(2) Consider the function f : R → R defined by Í 1 x E [-L,0) f(x + 2L) = f(z) -(x) f( 2L) o E l0,L) a. Graph f on the interval [-3L, 3L]. b. Compute Fi-L,Lf) c. Graph F-L(f) on the interval [-3L,3L]...
Question 6 Consider the function defined over the interval 0<x<L. Extend f(x) as a function of period 2L by using an odd half-range expansion 1) Sketch the extended function over the interval -3L<XS3L. 2) Calculate the coefficients for the Fourier Series representation of the extended function. 3) Write the first 5 non-zero terms of the Fourier Series. (10 marks)
1. Consider the function defined by (1 -2, 0 r< 1, f(x) 1 < |x2 (0. and f(r) f(x+ 4) (a) Sketch the graph of f(x) on the interval -6,61 (b) Find the Fourier seriess representation of f(x). You must show how to evaluate any integrals that are needed. 1. Consider the function defined by (1 -2, 0 r
1. Consider the function defined by f(x) 0, |x| < 2 1 and f(x) f(x 4) (a) Sketch the graph of f(x) on the interval -6,6 8 (b) Find the Fourier series representation of f(z). You must show how to evaluate any integrals that are needed 1. Consider the function defined by f(x) 0, |x|
1. Consider the function defined by 1- x2, 0< |x| < 1, f(x) 0, and f(r) f(x+4) (a) Sketch the graph of f(x) on the interval -6, 6] (b) Find the Fourier series representation of f(x). You must show how to evaluate any integrals that are needed 2. Consider the function 0 T/2, T/2, T/2 < T. f(x)= (a) Sketch the odd and even periodic extension of f(x) for -3r < x < 3m. (b) Find the Fourier cosine series...
3. Consider the function defined by f(x) = 1, 0 < r< a, | 0, a< x < T, where 0a < T (a) Sketch the odd and even periodic extension of f (x) on the interval -3n < x < 3« for aT/2 (b) Find the half-range Fourier sine series expansion of f(x) for arbitrary a. (e) To what value does the half-range Fourier sine series expansion converge at r a? [8 marks 3. Consider the function defined by...
using discrete structures 3. Consider the function F(x, y, z) for x, y, z z 0 defined as follows: a. F(x, y, 0)-y+1 b. F(x, 0, 1)-x c, F(x, 0, 2) = 0 d. F(x, 0, z+ 3)-1 e. F(x, y, z)-F(x, F(x, y-1, z), z-1) Using Induction, prove the following a. F(x, y, 1)-x +y b, F(x, y, 2) = xy c. F(x, y, 3)-xy 3. Consider the function F(x, y, z) for x, y, z z 0 defined...
1. Consider the function defined by 1-2, 0 < Ixl < 1, f(x) = and f(x) = f(x + 4). 1 (a) Sketch the graph of f(x) on the interval [-6,6). 8 (b) Find the Fourier series representation of f(x). You must show how to evaluate any integrals that are needed 1. Consider the function defined by 1-2, 0
1. Consider the function defined by 1-2, 0 < Ixl < 1, f(x) = and f(x) = f(x + 4). 1 (a) Sketch the graph of f(x) on the interval [-6,6). 8 (b) Find the Fourier series representation of f(x). You must show how to evaluate any integrals that are needed 1. Consider the function defined by 1-2, 0
3. Let f(r) be defined by and let F(x) be defined by F(x) = Í f() dt, a. Find F(x). 0 x 2. For what value of b in the definition of f is F(x) differentiable for all x E [0, 2)?
2. Consider the function f(x) defined on 0 <x < 2 (see graph (a) Graph the extension of f(x) on the interval (-6,6) that fix) represents the pointwise convergence of the Sine series. At jump discontinuities, identify the value to which the series converges (b) Derive a general expression for the coefficients in the Fourier Sine series for f(x). Then write out the Fourier series through the first four nonzero terms. Expressions involving sin(nt/2) and cos(nt/2) must be evaluated as...