Part1_PeriodicFunctions_A - Periodic Functions[1.2 of the...

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Periodic Functions [ § 1.2 of the Notes, Chapter 1 of Stewart is also useful ] There are many examples in nature of events repeating themselves over and over again. Nature is periodic! Example 1 If you plot the length of day against time the graph repeats itself at yearly intervals. 100 200 300 400 500 600 700 x 5 10 15 20 y MATH1011 [2012 - Part 1] 1
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Example 2 If you plot blood pressure against time for a healthy person at rest the graph repeats itself. A function, f , is periodic if for some number p (called the period ), f ( x + p ) = f ( x ) . MATH1011 [2012 - Part 1] 2
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From this we can also see that f ( x + 2 p ) = f (( x + p ) + p ) = f ( x + p ) = f ( x ) . Furthermore, f ( x + 3 p ) = f ( x ) and thus in general we have f ( x + kp ) = f ( x ) for all integers k . MATH1011 [2012 - Part 1] 3
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This means that any piece of the graph of length p gives you the graph of the function by repetition. -4 -2 2 4 x 0.2 0.4 0.6 0.8 1 y More examples -10 -5 5 10 x 2 4 6 8 10 12 14 16 y MATH1011 [2012 - Part 1] 4
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-4 -2 2 4 x -1 -0.5 0.5 1 y -4 -2 2 4 x -1 -0.5 0.5 1 y -4 -2 2 4 6 8 x 0.2 0.4 0.6 0.8 1 y MATH1011 [2012 - Part 1] 5
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Typically, the recipe for a periodic function looks like this 1 : f ( x ) = ( 1 - x 2 - 1 < x 1 f ( x + 2) for all x. To sketch the graph take the bit explicitly given by 1 - x 2 where - 1 < x 1 . Sketch this, -1 1 x 1 y and copy it along by ± 2 , ± 4 ,. . . etc. 1 You may want to review “Piecewise Defined Functions” in § 1.1 p17 of Stewart. MATH1011 [2012 - Part 1] 6
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-5 -4 -3 -2 -1 1 2 3 4 5 x 1 y To find values of the function f ( x ) = ( 1 - x 2 - 1 < x 1 f ( x + 2) all x you need to find an n such that - 1 < x + 2 n 1 MATH1011 [2012 - Part 1] 7
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and use the explicit part. f (25) = f (1 + 2 × 12) = f (1) = 0 f ( - 3) = f (1 + 2 × ( - 2)) = f (1) = 0 f (2 . 35) = f (0 . 35 + 2 × 1) = f (0 . 35) = 0 . 8775 . In other words you “ find the closest integer multiple of p to x and add or subtract to make the number smaller .” MATH1011 [2012 - Part 1] 8
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Trigonometric Functions [ Stewart § 1.2 pp32-33, § 1.3 pp37- 40, also revise Appendix D ] We are now going to study functions corresponding to “sine-like” (or sinusoidal) curves 2 : curves obtained by moving the curve y = sin x -8 -6 -4 -2 2 4 6 8 x -1 1 y 2 You would have studied sin , cos , tan and their graphs in high school. MATH1011 [2012 - Part 1] 9
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up -8 -6 -4 -2 2 4 6 8 x 1 2 3 y and down -8 -6 -4 -2 2 4 6 8 x -3 -2 -1 y MATH1011 [2012 - Part 1] 10
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right to left -8 -6 -4 -2 2 4 6 8 x -1 1 y and left to right -8 -6 -4 -2 2 4 6 8 x -1 1 y MATH1011 [2012 - Part 1] 11
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or stretching them horizontally -8 -6 -4 -2 2 4 6 8 x -1 1 y or vertically -8 -6 -4 -2 2 4 6 8 x -2 -1 1 2 y MATH1011 [2012 - Part 1] 12
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or a combination of all the above.
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