Chapter 10

Use the properties of limits to calculate the following limits: \(\lim _{(x, y) \rightarrow(0,2)}\left(4 x y^{2}-\frac{x+1}{y^{2}}\right)\)

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 (10.50). 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 \((10.50)\)

Locate the following points in a three-dimensional Cartesian coordinate system: (a) \((1,3,2)\) (b) \((-1,-2,1)\) (c) \((0,1,-1)\) (d) \((2,0,2)\)

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

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

In Problems \(1-8\), find the gradient of each function. $$ f(x, y)=\ln \left(\frac{x}{y}+\frac{y}{x}\right) $$

7\. Write down an expression for \(\frac{d z}{d t}\) where \(z=f(x, y)\) with \(x=u(t)\) and \(y=v(t) .\)

Find \(\partial f / \partial x\) and \(\partial f / \partial y\) for the given functions. \(f(x, y)=\sin ^{2}\left(y^{2} x-x^{3}\right)\)

Use the properties of limits to calculate the following limits: \(\lim _{(x, y) \rightarrow(1,1)} \frac{x y}{x^{2}+y^{2}}\)

Describe in words the set of all points in \(\mathbf{R}^{3}\) that satisfy the following expressions: (a) \(x=0\) (b) \(y=0\) (c) \(z=0\) (d) \(z \geq 0\) (e) \(y \leq 0\)

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

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

In Problems \(1-8\), find the gradient of each function. $$ f(x, y)=\cos \left(3 x^{2}-2 y^{2}\right) $$

8\. Write down an expression for \(\frac{d w}{d t}\) where \(w=e^{f(x, y)}\) with \(x=u(t)\) and \(y=v(t)\)

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

Use the properties of limits to calculate the following limits: \(\lim _{(x, y) \rightarrow(2.1)} \frac{x^{2}+y^{2}}{x^{2}-y^{2}}\)

Evaluate each function at the given point. \(f(x, y)=x^{2}+y\) at \((2,3)\)

The functions are defined for all \((x, y) \in R^{2} .\) 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 9-14, compute the directional derivative of \(f(x, y)\) at the given point in the indicated direction. $$ f(x, y)=\sqrt{x^{2}+2 y^{2}} \text { at }(-1,2) \text { in the direction }\left[\begin{array}{l} 1 \\ 1 \end{array}\right] $$

Find \(\frac{d y}{d x}\) if \(\sqrt{x^{2}+y^{2}}=1\)

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

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

Evaluate each function at the given point. \(f(x, y, z)=x^{2}-3 y+z\) at \((3,-1,1)\)

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

The functions are defined for all \((x, y) \in R^{2} .\) 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 9-14, compute the directional derivative of \(f(x, y)\) at the given point in the indicated direction. $$ f(x, y)=\sin (x+y) \text { at }(-1,0) \text { in the direction }\left[\begin{array}{r} 2 \\ -1 \end{array}\right] $$

Find \(\frac{d y}{d x}\) if \(\frac{x y}{x+y}=1\).

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 \(b>0 .\) )

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

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

Evaluate each function at the given point. \(f(x, y)=\sqrt{2 x+3 y^{2}}\) at \((-1,2)\)

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

In Problems 9-14, compute the directional derivative of \(f(x, y)\) at the given point in the indicated direction. $$ f(x, y)=\exp \left(x+y^{2}\right) \text { at }(0,0) \text { in the direction }\left[\begin{array}{r} 1 \\ -1 \end{array}\right] $$

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

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 \(P_{0}=30\).

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

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

Evaluate each function at the given point. \(f(x, y)=\frac{x^{2}}{y}\) at \((3,2)\)

Show that \(f(x, y)\) is differentiable at the indicated point. \(f(x, y)=x y+3 x^{2} ;(1,0)\)

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 9-14, compute the directional derivative of \(f(x, y)\) at the given point in the indicated direction. $$ f(x, y)=x^{3} y^{2} \text { at }(2,3) \text { in the direction }\left[\begin{array}{r} -2 \\ 1 \end{array}\right] $$

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

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 \(P_{0}=30\).

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

Use the properties of limits to calculate the following limits: \(\lim _{(x, y) \rightarrow(2,0)} \frac{2 x+4 y^{2}}{y^{2}+3 x}\)

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

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

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

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 \(a\) ?