Applications of Sine and Cosine Derivatives
Name: ______________________ Per: ___
Sinusoidal Functions general equation: y C A cos B( x D) C: Sinusoidal Axis A : Amplitude D: Phase Displacement B: Reciprocal of Horizontal Dilation (# of cycles in 2 ). Remember that Period B 2 Warm up
1. 1. A mass is bouncing up and down on a spring hanging from the ceiling. Its distance, y, in feet from the ceiling varies sinusoidally with time, t, in seconds, making a complete cycle every 1.6 s. At t = 0.4, y reaches its greatest value, 8 ft. The smallest y gets is 2 ft. a) Sketch a picture of the situation. b) Sketch the graph of the function y and write an equation for y in terms of t. c) Write an equation for y’(t)
d) How fast is the mass moving when t = 1? When t = 1.5? When t = 2.7? e) At t = 2.7, is the mass moving up or down? Justify your answer. f) What is the fastest the mass moves? Where is the mass when it is moving fast? 2. Suppose that a water wheel, 14 ft in diameter, rotates counterclockwise at 6 revolutions per minute. You start your stopwatch and point P, on the rim of the wheel (northeast part of the wheel), is "d" ft from the surface of the water. Two seconds later, point P is at its greatest height, 13 ft above the surface of the water. The center of the water wheel is 6 ft above the surface. a) Sketch a picture of the situation. b) Sketch the graph of d as a function of t, in seconds, since you started the stopwatch b) Model the distance d (Assuming that d is a sinusoidal function), of point P from the surface of the water in terms of the number of seconds, t, the stopwatch reads. c) When does point P first enter the water since you started the stopwatch? d) Write the equation of the derivative, d'. e) How fast is the point on the water wheel moving vertically when i. t = 5 ii. t = 9 f) At t = 5, is the wheel moving towards the water or away from the water? Justify your answer g) At t = 9, is the wheel moving towards the water or away from the water? Justify your answer
3. When you ride a Ferris wheel, your distance y(t) from the ground varies sinusoidally with time t (in seconds since the wheel started rotating. Suppose that the Ferris wheel has a diameter of 40 ft and that its axle is 25 ft above the ground. Three seconds after it starts, your seat is at its high point. The wheel makes 3 rev/min. a) Sketch a picture of the situation. b) Sketch the graph of the function y and figure out the particular equation for y(t) c) Write an equation for y’(t) d) When t = 15 is y(t) increasing or decreasing? How fast? e) What is the fastest y(t) changes? Where is the seat when y(t) is changing the fastest? 4. A pendulum hung from the ceiling makes a complete back‐and‐forth swing every 6 seconds. As the pendulum swings, its distance, d, in cm, from one wall of the room, depends on the time, t, in seconds, since it was set in motion. At t = 1.3, d is at its maximum of 110 cm from the wall. The closest the pendulum gets to the wall is 50 cm. Assume that d is a sinusoidal function of t. a) Sketch a picture of the situation. b) Sketch the graph of the function d and write an equation expressing d as a function of time. c) Write an equation for d’(t) d) How fast is the pendulum moving when t = 5? When t = 11? How do you explain the relationship between these two answers e) When t = 20, is the pendulum moving towards the wall or away from it? Explain. f) What is the fastest the pendulum swings? Where is the pendulum when it is swinging the fastest? g) What is the first positive value of t at which the pendulum is swinging 0 cm/s? Where is the pendulum at this time?
