Analysis1997Aug - Preliminary Examination Analysis 18...

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Unformatted text preview: Preliminary Examination Analysis 18 August 1997 . Let K C IR” be a compact set and let 6 > 0. Set J = {1: 6 R" | dist(z, K) S e}, where dist(a:, K) = inf{||:c-- yllg I y 6 K} and Iltllz is the usual norm in IR". Prove that J is compact. . Determine the convergence or divergence of the following sequences {xn}°° n=1' (a):c-1+ 2 +-+ n "—n2+1 n2+2 n2+n 0» x» = (a (2;) nu + _n n 1 n n = -————— 1 _ (c) x 2 + ( + 2n) . Determine whether or not E11143) converges uniformly on I, where u" and I are given in parts (a) and (b) below 0 ,|x]$nor|x|2n+l (a) I=1Rand un(:r) = nsin(1/n2) ,n < lxl < n + 1 (b) I = [1,00) and u,,(z) = [1’ e-"t’dt, :1: e I. . Let D+ and D" denote the operation of taking derivatives of real functions from the right and f(y)-f( left respectively, for example DJr f (x) = lim 9:) , D" is defined similarly. y—)z+ y — It (a) Give an example of a function for which D+ f (0), D“ f (0) both exist but are not equal. (b) Prove or disprove: if D+f(0), D" f (0) both exist then the function f is continuous at a: = 0. 0 ,OSI<1/2 . Suppose that f = :1: and 9(1):) = 1/2 , m = 1/2 , evaluate: 1 ,1/2<a:§1 (b) 0 gdf . For a nonnegative integer I let P,(:r) = 22:0 akx" for real numbers ak and a: 6 [—1,1]. Given a positive integer 72 set .7-‘(n) = {Pz(:c) | 0 S l S n and Iakl < 1 for k = 0,... ,1}. So .7-‘(n) is 1 the set of polynomials of degree less than or equal n whose coefficients all have absolute value less than 1. Prove or disprove, for each n the set .7:(n) is equicontinuous. . Let f(1:, y) = |:t:ll/2]y|l/2 + my be a. real function on R2. (3.) Find the partial derivatives of f at the origin. (b) Discuss the differentiability of f at the origin. . Let a: = 1' 003(0) sin(¢), y = r sin(9) sin(¢), and z = r cos(¢) define the map F(r, 0, 45) = (2:, y, z) from (130,45) 6 R3 to (x,y,z) E R3. (a) Prove or disprove, F has a global inverse on R3. 6 (b) Find $0(0, 1, 0). ...
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This note was uploaded on 06/19/2011 for the course MATH 600 taught by Professor Na during the Spring '11 term at University of North Carolina School of the Arts.

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Analysis1997Aug - Preliminary Examination Analysis 18...

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