9. La ste) defined as follows 9. Let f(x) defined as follows: f(x) = 0 if x < -1 = 2(x + 1)/27 if - 1<x<2 = 2/9 if 2 < x < 5 = 0 otherwise. Find F(u) = f(x)dx, where u E R.
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...
4. Let X and Y have joint probability density function f(x,y) = 139264 oray3 if 0 < x, y < 4 and y> 4-1, otherwise. (a) Set up but do not compute an integral to find E(XY). (b) Let fx() be the marginal probability density function of X. Set up but do not compute an integral to find fx(x) when I <r54. (c) Set up but do not compute an integral to find P(Y > X).
A periodic function f(x) with period 21 is defined by: X + -1<x< 0 2 f(x) = 0<x< 2 Determine the Fourier expansion of the periodic function f(x). X - TT
2. Consider the cubic spline for a function f on [0, 2] defined by S(x) = { ={ (z. 2x3 + ax2 + rx +1 if 0 < x <1 (x - 1)3 + c(x - 1)2 + d(x - 1) + ß if 1 < x < 2 where r, c and d are constants. Find f'(0) and f'(2), if it is a clamped cubic spline.
48 The function f is defined by f(x) = for 3 <x< 7. The function g is defined by g(x) = 2x - 4 for .X-1 a<x<b, where a and b are constants. (i) Find the greatest value of a and the least value of b which will permit the formation of the composite function gf. [2] It is now given that the conditions for the formation of gf are satisfied. (ii) Find an expression for gf(x). [1] (iii) Find...
Real Analysis question, give clear writing please Let h(x) be the function on (0, 1) defined by ſi x <1 h(x) = 2 X=1 (a) For any P, what is the value of L(f,P)? (b) Can you find a P such that U(f,P) is within 1/10 of L(f,P)? (c) Show that h is integrable.
The probability density function of X is given by 0 elsewhere Find the probability density function of Y = X3 f(r)-(62(1-x)for0 < x < 1
7. Consider the function f:R + R defined by f(x) = x < 0, 3 > 0. e-1/x2, Prove that f is differentiable of all orders and that f(n)(0) = 0 for all n e N. Conclude that f does not have a convergent power series expansion En Anx" for x near the origin. [We will see later in this class that this is impossible for holomorphic functions, namely being (complex) differentiable implies that there is always a convergent power...
Problem 29.1 Let X have the density function given by 0.2 -1<r<0 f(x) = 0.2 + cx 0 〈 x < 1 otherwise. (a) Find the value of c.