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Unformatted text preview: NOTE TO PRINTER CMM W“!
ﬁll/15W (These instructions are for the printer. They should not be duplicated.)
THIS EXAMINATION SHOULD BE PRINTED ON 8% x 14 PAPER, AND STAPLED WITH 3 SIDE STAPLES7 SO THAT IT OPENS LIKE
A LONG BOOK. There are FOUR versions of this examination. Each of them should be printed with a
DIFFERENT COLOURED COVER. IN THE EXAMINATION ROOM VERSIONS ##172
OF THE BOOKS SHOULD BE ALTERNATED IN ONE ROW (=STRIPE=OOLUMN)
AND VERSIONS ##BA IN THE ROWS (=STRIPES=COLUMNS) ON EITHER. SIDE,
WITH A DIFFERENT PAIR OF BOOKS USED IN ALTERNATE ROVVS, SO THAT NO
STUDENT IS NEXT ON ANY SIDE TO A STUDENT WRITING’THE SAIVIE COLOUR
OF EXAMINATION. NORMALLY THIS BOOK IS ENOUGH FOR ALL THE STUDENT’S WRITTEN
WORK, including rough work; STUDENTS SHOULD NOT NORMALLY BE GIVEN A
BLANK EXAMINATION BOOKLET IN ADDITION. VERSION 1 MCGILL UNIVERSITY
FACULTY OF SCIENCE
FINAL EXAMINATION MATHEMATICS 222 2005 09 CALCULUS 3 EXAMINER: Professor W. G. Brown DATE: Monday, December 19th7 2005
ASSOCIATE EXAMINER: Prof. P. Kassaei TIME: 14:00 — 17:00 hours FAMILY WWW
GIVEN NAMESDJJIDJJIJJEID 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, The examination booklet consists of this cover, Pages 1 through 7 containing questions; and Pages
8, 9, and 10, which are blank. Your neighbour’s version of this examination may not be the same
as yours. 5. There are two kinds of problems on this examination, each clearly marked as to its type. 0 Some of the questions on this paper require that you SHOW ALL YOUR WORK! I Their
solutions are to be written in the space provided on the page where the question is printed.
When that space is exhausted, you may write on the facing page. Any solution may be
continued on the last pages, or the back cover of the booklet, but you must indicate any
continuation clearly on the page where the question is printed! 0 Some of the questions on this paper require only BRIEF SOLUTIONS ; for these you are expected to write the correct answer in the box provided; you are not asked to show your
work, and you should not expect partial marks for solutions that are not completely correct. You are expected to simplify your answers wherever possible. You are advised to spend the first few minutes scanning the problems. (Please inform the invigilator
if you find that your booklet is defective.) 6. A TOTAL OF 70 MARKS ARE AVAILABLE ON THIS EXAMINATION. PLEASE DO NOT WRITE INSIDE THIS BOX 4 ( a) TOTAL Final Examination — Math 222 2005 09 — V61" SI OH 1 l
’) ’) 1. BRIEF SOLUTIONS [10 MARKS]
y‘ 2‘ Consider the function f(:13, y, z) = a: — g — E . (a) [3 MARKS] Give an equation for the level surface off passing through
the point (4, 2, 3). ANSWER ONLY (b) [4' MARKS] Give an equation for the tangent plane through 13(4, 2, 3)
to the level surface of f passing through P. ANSWER ONLY ((3) [3 MARKS] At the point (4,2,3) determine the rate of change of
f(x, y, z) with respect to distance in the direction of 2i +j — 2k. ANSWER ONLY F ina] Examination — Math 222 2005 09 — 81011 1 Ex} 2. [[710 MARKS] (a) BRIEF SOLUTIONS [2 MARKS] For the function 1 , 1+1: the Maclaurin series is ANSWER ONLY The interval of convergence of this series is ANSWER ONLY (b) SHOW ALL YOUR WORK! [4 .MARKS] Use your result «m the pre— ceding part to determine the Maclaurin series for : ln(1 — KL“).
Justify all steps in your derivation, and state without proof the in—
terval of convergence of the series you obtain. (c) SHOW ALL YOUR WORK! [4 MARKS] When a: = , carefully de— termine a partial sum of the Maclaurin series for f that could be
used to approximate 111% to Within an error of 0.01. [\DIH Final Examination — Math 222 2005 09 — V61" SI OH 1 3 alSHOMIAu_YOUR\NORKMpniuARKa
(a) [6 MARKS] For the curve mw=( 2 —Qi+ % er (—w<t<+m) 1 + t2 1 + t2
determine — simpliﬁed as much as possible — a formula for the
distance 5(t) from the point with parameter value t = —1 measured in the direction of increasing 15. Show all your work. (b) [4 MARKS] Showing all your work, determine — simpliﬁed as much
as possible — parametric equations for the tangent to the curve at
the point where t : 2. Final Examination — Math 222 2005 09 — VGISJ OH 1 'I 4. [10 MARKS] Let 12/ / 6—” _y dxdy.
‘0 y (a) BRIEF SOLUTIONS [2 MARKS] Sketch the region over which this iterated integral is taken. ANSWER ONLY (b) BRIEF SOLUTIONS [2 MARKS] Write I as one or more iterated integrals with the order of integration reversed from that of the given
iteration. ANSWER ONLY (c) BRIEF SOLUTIONS [2 MARKS] Write I as an iterated integral in polar coordinates. ANSWER ONLY (d) SHOW ALL YOUR WORK! [4 MARKS] Evaluate I. Final Examination — Math 222 2005 09 — V61” 81 OH 1 s 5. SHOW ALL YOUR WORK! [10 MARKS] Use the method of Lagrange multipliers — no other method will be accepted for this problem  to
ﬁnd all points on the surface 3:2 + 10y2 +22 2 5 Wherethe function 8.7; —42
has a global maximum, and all points Where it has a global minimum. 4 Final Examination — Math 222 2005 09 ~ VGFSI OH 1 ['i 6. SHOW ALL YOUR“"WORK!\ [10 MARKS] (a) [5 MARKS] Showing all your work, determine and classify all critical
points of the function
095,31): 3563/ — 9321/  5W2 as local maxima, local minima, or saddle points. (b) [5 MARKS] Determine the global extrema of f over the triangle with
vertices at (0,0), (0,4), and (4,0). Final Examination — Math 222 2005 09 — V81" SI on 1 7 7. SHOW ALL YOUR WORK! [10 MARKS] Suppose that 2: : ﬂu, o), where f has continuous second partial derivatives. If u 2 es cos t, and o =
e'5 sint, carefully determine the value of (2+ 2) 822 _ 8—22+82Z
“_ U at? +602 852 a2 and show that it is constant. Final Examination — Math 222 2005 09 — V8131 On 1 CONTINUATION PAGE FOR PROBLEM NUMBER You must refer to this continuation page on the page where the problem is
printed! 8 Final Examination 4 Math 222 2005 09 — SI On 1
CONTINUATIGN PAGE FOR PROBLEM NUMBER You must refer to this continuation page on the page Where the problem is
printed! 5} F ina] Examination — Alath 222 2005 09 —— 81 OH 1 L0
CONTINUATION PAGE FOR PROBLEM NUMBER You must refer to this continuation page on the page Where the problem is
printed! ...
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 Spring '08
 KarlPeterRussell
 Math

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