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B U Department of Mathematics
Math 101 Calculus I Fall 2002 Final Exam
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1. Show that
f
(
x
) =
x
2
+ 1
x

1
+
x
4
x

2
has at least one root in the open interval (1,2) (Hint:
Check continuity and the behaviour at the end points).
Solution:
lim
x
→
+
∞
f
(
x
) = +
∞
.
lim
x
→∞
f
(
x
) =
∞
.
Therefore there exists an element
c
in (1
,
2) such that
f
(
c
) = 0.
2. Compute
f
0
(0) by using the deﬁnition of derivative if
f
(
x
) =
±
1
x
2
R
x
0
sin(
t
2
)
dt
if
x
6
= 0
0
if
x
= 0
Solution:
f
0
(0) = lim
x
→
0
1
x
3
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This note was uploaded on 11/16/2011 for the course MATH 102 taught by Professor Soysal during the Winter '09 term at Boğaziçi University.
 Winter '09
 SOYSAL
 Calculus

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