Ask your homework questions to teachers and professors, meet other students, and be entered to win $600 or an Xbox Series X 🎉Join our Discord! ## Chapter 14 ## Partial Derivatives ## Educators ag JF  + 1 more educators ### Problem 1 In Example 2 we considered the function$ W = f(T,v) $, where$ W $is the wind-chill index,$ T $is the actual temperature, and$ v $is the wind speed. A numerical representation is given in Table 1 on page 889. (a) What is the value of$ f(-15,40) $? What is its meaning? (b) Describe in words the meaning of the question "For what value of$ v $is$ f(-20, v) = -30 $?" Then answer the question. (c) Describe in words the meaning of the question "For what value of$ T $is$ f(T, 20) = -49 $?" Then answer the question. (d) What is the meaning of the function$ W = f(-5, v) $? Describe the behavior of this function. (e) What is the meaning of the function$ W = f(T, 50) $? Describe the behavior of this function. Jacquelyn T. Numerade Educator ### Problem 2 The$ \textit{temperature-humidity index I} $(or humidex, for short) is the perceived air temperature when the actual temperature is$ T $and the relative humidity is$ h $, so we can write$ I = f(T, h) $. The following table of values of$ I $is an excerpt from a table compiled by the National Oceanic & Atmospheric Administration. (a) What is the value of$ f(95, 70) $? What is its meaning? (b) For what value of$ h $is$ f(90, h) = 100 $? (c) For what value of$ T $is$ f(T, 50) = 88 $? (d) What are the meanings of the functions$ I = f(80, h) $and$ I = f(100, h) $? Compare the behavior of these two functions of$ h $. ag Alan G. Numerade Educator ### Problem 3 A manufacturer has modeled its yearly production function$ P $(the monetary value of its entire production in millions of dollars) as a Cobb-Douglas function $$P(L, K) = 1.47L^{0.65}K^{0.35}$$ where$ L $is the number of labor hours (in thousands) and$ K $is the invested capital (in millions of dollars). Find$ P(120, 20) $and interpret it. Carson M. Numerade Educator ### Problem 4 Verify for the Cobb-Douglas production function $$P(L, K) = 1.01L^{0.75}K^{0.25}$$ discussed in Example 3 that the production will be doubled if both the amount of labor and the amount of capital are doubled. Determine whether this is also true for the general production function $$P(L, K) = bL^{\alpha}K^{1 - \alpha}$$ JF Jun F. Numerade Educator ### Problem 5 A model for the surface area of a human body is given by the function $$S = f(w, h) = 0.1091w^{0.425}h^{0.725}$$ where$ w $is the weight (in pounds),$ h $is the height (in inches), and$ S $is measured in square feet. (a) Find$ f(160, 70) $and interpret it. (b) What is your own surface area? Carson M. Numerade Educator ### Problem 6 The wind-chill index$ W $discussed in Example 2 has been modeled by the following function: $$W(T, v) = 13.12 + 0.6215T - 11.37v^{0.16} + 0.3965Tv^{0.16}$$ Check to see how closely this model agrees with the values in Table 1 for a few values of$ T $and$ v $. ag Alan G. Numerade Educator ### Problem 7 The wave heights$ h $in the open sea depend on the speed$ v $of the wind and the length of time$ t $that the wind has been blowing at that speed. Values of the function$ h = f(v, t) $are recorded in feet in Table 4. (a) What is the value of$ f(40, 15) $? What is its meaning? (b) What is the meaning of the function$ h = f(30, t) $? Describe the behavior of this function. (c) What is the meaning of the function$ h = f(v, 30) $? Describe the behavior of this function. ag Alan G. Numerade Educator ### Problem 8 A company makes three sizes of cardboard boxes: small, medium, and large. It costs$ \$2.50$ to make a small box, $\$ 4.00 $for a medium box, and$ \$4.50$ for a large box. Fixed costs are $\$ 8000 $. (a) Express the cost of making$ x $small boxes,$ y $medium boxes, and$ z $large boxes as a function of three variables:$ C = f(x, y, z) $. (b) Find$ f(3000, 5000, 4000) $and interpret it. (c) What is the domain of$ f $? ag Alan G. Numerade Educator ### Problem 9 Let$ g(x, y) = \cos(x + 2y) $. (a) Evaluate$ g(2, -1) $. (b) Find the domain of$ g $. (c) Find the range of$ g $. Bobby B. University of North Texas ### Problem 10 Let$ F(x, y) = 1 + \sqrt{4 - y^2} $. (a) Evaluate$ F(3, 1) $. (b) Find and sketch the domain of$ F $. (c) Find the range of$ F $. Jacquelyn T. Numerade Educator ### Problem 11 Let$ f(x, y, z) = \sqrt{x} + \sqrt{y} + \sqrt{z} + \ln (4 - x^2 - y^2 - z^2) $. (a) Evaluate$ f(1, 1, 1) $. (b) Find and describe the domain of$ f $. ag Alan G. Numerade Educator ### Problem 12 Let$ g(x, y, z) = x^3y^2z\sqrt{10 - x - y - z} $. (a) Evaluate$ g(1, 2, 3) $. (b) Find and describe the domain of$ g $. ag Alan G. Numerade Educator ### Problem 13 Find and sketch the domain of the function.$ f(x, y) = \sqrt{x - 2} + \sqrt{y - 1} $ag Alan G. Numerade Educator ### Problem 14 Find and sketch the domain of the function.$ f(x, y) = \sqrt{x - 3y} $ag Alan G. Numerade Educator ### Problem 15 Find and sketch the domain of the function.$ f(x, y) = \ln (9 - x^2 - 9y^2) $ag Alan G. Numerade Educator ### Problem 16 Find and sketch the domain of the function.$ f(x, y) = \sqrt{x^2 + y^2 - 4} $ag Alan G. Numerade Educator ### Problem 17 Find and sketch the domain of the function.$ g(x, y) = \dfrac{x - y}{x + y} $ag Alan G. Numerade Educator ### Problem 18 Find and sketch the domain of the function.$ g(x, y) = \dfrac{\ln(2 - x)}{1 - x^2 - y^2} $ag Alan G. Numerade Educator ### Problem 19 Find and sketch the domain of the function.$ f(x, y) = \dfrac{\sqrt{y - x^2}}{1 - x^2} $ag Alan G. Numerade Educator ### Problem 20 Find and sketch the domain of the function.$ f(x, y) = \sin^{-1}(x + y) $ Jacquelyn T. Numerade Educator ### Problem 21 Find and sketch the domain of the function.$ f(x, y, z) = \sqrt{4 - x^2} + \sqrt{9 - y^2} + \sqrt{1 - z^2} $ag Alan G. Numerade Educator ### Problem 22 Find and sketch the domain of the function.$ f(x, y, z) = \ln(16 - 4x^2 - 4y^2 - z^2) $ Jacquelyn T. Numerade Educator ### Problem 23 Sketch the graph of the function.$ f(x, y) = y $ Amrita B. Numerade Educator ### Problem 24 Sketch the graph of the function.$ f(x, y) = x^2 $ Amrita B. Numerade Educator ### Problem 25 Sketch the graph of the function.$ f(x, y) = 10 - 4x - 5y $ Amrita B. Numerade Educator ### Problem 26 Sketch the graph of the function.$ f(x, y) = \cos y $ Amrita B. Numerade Educator ### Problem 27 Sketch the graph of the function.$ f(x, y) = \sin x $ Jacquelyn T. Numerade Educator ### Problem 28 Sketch the graph of the function.$ f(x, y) = 2 - x^2 - y^2 $ Carson M. Numerade Educator ### Problem 29 Sketch the graph of the function.$ f(x, y) = x^2 + 4y^2 + 1 $ag Alan G. Numerade Educator ### Problem 30 Sketch the graph of the function.$ f(x, y) = \sqrt{4x^2 + y^2} $ Carson M. Numerade Educator ### Problem 31 Sketch the graph of the function.$ f(x, y) = \sqrt{4 - 4x^2 - y^2} $ Carson M. Numerade Educator ### Problem 32 Match the function with its graph (labeled I - VI). Give reasons for your choices. (a)$ f(x, y) = \dfrac{1}{1 + x^2 + y^2} $(b)$ f(x, y) = \dfrac{1}{1 + x^2y^2} $(c)$ f(x, y) = \ln(x^2 + y^2) $(d)$ f(x, y) = \cos\sqrt{x^2 + y^2} $(e)$ f(x, y) = | xy | $(f)$ f(x, y) = \cos(xy) $ag Alan G. Numerade Educator ### Problem 33 A contour map for a function$ f $is shown. Use it to estimate the values of$ f(-3, 3) $and$ f(3, -2) $. What can you say about the shape of the graph? ag Alan G. Numerade Educator ### Problem 34 Shown is a contour map of atmospheric pressure in North America on August 12, 2008. On the level curves (called isobars) the pressure is indicated in millibars (mb). (a) Estimate the pressure at$ C $(Chicago),$ N $(Nashville),$ S $(San Francisco), and$ V $(Vancouver). (b) At which of these locations were the winds strongest? ag Alan G. Numerade Educator ### Problem 35 Level curves (isothermals) are shown for the typical water temperature (in$^{\circ} $C) in Long Lake (Minnesota) as a function of depth and time of year. Estimate the temperature in the lake on June 9 (day 160) at a depth of 10 m and on June 29 (day 180) at a depth of 5 m. ag Alan G. Numerade Educator ### Problem 36 Two contour maps are shown. One is for a function$ f $whose graph is a cone. The other is for a function$ g $whose graph is a paraboloid. Which is which, and why? Jacquelyn T. Numerade Educator ### Problem 37 Locate the points$ A $and$ B $on the map of Lonesome Mountain (Figure 12). How would you describe the terrain near$ A $? Near$ B $? ag Alan G. Numerade Educator ### Problem 38 Make a rough sketch of a contour map for the function whose graph is shown. ag Alan G. Numerade Educator ### Problem 39 The$ \textit{body mass index} $(BMI) of a person is defined by $$B(m, h) = \dfrac{m}{h^2}$$ where$ m $is the person's mass (in kilograms) and$ h $is the height (in meters). Draw the level curves$ B(m, h) = 18.5 $,$ B(m, h) = 25 $,$ B(m, h) = 30 $, and$ B(m, h) = 40 $. A rough guideline is that a person is underweight if the BMI is less than 18.5; optimal if the BMI lies between 18.5 and 25; overweight if the BMI lies between 25 and 30; and obese if the BMI exceeds 30. Shade the region corresponding to optimal BMI. Does someone who weighs 62 kg and is 152 cm tall fall into this category? ag Alan G. Numerade Educator ### Problem 40 The body mass index is defined in Exercise 39. Draw the level curve of this function corresponding to someone who is 200 cm tall and weighs 80 kg. Find the weights and heights of two other people with that same level curve. Carson M. Numerade Educator ### Problem 41 A contour map of a function is shown. Use it to make a rough sketch of the graph of$ f $. ag Alan G. Numerade Educator ### Problem 42 A contour map of a function is shown. Use it to make a rough sketch of the graph of$ f $. ag Alan G. Numerade Educator ### Problem 43 A contour map of a function is shown. Use it to make a rough sketch of the graph of$ f $. Carson M. Numerade Educator ### Problem 44 A contour map of a function is shown. Use it to make a rough sketch of the graph of$ f $. ag Alan G. Numerade Educator ### Problem 45 Draw a contour map of the function showing several level curves.$ f(x, y) = x^2 - y^2 $ag Alan G. Numerade Educator ### Problem 46 Draw a contour map of the function showing several level curves.$ f(x, y) = xy $ag Alan G. Numerade Educator ### Problem 47 Draw a contour map of the function showing several level curves.$ f(x, y) = \sqrt{x} + y $ag Alan G. Numerade Educator ### Problem 48 Draw a contour map of the function showing several level curves.$ f(x, y) = \ln(x^2 + 4y^2) $ Jacquelyn T. Numerade Educator ### Problem 49 Draw a contour map of the function showing several level curves.$ f(x, y) = ye^x $ag Alan G. Numerade Educator ### Problem 50 Draw a contour map of the function showing several level curves.$ f(x, y) = y - \arctan x $ag Alan G. Numerade Educator ### Problem 51 Draw a contour map of the function showing several level curves.$ f(x, y) = \sqrt{x^2 + y^2} $ag Alan G. Numerade Educator ### Problem 52 Draw a contour map of the function showing several level curves.$ f(x, y) = y/(x^2 + y^2) $ Carson M. Numerade Educator ### Problem 53 Sketch both a contour map and a graph of the function and compare them.$ f(x, y) = x^2 + 9y^2 $ag Alan G. Numerade Educator ### Problem 54 Sketch both a contour map and a graph of the function and compare them.$ f(x, y) = \sqrt{36 - 9x^2 - 4y^2} $ag Alan G. Numerade Educator ### Problem 55 A thin metal plate, located in the$ xy $-plane, has temperature$ T(x, y) $at the point$ (x, y) $. Sketch some level curves (isothermals) if the temperature function is given by $$T(x, y) = \dfrac{100}{1 + x^2 + 2y^2}$$ Jacquelyn T. Numerade Educator ### Problem 56 If$ V(x, y) $is the electric potential at a point$ (x, y) $in the$ xy $-plane, then the level curves of$ V $are called$\textit{equipotential curves} $because at all points on such a curve the electric potential is the same. Sketch some equipotential curves if$ V(x, y) = c/\sqrt{r^2 - x^2 - y^2} $, where$ c $is a positive constant. Jacquelyn T. Numerade Educator ### Problem 57 Use a computer to graph the function using various domains and viewpoints. Get a printout of one that, in your opinion, gives a good view. If your software also produces level curves, then plot some contour lines of the same function and compare with the graph.$ f(x, y) = xy^2 - x^3 $(monkey saddle) ag Alan G. Numerade Educator ### Problem 58 Use a computer to graph the function using various domains and viewpoints. Get a printout of one that, in your opinion, gives a good view. If your software also produces level curves, then plot some contour lines of the same function and compare with the graph.$ f(x, y) = xy^3 - yx^3 $(dog saddle) ag Alan G. Numerade Educator ### Problem 59 Use a computer to graph the function using various domains and viewpoints. Get a printout of one that, in your opinion, gives a good view. If your software also produces level curves, then plot some contour lines of the same function and compare with the graph.$ f(x, y) = e^{-(x^2 + y^2)/3}(\sin(x^2) + \cos(y^2)) $ag Alan G. Numerade Educator ### Problem 60 Use a computer to graph the function using various domains and viewpoints. Get a printout of one that, in your opinion, gives a good view. If your software also produces level curves, then plot some contour lines of the same function and compare with the graph.$ f(x, y) = \cos x \cos y $ag Alan G. Numerade Educator ### Problem 61 Match the function (a) with its graph (labeled A-F below) and (b) with its contour map (labeled I-VI). Give reasons for your choices.$ z = \sin(xy) $ag Alan G. Numerade Educator ### Problem 62 Match the function (a) with its graph (labeled A-F below) and (b) with its contour map (labeled I-VI). Give reasons for your choices.$ z = e^x \cos y $ag Alan G. Numerade Educator ### Problem 63 Match the function (a) with its graph (labeled A-F below) and (b) with its contour map (labeled I-VI). Give reasons for your choices.$ z = \sin(x - y) $ag Alan G. Numerade Educator ### Problem 64 Match the function (a) with its graph (labeled A-F below) and (b) with its contour map (labeled I-VI). Give reasons for your choices.$ z = \sin x - \sin y $ag Alan G. Numerade Educator ### Problem 65 Match the function (a) with its graph (labeled A-F below) and (b) with its contour map (labeled I-VI). Give reasons for your choices.$ z = (1 - x^2)(1 - y^2) $ag Alan G. Numerade Educator ### Problem 66 Match the function (a) with its graph (labeled A-F below) and (b) with its contour map (labeled I-VI). Give reasons for your choices.$ z = \dfrac{x - y}{1 + x^2 + y^2} $ag Alan G. Numerade Educator ### Problem 67 Describe the level surfaces of the function.$ f(x, y, z) = x + 3y + 5z $ Jacquelyn T. Numerade Educator ### Problem 68 Describe the level surfaces of the function.$ f(x, y, z) = x^2 + 3y^2 + 5z^2 $ Jacquelyn T. Numerade Educator ### Problem 69 Describe the level surfaces of the function.$ f(x, y, z) = y^2 + z^2 $ag Alan G. Numerade Educator ### Problem 70 Describe the level surfaces of the function.$ f(x, y, z) = x^2 - y^2 - z^2 $ag Alan G. Numerade Educator ### Problem 71 Describe how the graph of$ g $is obtained from the graph of$ f $. (a)$ g(x, y) = f(x, y) + 2 $(b)$ g(x, y) = 2 f(x, y) $(c)$ g(x, y) = -f(x, y) $(d)$ g(x, y) = 2 - f(x, y) $ag Alan G. Numerade Educator ### Problem 72 Describe how the graph of$ g $is obtained from the graph of$ f $. (a)$ g(x, y) = f(x - 2, y) $(b)$ g(x, y) = f(x, y + 2) $(c)$ g(x, y) = f(x + 3, y - 4) $ag Alan G. Numerade Educator ### Problem 73 Use a computer to graph the function using various domains and viewpoints. Get a printout that gives a good view of the "peaks and valleys". Would you say the function has a maximum value? Can you identify any points on the graph that you might consider to be "local maximum points"? What about "local minimum points"?$ f(x, y) = 3x - x^4 - 4y^2 - 10xy $ag Alan G. Numerade Educator ### Problem 74 Use a computer to graph the function using various domains and viewpoints. Get a printout that gives a good view of the "peaks and valleys". Would you say the function has a maximum value? Can you identify any points on the graph that you might consider to be "local maximum points"? What about "local minimum points"?$ f(x, y) = xye^{-x^2 - y^2} $ag Alan G. Numerade Educator ### Problem 75 Graph the function using various domains and viewpoints. Comment on the limiting behavior of the function. What happens as both$ x $and$ y $become large? What happens as$ (x, y) $approaches the origin?$ f(x, y) = \dfrac{x + y}{x^2 + y^2} $ag Alan G. Numerade Educator ### Problem 76 Graph the function using various domains and viewpoints. Comment on the limiting behavior of the function. What happens as both$ x $and$ y $become large? What happens as$ (x, y) $approaches the origin?$ f(x, y) = \dfrac{xy}{x^2 + y^2} $ag Alan G. Numerade Educator ### Problem 77 Investigate the family of functions$ f(x, y) = e^{cx^2 + y^2} $. How does the shape of the graph depend on$ c $? ag Alan G. Numerade Educator ### Problem 78 Use a computer to investigate the family of surfaces $$z = \bigl(ax^2 + by^2\bigr) e^{-x^2 - y^2}$$ How does the shape of the graph depend on the numbers$ a $and$ b $? ag Alan G. Numerade Educator ### Problem 79 Use a computer to investigate the family of surfaces$ z = x^2 + y^2 + cxy $. In particular, you should determine the transitional values of$ c $for which the surface changes from one type of quadric surface to another. ag Alan G. Numerade Educator ### Problem 80 Graph the functions $$f(x, y) = \sqrt{x^2 + y^2}$$ $$f(x, y) = e^{\sqrt{x^2 + y^2}}$$ $$f(x, y) = \ln \sqrt{x^2 + y^2}$$ $$f(x, y) = \sin (\sqrt{x^2 + y^2})$$ and $$f(x, y) = \dfrac{1}{\sqrt{x^2 + y^2}}$$ In general, if$ g $is a function of one variable, how is the graph of $$f(x, y) = g\biggl(\sqrt{x^2 + y^2}\biggr)$$ obtained from the graph of$ g $? ag Alan G. Numerade Educator ### Problem 81 (a) Show that, by taking logarithms, the general Cobb-Douglas function$ P = bL^{\alpha}K^{1 - \alpha} $can be expressed as $$\ln \dfrac{P}{K} = \ln b + \alpha \ln \dfrac{L}{K}$$ (b) If we let$ x = \ln (L/K) $and$ y = \ln (P/K) $, the equation in part (a) becomes the linear equation$ y = \alpha x + \ln b $. Use Table 2 (in Example 3) to make a table of values of$ \ln (L/K) $and$ \ln (P/K) $for the years 1899 - 1922. Then use a graphing calculator or computer to find the least squares regression line through the points$ (\ln (L/K), \ln (P/K)) $. (c) Deduce that the Cobb-Douglas production function is$ P = 1.01L^{0.75}K^{0.25} \$. Jacquelyn T.
Numerade Educator