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Unformatted text preview: @ MAT 1322A W2009 Wednesday, Feb. 4th 8:30—9:50 Prof. Desjardins MIDTERM TEST 1 Max=20 Student Number: OTime: 80 min. 0 Only basic scientiﬁc calculators are permitted (non—graphing, nonprogrammable, no
integration or differentiation capabilities). Notes or books are not permitted. 0 Work all problems in the space provided. Use the backs of the pages for rough work if
necessary. Do not use any other paper. 0 Write only in nonerasable ink (ballpoint or pen), not in pencil. Cross out, if necessary,
but do not erase or overwrite. o The problems require complete and clearly presented solutions and carry part marks if
there is substantial correct work towards the solution. 0 There are ﬁve questions worth four marks each. ® 5
. . 2 . . .
1. (a) Cons1der the lutegral / (33—3— dm. Does it converge or d1verge? If it con
0 _ )4/3
verges, give its value. 5 .  5
f 0? 6/1 = J , 5/1 + 0?. 0/9:
0 (7013) W3 ‘ ~ VI} 3 . =‘/(3 0 (253) _ (1,3)
3 I:
f . a? / dx = Im/ / _ O? 4/2:
a (JO.3)” £93" 6' [7L~3)% 2 CX3 2 _
(b) Determine if the integral / ﬂ 2 + z d2: converges or diverges. Justify your claim.
1 x e M/ xa 4/1/4444 fé (many; 2. What is the area of the region bounded by the curves y = 4 — 172 and y : 3w ? @ , 3. Find the volume of the solid obtained when the region bounded by y = 1/15, 3; = 0, m z 1 and a: = 2 is rotated around the line y = 2. Include a sketch of the region and
a typical crosssection of the solid in your solution. CHIS} Jail/O}! Ar <21 odaJ’éf 0r I‘lp/j' .4;
5/ 71% lid/16f a); I} 0,, z :2 ~ 9)!
J76. auJé/ heal/(tar [f PM : 9’)  17 H
‘ 2
J}; m JCT/uh": N V: anIc/x
I
2 ® 4. A heavy rope of length 15 m has a density of 1.2 kg/m and is hanging over the edge
of a tall building. How much work is done pulling the rope to the top of the building?
The acceleration of gravity is g = 9.8 m/s2. Deﬁne clearly all the variables that enter
into your solution and provide a drawing which shows their meaning. 1 ca ”3/” M pm Mr Wm
M m gaze/I ﬁe?” 5““ W” ”5’ A1 1‘;
W wag/1;! é;ﬂ){9f)4x ’U : ”‘76/11 N ﬂit [1:306 0.2L JKL‘M’) Z,“ 11an /%€ flZé
47F % M77 awe a Mir/7g”, w Jim] ya 7% war4 ff W!” = [/LK'JPGAX— 'J n m
79H (.0sz I‘J’ £1} 5: 2U}; :. 2 (L791; A)! j ‘3’ L =I 7&4. [Ki/WU} a0 4.2—; 0 0, 14 .9 00 A, M Al= [Mien/z = Swazi/0”” 5. Solve the following differential equations: dy_2:1: (a) (772 + 1)dx y2 (b) ﬁ ”“3 dsc _ cos y MAT 1322A W2009. Wednesday, Feb. 4th 8:30—9:50 Prof. Desjardins MIDTERM TEST 1 Max=20 52¢ Mgr/Vin x4
74’ We‘re M14 Student Number: oTime: 80 min. 0 Only basic scientiﬁc calculators are permitted (nongraphing, nonprogrammable, no
integration or differentiation capabilities). Notes or books are not permitted. 0 Work all problems in the space provided. Use the backs of the pages for rough work if
necessary. Do not use any other paper. 0 Write only in non—erasable ink (ballpoint or pen), not in pencil. Cross out, if necessary,
but do not erase or overwrite. o The problems require complete and clearly presented solutions and carry part marks if
there is substantial correct work towards the solution. 0 There are ﬁve questions worth four marks each. @ 4
3
1. (a) Consider the integral / Wdas. Does it converge or diverge? If it con
0 _
verges, give its value.
v ‘9 2’ q 3
9 06¢ : / all + f 1 ‘ 4/76
a (x 2)"/~3 o (252)” .2 (2: W3 2 3 . ,9 0/
Z (75270 CA : jw / (2:4)?” )6 °° 1+sin2x b Dt ' 'fth 't 1
() e erminei em egra /1 m3+ex dd; converges or diverges. Justify your claim. 57nd /+J‘/73772z S Q 3
(25.3! 23‘6’1) )Z‘ ./+Jvi«‘>a 4 9 < .2.
27%" 1+6 00 ﬂ 2 oo .
51) / M 0/9: C :3 6/" Vllrc/ (meager
/ 3 n 2 ’
'2 4e I 2. What is the area of the region bounded by the curves y = 0:2 and y = a: + 2 ? : (3’ (3)2 9(a)  31(9)?)  (lg/.01 awr 5403} = ('2+°'§)(3‘1—2+:;> : ~§ —' ’
(a 3 3+2 '3‘
: $441
1 3
2 S“‘/3 3. Find the volume of the solid obtained when the region bounded by y = 1/22, 3/ = 0,
a; = 1 and x = 2 is rotated around the line y = —1 . Include a sketch of the region and a typical cross—section of the solid in your solution. 7A
A:l/"
I (y >x [/‘Jﬂ‘— “(ﬂ/IQ a we war/Len @ (Q 4. A heavy rope of length 12 in has a density of 1.5 kg/m and is hanging over the edge
of a tall building. How much work is done pulling the rope to the top of the building?
The acceleration of gravity is g = 9.8 m/s2. Deﬁne clearly all the variables that enter
into your solution and provide a drawing which shows their meaning. ”/ng 70 W 4),, 44:4" WW
().§)(7—d"‘)4x AJ '= N57 M M AmoééaWZ/m 9; a}; = /‘£7x,~‘/Jx J h
w; = :2 H.790” J’ 1—! 7%“ ,4k/ A} a): H‘N‘: 5. Solve the following differential equations:
4x (a) (x2 + 1);: = 92 — V1 4% 22+! k y3 : QA[22%/)+C ﬂan 5,5,1 : (bi/210+ Cy; dy 330033:
(b) E  cos y ijﬂ 4/; 3 760247 (AC (740 = Ctr/$41 +a’2m + C To ﬂ : Kira/i1 {kl/112+ (22.1% 1L C ) ...
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 Winter '10
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