5 (10 pts) Let b 0 be a number and f)for (o.b, Lt artition of [O, b, wherefor0, 1,2, ns bea 72 ( 1) Find the upper sum U(f, P) . (2) Find lim Uf. P). 5 (10 pts) Let b 0 be a number and f)for (o....
Problems 5) Let (X, M, u) be a measure space, and f e Lt. Assume that S fdu = 1. Prove that 00, 0<a<1, lim n ln (1 +(${2))a) du(x) = { 1, a = 1, 10. a 1. Hint: Use Fatou's lemma for a < 1 and LDCT for a > 1 (dominate by af). 1+00)
(5 Marks) ii. Let f:[0, ] → R be the function such that f(0) = cos z for all : € (0,7), and let P be the partition {0, 1} a. Find Ax;, m; and M for all y, where ; represents the number of intervals and my and M, represent the minimum and maximum on each interval. b. Hence find L(P)(Lower sum) and U(P)(Upper sum)
12) Let f(A) = x + 1 xxo 0 x=0 (ax+ 2 x>0 a) Find lim f(x). X-70 b) Find lim F(x). X70 c) Find lim f(x). 13) Let f(x) = x - x .O 12.1 x 0 a) Find lim f(x). 6) Find
1. (2 pts each) The graph of some unknown function f is given below. 10 6/ 8-64-2 624 10 12 Use the graph to estimate the following quantities: (0 f (9) (g) f(4) b) lim (a) lim (e) (d) lim ( 6) (e) lim f(x) (c) lim f(x) if g(x)f(x) 6) a value of r where f is continuous but not differentiable (k) a value of r where f"(x) 0 and f"(x)>0 (1) the location of a relative maximum value...
3. (25 pts) Let fe C2[a, b], for a < b, and let {p,}0 be Newton's method, where p,n E [a, b] for all n 2 0. Suppose pn Converges top E [a, b], where f(p) 0, f'(p) 0, and p #p for all n 2 0. Find an expression for X 2 0, where sequence generated by a. Pn+1 -p lim = Pn-pl2 3. (25 pts) Let fe C2[a, b], for a
(5) Recall that X ~Uniform(10, 1,2,... ,n - 1)) if if k E (0, 1,2,... ,n -1, P(x k)0 otherwise (a) Determine the MGF of such a random variable. (b) Let X1, X2, X3 be independent random variables with X1 Uniform(10,1)) X2 ~Uniform(f0, 1,2]) Xs~ Uniform(10, 1,2,3,4]). X3 ~ U x2 ~ Uniform(10, 1,2)) 13Uniform Find the laws of both Y1 X1 +2X2 +6X3 and Y2 15X1 +5X2 + X3. (c) What is the correlation coefficient of Yi and ½?...
Question 25 2 pts Let C be the path from (0,0) to (1, 0) to (1,2). The work done by the force F =< 0,5 > on C is O 10 00 05 O 15
Which of the following series diverges? n +2 2n -1 n1 n+3 O A. 2 B. O C. 1,3 O D. 1, 2 OE. 2, 3 F. None O G. O H. 1,2,3 Find the sum of the series A. B. OC. 1/10 D. 1/2 3/2 3/4 OE. 1 F. 5/12 OG. 1/4 H. Divergent Which of the following series converges? oo 2n 1.Σ n 1 23n nE1 (n+ 1)3 n+ 1 3. O A. None O B. 2 O...
Solve the following partial differential equations using the Laplace transform method. x〉o, 2 5 ,>0 lim u(x, ,) = 0, u(0,,)-1, 3) İhu Ot x〉o, 2 5 ,>0 lim u(x, ,) = 0, u(0,,)-1, 3) İhu Ot
4. Let 3 f(x, y, z) = x’yz-xyz3, 4 P(2, -1, 1), u =< 0, > 5 a). Find the gradient of f. b). Evaluate the gradient at the point P. c). Find the rate of change of f at the point of P in the direction of the vector u.