Definition 1. A function f(x) defined on (-L, L] is called piece-wise continuous if there are finitely many points xo =-L < x1 < x2 < < xn-L such that f is continuous on (xi, i+1) a...
Sub-problem 7. Plot f, the third partial sum of its Pourier serics sum of its Fourier series using Desnos. Print and staple this Make sure the interval [-3,3] is visible. ourier series and the sizth partical to the end of the eram Sub-problem 8. Evaluate f(2) Sub-problem 6. Let f(x) 1-) on (-2,2) and extended periodically to the line Compute the Fourier series f of f. l sum of its Fourier series and the sirth partial urier series using Desmos....
Fourier Series MA 441 1 An Opening Example: Consider the function f defined as follows: f(z +2n)-f(z) Below is the graph of the function f(x): 1. Find the Taylor series for f(z) ontered atェ 2. For what values of z is that series a good approximation? 3. Find the Taylor series for this function centered at . 4. For what values ofェis that series a good approximation? 5, Can you find a Taylor series for this function atェ-0?
Fourier Series...
please show all work
Consider the piece-wise continuous function k(x) as defined below: (Vx+1 k(x) = -* 10 -1<x< 0 0<x51 otherwise a) Find a valid PDF for random variable X, fx(x), that can be derived from k(x), then plot this PDF b) Find and plot the CDF for the random variable X, Fx(x) c) Find the expected value of X, E(X)
*5. A function f defined on an interval I = {x: a <x<b> is called increasing f(x) > f(x2) whenever xi > X2 where x1, x2 €1. Suppose that has the inter- mediate-value property: that is, for each number c between f(a) and f(b) there is an x, el such that f(x) = c. Show that a functionſ which is increasing and has the intermediate-value property must be continuous. This is from my real analysis textbook, We are establishing the...
This is for Stochastic Processes
Let Xo, Xi,... be a Markov chain whose state space is Z (the integers). Recall the Markov property: P(X, _ in l Xo-to, X1-21, , Xn l-an l)-P(Xn-in l x, i-İn 1), Vn, Vil. Does the following always hold: (lProve if "yes", provide a counterexample if "no")
Let Xo, Xi,... be a Markov chain whose state space is Z (the integers). Recall the Markov property: P(X, _ in l Xo-to, X1-21, , Xn l-an l)-P(Xn-in...
1. Let f(x) be the 2T-periodic function which is defined by f(xcos(x/4) for -<< (a) Draw the graph of y = f(x) over the interval-3r < x < 3π. Is f continuous on R? (b) Find the trigonometric Fourier Series (with L = π) for f(x). Does the series converge absolutely or conditionally? Does it converge uniformly? Justify your answer. (c) Use your result to obtain explicit values for these three series: and , and 162 16k2-1" 16k2 1)2 に1...
A continuous probability density fanction is a non-negati ve continuous function f with integral over its entire domain D Rn equal to unity. The domain D may have any number n of dimensions. Thus . . .lofdェ1 . . . drn-1, The mean, also called expectation, of a function q is denoted by尋or E(q) and defined by 1··· DG-f) d工1-.. drn. The same function fmay also represent a density of matter or a density of electrical charges Definition 1 The...
Let f(x) be the 27-periodic function which is defined by f(x)-cos(x/4) for-π < x < 1. π. (a) Draw the graph of y f(x) over the interval-3π < x < 3π. Is f continuous on R? (b) Find the trigonometric Fourier Series (with L π) for f(x). Does the series converge absolutely or conditionally? Does it converge uniformly? Justify your answer. (c) Use your result to obtain explicit values for these three series: 16k2 1 16k2 1 (16k2 1)2 に1...
A continuous probability density fanction is a non-negati ve continuous function f with integral over its entire domain D Rn equal to unity. The domain D may have any number n of dimensions. Thus . . .lofdェ1 . . . drn-1, The mean, also called expectation, of a function q is denoted by尋or E(q) and defined by 1··· DG-f) d工1-.. drn. The same function fmay also represent a density of matter or a density of electrical charges Definition 1 The...
1. If Ea) 2. The Fourier series expansion of the function f() which is defined over one period by , 1<zc2 is f(z) = ao + Find the coefficients an and simplify you answer. 1 z sin ax cos ar Jzcos az dz = Hint: f(x) cos(n") dz and a.-Th 3. The propagation of waves along a particular string is governed by the following bound- ary value problem u(0,t) 0 ue(8,t)0 u(x,0) = f(x) u(x,0) g(x) Use the separation of...