Consider the function f(t) = 0.01e2t for 0 ≤t≤ π). (a) Sketch the even extension, feven, and the odd extension fodd of the function f(t) over the range -37 ≤ t ≤ 3n, clearly indicating each case. (b) Now consider the function g(t) = feven (t) + fodd (t), which has Fourier series G(t) given by 8 ∞ G(t) = A + Σ An cos(nat) + Σ B, sin(nat). B₂ n=1 n=1 Calculate the constant term Ao.

Q/Determine the Fourier expansions of the periodic functions whose definitions in one period are: f1 + t 7 f(t) = -1

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 = 0

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