Exercise 1. Let f be defined on [0,3], by I f(x) = { i 1 0
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![Exercise 1. Let f be defined on [0,3], by
f(x) = {
X
1
Q≤x≤ 2,
<x<3
(1) Sketch the graph of the even and periodic extension of f with period 6.
(2) Sketch the graph of the odd and periodic extension of f with period 6.
(3) Find the Fourier series associated to f on [0, 3].
Choose the period appropriately.
Exercise 2. Similar questions of Exercise 1, for the function
f(x)=4-x², for 0<x< 1](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F7f1b4368-5a5b-4a6d-b39b-68f4db26d2cd%2F43aafbb2-1577-4309-9719-ebdf2a8079a8%2Fhzolmi_processed.jpeg&w=3840&q=75)
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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 = 0A periodic function, f(x) with period 4x is defined as - 2n sx<-1 - nSX<0 2n, %3D f(x) Osx<* 2n, Sketch the graph of f(x) on the interval [-5x, 57). Determine if f(x) is an even, odd or neither even nor odd function. a) b) Find the Fourier series of f(x).Consider the function f(x)=3−x² defined in (0,1] Find its Fourier series of sines of period 2.
- A periodic function f(x) is defined by f(x) = x² + 3 for –2period -π ≤ t ≤ π. The following f(t) is a periodic function of period T = 27, defined over the -2 2 when when 0 < t ≤π -2. Consider the function f(x) = 1 x on the interval [0, 1]. a) In two separate graphs, sketch i) the odd extension fodd on the interval [-1,1], ii) the Fourier series associated with fodd on the interval [-3,3]. b) Carefully state Fourier's theorem, including the definition of the Fourier series and the Fourier coefficients. c) Compute the Fourier coefficients of fodd and write down the Fourier series. Give full justification for your answer. d) By evaluating the Fourier series for an appropriate value of x, show that 1 2 2 2 2 3π 5TT 7π = 2|7 + +Determine the Fourier Series of f(x) = x², over the interval - < x < and has period 2 ! f(x) = 3r [c cos x + cos 3x + cos 5x + .] sina+sin 2x + sin 3x +... 2 1 1 f(x) sin x + sin 3x + sin 5x + 1 π 3 5 1 1 -{sin z + sin 2x + sin 3x + x ...} 2 2 3 $ 1 f(2)=-4 [cos 2-2008 22 +00832 - 008 42 + cos cos cos 4x 4] 3 4² = # f(x) 12 = T1. Express f(x) by the Fourier series where f(x)= {, 2, -7Recommended textbooks for youAdvanced Engineering MathematicsAdvanced MathISBN:9780470458365Author:Erwin KreyszigPublisher:Wiley, John & Sons, Incorporated
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