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exam1MATH262 - MCGILL UNIVERSITY FACULTY OF SCIENCE FINAL...

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Unformatted text preview: MCGILL UNIVERSITY FACULTY OF SCIENCE FINAL EXAMINATION MATHEMATICS 262 Winter 2006 DATE: April 11, 2006 TIME: 2:00 PM — 5: 00 PM EXAMINER: Prof. P. Kassaeifi/f/WA ’56 _ L ASSOCIATE EXAMINER: Pr abute INSTRUCTIONS 1. Fill in the above clearly. 2. DO NOT TEAR PAGES FROM THIS BOOK! All your writing — even rough work — must be handed in. You may do rough work anywhere in the booklet. 3. Calculators are NOT permitted. 4. You are allowed to use a regular or a translation dictionary, if you wish to. 5. This is a closed-book examination. 6. The examination booklet consists of this cover, Pages 1 through 11 containing questions; and Pages 12, 13, and 14, which are blank. Your neighbour’s version of this examination may not be the same as yours. 7. A TOTAL OF 80 MARKS ARE AVAILABLE ON THIS EXAMINATION. PLEASE DO NOT WRITE INSIDE THIS BOX 1 2(a) 5(2).) 5(1)) 6 7 /3 III...- ' TOTAL /80 Final Examination — Math 262, Winter 2006 — 1. (8 MARKS) Find all points a: for which the power series 00 3310—2" ZLngT) n=1 converges absolutely, converges conditionally, or diverges. Final Examination — Math 262, Winter 2006 — 2. (12 MARKS) Recall that the Maclaurin series of em is given by 00 :1: 2 :33” e = —l. TL. n=0 This question continues overleaf. (a) (6 MARKS) Write down the Maclaurin series of g(:r)=/ te‘tadt 0 and find its radius of convergence. Final Examination — Math 262, Winter 2006 — 1/2 3 / te‘t dt 0 with an error less than 10—3. (b) (6 MARKS) Estimate Final Examination — Math 262, Winter 2006 — 4 3. (12 MARKS) Consider the curve parameterized as follows r(t) = ti + t2j + t3k fOIOStSZ This question continues overleaf. (a) (3 MARKS) Write down an integral that calculates the length of this curve (DON’T evaluate the integral). (b) (5 MARKS) Find the unit tangent, the unit normal, and the unit binormal vectors at the point (1, 1, 1) on the curve. Final Examination — Math 262, Winter 2006 — (c) (4 MARKS) Find the curvature and torsion at (1, 1, 1). Final Examination —— Math 262, Winter 2006 — 5 4. (12 MARKS) Consider the function F(:c, y, z) = sin(:ryz) — :10 — 2y — 3,2. This question continues overleaf. (a) (2 MARKS) Write the equation of a level surface of F passing through (2, —1, 0). (b) (4 MARKS) Find the equation of the tangent plane through the point (2, —1,0) to the level surface in part (a). Final Examination — Math 262, Winter 2006 — 7 (c) (3 MARKS) Find the direction in which F (5c,y,z) increases most rapidly at the point (2, —1,0). (d) (3 MARKS) Find the directional derivative of F at (2, —-1,0) in the direction of the vector i — j + 3k. Final Examination — Math 262, Winter 2006 — 8 5. (14 MARKS) This question continues overleaf. (a) (8 MARKS) Showing all your work, determine and classify all critical points of the function f(rr, y> = as local maxima, local minima, or saddle points. Final Examination — Math 262, Winter 2006 — 9 (b) (6 MARKS) Determine the global extrema of the function f over the disc D = {(cc,y)|:v2 + y2 g 1}. Final Examination — Math 262, Winter 2006 — 10 6. (12 MARKS) Use the method of Lagrange multipliers to find all the max- imum and minimum values of f (:0, y, z) = xyz subject to the condition x2+2y2+322=6 and all the points at which those values are attained. (No other method Will be accepted) Final Examination — Math 262, Winter 2006 — 11 7. (10 MARKS) If 2 = f (33,31) has second partial derivatives, and cc 2 65, y = 8H5, show that 822 H3 (% 822 fizz _ s _ s+t asat ‘ 6 6g + axaye + 51,26 ) mwmpfia B 863ng mg 98:3 owdg mg no mmwg mofldsfipqg mEp 0p Hflg “5:: 50% EHMEDZ qumOMnH EOE m0<m ZOHB<DZHHZOO NH I robbm ~3an “mmm fin: l QQEQEEQNH ESE :Vmpfia E 98303 mg 95:3 owdg 9E no omdg qosdsfipmoo 3% Op 3on “SSE 50> mmszZ qumOMnH MOE m®<m ZOHH<DZHHZOO 2 l QQQN ARES» “Nam 53: I moflmflfidkm REE mwmpfia 2 86305 map 98;? mwda 23 no owwg mofldsfifiso 2% Op $me #4:: 50% mmmzbz qumoamm mow m0<m ZOHE<DZHBZOO E I QQQN mounts “mmm in: I: :oEmESmNm ESE ...
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