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Unformatted text preview: Name:
Student ID Number McGill Universigg
Faculﬁ of Engineering Final Examination: Math 262
Intermediate Calculus MAW/"J Examiner: Prof. N.Sancho Date: Thursday, Dec. 13, 2007
ELL—« Assoc. Examiner: Dr. D. Serbin Time: 9:00 — 12:00
Instructions Attempt all questions. No calculators allowed.
No dictionaries are permitted. This is a closed book
exam. Answer all questions on the pages provided.
If needed continue questions on PRECEDING page or in the blank pages at the end of the booklet. This examination comprises the cover and 12 pages with 9
questions and 3 blank pages Final Math 262 Fall 2007 — Dec. 13th — 9:00 to 12:00 1. (10 marks) Find the interval of convergence of the power series including the end
points: I”! Z (—1)“ 2" (x—3)"_ Final Math 262 Fall 2007 — Dec. 13th — 9:00 to 12:00 2(a) (6 marks) Find the Maclaurin series for the ﬁlnction etel
t dt. E(x) :3 How many terms of this series are needed to compute E(1) with error less than 0.001?
(b) (6 marks) Find the sum of the series i1 "=1 ne" Final Math 262 Fall 2007 — Dec. 13th — 9:00 to 12:00 2
3. (12 marks) Find the general solution of Z )2} = x y , in the form of a power series
x y = Zan(x—1)" with a0 and a1 arbitrary. n=0 Final Math 262 Fall 2007 — Dec. 13111 — 9:00 to 12:00 4. A curve is given by r(t)—  (cost +tsint)i + (sint— tcost) j + t2 k.
(a) (4 marks) Find the arc length between the points where t— — 0 and t— — 7r / 2 (b) (4 marks) Find the unit tangent vector T ,unit normal N and unit binormal B at
the point t— — 7z/2. (c) (4 marks) Find the curvature K at the point t = 7r/ 2 . Final Math 262 Fall 2007 — Dec. 13* — 9:00 to 12:00 5 (a) (5 marks) Find the parametric equation of the tangent line to the curve of
intersection of the surfaces x2 + y2 =1 and x + y + z — 1 = 0 at the point (1, 0, 0). (b) (5 marks) Let f(x, y) = 1n lrl where r = xi + yj. Show that V f = Ill;
1' Final Math 262 Fall 2007 — Dec..13th — 9:00 to 12:00 6 (a) (6 marks) Find the direction in which the function f (x, y, z) 2 x32 + y3z2 — xyz
has its maximum rate of increase at the point (1, 1, 1). What is the rate of increase at
( 1, 1, 1) in the direction of the point (1, 2, 2)?
(b) (4 marks) Find the tangent plane to the surface: x3 z + y3 z2 — 30/2 =1 at the
point (1, 1, 1). Final Math 262 Fall 2007 — Dec. 13th — 9:00 to 12:00 7. (12 marks) Let x, y, z, u, v be related by the equations
x=u+mv yzv—lnu
z=2u+v' Find % and .a—z atu=1,v=1.
6x 6y
y x Final Math 262 Fall 2007 — Dec. 13Th — 9:00 to 12:00 8. (12 marks) Find and classify the critical points of the ﬁmction
f(x,y) = x4 + y“ — (x +y)2 Final Math 262 Fall 2007 — Dec. 13th — 9:00 to 12:00 9. (10 marks) Use the method of Lagrange multipliers to ﬁnd the maximum and
minimum values of f(x, y, z) = xyz on the sphere x2 + y2 + z2 = 12. Final Math 262 Fall 2007 — Dec. 13th — 9:00 to 12:00 Continuation of # 10 Final Math 262 Fall 2007 — Dec. 13‘h — 9:00 to 12:00 WWI—0M 11 Continuation of # Final Math 262 Fall 2007 — Dec. 13th — 9:00 to 12:00 12 ...
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 Spring '09
 PAYNE
 Math, Calculus

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