(27) Consider the piecewise function f : [-L, L] –→ R given by Ja + L, if – L
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![(27) Consider the piecewise function f : [-L, L] –→ R given by
Ja + L, if – L <x < 0
f(x) :
x*,
if 0 < x < L.
What is the value of the Fourier series of f at
(а) х
(b) х —
(c) x =
-L
L
-
2
(c)
(d)
(e) x = L.
L
x =
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- Consider the function. f(x) = { -25 x+1, -7 < x < 0 1-x, 0≤x <7' (a) Sketch the graph of f(x) for three periods fx -20 -10 f(x)= [Choose one f(x+14) = f(x) -5 10 M (b) Find the Fourier series of f(x) 15 20 25 81. Find the Fourier transform of (a) f (x) = exp (-a.x²), (b) f (x) = exp (-a|x|), where a is a constant.Prove that the Fourier transform of f") (x), the nth derivative of f(x) is (- ip) times the Fourier transform of f(x) provided that the first (n – 1) derivatives of f(x) vanish as x→to.
- Find the degree three Taylor polynomial T3 centered at x = = 0 for f when 2. T3(x) 1. T3(x) = ln 3. 3. T3(x) 4. T3(x) 5. T3(x) f(x) 6. T3(x) = - X 3 = = = = In 3+ 4 3x = In 3 4 -X 3 + ln(3 – 4x). 4 -X 3 x² 4 8 io - X 3 9 22 - 9 + 8 9 64 81 + 8 610 شرح 4 z x + 7 x 64 81 X3 + 64 81 خرج 64 -x³ 81 32 -x³ 81 64 من مخ 81Graph and find the Fourier coefficients of the following functions f (x): - T/2 < x < Tn/2 1/2 < x < 3n/2 1 a) f(x) = { -1 b) f(x) = { "** -πSuppose a function f(x) = x between 0 and 1, and f(x) = 0 elsewhere. Solve for the fourier transform g(k) of the original function f(x).Which of the following are true for all polynomial functions? (1) If f'(x) = g'(x), then f(x) = g(x) %3D (2) If f(x) = g(x), then f'(x) = g'(x) %3D Select one: O a. Only (2) O b. Neither (1) or (2) O c. Only (1) O d. Both (1) and (2)It Let f(t) be defined as f(t) = t 7L -1. Express f(x) by the Fourier series where f(x)= {, 2, -7Let f be a function on R with Fourier transform 1 F[f](s) = f(s) = f(x) e** dx. irs (a) Show that the Fourier transform of the function ga(x) : f(ax), where a > 0 is a constant, is Flg.|(6) = Lf (). 1 (b) Show that the Fourier transform of the function Je-, r20 f(x) = 0, x < 0 is f(s) = 1 1+ is V2n s2 +1Let -T < x < 0 0 < x < T. T + x, f(x) = { TT – x, (a) Find the Fourier series of f on (-n,7)Recommended textbooks for youAdvanced Engineering MathematicsAdvanced MathISBN:9780470458365Author:Erwin KreyszigPublisher:Wiley, John & Sons, Incorporated
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