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MIT18_03S10_c07

MIT18_03S10_c07 - 18.03 Class 7 Exponential and Sinusoidal...

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18.03 Class 7 , Feb 17, 2010 Exponential and Sinusoidal input and output [1] Sinusoidal functions [2] Trig sum formula [3] Integration of complex valued functions [4] Linear equations with sinusoidal input signal [5] Complex replacement Euler: Re e^{(a+bi)t} = e^{at} cos(bt) Im e^{(a+bi)t} = e^{at} sin(bt) [1] Sinusoids A "sinusoidal function" f(t) is one whose graph is a (co)sine wave. I drew a large general sinusoidal function, f(t) . I drew the graph of cos(theta) ; this is our model example of a sinusoid. A sinusoidal function is entirely determined by just three measurements, or parameters, which determine it in terms of cos(theta) . A = Amplitude = height of its maxima = depth of its minima P = Period = elapsed time till it repeats (or, in spatial terms, lambda = wavelength = the distance between repeats) t_0 = Time lag = time of first maximum f(t) can be written in terms of cosine. Clearly, f(t) = A cos(theta). To work out how, express theta as a function of t . I started drew a t-axis horizontally and a theta-axis vertically. When t = t_0 , theta = 0 . When t = t_0 + P , theta = 2pi . I marked these data points on all three graphs. The graph of theta as a function of t is a straight line; otherwise the cosine would get distorted, bunched up. So: theta = (2pi/P) ( t - t_0 ) and so
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