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Unformatted text preview: MCGILL UNIVERSITY 1 FACULTY OF ENGINEERING FINAL EXAMINATION W  _ . INTERMEDIATE CALCULUS
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Exa ' er: Professor J. Labute A Date: Monday, December 12, 2002 Associate Exami e ‘ Professor JwJ. Xu Time: 9:00 A.M. _ 12 NOON INSTRUCTIONS Attempt all questions.
All questions are of equal value.
Justify all your statements.  This exam comprises the cover and 2 pages with 9 questions.  W Is ﬂax/17:0 mam from Final Examination MATH 262 December 12, 2005. 1. Find the interval of convergence of the series and determine whether the series converges absolutely or conditionally at the
endpoints. 2. (a) Find the Maclaurin expansion for the function
Fm = / sin(t2) art
0 and compute F(1) to 2 decimal places. (b) Find the Taylor series expansion of 23—1 flxl=m about a = 1 and use this to compute f(9)(1) and f (MG). 3. Find the solution of the differential equation (1 . $2)y” e my’ + 9y a 0 satisfying y(0) = 0, y’(0) : 1. 4. Given the curve r = tzi + t2j + t3k,
(a) ﬁnd the arc length of the curve from t m 0 to t = 1; (b) ﬁnd the unit tangent vector T, principal normal N, and binormal 13 of the
curve at the point t = 1; (c) ﬁnd the curvature K. and torsion r of the curve at point t n 1. Final Examination MATH 262 ~ December 12, 2005. 5. Show that the function me if (at) m (0,0),
0 if (m) s (0,0) is continuous at (0, 0) if a m 0 but is not continuous at (O, 0) if a “ft 0. 6. (a) Find the tangent plane and normal line to the surface
29y + yz + 5522 = 3
at the point (1,1,2). (1)) Find the direction in which the function . f (353,52) 2 x2 + my + 222 has
its maximum rate of increase at the point P(1,1,1). What is the rate of
increase at P in the direction from P to the point Q(3, 4, 5). 7. A surface 2 2 f (3;, y) has the parametric representation 2 3
) wzui—vg, yrrxu —o 3:21“! in a neighbourhood of the point (3,3,4) which corresponds to the point (2, 1)
in the uuplane. Find %(3,3), %(3,3), £23136, 3). 8. Find and classify the critical points of the function f (3:, y) : 2:33 7"“ 6mg + 31f.
What is the Taylor polynomial of degree 2 in the Taylor expansion of f (may)
about each of the critical points? 9. Using the Lagrange multiplier method, ﬁnd the maximum. and minimum values
of the function
f (it’ y) = 256 + 9 on the ellipse 2:2 + my + y2 2 1. ...
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 Spring '09
 GREGRIX
 Math, Calculus

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