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.SECTION 15.5 The Gradient and Directional Derivatives 845 45. Use the geostrophic flow model to explain the following: In the South- ern Hemisphere, winds blow with low pressure to the right, and the closer together the isobars, the stronger the winds. In particular, winds blow clockwise around low pressure systems and counterclockwise around high pressure systems. In Exercises 46-49, find an equation of the tangent plane to the surface at the given point. 46. x2 + 3y2 + 422- 20. P = (2. 2. 1) 47. xz + 2x2y +y223 =11. P = (2.1. 1) joint 48. x2 + 22ey-x = 13, P = (2.3.-) point 49. In(1 + 4x2 +94) - 0.122 -0. P - (3, 1.6.1876) 50. Verify what is clear from Figure 15: Every tangent plane to the cone rate x2 + y2 - z2 = 0 passes through the origin. in FIGURE 15 Graph of x2 + y2 - z2 = 0. an 51. CAS Use a computer algebra system to produce a contour plot of P), f(x, y) =x2 - 3xy + y - y2 together with its gradient vector field on the domain [-4, 4] x [-4, 4]. of 52. Find a function f(x, y, z) such that Vf is the constant vector (1, 3, 1). on 53. Find a function f(x, y, z) such that Vf = (2x, 1, 2). 54. Find a function f(x, y, z) such that Vf = (x, y2, z3). out Just word? 55. Find a function f (x, y, z) such that Vf = (z, 2y, x). (d) blad ion at 56. Find a function f (x, y) such that Vf = (y, x). 57. Show that there does not exist a function f(x, y) such that Vf = (y2, x). Hint: Use Clairaut's Theorem fxy = fyx. 58. Let Af = f(a + h, b + k) - f(a, b) be the change in f at P = (a, b). Set Av = (h, k). Show that the Linear Approximation can be written 6 Af ~ Vfp . AV 59. Use Eq. (6) to estimate oldshevinuM . Af = f(3.53, 8.98) - f(3.5, 9) 1 assuming that Vf(3.5,9) = (2, -1). OW 60. Find a unit vector n that is normal to the surface z2 - 2x4 - y4 = 16 at P = (2, 2, 8) that points in the direction of the xy-plane (in other words, if you travel in the direction of n, you will eventually cross the xy-plane). 61. Suppose, in the previous exercise, that a particle located at the point P = (2, 2, 8) travels toward the xy-plane in the direction normal to the surface.C X Z alt command option trol Uela 846 CHAPTER 15 DIFFERENTIATION IN SEVERAL VARIABLES (a) Through which point Q on the xy-plane will the particle pass? vector v = VFp x VGp is a direction vector for the tangent line to C (b) Suppose the axes are calibrated in centimeters. Determine the path r(t) at P. of the particle if it travels at a constant speed of 8 cm/s. How long will it 64. Let C be the curve of intersection of the spheres x2 + y2 + z2 = 3 and take the particle to reach Q? (x - 2)2 + (y -2)2 + z2 = 3. Use the result of Exercise 63 to find para- metric equations of the tangent line to C at P = (1, 1, 1). 62. Let f(x, y) = tan- - and u = 12 2 2 65. Let C be the curve obtained as the intersection of the two surfaces x3 + 2xy + yz =7 and 3x2 - yz = 1. Find the parametric equations of (a) Calculate the gradient of f. the tangent line to C at P = (1, 2, 1). (b) Calculate Duf (1, 1) and Duf(