SECTION 8.5 Simple Harmonic Motion; Damped Motion; Combining Waves 589 In Figure 50(b), notice that the graph of ( ) = + f x x x sin intersects the line = y x whenever = x sin 0. Also, notice that the graph of f is not periodic. The next example shows a periodic graph. Graphing the Sum of Two Sinusoidal Functions Graph ( ) ( ) = + f x x x sin cos 2 . Solution EXAMPLE 5 Graph f by adding the y -coordinates of = y x sin and ( ) = y x cos 2 . Table 3 shows the steps for computing several points on the graph of f. Figure 51 illustrates the graphs of the component functions, ( ) = = y f x x sin 1 (in blue), and ( ) ( ) = = y f x x cos 2 2 (in black), and the graph of ( ) ( ) = + f x x x sin cos 2 , which is shown in red. Figure 51 y 5 cos (2x) x 21 22 2 p–– 2 p 2p 1 2 p–– 2 3p –– 2 y f(x) 5 sin x 1 cos (2x) y 5 sin x x π − 2 0 π 2 π π3 2 π2 ( ) = = y f x x sin 1 −1 0 1 0 −1 0 ( ) ( ) = = y f x x cos 2 2 −1 1 −1 1 −1 1 ( ) ( ) = + f x x x sin cos 2 −2 1 0 1 −2 1 Point on graph of f π ( ) − − 2 , 2 ( ) 0, 1 π( ) 2 , 0 π( ) , 1 π ( ) − 3 2 , 2 π ( ) 2 , 1 Table 3 Notice that the function ( ) ( ) = + f x x x sin cos 2 is periodic, with period π2 . Check: Use a graphing utility to graph Y x Y x sin , cos 2 , and 1 2 ( ) = = Y x x sin cos 2 3 ( ) = + and compare the result with Figure 51. Now Work PROBLEM 27 ‘Are You Prepared?’ The answers are given at the end of these exercises. If you get a wrong answer, read the pages listed in red. 8.5 Assess Your Understanding 1. The amplitude A and period T of ( ) ( ) = f x x 5sin 4 are and . (pp. 430–432) 2. Approximate the angular speed of the second hand on a clock in rad/sec. (Round to three decimal places.) (pp. 390–391) 3. Write an equation for a sine function with period 12 and amplitude 7. (p. 436) 1. Now Work 1. Modeling 1.ExplainingConcepts Calculus Preview 1.InteractiveFigure
RkJQdWJsaXNoZXIy NjM5ODQ=