Math32B_midterm2solutions

Math32B_midterm2solutions - Friday 2/22 MATH 3113 A...

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Unformatted text preview: Friday 2/22 MATH 3113 A oémeiilz (if C Last name : First name : Section -. UCLA ID number : Signature : Instructions : full credit will be given only if your solution is given with all the details. Notes and calculators are not allowed. CREDIT ; SCORE : Problem l :10 points Problem 2 :10 point-S Problem 3 :20 point-e Problem 4 :10 points TOTAL : 40 points DER : cosgw +Sin2$ : 1. seegzr a 1 + ten2 :3. (i i a"; (arctan) z w. Comparison theorem. Consider two positive functions 0 S S on an interval I. Then, f{:r)d.:r. convergesl implies [f1 g{3:)dz converges]. in) g(;e)d:r divergesl implies [f] f{$)dcc diverges]. Absolute c0nvergence. converges] implies f(:c)d$ converges]. Probiem Find f < 1, 3x2 +2$+3 d3 2: +1){m2 +1) 35%»: “A ms gfifi‘ ? a”) gmgv .g ,2 ma EWan 4 5:: M £3; 9%?“ j $53415 “WM walem ’2. Find 1 ah. I $4 “xi-$2 +1 a :Ezgfswmgw «w 1: 3mm 5 y 3% gym? 65$ 57 j £235“ «:2 flag: as; f r <5 2* f3?» ‘1. ii Ami? {£53 a. jig? (£25; 1 r WW :2: M M j; 5%? fig; ‘ 5%? 4 “’3 g aw in.“ { éxf M 5359;? {(55% j 3*“ 553' WVW“ Sgh§ 25) (iii '2 w§ .‘3 ,_ 2 agfifié: asafififw’: 5”)” .5; 4 g fMfif/Q r: M I" g; c552”: fimwzigfih fiery @ jig-fig fig: M m» 9.» mg; + 3 $5: Probiem 3. (You are not asked :0 find the vaiue) a) Preve that I; 3mm J31"- dx converges. b} Prove that fix gigs—gait canverges‘ S‘MK < ai Q g M“ “‘ g; w; Z él “‘ £3) a g ’5’ i?§€ «Pg-2‘3 w {- Q g“ Q 0% C?" a? éfli f “1 cm L 5% i VSMX {gt CV LG? 2 m M 5 i%%g Thaw/9“; Probiem +OD ta—l Cflnsider the fmmtion [1(a) 2 a (if 1 + t “ a} Prove that H(a) converges for D < a < 1. b) W'hat. is Hfl/Q). c) Prove that HQ A a) : H(a), Va E (0,1). a} X, fféwfifi : «4 , j 5,: fl 5:: WP :afif 0/5” {E}; C: 6’ (i/ 5" 2M Mg: ass—vi 4 90$ rf’va s$éfl é? (fa «- Mai fl 15,5? g .1... G $0» aw: i} 1/,” {Law} 45/ 5+5?- f 4/ my- 1 ma “ £5; CV mg. iw>i 7»; they} CV 14;; 9’ 95;. ,4; a: W: «$4629 r“ J :1 W Q 34-5/5?” i} ggswiffiww H, j . . M W? ’36 y '3 fwd-a V3}: {fig ...
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This note was uploaded on 10/27/2008 for the course MATH 32B taught by Professor Rogawski during the Winter '08 term at UCLA.

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Math32B_midterm2solutions - Friday 2/22 MATH 3113 A...

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