Chapter 10

Let \(f(x, y)=x^{2}+y^{2}\) with \(x(t)=3 t\) and \(y(t)=e^{t} .\) Find the derivative of \(w=f(x, y)\) with respect to \(t\) when \(t=\ln 2\).

The tangent plane at the indicated point \(\left(x_{0}, y_{0}, z_{0}\right)\) exists. Find its equation. $$ f(x, y)=2 x^{3}+y^{2} ;(1,2,6) $$

Cardiac output (CO) is a physiological quantity that is calculated as the product of heart rate (HR) and stroke volume (SV). Write cardiac output as a function of heart rate and stroke volume. If heart rate is measured in beats per minute and stroke volume in liters per beat, what is the unit for cardiac output? Determine the domain and range of the function describing cardiac output.

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

In Problems \(1-10\), 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). \(f(x, y)=x^{2}+y^{2}-2 x\)

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

Let \(f(x, y)=e^{x} \sin y\) with \(x(t)=t\) and \(y(t)=t^{3}\). Find the derivative of \(w=f(x, y)\) with respect to \(t\) when \(t=1\).

Mean arterial blood pressure (MAP) is a function of systolic blood pressure (SP) and diastolic blood pressure (DP). At a resting heart rate, $$ \mathrm{MAP} \approx \mathrm{DP}+\frac{1}{3}(\mathrm{SP}-\mathrm{DP}) $$ If systolic pressure is greater than diastolic pressure and both are nonnegative, what is the range of the function describing mean arterial pressure?

The tangent plane at the indicated point \(\left(x_{0}, y_{0}, z_{0}\right)\) exists. Find its equation. $$ f(x, y)=x^{2}-3 y^{2} ;(-1,1,-2) $$

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

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). \(f(x, y)=-2 x^{2}-y^{2}+3 y\)

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

Let \(f(x, y)=\sqrt{x^{2}+y^{2}}\) with \(x(t)=t\) and \(y(t)=\sin t\). Find the derivative of \(w=f(x, y)\) with respect to \(t\) when \(t=\pi / 3\).

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

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

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

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). \(f(x, y)=x^{2} y-4 x^{2}-4 y\)

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

Let \(f(x, y)=\ln \left(x y-x^{2}\right)\) with \(x(t)=t^{2}\) and \(y(t)=t\). Find the derivative of \(w=f(x, y)\) with respect to \(t\) when \(t=5\).

When the initial parasitoid density is \(P_{0}=0\), the NicholsonBailey 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 \(b>0\).)

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 point \(\left(x_{0}, y_{0}, z_{0}\right)\) exists. Find its equation. $$ f(x, y)=\sin x+\cos y ;(0,0,1) $$

In Problems 1-16, find \(\partial f / \partial x\) and \(\partial f / \partial y\) for the given functions. $$ f(x, y)=\frac{y^{4}}{x^{3}}-\frac{1}{x^{3} y^{4}} $$

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). \(f(x, y)=x y-2 y^{2}\)

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

Evaluate the Nicholson-Bailey model for the first 15 generations when \(a=0.02, c=3\), and \(b=1.5 .\) For the initial host density, choose \(N_{0}=5\), and for the initial parasitoid density, choose \(P_{0}=5\).

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

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

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

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). \(f(x, y)=-2 x^{2}+y^{2}-6 y\)

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

Let \(f(x, y)=x e^{y}\) with \(x(t)=e^{t}\) and \(y(t)=t^{2} .\) Find the derivative of \(w=f(x, y)\) with respect to \(t\) when \(t=0\).

Evaluate the Nicholson-Bailey model for the first 25 generations when \(a=0.02, c=3\), and \(b=1.5 .\) For the initial host density, choose \(N_{0}=15\), and for the initial parasitoid density, choose \(P_{0}=8\).

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

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

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

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). \(f(x, y)=x(1-x+y)\)

In Problems 1-14, use the properties of limits to calculate the following limits: $$ \lim _{(x, y) \rightarrow(-5,1)} y\left(x y+x^{2} y^{2}\right) $$

Find \(\frac{d z}{d t}\) for \(z=f(x, y)\) with \(x=u(t)\) and \(y=v(t)\).

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 10 generations when \(a=0.02, c=3, k=0.75\), and \(b=1.5 .\) For the initial host density, choose \(N_{0}=5\), and for the initial parasitoid density, choose \(P_{0}=0\).

Evaluate each function at the given point. (a) \(f_{1}(x, y)=2 x-3 y^{2}\) at \((-1,2)\) (b) \(f_{2}(y, x)=2 x-3 y^{2}\) at \((-1,2)\)

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

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

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). \(f(x, y)=e^{-x^{2}-y^{2}}\)

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

Find \(\frac{d w}{d t}\) for \(w=e^{f(x, y)}\) with \(x=u(t)\) and \(y=v(t)\).

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 10 generations when \(a=0.02, c=3, k=0.75\), and \(b=0.5 .\) For the initial host density, choose \(N_{0}=15\), and for the initial parasitoid density, choose \(P_{0}=0\)

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

Evaluate each function at the given point. (a) \(f_{1}(x, y)=\frac{x}{y}\) at \((3,2)\) (b) \(f_{2}(y, x)=\frac{x}{y}\) at \((3,2)\) (c) \(f_{3}(y, x)=\frac{y}{x}\) at \((3,2)\)

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