2+sin^2(x+2)+cos^2(x+2) Δ ν =[1 1 2 ν 1, ν ) ( ( π (x α) ημ β α π ) ) +1 + a 2

Transcript

1 Parathr seic sta Jèmata Jetik c kai Teqnologik c KateÔjunshc tou ètouc 7 Peiramatikì LÔkeio Euaggelik c Sqol c SmÔrnhc 1 IounÐou 7 PerÐlhyh Oi shmei seic autèc anafèrontai sta jèmata Majhmatik n Jetik c kai Teqnologik c KateÔjunshc tou ètouc 7kai kat kôrio lìgo apeujônontai se majhmatikoôc. EpiqeireÐtai mða suz thsh twn jem twn pèran twn tupik n apant sewn. Gia dieukìlunsh oi ekfwn seic twn jem twn èqoun perilhfjeð wc pros rthma sto tèloc. 1 To Jèma Ta erwt mata A1, A, A3 Prìkeitai gia erwt mata pou oi apant seic up rqoun sto sqolikì biblðo. Qarakthristikì ìmwc eðnai ìti arketoð majhtèc wc ap nthsh sto A èdwsan thn akìloujh << DÔo sunart seic eðnai Ðsec an èqoun to Ðdio pedðo orismoô kai ton Ðdio tôpo >>. H ap nthsh aut eðnai esfalmènh gia polloôc lìgouc. Pr ta ap' ìla eis gei mða ènnoia pou den eðnai kal c orismènh sto sqolikì biblðo. EkeÐnh tou << tôpou >>. All akìmh kai an to prosper soume ja prèpei na sumfwn soume tð ennooôme lègontac << tôpoc >>. Profan c lðgo polô ennooôme k poia èkfrash gnwst n sumbìlwn pou mac dðnei th tim thc sun rthshc se k je x pou h sun rthsh orðzetai. Endeqomènwc na qrhsimopoiôntai diazeuktik diaforetikèc tètoiec ekfr seic ìpwc sumb inei me tic sunart seic pollaploô tôpou. Ac p roume th sun rthsh F (x) = tdt. Poiìc eðnai o tôpoc thc? O 1 x 1 tdt o 1 x 1? An poôme o deôteroc ti èqoume na poôme gia th sun rthsh F (x) = ημt 1 t dt gia thn opoða eðnai gnwstì ìti to olokl rwma den upologðzetai 1 stoiqeiwd c? Sto deôtero par deigma ja prèpei na arkesjoôme sthn èkfrash ημt 1 t dt pou piì polô apoteleð mða perigraf par èna tôpo ìpwc ton antilambanìmaste. En tèlei ìla ta Majhmatik mporoôn na grafoôn seiriak dhlad se mða eujeða gramm. 'Enac tôpoc eðnai mða sumboloseir pou akoloujeð bèbaia k poiouc suntaktikoôc kanìnec. DÔo tôpoi eðnai Ðdioi an gia na touc gr youme qrhsimopoi- same akrib c ta Ðdia sômbola me thn Ðdia akrib c seira. An m lista gr foume se grafomhqan upologist autì shmaðnei na pat soume ta Ðdia akrib c pl ktra me thn Ðdia bèbaia seir. Wstìso mporeð dôo Ðsec sunart seic na èqoun 1 Dhlad, ìpwc èqei apodeiqjeð, den up rqei trìpoc na grafeð to olokl rwma autì wc par stash twn gnwst n mac << stoiqeiwd n >> sunart sewn 1

2 7 1. Ta erwt mata Ba, Bb, Bg, Bd, Be tôpouc pou den eðnai me thn parap nw ènnoia Ðdioi. Oi tôpoi 3 kai +sin^(x+)+cos^(x+) eðnai diaforetikoð all ekfr zoun Ðsec sunart seic. 1. Ta erwt mata Ba, Bb, Bg, Bd, Be Ba EÐnai l joc. Apì to gegonìc ìti h sun rthsh f eðnai gnhsðwc aôxousa sto di sthma Δ mporoôme na sun goume mìno ìti ja èqei mh arnhtik par gwgo. Pr gmati an x eðnai eswterikì shmeðo tou Δ tìte lìgw thc monotonðac gia k je x x isqôei f(x) f(x ) x x >. 'Otan ìmwc per soume sto ìrio gia x x tìte apl c mporoôme na sumper noume eðnai ìti lim x x f(x) f(x ) x x dhlad f (x ). Klasikì par deigma apoteleð h sun rthsh f (x) =x 3 pou an kai gnhsðwc aôxousa h par gwgoc thc f (x) =3x eðnai pantoô jetik ektìc apì to ìpou eðnai. 'Ena ligìtero sunhjismèno par deigma eðnai h sun rthsh f orismènh sto [, 1) h opoða se k je upodi sthma Δ ν =[1 1 ν 1, 1 1 ν ) tou [, 1) (ta diast mata aut eðnai an dôo xèna kai kalôptoun to [, 1)) orðzetai na eðnai f (x) = β α me α =1 1 ν 1, β =1 1 ν ( ( π (x α) ημ β α π ) ) +1 + a H grafik par stash aut c thc sun rthshc prokôptei me epan lhyh enìc kommatioô pou eðnai ìmoio me to prohgoômeno tou kai me lìgo omoiìthtac

3 To Jèma Prìkeitai gia mða paragwgðsimh kai gnhsðwc aôxousa sun rthsh thc opoðac h par gwgoc mhdenðzetai se peira shmeða. H epìmenh prìtash mac dðnei kai èna krit rio gn siac monotonðac gia mða sun rthsh me mh arnhtik par gwgo: Prìtash 1.1 'Estw f :Δ R mða paragwgðsimh sun rthsh me f (x) gia ìla ta x. 'Estw M (f ) to sônolo twn shmeðwn tou Δ sta opoða mhdenðzetai h f. Ta ex c eðnai isodônama: 1. H f eðnai gnhsðwc aôxousa.. To M (f ) den perièqei di sthma. Apìdeixh. (1..) Upojètoume ìti h f eðnai gnhsðwc aôxousa. upotejeð ìti to M (f ) perièqei k poio di sthma J tìte jewroôme x 1 <x apì to J. Asfal c J Δ. Apì to je rhma mèshc tim c èqoume ìti f (x 1 ) f (x )=f (ξ)(x 1 x ) gia k poio ξ metaxô twn x 1,x. Ja eðnai ìmwc ξ J M (f ) kai epomènwc f (ξ) =. 'Ara f (x 1 ) = f (x ) ( topo). (. 1.) Upojètoume ìti to M (f ) den perièqei di sthma kai deðqnoume ìti h f eðnai gnhsðwc aôxousa. JewroÔme giautì x 1 <x kai jèloume f (x 1 ) <f(x ). 'Hdh apì th sunj kh f (x) èqoume ìti f (x 1 ) f (x ) kai epomènwc arkeð na apokleðsoume ìti f (x 1 )=f(x ). 'An ìmwc autì sunèbaine gia k je x me x 1 x x ja tan f (x 1 ) f (x) f (x ) kai epomènwc f (x) =f (x 1 ). Autì shmaðnei ìti h f eðnai stajer sto di sthma [x 1,x ]. 'Ara f (x) =gia ìla ta x [x 1,x ]. Autì sunep getai ìti [x 1,x ] M (f ) ( topo). 'Ara h f eðnai gnhsðwc aôxousa. o.e.d. Bg EÐnai l joc. H èkfrash << h h eðnai suneq c sto x >> emperièqei dôo paradoqèc pou prèpei na plhroôntai kai oi dôo: An H h orðzetai sto x To ìrio thc h sto x kaj c kai h tim thc sto x sumpðptoun Tìso h sunèqeia ìso kai h asunèqeia anafèrontai se shmeða tou pedðou orismoô. Sthn prokeimènh perðptwsh me h = g f h h mporeð na mhn orðzetai kan sto x na orðzetai all na apotugq nei na eðnai suneq c. DÔo paradeðgmata aut n twn peript sewn eðnai ta akìlouja: 1. f (x) =x 3, g (x) = x 1, x = { x +1, x < 3. g (x) = x, x 3, f (x) =x +1, x = Bd EÐnai swstì. ApoteleÐ sunduasmì dôo tôpwn parag gishc. Be EÐnai swstì. M lista isqôei to << an kai mìno an>>. To Jèma Sto jèma autì qrhsimopoi jhke ènac metasqhmatismìc Möbius k ti pou èqei sumbeð arketèc forèc kai sto pareljìn. Metasqhmatismìc Möbius eðnai k je

4 4 7 apeikìnish tou migadikoô f pou apeikonðzei migadikoôc se migadikoôc kai èqei th morf f (z) = αz + β γz + δ, α β γ δ (1) Gia to pedðo orismoô kai to sônolo tim n thc f isqôoun ta akìlouja: An γ = : Tìte ja eðnai anagkastik δ. To pedðo orismoô kaj c kai to sônolo tim n eðnai ìlo to C. An γ : To pedðo orismoô thc apartðzetai apì ìlouc touc migadikoôc pou eðnai di foroi apì to δ γ. To sônolo tim n thc perilamb nei p li ìlouc touc migadikoôc arijmoôc ektìc apì èna: ton α γ. Autì diapist netai eôkola. H exðswsh αz+β γz+δ = w èqei p nta akrib c mða lôsh ektìc an w = α γ opìte eðnai adônath. Se k je perðptwsh h apeikìnish f eðnai 1-1. Pèran toôtou èqei kai merikèc akìmh idiìthtec pou thn kajistoôn elkustik sth dhmiourgða jem twn. Ac doôme giatð: EÐnai gnwstì ìti k je eujeða sto migadikì epðpedo mporei na p rei th morf z z 1 = z z, z 1 z () Autì diìti k je eujeða eðnai kai mesok jetoc k poiou eujugr mmou tm matoc. Ac jewr soume t ra èna kôklo C. Epilègoume mða di metro KL tou C kai èna shmeðo tou A. Apì to A fèrnoume mða eujeða pou tèmnei thn di metro eswterik se k poio shmeðo G diaforetikì apì to kèntro tou kôklou. H summetrik eujeða thc AG wc proc thn AK tèmnei ton forèa thc KL sto B. Sto trðgwno ABG h AK eðnai eswterik diqotìmoc kai profan c h AL eðnai exwterik diqotìmoc. O kôkloc C den eðnai lloc apì ton Apoll nio kôklo tou ABG kai epomènwc gia k je shmeðo tou M ja isqôei MB MΓ = AB AΓ An onom soume z 1,z touc migadikoôc pou antistoiqoôn sta shmeða A, B kai λ = AB AΓ 1blèpoume ìti o kôkloc C perigr fetai apì th sqèsh z z 1 z z = λ (3) ApoteleÐ mða eôkolh skhsh na deðxoume ìti an ènac migadikìc ikanopoieð thn sqèsh (3) me λ 1tìte an kei se kôklo. H sqèsh (3) ìmwc mporeð na perigr yei

5 To Jèma 7 5 kai eujeðec arkeð na epitrèyoume sto λ kai thn tim 1 afoô h () eðnai eidik perðptwsh thc (3). EÐmaste t ra se jèsh na apodeðxoume thn epìmenh: Prìtash.1 'Enac metasqhmatismìc Möbius apeikonðzei kôklouc eujeðec se kôklouc eujeðec. Apìdeixh: K je kôkloc eujeða perigr fetai apì mða exðswsh thc morf c (3). Ac onom soume w = αz+β γz+δ. Ja broôme se tð gramm an kei o w. LÔnontac wc proc z brðskoume z = β wδ α wγ. Antikajist ntac sthn (3) brðskoume ìti: β wδ α wγ z 1 β wδ α wγ z = λ h opoða ìpwc diapist netai met apì lðgec pr xeic isodunameð me thn: w β+z1α δ+z 1γ w β+zα = λ δ + z γ δ + z 1 γ δ+z γ H teleutaða ìmwc sqèsh eðnai thc morf c (3). Epomènwc to h eikìna tou w an kei se eujeða kôklo. o.e.d. Sto sugkekrimèno jèma èqoume ton metasqhmatismì α iα+ α+i. MporoÔme na m joume - kai toôto anex rthta apì thn ekf nhsh pou mac to lèei- se ti gramm an kei o w = iα+ iw 1 α+i arkeð na lôsoume wc proc α brðskontac α = w i kai na apait soume α R pou isodunameð me thn iw 1 ( w i = iw 1 ) w i apì thn opoða brðkoume i w = i dhlad w = 1. Den eðnai dôskolo na diapist soume ìti k je shmeiì tou monadiaðou kôklou eðnai kai eikìna k poiou α ektìc apì to shmeðo pou eðnai h eikìna tou i. Me lla lìgia h pragmatik eujeða surrikn netai ste na gðnei èna anoiktì di sthma pl touc π pou telik ja tuliqjeð ston kôklo.

6 6 7.1 To Er thma a..1 To Er thma a..1.1 H sôntomh apìdeixh GÐnetai me meso upologismì: +αi a +i = +αi a +i = + α α + =1.1. Oudèn kakìn amigèc kaloô Merik paidi gia na apant soun sto er thma èkanan arketèc pr xeic. Pr ta èjesan ton migadikì arijmì se kanonik morf : z = +iα α +i = 4α α +4 + α 4 α +4 i (4) kai sth sunèqeia upolìgisan to mètro: ( ) ( ) 4α α 4 16α z α + +4 α = +(α 4) α4 +8α +4 (α +4) = +16 α 4 +8α +16 =1 Poll doulei gia to tðpota! Wstìso h sqesh (4) èqei endiafèron. Me merikèc epemb seic gr fetai: z = ( ) ( α α ) 1 ( α ) + ( +1 α ) i +1 An onom soume α = εϕ t èqoume: z = εϕ t 1+εϕ t kai apì gnwstoôc tôpouc brðskoume: 1 εϕ t 1+εϕ t i z = ημt iσυνt MporoÔme na èqoume kai mða gewmetrik ermhneða thc parap nw sqèshc an krat soume to monadiaðo kôklo sth jèsh tou all metakin soume thn pragmatik eujeða k nontac thn efaptomènh tou.

7 3 To Jèma EÔkoloi upologismoð sto trðgwno KLM mac deðqnoun ìti KMΛ = t kai epomènwc heikìna tou z èqei suntetagmènec συν ( π + t) kai ημ ( π + t) dhlad ημt, συνt. Gia na broôme pou antistoiqeð to α to topojetoôme ston xona apoktoôme to L to opoðo kai en noume me to M.ToshmeÐo ìpou h LM ja tm sei ton kôklo eðnai h eikìna tou z. Aut h di taxh mac epitrèpei ousiastik na apeikonðsoume thn eujeða ston kôklo. An t ra ìlo to sôsthma peristrafeð gôrw apì thn MK h eujeða KL ja mac d sei èna epðpedo. Ac fantasjoôme ìti eðnai to migadikì. O kôkloc ja mac d sei mða sfaðra pou ja eðnai monadiaða. To L ja eðnai plèon ènacmigadikìc arijmìc kai ja apeikonðzetai sthn epif neia thc sfaðrac. 'Ola ta shmeða thc sfaðrac ja eðnai antðstoiqa k poiou migadikoô ektìc apì ton Bìreio Pìlo thc dhlad to shmeðo M. Prìkeitai gia th gnwst di taxh thc sfaðrac tou Riemann me thn opoða mporoôme na apeikonðsoume alli c na doôme to migadikì epðpedo kai ta sumb nta tou p nw se mða sfaðra.. To Er thma b Gia α =brðskoume z 1 = i kai gia α =brðskoume z =1...1 To Er thma bi. EÐnai z 1 z = i 1 = k ti pou faðnetai bèbaia kai apì to sq ma... To Er thma bii. EÐnai z1 ν = ( ) z1 ν ν =( 1) kai ( z ) ν =( 1) ν epomènwc z1 ν =( z ) ν. 3 To Jèma 3 Sto jèma autì zht jhkan apl all ousi dh kaj konta se mða tritob jmia kampôlh. H morf thc kampôlhc f (x) =x 3 3x ημ θ mpèrdeye k poiouc majhtèc oi opoðoi parag gisan kai to hmðtono wc na tan kai h θ metablht. Sthn ousða prìkeitai gia mða parmetrik oikogèneia kampul n me par metro to θ. Ta pr gmata aplousteôontai polô an l boume up' ìyin ìti afoô θ kπ + π to ημ θ diatrèqei to (, 1] kai epomènwc h oikogèneia mac eðnai thc morf c f (x) =x 3 3x m, m (, ] Sto sq ma pou akoloujeð emfanðzontai merik mèlh thc oikogèneiac. H sun rthsh pou brðsketai yhlìtera eðnai h f (x) =x 3 3x kai perilamb netai sthn oikogèneia.

8 To Er thma a. H sun rthsh pou brðsketai qamhlìtera eðnai h f (x) =x 3 3x perilamb netai sthn oikogèneia. kai den 3.1 To Er thma a. 'Eqoume f (x) =x 3 3x m, f (x) =3x 3=3(x 1) (x +1), f (x) =6x kai h f èqei ton akìloujo pðnaka metabol c: apì ton opoðo prokôptei kai h ap nthsh sto èr thma. 3. To Er thma b. Epeid to ìriothcf sto eðnai ja up rqei x 1 < 1 ste f (x 1 ) <. EÐnai f ( 1) = m> kai epomènwc ja up rqei tim tou x metaxô twn x 1, 1 dhlad sto di sthma (, 1) pou ja mhdenðzei thn f. Lìgw monotonðac h tim aut eðnai h monadik sto di sthma (, 1). 'Ara h f èqei akrib c mða rðza se autì to di sthma. Epiqeirhmatolog ntac an loga sun goume ìti h f èqei llec dôo rðzec mða se k je èna apì ta diast mata ( 1, 1), (1, + ). 3.3 To Er thma g. Me to sumbolismì pou qrhsimopoi same sta prohgoômena h eujeða gr fetai y = x m kai eðnai entel c aplì na epalhjeôsoume ìti ta shmeða ( 1, m), (, m), (1, m) an koun se aut thn eujeða.

9 3 To Jèma Sto er thma autì axiopoieðtai h summetrða thc kubik c kampôlhc k ti pou èqei gðnei kai se èna jèma tou 'Estw h sun rthsh f (x) =αx 3 + βx + γx + δ me α. H f mporeð na èqei na mhn èqei topik akrìtata (autì exart tai apì to anβ ( 3αγ ( > )) anβ 3αγ ) all se k je perðptwsh èqei shmeðo kamp c to β 3α,f β 3α to opoðo eðnai kai kèntro summetrðac thc grafik c thc par - stashc. Gia na to epalhjeôsoume autì arkeð ( na apodeðxoume ( ìti to )) summetrikì tou tuqìntoc shmeðou thc C f, ac poôme tou β 3α h, f β 3α h wc proc to ( ( )) β 3α,f β 3α eðnai epðshc shmeðo thc C f. An eðnai (p, q) to summetrikì tìte apì tic sqèseic brðskoume ìti β 3α h + p = β 3α, f ( β 3α h ) + q = f ( β ) 3α p = β ( + h, q =f β ) ( f β ) 3α 3α 3α h kai prèpei na diapist soume ìti isqôei f (p) =q me lla lìgia ìti: ( f β ) ( 3α + h =f β ) ( f β ) 3α 3α h K nontac k poiec pr xeic diapist noume ìti h (5) alhjeôei. AfoÔ h (5) alhjeôei gia k je h mporoôme na thn paragwgðzoume. Ja broôme: ( f β ) ( 3α + h = f β ) 3α h (6) H sqèsh (6) mac plhroforeð ìti h f èqei xona summetrðac thn eujeða x = β 3α. Lìgw thc (6) oi tetmhmènec twn riz n thc f, ef' ìson up rqoun, eðnai shmeða tou x x sumetrik wc proc to shmeðo tou me tetmhmènh β 3α kai lìgw thc (5) ta shmeða thc C f pou antistoiqoôn se autèc tic rðzec dhlad ta shmeða akrot twn thc f ja eðnai summetrik wc proc to shmeðokamp c. (5)

10 To Er thma d. 3.4 To Er thma d. Me m = ημ θ, ìpwc kai prin, h sun rthsh gr fetai f (x) =x 3 3x m kai h exðswsh thc eujeðac eðnai y = x m. H diafor touc eðnai x 3 3x m ( x m) =x (x 1) (x +1). EujeÐa kai kampôlh èqoun ìpwc anamèntai trða koin shmeða kai h diafor metaxô twn -1, eðnai jetik èn metaxô twn,1 eðnai arnhtik. To zhtoômeno embadìn eðnai: E = 1 f (x) ( x m) dx = x (x 1) (x +1) dx = = x (x 1) (x +1)dx 1 1 x (x 1) (x +1)dx = = [ 1 4 x4 1 ] [ 1 x 1 4 x4 1 ] 1 x = 1 4 To Jèma To Er thma a. 'Htan to eôkolo er thma tou jèmatoc kai sugkèntrwne perðpou to 1/3 twn mon dwn. H f eðnai gnhsðwc aôxousa kai afoô h arqik tim thc f() eðnai jetik ja eðnai kai ìlec oi llec timèc. H g eðnai pantoô jetik kai epomènwc h fg eðnai pantoô jetik. Oloklhr noume mða jetik sun rthsh sto di sthma [,x] kai epomènwc to apotèlesma ja eðnai jetikì. Gewmetrika autì den lèei tðpote perissìtero apì to ìti ta embad didi statwn qwrðwn eðnai p nta jetik!

11 4 To Jèma Asfal c mporoôn na dojoôn kai llec apodeðxeic pou qrhsimopoioôn parag gish. 4. To Er thma b. To er thma autì tan to duskolìtero. H duskolða tou ofeðletai sto ìti zhteðtai na apodeiqjeð mða anisìthta ìpou k poio mèloc (to pr to) den eðnai kat' an gkhn paragwgðsimh sun rthsh. Kai epomènwc de mporoôn na efarmosjoôn oi tupikèc teqnikèc melèthc sun rthshc. Me to er thma autì oi exetastèc zhtoôsan apì ta paidi to autonìhto: na skefjoôn. Af nontac bèbaia sthn krh tic diabìhtec << mejodologðec >> MÐa atuq c ap nthsh An h sun rthsh f tan paragwgðsimh, lème n, tìte onom zontac kai paragwgðzontac ja èqoume apì thn opoða prokôptei dhlad h (x) =f (x) G (x) F (x) h (x) =f (x) G (x)+f (x) G (x) F (x) h (x) =f (x) G (x)+f (x) g (x) f (x) g (x) h (x) =f (x) G (x) AfoÔ h sun rthsh f eðnai gnhsðwc aôxousa ja eðnai f (x) kai dedomènou ìti g (x) > sto (,x] èqoume ìti h (x). Autì mac epitrèpei apl c na sun goume ìti h (x) h () =. Wstìso mporoôme na exasfalðsoume kai thn gn sia anisìthta h (x) > wc ex c: An sunèbaine h (x) =tìte sto di sthma [,x] h h ja tan afoô eðnai aôxousa kai oi timèc sta kra sumpðptoun. 'Ara

12 To Er thma b. sto (,x) h par gwgoc thc h dhlad h f (x) G (x) ja tan mhdèn kai epomènwc ja tan anagkastik h f mhdèn dedomènou ìti h G eðnai p nta jetik. Autì ja s maine ìti h f eðnai stajer sto di sthma [,x] pr gma adônato afoô h f eðnai gnhsðwc aôxousa. Fusik mða tètoia ap nthsh proôpojètei ìti h f eðnai paragwgðsimh kai epomènwc katapi netai me mða polô eidik perðptwsh. Throumènwn twn analìgi n eðnai wc e n na zhteðtai na apodeiqjeð ìti ta Ôyh enìc trig nou sumpðptoun kai na to apodeiknôei k poioc gia orjog nia! Wc ap nthsh sto er thma akìma kai se pl rh an ptuxh eðnai esfalmènh. 4.. MÐa Gewmetrik ErmhneÐa Prin per soume stic apant seic sto er thma autì ac doôme ti mac lèei h proc apìdeixh anisìthta gewmetrik : 'Estw èna x metaxô, 1. H eujeða pou eðnai k jeth ston x x sto shmeðo tou me tetmhmènh x kai o xonac y y orðzoun mða tainða. Mèsa se aut thn tainða kai p nw apì ton x x brðskontai mèrh twn grafik n parast sewn twn sunart sewn f, g kai fg pou an tic periorðsoume sto [,x] èqoun tôpouc f(t), g(t) kai f(t)g(t). H g(t) mèsa sthn tainða kai p nw apì ton x x sqhmatðzei èna qwrðo embadoô E 1 en antðstoiqo embadì E sqhmatðzei kai h f(t)g(t). T ra gia na prokôyei h grafik par stash thc sun rthshc f (t) g (t), t [,x] paðrnoume k je shmeðo (t, g(t)) thc grafik c par stashc thc g kai pollaplasi zoume thn tetagmènh tou me f(t). An ìmwc pollaplasi soume ta g(t) me to f(x) pou eðnai, lìgw thc monotonðac thc f pio meg lo apì ìla ta llaf(t) ja prokôyei h grafik par stash thc f(x)g(t) pou ja eðnai pio p nw apì thc f(t)g(t) kai ja sqhmatðzei embadìn E 3. 'Ara E 3 E. 'Omwc pollaplasiasmìc thc g(t) me f(x) epifèrei an logh metabol sto embadìn dhlad E 3 = f (x) E 1 kai epomènwc f (x) E 1 E.

13 4 To Jèma MÐa pr th ap nthsh sto 4b. MÐa tupik apìdeixh thc anisìthtac ja mporoôse na eðnai h akìloujh: f (x) f (x) f (x) G (x) >F(x) g (t) dt > g (t) dt f (x) g (t) dt f (t) g (t) dt f (t) g (t) dt > f (t) g (t) dt > (f (x) g (t) f (t) g (t)) dt > (f (x) f (t)) g (t) dt > H teleutaða anisìthta isqôei diìti oloklhr noume se di sthma kai o oloklhrwtèoc (f (x) f (t)) g (t) gia <t<xeðnai jetikìc lìgw thc monotonðac thc f kai tou jetikoô pros mou thc g kai gðnetai mhdèn mìno gia t = x ParagwgÐzontac all swst 'Omwc h anisìthta f (x) G (x) >F(x) mporeð na apodeiqjeð kai me parag gish. H parak tw eufu c lôsh, sqedìn autoôsia, dìjhke apì thn teleiìfoith tou PeiramatikoÔ LukeÐou thc Euaggelik c Sqol c SmÔrnhc Alex ndra Zampet kh: Gia dojèn x (, 1] jewroôme th sun rthsh H (ρ) =f (x) G (ρ) F (ρ), ρ [,x] Profan c h H eðnai paragwgðsimh kai isqôei: H (ρ) =f (x) G (ρ) F (ρ) =f (x) g (ρ) f (ρ) g (ρ) =g (ρ)(f (x) g (ρ)) Lìgw tou jetikoô pros mou thc g kai thc monotonðac thc f eðnai H (ρ) gia k je ρ (,x) kai epomènwc h H eðnai gnhsðwc aôxousa. Epomènwc H (ρ) >H() = gia k je ρ (,x]. Eidik gia ρ = x ja eðnai H (x) > kai epomènwc H (x) =f (x) G (x) F (x) > 4.3 To Er thma g. To er thma autì duskìleye ligìtero apì to b. touc exetazìmenouc diìti eðqan th dunatìthta na ergasjoôn me parag gouc.

14 To Er thma g MÐa pr th ap nthsh MÐa sqedìn mhqanik ap nthsh prokôptei an thn apodeiktèa gr youme sthn isodônamh morf : F (x) G (x) F (1) G (1) F (x) G (1) F (1) G (x) (7) Onom zoume h (x) =F (x) G (1) F (1) G (x) x [, 1] ParagwgÐzontac brðskoume: h (x) =f (x) g (x) G (1) g (x) F (1) Dhlad ìti: h (x) =g (x)(f (x) G (1) F (1)) Lìgw thc monotonðac thc f eðnai f (x) G (1) F (1) >f() G (1) F (1) = 1 (f () f (t)) g (t) dt < diìti stoolokl rwma 1 (f () f (t)) g (t) dt ooloklhrwtèoc (f () f (t)) g (t) eðnai arnhtikìc gia ìlec tic timèc tou t ektìc apì thn t =. Sunep c h h eðnai gnhsðwc fjðnousa kai gia k je x [, 1] isqôei h (x) h () = MÐa deôterh kai pio endiafèrousa ap nthsh Jèloume JewroÔme th sun rthsh F (x) G (x) F (1) G (1) R (x) = F (x) G (x) x (, 1] kai thn meletoôme wc proc th monotonða. 'Eqoume R (x) = ( ) F (x) = F (x) G (x) F (x) G (x) G (x) G = (x) = f (x) g (x) G (x) F (x) g (x) G (x) = g (x)(f (x) G (x) F (x)) G (x) Apì to er thma b. èqoume ìti R (x) > kai epomènwc h R eðnai gnhsðwc aôxousa. Gia <x 1 ja isqôei R (x) R (1) kai m lista toðson jaisqôei mìno gia x =1. H apodeiktèa profan c isqôei.

15 4 To Jèma Perissìteroc kìpoc all megalôterh antamoib An ergasjoôme ìpwc prin all de qrhsimopoi soume to er thma 4b. ja èqoume: R (x) = f (x) g (x) g (t) dt g (x) f (t) g (t) dt G = (x) = f (x) g (x) g (t) dt G (x) g (x) f (t) g (t) dt = g (x)(f (x) f (t)) g (t) dt G (x) ParathroÔme ìti gia t [,x] eðnai g (x)(f (x) f (t)) g (t) kai to Ðson isqôei mìno gia t = x. Autì exasfalðzei ìti R (x) > kai p li ìpwc prin h R eðnai gnhsðwc aôxousa opìte gia <x 1 eðnai R (x) R (1). 'Ara R (x) F (1) G(1) kai m lista to Ðson ja isqôei mìno gia x =1. Bèbaia o anagn sthc asfal c ja prìsexe ìti h parap nw apìdeixh emperièqei kai èna epiqeðrhma parìmoio me ekeðno pou qrhsimopoi jhke gia thn apìdeixh tou 4b. Ed apodeðxame to er thma 4g. autotel c qwrðc na qrhsimopoi soume to 4b. H krðsimh sqèsh tan ( ) F (x) > (8) G (x) Ac thn doôme xan : ( ) F (x) > G (x) F (x) G (x) F (x) G (x) G (x) > F (x) G (x) F (x) G (x) > F (x) G (x) >F(x) G (x) F (x) G (x) > F (x) G (x) f (x) g (x) g (x) > F (x) G (x) f (x) G (x) >F(x) Epomènwc ìqi mìno to g. mporeð na prokôyei to b. all ex' Ðsou kal mporeð k poioc na apodeðxei pr ta, autotel c, to g. kai met na apodeðxei to b MÐa gewmetrik ermhneða thc sqèshc (8) Oi sunart seic FGekfr zoun ìpwc eðdame embadìn. To phlðko twn parag gwn touc fg, g eðnai h gnhsðwc aôxousa sun rthsh f. Epeid o lìgoc twn sunart sewn fg, g to susswreuìmeno embadìn aux nei:

16 To Er thma d. 4.4 To Er thma d. 'Eqoume aprosdiìristh morf. 'Amesh efarmog tou kanìna tou De l Hospital odhgeð sthn anaz thsh tou orðou: f (x) g (x) lim x + ( ) ημt dt +xημx ( 4 x f (t) g (t) dt) g (x) x 5 +5x ( 4 x g (t) dt) pou eðnai arketì gia na apojarrônei thn peraitèrw sunèqish tou egqeir matoc. H di spash thc sun rthshc thc opoðac zhteðtai to ìrioeðnai epibeblhmènh: ( ( x ) f (t) g (t) dt) x ημt dt x ( f (t) g (t) dt x g (t) dt) = ημt dt x 5 x g (t) dt x 5 To pr to komm ti parousi zei mða kal sumperifor : An to deð kaneðc wc F (x) G(x) sun gei, sthrizìmenoc sto er thma b., ìti f (t) g (t) dt < <f(x) g (t) dt An to dei kaneðc wc èna phlðko tou opoðou èqei kaj kon na breð to ìrio gia x + tìte epeid apoteleð aprosdiìristh morf ja breð lim x + f (t) g (t) dt = g (t) dt lim f (x) =f () > x + O deôteroc par gontac èqei akìmh kalôterh sumperifor diìti lim x + ημt dt x 5 = ( ) lim xημx 4 x + 5x 4 = ( )

17 4 To Jèma ημx 4 = lim x + 5x 3 = ( ) lim ημx 4 x + 5x 3 = lim x + 5 ημx4 x 4 x = 5 1 = Epomènwc gia na broôme to zhtoômeno ìrio ja parathr soume ìti < f (t) g (t) dt g (t) dt ημt dt x 5 <f(x) ημt dt x 5 kai ja sun goume ìti to zhtoômeno ìrio eðnai apì to krit rio thc parembol c eðte ja katal xoume sto Ðd io sumpèrasma gr fontac: lim x + f (t) g (t) dt g (t) dt ημt dt x 5 = f () = 4.5 Xan sth deôterh lôsh tou g. EÐdame ìti h kardi tou jèmatoc eðnai h sqèsh (8). Ti Majhmatik ìmwc perièqontai se autì to er thma? Gia na ta doôme kalôtera ac xeq soume proc stigm n pwc orðsjhkan oi F kai G. SÐgoura: 1. EÐnai dôo paragwgðsimec sunart seic orismènec se k poio d i sthma[α, β] (to ìti eðqame [, 1] den èpaixe k poio rìlo).. Sto α eðnai Ðsec me kai sta upìloipa shmeða tou [α, β] paðrnoun jetikèc timèc. 3. Oi par gwgoi touc F (x), G (x) eðnai suneqeðc kai paðrnoun jetikèc timèc. 4. To phlðko twn parag gwn touc eðnai mða gnhsðwc aôxousa sun rthsh. Antistrìfwc an èqoume dôo sunart seic F, G pou ikanopoioôn tic upojèseic 1-4 tìte to prìblhma mporeð na anasustajeð arkeð na onom soume: 1. g (x) =G (x). f (x) = F (x) G (x) Diìti tìte h G ja eðnai h monadik par gousa thc g pou gia x = gðnetai epomènwc G (x) = g (t) dt. Akìmh F (x) =f (x) G (x) dhlad F (x) = f (x) g (x) opìte h F ja eðnai h monadik par gousa thc fg pou sto gðnetai dhlad den eðnai llh apì thn F (x) = f (t) g (t) dt. To krðsimo sumpèrasma pou prokôptei apì ta 1,,3,4eÐnai ìti h F (x) G(x) eðnai gnhsðwc aôxousa. Epomènwc k poioc lônontac to 4o jèma metaxô llwn èqei apodeðxei kai to akìloujo: Je rhma 4.1 'Estw F :[α, β] R kai G :[α, β] R dôo paragwgðsimec sunart seic me G (α) =F (α) =oi opoðec èqoun jetikèc kai suneqeðc parag gouc. An h sun rthsh F G eðnai gnhsðwc aôxousa tìte kai h sun rthsh F G orismènh sto [α, β] eðnai epðshc gnhsðwc aôxousa. To parap nw je rhma eðnai eidik perðptwsh tou epomènou:

18 Xan sth deôterh lôsh toug. Je rhma 4. (O kanìnac monotonðac tôpou De l Hospital) 'Estw f kai g dôo suneqeðc sunart seic orismènec sto di sthma [α, β] oi opoðec eðnai paragwgðsimec sto (α, β) me g (x) gia k je x (α, β). An hsun rthsh eðnai gnhsðwc aôxousa sto (α, β) tìte kai h sun rthsh eðnai gnhsðwc aôxousa sto (α, β). f g f (x) f (α) g (x) g (α) To Je rhma 4. to opoðo dustuq c den anadeiknôetai sta biblða ApeirostikoÔ LogismoÔ mac epitrèpei prokeimènou na broôme th monotonða enìc phlðkou na k noume ìti k noume kai me ta ìria: Na katafôgoume sto phlðko twn parag gwn. MporoÔme na d soume mða autotel apìdeixh tou Jewr matoc 4.. 'Omwc eðnai protimìtero na apodeðxoume prohgoumènwc dôo endi mesa jewr mata diìti ètsi faðnontai kalôtera oi idèec thc apìdeixhc: Je rhma 4.3 (To je rhma mèshc tim c tou Cauchy) Upojètome ìti oi sunart seic f kai g eðnai orismènec sto [α, β], eðnai suneqeðc kai epiplèon eðnai paragwgðsimec sto (α, β) me g (x). Tìte up rqei ξ (α, β) ètsi ste f (β) f (α) g (β) g (α) = f (ξ) g (ξ) Apìdeixh: Kat' arq n parathroôme ìti afoô g (x) sto (α, β) ja eðnai g (β) g (α) (diaforetik me efarmog tou jewr matoc mèshc tim c katal goume se topo). JewroÔme th sun rthsh h (x) =(f (β) f (α)) g (x) (g (β) g (α)) f (x) sto [α, β] h opoða eðnai suneq c sto [α, β] kai paragwgðsimh sto (α, β). Epiplèon me lðgec pr xeic brðskoume ìti h (α) =h (β). To apodeiktèo prokôptei wc efarmog tou jewr matoc tou Rolle. o.e.d. Je rhma 4.4 (To je rhma endi meshc tim c tou Darboux) An mða sun rthsh f eðnai paragwgðsimh se èna di sthma Δ tìte h f èqei thn idiìthta thc endi meshc tim c dhlad k je tim metaxô dôo tim n thc f eðnai kai aut tim thc f. Apìdeixh: 'Estw y 1 = f (x 1 ) kai y = f (x ) dôo diaforetikèc timèc thc f. Asfal c ja eðnai x 1 x kai to di sthma me kra ta x 1,x perièqetai sto Δ. 'Estw akìmh c mða tim metaxô twn y 1 kai y. Ja deðxoume ìti f (x) =c gia k poio x metaxô twn x 1,x. Gia to skopì autì jewroôme th sun rthsh g (x) =f (x) cx hopoðaeðnai paragwgðsimh sto Δ. EÐnai g (x) =f (x) c Molonìti merikoð suggrafeðc ton onom zoun <<Kanìna monotonðac tou De l Hospital>> den katìrjwsa na br mða sôndesh tou jewr matoc me ton De l Hospital giautì protðmhsa aut thn onomasða pou epðshc qrhsimopoieðtai

19 4 To Jèma kai afoô to c eðnai metaxô twn y 1, y oi timèc g (x 1 )=f (x 1 ) c = y 1 c kai g (x )=f (x ) c = y c eðnai eterìshmec. 'Ara k poio apì ta ìria g (x) g (x 1 ) lim x x 1 x x 1 g (x) g (x ) lim (9) x x x x eðnai jetikì kai k poio arnhtikì. Ja up rqoun x 1 kai x metaxô twn x 1,x ètsi ste ta ìria (9) na eðnai omìshma, antistoðqwc, me touc arijmoôc: g (x 1 ) g (x 1) x 1 x 1 g (x ) g (x ) x x (1) 'Ara kai oi arijmoð (1) eðnai eterìshmoi dhlad : (g (x 1 ) g (x 1)) (g (x ) g (x )) (x 1 x 1)(x x ) < (11) Epeidh ta x 1,x eðnai metaxô twn x 1,x oparonomast c tou kl smatoc (11) eðnai arnhtikìc. Epomènwc o arijmht c eðnai jetikìc. Autì shmaðnei ìti ja isqôei g (x 1 ) >g(x 1) g (x ) >g(x ) (1) eðte ja isqôei: g (x 1 ) <g(x 1) g (x ) <g(x ) (13) H g wc suneq c sto kleistì di sthma me kra ta x 1,x ja èqei mègisto kai el qisto. An isqôoun oi (1) kanèna apì ta g (x 1 ),g(x ) den apoteleð mègisto thc g ra h mègisth tim thc g ja parousi zetai se k poio shmeðo ξ sto eswterikì tou diast matoc me kra ta x 1,x. Apì to je rhma tou Fermat ja isqôei g (ξ) =. An isqôoun oi (13) tìte h el qisth tim thc g ja parousi zetai se k poio shmeðo ξ sto eswterikì tou diast matoc me kra ta x 1,x. P li apì to je rhma tou Fermat ja isqôei g (ξ) =. Se k je perðptwsh ja up rqei ξ metaxô twn x 1,x ètsi ste g (ξ) =. 'Omwc g (ξ) =f (ξ) c. 'Ara h par gwgoc thc f sto ξ ja eðnai Ðsh me c. o.e.d. H Apìdeixh tou Jewr matoc 4. Gia na apodeðxoume ìti h sun rthsh f (x) f (α) g (x) g (α) eðnai gnhsðwc aôxousa arkeð na apodeðxoume ìti h par gwgoc thc eðnai jetik. 'Eqoume: kai epomènwc ( ) f (x) f (α) = f (x)(g(x) g (α)) g (x) f (x) f (α) g (x) g (α) (g (x) g (α)) ( ) ( ) f (x) f (α) g (x)(g(x) g (α)) f (x) g = (x) f(x) f(α) (g(x) g(α)) g (x) g (α) (g (x) g (α))

20 7 Anaforèc ParathroÔme ìti apì to je rhma mèshc tim c tou Cauchy ja up rqei ξ (α, x) ste f (x) f (α) g (x) g (α) = f (ξ) g (ξ) < f (x) g (x) Sthn par stash {( }} ){ T {}}{ f (x) g g (x)(g (x) g (α)) (x) f(x) f(α) (g(x) g(α)) (g (x) g (α)) to kl smas eðnai jetikì. H apìdeixh ja èqei telei sei an eðnai kai to T jetikì. Apì to je rhma mèshc tim c eðnai g (x) g (α) =(x α) g (m) kai epomènwc afoô to x α eðnai jetikì arkeð to g (x) g (m) na eðnai jetikì. An tan arnhtikì tìte ta g (x),g (m) ja san eterìshma kai epomènwc apì to je rhma tou Darboux h par gwgoc ja èprepe na mhdenðzetai pr gma adônato. 'Ara telik kai to T eðnai jetikì. o.e.d. S Anaforèc [1] G.D. Anderson, M.K. Vamanarthy kai M. Vuorinen. Inequalities for Quasiconformal Mappings in Space. Pacific Journal of Mathematics, 16,1-18,1993. [] Michael J. Cloud kai Byron C. Drachman. Inequalities. With applications to Engineering. Springer, [3] Richard R. Goldberg. Methods of Real Analysis. Blaisdel, [4] William R. Parzynski kai Philip W. Zipse. Introduction to Mathematical Analysis. Mc Graw-Hill, 198. [5] Dan Pedoe. Geometry. A Comprehencive Course. Dover, 1988 (197). [6] Iosif Pinelis. L Hospital-Type Rules for Monotonicity: Include Them into Calculus Texts! ipinelis/, q.q. [7] Iosif Pinelis. L Hospital Rules for Monotonicity and the Wilker-Anglesio Inequality. American Mathematical Monthly, 111, 95-99, 4.

21 ΑΡΧΗ 1ΗΣ ΣΕΛΙ ΑΣ ΑΠΟΛΥΤΗΡΙΕΣ ΕΞΕΤΑΣΕΙΣ Γ ΤΑΞΗΣ ΗΜΕΡΗΣΙΟΥ ΓΕΝΙΚΟΥ ΛΥΚΕΙΟΥ ΠΕΜΠΤΗ 4 ΜΑΪΟΥ 7 ΕΞΕΤΑΖΟΜΕΝΟ ΜΑΘΗΜΑ: ΜΑΘΗΜΑΤΙΚΑ ΘΕΤΙΚΗΣ ΚΑΙ ΤΕΧΝΟΛΟΓΙΚΗΣ ΚΑΤΕΥΘΥΝΣΗΣ ΣΥΝΟΛΟ ΣΕΛΙ ΩΝ: ΠΕΝΤΕ (5) ΘΕΜΑ 1 o A.1 Αν z 1, z είναι μιγαδικοί αριθμοί, να αποδειχθεί ότι: z 1 z = z1 z. Μονάδες 8 Α. Πότε δύο συναρτήσεις f, g λέγονται ίσες; Μονάδες 4 Α.3 Πότε η ευθεία y = λέγεται οριζόντια ασύμπτωτη της γραφικής παράστασης της f στο + ; Μονάδες 3 B. Να χαρακτηρίσετε τις προτάσεις που ακολουθούν, γράφοντας στο τετράδιό σας δίπλα στο γράμμα που αντιστοιχεί σε κάθε πρόταση, τη λέξη Σωστό, αν η πρόταση είναι σωστή, ή Λάθος, αν η πρόταση είναι λανθασμένη. α. Αν f συνάρτηση συνεχής στο διάστημα [α,β] και για κάθε x [ α, β] ισχύει f(x) τότε f(x) dx >. α β Μονάδες β. Έστω f μια συνάρτηση συνεχής σε ένα διάστημα και παραγωγίσιμη σε κάθε εσωτερικό σημείο x του. Αν η συνάρτηση f είναι γνησίως αύξουσα στο τότε f (x) > σε κάθε εσωτερικό σημείο x του. ΤΕΛΟΣ 1ΗΣ ΣΕΛΙ ΑΣ Μονάδες

22 ΑΡΧΗ ΗΣ ΣΕΛΙ ΑΣ ΘΕΜΑ ο γ. Αν η συνάρτηση f είναι συνεχής στο x και η συνάρτηση g είναι συνεχής στο x, τότε η σύνθεσή τους gof είναι συνεχής στο x. Μονάδες δ. Αν f είναι μια συνεχής συνάρτηση σε ένα διάστημα και α είναι ένα σημείο του, τότε g(x) ( f(t) dt) = f ( g(x) ) g (x) α με την προϋπόθεση ότι τα χρησιμοποιούμενα σύμβολα έχουν νόημα. Μονάδες ε. Αν α > 1 τότε lim α =. x ίνεται ο μιγαδικός αριθμός x + αi z = με α IR. α + i Μονάδες α. Να αποδειχθεί ότι η εικόνα του μιγαδικού z ανήκει στον κύκλο με κέντρο Ο(,) και ακτίνα ρ =1. Μονάδες 9 β. Έστω z 1, z οι μιγαδικοί που προκύπτουν από τον τύπο z = α + + αi i για α = και α = αντίστοιχα. i. Να βρεθεί η απόσταση των εικόνων των μιγαδικών αριθμών z 1 και z. Μονάδες 8 ΤΕΛΟΣ ΗΣ ΣΕΛΙ ΑΣ

23 ΑΡΧΗ 3ΗΣ ΣΕΛΙ ΑΣ ii. Να αποδειχθεί ότι ισχύει: ν (z1) = ( z ) ν για κάθε φυσικό αριθμό ν. Μονάδες 8 ΘΕΜΑ 3 ο ίνεται η συνάρτηση: f(x) = x 3 3x ημ θ π όπου θ IR μια σταθερά με θ κπ +, κ Z. α. Να αποδειχθεί ότι η f παρουσιάζει ένα τοπικό μέγιστο, ένα τοπικό ελάχιστο και ένα σημείο καμπής. Μονάδες 7 β. Να αποδειχθεί ότι η εξίσωση f(x) = έχει ακριβώς τρεις πραγματικές ρίζες. Μονάδες 8 γ. Αν x 1, x είναι οι θέσεις των τοπικών ακροτάτων και x 3 η θέση του σημείου καμπής της f, να αποδειχθεί ότι τα σημεία Α(x 1, f(x 1 )), B(x, f(x )) και Γ(x 3, f(x 3 )) βρίσκονται στην ευθεία y = x ημ θ. Μονάδες 3 δ. Να υπολογισθεί το εμβαδόν του χωρίου που περικλείεται από τη γραφική παράσταση της συνάρτησης f και την ευθεία y = x ημ θ. Μονάδες 7 ΤΕΛΟΣ 3ΗΣ ΣΕΛΙ ΑΣ

24 ΑΡΧΗ 4ΗΣ ΣΕΛΙ ΑΣ ΘΕΜΑ 4 ο Έστω f μια συνεχής και γνησίως αύξουσα συνάρτηση στο διάστημα [, 1] για την οποία ισχύει f() >. ίνεται επίσης συνάρτηση g συνεχής στο διάστημα [, 1] για την οποία ισχύει g(x) > για κάθε x [, 1]. Ορίζουμε τις συναρτήσεις: F(x) = f(t) g(t) dt, x [, 1], G(x) = g(t) dt, x [, 1 ]. α. Να δειχθεί ότι F(x) > για κάθε x στο διάστημα (, 1]. Μονάδες 8 β. Nα αποδειχθεί ότι: f(x) G(x) > F(x) για κάθε x στο διάστημα (, 1]. γ. Nα αποδειχθεί ότι ισχύει: F(x) F(1) G(x) G(1) για κάθε x στο διάστημα (, 1]. Μονάδες 6 Μονάδες 4 δ. Να βρεθεί το όριο: lim x + x f(t) g(t) dt x x g(t) dt x ημt 5 dt. Μονάδες 7 ΤΕΛΟΣ 4ΗΣ ΣΕΛΙ ΑΣ