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Department of Mathematics
Final Exam
Mathematics 9A  First Year of Calculus
Sample 1
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
:::
.
(a) (6 points)
lim
x
3
x
3
9
x
6 + 2
x
(b) (7 points)
lim
x
!
0
+
sin(2
x
)
x
2
(c) (7 points)
lim
x
3
x
p
4
x
2
+
x
+ 5
f
(
x
) =
(
x
+ 5
if
x <
3
4
p
x
+ 1
if
x
±
3
(a) (6 points) Show that
f
(
x
)
is continuous at
x
= 3
.
(b) (14 points) Using the limit de&nition of the derivative, and com
puting the limits from both sides, show that
f
(
x
)
is di/erentiable
at
x
= 3
.
1
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 Spring '09
 Nouh
 Calculus

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