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Midterm 1 Spring 2005 1

Midterm 1 Spring 2005 1 - Ill-MEI“ I EDLLTITDHE MATH...

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Unformatted text preview: Ill-MEI“ I EDLLTITDHE MATH Ilfl. FEBBLMIH' l3. m5 Pruhlerl l. lffifiel .1 gmcn' Herr 51:41me “HI-”WET Mdflfl'fl'l'l'm' nu- r-mnrr mirfirrfl'nn. Tili- ram-fl: rIr-r Palm-raid lard-1H: Making Illi- Hui-mm d'rm'n by firm many Minn dihedral-5.7 film- did n'irl': Ilia: mn- ram-fr! HEEL .I'L'5LI'J' n:- flfllfl. mud flue-r raw}.- _| LI'I'lrl'E'PIEI'JI'J‘ Esme-mm Gilerfllfll 4 in} Find Hir- pmfinhill'ry Hm: n mineral-r {Mi-er: rlnflmr n'i'rl' dimmer: Imam Elmnflnmiflem with me new. Eb} Gr'vm Hull :l're rummrr did mm Hear! £500 nJI' We: will: :l're slim. M m: mm? mm m: nm‘rmrer gaw- nn "Awmge" ml- "13. {1:} Deli-4min: flirt-Fm rlrwr rural! 'Errrfl'rnl' Md ‘flwr flUUU' rm" in- drpmn'mr. Explain rmrrccrmrlrlg.’ PM It'sdgd-nd idea t1:- Iill iI1 Ih: marginal value: dr'lln: LuflHnlll'Idugh we won‘l need all mm valum}; thermult Iddtfi like _lm ”WEN” {a} We see dul 1I5 euelemeni did hem-em 35m and EIIIIIEI nF Maine-e: Willi II'E minimally-.51: Il"IE: probability in: 2I5Jl'5lili = (HE. [b]: There are 2I5 + I135 = 3.1!] moment who did "Lien: Ilia-1. 15mm hufiiuem with ll'E more." [Time are Il"IB 21312th and Ihir'd numieel eelunm in Ih: IalJleJ This in me new mime. Dr" mm we at: the: [El] + 45 = II55 gain: an 'flverage‘ “ting; met-drum Il1e pmhnhllily lfi lfifi _ 33 E _ 5' Tan should elm-5: mint, ntiuml numb-uni td lowest terms; 15}- Ill: lime ll1i: final exam mllfi amund. prim may he mien nflfirji'ir'l'ing Muir: m. finddmrndcqiallle 3mm isfljlfi. Almmgll you wuw‘l allowed dulmlnlm. if: may endth to dun-by hand. {1:} Nd. llieym no! independent (“all Ill: mini: E {for ‘Eldcllcnl'l- and 0 {[dl ‘Dveriltlll'l- In drdfl H:- be independent we must have FEEI’TG‘] = FIE] - FIG]. But PiEf‘I D} = 4535M while FEE] = ll]5.-'5|IHJ. and it is nnr nu: ll1nl 45 _ I65 I05 5m _ 5m 5m- Fdr example. multiplying d1: last equalim I13- SIIF m: mld dbuin [I] 45x5fl5=ldjxlfli Ifififfiflfl and Pifi'il = Le. 11500 = I'J'315. Annular way In see diet [I]: is impossible in. In none Il‘mi I I divides [155. and hence. the. right side qu[|}. IJuI. Ind: 115. ll": lefl Hid-d El Problem 1. [20%.] Swain-1w l'le' 15% ry'flre madame lam' Rah-uglier Few: A men- Efifflgm'l'llfl' rur :1 derived n-Irl: due J‘lflmflflg i'fiul‘flrlfll'l'l‘l'fl‘l.‘ + flfpmfflr inim- .l'rmne HMIrnm Fri-lei: 93% Mr! msflr'w and 29':- rm ”emit-e: + flfpmfflrwhndmrwl'nwr Hemulim Fm: 5% lrxrmxirhlrrmd 335% m: uni-germ. FMS-e1- up :l’re flflh’lflfillfl' rm fin-r rill: airmail-m. {h} HWHW i: nerdnmfiv pirtedfmn :l’re papilla-Jinn and rem .Im'irhle. wire: i: die thifl'fl' Hm: FEW l’mw Rmulauifflw? Pun-y“. Shudder. Drawing graphs like ll‘lifi in Le'lhll im'l. easy. But here's what then-nee in Inn [:1 Inch. lilda: Frem Illie wemduee dial. Haw wu'min l. paailiun In Eli-WP.“ I11]. TIII: pmflml Int: fur PierlH. FIGURE l. Tm: diagram PERM I'"I +1 = fl.lfi 419$ = Ill-1'? PERM I'"I -II = [1.55 419$ = [1.533 PERM fl +3 = [1.85 41.05 = [1.0415 PERM I'"I -:I = 0.55 41.95 = 0.5335. and by dnfinit'mn this is filmmlhfl PI[+_'I I FERN-III “+1 + PIE-'31! I"| +_'I I 13.147 +{Ifl-flfi I (”395. hmflm HRMIH I Pianl+] = HRMH+J PH} Ill-1? {HESS = ET??? . sum [Mmmptniflnm fl.l5 :«I 0.95 0.15 x um + mas x nm‘ but [ rcgnfl Jul-ring Inld pmpl: IJIiII farm “mild tn: DE. Than: iIIIJ‘t THAT ml: W111: in cumming Ihjn wllhunl lualflflam: El thkm 3-. {511%} 1'1” Buhhfiafip Err-da (my ha: a Milling fl'l'e'l'l'hliflfi whirl: undrlj'ifl'x 5% qf'rfi'r birth-'5 III-kirk pm; I'llmuak it. Air-twirl; .I'fi'm‘. umicflit fl'L‘l'Ilr' mm'fmlyr [a] 0m nfn hawk $13.1 Mm. what it the award man-Illa!" Ifmdflh Mai Writs? {In mqnbmrhq'ifihnulm whalirrfirrpmimfihh'rhmi arm-ml" lurker will! brundurfl'ifluf? {1:} flu: rzj'n butch nj'fi mm. 1.4:... L: dummy. :hmr mun rfl'flu Milena!- WWW? PM This LI :1 binnmial pmhat-ilily pthtm: mm; ‘mm' on mean In:- dernlling a Mule. III: pmhfl-ililar chums wiIh my one haul:- L: 0.35. while [111?— pmbiJiu' ut’ I’Iu'JIIre l!- IJBS. Ha have n - 25. A: the back mm..- mm nwnrmmhfliwfimflpmhfllflirvhflcfirn I flmfifld with WI nI' It: hook, {III 1h: “panned value is up - 2.5 1-: I105 - 1.15 hauler. {I1} Letting 1' dfll'IEIEE Lha- number at' “amen—M... I113 I'I1I'I1IIBE unfun- Ihrl’IlJBd bmflm—fmm IhB Inch Inf 1}. we are: filing fir HI 5 2]. This can he: lacked I11 t'mm in Idfla; map-MIN Ell-.313. MIME. andalm accepted. in (311.05%.95“ + (7303419514 + (gill-5:035:3- pram-fad you nude. uh: :impLilinIims 15 25 25x24 (15):“ (1):“ [2): Ix: =3m‘ mn.nfi"=1. [a]:1h:min9.95“»:mly;mmmirmdidmumem-up. El Pruhhu 1-. (IRE-J Can arm-m3 n: the Fla-name fin: Swim Maw r: Hm diflrfhurfi-m, 1dr}! an «Items: qr"! mm per Muir. {I1} III-Tu: lg Mpnflaflm} I'M: 3 I'll‘fl'llfl? rdmdntwbr Hymn-1m? mm [l1] Wild: 5.: the pnflafim} their l5- arm-er can: arm in the nu: Jamar? Pm-y’. Again. I'J'IL'I. LII. a. Labia. tuning. {a} PE}: :3 3:. = I — Pix 5 1}Enml—FEX 5 3w]. Nmrormn Pnimm diminution wim u = 4 w: ham: Pr}: = a} = run“. Whammgwig l-F{I=D]- HI: Il-FEX=1} 4| ‘2 - l —r_' —:4fi—r"i - I — Br". Tim’s Ih: mama: and mammabl: {ind parhapfi unread}. But a tafltcfl'mmflflimmdjfiMMIMngm in “1:: tract nflhnh-u-nk; by lunl'upt'mm Ih: bible. FIX 5 '3) = ELISEJIJd mm L‘rlcuurrmumwjnl — 11233 = DIME. {b} SiTnjlarljr.wcm get succinct mm for part lb}: PIIEEII HA:-+}}+Ptx-1}+P{E-2}+Ptx-3}+ PEI I4J+ FIE-5}+PEE I15] ‘I'I 4| ‘1 ‘3 _1 -1. -4 _1 =r.‘ —C. +c —” +¢r —11+r —3!+ 4" 11:5 II" _4. _.1 _.1. E It +-E —5! +r.‘ —E!. Bul ll1ifi than :im‘lgcrnd mgh;ruummdtucvaluhcflr=pnwcmut' 4mdfllnfamillfi. Wuymdnlhmlhcannmmmuutluhc d 1' H1: 5 5;. = fl: -"_ An umly ugly uprafldnn. and mm whim involve; Fultlng a Jul at' t'raclinm men- 3 mm damnfiualnt'. The: table loot-up mini: from “1:: Incl nflhc huntincanifl! Pmsfi) = 11:339. flute marmaugwuinm PHIL" SW+FEIEI1+PEI E 1}+---+ PEIEfiJ. (Julie anuntuufpwpleaddedup dramha'sflfllfl. 11192.1}.233. 0.433. am. 11135. 0.339 to gen 3.13611. Bu: :1 pmbfl-iliw tan': 13¢ graterlhmm1Thigghnuldme warnedlhm: Linnea-rem! the mbmmy’PEI = 1}. 5:1!ka wiman{I 5 1'). El Problem 5. NEE-M. musk; m rzf'duarmbrmh. Ina-Mum.- 5!] m n hm; a... m:- tamed rum-uh? nj’rru'h path-1: urn-flex. wlrh WM fl-m-I- burfi-Ht NW1.- —---l:u —l2:I--- {a} findflw fipwtud mmmdmflard dwinliunfwl'flrnumherrf Mark: in n hm. [by Am [I'm bum with 51} a: man: Ihumhmks.find flu: warm-f mink: drill.- "HM-hr! qfl'hmfi'nmh. {fl-u: mil-din!!! Err pmhfiflitr dfnrihHim JIM rfumgrdf In mnrrm the Md mm: find hmrfir 5H nrmfl} Fwy“. fimmhle. The. :rilhmacl: mm mm hard. {3] Bydal‘mllinu u --1-E rum-+49 xfl.2‘!+5fl x1}.35+filx1}.25+52 :1: {Hi}. I' you find WE jnupab]: ul' multiplying a two-digit TI'I.I'I1Ih:.|.' limaIMJ-djgitmnmnr. than mmnmymhflfin: jl lunhilih: “1|: daumnlflmm in. sci-we‘ll do L‘rlnumnpumlunalmm 51}: #=f5fl—1}Hfl.flfl+flfl—I}ufl.21+mufl.3~5+ {50+ |}Kfl.11+fm+1)3ffl.lfl =f5flia!flfl3+5flxI].ITI'+5EI:-rfl.35+lflrfl.fi+lflxfl.lm+ +{—‘3Hfl.fl3— I xfl.fi+l KELE+IRI1IDJL Examining the Jan mu: I'mas timely. we see Ital u =51] :r: I +{—fl.flfi—fl.1’i'+l].15+|11fl-} = SDJ‘J. Thawdmmis duret'm Farm = 431 x 11.03 + 491 :«c 0.1? + 5t]:E x 1135+ 5|2 gags-+521 gum—50.111 = 13M:l].flfl+1-=lfll xnanzsm 1035+Efifllx 1115+ 2704 n ma — 5cm! = Lil-25E- Th: nmdard dwialinn B: flnmt'm'cd = fFarEIi = 41.1115 = lflll. sum —E:IEJ- “El-[El [by The Iable I’m-bout: ut’fifl Drum mum: is but unit is nulunger a pmhhfliu' table; ch: sum ut' Ihu mbahillliui is fli'. Tu mil-5: :ll :1 pl'un‘tuiluiLily.I LEHB. we mun difldn all [IE pmbahdlidafi by 11?: l].35ffl.‘." = 11.5. 0.25311} = 1139.6. [LINE = Ill-Ill Thus the. a:- Nwmr-m Infill}: mmvuluula pl :51: I15+5l xfl.33|5-+ 5110143 =Sfl.l5-l. Allnmnliu'uly. Dill: uuuld dim-id: hyfl-f." a El": llfil: fitup: Sflflfl.35+51 Hfl.15+.512.lfl 36.45 “ ' T 'W'W- Ilul .51] 1-: 11.35 + 51 1-: 0.15 + .51 a: lllfl Lt Jammy wan-:3. El Prulflumi. HE‘EH mun-fer ch}:- muajfilflurlflg mummy llflffl mm width pm“ dam-lire chip mar flewnlfiw. 31W a Mull after: claim mum: curred by .rlu'u mum-lime in? mm {a} ll'lw is! I'll: Fri-militia} Hm: nun: ry'rflecflfl'u .I'.I:' defifllw ? lb} ll'hm' l: [kph-WWW I‘llfll' finally turn when!” uni defmlw? {u} III-Tu: L: I'll: pmfilfljl Hun: lira ur nun: raffle clrwmdefeflhu'? inal'. Thin in again a binomial pruhnhiJirf: again. wt: uEHIrIt adufuufiw uhip :H- a men. The. pruhahilily uf mucus in interim: us - 0.2. Ind-file the pmhubijiu' ul’I’aiJure is 4:5 - 11.3. m It: hm: a “rm"uflm fIiJqufi ha: piuhfln'lir: M'". This m mum. bum in: pruru'fllll: bu luv: :il in m: nun than. ll'yuu lhfiullluljr mam mlluulu it. I suggest ll“: fullcmhg {which 1's useful in Tl'lillilj' Himalium hum-alum binary uumbam: “u have 2'“ - mm s: mm - lflfiLaud {IL-BID: {21: 3|! [El—Ill“ = 11-" '5' Ifl—IEI = [IIDJJ- 3H! I'D—l“ in: Illa-aim apprullnumly {I113}? u m-I“ = :11- {I1} Since: [b.L‘I. i3 3. binumlal pmhhlJiry Iha “fiwlfi HI = 1]- = ('3) x nu? 1-: us". But as [ wan-m1 incur-L yuu haul lusm'lplit’y 1 may {20)— [:1 "5' The mm: 45x11? will“ was Manly accumulate. NWally. I11: value is drum 13.3112. [1:] HEEHI l—Hfl'fi I}- I—HI HIE—HE. minim: Muefnmum m: = u} = (Bufuam = uu'“. ll} H1: = 1]. = ( l)u.1'u.s“1u u: :11 a»: nu“. Thus dramswuris 1 - u.3“‘ - 1 x m“. [1‘3 Pullulm ID Ll'l'lr In dn a. table luuhp for min problem: l'qlru'nt'I pun-fink u mull-r tr. lwag mullet-anal} trying nu get you [u use: binumlal pt’uhhlJiIL-uu and run sinplily Eli Hut-lulu] muflkim- El ...
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