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Making Connections and Instantaneous Rate of Change Sinusoidal models apply to many real-world phenomena that do not necessarily involve angles. In a vehicle that uses a steering system that moves back and forth easily, an oscillation known as a speed wobble, or shimmy, can occur if the amplitude of the oscillations is not reduced. Dangerous speed wobbles can occur in motorcycles, bicycles, tricycle-geared aircraft, jeeps, skateboards, in-line skates, and grocery carts. The speed wobble can be modelled using a sinusoidal function. Investigate How can you determine average and instantaneous rates of change for a sinusoidal function? 1. a) Open The Geometer’s Sketchpad® . From the Edit menu, choose Preferences , and click on the Units tab. Set the angle units to radians and the precision to ten thousandths for all measurements. b) Turn on the grid, and plot the function f ( x ) sin x . c) Plot points A and B on the function. d) Measure the x - and y -coordinates of points A and B. e) Construct a secant from B to A. f) Measure the slope of secant BA. 2. a) Drag point A until its x -coordinate is approximately π _ 4 , rounded to two decimal places. Drag point B until its x -coordinate is approximately π _ 6 , rounded to two decimal places. b) Calculate the average rate of change of the sine function from x π _ 6 to x π _ 4 . c) Drag point B toward point A. What type of line does secant AB approach as point B approaches point A? How is the slope of this line related to the rate of change of the function? d) What is the instantaneous rate of change of the sine function when x π _ 4 ? Tools • computer with The Geometer’s Sketchpad ® Optional • graph of sine function and mathematics construction set 5.5 290 MHR • Advanced Functions • Chapter 5

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3. a) Copy the table and record your results. One line has already been filled in for you. b) In a similar manner, determine the instantaneous rate of change of the sine function for the other values of x in the table. c) Extend the table to multiples of the first quadrant angles up to, and including, 2 π . 4. Add the instantaneous rates of change to the graph of the sine function. Sketch a smooth line through these points. 5. Reflect a) Inspect the rate of change of the sine function at x π _ 4 . Explain why this value makes sense. b) Inspect the rate of change of the sine function at x π _ 2 . Explain why this value makes sense. c) Inspect the rate of change of the sine function at x 3 π _ 4 . Explain why this value makes sense. d) Note that the graph of the instantaneous rate of change of the sine function is also periodic. Explain why this makes sense. Angle x f ( x ) sin x Instantaneous Rate of Change 0 π _ 6 π _ 4 0.71 0.71 π _ 3 π _ 2 5.5 Making Connections and Instantaneous Rate of Change • MHR 291
Example 1 Height of a Car on a Ferris Wheel The height, h , in metres, of a car above the ground as a Ferris wheel turns can be modelled using the function h 20 sin ( π t _ 60 ) 25, where t is the time, in seconds.

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