k:sc:2
Spring 2012
MATH 1220-002: Final Exam
N a me:
~
Robbie Sneliman
[ustructions: The following questions are intended to assess your abilities on the basic concepts that we
have covered this semester
Answer all questions to the best of your ability and simplify all solutions as
much as possible. Regardless of the simplicity of the problem a/I work must be shown in order to receive full
credit, otherwise no credit will be awarded.
10~
(a)
th
Problem 1: Compute ~ for the following functions (you may use any method):
y
=
sin2(x3)
=
(b)
y
=
3 ln(e5’ + 1)
~
c3X
(c)
y
xtan’(~)
I
1
2-
Problem 2: Compute the following integral:
J
t~
S
(0 ~
10
(
U~
-
-
fC
9
10
Problem 3: Compute the following integral:
f
r~4
c~
~1-
~ (~ ~
X
~
x
Coc(~~~
~
2-
~~Coc(~)
j
1O4~-N
~
Lf
/~
0
3
10
[o?~
Problem -1: Find the indicated limit, or stat-c that the limit does not exist:
3:c
ln(lOOx +
e’)
~
L~ ~ ~
©c.
~ ~O
JL~I~O~
~
o
~
~&
C~c~c~
(1C)~-~
~
4
l0
Problem 5: Evaluate the following integral:
Ji
‘~
+
(i:c
8
di
c~c~c~
~
x
~
~
A~.
~
-
7
(3
~(~-
~
c~s~
~ I
)~
-
5
____
2~(~
~
—
Problem 6: Does the series ~
‘On~
~ converge or diverge? \Vhv or why not?
=
3
~Th
~°
-
I
2o~
3
_________
6
‘
I ~OV\
T~
Problem 7: Find the sum of the following series:
~)fl~I
~
‘I =
1
5fl
~
C~C~
~
10
~o
~-S
Problem 8: Find the power series representation for
f(i)
(it c)9
-
~zz~
=
‘f\ ~-o
2
(i~>cN
8
I0
Problem 9:
~ (a) Sketch the graph of tI’e equation
x2 + ~i~2
—
2~ + l6y ± I
=
0.
2-
$
~\
=~.
(x-~_/
—
~
/
2-
2*
(~)
(b) Make a translation of axes from (x. y)-coordinates to
_—
~
p
terms in the above equation. Rewrite the above equation in terms of u and v.
~Cc~
)
9
~0
Problem 10: Consider the parametric curve given by the following equations:
j:=
Find
1
ens.
p
1 + sin(t).
and ~~vithout eliminating the parameter (Assume t ~ u~).
-
Ccc~ ~)
—
10
to
3+
~o
3~p~t~-)
~
~oblern 11: Find the area of the polar region given by r
c~ e ~
C
‘~
3
=
~-
3 sin(O
21~
~ ~\ ~m (
()
~
.~
t~
~
2
1’
21U
c~
2.
~L
4-
I ~ ¶~ I ~i
I
—
2TC
2
4
—
fI.
~1o(~
~
Cos(~N
~
—
b
o-4 -o~
11
0
12: Find the area of the region inside the smaller loop of the limaçon whose equation is r
Problem
2
—
=
4 cos(’O).
P~c~ure
t—c,o)
Q(~c~c~
(Lr~z~
3
C
:3
~
/3,
—
J~f
‘3
~1V~
-
C
~
‘3
2
Co~ (z~
4
—
c
\z~—
~
L~
~
3
~
—
I
(~
—
-~
/
12