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vector01 - Lakehead University Department of Mathematical...

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Unformatted text preview: Lakehead University Department of Mathematical Sciences Nlath 4012 Final Exam December 14, 2001 Starting Time: 9:00am Duration: 3 hours Instructions: Show and explain all work. Do all questions. Give your answers in the space provided: using the back of the preceding page if nec- essary. Hand in all pages of this booklet. No calculators or other aids are permitted. There are 11 pages in this booklet, including this one. Famin Name: Question Given Name: Student Number: 1. [3 marks) Do the lines fut) 2< 1, U, 1 > +t < 2,3,1 > and F2(3) :< 3, 3, 2 > +3 < 2,4,1 > intersect? If they do, at what. point? 2. (3 marks} A particle moves with position function Ht) =< t, cos t, 1+t > . Find the position1 velocity, and acceleration of the particle when t = 7r/2. 3. [3 marks] Find the limit, if it exists, 01' Show that. the limit. does not exist , my — y + 2:1: — 2 11111 ~—9—— [am—111,1) 3“ _ 1 4. {5 marks) Use the change of variables 3: : 2-u—t—3v, y : 3-u— 21.1 to evaluate the integral Hell" + y) UTA, where R is the Square with vertieee (0, fl), (2, 3), (5.1), and [3, —2]. 5. (4 marks] Find the mass of an object that occupies the region in the first octant bounded by the planes 3: = U, y = 0, z : 0, and the sphere 932 + 312 + Z2 = 1 if the density is given by p(:::_. y, z] : 1 + 3:2 + y? + 22. 6. (3 marks) Compute f0 31:2 dm + 4my2 d-y, where C is the segment. of the curve 3; : 3:3 from (1,1) to (2,8). 7. (4 'r’narks) Cmnpute 390 my 033: + arms 3,: d-y: where C is the triangle with vertices [0, 0], [110), and (U, 1), 8. (5 marks) Compute the surface integral HS 222 d5: where S is the part of the cylinder :62 + y? = 2 that lies between the planes z : U and 2 = 1. 9 9. (5 marks] Evaluate Ifl'flcurlfi-fiyiS, where Flag-y, z) : yi—mj+111(1+z2)f< and S is the part of the sphere x2 + y2 + [z + 1)2 : 4 that lies above the )<:y-};)l::1ne1 oriented upward. 10 10. (5 marks) Evaluate ffsflg: - fide, where S is the surface of the solid bounded by the planes 3: = 0, y = 0, z = 0, and E + y + z = 1, f1 points OutwaUL and fi‘{:::, 3;, z] z [y + z]i + 3:223 + [$2 + yfic. 11 ...
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