If f is the Fourier series of g(x) ==f(x) =32(-1)"+1n²-²[16-r²,What does f(-4) equal? f(-4)What does f(-2) equal? f(-2)=What does f(0) equal? f(0) n²What does f(1) equal? f(1)What does f(4) equal? ƒ(4)FIT4-4< <00

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Answered: If f is the Fourier series of g(x) = =…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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5. Convert the function f(x)=eux and f(x+2) = f(x) defined in the interval (-π < x < π) to the complex Fouriér Series.
1-x? 5 Find f(a) if f(x)=- |15 | |* >1| by Fourier transform method
Please answer all part correctly
- Ž 2(1 − (−1)") sin( 21² ) In n=1 Sf(x) = f(x)=x on [-1,1] Sf(x) = + -4 (-1)" z²n² n=1 Match the Function with its Fourier Series Look here to see if you are correct. f(x) = x² on [−1,1] cos(xnx) - Σ - ²(-1)" An n=1 Sf(x) = f(x) = |x| on [-2,2] sin(xnx) Sf(x) = 1 + ((−1)ª −¹) cos( z** ) n=1 f(x) = 1 on [0,2] and f(x) = -1 on [-2,0)
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Question 4 Let f be a function that has derivatives of all orders on the interval (3, 5). Use the values in the table below and the formula for Taylor polynomials to give the 4" degree Taylor polynomial for f centered at x = = 4: f(4) f'(4) f"(4) f"(4) f(4)(4) -4 10 8 -5 a) O-4 + 6 (x – 4) + 10(x – 4)² + 8(x – 4)³ – 5(x – 4)* b) O-5x4 + 8x³ + 10x² + 6x – 4 c) O-4 + 6 (x + 4) + 10(x + 4)² + 8(x + 4)' – 5(x + 4)* d) O-4 + 6 (x – 4) + 5(x – 4)° +x – 4° –- – 4) 5 (x- 4)* 24 4 e) O-4 + 6 (x + 4) + 5(x + 4)² + 5 x +4)° – x+4* 24 5 f) O- 24 4 + '+ 5x? + 6x – 4 3
= 1) The function f(x) periodic on the interval [0, 2л] has complex Fourier series f(x): Σ(1/n²) einx where the sum over n goes from - infinity to infinity. Convert this to cosine and sine Fourier Series by finding the values of A's and B's in the expression Ao + ΣAn cos(nx) + Σ Bn sin(nx) where each sum goes from 1 to infinity. Hint: consider the n and -n term together in the complex Fourier Series or use Euler's identity.
Consider the following. (A computer algebra system is recommended.) -23- . -3 Sxs 0, (x + 6) - (x) O
If f is the Fourier series of g(x)= √3, [16-², -4 < x < 0 then 0≤ < 4 f(2)=¯ + 2 [(0) cos (1 x) + ( ) sin (7-²)] 2 What does f(-4) equal? f(-4) What does f(-2) equal? f(-2) = What does f(0) equal? What does f(1) equal? What does f(4) equal? (0) f(1) = ƒ(4) = *
5) a) Construct a Fourier Series for f(x) = (x² - 0x)for 0 < x < 3t b) Briefly explain the application of half – range Fourier series in your Specialization
(a) (i) Simplify this function and plot it on the interval -2.5 ≤x≤ 2.5. (ii) Consider the periodic function defined within one period by the formula: P(x)=x+1-(x+1) it - sxs if 2 Determine whether the function is odd, even or neither. (iii) Determine the average value of the function on the period. (iv) Find coefficients of the Fourier series for this function. (V) Present the Fourier series of the function in terms of the E-notation.

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If f is the Fourier series of g(x) = = f(x) = 32(-1)"+1 n²-² [16-z², What does f(-4) equal? f(-4) What does f(-2) equal? f(-2)= What does f(1) equal? f(1) What does f(0) equal? f(0) n² What does f(4) equal? ƒ(4) -1%2 FIT 4 -4 < a <0 0