Question

Problem #14 a.) sketch the graph of the given function for three periods

b.) Find the fourier series for the given function

(& #18 if can do)

12. In each of Problems 13 through 18: Verify equations (6) and (7) in this section by direct integration. a. Sketch the graph of the given function for three periods. b. Find the Fourier series for the given function. f(x) =-x, -L < x < L; f(x + 2L) = f(x) 13. 1, -L x <0, x < L, -π < x < 0, 0 x<r; f(x + 2L) = f(x f(x + 2π) = f(x) f(x +2)=f(x) 14. f(x)= 0 rx, 10, [1-x, 0S1<!; 17, f(x)=İL, 0 < x < L, f(x + 2L) = f(x) 一2 < x <-1, -1 < x < 1, 0, 18. f(x) = f(x + 4) = f(x)

Answer 1

(a) graph of f(x) is as shown below (b) f(x) = a_0/2 + sigma^infinity_n = 1 a_n cos(n pi x/L) + b_n sin(n pi x/L)) a_0 = 1/L integral^L_- L f(x)dx = 1/L integral^0_- L 1dx = 1/L times L = 1 a_n = 1/L integral^L_- L f(x) cos (n pi x/L)dx = 1/L integral^0_- L cos(n pi x/L)dx = [sin(n pi x/L)/n pi/L]^0_- L = 0 b_n = 1/L integral^L_- L f(x) sin (n pi x/L)dx = 1/L integral^0_- L sin (n pi x/L)dx = [- cos(n pi x/L)/n pi/L]^0_- L = -[1 - cos(n pi/n pi/L]