588 CHAPTER 8 Applications of Trigonometric Functions NOTE The first Tacoma Narrows Bridge, in Washington state, collapsed in 1940. The collapse has been attributed to oscillations created by resonance. j Graphing the Sum of Two Functions Graph ( ) = + f x x x sin . Solution EXAMPLE 4 First, graph the component functions, y f x x y f x x sin 1 2 ( ) ( ) = = = = on the same coordinate axes. See Figure 50(a). Now, select several values of x, say π π π π = = = = = x x x x x 0 2 3 2 and 2 and use them to compute ( ) ( ) ( ) = + f x f x f x . 1 2 Table 2 shows the computations. Plot these points and connect them to get the graph, as shown in Figure 50(b). x 0 π 2 π π3 2 π2 ( ) = = y f x x 1 0 π 2 π π3 2 π2 ( ) = = y f x x sin 2 0 1 0 −1 0 ( ) = + f x x x sin 0 π + ≈ 2 1 2.57 π π − ≈ 3 2 1 3.71 π2 Point on graph of f ( ) 0, 0 π( ) 2 , 2.57 π π ( ) , π ( ) 3 2 , 3.71 π π ( ) 2 , 2 Table 2 Situations also exist where external forces cause a vibrating system to oscillate at larger and larger amplitudes. This phenomenon, known as resonance from the Latin resonare (meaning “resound”) or resonantia (meaning “echo”), occurs when external vibrations match the natural frequency of the vibrating system. Resonance can be destructive to bridges, buildings, or even mechanical devices. For example, bridges can be affected by soldiers marching in step, buildings can be affected by blowing winds, and automobiles can be affected by the vibrations of its tires. Engineers account for expected external vibrations in their designs and incorporate shock absorbers or dampers to counter the effect of resonance. 4 Graph the Sum of Two Functions Many physical and biological applications require the graph of the sum of two functions, such as ( ) ( ) ( ) = + = + f x x x g x x x sin or sin cos 2 For example, if two tones are emitted, the sound produced is the sum of the waves produced by the two tones. See Problem 51 in Section 7.7 for an explanation of Touch-tone phones. To graph the sum of two (or more) functions, add the y-coordinates that correspond to equal values of x. Check: Graph = = Y x Y x , sin , 1 2 and = + Y x x sin 3 and compare the result with Figure 50(b). Use INTERSECT to verify that the graphs of Y1 and Y3 intersect when = x sin 0. Figure 50 x 21 2 p–– 2 p 2p (a) (b) 1 2 3 4 5 6 p–– 2 3p –– 2 y y 5 x y 5 x x 21 2 p–– 2 p 2p 1 1 , 2.57 (p, p) (2p, 2p) ) ( , 3.71) ( 1 2 3 4 5 6 p–– 2 p–– 2 3p –– 2 3p –– 2 y y 5 sin x f(x) 5 x 1 sin x y 5 sin x
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