Consider the function f(x) = = −1, X, 1, −2 < x < −1, −1 ≤ x < 1, 1 ≤ x < 2. (a) Sketch the graph of f. (b) Find the Fourier series of f over the interval (–2, 2).
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- Find the Fourier series of the following functions (on [-π, π]). (1) sin²x. (2) r. Also sketch the curve of the Fourier series (on [-37, 3π]). Repeat the above problem on [-L, L].The Fourier series to represent the function f(x)=2x in the range - T to +n is 1 2x = 4 sin c,x - sin c,x + sin c,x - sin c̟x + sin c,x - sin cx + .. sin c̟x + Find: c,, C2, C2, C, C, Ce- C1= C2= C3= C4= C5= C6=(b) Using the answer to Part (a) and without using Theorem 2.2.4, find the real Fourier series of the function f(x) = -2x 2r-8 if 0 < x < 2 if 2 < x < 4 and f(x+4)=f(x).
- Let f be a function that has derivatives of all orders on the interval (5, 7). 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 = = 6: f(6) f'(6) f"(6) | f"(6) | f(4)(6) -3 4 8 9. -3 3 x³ + 4x² + 4x – 3 1 b) O-3 + 4 (x + 6) +4(x + 6)° + 2(x + 6)3. 8 3 c) O-3 + 4 (x – 6) + 4(x – 6)² + x – 6)³ - (x – 6)* d) O-3x* + 9x³ + 8x2 + 4x – 3 e) O-3 +4 (x – 6) + 8(x – 6)² + 9(x – 6)³ – 3(x – 6)* f) O-3 + 4 (x + 6) + 8(x + 6)² + 9(x + 6) – 3(x + 6)* g) None of the above.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 3If 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) = *
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