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.