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Unformatted text preview: MATH241 FINAL EXAM DEC. 14, 2006 W. .
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Directions : 1) There are eight problems, each worth 25 points. You must answer all of them.
2) Use a separate sheet of paper for each problem. 3) Write your name, TA’S name and section number on every page. 4) Show all your work and explain your calculations. 1
W I : Find the equation of the plane that contains both the point (1,2,3) and the line parametrized by the equations :3 = 3 + 2t, y = 1 — 2t and z = 5 — t.
W 2 : Consider the curve
r(t) = (2 + 382t)i + (5 — 62t)j + (3 + e2t)k. Compute the curvature of this curve, K, = “V X a/Hv3, where v is the velocity and a is the acceleration of the curve.
———_—._____________________________ 3 : Consider the surface f (m, y) = xiyg. Compute the equation of the tangent plane
to this surface at the point (16,16,16). Compute the directional derivative Duf(16, 16),
where u. = %i + % j. In what direction is f increasing the most rapidly at (16, 16, 16)? 4 : A rectangular box without a top has a volume of 32 m3. Find the dimensions of the
box with minimum surface area. 5 : Use repeated integration (Fubini) to compute the integral / I! mde where D is the region bounded above by the circle 302 + y2 = 4 and below by the line
x+y=2 6 : Compute the line integral /Fdr, C where the vector ﬁeld is given by F = —yi + azj + zk, and C' is the curve (Helix) described
by r(t) = 2005(t)i + 2sin(t)j + tk Where 0 g t g 27r . 7 : Use Green’s theorem to compute the line integral /Fdr C where the vector ﬁeld is given by F = (m2 + 2y)i + (4x  312) j and C is the curve described
in polar coordinates by 7" = 200s(0) Where —7r/ 2 g 0 3 7r/ 2.
Hint : The inside of this curve is the disc described by {(33 — D2 + y2 S I}. 8 : Consider the vector ﬁeld B1: (I: y . Z —__._.._' —————————, ———————————————————. k.
(x2 + y2 + Z2)3/21 + (£172 + y2 + z2)3/2J + ($2 + 3/2 + Z2)3/2 Use Gauss’s theorem to compute the ﬂux through the closed surface 2, i.e. fz/Fmds where Z is the surface Where the top part is a portion of the sphere x2 + y2 + z2 = 9 and
the bottom part is the plane 2: = I. ...
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 Spring '08
 Wolfe
 Calculus

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