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Applied 40S May 26, 2009


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Applications of Periodic Functions.

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Applied 40S May 26, 2009

  1. 1. Applications of Periodic Functions or Bugs On Wheels Suicidal Shield Bug by flickr user ChinchillaVilla
  2. 2. How many periods are illustrated in each graph? HOMEWORK How many revolutions (in radians and degrees) are illustrated in each graph? Periods = 1.5 Radians Rotated = 3π Degrees Rotated = 540° Periods = 3.25 Radians Rotated = 6.5π Degrees Rotated = 1170° Periods = 0.75 Radians Rotated = 1.5π Degrees Rotated = 270°
  3. 3. Determine approximate values for the parameters 'a', 'b', 'c', and 'd' from the graphs, and then write the equations of each graph as a sinusoidal function in the form: y = a sin b(x - c) + d HOMEWORK Re quot;D me y = 6sin(πx) + 7 AB mb 3 C! er quot; y = 4sin(πx) - 4 y = 2sin(π(x - 1)) 3
  4. 4. State the amplitude, period, horizontal shift, and vertical shift for each of the following: HOMEWORK amplitude: 2 amplitude: 4 period: 2π period: 2 3 horizontal shift: +4 horizontal shift: 0 vertical shift: +1 vertical shift: 0
  5. 5. Properties and Transformations of the sine function ... Let's look at some graphs ... ƒ(x) = AsinB(x - C) + D
  6. 6. The Role of Parameter D D is the sinusoidal axis, average value of the function, or the vertical shift. D > 0 the graph shifts up D units. D < 0 the graph shifts down D units.
  7. 7. The Role of Parameter A The amplitude is the absolute value of A; |A|. It is the distance from the sinusoidal axis to a maximum (or minimum). If it is negative, the graph is reflected (flips) over the sinusoidal axis. y = 2sin(x) y = 1 sin(x) 2 y = -3sin(x)
  8. 8. The Role of Parameter B B is not the period; it determines the period according to this relation: or y = sin(2x) y = sin(3x)
  9. 9. The Role of Parameter C C is called the phase shift, or horizontal shift, of the graph. y = sin(x + π ) 4 y = sin(x - π ) 4
  10. 10. In general form, the equation and graph of the basic sine function is: ƒ(x) = AsinB(x - C) + D -2π -π π 2π A=1, B=1, C=0, D=0 The quot;starting point.quot; Note that your calculator displays: ƒ(x) = asin(bx - c) + d Which is equivalent to: ƒ(x) = AsinB(x - c/b) + D
  11. 11. In general form, the equation and graph of the basic cosine function is: ƒ(x) = AcosB(x - C) + D The quot;starting point.quot; -2π 2π A=1, B=1, C=0, D=0 -π π Since these graphs are so similar (they differ only by a quot;phase shiftquot; of π units) 2 we will limit our study to the sine function.
  12. 12. State the amplitude, period, horizontal shift, and vertical shift for each of the following: amplitude: amplitude: period: period: horizontal shift: horizontal shift: vertical shift: vertical shift:
  13. 13. Jud was working with sinusoidal data, but lost all of it except for two points. A maximum point was (3, 13) and a minimum point next to it was (7, 1). Write a sinusoidal equation that matches Jud's data. ƒ(x) = AsinB(x - C) + D
  14. 14. Enter the values into your calculator, and use a sinusoidal regression to determine the equation. Round the values of the parameters to one decimal place. HOMEWORK x -1 -0.5 0 0.5 1 1.5 2 2.5 y 1 -2.6 -5.6 -5.4 -2 1.4 1.6 -1.4
  15. 15. At a sea port, the depth of the water, h meters, at time t hours after midnight, during a certain day is given by this formula: HOMEWORK (a) State the: (i) period (ii) amplitude (iii) phase shift. (b) What is the maximum depth of the water? When does it occur? (c) Determine the depth of the water at 5:00 am and at 12:00 noon. (d) Determine one time when the water is 2.25 meters deep.
  16. 16. The Bug on the Water Wheel HOMEWORK A water wheel with a 7.0 ft radius has 1.0 ft. submerged in the water as shown, and rotates counterclockwise at 6.0 revolutions per minute. A bug is sitting on the wheel at point B. You start your stopwatch, and two seconds later the bug at point B is at its greatest height above the water. You are to model the distance 'h' of the bug from the surface of the water in terms of the number of seconds 't' the stopwatch reads. (a) Sketch the graph. (b) Write the algebraic equation of the sinusoid. (c) How far is the bug above the water when t = 5.5 seconds?
  17. 17. Apogee and Perigee HOMEWORK A spacecraft is in an elliptical orbit around the earth as shown in the diagram. At its perigee the spacecraft is at its lowest point above the earth's surface, and at its apogee it is at its highest point above the earth. The equation: describes the spacecraft's distance from the earth over time, where 'x' represents the time in minutes, and 'y' the distance in kilometres. Write answers for the following questions: (a) How long does it take for the spacecraft to fly around the earth? (b) What are the distances at apogee and perigee?
  18. 18. The Water Heater and the Hot Tub HOMEWORK An electric heater cuts in and out on a cyclical basis as it heats the water in a hot tub. The water temperature 'C' in degrees Celsius varies sinusoidally with time 't'. The thermostat is set to turn the heater on when the temperature drops to 36°C and off when it reaches 42°C. In order to measure the period, a timer was started at 0 when the heater cut in. The heater turned off 30 minutes later and turned on again after another 30 minutes. (a) Sketch the graph of the function for 0 ≤ t ≤ 60. (b) Write the equation expressing temperature, C, in terms of time, t. (c) Use your equation to determine the temperature 10 minutes after the heater turns on.