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110
Chapter 5
•
Graphing and Inverse Functions
§5.2
The above example made use of the graph of sin 2
x
, but the period can be found analyt
ically. Since sin
x
has period 2
π
,
1
we know that sin (
x
+
2
π
)
=
sin
x
for all
x
. Since 2
x
is a
number for all
x
, this means in particular that sin (2
x
+
2
π
)
=
sin 2
x
for all
x
. Now deFne
f
(
x
)
=
sin 2
x
. Then
f
(
x
+
π
)
=
sin 2(
x
+
π
)
=
sin (2
x
+
2
π
)
=
sin 2
x
(as we showed above)
=
f
(
x
)
for all
x
, so the period
p
of sin 2
x
is
at most
π
, by our deFnition of period. We have to show
that
p
>
0 can not be smaller than
π
. To do this, we will use a
proof by contradiction
. That
is, assume that 0
<
p
<
π
, then show that this leads to some contradiction, and hence can not
be true. So suppose 0
<
p
<
π
. Then 0
<
2
p
<
2
π
, and hence
sin 2
x
=
f
(
x
)
=
f
(
x
+
p
)
(since
p
is the period of
f
(
x
))
=
sin 2(
x
+
p
)
=
sin (2
x
+
2
p
)
for all
x
. Since any number
u
can be written as 2
x
for some
x
(i.e
u
=
2(
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This note was uploaded on 01/21/2012 for the course MAC 1130 taught by Professor Dr.cheun during the Fall '11 term at FSU.
 Fall '11
 Dr.Cheun
 Calculus, Inverse Functions

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