Obtained from Ryerson's Mechanical Engineering Course Union (MECU)
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MTH 240 Test 1
1
DEPARTMENT OF MATHEMATICS
MIDTERM TEST #1
MTH 240 - Calculus II
Last Name (Print): _ _ _ _ _ _ _ . First Name: _ _ _ _ _ _ _ . Student Number: - - - - Signature: _ _ _ _ _ _ __
Section (circle one)
Date: May 31, 2012, 6:00 pm
Instructions:
Duration: 90 minutes
1. This is a closed-book test. Notes, calculators and other aids are
Dr. B. Tasic :
1 2 3 7
Dr. A. Alvarez :
4 5 6 8
For Instructor's use
not permitted.
only.
2. Verify that your test has pages 1-6.
3. (a) Unless otherwise instructed, make sure you include all signif-
Page
Mark
U
icant steps in your solution, presented in the correct order. Unjustified answers will be given little or no credit.
2
I
3
I
4
I
5
I
6
I
EC
Cross out or erase all rough work not relevant to your
solution. Put a box around your final. Ianswer. I
(b) Write your solutions in the space provided. If you need more
space, use the back of the page. Indicate this fact on the original
M
page, making sure that your solution cannot be confused with any
rough work which may be there. Marks (out of 50) are shown in
brackets.
4. Do not separate the sheets.
5. Have your student card available on your desk.
MECU
Page 1 of 6
Total
/50
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Obtained from Ryerson's Mechanical Engineering Course Union (MECU)
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MTH 240 Test 1
2
1. [7 marks] Evaluate the following integral.
fo7r t 2 sin t dt
t? = U.
Pu.J·
a.uJ
Sin
t df = dv .
Then.
21:: Jt = du. and
V:::-cost
1r
1Jrt s.:n.t dt
0
Now
')'(
- - i}coscl +2Jtcost dt
0
0
i. ~ u.- and c.ost dt :::dv w~c.l.
2
~
v
= - t~l:, / +2
1T"
t2
cost
0
/II
D
-11"~ (-t) +
I T+ !2 ClJst IT
~
0
t-
2 (-1
o
-!)
M
EC
:=
+ 2 t.stnt
0
U
- -
.,.
{-t.stnt / - jsmt cit)
0
-
EJI'ue:J
2. [3 marks] Write out the form of the partial fraction decomposition of the function:
x(x- l)(x 2 + x
A
:c.
MECU
-+
8
+ l)(x2 + 1)3
+
x-1
Page 2 of 6
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Obtained from Ryerson's Mechanical Engineering Course Union (MECU)
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3
MTH 240 Test 1
3. [10 marks] Evaluate the following integral.
: sec x.
ho..nx -
J secx. ( sec. x. -1.) dx.
-:. se.c :t
t-a..n ~c
J sec 3x dx.. + Jsecx. c:it: .
J sec'3.ocdx.
Flna.lllf
MECU
dv
= .sec x
-
2
ht.n x + J
Bec:,cdJC ""
=
Page 3 of 6
HenCQ.
.sec:cl: a.nx + 1n J secx -1- f:<m::c/
J Sec"~>oc doc = ±~=ton"' + f k.i sec:.:+ tlll\:r I
M
·2
1
~e c 2 e>c. d-:>c. ::
EC
sec ::c =- u..
U
Jtan2x sec3 x dx
}
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~
Obtained from Ryerson's Mechanical Engineering Course Union (MECU)
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MTH 240 Test 1
4. [10 marks] Evaluate the following integral.
.! sec. e
2
: .29 -$c:n.28 + C =
Z sec-' {Zx)
- /
4~'2- -1
+C
DR
co~-· ( .!.
2x.
)-
1'1 J:}·- 1 + c
2~2
M
-= 2
EC
2x,2.
MECU
.'4,.~~-
U
X. ::
su.e =2x. ='> eose
=>
I
-= -
2X
S1:ne : ~1
2x
He,...,c.e ~n.?.E9 =2SiJ18 tosS :: -IZiJ:.~1
}
J.x,2..
e =s~c-• (2.x)
Page 4 of 6
OR
8 = cos-1 ( ~x.) }
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Obtained from Ryerson's Mechanical Engineering Course Union (MECU)
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5
MTH 240 Test 1
5. [10 marks] Evaluate the following integral
4x
d
Jx3 +x2 +x + 1 x
x. 2 (:X.+1
'-1-x.
(.x+1 )(:X.2 +1)
-=
)+ X-t-1
(.x+1)(X."L-+i)
A+B: o
B+C.= 4
A+c. =o
= ~
}
Ax'+A+ "E>::r."2.+6xtCx1- c _ "'+B):~?+(B+-c.):x: +A+C}
4x..
-=
-=?>
-
+
X+1
f3x+
c
:X}+1
(X+1)(.x.'+1)
B=-A
-A+c.=4
A-+C=O
{C-=2.]
fA=-2(
U
2C.=4 =j
IB-=-2/
J x>~:~+:x+1 d:x : J(::-1 ~~+2) dx = -2.J dx. + J 2xd;(
+-<j d:c
2.+
EC
+
t1
.X+1
~
1
M
X
MECU
Page 5 of 6
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:X:2t1
}
Obtained from Ryerson's Mechanical Engineering Course Union (MECU)
www.ryemecu.com
MTH 240 Test 1
6
6. [10 marks] Determine whether the following integral is convergent or divergent. H it is convergent, find its
value.
dx
Jo ~x 1
{9
}
( I
= -12 + "
(
)
1
~
!.
t.-?1+
= I IYY)
::
~
2
MECU
9
8
ck.
M
I
j .~ A_
EC
U
}
~
Inn
t-+1+
~
.X.-j
[
:z.
:
Ia~')')
t-~1+
~
J -u..-~k
i-i
B l _ ~ --t)i J =~ . 4 ::b
Page 6 of 6
=
/,rn
t-+ 1'*"
2.
a
~ ~3 J
'=
-t.-1
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