Answer: Forestry The Doyle Log Rule is one of several methods used to determine the

In Exercises 1 and 2, use the graph to determine whether z is a function of x and y Explain.

In Exercises 1 and 2, use the graph to determine whether z is a function of x and y Explain.

In Exercises 3–6, determine whether z is a function of x and y. \(x^{2} z+3 y^{2}-x y=10\) Text Transcription: x^2 z + 3y^2 - xy = 10

In Exercises 3–6, determine whether z is a function of x and y. \(x z^{2}+2 x y-y^{2}=4\) Text Transcription: xz^2 + 2xy - y^2 = 4

In Exercises 3–6, determine whether z is a function of x and y. \(\frac{x^{2}}{4}+\frac{y^{2}}{9}+z^{2}=1\) Text Transcription: x^2/4 + y^2/9 + z^2 = 1

In Exercises 3–6, determine whether z is a function of x and y. z + x ln y - 8yz = 0

In Exercises 7–18, find and simplify the function values. f(x, y)=xy (a) (3, 2) (b) (-1. 4) (c) (30, 5) (d) (5, y) (e) (x, 2) (f) (5, t)

In Exercises 7–18, find and simplify the function values. \(f(x, y)=4-x^{2}-4 y^{2}\) (a) (0, 0) (b) (0, 1) (c) (2, 3) (d) (1, y) (e) (x, 0) (f) (t, 1) Text Transcription: f(x, y) = 4 - x^2 - 4y^2

In Exercises 7–18, find and simplify the function values. \(f(x, y)=x e^{y}\) (a) (5, 0) (b) (3, 2) (c) (2, -1) (d) (5, y) (e) (x, 2) (f) (t, t) Text Transcription: f(x, y)=xe^y

In Exercises 7–18, find and simplify the function values. g(x, y)=ln |x+y| (a) (1, 0) (b) (0, -1) (c) (0, e) (d) (1, 1) (e) (e, e/2) (f) (2, 5)

In Exercises 7–18, find and simplify the function values. \(h(x, y, z)=\frac{x y}{z}\) (a) (2, 3, 9) (b) (1, 0, 1) (c) (-2, 3, 4) (d) (5, 4, -6) Text Transcription: h(x, y, z)=xy/z

In Exercises 7–18, find and simplify the function values. \(f(x, y, z)=\sqrt{x+y+z}\) (a) (0, 5, 4) (b) (6, 8, -3) (c) (4, 6, 2) (d) (10, -4, -3) Text Transcription: f(x, y, z)=sqrt x+y+z

In Exercises 7–18, find and simplify the function values. f(x, y)=x sin y (a) \((2, \pi / 4)\) (b) (3, 1) (c) \((-3, \pi / 3)\) (d) \((4, \pi / 2)\) Text Transcription: (2, pi/4) (-3, pi/3) (4, pi/2)

In Exercises 7–18, find and simplify the function values. \(V(r, h)=\pi r^{2} h\) (a) (3, 10) (b) (5, 2) (c) (4, 8) (d) (6, 4) Text Transcription: V(r, h)=pi r^2 h

In Exercises 7–18, find and simplify the function values. \(g(x, y)=\int_{x}^{y}(2 t-3) \ d t\) (a) (4, 0) (b) (4, 1) (c) \(\left(4, \frac{3}{2}\right)\) (d) \(\left(\frac{3}{2}, 0\right)\) Text Transcription: g(x, y)=int_x^y (2t-3) dt (4, 3/2) (3/2, 0)

In Exercises 7–18, find and simplify the function values. \(g(x, y)=\int_{x}^{y} \frac{1}{t} \ d t\) (a) (4, 1) (b) (6, 3) (c) (2, 5) (d) \(\left(\frac{1}{2}, 7\right)\) Text Transcription: g(x, y)=int_x^y 1/t dt (1/2, 7)

In Exercises 7–18, find and simplify the function values. \(f(x, y)=2 x+y^{2}\) (a) \(\frac{f(x+\Delta x, y)-f(x, y)}{\Delta x}\) (b) \(\frac{f(x, y+\Delta y)-f(x, y)}{\Delta y}\) Text Transcription: f(x, y)=2x + y^2 f(x+Delta x, y) - f(x, y) / Delta x f(x, y+Delta y) - f(x, y) / Delta y

In Exercises 7–18, find and simplify the function values. \(f(x, y)=3 x^{2}-2 y\) (a) \(\frac{f(x+\Delta x, y)-f(x, y)}{\Delta x}\) (b) \(\frac{f(x, y+\Delta y)-f(x, y)}{\Delta y}\) Text Transcription: f(x, y)=3x^2 - 2y f(x+Delta x, y) - f(x, y) / Delta x f(x, y+Delta y) - f(x, y) / Delta y

In Exercises 19–30, describe the domain and range of the function. \(f(x, y)=x^{2}+y^{2}\) Text Transcription: f(x, y)=x^2 + y^2

In Exercises 19–30, describe the domain and range of the function. \(f(x, y)=e^{x y}\) Text Transcription: f(x, y)=e^xy

In Exercises 19–30, describe the domain and range of the function. \(g(x, y)=x \sqrt{y}\) Text Transcription: g(x, y)=x sqrt y

In Exercises 19–30, describe the domain and range of the function. \(g(x, y)=\frac{y}{\sqrt{x}}\) Text Transcription: g(x, y)=y / sqrt x

In Exercises 19–30, describe the domain and range of the function. \(z=\frac{x+y}{x y}\) Text Transcription: z=x+y / xy

In Exercises 19–30, describe the domain and range of the function. \(z=\frac{x y}{x-y}\) Text Transcription: z=xy/x-y

In Exercises 19–30, describe the domain and range of the function. \(f(x, y)=\sqrt{4-x^{2}-y^{2}}\) Text Transcription: f(x, y)=sqrt 4-x^2 - y^2

In Exercises 19–30, describe the domain and range of the function. \(f(x, y)=\sqrt{4-x^{2}-4 y^{2}}\) Text Transcription: f(x, y)=sqrt 4-x^2 - 4y^2

In Exercises 19–30, describe the domain and range of the function. f(x, y)=arccos (x+y)

In Exercises 19–30, describe the domain and range of the function. f(x, y)=arcsin (y/x)

In Exercises 19–30, describe the domain and range of the function. f(x, y)=ln (4-x-y)

In Exercises 19–30, describe the domain and range of the function. f(x, y)=ln (xy-6)

Think About It The graphs labeled (a), (b), (c), and (d) are graphs of the function \(f(x, y)=-4 x /\left(x^{2}+y^{2}+1\right)\). Match the four graphs with the points in space from which the surface is viewed. The four points are (20, 15, 25), (-15, 10, 20), (20, 20, 0) and (20, 0, 0). Text Transcription: f(x, y)=-4x (x^2 + y^2 + 1)

Think About It Use the function given in Exercise 31. (a) Find the domain and range of the function. (b) Identify the points in the plane at which the function value is 0. (c) Does the surface pass through all the octants of the rectangular coordinate system? Give reasons for your answer.

In Exercises 33– 40, sketch the surface given by the function. f(x, y)=4

In Exercises 33– 40, sketch the surface given by the function. f(x, y)=6-2x-3y

In Exercises 33– 40, sketch the surface given by the function. \(f(x, y)=y^{2}\) Text Transcription: f(x, y)=y^2

In Exercises 33– 40, sketch the surface given by the function. \(g(x, y)=\frac{1}{2} y\) Text Transcription: g(x, y)=1/2 y

In Exercises 33– 40, sketch the surface given by the function. \(z=-x^{2}-y^{2}\) Text Transcription: z=-x^2 - y^2

In Exercises 33– 40, sketch the surface given by the function. \(z=\frac{1}{2} \sqrt{x^{2}+y^{2}}\) Text Transcription: z=1/2 sqrt x^2 + y^2

In Exercises 33– 40, sketch the surface given by the function. \(f(x, y)=e^{-x}\) Text Transcription: f(x, y)=e^-x

In Exercises 33– 40, sketch the surface given by the function. \(f(x, y)=\left\{\begin{array}{rr}x y, & x \geq 0, y \geq 0 \\0, & x<0 \text { or } y<0\end{array}\right.\) Text Transcription: f(x, y)={x y, & x\geq 0, y geq 0 over 0, & x<0 or y<0

In Exercises 41–44, use a computer algebra system to graph the function. \(z=y^{2}-x^{2}+1\) Text Transcription: z=y^2 - x^2 + 1

In Exercises 41–44, use a computer algebra system to graph the function. \(z=\frac{1}{12} \sqrt{144-16 x^{2}-9 y^{2}}\) Text Transcription: z=1/12 sqrt 144 - 16x^2 - 9y^2

In Exercises 41–44, use a computer algebra system to graph the function. \(f(x, y)=x^{2} e^{(-x y / 2)}\) Text Transcription: f(x, y)=x^2 e^(-xy/2

In Exercises 41–44, use a computer algebra system to graph the function. f(x, y)=x sin y

In Exercises 45–48, match the graph of the surface with one of the contour maps. [The contour maps are labeled (a), (b), (c), and (d).] \(f(x, y)=e^{1-x^{2}-y^{2}}\) Text Transcription: f(x, y)=e^1 - x^2 - y^2

In Exercises 45–48, match the graph of the surface with one of the contour maps. [The contour maps are labeled (a), (b), (c), and (d).] \(f(x, y)=e^{1-x^{2}+y^{2}}\) Text Transcription: f(x, y)=e^1-x^2 + y^2

In Exercises 45–48, match the graph of the surface with one of the contour maps. [The contour maps are labeled (a), (b), (c), and (d).] \(f(x, y)=\ln \left|y-x^{2}\right|\) Text Transcription: f(x, y)=\ln |y-x^2|

In Exercises 45–48, match the graph of the surface with one of the contour maps. [The contour maps are labeled (a), (b), (c), and (d).] \(f(x, y)=\cos \left(\frac{x^{2}+2 y^{2}}{4}\right)\) Text Transcription: f(x, y)=cos (x^2 + 2y^2 / 4)

In Exercises 49–56, describe the level curves of the function. Sketch the level curves for the given c-values. z=x+y, c=-1,0,2,4

In Exercises 49–56, describe the level curves of the function. Sketch the level curves for the given c-values. z=6-2x-3y, c=0,2,4,6,8,10

In Exercises 49–56, describe the level curves of the function. Sketch the level curves for the given c-values. \(z=x^{2}+4 y^{2}\), c=0,1,2,3,4 Text Transcription: z=x^2 + 4y^2

In Exercises 49–56, describe the level curves of the function. Sketch the level curves for the given c-values. \(f(x, y)=\sqrt{9-x^{2}-y^{2}}\), c=0,1,2,3 Text Transcription: f(x, y)=sqrt 9-x^2 - y^2

In Exercises 49–56, describe the level curves of the function. Sketch the level curves for the given c-values. f(x, y)=xy, \(c=\pm 1, \pm 2, \ldots . \pm 6\) Text Transcription: c=pm 1, pm 2, ldots . pm 6

In Exercises 49–56, describe the level curves of the function. Sketch the level curves for the given c-values. \(f(x, y)=e^{x y / 2}\), \(c=2,3,4, \frac{1}{2}, \frac{1}{3}, \frac{1}{4}\) Text Transcription: f(x, y)=e^xy/2 c=2,3,4, 1/2, 1/3, 1/4

In Exercises 49–56, describe the level curves of the function. Sketch the level curves for the given c-values. \(f(x, y)=x /\left(x^{2}+y^{2}\right)\), \(c=\pm \frac{1}{2}, \pm 1, \pm \frac{3}{2}, \pm 2\) Text Transcription: f(x, y)=x/(x^2 + y^2) c=pm 1/2, pm 1, pm 3/2, pm 2

In Exercises 49–56, describe the level curves of the function. Sketch the level curves for the given c-values. f(x, y)=ln (x-y), \(c=0, \pm \frac{1}{2}, \pm 1, \pm \frac{3}{2}, \pm 2\) Text Transcription: c=0, pm 1/2, pm 1, pm 3/2, pm 2

In Exercises 57–60, use a graphing utility to graph six level curves of the function. \(f(x, y)=x^{2}-y^{2}+2\) Text Transcription: f(x, y)=x^2 - y^2 + 2

In Exercises 57–60, use a graphing utility to graph six level curves of the function. f(x, y)=|xy|

In Exercises 57–60, use a graphing utility to graph six level curves of the function. \(g(x, y)=\frac{8}{1+x^{2}+y^{2}}\) Text Transcription: g(x, y)=8 / 1 + x^2 + y^2

In Exercises 57–60, use a graphing utility to graph six level curves of the function. h(x, y)=3 sin (|x|+|y|)

What is a graph of a function of two variables? How is it interpreted geometrically? Describe level curves.

All of the level curves of the surface given by z = f(x, y) are concentric circles. Does this imply that the graph of f is a hemisphere? Illustrate your answer with an example.

Construct a function whose level curves are lines passing through the origin.

Consider the function f(x, y) = xy for \(x \geq 0\) and \(y \geq 0\) (a) Sketch the graph of the surface given by f. (b) Make a conjecture about the relationship between the graphs of f and g(x, y) = f(x, y) - 3. Explain your reasoning. (c) Make a conjecture about the relationship between the graphs of f and g(x, y) = - f(x, y). Explain your reasoning. (d) Make a conjecture about the relationship between the graphs of and \(g(x, y)=\frac{1}{2} f(x, y)\). Explain your reasoning. (e) On the surface in part (a), sketch the graph of z = f(x, x). Text transcription: x geq 0 y geq 0 g(x, y)=1/2 f(x, y)

Writing In Exercises 65 and 66, use the graphs of the level curves ( c-values evenly spaced) of the function f to write a description of a possible graph of f. Is the graph of f unique? Explain.

Writing In Exercises 65 and 66, use the graphs of the level curves ( c-values evenly spaced) of the function f to write a description of a possible graph of f. Is the graph of f unique? Explain.

Investment In 2009, an investment of $1000 was made in a bond earning 6% compounded annually. Assume that the buyer pays tax at rate R and the annual rate of inflation is I. In the year 2019, the value V of the investment in constant 2009 dollars is \(V(I, R)=1000\left[\frac{1+0.06(1-R)}{1+I}\right]^{10}\) Use this function of two variables to complete the table. Text Transcription: V(I, R)=1000[1+0.06(1-R) / 1+I]^10

Investment A principal of $5000 is deposited in a savings account that earns interest at a rate of r (written as a decimal), compounded continuously. The amount A(r, t) after t years is \(A(r, t)=5000 e^{r t}\). Use this function of two variables to complete the table. Text Transcription: A(r, t)=5000 e^rt

In Exercises 69–74, sketch the graph of the level surface f(x, y, z) = c at the given value of c. f(x, y, z)=x-y+z, c=1

In Exercises 69–74, sketch the graph of the level surface f(x, y, z) = c at the given value of c. f(x, y, z)=4x+y+2z, c=4

In Exercises 69–74, sketch the graph of the level surface f(x, y, z) = c at the given value of c. \(f(x, y, z)=x^{2}+y^{2}+z^{2}\), c=9 Text Transcription: f(x, y, z)=x^2 + y^2 + z^2

In Exercises 69–74, sketch the graph of the level surface f(x, y, z) = c at the given value of c. \(f(x, y, z)=x^{2}+\frac{1}{4} y^{2}-z\), c=1 Text Transcription: f(x, y, z)=x^2 + 1/4 y^2 - z

In Exercises 69–74, sketch the graph of the level surface f(x, y, z) = c at the given value of c. \(f(x, y, z)=4 x^{2}+4 y^{2}-z^{2}\), c=0 Text Transcription: f(x, y, z)=4x^2 + 4y^2 - z^2

In Exercises 69–74, sketch the graph of the level surface f(x, y, z) = c at the given value of c. f(x, y, z)=sin x-z, c=0

Forestry The Doyle Log Rule is one of several methods used to determine the lumber yield of a log (in board-feet) in terms of its diameter d (in inches) and its length L (in feet). The number of board-feet is \(N(d, L)=\left(\frac{d-4}{4}\right)^{2} L\) (a) Find the number of board-feet of lumber in a log 22 inches in diameter and 12 feet in length. (b) Find N(30, 12). Text Transcription: N(d, L)=(d-4/4)^2 L

Queuing Model The average length of time that a customer waits in line for service is \(W(x, y)=\frac{1}{x-y}\), x>y where y is the average arrival rate, written as the number of customers per unit of time, and x is the average service rate, written in the same units. Evaluate each of the following. (a) W(15, 9) (b) W(15, 13) (c) W(12, 7) (d) W(5, 2) Text Transcription: W(x, y)=1/x-y

Temperature Distribution The temperature T (in degrees Celsius) at any point (x, y) in a circular steel plate of radius 10 meters is \(T=600-0.75 x^{2}-0.75 y^{2}\) where x and y are measured in meters. Sketch some of the isothermal curves. Text Transcription: T=600 - 0.75x^2 - 0.75y^2

Electric Potential The electric potential V at any point (x, y) is \(V(x, y)=\frac{5}{\sqrt{25+x^{2}+y^{2}}}\) Sketch the equipotential curves for \(V=\frac{1}{2}\), \(V=\frac{1}{3}\) and \(V=\frac{1}{4}\). Text Transcription: V(x, y)=5 / sqrt 25 + x^2 + y^2 V=1/2 V=1/3 V=1/4

Cobb-Douglas Production Function Use the Cobb-Douglas production function (see Example 5) to show that if the number of units of labor and the number of units of capital are doubled, the production level is also doubled.

Cobb-Douglas Production Function Show that the CobbDouglas production function \(z=C x^{a} y^{1-a}\) can be rewritten as \(\ln \frac{z}{y}=\ln C+a \ln \frac{x}{y}\) Text Transcription: z=Cx^a y^1-a ln z/y = ln C + a ln x/y

Construction Cost A rectangular box with an open top has a length of x feet, a width of y feet, and a height of z feet. It costs $1.20 per square foot to build the base and $0.75 per square foot to build the sides. Write the cost C of constructing the box as a function of x, y and z.

Volume A propane tank is constructed by welding hemispheres to the ends of a right circular cylinder. Write the volume V of the tank as a function of r and l, where r is the radius of the cylinder and hemispheres, and l is the length of the cylinder.

Ideal Gas Law According to the Ideal Gas Law, PV = kT where P is pressure, V is volume, T is temperature (in Kelvins), and k is a constant of proportionality. A tank contains 2000 cubic inches of nitrogen at a pressure of 26 pounds per square inch and a temperature of 300 K. (a) Determine k (b) Write P as a function of V and T and describe the level curves.

Modeling Data The table shows the net sales x (in billions of dollars), the total assets y (in billions of dollars), and the shareholder’s equity z (in billions of dollars) for Wal-Mart for the years 2002 through 2007. (Source: 2007 Annual Report for Wal-Mart) A model for these data is z=f(x, y)=0.026x + 0.316y + 5.04 (a) Use a graphing utility and the model to approximate z for the given values of x and y (b) Which of the two variables in this model has the greater influence on shareholder’s equity? (c) Simplify the expression for f(x, 95) and interpret its meaning in the context of the problem.

Meteorology Meteorologists measure the atmospheric pressure in millibars. From these observations they create weather maps on which the curves of equal atmospheric pressure (isobars) are drawn (see figure). On the map, the closer the isobars the higher the wind speed. Match points A, B and C with (a) highest pressure, (b) lowest pressure, and (c) highest wind velocity

Acid Rain The acidity of rainwater is measured in units called pH. A pH of 7 is neutral, smaller values are increasingly acidic, and larger values are increasingly alkaline. The map shows curves of equal pH and gives evidence that downwind of heavily industrialized areas the acidity has been increasing. Using the level curves on the map, determine the direction of the prevailing winds in the northeastern United States.

Atmosphere The contour map shown in the figure was computer generated using data collected by satellite instrumentation. Color is used to show the “ozone hole” in Earth’s atmosphere. The purple and blue areas represent the lowest levels of ozone and the green areas represent the highest levels. (Source: National Aeronautics and Space Administration) (a) Do the level curves correspond to equally spaced ozone levels? Explain. (b) Describe how to obtain a more detailed contour map.

Geology The contour map in the figure represents color coded seismic amplitudes of a fault horizon and a projected contour map, which is used in earthquake studies. (Source: Adapted from Shipman/Wilson/Todd, An Introduction to Physical Science, Tenth Edition) (a) Discuss the use of color to represent the level curves. (b) Do the level curves correspond to equally spaced amplitudes? Explain.

True or False? In Exercises 89–92, determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If \(f\left(x_{0}, y_{0}\right)=f\left(x_{1}, y_{1}\right)\), then \(x_{0}=x_{1}\) and \(y_{0}=y_{1}\). Text Transcription: f(x_0, y_0) = f(x_1, y_1) x_0 = x_1 y_0 = y_1

True or False? In Exercises 89–92, determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If f is a function, then \(f(a x, a y)=a^{2} f(x, y)\). Text Transcription: f(ax, ay)=a^2 f(x, y)

True or False? In Exercises 89–92, determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. A vertical line can intersect the graph of z=f(x, y) at most once.

True or False? In Exercises 89–92, determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. Two different level curves of the graph of z=f(x, y) can intersect.