Let z=5 where x, y, z E R. Prove that z? +z2+z?>
Let U be an open subset of R". Let f: UCR" ->Rm. (a) Prove that f is continuously differentiable if and only if for each a e U, for eache > 0, there exists o > 0 such that for each xe U, if ||x - a| << ô, then |Df (x) Df(a)| < e.
3. (a) (5 points) On the set A= R\{0}, let x ~ y if and only if x · y > 0. Is this relation an equivalence relation? Prove your answer. (b) (5 points) Let B = {1, 2, 3, 4, 5} and C = {1,3}. On the set of subsets of B, let D ~ E if and only if DAC = EnC. Is this relation an equivalence relation? Prove your answer.
Consider the vector field F (x, y, z) = <y?, z2, x?>. Compute the curl (F). Use Stokes' Theorem to evaluate S. F. dr where C is the triangle (0,0,0), (1,0,0), and (0, 1, 1) oriented counter-clockwise when viewed from above.
The cumulative distribution function of the random variable X is given by F(x) = 1-e-r' (z > 0). Evaluate a) P(X > 2) b) P(l < X < 3 c) P(-1 〈 X <-3). d) P(-1< X <3)
(2) Prove that the following are equivalent for x ER and A CR. (a) X E A. Here A denotes the closure of A. (b) For every e > 0, N(x; e) n A +0. (c) For every open set U, if r EU then UNA+.
Exercise 7. Let X be a standard normal random variable. Prove that for any integer n > 0, ELY?"] = 1207) and E[x2n+] = 0.
where 7 is the region defined by >0, y >0, >0, r+y+z<3.
Evaluate SS Pods where È CM,y,z) = 5 zex, sy, 2-yz?> s is the where s suface of the unit cube in IR R? not including the face Z-0.
7. Let X1,... , Xn be iid based on f(x; 6) -22e-z?/e where x > 0. Show that θ=-yx? is efficient
1. Let x, a € R. Prove that if a <a, then -a < x <a.