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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: : 2u—t—3v, y : 3u— 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 dy, 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,: dy: 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'ﬂcurlﬁﬁyiS, where Flagy, 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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