Pre-Calc Exam Notes 113

Pre-Calc Exam Notes 113 - namely 2. To see why, note that...

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Properties of Graphs of Trigonometric Functions Section 5.2 113 Example 5.8 Find the amplitude and period of y = 2 sin ( x 2 ). Solution: This is not a periodic function, since the angle that we are taking the sine of, x 2 , is not a linear function of x , i.e. is not of the form ax + b for some constants a and b . Recall how we argued that sin 2 x was “twice as fast” as sin x , so that its period was π instead of 2 π . Can we say that sin ( x 2 ) is some constant times as fast as sin x ? No. In fact, we see that the “speed” of the curve keeps increasing as x gets larger, since x 2 grows at a variable rate, not a constant rate. This can be seen in the graph of y = 2 sin ( x 2 ), shown in Figure 5.2.6: 2 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 0 π 2 π 3 π 2 2 π y x Figure 5.2.6 y = 2 sin ( x 2 ) Notice how the curve “speeds up” as x gets larger, making the “waves” narrower and narrower. Thus, y = 2 sin ( x 2 ) has no period. Despite this, it appears that the function does have an amplitude,
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Unformatted text preview: namely 2. To see why, note that since | sin | 1 for all , we have | 2 sin ( x 2 ) | = | 2 | | sin ( x 2 ) | 2 1 = 2 . In the exercises you will be asked to nd values of x such that 2 sin ( x 2 ) reaches the maximum value 2 and the minimum value 2. Thus, the amplitude is indeed 2. Note: This curve is still sinusoidal despite not being periodic, since the general shape is still that of a sine wave, albeit one with variable cycles . So far in our examples we have been able to determine the amplitudes of sinusoidal curves fairly easily. This will not always be the case. 2 This graph was created using Gnuplot, an open-source graphing program which is freely available at http:// gnuplot.info . See Appendix B for a brief tutorial on how to use Gnuplot....
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