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WORKSHEET 21  Fall 1995
1. Find
f
0
(
x
)intermso
f
g
(
x
)and
g
0
(
x
), where
g
(
x
)
>
0 for all
x
. (Recall: If
c
is a constant, then
g
(
c
)is
a constant.)
a)
f
(
x
)=
g
(
x
)(
x
−
a
)
b)
f
(
x
)=
g
(
a
)(
x
−
a
)
c)
f
(
x
)=
g
(
x
+
g
(
x
))
d)
f
(
x
)=
g
(
x
)
x
−
a
e)
f
(
x
)=
1
g
(
x
)
f)
f
(
x
)=
g
(
xg
(
a
))
g)
f
(
x
)=
p
g
(
x
)
2
h)
f
(
x
)=
p
g
(
x
2
)
i)
f
(2
x
+3)=
g
(
x
2
)
2. Suppose that
f
(
a
)=
g
(
a
)=
h
(
a
), that
f
(
x
)
≤
g
(
x
)
≤
h
(
x
) for all
x
,andthat
f
0
(
a
)=
h
0
(
a
).
a) Prove that
g
(
x
) is di±erentiable at
a
,andthat
f
0
(
a
)=
g
0
(
a
)=
h
0
(
a
).
(Hint: Begin with the de²nition of
g
0
(
a
). The Squeeze Theorem may be useful.)
b) Show that the conclusion does not follow if we omit the hypothesis
f
(
a
)=
g
(
a
)=
h
(
a
).
c) Draw a graph illustrating part a).
d) Draw a graph illustrating part b).
e) Useparta)toshowthati
f
g
(
x
)=
±
x
2
sin
1
x
,x
6
=0;
0
,x
=0
then
g
0
(0) = 0.
(Hint: Use
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This note was uploaded on 04/11/2011 for the course MATH 1400 taught by Professor Grether during the Spring '08 term at North Texas.
 Spring '08
 Grether

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