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15.1 Graphs and Level Curves 927 (a) Figure 15.18 SECTION 15.1 EXERCISES 10. Katie and Zeke are standing on the surface above D(1,0). Katie hikes on the surface above the level curve containing D(1,0) o B(2.1) and Zeke walks cast along the surface to E(2. 0). What can Getting Started y-y dentify the independent 1. A function is defined by and dependent variables. be said about the elevations of Katie and Zeke during their hikes? What is the domain of f(x, y) y-y 2. Describe in woeds the level curves of the paraboloid z-+ 12. How many axes (or how many dimensions) are needed to graph the level surfaces of wf(x.y, )? Explain. 11. What is the domain of g(x, y) V-y 4. What is the domain of x, y) 13. The domain of Q-fla, v, w, x, Y. z ) lies in R" for what valoe of w? Explain 5. How many axes (or how many dimensions) are needed to graph the function : f(x. y)? Explain. 14. Give two methods for graphically repeesenting a function withe three independent variables. Explain how to graph the level curves of a surfacez-f(x. y) Given the function f(x, y)-V10-x+ y,evaluate f(2, 1) and f(-9,-3). 7 Practice Exercises 15-24. Domains Find the domoin of the following fnctions: , evaluate g(1, 5. 3) and Given the function g(x. y,) 16. fx, )cos (-) 15. (xy)-2ay 3x+4y s(3,7.2) 9-10. The function:= f(x. y) gives the elevation : (in hundreds of feet) of a hillside above the point (x, y) Use the level curves of f to- answer the following questions (see figure) 17. f(x.y)-V25-- y 18. f(x y) - Ve+-25 12 2. f(xy) - 19. f(x, y)- sin 21. (xy)-In (x -y) 22. f(xy)-sin(y-) 23 y) 24. x y)v-2y + 4 25-33. Graphs of familiar functions Use what you leamed abowt surfaces in Secions 13.5 and 13.6 to sketch a graph of the following functions. In each case, identify the surface and state the domain and range of the function 25. fxy)6-x-2y 26. (y)4 27. py) - y 29, Gxy)-VI++ 28. x y) 2+3y 9. Katie and Zeke are standing on the surface above the point A(2, 2) 30, Fxy)-V1-2-Y a At what ckevation are Katie and Zeke standing? b. Katie hikes south to the point on the surface above B(2, 1) and Zeke hikes east to the point on the surface above C(3, 2). Who experienced the greater elevation change and what is the differ- 31. Pxy)-V+y? = I 32. Hxy)-VR+ y 33. (y)-V16-4 ence in their elevations? 928 CHAPTER 15 FUNCTIONS OF SEVERAL VARIABLES 34. Matching level curves with surfaces Match surfaces a-f in the figure with level curves A-F (D) (E) (F) 35. Matching surfaces Makch functions a-d with surfaces A-D in thet figure a. f(x.y)-cos ay b. xy)-(x+) d. Py) + c. x, ) (B) tel (C) (Di 36-43. Level curves Graph several level carves of the following functions asing the given window. Label at least eo level curves with their 2rales (A) B) 36 1-4,4) x-4,4 37. x-y0,4) x -2.2 38 2x-y-2,21 x 1-2.21 1s.1 Graphs and Level Curves 929 39. V+4y-8, 81 x1-8, 8 40. -2,21 x-2,2 41. V25-d- ya (-6,6) x [-6,6 42. Vy- - ; [ -5,5 ) x [-5.5 43.3 cos (2x+ y):[-2,2] x [-2,2 48. Level curves of a savings account Suppose you make a one-time deposit of P dollars into a savings account that earns interest at an annual rate of pcompounded continuously. The balance in the account after t years is 8(P.r.r)- Pe wherer-p/100 (for example, if the aneual inmerest rate is 4%, then r-0.04). Let the interest rata be fixed atr004, a. With a target balance of $2000, find the set of all points (P.) that satisfy B2000 This curve gives all deposits P and times r that result in a balance of $2000 Repeat part (a) with $500, S1000, $1500, and $2500, and draw the resalting level curves of the balance function e. In general, on one level curve, if r increases, does P increase or 44. Earned run average A baseball pitcher's camed run average (ERA) is A(e. )-9e/, where e is the number of earned runs given up by the pitcher and i is the number of innings pitched Good pischers have low ERAS. Assumee z 0and i> 0are real numbers decrease? aThe single-season major league record for the lowest ERA was 49, set by Dutch Lecnard of the Detroit Tigers in 1914. During that season, Dutch pilched a total of 224 innings and gave upe just 24 earmed runs. What was his ERA? b. Determine the ERA of a relief pitcher who gives up 4 eamed runs in cne-third of an inning c. Graph the level curve A(e, i)-3 and describe the relation ship between e and i in this case. Level curves of a savings plan Suppose you make monthly deposits of P dollars into an account that earms interest at a monsly rae of p%. The balance in the account after r years is (1+)e where r (for example, B(Pr,) if the annual interest rate is 9%, then p 0.75 and 12 0,0075) Let the time of investment be fiscd at r 20 years 45. Electric potential function The clectric posential function for two positive charges, one at (0. 1) with twice the strength of the charge at (0,-1). is given by a. With a target balance of $20,000, find the set of all points (P.r) that satisfy B = 20,000. This curve gives all deposits Pand monthly interest rates r that result in a balance of $20,000 after 20 years. Repeat part (a) with #$5000, S10,000, S15,000, and $25,000, and draw the resulting level curves of the balance function. elx, y) V y+1 aGraph the electric potential using the window [-5,5] x[-5,5) x (a, 10). b. For what values of x and y is the potential e defined? . Is the electric potential greater at (3, 2) or (2.3)? d. Describe how the clectric potential varies along the liney=x. 50-56. Domains of functions of three or more variables Find the domain of the following functions. If possible, give a descripeion of the domains (for example, all points outside a sphere of radias 1 centered at the origin 46. Cobb-Douglas production function The output Q of an eco- nomic system subject to two inputs, such as labor L and capital K, is often modeled by the Cobb-Douglas production function QL K)cK, where a, h, and e are positive real numbers. When a+ b 1. the case is called constaw trs to scale Suppose a 1/3,b-2/3, and c 40. s0. f(x.y.) 2ry 3+4y 51. y) 52. Py.z)-V+ y? + z2 = 9 53. f(xy.)Vy- a Geaph the output function using the window [o, 201 x [0, 201 x [0, S00 b. Ifis hed constant at / 10, write the function that gives 10 54 Qy.) the dependence of Q on K. . If K is held comstant at K-15, write the function that gives the dependence ofQon L 4 55. Fxy) Vr- 47. Resistors in parallel Two resistors wired in parallel in an electri- 56. f(w,x.y.)-VI- cal circuit give an effective resistance of R(x, y) where +y 57. Explain why or why not Determine whether the following state- ments are true and give an explanation or counterexample. a. The domain of the function f(x. y)-1-x-y is ,y)x y The domain of the function Q- g(w.x.y. ) is a region in R e. All level curves of the planez-2-3y are lines Quarterback passer ratings One measurement of the qual and y are the positive resistances of the individaal resistors (typically measured in ohm) 58. ity of a quarterback in the National Foothall League is known as the quarterback passer rating. The rating formula is a Graph the resistance function using the window 0, 10x 0, 10] x [0, 5). b. Estimate the maximum valae of R, for 0

Answer 1

28. h(x, y) 2x2 +3y The function h is defined for all x and y Domain (h) (, 0o)x(0, ) Range [0, c) 36. =x 4,4]x [-4,4 1 y =1 2x 4 3x 9 The level curves on the domain -4, 4]x[-4,4] are shown below +

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