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Name: ____________________________ PID: ________________
TA: ______________________ Sec. No: _____ Sec. Time: ______
Math 20B.
Final Examination
December 12, 2007
Turn off and put away your cell phone
.
No calculators or any other devices are allowed on this exam
.
You may use one page of notes, but no books or other assistance on this exam
.
Read each question carefully
,
and answer each question completely
.
Show all of your work; no credit will be given for unsupported answers
.
Write your solutions clearly and legibly
;
no credit will be given for illegible solutions
.
If any question is not clear
,
ask for clarification
.
#
Points
Score
18
26
36
48
56
66
71
0
86
96
10
10
72
Σ
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View Full Document 1.
Let
2
2
1
()
(2
)
(
1
)
xx
fx
+−
=
−+
.
(a)
(4 points) Find the partial fraction decomposition of
f
(
x
).
22
2
1(
1
)
(
)
(
2
)
)
(
1
)
2
1
)
(
1
)
A
B
x
CA
x
B
x
C
x
x
x
+
++
+
−
=+
=
−
+
So, we see that
x
2
+
x
– 1 =
A
(
x
2
+ 1) + (
Bx
+
C
)(
x
– 2).
Letting
x
= 2, we have: 5 = 5
A
, i.e.
A
= 1.
Letting
x
= 0, we have: 1 =
A
– 2
C
, i.e.
C
= 1.
Letting
x
= 1, we have: 1 = 2
A
–
B
–
C
, i.e.
B
= 0
So, we end up with
2
111
)
(
1
)
2
1
x
x
−
+
.
(b)
(4 points) Find
f xdx
∫
.
2
ln
2
arctan( )
)
(
1
)
2
1
dx
dx
dx
x
x
C
x
x
=
+=
−
+
+
−
+
∫∫
∫
2.
(6 points) Find the area of the region that lines
inside
the circle
r
= 1 and
outside
the
curve
r
= 1 – sin(
q
).
Notice that the two curves intersect at 0 and
p
. That is, we solve for when 1 =
r
=
1 – sin(
q
) and the problem boils down to when sin(
q
) = 0. (The picture helps to
confirm our intuition.)
The area will be given by
()
22
2
00
2
0
0
0
11
1(
1
s
i
n
)
1
(
12
s
i
n s
i
n
1
2sin
sin
2
c
o
s
2
s
i
n
2
2cos
4
1
1
sin 2
1
sin 2(0)
2cos(0)
(0)
4
2
4
20
2
0
dd
d
d
ππ
π
θθ
θ
−−
=
+
=−
⎛−
⎞
⎛⎞
⎜⎟
⎝⎠
⎡⎤
−
+
⎢⎥
⎣⎦
⎡
⎤
⎛
⎞
−
+
−
−
−
+
⎜
⎟
⎢
⎥
⎝
⎠
⎣
⎦
+
+
+
∫∫
∫
∫
0
2
4
−
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View Full Document 3.
(6 points) Using
complex exponentials
, compute
2
cos ( )
x
ex
d
x
∫
. You need not
simplify the result and may leave it in complex exponential form.
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This note was uploaded on 04/29/2010 for the course MATH MATH 20B taught by Professor Takeda during the Spring '07 term at UCSD.
 Spring '07
 Takeda
 Math

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