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- Suppose that f(t) is periodic with period -T, 7) and has the following real Fourier coefficients: a2 = 3, аз — 3, b2 = -3, b3 = 0, ao 4, a1 = 1, b, = 2, (A) Write the beginning of the real Fourier series of f(t) (through frequency 3): f(t) = 2+cost+2sint+3cos2t-3sin2t+3cos3t+0+. (B) Give the real Fourier coefficients for the following functions: (i) The derivative f'(t) ao = , a1 = -1 , az = -6 , az = -9 b, = 2 , b2 = -6 bz = (ii) The function f(t) – 2 ao = , a1 = , az = 3 , аз — 3 b1 = 2 , b2 = -3 bz = (iii) The antiderivative of (f(t) 2) (with C 0) ao = , a1 , a2 = 3/2 , аз 1 b1 =-2 b2 = 3/2 b3 = (iv) The function f(t) + 3 sin(3t) + 3 cos(2t) ao = 2 , a1 = 1 , a2 = 6 , аз — 3 b, = 2 b2 = -3 b3 = 3 (iv) The function f(2t) an =2 , a1 , a2 3 , a3 3 b, = 2 , b2 = -3 b3 = 0Which of the following is NOT a valid expression for a Fourier coefficient of function f(t) with period 4m? Ⓒao=ff(t)dt be=ff(t) sin(6t)dtas= f(t) cos(3t)dt ao=/f f(t)dt Which of the following are valid for Fourier coefficients of 1 t ≤ 4T f(t)= = {1 0 4 < t < 8T with period 16. [Tick all that apply - points will be deducted for wrong answers.] 1 Ⓒao = 16-80 fdt = □bk=0ak = cos(kt)dt 14 -4T ak cos()dt 8m ak = 8x 8K -8T cos(k)dt Which of the following expressions is NOT equal to zero? ○ 813 Of sin(3x) sin(a)da of sin(3x) cos(x) dx 12m Of* sin(2x) dx of cos(3x) cos(3x)da5) a) Construct a Fourier Series for f(x) (x2 . 12x)for 12 < x < 3t b) Briefly explain the application of half - the range Fourier series in your Specialization
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