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# Exam2pdf - Exam 2 Real Analysis 1 11 0/2008 Each problem is...

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Unformatted text preview: Exam 2 Real Analysis 1 11!] 0/2008 Each problem is worth 10 points. Show adequate justiﬁcation for full credit. Don’t panic. 1. State the deﬁnition of compactness. 4 98 T' 8 3 r: o f’?""/iT'/’r7&—f 9 Q ‘1')“ (9pﬁm Cos/é” 0'7"" fiﬁxi a? “977‘ <— 5)“ b (o [/96 W, / 2. State the (local) deﬁnition of continuity. L‘E‘x‘i' rib-'7nz M (1.5!). U-K 6001 C I5 (oﬂ4inuoxosﬂ7l'ai'g' Cor vex» 33m 5+. )x—QIAS and m) =9 )chb—kag' p? 3. Give an example of a function ﬁR - R that fails to be differentiable at exactly one point. ’FL {5 atmwuw 8.134- ‘ ng. 4. State the Intemiediate Value Theorem. "Us {he buncfioh'kiS Conh'nuous on [(1);] and K 13 a man number be..meen Madonna! {50:}; Adhere wig’ts q qul number ceCa,b) sum/la "(lamb [703)”;1'Q- M / ' €43 cm» ArﬁkaaW of o. owl £1643 to 5. State and prove the Quotient Rule for Derivatives. Hull “*- fl 3%?- l I 1 "\$015 Lcmm. m MW. m: f raw (rm = a? EOE; won, 12m (M _ N is”) * "ng X, —- f x- >A 0‘ xga X_0\ (00— (x) :: PM? U. . J1 I 3mm du -.: I:.m.« @519 ,1...— r I x—vox XW'Q ' gum) 5 '30“) ‘ﬂim -"'"'” a z. xaa net. ‘11“ detin “Lgkg M W 39:, Mnums 0* 0k Luv/u; “‘3‘” ‘H’t UJA 4L, ' by TEAM Nu m ﬁxi' «(100)3ch + #145 gusséﬁm ’5 gab - ' ‘C ’3“) “O I 395 4 (as er- ; (03 ﬂ mﬁm ’ am?- 5:03 3—3637. = \$565 5 - EasFms 530351 Tbs 1d»; (from? '13 cowgdu, 4/ 6. Provo directly from the deﬁnition that ﬁx) = 5x + 2 is continuous at x = 3. 16(5):- . I ‘z’ﬂ' Mj gimp £>0/ we always >H’“*4m} <8 [5x {(2 -H} (e am ﬁnd 5‘ 9/5 ma“ 1hr" 9'] Isms” (a Hmm g / WWW/\$.ch ‘ I XFBI [Aha/<5) x60 IE fftg/ < 9/5 5 5/x-.%/ < a /5x-15/ (E jag—H" (8 _ M =7 Manhaige . / -? iijMAQﬁMW 400013 con/I’M éf‘xgz , 7. State and prove the Mean Value Theorem. 1‘5 OI BJHCHDY“) B Conﬁhqoq‘s on (O.le and dlﬁemnh‘ame 0n Lq,b)_ there was“ c: eta-(Ado) Suck Nab HO: know- {5(9) WEsz T’mv [6% (b) ~— 15 (a) _ ’ 39x): (’70:) - _ ué‘v 5') Ewan . b _ cwi 9'04?) :: B (7") - 5.2L J“ lorq Ga Kane's heomm, we Khan)! Mere axis» 01 (“é—Ca b) SUCH J’lflc‘vi— -_ 0 star 9E6) -: (s'cc > (if? Leif” Jun-q m” 5 36 ) 2 5:11.252 174:1 I 8. Prove that the complement of a closed subset of IR is open. Le)? E be; M cAoSad 80V) mg'R _ - S'gose, TE *5 was) “dc QQQJQ. W 3x 625%”; '5A. no MR%V\\OO(V¥DO§\$ o£ x 0% comwckﬁ uﬁ‘WYt“ 13’? o C {\QACbWbof “GOA (SQ u W38 <1...(\ Q_kg..w\.Q—A¥ arc {El . TM 51 ts OAK 0.6 cwmu\a.\~?0(\ ""“c 6‘ BU“ am 5 2% C/{Oﬂ/ x1 M—‘r' \CLQ G: “W6 Coﬁjffaokic}: Mac‘r x-e; E437 5;; “2'5 MAB-'er. open. / Mica? _ 0 ifx ¢ 0 9. Is the functlo'n f(x) = _ 0 a rational ﬁmction? Support your answer. a radiancﬂ Rbﬁczh-Oh COL“ he LOHHUA (11.3; (Jolﬂﬂbme €dgmmmi v O po’YnQMialS (we, ConfinUOuLS and dr'JrLtféhhhb‘e on )2 {Ct-hoan mncb'ons we. df’tiexcenh'alole on I WM medms w"? g/ {’93) i5 mg Al‘kamhab’eag’; 2399 m, ()6 a? m: Jermain o? W) b!) (N) .6 mg!- a rah‘onod'FUthbYﬁx 10. Prove or give a counterexample: A function f which is differentiable at x = a must be differentiable on an interval ofthe form (x - 8, x + e) for some a > 0. I Cbuwis Er examp ‘1‘.) 5 {X0 X is r0x+1‘0“0\ i i: 0 iii: ‘>< i=3 !-Vf0\+-I:OV1CL\ 03' One. 00‘me which ‘5 K 3 ,__—_ Sc. ii is 391 dimwmhﬁble on m; {ml-um} ...
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