University of California, Riverside
Department of Mathematics
Final Exam
Mathematics 9A  First Year of Calculus
Sample 5
Instructions:
This exam has a total of 200 points. You have 3 hours.
You must show all your work to receive full credit. You may use any result
done in class. The points attached to each problem are indicated beside the
problem. You are not allowed to use books, notes, or calculators. Answers
should be written as
p
2
as opposed to
1
:
4142135
:::
.
1. Find each of the following limits if it exists. If you think the limit does
not exist provide a reason.
(a) (5 points)
lim
x
!
0
sin (5
x
)
1
°
p
1
°
x
(b) (5 points)
lim
t
!
9
+
5
t
3
°
p
t
(c) (5 points)
lim
x
!
8
f
(
x
)
;
given that
lim
x
!
8
xf
(
x
)
3
=
°
2
(d) (5 points)
lim
x
!°1
p
9
x
6
°
x
3
x
3
+ 4
x
2. Find the derivative of the following functions:
1
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(a) (5 points)
g
(
°
) =
±
2
(sec
°
°
sin 2
°
)
2
(b) (5 points)
f
(
x
) =
x
2
(5
°
x
)
7
(c) (5 points)
y
=
1
2
p
x
°
5
3
x
°
cos (3
±
)
(d) (5 points)
f
(
t
) =
tan
t
±
°
cos 2
t
3. (20 points) Find the derivative of the following function using the limit
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 Spring '09
 Nouh
 Calculus, lim, Rolle's theorem

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