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Unformatted text preview: AEMA 102, Calculus II FINAL EXAMINATION Date: April 27, 2010 Examiners: Time: 14:00  17:00 Profs. J.F. Ha es and RJ. Cue Use of calculators is prohibited. MW 1. Determine if each of the following sequences {an} converges or diverges. Show clearly how
you arrive at your answer. 9 marks (a) { \/n2+2n ~n } (b) <c> {"23"} 2. Determine if each of the following series converges or diverges. Show clearly how you arrive  p
at your answer. ' 15 marks 3 (a) )3 5’33
n22 ’1 (b) 2 3
"=3 n24 °° (1)"*’<n+1)
(0) mill ln(n+l) 0° 24:1
(d) "20 (2n+1)l (e) E ("0” n21 ("W2 * 3. Determine if each of the following series converges absolutely, conditionally, or diverges.
Show clearly how you arrive at your answer. 9 marks °° (—1)"
(a) X
n = l 1171"
(b) 2 100003 (n 11:)
n 3 1 2n + 3
00 n!
(c) 2 n
n = l ("1 00)
4. Evaluate the following improper integrals. ' 1“ 12 marks (a) I: cosxdx 5. Evaluate the following integrals. 15 marks (a) [an x)3 dx (b) I sinzx coszx dx 1dx
“4 (a) [F 6. Find the area of the region in the ﬁrst quadrant that lies inside both the circle r 2 l and the
circle r 3 2 cos 9. 8 marks 7. Find the arc length ofthe graph of 1* =1 ~0058, 0 g 6 3% . 8 marks 8. Find all the points of horizontal and vertical tangency to the graph of
xzcast, yzsin(2t), OstsZn. 7marks 9. The x— and y coordinates of an object moving on a path are given by
" x =cost +tsint, y :sint  tcost, 0 s is 7: (a) What is the initial speed and what is the ﬁnal speed of the object? 3 marks b) What is the distance travelled b the ob'ect? 6 marks
( y J r 10. A portion of a sphere of radius 2 is removed by cutting out a circular cone with its vertex at , ‘ the centre of the sphere. Find the surface area removed from the sphere if the vertex of the cone forms an angle of 8126 —— . a _ 8 marks 3 ...
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