5. Let A = P(R). Define f : R → A by the formula f(x) = {y E RIy2 < x). (a) Find f(2). (b) Is f injective, surjective, both (bijective), or neither? Z given by f(u)n+l, ifn is even n - 3, if n is odd 6. Consider the function f : Z → Z given by f(n) = (a) Is f injective? Prove your answer. (b) Is f surjective? Prove your answer
8. (10 pts.) The moment generating functions of X and Y are given by Mx(e) = (3x + 3) * and My (0) = + bene + cena respectively. Assuming that X and Y are independent, find (a) P{XY = 0} (b) P{XY >0} (c) Var (3X - 6Y + 2). (d) EXY.
3. [Total: 24 pts] a) (8 pts) Calculate the probability of finding a particle in the classically forbidden regime for the ground state of the 1D harmonic oscillator. Simplify the integral expression for the probability as much as possible - the integral can only be solved numerically. b) (8 pts) For the 1D harmonic oscillator, the energy eigenstates are either even or odd. This is indeed a special case of a more general statement: If V(x) is an even function...
10 [6 pts] True or False. Explain yo ur answer Vz E Z, Vy E Y,Hr X, P(,y, z (a)E X,Vy e Y,Vz E Z, P(x,y, z) (b) V E X,y E Y,Vz e Z, P(æ, y, z) Vz E Z, y E Y, Vax € X, P(æ, y, z) 10 [6 pts] True or False. Explain yo ur answer Vz E Z, Vy E Y,Hr X, P(,y, z (a)E X,Vy e Y,Vz E Z, P(x,y, z) (b) V E...
6.6a b 6.7c d e of the following irrational z transforms. (a) x(z) = ea", Izl > 0, o)X(z)log (1- az), zl < 1/lal. . 1 Show the following p roperties for the z transforms of even and odd 6.7 discrete-time functions. (a) If x[n] is even, that is, x[n-x(-n], then X(z) = X(z-1) (b) If x[n] is odd, that is, x[n] =-x|-n], then X(z)--X(2-1). (c) If x[n] is odd, then there is a zero in X(z) at z 1.
. xydV SIS xydzdxdy E where E is bounded by the parabolic cylinders y = x2 and x = y2 and the planes z = 0 and z = x + y. Find a,b,c,d,e,f. Be careful with the order of the variables! 1. y 2. x + y e = < 3. y2 a = V 4. Vý b= 5. "Vx d = 6. X 1. y 2. x + y e = 3. y2 a = 4. b= 5....
all i need is Q.3 1) (30 pts total, Ch11.1 and 11.6) For the function f(x) = 1 for <x< and 0 for the rest of the period: a) Draw a sketch of the function. Is it even or odd? b) (10 pts) Find the Fourier series for f(x) which has a period of 2nt for the terms up to sin5x and cos5x c) (10 pts) Find the error of your approximation 2) (30 pts total, Ch11.2 and 11.6) For...
4. Simplify and state the restrictions. 2x+8 4x+16 a) 3x 6x2–5x+1 b) x2-4 Х x2-x-2 x2-3x 2x2 4 1 c) x2+3x+2 + 1 x2+4x+3 11x d)- x2+3x-28 X-4
Use spherical coordinates to find the mass m of a solid Q that lies between the spheres x2 + y2 +z" 1 and x2 + y2 + z2-4 given that the density at each point P(x, y, z) is inversely proportional to the distance from P to the origin and 8(o, 3,02 15 pts] (0, 1,0)-2/m3 from P to the origin and Use spherical coordinates to find the mass m of a solid Q that lies between the spheres x2...
2. (20 pts) The following price demand for three goods x, y and z are given: Px = 130 - 3x-y-2z Py = 200 - x +4y P, = 150 - y-3z The total cost is given by the following function: TC = x2 + xy + y2 + yz + z2 (2 pts) Establish the profit function (3 pts) Use the F.O.C. in the profit function with respect to each of the three variables (12 pts) Now that you...