Mathematics sinusoidal functions:
1. Find a possible formula for each of the following sinusoidal functions.
2. A population of animals oscillates annually from a low of 1300 on January 1st to a high of 2200 on July 1st and back to a low of 1300 on the follow January 1st. Assume that the population is well-approximated by a sine or a cosine function.
(a) Find a formula for the population,P, as a function of time t. Let t represent the number of months after January 1st.Hint: First make a rough sketch of the population and then use the sketch to nd the amplitude, period, and midline.
(b) Estimate the animal population on May 15th.
(c) On what dates will the animal population be halfway between the maximum and minimum populations?
3. Lety= 2 sin(2x ) + 2. Find the amplitude, period, and phase shift. Then,carefullysketch a graph of the function. Be sure to clearly label all important aspects of this graph.
4. The pointPis rotating around a circle of radius 5 shown in the gure below. The angle, in radians, is given as a function of time, t, by the graph shown below.
(a) Estimate the coordinates ofPwhent= 1:5.
(b) Describe in words the motion of the pointPon the circle.
(c) The pointPhas ax-coordinate at each time t. Plot this x-coordinate against time t for 0t2:5. You will need a graphing utility to do this (e.g. calculator,Maple, etc.). Hint: First, determine a formula for(t).
5. The Bay of Fundy in Canada has the largest tides in the world. The di erence between low and high water levels is 15 meters (nearly 50 feet). At a particular point the depth of the water,y meters, is given as a function of time,t, in hours since midnight by y=D+Acos(B(t C)):
(a) What is the physical meaning ofD?
(b) What is the value ofA?
(c) What is the value ofB? Assume the time between successive high tides is 12.4 hours.
(d) What is the physical meaning ofC?