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271E3-F2003 - MATHEMATICS 271 TEST III FALL 2002 8(10 pts 1...

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Unformatted text preview: MATHEMATICS 271 TEST III FALL 2002 8 (10 pts) 1) Find 6—: at (m,y,z)=(2,1,—1) if w=p;an=—x+y+za q:m—y+z, andr=x+y+z. (10 pts) 2) If f(:z:,y,z) = y2 +zlnx find a) Vf at (1,171)) b) the direction in which f changes most rapidly at (1,1,1), c) the equation of the tangent plane to y2 + zlnx = 1 at (1, 1, 1). (10 pts) 3) Find all maxima, minima, and points of inflection of f (:1), y) = x3 — y3 — 21:31 + 6. (10 pts) E)va1uate by reversing the order of integration. He My (10 pts) 5) Find the average value of f(ac, y) = ycos a: in the area bounded by y = O, y = sin :3, 0 S a: g 7r. (10 pts) 6) If f(:r,y) = ysinm find a) The linear approximation near (0,0). b) The quadratic approximation near (0,0). c) An estimate of the error made if f is replaced by its quadratic approximation. Assume |Aac| < 10‘2 and |Ay| < 10—2. (10 pts) 7) Find the absolute maximum and minimum of f (51:, y) = x2 + my + y2 — 6:1: + 2 on {(117,y)|0 S A S 57 _3 S y S 0} (10 pts) 8) Find the largest product of the positive numbers 2:, y, and z if m + y + z2 = 16. (20 pts) 9) Set up but do not evaluate integrals for the following a) The area inside t = 2(1 + sin 0) and outside 7' = 1, b) I z (the moment of inertia with respect to the z~axis) of the tetrahedan with corners (0,0,0), (1,0,0), (0,2,0), and (0,0,2) if the density 6 = my. c) The volume between the cylinders 132 + y2 = 4 and :62 + y2 = 1 inside 3:2 + y2 + 22 = 9. d) The volume inside 2 = \/$2 + :92 and m2 + 11/2 + 22 = 9. e) The area for the smaller part of the region cut from m2 + 4312 = 12 by a: = 4:92. ...
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