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Unformatted text preview: McGill University
F aculﬂ of Engineering F inal Examination Math 262 Intermediate Calculus Examiner: Prof. N.Sancho ij/XFW‘LO
AssExaminer: Prof. W.Jonsson W Date: Monday Dec. 11 2006
Time: 9:00 am — 12:00 pm Instructions Please answer all questions in the exam booklet provided.
This is a closed book exam. Calculators are not permitted Dictionaries are not allowed This exam comprises the cover page and 1 page of 9 questions Math 262 Fall 2006 11th Dec. 9:00 — 12:00 1 (10 marks) Find the radius of convergence and interval of convergence (including end points) of the series
2 (x +1)"
n(n +1) n=1 x3 2(a) (6 marks) Find the MacLaurin series expansion of, f(x) = 2
x + . Find also the higher derivative f (1 0) (0) . 0.1
(b) (6 marks) Use a series to approximate the deﬁnite integral: I sin (x2)dx, with an 0
error 3 10'5 . 3 (12 marks) Find a series solution in powers of X of y” — xy = 0, y(0) = 1, y’(0) = 0. 4 (12 marks) Find the radius of curvature of r=(2/ 3)t3i + t2 j + tk at the point t = 1. Find also the torsion 2' , the unit binomial vector B and the unit normal vector N at
t = 1. ﬁzz
atas . 5 (a) (6 marks) If x=st,y=s+t and z=u(x,y)=v(s,t).Find (b) (6 marks) Find the equation of tangent plane and normal line to the surface
4x2 — y2 + 322: 10 at the point (2, 3, 1) 6 (10 marks) The 2intensity of light 1n the xy— plane 15 given by a ﬁmction of the form I(x, y)= A— 2x2 — y2( A 1s a constant) Find the path followed by a light seeking
particle, 1f the particle originates at the point (2, 1) 7 ( 12 marks). Show that the equations :
' xu+y2v+uvxy=3 x3 v + u v2 — xy = 1 can be solved for y, v as functions of x and u near the point ( 1, 1, 1, 1) . Find (6%”) and (5%)” at that point. 8 (10 marks) For the ﬁJnction f(x,y) = —xye‘("2+y2)’2 . Find the local max, local min. and
saddle points. 9 (130 ma3rks)3 Find the maximum of f(x, y, z)= xyz subject to the side condition
x3 +y3 +z3 =lwith x>0 y>0, z>0. ...
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 Spring '09
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 Math, Calculus

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