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271E2-F2002

# 271E2-F2002 - MA 271 EXAM 2 FALL 2002 Name 10 11 12 13 3...

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Unformatted text preview: MA 271 EXAM 2 FALL 2002 Name 10. 11. 12. 13. 3 Let f(x,y) = (3% +34) s. Find 2% (4, —5). (9w . Find — when r = 1, s = —1ifw = em+y+z, :1: = T—S, y = 00504—3), z = sin(7‘+s). 87' 2 . Find the equation of the tangent plane to the surface y + z — x— = 1 at (2,1,2). 2 . Let f(:r, y, z) = xzy + yzz + 22:3, and P0 = (1,1,—1). Find the directional derivative of f at P0 in the direction that the function increases most rapidly. . Classify the critical points of f(m, y) = 4333; — x4 —— 3/4. . Find the extreme value of f (cc, 3/, z) = a: + 2y + 22 subject to the constraint m2+y2+22=9. 1 f . Evaluate / / sec4 :5 dx dy. 0 tan—1 y . Let D be the solid region bounded below by the paraboloid z = x2 + y2 and above by z = 4. Evaluate the total mass of D if the density function is 5(33, y, z) = \/.Z'2 + 3/2. . Evaluate the integral /// zdv, where D is the solid region bounded above by the D 71' sphere p = 1 and below by the cone (15 = E. Let D be the image of the rectangle {(u,v)| — 1 S u g 2, 0 s v S 2} under the —- 2 transformation :1: = 2a + 311, y : u ~— 1). Evaluate // x 5 y dA. D The mass density at a given point of a thin wire (3' is 6(33, y, z) = x. If C is parametrized by r(t) = (6‘, 2t, 26_t>, 0 g t g 1, ﬁnd the mass of the wire. Let F be a conservative vector ﬁeld given by F = (2:1:y,:v2 + cos(y + 22).,"322 + 22 cos(y + 22)). Find f such that Vf = F. Let R be the region bounded by y = x2 and a: = 312, and C the boundary of R. Compute f ]F - dr for IF 2 (2y + ea”, 33: + sin y). C ...
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