Math 1310-001
Final Exam
Spring 2013
April 30, 2013
Final Exam: Calculus I
Instructions: You may use a calculator to assist you with any arithmetic, a writing
utensil, and your 3x5 reference card that you have prepared. No other materials
are permitted. Please show all of your work to the level of detail that is specified
in the instructions for that particular problem. You MUST write your final answer
in the space provided to receive full credit. The point value of each problem is
specified in the problem statement. Most of the credit will be awarded for doing
the calculus correctly and showing the appropriate steps. A correct final answer
with no work supporting it will receive very few points!! You may turn in your
exam and leave when you have finished.
Recommendation: You do not need to work on these problems in order. I would
suggest first answering the ones you find the easiest. Then go back and work on
the harder ones at the end. Also, if you finish early I would suggest double
checking all of your work, just in case you made any very simple mistakes.
Name:
Math 1310-001
Final Exam
Spring 2013
April 30, 2013
A balloon is being inflated so that its volume is changing at a rate of 8 cubic centimeters
per second. How fast is the surface area of the balloon changing at the moment the
radius is 10 centimeters? Show all of your work and write your final answer in the box
1.
below. (15 points) [Recall that the volume of a sphere is V
is S
=
=
and the surface area
]
2
4irr
Rateofchangeofsurfacearea=
riO
S
C
(
FE
Jd_
)2
L/QO
VI
SI
z
IO’
Spring 2013
April 30, 2013
Math 1310-001
Final Exam
2.
A rectangular storage container with an open top is to have a volume of 10 cubic
meters. The base of the container must be a square. The material used for the base
costs $10 per square meter while the material used for the lateral sides of container
cost $6 per square meter. Find the cost of the materials for the cheapest such
container. Show all of your work and write your final answer in the box below.
(20 points)
Cost of materials=
1
[ 1c7
i—i
-
o O
2OL
5
v L]14
/0 (cJ1’)
2
)OL
+
{LiI)$i)
Si
I
I2
(iOLEO)
= 12
3
L
LE
((J
(flJL
0
Spring 2013
April 30, 2013
Math 1310-001
Final Exam
Use the limit definition of the definite integral to evaluate the following definite
3.
integral. Show all of your work and write your final answer in the box below. (15 points) [In
this problem you need to take the limit of the sum, rather than using the fundamental theorem
of calculus]..,’
t.’
2
L
x
1
dx
=
limZf(xk)Ax
=
V)
)(
O
ii
-
IL?
h
1
3
x
3)o
fr i
3
- I
1
c1k’
•—
12
,
-I
h-’
1W’
3.
)
(0
3
iJ]
Spring 2013
April 30, 2013
Math 1310-001
Final Exam
4. Use integration by parts to evaluate the following indefinite integrals. Show all of your work
and write your final answers in the boxes below. (15 points) [Don’t forget about the “+C”]
v.
ln(x)dx=
3
j’x
(a)
—
xJ
i(x)
9
x
—‘s
x
(b)
L)
j
—
f
X
2
e
sin(3x) dx
=
—
3(QS(3)
I
Z
LX
it3)
1
ç
(c)
e.
zx
(x)dx
1
f sin
=
2
u I—x
x
vi6()
4u
-t
_ax(x
)(OX
I
-
C “-
xc
Iu
.
x
C
ci
mm
I—
-I
0
0
‘-I
m
H
I—
0
•—
(Jo
w >
o
>,
Lf)
C
>- 0
>
c’
G)
Z
°
Ow
E
w
ci
Q)
G)2j
cii
-—
cci
Cici
cii
-
D>0
Gioci
9-
-4-.
z v
>.cci
.C
UGJC
_C
Lr
‘
D
>
D
‘.
2
‘I
4
I’
iN’
\)
cN
4-.
U
C-)
-Th
I
—
—\
N
—
r
_[‘
c_i
__-_J
I
LL.
5
•‘—.--‘
\j
ct
‘
—‘
2
-.
—
—‘‘
Spring 2013
April 30, 2013
Math 1310-001
Final Exam
Use partial fraction decomposition to evaluate the following integrals. Show all of your
work and write your final answers in the boxes below. (20 points)
6.
.
I
(a)
3I\1j)
+i)(X3X
3
I ±I
-(Ix’I
-3)
i3k
1
+
I-
3’.
-
f
(b)
-
j
(X_1)(X2+1)
—
=
-
tLj:_c
C(l)
At))
-(
fr
it,
L(
—,
(&
U + (213 —
+)(i)
-‘
I
xl
1i
i
)
±
()(x-’)
A(XL÷i)
f
xL
dx
I
I -4
(I
‘I
j
-