University of California, Riverside
Department of Mathematics
Final Exam
Mathematics 9A  First Year of Calculus
Sample 2
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. Evaluate the following limits.
(a) (5 points)
lim
x
!
1
x
°
1
p
3
x
+1
°
2
(b) (5 points)
lim
x
!
3
x
2
°
6
x
+9
x
2
°
9
(c) (5 points)
lim
x
!
0
sin(4
x
)
3
x
cos(5
x
)
(d) (5 points)
lim
x
!1
2+3
x
2
5
x
2
°
4
x
2. Find the derivatives of the following functions.
(a) (6 points)
f
(
x
) =
x
2
°
1
x
2
+1
(b) (7 points)
g
(
x
) = 3 sin(4
x
) + 4 cos(3
x
)
(c) (7 points)
h
(
x
) = tan(
p
1 +
x
3
)
3. Given the function
f
(
x
) = 5
x
°
2 + sin(
x
)
,show that the equation
f
(
x
) = 0
has exactly one real root with the following two steps.
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 Spring '09
 Nouh
 Calculus, Derivative, Mathematical analysis, Convex function

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