B U Department of Mathematics
Math 101 Calculus I
Summer 1999 Final Exam
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1. Using the definition of the derivative evaluate
f
(0) if
f
(
x
) =
xe

1
/x
2
if
x
= 0
0
if
x
= 0
Solution:
f
(0)
=
lim
h
→
0
f
(0 +
h
)

f
(0)
h
=
lim
h
→
0
he

1
/h
2

0
h
=
lim
h
→
0
1
e
1
/h
2
=
0
2. Let
f
:
R
→
R
be a function such that, for all
x, y
∈
R
f
(
x
+
y
) =
f
(
x
) +
f
(
y
)
.
a)
Show that
f
(0) = 0.
b)
Show that if
f
is continuous at 0, then
f
must be continuous at every
x
in
R
.
Solution:
a)
f
(0) =
f
(0 + 0) =
f
(0) +
f
(0)
⇒
f
(0) =
f
(0)

f
(0) = 0
.
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 Calculus, Mathematical Series, lim

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