caps lock X C Z shift alt Figure 21. 22. Match the functions (a)-(d) with their contour maps (A)-(D) in (a) f(x, y) = 3x + 4y (c) h(x, y ) = 4x - 3y (b) 8 (x, y ) = x3 - y (d) k (x, y) = x2 - y 104 104 5 - 5 0 - -5+ -5 40. -10 4 -10 + cha -10 -5 0 5 10 -10 -5 0 5 10 (a) (A) (B) 104 104 5 - 5 0 0 -5 -5 -10 10-10 -10 -5 0 5 10 -10 -5 10 (C) ed sovig ed sourpert (D) nibeoge FIGURE 21 In Exercises 23-28, sketch the graph and draw several vertical and hori- zontal traces. 23. f (x, y) = 12 - 3x - 4y 24. f (x, y) = V4-x2 - yz 25. f (x, y) = x2 +4y2 26. f(x, y) = 12 27. f(x, y) = sin(x - y) 28. f ( x, y ) = x2 + y2 + 1 29. Sketch contour maps of f(x, y) = x + y with contour intervals m = 1 and 2. 30. Sketch the contour map of f (x, y) = x2 + y2 with level curves c = 0, 4, 8, 12, 16. In Exercises 31-38, draw a contour map of f(x, y) with an appropriate contour interval, showing at least six level curves. 31. f (x, y) = x2 - y 32. f ( x, y) = = 33. f (x, y) = = 34. f ( x, y) = xy 35. f (x, y ) = x2 + 4y2 36. f (x, y) = x+2y-1 37. f(x, y) =x2 20 0 38. f(x, y) = 3x2 - y2 39. Find the linear function whose contour map (with contour in terval m = 6) is shown in Figure 22. What is the linear function if m = (and the curve labeled c = 6 is relabeled c = 3)?SECTION 15.1 Functions of Two or More Variables 805 in 6 C =6 C=0 FIGURE 22 Contour map with contour interval m = 6. change: 40. Use the contour map in Figure 23 to calculate the average rate of (a) from A to B. (b) from A to C. C=-3 C= 0 -4 -2 *x 2 4 6 FIGURE 23 Exercises 41-43 refer to the map in Figure 24. 41. (a) At which of A-C is pressure increasing in the northern direction? (b) At which of A-C is pressure increasing in the westerly direction? 42. For each of A-C indicate in which of the four cardinal directions, N, S, E, or W, pressure is increasing the greatest. 43. Rank the following states in order from greatest change in pressure across the state to least: Arkansas, Colorado, North Dakota, Wisconsin. 1020 1032-1032 -1024 1028 1008 1012 1016 1020 1028 -1024 1024 (B)1016 1020 -1016 1012 A 1008 1012 1016 1004 1016 1008 1004 1000 1016 FIGURE 24 Atmospheric pressure (in millibars) over North America on March 26, 2009. In Exercises 44 47, let T(x, y, z) denote temperature at each point in space. Draw level surfaces (also called isotherms) corresponding to the fixed temperatures given. 44. T(x, y, z) = 2x + 3y - z, T = 0, 1,2 45. T(x, y. z) = x - y + 2z, T = 0, 1,2A S caps lock 806 CHAPTER 15 DIFFERENTIATION IN SEVERAL VARIABLES 46. T (x, y, 2) = x2 + 12 - z, T = 0, 1, 2 In Exercises 52-55, refer to Figure 26. 47. T (x, y, z ) = 12 - y2 + 22, 1 = 0, 1, 2, -1, -2 52. Find the change in seawater density from A to B. In Exercises 48-51, p(S, T) is seawater density (kilograms per cubic me- 53. Estimate the average rate of change from A to B and from A ter) as a function of salinity S (parts per thousand) and temperature T to C. (degrees Celsius). Refer to the contour map in Figure 25. 54. Estimate the average rate of change from A to points i, ii, and iii. 48. Calculate the average rate of change of p with respect to T from B O A. 55. Sketch the path of steepest ascent beginning at D. 49. Calculate the average rate of change of p with respect to S from B to C. 50. At a fixed level of salinity, is seawater density an increasing or a de- creasing function of temperature? 540 B 51. Does water density appear to be more sensitive to a change in temper- 500 ature at point A or point B? 400 D iji 1.0230 20 1.0235 1.0240 Contour interval = 20 m 2 km 1.0245 0 15 1.0250 Temperature T (C 1.0255 FIGURE 26 10 1.0260 1.02657 56. Let temperature in 3-space be given by T(x, y, z) = x2 + y2 - z. 5 1 0270 Draw isotherms corresponding to temperatures T = -2, -1, 0, 1, 2. 57. Let temperature in 3-space be given by T(x, y, z) = = + + z2. 0+ 31.5 32.0 32.5 33.0 33.5 Draw isotherms corresponding to temperatures T = 0, 1, 2. 34.0 34.5 Salinity (ppt) 58. Let temperature in 3-space be given by T(x, y, z) = x2 - y2 - z. Draw isotherms corresponding to temperatures T = -1, 0, 1. FIGURE 25 Contour map of seawater density p(S, T) (kilograms per cubic meter). 59. Let temperature in 3-space be given by T(x, y, z) = x2 - y2. z2 Draw isotherms corresponding to temperatures T = -2, -1, 0, 1, 2