Functions Modeling Change A Preparation for Calculus Third Edition

CHAPTER 6 TRIGONOMETRIC FUNCTIONS SECTION 6.1 INTRODUCTION TO PERIODIC FUNCTIONS

The World's Largest Ferris Wheel To celebrate the millennium, British Airways funded construction of the London Eye, the world's largest ferris wheel. The wheel is located on the south bank of the river Thames, in London, England, measures 450 feet in diameter, and carries up to 800 passengers in 32 capsules. It turns continuously, completing a single rotation once every 30 minutes. This is slow enough for people to hop on and off while it turns. Page 244 3

Ferris Wheel Height as a Function of Time Suppose you hop on this ferris wheel at time t = 0 and ride it for two full turns. Let f(t) be your height above the ground (measured in feet as a function of t, the number of minutes you have been riding). Page 244 4

Ferris Wheel Height as a Function of Time Let's imagine that the wheel is turning in the counterclockwise direction. At time t = 0 you have just boarded the wheel, so your height is 0 ft above the ground (not counting the height of your seat). Thus, f(0) = 0. Page 244 5

Ferris Wheel Height as a Function of Time Since the wheel turns all the way around once every 30 minutes, after 7.5 minutes the wheel has turned one-quarter of the way around. Thinking of the wheel as a giant clock, this means you have been carried from the 6 o'clock position to the 3 o'clock position, as shown in Figure 6.1. You are now halfway up the wheel, or 225 feet above the ground, so f(7.5) = 225. Page 244 6

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After 15 minutes, the wheel has turned halfway around, so you are now at the top, in the 12 o'clock position. Thus, f(15) = 450. Page 244 8

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And after 22.5 minutes, the wheel has turned three quarters of the way around, bringing you into the 9 o'clock position. You have descended from the top of the wheel halfway down to the ground, and you are once again 225 feet above the ground. Thus, f(22.5) = 225. Page 244 10

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Finally, after 30 minutes, the wheel has turned all the way around, bringing you back to ground level. This means that f(30) = 0. Page 244 12

30 Page 245 13

Since the wheel turns without stopping, at time t = 30 it begins its second turn. Thus, at time t = 37.5, the wheel has again turned one quarter of the way around. This means that f(37.5) = 225. Page 245 14

Similarly, at time t = 45 the wheel has again turned halfway around, bringing you back up to the very top. This means that f(45) = 450. Page 245 15

Likewise, at time t = 52.5, the wheel has carried you halfway back down to the ground. This means that f(52.5) = 225. Page 245 16

Finally, at time t = 60, the wheel has completed its second full turn and you are back at ground level. This means that f(60) = 0. Page 245 17

t (minutes) 0 7.5 15 22.5 30 37.5 45 52.5 f(t) (feet) 0 225 450 225 0 225 450 225 t (minutes) 60 67.5 75 82.5 90 97.5 105 112.5 120 f(t) (feet) 0 225 450 225 0 225 450 225 0 Page 245 18

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It's tempting to connect the points with straight lines... But this is not correct... Page 246 20

Consider the first 7.5 minutes of your ride, starting at the 6 o'clock position and ending at the 3 o'clock position. Halfway through this part of the ride, the wheel has turned halfway from the 6 o'clock to the 3 o'clock position. Page 246 21

However, as you can see, your seat rises less than half the vertical distance from y = 0 to y = 225. Page 246 22

At the same time, the seat glides more than half the horizontal distance. Page 246 23

If the points were connected with straight lines, f(3.75) would be halfway between f(0) and f(7.5), which is incorrect. Page 246 24

The graph of f(t) in Figure 6.6 is a smooth curve that repeats itself. It looks the same from t = 0 to t = 30 as from t = 30 to t = 60, or from t = 60 to t = 90, or from t = 90 to t = 120. Page 246 25

STAT ENTER (for 1: Edit) Under L1 Key in: t (time: minutes), Under L2 Key in: f(t) (distance: feet) Page N/A 26

t (minutes) Let's key this data into our calculator. 0 7.5 15 22.5 30 37.5 45 52.5 f(t) (feet) 0 225 450 225 0 225 450 225 t (minutes) 60 67.5 75 82.5 90 97.5 105 112.5 120 f(t) (feet) 0 225 450 225 0 225 450 225 0 Page 245 27

2nd Y= (for Stat Plots) ENTER (for 1:) ENTER (to turn plot to "On") Down arrow ENTER (for scatterplot) XList: L1 YList: L2 Page N/A 28

Y= clear any equations If necessary, put cursor at: \Y 1 = Page N/A 29

Zoom 9 (Zoomstat): will show graph If necessary, adjust the window: Page N/A Window Value Xmin -12 Xmax 132 Xscl 10 Ymin -76.5 Ymax 526.5 Yscl 50 30

Zoom 9 (Zoomstat): will show graph Page N/A 31

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Zoom 9 (Zoomstat): will show graph Trace (will go from point to point) Page N/A 33

Zoom 9 (Zoomstat) Trace (will go from point to point) 2nd 0 (zero) (Catalog) x -1 will give the "d's" Down to DiagnosticOn Enter Enter (you'll see "Done") Page N/A 34

STAT (for Calc) C :SinReg Enter will give the regression equation on the main screen: Page N/A 35

STAT (for Calc) C :SinReg Enter will give the regression equation on the main screen: y=a * sin(bx+c)+d a=225 b=.2094395102 c=-1.570796327 d=225 Page N/A 36

STAT (for Calc) C :SinReg Enter will give the regression equation on the main screen Y= Page N/A 37

STAT (for Calc) C :SinReg Enter will give the regression equation on the main screen Y= Vars 5 :Statistics (for EQ) 1: RegEQ (will fill Y= screen with the regression equation. Graph will display the scatterplot with the sin regression result filled in. Page N/A 38

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TRACE and up/down arrows will jump between data points and the sin regression curve. Let's verify the accuracy of the fit: Page N/A 40

t (minutes) 0 7.5 15 22.5 30 37.5 45 52.5 f(t) (feet) 0 225 450 225 0 225 450 225 t (minutes) 60 67.5 75 82.5 90 97.5 105 112.5 120 f(t) (feet) 0 225 450 225 0 225 450 225 0 Page 245 41

To recap: Here we utilzed the sin regression function to fit a sin curve to the set of points. Here is the result: y=a * sin(bx+c)+d y = 225sin(.2094395102x -1.570796327)+225 Page N/A 42

Periodic Functions: Period, Midline, and Amplitude The ferris wheel function, f, is said to be periodic. The smallest time interval during which a function completes one full cycle is called its period and is represented as a horizontal distance in Figure 6.7. Page 246 43

We can think about the period in terms of horizontal shifts. If the graph of f is shifted to the left by 30 units, the resulting graph looks exactly the same. That is... Page 246 44

Graph of f shifted left by 30 units = original graph f(t+30) Page 246 45

In General: A function f is periodic if its values repeat at regular intervals. If the graph of f is shifted horizontally by c units, the new graph is identical to the original. In function notation, periodic means that, for all t in the domain of f, f(t + c) = f(t) Page 247 Blue Box 46

In General: In function notation, periodic means that, for all t in the domain of f, f(t + c) = f(t) The smallest positive constant c for which this relationship holds for all values of t is called the period of f. Page 247 Blue Box 47

Below, the dashed horizontal line is the midline of the graph of f. Page 247 48

The amplitude of a wave-like periodic function is the distance between its maximum and the midline (or the distance between the midline and the minimum). Page 247 49

Thus the amplitude of f is 225 because the ferris wheel's maximum height is 450 feet and its midline is at 225 feet. The amplitude is represented graphically as a vertical distance. Page 247 50

In general: The midline of a periodic function is the horizontal line midway between the function's maximum and minimum values. The amplitude is the vertical distance between the function's maximum (or minimum) value and the midline. Page 247 51

End of Section 6.1 52