SECTION 15.5 24. f (x, y) = sin(x - y), v= (1,1), P = (5, 6) 45. Use the geostrophi 25. f (x, y ) = tan-1 (xy), v = (1,1), P = (3,4) ern Hemisphere, winds together the isobars, 26. f (x, y ) = exy -12, v = (12, -5), P = (2, 2) clockwise around low pressure systems. 27. f ( x, y ) = In(x2 + 12 ), v = 3i - 2j, P = (1, 0) In Exercises 46-49, fi 28. 8 (x, y, z) = z2 - xy + 2y2, v= (1, -2, 2), P = (2, 1, -3) the given point. 29. 8 (x, y, z) = xe-yz, v = (1, 1, 1), P = (1, 2, 0) 46. x2 + 32 + 422 30. g (x, y, z) = x In(y + z), v= 2i-j+k, P = (2, e, e) 47. xz+ 2x2y + 2 31. Find the directional derivative of f(x, y) = x2 + 4y2 at the point 48. x2 + z2 ey -x = P = (3,2) in the direction pointing to the origin. 32. Find the directional derivative of f(x, y, z) = xy + 23 at the point 49. In(1 + 4x2 + 9 P = (3, -2, -1) in the direction pointing to the origin. 50. Verify what is In Exercises 33-36, determine the direction in which f has maximum rate x2 + y2 - z2 =0 of increase from P, and give the rate of change in that direction. 33. f ( x, y ) = xe-y, P = ( 2, 0) 34. f ( x, y ) = x2 - xy+ y2, P= (-1,4) 35. f ( x, y, z ) = -, P = (1, -1, 3) 36. f (x, y, z) = x2yz, P = (1, 5,9)(or hog off Hi 37. Suppose that Vfp = (2, -4, 4). Is f increasing or decreasing at P in the direction v = (2, 1, 3)? 38. Let f(x, y) = xex-y and P = (1, 1). (a) Calculate || Vfell. (b) Find the rate of change of f in the direction Vfp. (c) Find the rate of change of f in the direction of a vector making an angle of 450 with Vfp. 51. CAS 39. Let f(x, y, z) = sin(xy + z) and P = (0, -1, ). Calculate Du f(P), f (x, y ) =x where u is a unit vector making an angle 0 = 30 with V fp. domain [- 40. Let T(x, y) be the temperature at location (x, y) on a thin sheet of 52. Find a metal. Assume that VT = (y - 4, x + 2y). Let r(t) = (12, t) be a path on 53. Find the sheet. Find the values of f such that 54. Find a T (r(1 ) ) = 0 55. Find 41. Find a vector normal to the surface x2 + y2 - z2 = 6 at 56. Fin P = (3,1,2). 57. Sh 42. Find a vector normal to the surface 323 + x2y - y2x = 1 at P = ( y 2 , x). (1, -1, 1). 58. L 43. Find the two points on the ellipsoid Set A x2 5+22 =1 59. where the tangent plane is normal to v = (1, 1, -2). slosh 44. Assume we have a local coordinate system at latitude L on the earth's surface with east, north, and up as the x, y, and z directions, respec- as tively. In this coordinate system, the earth's angular velocity vector is $2 = (0, w cos L, w sin L). Let w = (w1, w2,0) be a wind vector. (a) Determine the components of the Coriolis force vector Fo = -2m $2 x w. (b) The equation -VVp + Fe = 0 results from balancing the pressure gradient force, -VVp, and the Coriolis force. Show that the x- and y- components of this equation result in Eq. 5
