664 CHAPTER 6 The Circular Functions and Their Graphs SOLUTION (a) Figure 67 shows a TI-84 Plus graph of y1, y2, and y3 in the window 30, 34 by 3-1, 14. (b) The graph of y3 is bounded above by the graph of y1 and below by the graph of y2. (The graphs of y1 and y2 are referred to as envelopes for the graph of y3.) (c) When 2px = 0, 2p, 4p, and 6p, cos 2px = 1. Thus, the value of e-x cos 2px is the same as the value of e-x when 2px = 0, 2p, 4p, and 6p—that is, when x = 0, 1, 2, and 3. (d) When 2px = p 2 , 3p 2 , 5p 2 , 7p 2 , 9p 2 , and 11p 2 , cos 2px = 0. Thus, the graph of y3 intersects the x-axis when x = 1 4, 3 4, 5 4, 7 4, 9 4, and 11 4 . S Now Try Exercise 39. y3 =e –x cos 2px −1 0 1 3 y2 =−e –x y1 =e –x Figure 67 CONCEPT PREVIEWAn object in simple harmonic motion has position function s(t), in inches, from an equilibrium point as follows, where t is time in seconds. s1t2 = 5 cos 2t 1. What is the amplitude of this motion? 2. What is the period of this motion? 3. What is the frequency? 4. What is s102? 5. What is sAp 2 B? 6. What is the range of the graph of this function? (Modeling) Solve each problem. See Examples 1 and 2. 7. Spring Motion An object is attached to a coiled spring, as in Figure 65 (repeated here). It is pulled down a distance of 4 units from its equilibrium position and then released. The time for one complete oscillation is 3 sec. (a) Give an equation that models the position of the object at time t. (b) Determine the position at t = 1.25 sec to the nearest hundredth. (c) Find the frequency. 8. Spring Motion Repeat Exercise 7, but assume that the object is pulled down a distance of 6 units and that the time for one complete oscillation is 4 sec. 9. Voltage of an Electrical Circuit The voltage E in an electrical circuit, where t is time measured in seconds, is modeled by E = 5 cos 120pt. (a) Find the amplitude and the period. (b) Find the frequency. (c) Find E, to the nearest thousandth, when t = 0, 0.03, 0.06, 0.09, 0.12. (d) Graph E for 0 … t … 1 30. 10. Voltage of an Electrical Circuit The voltage E in an electrical circuit, where t is time measured in seconds, is modeled by E = 3.8 cos 40pt. (a) Find the amplitude and the period. (b) Find the frequency. (c) Find E, to the nearest thousandth, when t = 0.02, 0.04, 0.08, 0.12, 0.14. (d) Graph E for 0 … t … 1 20. 6.7 Exercises a y 0 –a A. B. C.
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