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Unformatted text preview: MCGILL UNIVERSITY FACULTY OF SCIENCE FINAL EXAMINATION MATH 222 Calculus 3 Examiner: Gil Alon Date: Wednesday April 27, 2005 Associate Examiner: Professor Wilbur Jonsson Time: 2:00PM — 5:00PM INSTRUCTIONS Please answer all questions in exam booklets provided. This is a closed book exam. Calculators are not permitted. Regular or Translation dictionaries are not permitted. This exam comprises the cover page, and 1 page of 8 questions. McGill University — Dept. of Mathematics and Statistics Math 222- Calculus 3 Final Examination - 2005 Winter term Examiner: Dr. G. Alon Associate examiner: Prof. W. Jonsson Answer the following 8 questions. You do not have to simplify numerical answers (e.g, sin 5 —10g4 is an acceptable answer). 1. (13 points) Find the radius of convergence of each of the following power series: (a) i 73(5): — 1)” 71:1 00 271x211 (b) 71. Z 1 n 2. (13 points) (a) Find the first two nonzero terms in the Maclaurin series of f = cos5w and find an estimate for the sum of the remaining terms of the series, for 0 S a: S 0.1. 1 1+2:2 (b) Find the Maclaurin series of arctant by integrating find ,f‘10)(0) where = arctan 5x2. . Use this expansion to 3. (13 points) Find the equations of the tangent plane and the normal vector to the given surface at the given point, for: (a) (:v,y,z) : (2set,coss+t,§), at s = t = 2 (b) ($2+y2+22)2+‘/yz=10atx:y:z:1. 4 12 points) - ( (a) Find the arc length of the curve r(t) = (3752, 4x/2t1‘5, fit) between 7(0) and r(1). ( b) Write the arc length parametrization of this curve. The answer should be a function of the parameter s. (c) If P : (12,16, 12) and f(:r,y, z) is a differentiable function defined in a neighbour— d hood of P and satisfying Vf(P) = (1, 1, 2), find Ef(r(t))|t:2. (Note: r(t) is the same curve as in part (a)). 01 . (13 points) Let C be the curve of intersection of the surfaces defined by the equations: 322 + y2 + 22 = 12 and zewfl’ = 2. Find the equation of the tangent line to C at the point (2, 2, 2). 12 points) Let f(;r, y) = 3:232 + 2933113 + 3y2. a) Find and classify the critical points of f in the entire plane. b) Find the minimum and maximum value of f in the region *2 g a: S 2, “2 g y S 2. DJ 1 2 2 *— ' = : < l/fljm2+y2+5drdy whereD {(x,y) :6 +9 _16} b) f / mdmdy where D is the triangle whose vertices are the points (0,1) , (1, 2) and D ( ( ( 7. (12 points) Calculate the integrals: ( ( (2» 0)- y 8. (12 points) Let : {Jr—0:33?— y direction of the vector (4, 3). Find the direction of fastest ascent for f (the answer should be a unit vector). . Find the directional derivative of f at (0,0) in the ...
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This test prep was uploaded on 04/19/2008 for the course MATH 222 taught by Professor Karlpeterrussell during the Spring '08 term at McGill.

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