c\), together with the possible values of \(c .\) \(f(x, y)=x^{2}+y^{2}\)

In Problems 1-16, find \(\partial f / \partial x\) and \(\partial f / \partial y\) for the given functions. $$ f(x, y)=\ln (2 x+y) $$

In Problems \(13-16\), the functions are defined on the rectangular domain $$ D=\\{(x, y):-1 \leq x \leq 1,-1 \leq y \leq 1\\} $$ Find the absolute maxima and minima of \(f\) on \(D\). \(f(x, y)=2 x-y\)

In Problems 1-14, use the properties of limits to calculate the following limits: $$ \lim _{(x, y) \rightarrow(2,0)} \frac{2 x+4 y^{2}}{y^{2}+3 x} $$

Find \(\frac{d y}{d x}\) if \(y=\arctan x\)Find \(\frac{d y}{d x}\) if \(y=\arctan x\).

In the negative binomial model, the fraction of hosts escaping parasitism is given by $$ f(P)=\left(1+\frac{a P}{k}\right)^{-k} $$ (a) Graph \(f(P)\) as a function of \(P\) for \(a=0.1\) and \(a=0.01\) when \(k=0.75\) (b) For \(k=0.75\) and a given value of \(P\), how are the chances of escaping parasitism affected by increasing \(a\) ?

Show that \(f(x, y)\) is differentiable at the indicated point. $$ f(x, y)=e^{x-y} ;(0,0) $$

Find the largest possible domain and the corresponding range of each function. Determine the equation of the level curves \(f(x, y)=c\), together with the possible values of \(c .\) \(f(x, y)=\sqrt{9-x^{2}-y^{2}}\)

In Problems 1-16, find \(\partial f / \partial x\) and \(\partial f / \partial y\) for the given functions. $$ f(x, y)=\ln \left(3 x^{2}-x y\right) $$

The functions are defined on the rectangular domain $$ D=\\{(x, y):-1 \leq x \leq 1,-1 \leq y \leq 1\\} $$ Find the absolute maxima and minima of \(f\) on \(D\). \(f(x, y)=3-x+2 y\)

In Problems 1-14, use the properties of limits to calculate the following limits: $$ \lim _{(x, y) \rightarrow(1,-2)} \frac{2 x^{2}+y}{2 x y+3} $$

The growth rate \(r\) of a particular organism is affected by both the availability of food and the number of competitors for the food source. Denote the amount of food available at time \(t\) by \(F(t)\) and the number of competitors at time \(t\) by \(N(t)\). The growth rate \(r\) can then be thought of as a function of the two time-dependent variables \(F(t)\) and \(N(t)\). Assume that the growth rate is an increasing function of the availability of food and a decreasing function of the number of competitors. How is the growth rate \(r\) affected if the availability of food decreases over time while the number of competitors increases?

Show that \(f(x, y)\) is differentiable at the indicated point. $$ f(x, y)=x+y^{2}-2 x y ;(-1,2) $$

Find the largest possible domain and the corresponding range of each function. Determine the equation of the level curves \(f(x, y)=c\), together with the possible values of \(c .\) \(f(x, y)=\ln \left(y-x^{2}\right)\)

In Problems 1-16, find \(\partial f / \partial x\) and \(\partial f / \partial y\) for the given functions. $$ f(x, y)=\log _{3}\left(y^{2}-x^{2}\right) $$

The functions are defined on the rectangular domain $$ D=\\{(x, y):-1 \leq x \leq 1,-1 \leq y \leq 1\\} $$ Find the absolute maxima and minima of \(f\) on \(D\). \(f(x, y)=x^{2}-y^{2}\)

Show that $$ \lim _{(x, y) \rightarrow(0,0)} \frac{x^{2}-2 y^{2}}{x^{2}+y^{2}} $$ does not exist by computing the limit along the positive \(x\) -axis and the positive \(y\) -axis.

The negative binomial model can be reduced to the Nicholson-Bailey model by letting the parameter \(k\) in the negative binomial model go to infinity. Show that $$ \lim _{k \rightarrow \infty}\left(1+\frac{a P}{k}\right)^{-k}=e^{-a P} $$ (Hint: Use l'Hospital's rule.)

Show that \(f(x, y)\) is differentiable at the indicated point. $$ f(x, y)=\tan \left(x^{2}+y^{2}\right) ;\left(\frac{\pi}{4},-\frac{\pi}{4}\right) $$