Application - Physics An oscillating spring has a frequency of 1.16 Hz (here is some information about the frequency unit Hertz) and amplitude 3 centimeters. You are testing the damping effect of a frictional force on the spring, and find that it reduces the frequency to 0.73 Hz and the amplitude to 2.1 centimeters. You want to model this behavior with a piecewise function of the form: f(t) = { Here each value of A represents an amplitude and each value of w ("omega") represents a frequency. (It may help to recall that the period of a sinusoidal function is the length of time for one cycle to occur, so frequency and period have a reciprocal relationship, and that the function A cos(Bt) has period when interpreting the pieces of this function.) A₁ сos(2πw₁t), A2 cos(2πw₂t+C), 1 0 < t < 5 t≥ 5 = 5. (Recall C controls the Find the value of the parameter C that makes the piecewise function continuous at t horizontal shift of a sinusoidal function; this shifts the t = 0 point to t This will help with interpretation, but you should be able to algebraically solve for C even if this interpretation isn't clear at first.) Use the limit definition of continuity in your computation, and be sure to tell the story of your thought process - how you choose each limit computation you do. 2 3 If it helps to visualize the process, here is the graph you get for the piecewise function if C = 0, illustrating the need for a horizontal shift to create continuity: X y = {0≤x<5:3 cos (2n(1.16)x)} 3 y= {x≥ 5:2.1 cos(2n(0.73)x)} X 2 -1- 0 -1- -2 27 B 2 3 B . 5 6 7 8 9 10

Application - Physics An oscillating spring has a frequency of 1.16 Hz (here is some information about the frequency unit Hertz) and amplitude 3 centimeters. You are testing the damping effect of a frictional force on the spring, and find that it reduces the frequency to 0.73 Hz and the amplitude to 2.1 centimeters. You want to model this behavior with a piecewise function of the form: f(t) = { Here each value of A represents an amplitude and each value of w ("omega") represents a frequency. (It may help to recall that the period of a sinusoidal function is the length of time for one cycle to occur, so frequency and period have a reciprocal relationship, and that the function A cos(Bt) has period when interpreting the pieces of this function.) A₁ сos(2πw₁t), A2 cos(2πw₂t+C), 1 0 < t < 5 t≥ 5 = 5. (Recall C controls the Find the value of the parameter C that makes the piecewise function continuous at t horizontal shift of a sinusoidal function; this shifts the t = 0 point to t This will help with interpretation, but you should be able to algebraically solve for C even if this interpretation isn't clear at first.) Use the limit definition of continuity in your computation, and be sure to tell the story of your thought process - how you choose each limit computation you do. 2 3 If it helps to visualize the process, here is the graph you get for the piecewise function if C = 0, illustrating the need for a horizontal shift to create continuity: X y = {0≤x<5:3 cos (2n(1.16)x)} 3 y= {x≥ 5:2.1 cos(2n(0.73)x)} X 2 -1- 0 -1- -2 27 B 2 3 B . 5 6 7 8 9 10