( a). FUNCTIONS OF TWO VARIABLES 67 G (,, ) ( c) (a, b, c) P E ( b) Figure.: The diagonal PGgives the distance between the points (,, ) and (a, b, c) F Using Pthagoras theorem twice gives (PG) =(PF) +(FG) =(PE) +(EF) +(FG) =( a) +( b) +( c). Thus, a formula for the distance between the points (,, ) and (a, b, c) in -space is Distance = ( a) +( b) +( c). Eample Find the distance between (,, ) and (,, ). Solution Distance = ( ) +( ) +( ) = 8 =.. Eample Solution Find an epression for the distance from the origin to the point (,, ). The origin has coordinates (,, ), so the distance fromthe origin to (,, ) is given b Distance = ( ) +( ) +( ) = + +. Eample Solution Find an equation for a sphere of radius with center at the origin. The sphere consists of all points (,, ) whose distance from the origin is, that is, which satisf the equation + + =. This is an equation for the sphere. If we square both sides we get the equation in the form + + =. Note that this equation represents the surface of the sphere. The solid ball enclosed b the sphere is represented b the inequalit + +. Eercises and Problems for Section. Eercises. Which of the points P =(,, ) and Q =(,, ) is closest to the origin?. Which two of the three points P = (,, ), P = (,, ) and P =(,, ) are closest to each other?
67 Chapter Twelve FUNCTIONS OF SEVERAL VARIABLES. Which of the points A = (.,.7, ), B = (.9,,.), C =(.5,.,.) is closest to the plane? Which one lies on the -plane? Which one is farthest from the -plane?. You are at the point (,, ), standing upright and facing the -plane. You walk units forward, turn left, and walk another units. What is our final position? From the point of view of an observer looking at the coordinate sstem in Figure. on page 668, are ou in front of or behind the -plane? To the left or to the right of the -plane? Above or below the -plane? 5. On a set of, and aes oriented as in Figure.5 on page 669, draw a straight line through the origin, ling in the -plane and such that if ou move along the line with our -coordinate increasing, our -coordinate is increasing. 6. What is the midpoint of the line segment joining the points (,, 9) and (5, 6, )? In Eercises 7 sketch graphs of the equations in -space. 7. = 8. = 9. =. =and =. With the -ais vertical, a sphere has center (,, 7) and lowest point (,, ). What is the highest point on the sphere?. Find an equation of the sphere with radius 5 centered at the origin.. Find the equation of the sphere with radius and centered at (,, ).. Find the equation of the vertical plane perpendicular to the -ais and through the point (,, ). Eercises 5 7 refer to the map in Figure. on page 666. 5. Give the range of dail high temperatures for: (a) Pennslvania (b) North Dakota (c) California 6. Sketch a possible graph of the predicted high temperature T on a line north-south through Topeka. 7. Sketch possible graphs of the predicted high temperature on a north-south line and an east-west line through Boise. For Eercises 8, refer to Table. on page 667, where p is the price of beef and I is annual household income. 8. Give tables for beef consumption as a function of p, with I fied at I =and I =. Give tables for beef consumption as a function of I, with p fied at p =. and p =.. Comment on what ou see in the tables. 9. Make a table of the proportion, P, of household income spent on beef per week as a function of price and income. (Note that P is the fraction of income spent on beef.). How does beef consumption var as a function of household income if the price of beef is held constant? Problems. The temperature adjusted for wind chill is a temperature which tells ou how cold it feels, as a result of the combination of wind and temperature. See Table.. Table. Temperature adjusted for wind chill ( F) as a function of wind speed and temperature Wind Speed (mph) Temperature ( F) 5 5 5 5 5 5 9 7 5 7 5 9 6 5 5 9 6 7 9 7 9 5 5 6 9 7 (a) If the temperature is F and the wind speed is 5 mph, how cold does it feel? (b) If the temperature is 5 F, what wind speed makes it feel like F? (c) If the temperature is 5 F, what wind speed makes it feel like F? Data from www.nws.noaa.gov/om/windchill, accessed on Ma,. (d) If the wind is blowing at mph, what temperature feels like F? In Problems, use Table. to make tables with the given properties.. The temperature adjusted for wind chill as a function of wind speed for temperatures of Fand F.. The temperature adjusted for wind chill as a function of temperature for wind speeds of 5 mph and mph.. A car rental compan charges $ a da and 5 cents a mile for its cars. (a) Write a formula for the cost, C, of renting a car as a function, f, of the number of das, d, and the numberofmilesdriven,m. (b) If C = f(d,m),find f(5, ) and interpret it. 5. The gravitational force, F newtons, eerted on an object b the earth depends on the object s mass, m kilograms, and its distance, r meters, from the center of the earth, so F = f(m, r). Interpret the following statement in terms of gravitation: f(, 7) 8.
. FUNCTIONS OF TWO VARIABLES 67 6. Consider the acceleration due to gravit, g, at a distance h from the center of a planet of mass m. (a) If m is held constant, is g an increasing or decreasing function of h? Wh? (b) If h is held constant, is g an increasing or decreasing function of m? Wh? 7. A cube is located such that its top four corners have the coordinates (,, ), (,, ), (,, ) and (,, ). Give the coordinates of the center of the cube. 8. Describe the set of points whose distance from the -ais is. 9. Describe the set of points whose distance from the -ais equals the distance from the -plane.. Find a formula for the shortest distance between a point (a, b, c) and the -ais.. Find the equations of planes that just touch the sphere ( ) +( ) +( ) =6and are parallel to (a) The -plane (b) The -plane (c) The -plane. Find an equation of the largest sphere contained in the cube determined b the planes =,=6; =5, = 9; and =, =.. A cube has edges parallel to the aes. One corner is at A = (5,, ) and the corner at the other end of the longest diagonal through A is B =(, 7, ). (a) What are the coordinates of the other three vertices on the bottom face? (b) What are the coordinates of the other three vertices on the top face?. Which of the points P = (,, 5), P = (,, ), P = ( 6, 5, ) and P = (,, 7) is closest to P =(6,, )? 5. (a) Find the equations of the circles (if an) where the sphere ( ) +( +) +( ) =intersects each coordinate plane. (b) Find the points (if an) where this sphere intersects each coordinate ais. 6. A rectangular solid lies with its length parallel to the - ais, and its top and bottom faces parallel to the plane =. If the center of the object is at (,, ) and it has a length of, a height of 5 and a width of 6, give the coordinates of all eight corners and draw the figure labeling the eight corners. 7. An equilateral triangle is standing verticall with a verte above the -plane and its two other vertices at (7,, ) and (9,, ). What is its highest point? 8. (a) Find the midpoint of the line segment joining A = (, 5, 7) to B =(5,, 9). (b) Find the point one quarter of the wa along the line segment from A to B. (c) Find the point one quarter of the wa along the line segment from B to A. Strengthen Your Understanding In Problems 9, eplain what is wrong with the statement. 9. In -space, =is a line parallel to the -ais.. The -plane has equation =.. The distance from (,, ) to the -ais is. In Problems, give an eample of:. A formula for a function f(, ) that is increasing in and decreasing in.. A point in -space with all its coordinates negative and farther from the -plane than from the plane = 5. Are the statements in Problems 57 true or false? Give reasons for our answer.. If f(, ) is a function of two variables defined for all and,thenf(,) is a function of one variable. 5. The volume V of a bo of height h and square base of side length s is a function of h and s. 6. If H = f(t, d) is the function giving the water temperature H C of a lake at time t hours after midnight and depth d meters, then t is a function of d and H. 7. A table for a function f(, ) cannot have an values of f appearing twice. 8. If f() and g() are both functions of a single variable, then the product f() g() is a function of two variables. 9. The point (,, ) lies above the plane =. 5. The graph of the equation =is a plane parallel to the -plane. 5. The points (,, ) and (,, ) are the same distance from the origin. 5. The point (,, ) lies on the graph of the sphere ( ) +( +) +( ) =5. 5. There is onl one point in the -plane that is a distance from the point (,, ). 5. There is onl one point in the -plane that is distance 5 from the point (,, ). 55. If the point (,b,) has distance from the plane =, then b must be. 56. A line parallel to the -ais can intersect the graph of f(, ) at most once. 57. A line parallel to the -ais can intersect the graph of f(, ) at most once.
678 Chapter Twelve FUNCTIONS OF SEVERAL VARIABLES shaped surface shown in Figure.5. The cross-sections with fied are horiontal lines obtained b cutting the surface b a plane perpendicular to the -ais. This surface is called a parabolic clinder, because it is formed from a parabola in the same wa that an ordinar clinder is formed from a circle; it has a parabolic cross-section instead of a circular one. Figure.5: A parabolic clinder = Figure.6: Circular clinder + = Eample 5 Solution Graph the equation + =in -space. Although the equation + =does not represent a function, the surface representing it can be graphed b the method used for =. The graph of + =in the -plane is a circle. Since does not appear in the equation, the intersection of the surface with an horiontal plane will be the same circle + =. Thus, the surface is the clinder shown in Figure.6. Eercises and Problems for Section. Eercises. Figure.7 shows the graph of = f(, ). (a) Suppose is fied and positive. Does increase or decrease as increases? Graph against. (b) Suppose is fied and positive. Does increase or decrease as increases? Graph against. Figure.7 (V) Figure.8. Without a calculator or computer, match the functions with their graphs in Figure.8. (a) =+ + (b) = (c) =( + ) (d) =+ (e) =. Without a calculator or computer, match the functions with their graphs in Figure.9. (a) = + (b) = e (c) = + + (d) = (e) = sin.
. GRAPHS AND SURFACES 679 In Eercises, sketch a graph of the surface and briefl describe it in words.. = 5. + + =9 6. = + + 7. =5 8. = 9. + + =. + =. + = (V) In Eercises, find the equation of the surface.. A clinder of radius 7 with its ais along the -ais. Problems Figure.9. A sphere of radius centered at (, 7, ).. The paraboloid obtained b moving the surface = + so that its verte is at (,, 5), its ais is parallel to the -ais, and the surface opens towards negative values. 5. Consider the function f given b f(, ) = +. Draw graphs of cross-sections with: (a) fied at =, =,and =. (b) fied at =, =,and =. Problems 6 8 concern the concentration, C, inmg per liter, of a drug in the blood as a function of, the amount, in mg, of the drug given and t, the time in hours since the injection. For and t, wehavec = f(, t) =te t(5 ). 6. Find f(, ). Give units and interpret in terms of drug concentration. 7. Graph the following single-variable functions and eplain their significance in terms of drug concentration. (V) (VI) (a) f(,t) (b) f(, ) 8. Graph f(a, t) for a =,,, on the same aes. Describe how the graph changes as a increases and eplain what this means in terms of drug concentration. 9. Without a computer or calculator, match the equations (a) (i) with the graphs (IX). (VII) (VIII) (a) = e ( + ) (b) =cos( + ) (c) =sin (d) = + (e) =cos cos (f) = sin( + ) + (g) =cos() (h) = (i) =( + )e (IX)
688 Chapter Twelve FUNCTIONS OF SEVERAL VARIABLES The Cobb-Douglas Production Model In 98, Cobb and Douglas used a similar function to model the production of the entire US econom in the first quarter of this centur. Using government estimates of P, the total earl production between 899 and 9, of K, the total capital investment over the same period, and of L, the total labor force, the found that P was well approimated b the Cobb-Douglas production function P =.L.75 K.5. This function turned out to model the US econom surprisingl well, both for the period on which it was based, and for some time afterward. Eercises and Problems for Section. Eercises In Eercises, sketch a possible contour diagram for each surface, marked with reasonable -values. (Note: There are man possible answers.)... Let f(, ) = +7 +. Find an equation for the contour that goes through the point (5, ). 5. (a) For = f(, ) =, sketch and label the level curves = ±, = ±. (b) Sketch and label cross-sections of f with = ±, = ±. (c) The surface = is cut b a vertical plane containing the line =. Sketch the cross-section. 6. Match the surfaces (a) (e) in Figure.8 with the contour diagrams (V) in Figure.9... (a) (b) (c) (d) In Eercises 5, sketch a contour diagram for the function with at least four labeled contours. Describe in words the contours and how the are spaced. 5. f(, ) = + 6. f(, ) = + (e) 7. f(, ) = + 8. f(, ) = + 9. f(, ) =. f(, ) =. f(, ) = +. f(, ) = +. f(, ) =cos + Figure.8
. CONTOUR DIAGRAMS 689 6 Table.7 \ Table.8 \ (V) 6 Figure.9 7. Match Tables.5.8 with contour diagrams in Figure.5. Figure.5 8. Figure.5 shows a graph of f(, ) = (sin )(cos ) for π π, π π. Use the surface =/ to sketch the contour f(, ) =/. Table.5 Table.6 \ \ Problems 9. Total sales, Q, of a product are a function of its price and the amount spent on advertising. Figure.5 shows a contour diagram for total sales. Which ais corresponds to the price of the product and which to the amount spent on advertising? Eplain. 6 5 Q = Q = Q = 5 5 6 Figure.5 Q = Figure.5. Figure.5 shows contours of f(, ) = e 5. Find the values of f on the contours. The are equall spaced multiples of..8.6.....6.8 Figure.5
69 Chapter Twelve FUNCTIONS OF SEVERAL VARIABLES. Match the functions (a) (f) with the level curves (VI): (a) f(, ) = + (b) g(, ) = + +5 (c) h(, ) = + + 8 (d) j(, ) = + + + (e) k(, ) = ( ) +( ) (f) l(, ) = ( ) +( ) wind speed, mph 8 6 6 Figure.5 air temp, F. Figure.55 shows contour diagrams of f(, ) and g(, ). Sketch the smooth curve with equation f(, ) =g(, )..5 6 8 6 5 6 7.5 8 6 8 6 Figure.55: Black: f(, ).Blue:g(, ) (V) (VI). Figure.56 shows the level curves of the temperature H in a room near a recentl opened window. Label the three level curves with reasonable values of H if the house is in the following locations..5.5 (a) Minnesota in winter (where winters are harsh). (b) San Francisco in winter (where winters are mild). (c) Houston in summer (where summers are hot). (d) Oregon in summer (where summers are mild). Room. The wind chill tells ou how cold it feels as a function of the air temperature and wind speed. Figure.5 is a contour diagram of wind chill ( F). (a) If the wind speed is 5 mph, what temperature feels like F? (b) Estimate the wind chill if the temperature is Fand the wind speed is mph. (c) Humans are at etreme risk when the wind chill is below 5 F. If the temperature is F, estimate the wind speed at which etreme risk begins. (d) If the wind speed is 5 mph and the temperature drops b F, approimatel how much colder do ou feel? Window Figure.56 5. You are in a room feet long with a heater at one end. In the morning the room is 65 F. You turn on the heater, which quickl warms up to 85 F. Let H(, t) be the temperature feet from the heater, t minutes after the heater is turned on. Figure.57 shows the contour diagram for H. How warm is it feet from the heater 5 minutes after it was turned on? minutes after it was turned on?
. CONTOUR DIAGRAMS 69 t (minutes) 6 5 85 8 75 7 65 5 5 5 Figure.57 (feet) 6. Using the contour diagram in Figure.57, sketch the graphs of the one-variable functions H(, 5) and H(, ). Interpret the two graphs in practical terms, and eplain the difference between them. 7. Figure.58 shows a contour map of a hill with two paths, A and B. (a) On which path, A or B, will ou have to climb more steepl? (b) On which path, A or B, will ou probabl have a better view of the surrounding countrside? (Assume trees do not block our view.) (c) Alongside which path is there more likel to be a stream? loan amount ($) 8, 7, 6, 5,,,, 6 8 5 7 9 5 Figure.59 interest rate (%) 9. Hiking on a level trail going due east, ou decide to leave the trail and climb toward the mountain on our left. The farther ou go along the trail before turning off, the gentler the climb. Sketch a possible topographical map showing the elevation contours.. Match the functions (a) (d) with the shapes of their level curves. Sketch each contour diagram. (a) f(, ) = (b) f(, ) = + (c) f(, ) = (d) f(, ) = I. Lines II. Parabolas III. Hperbolas IV. Ellipses. Figure.6 shows the densit of the fo population P (in foes per square kilometer) for southern England. Draw two different cross-sections along a north-south line and two different cross-sections along an east-west line of the population densit P. Goal = = kilometers north 5 A B =.5 5.5.5.5 North.5.5 Figure.58.5 6 8 kilometers east 8. Figure.59 is a contour diagram of the monthl pament on a 5-ear car loan as a function of the interest rate and the amount ou borrow. The interest rate is % and ou borrow $6 for a used car. (a) What is our monthl pament? (b) If interest rates drop to %, how much more can ou borrow without increasing our monthl pament? (c) Make a table of how much ou can borrow, without increasing our monthl pament, as a function of the interest rate. Figure.6. A manufacturer sells two goods, one at a price of $ a unit and the other at a price of $, a unit. A quantit q of the first good and q of the second good are sold at a total cost of $ to the manufacturer. (a) Epress the manufacturer s profit, π, as a function of q and q. (b) Sketch curves of constant profit in the q q -plane for π =,, π =,, andπ =, and the break-even curve π =.
69 Chapter Twelve FUNCTIONS OF SEVERAL VARIABLES. Match each Cobb-Douglas production function (a) (c) with a graph in Figure.6 and a statement (D) (G). (a) F (L, K) =L.5 K.5 (b) F (L, K) =L.5 K.5 (c) F (L, K) =L.75 K.75 (D) Tripling each input triples output. (E) Quadrupling each input doubles output. (G) Doubling each input almost triples output. K F = K F = F = L K Figure.6 F =.5 F = L F = F = F = F = L 6. Match the functions (a) (d) with the contour diagrams in Figures I IV. (a) f(, ) =.7ln +.ln (b) g(, ) =.ln +.7ln (c) h(, ) =. +.7 (d) j(, ) =.7 +.. A Cobb-Douglas production function has the form P = cl α K β with α, β >. What happens to production if labor and capital are both scaled up? For eample, does production double if both labor and capital are doubled? Economists talk about increasing returns to scale if doubling L and K more than doubles P, constant returns to scale if doubling L and K eactl doubles P, decreasing returns to scale if doubling L and K less than doubles P. 7. Figure.6 is the contour diagram of f(, ). Sketch the contour diagram of each of the following functions. (a) f(, ) (b) f(, ) (c) f(, ) (d) f(, ) What conditions on α and β lead to increasing, constant, or decreasing returns to scale? 5. (a) Match f(, ) =..8 and g(, ) =.8. with the level curves in Figures and. All scales on the aes are the same. (b) Figure shows the level curves of h(, ) = α α for <α<. Find the range of possible values for α. Again, the scales are the same on both aes. Figure.6