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let f(x) be a function of period 2pi such that f(x) = x^2 over the interval of -pi <x< pi
[EUM 114 1. Let f(x) be a function of period 2 (a) over the interval 0<x<2 such that f(x) = - f(x)pada selang Diberikan f(x) sebagai fungsi dengan tempoh 2t yang mana 0<x<2m Sketch a graph of f (x) in the interval of 0 <x< 4 (1 marks/markah) Demonstrate that the Fourier Series for f(x) in the interval 0<x< 2n is (ii) 1 2x+-sin 3x + 1 sin x + (6 marks/markah) Determine the half range cosine Fourier series expansion...
let f:[-pi,pi] -> R be definded by the function f(x) { -2
if -pi<x<0 2 if 0<x<pi
a) find the fourier series of f and describe its convergence
to f
b) explain why you can integrate the fourier series of f term
by term to obtain a series representation of F(x) =|2x| for x in
[-pi,pi] and give the series representation
DO - - - 1. Let f: [-T, 1] + R be defined by the function S-2 if-A53 <0...
Question 3 Let f(x)be the function of period 4 which is given on the interval (-2, 2) by f(z) = Find the Trigonometric Fourier Series of f(x) 0.2<0
Let f (x) be a periodic function on R with period 21. On the interval (-11,), f(x) is given by f(x)=sin(x) 0<x51, = Let F(x) be the Fourier series of f(x). Select all correct statements from below. The Fourier series of -f (x) is -F(x). F(-1) = 0.
Let f(t) be a 2L- periodic wave function with one period on -pi<= t <= pi defined as f(t) = 1 if |t| <= T and 0 if T < |t| <= pi Find the real fourier series of f(x) first and then convert to complex form
1. Consider the uniform distribution X defined over the interval [0, 2pi]. Now let Y = sin(X) (a) Calculate the CDF FY(y) of Y. (b) Calculate the PDF f(y) of Y. In particular, in what interval [a, b] is Y defined? (this mean f(y) = 0 for y < a and for y > b). (c) Verify that f(y) is a PDF.
Let f(t) be periodic function with period T = 1 defined over 1 period as f(t) = {t -1/2 < t < 1/2} (a) Plot f(t) and find its Fourier series representation. (b) Find the first four terms of the fourier series.
2. Let I be an interval and let f be a function which is differentiable on I. Prove that if f' is bounded on I then f is uniformly continuous on I. 3. Give an example to show that the converse of the result in the previous question is false, i.e., give an example of a function which is differentiable and uniformly continuous on an interval but whose derivative is not bounded. (Proofs for your assertions are necessary, unless they...