ΣΗΜΕΙΩΣΕΙΣ-ΜΑΘΗΜΑΤΙΚΑ 1 ΠΛΗΡΟΦΟΡΙΚΗ ΑΣΟΕΕ 1 ΕΞΑΜΗΝΟ

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ΜΑΘΗΜΑΤΙΚΑ 1 ΠΛΗΡΟΦΟΡΙΚΗ ΑΣΟΕΕ 1 ΕΞΑΜΗΝΟ

Transcript of ΣΗΜΕΙΩΣΕΙΣ-ΜΑΘΗΜΑΤΙΚΑ 1 ΠΛΗΡΟΦΟΡΙΚΗ ΑΣΟΕΕ 1 ΕΞΑΜΗΝΟ

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OIKONOMIKO PANEPISTHMIO AJHNWN, TMHMA PLHROFORIKHS

MAJHMATIKA I

Shmei¸seic

StaÔroc Toump c

OPA, 2013

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OdhgÐec Qr shc

To parìn DEN eÐnai didaktikì biblÐo. EÐnai oi shmei¸seic tou maj matoc {Ma-jhmatik� I}, ìpwc to did�skw sto pr¸to ex�mhno tou Tm matoc Plhroforik c touOikonomikoÔ PanepisthmÐou Ajhn¸n. Stic shmei¸seic èqoun enswmatwjeÐ orismènecaplèc ènnoiec pou den up�rqei lìgoc na anafèrontai sto m�jhma, all� paratÐjentai e-d¸ gia lìgouc plhrìthtac (p.q. gnwstèc trigwnometrikèc idiìthtec.) Up�rqoun epÐshcelafr¸c perissìtera paradeÐgmata apì ìsa prolabaÐnw na kalÔyw stic dialèxeic.

Eutuq¸c, gia to m�jhma up�rqoun diajèsima poll� exairetik� biblÐa grammèna staEllhnik�,   se kalèc metafr�seic apì ta Agglik�. Merik� apì aut� anafèrontai sthnbibliografÐa. SÔmfwna me ton nìmo, stouc foithtèc prosfèretai dwre�n èna biblÐo,me dunatìthta epilog c an�mesa se dÔo. Sto sugkekrimèno m�jhma ta biblÐa aut�eÐnai tou Spivak kai tou Thomas, kai ta dÔo diajèsima se kalèc metafr�seic apì ticPanepisthmiakèc Ekdìseic Kr thc. Den up�rqei upokat�stato thc eic b�joc melèthcenìc kaloÔ biblÐou gia thn katanìhsh tou antikeimènou. Den sunist¸ stouc foithtèc,eidik� se autoÔc pou den parakoloÔjhsan tic dialèxeic, na qrhsimopoi soun to parìnwc biblÐo, kaj¸c den èqei stìqo na upokatast sei ta biblÐa pou moir�zontai. O skopìcaut¸n twn shmei¸sewn eÐnai

1. na dieukolÔnoun thn parakoloÔjhsh twn foitht¸n, pou den qrei�zetai na anti-gr�foun ì,ti gr�fetai ston pÐnaka, sun jwc upì dusmeneÐc sunj kec,

2. na dieukolÔnoun thn melèth thc Ôlhc pou did�qthke se sunduasmì me to biblÐo,

3. na bohjoÔn touc foithtèc pou den parakoloujoÔn k�poiec dialèxeic (  kai ìlec)na meÐnoun se epaf  me to m�jhma,

4. na apojarrÔnoun touc foithtèc apì to na èrqontai stic dialèxeic apl¸c kai mìnogia na {p�roun tic shmei¸seic}.

Epiplèon, h suggraf  twn shmei¸sewn me bo jhse na katal�bw (kat� to dunatìn)pwc prèpei na did�sketai o Logismìc se prwtoeteÐc.

To antikeÐmeno tou maj matoc eÐnai o Oloklhrwtikìc Logismìc miac metablht c.H diamìrfwsh thc Ôlhc èqei gÐnei lamb�nontac up' ìyin tic ufist�menec gn¸seic, to

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gegonìc ìti to m�jhma gÐnetai se tm ma Plhroforik c, kai to ìti den up�rqei sto tm ma�llo m�jhma LogismoÔ, pèran apì k�poia stoiqeÐa LogismoÔ poll¸n metablht¸n wcuposÔnolo tou maj matoc {Majhmatik� II} tou deutèrou exam nou. 'O,ti up�rqei ed¸pisteÔw mporeÐ na kalufjeÐ se 13 ebdom�dec didaskalÐac. Prosp�jhsa na dom swètsi thn Ôlh ¸ste oi foithtèc pou parakoloujoÔn to m�jhma na mporoÔn na diab�zountautìqrona opoiod pote apì ta dÔo biblÐa pou touc parèqontai (Spivak kai Thomas)qwrÐc poll� mproc - pÐsw. AkoloÔjhsa epÐshc ìso mporoÔsa touc sumbolismoÔcaut¸n twn biblÐwn (kai ìpou up rqe apìklish metaxÔ touc, touc sumbolismoÔc touSpivak).

Oi �mesoi stìqoi tou maj matoc eÐnai:

1. Oi foithtèc na embajÔnoun sthn Ôlh pou  dh xèroun, dhlad  tic basikèc ènnoiectwn parag¸gwn kai twn oloklhrwm�twn. Ex' ou kai dÐnetai èmfash ston orismìtou orÐou, sthn ènnoia tou supremum, ton austhrì orismì (kat� Darboux) touoloklhr¸matoc, k.o.k.

2. Oi foithtèc na m�joun orismèna nèa komm�tia jewrÐac (p.q. sunèqeia Lipschitz)kai nèec efarmogèc twn  dh gnwst¸n touc ennoi¸n (p.q. upologismoÐ di�forwnìgkwn).

3. Oi foithtèc na ektejoÔn se perioqèc Ôlhc ìpwc ta polu¸numa Taylor kai di�forecarijmhtikèc mejìdouc, pou argìtera ja antimetwpÐsoun diexodikìtera, sta plaÐsia�llwn majhm�twn.

Pèran apì touc �mesouc didaktikoÔc stìqouc tou maj matoc, sthn diamìrfwsh kaiparousÐash thc Ôlhc kai twn ask sewn sunupologÐsthkan kai oi akìloujoi ap¸teroistìqoi:

1. H Ôlh na eÐnai sqetik� meg�lh, ¸ste oi foithtèc na {m�joun na majaÐnoun gr -gora}.

2. H Ôlh na parousi�zetai kat� to dunatìn austhr�, ¸ste na enisqujeÐ h ikanìthtatwn foitht¸n gia logikèc domhmènec skèyeic.

3. Oi ask seic na mporoÔn na lujoÔn apì ìsouc èqoun katal�bei thn Ôlh kai ìqiapì ìsouc èqoun apomnhmoneÔsei k�poiec mhqanistikèc mejìdouc, ¸ste na dÐne-tai èmfash sthn katanìhsh twn 5-6 apl¸n ennoi¸n tou LogismoÔ, kai ìqi sthnapomnhmìneush.

H entÔpwsh pou èqw apokomÐsei apì arket� qrìnia didaskalÐac se prwtoeteÐc, eÐnaiìti h didaskalÐa twn majhmatik¸n sto LÔkeio, kai eidik� to sÔsthma exet�sewn, èqounperÐpou antÐjetouc stìqouc.

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To parìn keÐmeno eÐnai upì diark  exèlixh. Ja ektim sw idiaitèrwc opoiad poteprìtash gia th beltÐws  tou, kai tuqìn parathr seic sqetik� me orjografik� kaitupografik� sf�lmata, l�jh stic ask seic, kai k�je fÔsewc �llo prìblhma. JaeÐnai qar� mou epÐshc an to qrhsimopoi soun autoÔsio   me tropopoi seic kai �lloidid�skontec.

StaÔroc Toump c, [email protected]

c© 2010-2013 StaÔroc Toump c

Epitrèpetai h qr sh mèrouc   ìlou tou parìntockaj¸c kai h anaparagwg  tou me opoiond pote trìpo

gia opoiad pote mh kerdoskopik  kai ìqi emporik 

didaktik    ereunhtik  drasthriìthta

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Perieqìmena

1 ArijmoÐ 11.1 Axi¸mata PedÐou . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Axi¸mata Di�taxhc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.3 AxÐwma Plhrìthtac . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.4 Sunèpeiec tou Axi¸matoc thc Plhrìthtac . . . . . . . . . . . . . . . . . . . . . . . . 141.5 Diast mata . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161.6 Sunart seic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171.7 Trigwnometrikèc Sunart seic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2 'Orio 252.1 Basik  Idèa OrÐou . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252.2 Orismìc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302.3 Idiìthtec OrÐou . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 352.4 'Oria Trigwnometrik¸n Sunart sewn . . . . . . . . . . . . . . . . . . . . . . . . . . 412.5 'Oria sto 'Apeiro, ApeirÐzonta 'Oria . . . . . . . . . . . . . . . . . . . . . . . . . . . 442.6 'Orio AkoloujÐac . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

3 Sunèqeia 553.1 Orismìc kai Basikèc Idiìthtec Sunèqeiac . . . . . . . . . . . . . . . . . . . . . . . . 553.2 Sunèpeiec Sunèqeiac se Kleistì Fragmèno Di�sthma . . . . . . . . . . . . . . . . . . 593.3 Mèjodoc Diqotìmhshc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 683.4 AntÐstrofh SuneqoÔc Sun�rthshc . . . . . . . . . . . . . . . . . . . . . . . . . . . . 703.5 Sunèqeia Lipschitz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

4 Par�gwgoc 774.1 Orismìc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 774.2 Basikèc Idiìthtec Parag¸gwn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 844.3 Par�gwgoc AntÐstrofhc Sun�rthshc . . . . . . . . . . . . . . . . . . . . . . . . . . 894.4 Je¸rhma Mèshc Tim c . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 944.5 Par�gousec . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

5 Efarmogèc Parag¸gou 1035.1 ProseggÐseic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1035.2 H Mèjodoc tou NeÔtwna . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1055.3 MonotonÐa kai Akrìtata Sunart sewn . . . . . . . . . . . . . . . . . . . . . . . . . 1095.4 Kurtèc Sunart seic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1125.5 Kanìnac tou L’Hopital . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

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6 Olokl rwma 1196.1 Orismìc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1196.2 Krit ria Oloklhrwsimìthtac . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1256.3 Idiìthtec Oloklhr¸matoc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1326.4 Olokl rwma Riemann . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

7 Olokl rwsh 1437.1 Jemeli¸dh Jewr mata LogismoÔ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1437.2 Logarijmik  kai Ekjetik  Sun�rthsh . . . . . . . . . . . . . . . . . . . . . . . . . . 1517.3 Ask seic me thn Logarijmik  kai thn Ekjetik  Sun�rthsh . . . . . . . . . . . . . . 1627.4 Kataqrhstik� Oloklhr¸mata . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166

8 Efarmogèc Oloklhrwm�twn 1738.1 Upologismìc EmbadoÔ se Kartesianèc Suntetagmènec . . . . . . . . . . . . . . . . . 1738.2 Upologismìc EmbadoÔ se Polikèc Suntetagmènec . . . . . . . . . . . . . . . . . . . . 1788.3 Upologismìc 'Ogkou me th Mèjodo twn DÐskwn . . . . . . . . . . . . . . . . . . . . 1878.4 Upologismìc 'Ogkou me th Mèjodoc twn KelÔfwn . . . . . . . . . . . . . . . . . . . 1918.5 Upologismìc M kouc KampÔlhc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196

9 Diaforikèc Exis¸seic 2059.1 Diaforikèc Exis¸seic Pr¸thc T�xewc . . . . . . . . . . . . . . . . . . . . . . . . . . 2059.2 Grammikèc Diaforikèc Exis¸seic Pr¸thc T�xewc . . . . . . . . . . . . . . . . . . . . 2089.3 DiaqwrÐsimec Diaforikèc Exis¸seic Pr¸thc T�xewc . . . . . . . . . . . . . . . . . . 2129.4 H Mèjodoc tou Euler . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217

10 Polu¸numo Taylor 22510.1 Orismìc kai Basikèc Idiìthtec . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22510.2 Upìloipo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231

11 Seirèc 23711.1 Seirèc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23711.2 Seirèc mh Arnhtik¸n 'Orwn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24111.3 ParadeÐgmata . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248

12 DianÔsmata 25112.1 Rn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25112.2 R2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26212.3 R3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 270

13 Kwnikèc Tomèc 28113.1 Kwnikèc Tomèc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28113.2 Allag  Suntetagmènwn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28413.3 Parabol  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28913.4 'Elleiyh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29213.5 Uperbol  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29613.6 Ax2 + Cy2 +Dx+ Ey + F = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30013.7 Ax2 +Bxy + Cy2 +Dx+ Ey + F = 0 . . . . . . . . . . . . . . . . . . . . . . . . . 304

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14 Par�rthma 30914.1 BibliografÐa . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30914.2 SumbolismoÐ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31014.3 Lexikì . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311

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Kef�laio 1

ArijmoÐ

1.1 Axi¸mata PedÐou

Orismìc 1.1. (Axi¸mata PedÐou): 'Estw R to sÔnolo twn pragmatik¸n arijm¸nkai èstw x, y, z opoioid pote pragmatikoÐ arijmoÐ. ApaitoÔme na isqÔoun oi akìloujecidiìthtec, pou kaloÔntai axi¸mata pedÐou:

1. (Antimetajetik  idiìthta) x+ y = y + x, xy = yx.

2. (Prosetairistik  idiìthta) x+ (y + z) = (x+ y) + z, x(yz) = (xy)z.

3. (Epimeristik  idiìthta) x(y + z) = xy + xz.

4. ('Uparxh oudèterwn stoiqeÐwn) Up�rqoun dÔo diaforetikoÐ metaxÔ touc pragmati-koÐ arijmoÐ, oi 0 kai 1, tètoioi ¸ste na èqoume 0+x = x kai 1·x = x gia k�je x ∈ R.O 0 kaleÐtai to oudètero stoiqeÐo thc prìsjeshc, kai o 1 to oudètero stoiqeÐo toupollaplasiasmoÔ.

5. ('Uparxh antÐjetwn) Gia k�je pragmatikì arijmì x, up�rqei pragmatikìc arijmìcy tètoioc ¸ste x+ y = 0. O y sumbolÐzetai −x kai kaleÐtai antÐjetoc tou x.

6. ('Uparxh antÐstrofwn) Gia k�je pragmatikì arijmì x 6= 0, up�rqei pragmatikìcarijmìc y tètoioc ¸ste xy = 1. O y sumbolÐzetai x−1 kai kaleÐtai antÐstrofoctou x.

Parat rhsh: Opoiod pote sÔnolo S efodiasmèno me dÔo pr�xeic ikanopoieÐ ta�nw axi¸mata kaleÐtai pedÐo   s¸ma. Up�rqoun poll� pedÐa: oi rhtoÐ, oi pragmatikoÐ,oi migadikoÐ, kai �lla, akìma pio asun jista. DeÐte to akìloujo par�deigma.

Par�deigma 1.1. (GF(5)) 'Estw to sÔnolo GF(5) = {0, 1, 2, 3, 4}, gia to opoÐoèqoume orÐsei dÔo pr�xeic wc ex c:

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2 KEF�ALAIO 1. ARIJMO�I

1. Thn prìsjesh modulo 5, pou th sumbolÐzoume me ⊕, kai sÔmfwna me thn opoÐato x⊕ y isoÔtai me to upìloipo thc diaÐreshc tou x+ y me to 5. IsodÔnama,

x⊕ y = x+ y − 5k,

ìpou k eÐnai o monadikìc akèraioc gia ton opoÐo 0 ≤ x+y−5k < 5, kai profan¸ck ∈ {0, 1}.

2. Ton pollaplasiasmì modulo 5, pou ton sumbolÐzoume me �, kai sÔmfwna me tonopoÐo to x� y isoÔtai me to upìloipo thc diaÐreshc tou xy me to 5. IsodÔnama,

x� y = xy − 5k,

ìpou k eÐnai o monadikìc akèraioc gia ton opoÐo 0 ≤ xy − 5k < 5, kai profan¸ck ∈ {0, 1, 2, 3}.

To sÔnolo GF(5), efodiasmèno me tic �nw pr�xeic, ikanopoieÐ ta axi¸mata pedÐou.Pr�gmati, h prosetairistik  idiìthta thc prìsjeshc prokÔptei parathr¸ntac ìti:

x⊕ (y ⊕ z) = x+ (y + z − 5k1)− 5k2 = x+ y + z − 5(k1 + k2),

(x⊕ y)⊕ z = (x+ y − 5k3) + z − 5k4 = x+ y + z − 5(k3 + k4).

Sta �nw, oi akèraioi k1, k2, k3, k4 eÐnai tètoioi ¸ste to antÐstoiqo �jroisma na eÐnaip�ntote an�mesa sto 0 kai to 4. Parathr ste t¸ra ìti up�rqei mìno ènac k tètoioc¸ste to x + y + z − 5k na eÐnai an�mesa sto 0 kai to 4. 'Ara k1 + k2 = k3 + k4, kaitelik�

x⊕ (y ⊕ z) = (x⊕ y)⊕ z.H prosetairistik  idiìthta gia ton pollaplasiasmì, h antimetajetikìthta tou polla-plasiasmoÔ kai thc prìsjeshc kai h epimeristik  idiìthta prokÔptoun an�loga. (Mpo-reÐte na gr�yete tic apodeÐxeic?)

Sqetik� me thn Ôparxh oudetèrwn, eÐnai profanèc ìti to 0 eÐnai kai p�li to oudèterostoiqeÐo thc prìsjeshc kai to 1 eÐnai to oudètero stoiqeÐo tou pollaplasiasmoÔ.

Sqetik� me thn Ôparxh antÐjetwn, eÐnai profanèc ìti

0 + 0 = 0, 1 + 4 = 0, 2 + 3 = 0,

�ra up�rqei antÐjetoc gia k�je stoiqeÐo sto GF(5).Sqetik� me touc antÐstrofouc, me lÐgec dokimèc brÐskoume ìti

1� 1 = 1, 2� 3 = 1, 4� 4 = 1,

�ra ìloi oi arijmoÐ ektìc tou 0 èqoun antÐstrofo.Pr�gmati, loipìn, to GF(5) eÐnai pedÐo. An, antÐ na qrhsimopoioÔsame to 5, qrh-

simopoioÔsame to 6, ja  tan to prokÔpton sÔnolo pedÐo?

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1.1. AXI�WMATA PED�IOU 3

Orismìc 1.2. (Akèraiec dun�meic)

1. Gia k�je a ∈ R, n ∈ N, orÐzoume an , a · a · · · · · a︸ ︷︷ ︸n φορές

.

2. Gia k�je a ∈ R, a 6= 0, n ∈ N orÐzoume a−n , (a−1)n.

3. Gia k�je a ∈ R, orÐzoume a0 = 1.

Parat rhsh: Sqedìn ìlec oi gnwstèc idiìthtec twn pragmatik¸n arijm¸n pouemplèkoun mìno isìthtec mporoÔn na apodeiqjoÔn b�sei twn axiwm�twn pedÐou. DeÐteto akìloujo par�deigma gia merikèc apì autèc

Par�deigma 1.2. (Idiìthtec pou aporrèoun apì ta axi¸mata pedÐou) Qrhsimopoi¸n-tac mìno ta axi¸mata pedÐou, mporoÔme na apodeÐxoume ìti gia k�je a, b, c, d ∈ RisqÔoun ta akìlouja:

1. (Kanìnac thc Apaloif c) An a + b = a + c, tìte b = c. 'Ara, to 0 eÐnai omonadikìc arijmìc me thn idiìthta 0 + x = x gia ìla ta x, kai epÐshc o antÐjetoc(−a) enìc arijmoÔ a eÐnai monadikìc.

2. (Orismìc AfaÐreshc) Up�rqei akrib¸c ènac x tètoioc ¸ste a+x = b. O x isoÔtaime x = b+(−a), kai sumbolÐzetai me b−a. Parathr ste pwc 0−a = 0+(−a) =−a.

3. −(−a) = a.

4. 0a = 0.

5. (−a)b = −(ab), (−a)(−b) = ab.

6. a(b− c) = ab− ac.7. An ab = ac kai a 6= 0, tìte b = c. 'Ara to 1 eÐnai o monadikìc arijmìc me thn

idiìthta 1 ·x = x gia ìla ta x, kai epÐshc o antÐstrofoc a−1 enìc arijmoÔ a eÐnaimonadikìc.

8. (Orismìc DiaÐreshc) An a 6= 0, up�rqei akrib¸c ènac x tètoioc ¸ste ax = b. Oarijmìc autìc kaleÐtai to phlÐko twn b, a, sumbolÐzetai me b/a   b

a , kai isoÔtai meb/a = b · a−1. Parathr ste pwc 1/a = 1 · a−1 = a−1.

9. (a−1)−1 = a.

10. An ab = 0, tìte eÐte a = 0 eÐte b = 0, eÐte kai ta dÔo.

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4 KEF�ALAIO 1. ARIJMO�I

11. (ab)−1 = a−1b−1, efìson a, b 6= 0.

12. (a/b)(c/d) = (ac)/(bd), efìson b, d 6= 0.

13. a/b+ c/d = (ad+ bc)/(bd), efìson b, d 6= 0.

14. −(a+ b) = −a− b.

15. (a− b) + (b− c) = a− c.

16. a−n = (an)−1.

Ja deÐxoume mìno tic pr¸tec tèsseric idiìthtec. Oi apodeÐxeic twn upoloÐpwn pa-raleÐpontai.

1. 'Estw y to antÐjeto tou a: y+a = 0. To y up�rqei lìgw tou sqetikoÔ axi¸matocpedÐou. 'Eqoume:

a+ b = a+ c

⇒ y + (a+ b) = y + (a+ c)

⇒ (y + a) + b = (y + a) + c (Prosetairistik  idiìthta)

⇒ 0 + b = 0 + c (Upìjesh)

⇒ b = c. (Oudètero stoiqeÐo)

'Estw t¸ra pwc up�rqoun �nw tou enìc oudètera stoiqeÐa. 'Estw gia par�deigmapwc ta b, c eÐnai oudètera stoiqeÐa. Tìte, me efarmog  tou kanìna thc apaloif cgia ta b, c kai èna opoiod pote a, prokÔptei pwc b = c.

Parìmoia, prokÔptei pwc up�rqei mìno ènac antÐjetoc gia k�je arijmì a. Pr�g-mati, èstw pwc gia k�poion a up�rqoun dÔo, oi x, y. Ja èqoume x + a = 0 kaiy + a = 0, �ra x+ a = y + a, kai apì ton kanìna thc apaloif c, x = y.

2. 'Estw −a o antÐjetoc tou a, dhlad  o arijmìc y gia ton opoÐo y + a = 0.ParathroÔme pwc gia ton arijmì x = b+ (−a) isqÔei:

a+ x = a+ (b+ (−a)) = a+ ((−a) + b) = (a+ (−a)) + b = 0 + b = b.

'Ara br kame ènan x gia ton opoÐo a + x = b. 'Estw pwc up rqe kai deÔterocx′ 6= x me thn Ðdia idiìthta. Tìte a + x′ = b kai apì ton kanìna thc apaloif cx = x′, dhlad  �topo.

3. ProkÔptei �mesa apì ton orismì tou antÐjetou. Pr�gmati, o a ikanopoieÐ thnsunj kh gia na eÐnai antÐjetoc tou (−a), �ra eÐnai o −(−a).

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1.1. AXI�WMATA PED�IOU 5

4. Parathr ste pwc

0 + 0a = 0a = (0 + 0)a = 0a+ 0a⇒ 0a = 0.

Stic pr¸tec dÔo isìthtec efarmìsame to gegonìc ìti to 0 eÐnai to oudèterostoiqeÐo thc prìsjeshc. Sthn trÐth isìthta qrhsimopoi same thn epimeristik idiìthta, kai sthn sunepagwg  ton kanìna thc apaloif c.

Parathr ste ìti oi �nw idiìthtec isqÔoun gia k�je pedÐo, afoÔ prokÔptoun apo-kleistik� me qr sh twn axiwm�twn pedÐou, �ra, gia par�deigma, kai gia to pedÐo touParadeÐgmatoc 1.1. MporeÐte na apodeÐxete tic upìloipec idiìthtec? Mhn k�nete tol�joc na qrhsimopoi sete stic apodeÐxeic sac idiìthtec pou den èqoun apodeiqjeÐ  dh!

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6 KEF�ALAIO 1. ARIJMO�I

1.2 Axi¸mata Di�taxhc

Orismìc 1.3. (UposÔnola) 'Estw sÔnola A,B pragmatik¸n arijm¸n. Gr�foumeA ⊆ B kai lème pwc to A eÐnai uposÔnolo tou B an ∀x ∈ A ja èqoume kai x ∈ B.Gr�foume A ⊂ B kai lème pwc to A eÐnai gn sio uposÔnolo tou B an eÐnai uposÔnolotou B kai epiplèon ∃x ∈ B : x 6∈ A.

Orismìc 1.4. (Axi¸mata Di�taxhc): Up�rqei uposÔnolo R+ ⊂ R, pou kaleÐtai tosÔnolo twn jetik¸n pragmatik¸n arijm¸n, me tic akìloujec idiìthtec:

1. An x, y ∈ R+, tìte x+ y, xy ∈ R+.

2. An x 6= 0, tìte eÐte x ∈ R+, eÐte −x ∈ R+, all� ìqi kai ta dÔo.

3. 0 6∈ R+.

Orismìc 1.5. (ArnhtikoÐ, anisìthtec kai apìluth tim ) OrÐzoume ta akìlouja:

1. 'Enac arijmìc x kaleÐtai arnhtikìc an o −x eÐnai jetikìc.

2. 'Estw R− to sÔnolo twn arnhtik¸n arijm¸n.

3. x < y shmaÐnei ìti o y − x eÐnai jetikìc.

4. x > y shmaÐnei ìti y < x.

5. x ≤ y shmaÐnei ìti x < y   x = y.

6. x ≥ y shmaÐnei ìti y ≤ x.

7. H apìluth tim  |a| enìc pragmatikoÔ a orÐzetai wc

|a| ,{a, a ≥ 0,

−a, a ≤ 0.

Parathr seic

1. Apì ta �nw, prokÔptei pwc

a eÐnai jetikìc⇔ a > 0, a eÐnai arnhtikìc⇔ a < 0,

2. Sqedìn ìlec oi gnwstèc idiìthtec twn pragmatik¸n arijm¸n pou emplèkoun ani-sìthtec prokÔptoun apì ta axi¸mata pedÐou kai di�taxhc. (DeÐte to parak�twpar�deigma.)

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1.2. AXI�WMATA DI�ATAXHS 7

3. Ta axi¸mata pedÐou kai di�taxhc den arkoÔn gia na prosdiorÐzoun pl rwc toucpragmatikoÔc. (Parathr ste ìti ikanopoioÔntai kai apì touc rhtoÔc.)

4. Gia na prosdiorÐzoume pl rwc tic idiìthtec twn pragmatik¸n, kai na touc diaqw-rÐzoume apì touc rhtoÔc, qreiazìmaste to AxÐwma thc Plhrìthtac, pou ja doÔmese lÐgo.

Par�deigma 1.3. (Idiìthtec pou aporrèoun apì ta Axiwm�twn Di�taxhc) Gia k�jea, b, c, d ∈ R kai n ∈ N, isqÔoun ta akìlouja:

1. (Kanìnac thc Triqotìmhshc) IsqÔei akrib¸c èna apì ta akìlouja: (i) a < b, (ii)a > b, (iii) a = b.

2. An a < b kai b < c, tìte a < c.

3. a < b, c > 0⇒ ac < bc.

4. a 6= 0⇒ a2 > 0.

5. a < b, c ≤ d⇒ a+ c < b+ d.

6. a < b⇒ −b < −a.

7. a < b, c > d⇒ a− c < b− d

8. a < b, c < 0⇒ ac > bc.

9. a > 1⇒ a2 > a.

10. 0 < a < 1⇒ a2 < a.

11. 0 ≤ a < b, 0 ≤ c < d⇒ ac < bd.

12. 0 ≤ a < b⇒ a2 < b2.

13. a, b ≥ 0, a2 < b2 ⇒ a < b.

14. 0 ≤ a < b⇒ an < bn.

15. a < b, n = 2k + 1⇒ an < bn.

16. ||a| − |b|| ≤ |a+ b| ≤ |a|+ |b| (Trigwnik  anisìthta).

Ja apodeÐxoume mìno tic pr¸tec treic idiìthtec. Oi apodeÐxeic twn upoloÐpwn pa-raleÐpontai.

1. 'Estw o arijmìc x = a− b. DiakrÐnoume dÔo peript¸seic, x = 0 kai x 6= 0.

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8 KEF�ALAIO 1. ARIJMO�I

(aþ) 'Estw pwc èqoume thn pr¸th perÐptwsh: x = a − b = 0 ⇒ a = b. 'Estwpwc epiplèon a > b. Ex orismoÔ thc anisìthtac, èqoume ìti o x = a − beÐnai jetikìc, pou eÐnai �topo giatÐ o x = 0 den an kei stouc jetikoÔc, apìto trÐto axÐwma thc di�taxhc. 'Estw pwc a < b. Tìte o b − a eÐnai jetikìc,kai èqw p�li �topo, giatÐ eÔkola prokÔptei ìti a − b = 0 ⇒ b − a = 0 apìta axi¸mata tou pedÐou.

(bþ) 'Estw t¸ra h deÔterh perÐptwsh: x 6= 0. Tìte apokleÐetai a = b, giatÐ ft�nwse �topo. Epiplèon, apì to deÔtero axÐwma thc di�taxhc, eÐte ja èqw x ∈ R+,dhlad  a > b, eÐte −x ∈ R+, dhlad  −(a − b) > 0 ⇔ b − a > 0 ⇔ b > a,all� ìqi kai ta dÔo.

2. Ex orismoÔ, a < b shmaÐnei ìti o b − a eÐnai jetikìc, kai b < c shmaÐnei ìti oc − b eÐnai jetikìc. 'Ara, apì to pr¸to axÐwma di�taxhc, ja eÐnai jetikìc kai o(b− a) + (c− b) = c− a, dhlad  a < c.

3. a < b shmaÐnei ìti o b − a eÐnai jetikìc, �ra apì to pr¸to axÐwma di�taxhc, jaeÐnai jetikìc kai o c(b− a) = cb− ca, dhlad  ca < cb.

MporeÐte na apodeÐxete tic upìloipec idiìthtec? Mhn k�nete to l�joc na qrhsimopoi- sete stic apodeÐxeic sac idiìthtec pou den èqoun apodeiqjeÐ  dh!

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1.3. AX�IWMA PLHR�OTHTAS 9

1.3 AxÐwma Plhrìthtac

Orismìc 1.6. (Fr�gmata)'Estw mh kenì S ⊆ R. An up�rqei b tètoio ¸ste x ≤ b gia k�je x ∈ S, tìte to b ka-leÐtai �nw fr�gma tou S kai to S kaleÐtai �nw fragmèno. An, epiplèon, b ∈ S, to b ka-leÐtai to mègisto stoiqeÐo (  maximum) tou S, kai gr�foume b = maxS. AntÐstoiqaorÐzoume tic ènnoiec k�tw fr�gma, k�tw fragmèno, kai el�qisto stoiqeÐo (  minimum)minS. Ta �nw kai k�tw fr�gmata kaloÔntai apì koinoÔ fr�gmata. An èna sÔnolo eÐnaikai �nw fragmèno, kai k�tw fragmèno, kaleÐtai fragmèno.

Parat rhsh: Den èqoun ìla ta sÔnola mègisto  /kai el�qisto stoiqeÐo. Giapar�deigma, ti isqÔei gia ta N = {1, 2, . . . }, S = (0, 1)?

Orismìc 1.7. (Supremum, infimum)'Enac arijmìc kaleÐtai el�qisto �nw fr�gma   supremum enìc mh kenoÔ sunìlou S,kai sumbolÐzetai supS an:

1. To supS eÐnai �nw fr�gma tou S, kai

2. Kanènac arijmìc mikrìteroc tou supS den eÐnai �nw fr�gma tou S.

AntÐstoiqa orÐzetai h ènnoia tou mègistou k�tw fr�gmatoc,   infimum, inf S.

Par�deigma 1.4. MporeÐte na breÐte ta supremum, infimum, minimum, kai ma-ximum twn akìloujwn sunìlwn, ìpou up�rqoun?

1. (0, 1),

2. (0,∞),

3.{

1, 12 ,

13 ,

14 , . . .

},

4. {x ∈ R : 0 < x2 − 1 ≤ 2},

5. {x ∈ Q : x ≥ 0, 0 < x2 − 1 ≤ 2}.Par�deigma 1.5. 'Estw oi diadoqikèc dekadikèc proseggÐseic tou π, dhlad  tosÔnolo twn rht¸n

{3, 3.1, 3.14, 3.141 , 3.1415, . . . }.'Eqei to sÔnolo autì supremum? Poio eÐnai? 'Eqei supremum an agno soume thnÔparxh twn �rrhtwn?

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10 KEF�ALAIO 1. ARIJMO�I

L mma 1.1. (maximum=supremum) An èna sÔnolo S èqei maximum maxS, tì-te èqei kai supremum supS, kai supS = maxS. Epiplèon, an èna sÔnolo S èqeiminimum minS, tìte èqei kai infimum inf S, kai inf S = minS.

Apìdeixh. Profan c.

L mma 1.2. (Monadikìthta supremum/infimum) 'Ena sÔnolo den mporeÐ na èqei dÔodiaforetik� supremum   infimum.

Apìdeixh. Profan c.

L mma 1.3. (Basikèc idiìthtec supremum/infimum)

1. An to S èqei supremum, tìte gia k�je h > 0, up�rqei x ∈ S tètoio ¸stex > supS − h.

2. An to S èqei infimum, tìte gia k�je h > 0, up�rqei x ∈ S tètoio ¸ste x <

inf S + h.

Apìdeixh. 1. Ja qrhsimopoi soume eic �topo apagwg . 'Estw loipìn pwc den isqÔeih dosmènh prìtash. H �rnhs  thc mac lèei pwc up�rqei k�poio h > 0 gia to opoÐoden up�rqei x tètoio ¸ste x > supS − h. Dhlad , ∀x, x ≤ supS − h. Tìteto supS − h eÐnai �nw fr�gma tou S, kai m�lista mikrìtero tou supS, k�ti poueÐnai adÔnato.

2. An�loga.

Par�deigma 1.6. (Tr�gkac/Makr ) Gia ex�skhsh se arn seic, ìpwc aut  pou qrh-simopoi same sthn �nw apìdeixh, ac exet�soume tic akìloujec stiqomujÐec, parmènecapì prìsfata deltÐa twn 8, pou apoteloÔntai apì mia prìtash kai mia �rnhs  thc.

1. (aþ) Gi¸rgoc Tr�gkac: {Den up�rqei upourgìc thc kubèrnhshc me ligìtero apì400 sumboÔlouc}.

(bþ) K�tia Makr : {Up�rqei toul�qiston ènac upourgìc thc kubèrnhshc pou èqeiligìterouc apì 400 sumboÔlouc}.

2. (aþ) K�tia Makr : {Se ìlec tic prohgoÔmenec kubern seic p�nw apì touc misoÔcupourgoÔc eÐqan apotÔqei}.

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1.3. AX�IWMA PLHR�OTHTAS 11

(bþ) Gi¸rgoc Tr�gkac: {Up�rqei toul�qiston mia kubèrnhsh apì tic prohgoÔme-nec thc opoÐac eÐqan apotÔqei oi misoÐ   ligìteroi upourgoÐ}.

3. DÐnontai stajer� C, sun�rthsh f : R→ R, kai sÔnolo I ⊆ R.

(aþ) Gi¸rgoc Tr�gkac: {∀x, y ∈ I, |f(y)− f(x)| ≤ C|y−x|}. Me lìgia, gia ìlata zeÔgh arijm¸n sto I, ikanopoieÐtai h |f(y)− f(x)| ≤ C|y − x|.

(bþ) K�tia Makr : {∃x, y ∈ I : |f(y)−f(x)| > C|y−x|}. Me �lla lìgia, up�rqeizeÔgoc x, y gia to opoÐo den ikanopoieÐtai h |f(y)− f(x)| ≤ C|y − x|.

4. 'Estw sun�rthsh f : R→ R, kai èstw x0, L ∈ R.

(aþ) Gi¸rgoc Tr�gkac: {∀ε > 0, ∃δ > 0 : 0 < |x− x0| < δ ⇒ |f(x)− L| < ε}.Me lìgia, gia k�je ε > 0, up�rqei k�poio δ > 0 ètsi ¸ste an 0 < |x−x0| < δ,tìte isqÔei |f(x)− L| < ε.

(bþ) K�tia Makr : {∃ε > 0 : ∀δ > 0,∃x : 0 < |x− x0| < δ, |f(x)− L| ≥ ε. Melìgia, up�rqei èna ε > 0 tètoio ¸ste ìso mikrì δ kai na p�roume, ja broÔmeèna x gia to opoÐo nai men 0 < |x − x0| < δ, all� h |f(x) − L| < ε den jaisqÔei, dhlad  |f(x)− L| ≥ ε.

Orismìc 1.8. (AxÐwma Plhrìthtac): K�je mh kenì, fragmèno �nw sÔnolo S èqeisupremum.

L mma 1.4. ('Uparxh infimum)K�je mh kenì, fragmèno k�tw sÔnolo S èqei infimum.

Apìdeixh. 'Estw sÔnolo S mh kenì kai fragmèno k�tw, dhlad  up�rqei L tètoio ¸stegia k�je x ∈ S, x ≥ L. 'Estw to sÔnolo

−S , {y : −y ∈ S}.To sÔnolo −S eÐnai profan¸c mh kenì. EÐnai epÐshc kai fragmèno �nw apì to −L.Pr�gmati, èstw opoiod pote stoiqeÐo tou y ∈ −S. To −y an kei sto S, �ra −y ≥L ⇒ y ≤ −L kai to −L eÐnai �nw fr�gma. 'Ara, apì to axÐwma thc plhrìthtac, to−S èqei supremum, pou to sumbolÐzw me sup(−S).

Ja deÐxoume ìti to S èqei gia infimum to − sup(−S).Katarq n, to − sup(−S) eÐnai k�tw fr�gma, dhlad  ∀x ∈ S, x ≥ − sup(−S).

Pr�gmati, èstw ìti up�rqei k�poio x sto S me x < − sup(−S). Tìte −x > sup(−S)kai èqw �topo, afoÔ −x ∈ −S.

EÐnai ìmwc kai to mègisto k�tw fr�gma. 'Estw pwc den eÐnai, kai pwc up�rqei ε > 0tètoio ¸ste gia k�je x ∈ S, x ≥ − sup(−S) + ε. 'Omwc, up�rqei y ∈ (−S) tètoio

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12 KEF�ALAIO 1. ARIJMO�I

¸ste y > sup(−S) − ε, alli¸c to sup(−S) den ja  tan supremum tou −S. 'Ara−y < − sup(−S) + ε. Epeid  to −y an kei sto S, katal xame se �topo.

Par�deigma 1.7. (Supremum/infimum uposunìlou) 'Estw B ⊆ A, me to B mhkenì, kai to A fragmèno. Ja deÐxoume ìti

inf A ≤ inf B ≤ supB ≤ supA.

Katarq n parathr ste ìti to A eÐnai mh kenì giatÐ èqei èna mh kenì uposÔnolo, kaiafoÔ eÐnai kai fragmèno, ja èqei supremum kai infimum. EpÐshc, profan¸c kai toB ja eÐnai fragmèno, kai afoÔ ja eÐnai kai mh kenì, ja èqei supremum kai infimum.ParathroÔme diadoqik�:

1. An inf A > inf B dhmiourgeÐtai �topo. Pr�gmati, èstw

h =inf A− inf B

2.

Ex' orismoÔ tou inf B, up�rqei x ∈ B, �ra kai x ∈ A, gia to opoÐo

x < inf B + h =inf A+ inf B

2< inf A,

pou eÐnai �topo, afoÔ to inf A eÐnai k�tw fr�gma tou A.

2. 'Estw opoiod pote b ∈ B. AfoÔ b ≥ inf B kai b ≤ supB, tìte inf B ≤ supB.

3. An supB > supA prokÔptei �topo ìpwc kai sto pr¸to skèloc.

Par�deigma 1.8. (Supremum/infimum ènwshc) An ta A, B èqoun supremum kaiinfimum, ja èqei kai h ènws  touc A ∪B, kai m�lista:1. sup(A ∪B) = max{supA, supB}.2. inf(A ∪B) = min{inf A, inf B}.Ja apodeÐxoume to pr¸to skèloc mìno, kaj¸c to deÔtero prokÔptei entel¸c an�-

loga.'Estw loipìn èna x ∈ A ∪ B. EÐte x ∈ A, opìte x ≤ supA, eÐte x ∈ B, opìte

x ≤ supB, opìte p�nta x ≤ max{supA, supB}. 'Ara to max{supA, supB} eÐnai�nw fr�gma tou A ∪B, opìte gia to el�qisto �nw fr�gma sup(A ∪B) ja isqÔei

sup(A ∪B) ≤ max{supA, supB}. (1.1)

Apì thn �llh, parathr ste pwc, epeid  A ⊆ A ∪B, ja èqoume

supA ≤ sup(A ∪B).

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1.3. AX�IWMA PLHR�OTHTAS 13

To �nw eÐnai diaisjhtik� profanèc, kai èqei apodeiqjeÐ austhr� kai sto Par�deigma 1.7.Gia ton Ðdio lìgo èqoume kai

supB ≤ sup(A ∪B),

opìte sundu�zontac tic dÔo anisìthtec èqoume

max{supA, supB} ≤ sup(A ∪B). (1.2)

Sundu�zontac tic (1.1), (1.2), prokÔptei h zhtoÔmenh isìthta.

Par�deigma 1.9. (SÔgkrish sunìlwn) 'Estw mh ken� sÔnola A,B ⊆ R. An giak�je a ∈ A, kai gia k�je b ∈ B, èqoume a ≤ b, tìte ja deÐxoume ìti to A èqeisupremum, to B èqei infimum, kai supA ≤ inf B.

Pr�gmati, to A eÐnai mh kenì kai fragmèno �nw apì k�poio tuqaÐo b ∈ B, �raèqei supremum. OmoÐwc, to B eÐnai mh kenì kai fragmèno k�tw apì k�poio tuqaÐoa ∈ A, �ra èqei infimum. 'Estw t¸ra ìti supA > inf B. 'Ara ja up�rqei ε > 0tètoio ¸ste supA = inf B + ε. 'Omwc apì thn kataskeu  tou supremum kai touinfimum, ja up�rqoun a ∈ A, b ∈ B, me a > supA − ε

2 , b < inf B + ε2 . 'Ara,

a − b > supA − inf B − ε = 0 ⇒ a > b, pou eÐnai �topo ex' upojèsewc. 'Ara,supA ≤ inf B.

Par�deigma 1.10. 'Estw A, B dÔo mh ken� kai fragmèna uposÔnola tou R. AnsupA = inf B, ja deÐxoume ìti gia k�je ε > 0 up�rqoun a ∈ A kai b ∈ B ¸ste0 ≤ b− a < ε.

'Estw opoiod pote ε > 0. Tìte kai ε/2 > 0, kai kat� ta gnwst� gia ta supremum,infimum ja up�rqoun a ∈ A, b ∈ B, tètoia ¸ste

supA− ε

2< a ≤ supA⇒ − supA ≤ −a < ε

2− supA,

inf B ≤ b < inf B +ε

2

Prosjètontac kat� mèlh tic diplèc anisìthtec, kai parathr¸ntac pwc inf B = supA,prokÔptei telik� to zhtoÔmeno.

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14 KEF�ALAIO 1. ARIJMO�I

1.4 Sunèpeiec tou Axi¸matoc thc Plhrìthtac

Pollèc apì tic idiìthtec twn pragmatik¸n pou paÐrnoume gia dedomènec den mporoÔnna apodeiqjoÔn mìno me ta Axi¸mata PedÐou kai Di�taxhc, all� prèpei sthn apìdeix touc na gÐnei qr sh kai tou Axi¸matoc Plhrìthtac. Gia par�deigma:

Je¸rhma 1.1. (Arqim deia idiìthta)

1. Oi fusikoÐ arijmoÐ den eÐnai fragmènoi �nw.

2. Gia k�je pragmatikì x up�rqei fusikìc n me n > x.

3. An x, y eÐnai pragmatikoÐ me x > 0, tìte up�rqei fusikìc n tètoioc ¸ste nx > y.

Je¸rhma 1.2. (Pukn� sÔnola)

1. 'Estw dÔo pragmatikoÐ x, y, me x < y. Up�rqei an�mes� touc toul�qiston ènacrhtìc. (Dhlad  oi rhtoÐ eÐnai puknoÐ stouc pragmatikoÔc.)

2. 'Estw dÔo pragmatikoÐ x, y, me x < y. Up�rqei an�mes� touc toul�qiston ènac�rrhtoc. (Dhlad  oi �rrhtoi eÐnai puknoÐ stouc pragmatikoÔc.)

3. 'Estw x pragmatikìc. Up�rqoun aujaÐreta kont� tou, kai apì arister� kai apìdexi�, ènac rhtìc kai ènac �rrhtoc. Dhlad :

∀x ∈ R, ∀δ > 0, ∃y ∈ Q : x− δ < y < x,

∀x ∈ R, ∀δ > 0, ∃y ∈ Q : x < y < x+ δ,

∀x ∈ R, ∀δ > 0, ∃y ∈ R−Q : x− δ < y < x,

∀x ∈ R, ∀δ > 0, ∃y ∈ R−Q : x < y < x+ δ.

Je¸rhma 1.3. ('Uparxh riz¸n)

1. 'Estw n perittìc jetikìc akèraioc kai x 6= 0. Up�rqei monadikìc y me thn idiìthta

yn = x. (1.3)

O y sumbolÐzetai wc n√x   x

1n . EpÐshc, orÐzoume

x−1n , (x−1)

1n .

2. 'Estw n �rtioc jetikìc akèraioc kai x > 0. Up�rqei monadikìc jetikìc y pou naikanopoieÐ thn exÐswsh (1.3). O y sumbolÐzetai wc n

√x   x

1n . O monadikìc �lloc

arijmìc pou ikanopoieÐ thn exÐswsh (1.3) eÐnai o arnhtikìc − n√x.

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1.4. SUN�EPEIES TOU AXI�WMATOS THS PLHR�OTHTAS 15

x

y

y = x1n

f(y) = yn

x

y

y = x1n−y = −x

1n

f(y) = yn

Sq ma 1.1: Oi peript¸seic tou perittoÔ kai tou �rtiou ekjèth n tou Jewr matoc 1.3.

EpÐshc, orÐzoume

x−1n , (x−1)

1n .

3. Gia k�je n jetikì akèraio, an x = 0 tìte o monadikìc y pou ikanopoieÐ thn (1.3)eÐnai o 0, kai gr�foume n

√0 = 0.

Orismìc 1.9. (Rhtèc dun�meic) 'Estw r = pq , ìpou oi akèraioi p, q eÐnai jetikoÐ kai

pr¸toi metaxÔ touc (dhlad  o mègistoc koinìc diairèthc touc eÐnai h mon�da), kai èstwpwc isqÔei èna apì ta akìlouja:

1. q �rtioc kai x ≥ 0,

2. q perittìc kai x ∈ R.

OrÐzoume xr , (x1q )p, kai, an x 6= 0, x−r , (x−1)r = ((x−1)

1q )p.

Parat rhsh: 'Otan x ≥ 0, isqÔoun kai sthn perÐptwsh twn rht¸n ekjet¸n ìlecoi sun jeic idiìthtec twn fusik¸n ekjet¸n, gia par�deigma ∀r1, r2 ∈ Q,

xr1xr2 = xr1+r2, (xr1)r2 = xr1r2, xr1yr1 = (xy)r1.

'Otan x < 0, up�rqoun exairèseic. Gia par�deigma:

5 = ((−5)23 )

32 6= (−5)

23× 3

2 = (−5)1 = −5.

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16 KEF�ALAIO 1. ARIJMO�I

1.5 Diast mata

Orismìc 1.10. (Diast mata)

1. Ta sÔnola thc akìloujhc morf c kaloÔntai diast mata:

(a, b) , {x : a < x < b}, (a, b] , {x : a < x ≤ b}, [a, b) , {x : a ≤ x < b},[a, b] , {x : a ≤ x ≤ b}, (a,∞) , {x : a < x}, [a,∞) , {x : a ≤ x},(−∞, b) , {x : x < b}, (−∞, b] , {x : x ≤ b}, (−∞,∞) , R,

ìpou a, b ∈ R me a < b.

2. Ta (a, b), (a,∞), (−∞, b), (−∞,∞) kaloÔntai anoikt�.

3. Ta [a, b], [a,∞), (−∞, b], (−∞,∞) kaloÔntai kleist�.

4. An I di�sthma kai c ∈ I, to I kaleÐtai geitoni� tou c.

5. To eswterikì tou diast matoc [a, b], int[a, b] orÐzetai wc int[a, b] , (a, b). AntÐ-

stoiqa, int(a, b] , (a, b), int[a, b) , (a, b), int(−∞, b] , (−∞, b], k.o.k. TashmeÐa pou an koun sto eswterikì kaloÔntai eswterik�.

6. Ta �kra tou (a, b) orÐzontai wc ta a (aristerì �kro) kai b (dexiì �kro). ParìmoioiorismoÐ isqÔoun kai gia ta �lla diast mata.

Parathr seic

1. Genik¸c ìtan gr�foume I ja ennooÔme èna di�sthma.

2. GiatÐ to R eÐnai kai anoiktì kai kleistì? H ex ghsh basÐzetai sto orismì para-k�tw.

Orismìc 1.11. (SÔnola) Genikìtera:

1. To eswterikì enìc sunìlou S ⊂ R eÐnai to sÔnolo twn shmeÐwn x tou S giata opoÐa up�rqei anoiqt  geitoni� I tètoia ¸ste x ∈ I ⊂ S. Ta shmeÐa toueswterikoÔ enìc sunìlou kaloÔntai eswterik�.

2. OrÐzoume to sumpl rwma enìc sunìlou S wc to sÔnolo Sc , {x ∈ R : x 6∈ S}.

3. KaloÔme èna sÔnolo S anoiktì ìtan ìla ta shmeÐa tou eÐnai eswterik�, kai kleistììtan to sumpl rwma tou Sc eÐnai anoiktì.

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1.6. SUNART�HSEIS 17

1.6 Sunart seic

Orismìc 1.12. (Sun�rthsh)OrÐzoume wc sun�rthsh f : A → R mia apeikìnish apì to pedÐo orismoÔ domf , A

sto R. To f(A) , {y : f(x) = y gia k�poio x ∈ A} kaleÐtai pedÐo tim¸n thc f .

Orismìc 1.13. (Eidikèc peript¸seic sunart sewn)

1. Mia sun�rthsh kaleÐtai periodik  me perÐodo p > 0 an o p eÐnai o mikrìterocjetikìc arijmìc gia ton opoÐo (i) to pedÐo orismoÔ thc perilamb�nei to x+ p ìpoteperilamb�nei to x, kai epiplèon (ii) f(x+ p) = f(x) gia ìla ta x sto domf .

2. Mia sun�rthsh f kaleÐtai �rtia an to pedÐo orismoÔ thc perilamb�nei to −x ìpoteperilamb�nei to x, kai epiplèon f(−x) = f(x) gia ìla ta x sto domf .

3. Mia sun�rthsh f kaleÐtai peritt  an to pedÐo orismoÔ thc perilamb�nei to −xìpote perilamb�nei to x, kai epiplèon f(−x) = −f(x) gia ìla ta x sto domf .

4. Mia sun�rthsh f kaleÐtai gnhsÐwc aÔxousa an x1 < x2 ⇒ f(x1) < f(x2).

5. Mia sun�rthsh f kaleÐtai aÔxousa an x1 < x2 ⇒ f(x1) ≤ f(x2).

6. AntÐstoiqa orÐzontai oi gnhsÐwc fjÐnousec kai fjÐnousec sunart seic.

7. Oi (gnhsÐwc) aÔxousec kai (gnhsÐwc) fjÐnousec sunart seic kaloÔntai apì koinoÔ(gnhsÐwc) monìtonec.

8. Mia sun�rthsh f kaleÐtai �nw fragmènh an up�rqei U ∈ R tètoio ¸ste f(x) ≤ Ugia ìla ta x sto pedÐo orismoÔ thc f , dhlad  to pedÐo tim¸n thc eÐnai fragmèno�nw apì to U .

9. Mia sun�rthsh f kaleÐtai k�tw fragmènh an up�rqei L ∈ R tètoio ¸ste f(x) ≥ Lgia ìla ta x sto pedÐo orismoÔ thc f , dhlad  to pedÐo tim¸n thc eÐnai fragmènok�tw apì to L.

10. Mia sun�rthsh kaleÐtai fragmènh an eÐnai �nw kai k�tw fragmènh.

11. Mia sun�rthsh f : A→ R kaleÐtai èna proc èna (1-1) an

f(x1) = f(x2)⇒ x1 = x2.

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18 KEF�ALAIO 1. ARIJMO�I

Orismìc 1.14. (Olik� akrìtata sun�rthshc)'Estw sun�rthsh f : A → R. OrÐzoume ta supremum, infimum, maximum ( 

(olikì) mègisto), minimum (  (olikì) el�qisto) thc f se èna sÔnolo B ⊆ A wc

supx∈B

f , sup{f(x) : x ∈ B}, infx∈B

f , inf{f(x) : x ∈ B},

maxx∈B

f , max{f(x) : x ∈ B}, minx∈B

f , min{f(x) : x ∈ B},

efìson ta dexi� mèlh up�rqoun. Ta �nw kaloÔntai apì koinoÔ olik� akrìtata,   apl¸cakrìtata sto B. 'Otan den dieukrinÐzetai to B, dhlad  gr�foume supB, maxB, inf B,  min f , ennoeÐtai ìti B = A.

An c ∈ B kai f(c) = maxx∈B

f , tìte to c kaleÐtai jèsh (olikoÔ) mègistou sto B. An

c ∈ B kai f(c) = minx∈B

f , tìte to c kaleÐtai jèsh (olikoÔ) el�qistou sto B.

Orismìc 1.15. (Topik� akrìtata sun�rthshc) 'Estw sun�rthsh f : A → R.H f èqei topikì mègisto (topikì el�qisto) to f(c) an up�rqei anoiktì di�sthma I mec ∈ I tètoio ¸ste

f(x) ≤ f(c) (f(x) ≥ f(c)) ∀x ∈ I ∩ A.

To c kaleÐtai jèsh topikoÔ megÐstou (jèsh topikoÔ el�qistou).Ta topik� mègista kai el�qista kaloÔntai apì koinoÔ topik� akrìtata, kai oi jèseic

topik¸n mègistwn kai el�qistwn kaloÔntai apì koinoÔ jèseic topik¸n akrìtatwn.

Parathr seic

1. 'Otan den dieukrinÐzetai to eÐdoc tou megÐstou   el�qistou, ennoeÐtai olikì.

2. MporeÐ mia sun�rthsh na èqei �peira topik� mègista/el�qista, an èqei orisjeÐ seèna fragmèno di�sthma A?

3. MporeÐ na mhn up�rqei c ∈ B tètoio ¸ste f(c) = supx∈B

f   f(c) = infx∈B

f !

Par�deigma 1.11. Gia tic epìmenec sunart seic, prosdiorÐste se poiec eidikècpeript¸seic sunart sewn tou OrismoÔ 1.13 an koun, kai epiplèon prosdiorÐste ìla tatopik� kai olik� akrìtat� touc.

1. f1(x) = sinx, x ∈ (0, π/2).

2. f2(x) = sinx, x ∈ (π/2, π).

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1.6. SUNART�HSEIS 19

x

f7(x)

Sq ma 1.2: H sun�rthsh f7(x) tou ParadeÐgmatoc 1.11.

3. f3(x) = sinx, x ∈ R.

4. f4(x) = mx+ b, x ∈ R.

5. f5(x) = sin(1/x), x > 0.

6. H sun�rthsh tou Dirichlet:

f6(x) =

{1, x ∈ Q,0, x 6∈ Q.

7. H sun�rthsh f7(x) tou Sq matoc 1.2.

Orismìc 1.16. 'Estw sunart seic f : A → R kai g : B → R me g(B) ⊆ A.OrÐsoume thn sÔnjesh   sÔnjeth sun�rthsh f ◦ g : B → R twn sunart sewn f, g wc

th sun�rthsh f ◦ g(x) , f(g(x)) gia k�je x ∈ A.

Par�deigma 1.12. Pìsec sunjèseic mporeÐte na k�nete me tic akìloujec?

1. f1(x) = x2, x ∈ R.

2. f2(x) = sinx, x ∈ R.

3. f3(x) = sinx, x ∈ [0, π/2].

4. f4(x) = 1/x, x ∈ R− {0}.5. f5(x) =

√x, x ∈ [0,∞).

6. f6(x) = (x+ 1)/(x− 1), x ∈ R− {1}.

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20 KEF�ALAIO 1. ARIJMO�I

1.7 Trigwnometrikèc Sunart seic

Orismìc 1.17. (Trigwnometrikèc sunart seic)'Estw monadiaÐoc kÔkloc me kèntro to shmeÐo O = (0, 0). 'Estw tìxo AB m kouc|x| me arq  to shmeÐo A = (1, 0), pou diagr�fetai antÐjeta apì th for� tou rologioÔan x > 0 kai sÔmfwna me th for� tou rologioÔ an x < 0. To sunhmÐtono cosx kaito hmÐtono sinx orÐzontai wc h tetmhmènh kai h tetagmènh tou shmeÐou B antÐstoiqa.

EpÐshc, orÐzoume thn efaptomènh tanx , sinxcosx kai thn sunefaptomènh cotx , cosx

sinx ,efìson oi paronomastèc eÐnai di�foroi tou mhdenìc.

Parat rhsh: Ex' orismoÔ, h epÐkentrh gwni� AOB pou baÐnei sto tìxo AB m kouc|x| tou monadiaÐou kÔklou isoÔtai me x.

L mma 1.5. (Basikèc idiìthtec trigwnometrik¸n sunart sewn)Oi akìloujec idiìthtec prokÔptoun �mesa apì ton orismì:

1.sin 0 = 0, sin π

2 = 1, sin π = 0, sin 3π2 = −1,

cos 0 = 1, cos π2 = 0, cos π = −1, cos 3π

2 = 0.

2. Oi sunart seic sinx, tanx, cotx eÐnai perittèc, en¸ h cosx eÐnai �rtia. Sunep¸c,gia k�je x sto pedÐo orismoÔ touc,

sin(−x) = − sinx, cos(−x) = cos x, tan(−x) = − tanx, cot(−x) = − cotx.

3. sin2 x+ cos2 x = 1.

4. Oi sinx, cosx eÐnai periodikèc me perÐodo 2π, en¸ oi tanx, cotx eÐnai periodikècme perÐodo π. Sunep¸c, gia k�je x sto pedÐo orismoÔ touc,

sin(x+2π) = sin x, cos(x+2π) = cos x, tan(x+π) = tan x, cot(x+π) = cot x.

5.sin(x+ π) = − sinx, cos(x+ π) = − cosx.

6.

sin(π

2− x)

= cosx, cos(π2 − x

)= sinx,

tan(π

2− x)

= cotx, cot(π2 − x

)= tanx.

7.| sinx| < |x|, x 6= 0.

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1.7. TRIGWNOMETRIK�ES SUNART�HSEIS 21

B=(cosx, sinx)

A=(1,0)O=(0,0) (cosx,0)

(0,sinx)

x

|x|

Sq ma 1.3: Orismìc tou hmitìnou kai tou sunhmitìnou.

Apìdeixh. ProkÔptei me apl� gewmetrik� epiqeir mata.

L mma 1.6. (SunhmÐtono diafor�c)Gia k�je x, y ∈ R isqÔei:

cos(y − x) = cos x cos y + sinx sin y.

Apìdeixh. Ja deÐxoume thn apìdeixh mìno gia thn perÐptwsh 0 ≤ x < y ≤ π/2.(H genik  perÐptwsh mporeÐ na apodeiqjeÐ me qr sh thc eidik c aut c perÐptwshc kaiidiot twn tou Jewr matoc 1.5, all� h apìdeixh paraleÐpetai.)

Apì to orjog¸nio trÐgwno AQP tou Sq matoc 1.4 èqoume:

(cosx− cos y)2 + (sin y − sinx)2 = d2.

OmoÐwc, apì to trÐgwno BQP èqoume:

sin2(y − x) + (1− cos(y − x))2 = d2.

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22 KEF�ALAIO 1. ARIJMO�I

Q=(cosy,siny)

P=(cosx,sinx)A

B

xy

O=(0,0) (1,0)

(0,1)

d

Sq ma 1.4: H apìdeixh tou L mmatoc 1.6.

(Gia na katal�bete thn teleutaÐa isìthta, upologÐste tic suntetagmènec twn shmeÐwnP , Q se èna sÔsthma suntetagmènwn peristrammèno kat� gwnÐa x.) Exis¸nontac tadÔo skèlh, kai anaptÔssontac to tetr�gwno, prokÔptei to zhtoÔmeno.

L mma 1.7. (Trigwnometrikèc idiìthtec)

cos(x+ y) = cos x cos y − sinx sin y, sin(x± y) = sin x cos y ± cosx sin y,

sin 2x = 2 sin x cosx, cos 2x = cos2 x− sin2 x = 2 cos2 x− 1 = 1− 2 sin2 x,

sin(x

2

)= ±

√1− cosx

2, cos

(x2

)= ±

√1 + cos x

2,

sinx+ sin y = 2 sin

(x+ y

2

)cos

(x− y

2

),

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1.7. TRIGWNOMETRIK�ES SUNART�HSEIS 23

cosx+ cos y = 2 cos

(x+ y

2

)cos

(x− y

2

),

sinx− sin y = 2 cos

(x+ y

2

)sin

(x− y

2

),

cosx− cos y = −2 sin

(x+ y

2

)sin

(x− y

2

),

sinx sin y =1

2[cos(x− y)− cos(x+ y)] ,

cosx cos y =1

2[cos(x+ y) + cos(x− y)] ,

sinx cos y =1

2[sin(x+ y) + sin(x− y)] .

Apìdeixh. Oi exis¸seic prokÔptoun me apl  efarmog  twn basik¸n idiot twn twn Po-rism�twn 1.5, 1.6.

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24 KEF�ALAIO 1. ARIJMO�I

Page 35: ΣΗΜΕΙΩΣΕΙΣ-ΜΑΘΗΜΑΤΙΚΑ 1 ΠΛΗΡΟΦΟΡΙΚΗ ΑΣΟΕΕ 1 ΕΞΑΜΗΝΟ

Kef�laio 2

'Orio

2.1 Basik  Idèa OrÐou

{Orismìc}: Mia sun�rthsh f èqei ìrio to L,   teÐnei sto L, kaj¸c to x teÐneisto c, an, ìtan to x eÐnai {kont�} sto c, all� diaforetikì apì to c, tìte h f(x) eÐnai{kont�} sto L. Gr�foume

limx→c

f(x) = L   enallaktik� f(x)→ L kaj¸c x→ c.

Parathr seic

1. Up�rqoun polloÐ lìgoi gia touc opoÐouc mac endiafèrei to ìrio san ènnoia:

(aþ) H par�gwgoc eÐnai èna ìrio. Gia par�deigma, sto Sq ma 2.1, kaj¸c ta xiteÐnoun sto a, h klÐsh tou eujÔgrammou tm matoc pou en¸nei ta shmeÐa(xi, f(xi)) kai (a, f(a)) teÐnei sthn par�gwgo f ′(a) sto a.

(bþ) H proshmasmènh epif�neia pou perikleÐetai apì mia kampÔlh kai ton �xona twnx metaxÔ dÔo orÐwn (dhlad  to aplì thc olokl rwma metaxÔ aut¸n twn orÐwn)eÐnai èna ìrio. Gia par�deigma, sto Sq ma 2.2, h epif�neia thc klimakwt csun�rthshc teÐnei sthn epif�neia tou hmitìnou, kaj¸c to pl�toc twn bhm�twnteÐnei sto 0.

(gþ) To m koc miac kampÔlhc eÐnai èna ìrio. Gia par�deigma, sto Sq ma 2.3 tom koc thc tmhmatik� eujÔgrammhc, diakekommènhc kampÔlhc teÐnei sto m kocthc suneqoÔc kampÔlhc kaj¸c to m koc twn eujÔgrammwn tmhm�twn teÐneisto 0.

(dþ) O proshmasmènoc ìgkoc pou perikleÐetai apì mia epif�neia kai èna qwrÐo touR2 (dhlad  to diplì olokl rwma thc epif�neiac sto qwrÐo) eÐnai èna ìrio. Giapar�deigma, sto Sq ma 2.4 o ìgkoc thc klimakwt c sun�rthshc teÐnei ston

25

Page 36: ΣΗΜΕΙΩΣΕΙΣ-ΜΑΘΗΜΑΤΙΚΑ 1 ΠΛΗΡΟΦΟΡΙΚΗ ΑΣΟΕΕ 1 ΕΞΑΜΗΝΟ

26 KEF�ALAIO 2. �ORIO

a x1x2x3

y0

y3

y2

y1

f(x)y

x

tanθ=f’(a)

θ

Sq ma 2.1: H par�gwgoc wc ìrio.

ìgko tou stereoÔ metaxÔ thc f(x, y) = x2 +y2 kai tou qwrÐou [0, 10]× [0, 10]kaj¸c h epif�neia thc b�shc k�je klÐmakac teÐnei sto 0.

(eþ) To ìrio emfanÐzetai kai se polloÔc �llouc orismoÔc.

2. Sumperasmatik�, mporoÔme na poÔme pwc olìklhroc o Majhmatikìc LogismìceÐnai h melèth orÐwn.

Par�deigma 2.1. MporeÐte na breÐte tic timèc twn akìloujwn orÐwn, efìson up�r-qoun?

limx→3

(2x− 5), limx→0

sinx

xlimx→2bxc lim

x→0sin

(1

x

)(H sun�rthsh bxc tou akèraiou mèrouc tou x eÐnai o megalÔteroc akèraioc mikrìteroc  Ðsoc tou x.)

MporeÐte na breÐte ta x ìpou èqoun ìrio oi akìloujec sunart seic?

f(x) =

{1, x ∈ Q,0, x ∈ R−Q,

g(x) =

{|x|, x ∈ Q,0, x ∈ R−Q.

'Opwc mporeÐte na diapist¸sete, to hjikì dÐdagma tou paradeÐgmatoc eÐnai ìti qwrÐcausthroÔc orismoÔc den mporoÔme na antimetwpÐsoume dÔskolec peript¸seic.

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2.1. BASIK�H ID�EA OR�IOU 27

0 1 2 30

0.2

0.4

0.6

0.8

1

x

y

0 1 2 30

0.2

0.4

0.6

0.8

1

x

y

0 1 2 30

0.2

0.4

0.6

0.8

1

x

y

0 1 2 30

0.2

0.4

0.6

0.8

1

x

y

Sq ma 2.2: To olokl rwma wc ìrio.

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28 KEF�ALAIO 2. �ORIO

Sq ma 2.3: To m koc miac kampÔlhc wc ìrio.

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2.1. BASIK�H ID�EA OR�IOU 29

Sq ma 2.4: To diplì olokl rwma wc ìrio.

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30 KEF�ALAIO 2. �ORIO

2.2 Orismìc

Orismìc 2.1. ('Orio sun�rthshc) 'Estw sun�rthsh f : A → R, orismènh pantoÔse mia anoikt  geitoni� tou c, ektìc Ðswc apì to c. H f èqei ìrio to L,   teÐnei sto L,kaj¸c to x teÐnei sto c, kai gr�foume lim

x→cf(x) = L, an

∀ε > 0 ∃δ > 0 : 0 < |x− c| < δ ⇒ |f(x)− L| < ε.

Parathr seic

1. O Orismìc 2.1 shmaÐnei qondrik� ìti egguhmèna mporoÔme na krat soume ìsokont� jèloume to f(x) sto L, arkeÐ na perioristoÔme se aut� ta x pou eÐnaiarket� kont� sto c.

2. H tim  thc f gia x = c den mac endiafèrei! (MporeÐ to f(c) na mhn orÐzetai kan.)

3. An h f eÐnai orismènh pantoÔ se mia anoikt  geitoni� tou c, ektìc Ðswc apì to c,tìte up�rqoun a, b, tètoia ¸ste (a, c) ∪ (c, b) ⊆ A.

4. H apaÐthsh na eÐnai h f orismènh se mia anoikt  geitoni� (ektìc Ðswc tou c) eÐnaianagkaÐa, alli¸c an autì den isqÔei h sunepagwg  0 < |x−c| < δ ⇒ |f(x)−L| <ε den èqei nìhma, anex�rthta tou δ, afoÔ h f(x) den ja orÐzetai gia k�poia x.

5. Se orismèna biblÐa h �nw apaÐthsh uponoeÐtai qwrÐc na anafèretai rht¸c.

6. H tim  tou δ pou antistoiqeÐ se k�je ε den eÐnai monadik ! An h sunepagwg ikanopoieÐtai gia èna δ, ja ikanopoieÐtai kai gia k�je jetikì δ′ < δ.

7. To mègisto epitreptì δ eÐnai sun jwc sun�rthsh tou c kai tou ε, kai bèbaiaexart�tai kai apì thn f(x).

8. O �nw orismìc eÐnai to mìno pou axÐzei na apomnhmoneÔsete se autì to m�jhma!

9. O orismìc den eÐnai kajìlou profan c. O Gauss (1777-1855) den ton skèfthkekan, o Cauchy (1789-1857) èftiaxe mia lanjasmènh morf , kai o pr¸toc pou tonbr ke  tan o Weierstrass (1815-1897).

10. Mia gewmetrik  perigraf  tou orismoÔ emfanÐzetai sto Sq ma 2.5.

Par�deigma 2.2. Ja deÐxoume ìti limx→3

(2x− 5) = 1.

'Estw ε > 0. Parathr¸ pwc:

|(2x− 5)− 1| < ε⇔ |2x− 6| < ε⇔ |x− 3| < ε

2. (2.1)

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2.2. ORISM�OS 31

x

f(x)

c+δc-δ c

L

L+ε

L-ε

BHMA 1

BHMA 2

y

Sq ma 2.5: Gewmetrik  perigraf  tou orismoÔ tou orÐou. Gia k�je p�qoc 2ε orizìntiac lwrÐdac perÐthn eujeÐa y = L (BHMA 1), up�rqei èna antÐstoiqo p�qoc 2δ k�jethc lwrÐdac perÐ thn eujeÐax = c (pou den perilamb�nei th gramm  x = c) (BHMA 2) tètoio ¸ste an to (x, f(x)) eÐnai entìc thck�jethc lwrÐdac, na eÐnai kai entìc thc orizìntiac lwrÐdac.

'Ara gia to sugkekrimèno ε jètw δ = ε2 . 'Estw t¸ra 0 < |x − 3| < δ. Tìte kai

|x− 3| < δ kai apì thn (2.1) èqoume |(2x− 5)− 1| < ε, �ra prokÔptei to zhtoÔmeno.

Par�deigma 2.3. Ja deÐxoume ìti limx→c

(ax+ b) = ac+ b.

Katarq n, èstw a 6= 0. 'Estw ε > 0. Parathr¸ pwc:

|(ax+ b)− (ac+ b)| < ε⇔ |a(x− c)| < ε⇔ |x− c| < ε

|a| . (2.2)

'Ara gia to sugkekrimèno ε > 0 jètw δ = ε|a| > 0. 'Estw t¸ra 0 < |x− c| < δ. Tìte

apì thn (2.2) èqoume |(ax+ b)− (ac+ b)| < ε, �ra prokÔptei to zhtoÔmeno.An a = 0, tìte prèpei na deÐxoume ìti lim

x→cb = b. Pr�gmati, èstw ε > 0. An

epilèxoume opoiod pote δ > 0, gia par�deigma to δ = 1, parathroÔme pwc pr�gmati an0 < |x− c| < δ tìte kai |f(x)− b| = |b− b| = 0 < ε.

Par�deigma 2.4. Ja deÐxoume ìti limx→c

(1x

)= 1

c , an c 6= 0.

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32 KEF�ALAIO 2. �ORIO

Katarq n parathroÔme pwc:∣∣∣∣1x − 1

c

∣∣∣∣ =

∣∣∣∣c− xcx

∣∣∣∣ =1

|c|1

|x||c− x|.

Jètoume δ = min{|c|2 , ε

c2

2

}, kai èstw pwc 0 < |x− c| < δ. 'Eqoume:

|x− c| < δ ⇒ ||x| − |c|| < |c|2⇒ |c| − |x| < |c|

2⇒ |x| > |c|

2⇒ 1

|x| <2

|c| , (2.3)

|x− c| < δ ⇒ |c− x| < δ < εc2

2. (2.4)

Sta �nw qrhsimopoi same thn idiìthta

a < min(b, c)⇒ a < b kai a < c.

Qrhsimopoi¸ntac tic dÔo anisìthtec (2.3), (2.4), èqoume, ìtan |x− c| < δ:∣∣∣∣1x − 1

c

∣∣∣∣ =1

|c|1

|x||c− x| <1

|c| ×2

|c| × εc2

2= ε,

kai h apìdeixh oloklhr¸jhke.

Par�deigma 2.5. Ja deÐxoume ìti limx→c√x =√c, an c > 0.

'Estw opoiod pote ε > 0 kai èstw δ = min{c, ε√c}. Tìte an 0 < |x − c| < δ jaèqoume x ≥ 0, kai epiplèon

|√x−√c| =∣∣∣∣(√x−√c)(√x+

√c)√

x+√c

∣∣∣∣ =

∣∣∣∣ x− c√x+√c

∣∣∣∣=|x− c|√x+√c≤ |x− c|√

c<

δ√c≤ ε,

kai h apìdeixh oloklhr¸jhke.

Par�deigma 2.6. An f(x) ≥ 0 kai limx→c

f(x) = 0, tìte limx→c

√f(x) = 0.

Pr�gmati, èstw ε > 0. Epeid  ε2 > 0, up�rqei δ > 0 tètoio ¸ste

0 < |x− c| < δ ⇒ |f(x)− 0| < ε2 ⇒ 0 ≤ f(x) < ε2 ⇒ |√f(x)− 0| < ε,

kai h apìdeixh oloklhr¸jhke.

Par�deigma 2.7. An limx→c

f(x) > 0, tìte limx→c

√f(x) =

√limx→c

f(x). H apìdeixh ja

dojeÐ argìtera, wc efarmog  tou Jewr matoc 3.3.

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2.2. ORISM�OS 33

Par�deigma 2.8. Tèloc, ja doÔme kai mia sun�rthsh pou den èqei ìrio se k�poioc. Sugkekrimèna, èstw h f(x) = bxc. Ja deÐxoume ìti den èqei ìrio sto c = 2,qrhsimopoi¸ntac apagwg  se �topo. 'Estw loipìn pwc èqei ìrio, to L, kai èstw pwcL ≥ 3

2 . (H perÐptwsh L < 32 prokÔptei an�loga, kai gia autì thn paraleÐpoume.) 'Estw

ε = 14 . Apì ton orismì tou orÐou, ja up�rqei k�poio δ ètsi ¸ste ìpote 0 < |x−2| < δ,

na èqoume |f(x) − L| < 14 . Parathr ste ìmwc, ìti ìpoia kai na eÐnai h tim  tou δ,

mporoÔme na broÔme k�poion arijmì y ∈ (2 − δ, 2) gia ton opoÐo f(y) = 1, kai �ra(afoÔ L ≥ 3

2) ja èqoume |f(y)− L| ≥ 12 >

14 = ε. 'Ara, èqoume �topo.

Orismìc 2.2. (Pleurik� ìria) 'Estw sun�rthsh f : A→ R, kai c ∈ R.

1. 'Estw pwc (c, b) ⊆ A gia k�poio b > c. H f èqei (pleurikì) ìrio to L kaj¸c to xteÐnei sto c apì dexi�, kai gr�foume lim

x→c+f(x) = L an

∀ε > 0 ∃δ > 0 : c < x < c+ δ ⇒ |f(x)− L| < ε.

2. 'Estw pwc (a, c) ⊆ A gia k�poio a < c. H f èqei (pleurikì) ìrio to L kaj¸c tox teÐnei sto c apì arister�, kai gr�foume lim

x→c−f(x) = L an

∀ε > 0 ∃δ > 0 : c− δ < x < c⇒ |f(x)− L| < ε.

Parat rhsh: Mia gewmetrik  perigraf  tou orismoÔ tou orÐou apì ta dexi� em-fanÐzetai sto Sq ma 2.6. H gewmetrik  perigraf  tou orÐou apì ta arister� eÐnaiantÐstoiqh kai paraleÐpetai.

Par�deigma 2.9. Ja deÐxoume ìti limx→0+

√x = 0.

'Estw ε > 0. Jètw δ = ε2 > 0. 'Eqoume:

0 < x < 0 + δ ⇒ 0 <√x < ε⇒ |√x− 0| < ε,

kai h apìdeixh oloklhr¸jhke.

Par�deigma 2.10. Ja deÐxoume ìti limx→2+bxc = 2, en¸ lim

x→2−bxc = 1.

'Estw ε > 0. Jètw δ = 12 > 0. Tìte:

2 < x < 2 + δ ⇒ bxc = 2⇒ |bxc − 2| = 0 < ε.

'Ara, gia k�je ε > 0 up�rqei èna δ > 0 ¸ste na isqÔei h sunepagwg  tou orismoÔ.M�lista, se aut  thn perÐptwsh br kame èna δ pou k�nei gia k�je ε. (Poia �llaup�rqoun?) 'Ara apodeÐxame thn Ôparxh tou dexi� orÐou. To arister� ìrio upologÐzetaientel¸c an�loga.

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34 KEF�ALAIO 2. �ORIO

x

f(x)

c+δc

L

L+ε

L-ε

BHMA 1

BHMA 2

y

Sq ma 2.6: Gewmetrik  perigraf  tou orismoÔ tou orÐou apì ta dexi�. Gia k�je p�qoc 2ε orizìntiaclwrÐdac perÐ thn eujeÐa y = L (BHMA 1), up�rqei èna antÐstoiqo p�qoc δ k�jethc lwrÐdac metaxÔtwn eujei¸n x = c kai x = c + δ (qwrÐc autèc na perilamb�nontai) (BHMA 2) tètoio ¸ste an to(x, f(x)) eÐnai entìc thc k�jethc lwrÐdac, na eÐnai kai entìc thc orizìntiac lwrÐdac.

Par�deigma 2.11. Ja upologÐsoume ta pleurik� ìria

limx→0+

|x|x, lim

x→0−

|x|x.

ParathroÔme pwc ìtan x > 0, tìte h sun�rthsh f(x) = |x|x = 1. 'Estw èna

opoiod pote ε > 0. Tìte gia δ = 1, èqoume ìti an 0 < x < δ ⇒ |x|x = 1 ⇒

|f(x) − 1| = 0 < ε. 'Ara, to dexÐ pleurikì ìrio eÐnai to 1. Parathr ste ìti ed¸ jamporoÔsame na epilèxoume opoiad pote tim  gia to δ.

Me an�logo trìpo, prokÔptei pwc to arister� pleurikì ìrio eÐnai to −1.

Parat rhsh: 'Opwc mporeÐte na diapist¸sete, me touc orismoÔc den p�me makru�!Qreiazìmaste pollèc pr�xeic kai skèyh, akìma kai gia profan  apotelèsmata. H lÔshsto prìblhma eÐnai ta jewr mata thc epìmenhc paragr�fou.

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2.3. IDI�OTHTES OR�IOU 35

2.3 Idiìthtec OrÐou

L mma 2.1. (Monadikìthta orÐou) An limx→c

f(x) = L kai limx→c

f(x) = M , tìte L = M .

Dhlad , k�je sun�rthsh èqei se èna shmeÐo to polÔ èna ìrio.

Apìdeixh. 'Estw ìti limx→c

f(x) = L kai limx→c

f(x) = M , kai ìti L 6= M . Ja deÐxoume ìti

tìte prokÔptei �topo. QwrÐc bl�bh thc genikìthtac, èstw L > M . (H apìdeixh giaL < M eÐnai entel¸c an�logh.) 'Estw ε = L−M

2 > 0. AfoÔ to L eÐnai ìrio, up�rqeiδ1 tètoio ¸ste

0 < |x− c| < δ1 ⇒ |f(x)− L| < ε =L−M

2

⇒ f(x)− L > M − L2

⇒ f(x) >L+M

2.

AfoÔ ìmwc kai to M eÐnai ìrio, up�rqei δ2 tètoio ¸ste

0 < |x− c| < δ2 ⇒ |f(x)−M | < ε =L−M

2

⇒ f(x)−M <L−M

2⇒ f(x) <

L+M

2.

An epilèxw δ = min{δ1, δ2}, tìte gia ìla ta x me 0 < |x−c| < δ prèpei na isqÔoun dÔoamoibaÐwc apokleiìmenec anisìthtec. 'Ara prokÔptei �topo, kai prèpei L = M .

L mma 2.2. (Up�rqei ìrio ⇔ up�rqoun pleurik� ìria) limx→c

f(x) = L ann up�rqoun

kai ta dÔo pleurik� ìria kai eÐnai Ðsa, dhlad  limx→c+

f(x) = L kai limx→c−

f(x) = L.

Apìdeixh. Kat' arq�c, èstw pwc limx→c

= L. 'Estw ε > 0. Tìte up�rqei δ > 0 tètoio

¸ste0 < |x− c| < δ ⇒ |f(x)− L| < ε.

'Ara c < x < c+ δ ⇒ 0 < |x− c| < δ ⇒ |f(x)−L| < ε kai �ra up�rqei to dexi� ìrio,en¸ epÐshc c − δ < x < c ⇒ 0 < |x − c| < δ ⇒ |f(x) − L| < ε, kai �ra up�rqei kaito arister� ìrio.

Antistrìfwc, èstw pwc up�rqoun kai ta dÔo pleurik� ìria. 'Estw ε > 0. Jaup�rqoun δ1, δ2 > 0 tètoia ¸ste

c < x < c+ δ1 ⇒ |f(x)− L| < ε,

c− δ2 < x < c ⇒ |f(x)− L| < ε.

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36 KEF�ALAIO 2. �ORIO

'Ara, an epilèxw δ = min{δ1, δ2}, tìte ìpote 0 < |x− c| < δ, ja isqÔei eÐte h pr¸th,eÐte h deÔterh perÐptwsh, kai ètsi sÐgoura |f(x)− L| < ε, �ra lim

x→cf(x) = L.

Par�deigma 2.12. Me qr sh tou �nw l mmatoc kai tou ParadeÐgmatoc 2.10, pro-kÔptei pwc h sun�rthsh f(x) = bxc den èqei ìrio sto 2. Akìma, me qr sh tou �nw

l mmatoc kai tou ParadeÐgmatoc 2.11, prokÔptei pwc h sun�rthsh f(x) = |x|x epÐshc

den èqei ìrio sto 0.AntÐjeta, me qr sh tou �nw l mmatoc mporoÔme na deÐxoume ìti h f(x) = |x| èqei

ìrio to 0 sto 0, afoÔ ta pleurik� ìri� thc sto 0 eÐnai Ðsa me 0. (Profan¸c, apeujeÐacapì ton orismì prokÔptei pwc h f(x) = |x| èqei ìrio sto c to |c| gia ìla ta c 6= 0.)

L mma 2.3. (Sunj kec mh Ôparxhc orÐou) 'Estw sun�rthsh f : A → R, orismènhpantoÔ se mia anoikt  geitoni� tou c, ektìc Ðswc apì to c.

1. H f den èqei ìrio to L ann up�rqei k�poio ε > 0 tètoio ¸ste

(aþ) na mhn up�rqei δ > 0 ¸ste na isqÔei h sunepagwg  0 < |x − c| < δ ⇒|f(x)− L| < ε,   isodÔnama,

(bþ) gia k�je δ > 0 na up�rqei èna x tètoio ¸ste 0 < |x−c| < δ kai |f(x)−L| ≥ ε.

2. H f(x) den èqei ìrio sto c ann up�rqei k�poio ε > 0 tètoio ¸ste gia k�je δ > 0,na up�rqoun x1, x2 ∈ (c− δ, c) ∪ (c, c+ δ) me |f(x1)− f(x2)| > ε.

Apìdeixh. ParaleÐpetai.

Par�deigma 2.13. B�sei tou �nw l mmatoc, eÔkola prokÔptei (mporeÐte na sum-plhr¸sete tic leptomèreiec?) pwc

1. H sun�rthsh Dirichlet

f(x) =

{1, x ∈ Q,0, x ∈ R−Q,

den èqei poujen� ìrio.

2. H sun�rthsh

f(x) =

{|x|, x ∈ Q,0, x ∈ R−Q,

den èqei ìrio se kanèna x 6= 0.

3. H sun�rthsh sin(

1x

)den èqei ìrio sto 0.

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2.3. IDI�OTHTES OR�IOU 37

Je¸rhma 2.1. (Algebrikèc pr�xeic orÐwn) 'Estw sunart seic f , g me ìria L =limx→c

f(x) kai M = limx→c

g(x) sto c. 'Estw n jetikìc akèraioc, kai k ∈ R stajer�.

IsqÔoun oi akìloujec isìthtec:

1. limx→c

k = k.

2. limx→c

x = c.

3. limx→c

kf(x) = klimx→c

f(x).

4. limx→c

[f(x) + g(x)] = limx→c

f(x) + limx→c

g(x).

5. limx→c

[f(x)− g(x)] = limx→c

f(x)− limx→c

g(x).

6. limx→c

[f(x)g(x)] = limx→c

f(x)limx→c

g(x).

7. limx→c

f(x)g(x) =

limx→c

f(x)

limx→c

g(x) , efìson limx→c

g(x) 6= 0.

8. limx→c

[f(x)]n =[limx→c

f(x)]n.

Ta skèlh 3-8 prèpei na ermhneutoÔn wc ex c: an up�rqoun ìla ta ìria sta dexi�,up�rqei kai to antÐstoiqo ìrio sta arister�.

Apìdeixh. Oi apodeÐxeic twn skel¸n 1,2 prokÔptoun apì to Par�deigma 2.3. H apì-deixh tou skèlouc 3 eÐnai apl  kai paraleÐpetai.

H apìdeixh tou skèlouc 4 èqei wc ex c: 'Estw ε > 0. Tìte kai ε2 > 0, kai ex

upojèsewc ja up�rqei δ1 > 0 tètoio ¸ste

0 < |x− c| < δ1 ⇒ |f(x)− L| < ε

2.

OmoÐwc, ja up�rqei δ2 > 0 tètoio ¸ste

0 < |x− c| < δ2 ⇒ |g(x)−M | < ε

2.

Epilègoume δ = min{δ1, δ2}, kai parathroÔme pwc:

|f(x) + g(x)− L−M | ≤ |f(x)− L|+ |g(x)−M | < ε

2+ε

2= ε.

H pr¸th anisìthta prokÔptei apì thn trigwnik  anisìthta. H deÔterh isqÔei ìtan0 < |x − c| < δ ≤ δ1, δ2. 'Ara gia to sugkekrimèno ε > 0 br kame to δ > 0 pouqreiazìmastan, kai h apìdeixh oloklhr¸jhke.

Oi apodeÐxeic twn upìloipwn paraleÐpontai.

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38 KEF�ALAIO 2. �ORIO

Par�deigma 2.14. Ja broÔme to limx→2

x2+62x .

limx→2

x2 + 6

2x=

limx→2

(x2 + 6)

limx→2

2x=

limx→2

(x2) + limx→2

6

2limx→2

x=

1

4

[(limx→2

x)2

+ 6

]=

5

2.

H pr¸th isìthta proèrqetai apì to skèloc 7 tou Jewr matoc, h deÔterh apì ta skèlh4 kai 3, h trÐth apì ta skèlh 1,2,8, kai h tètarth apì to skèloc 2.

Je¸rhma 2.2. ('Oria poluwnumik¸n kai rht¸n sunart sewn) An h f eÐnai poluw-numik    rht  sun�rthsh, tìte

limx→c

f(x) = f(c),

efìson o paranomast c thc f(c) den mhdenÐzetai (ìtan h f eÐnai rht ).

Apìdeixh. ProkÔptei �mesa me efarmog  tou Jewr matoc 2.1.

Par�deigma 2.15. Ja upologÐsoume to ìrio limx→1

4x4+2x3−2x+6x5+x2−x−10 . 'Eqoume:

limx→1

4x4 + 2x3 − 2x+ 6

x5 + x2 − x− 10=

4× 14 + 2× 13 − 2× 1 + 6

15 + 12 − 1− 10= −10

9.

Par�deigma 2.16. 'Estw h sun�rthsh Dirichlet tou ParadeÐgmatoc 2.13. 'EstwepÐshc h g(x) = −f(x). Parathr ste ìti en¸ to ìrio thc f(x) +g(x) = 0 up�rqei giak�je c, ta ìria thc f(x) kai thc g(x) den up�rqoun gia kanèna c, kai �ra den mporoÔmena gr�youme lim

x→c[f(x) + g(x)] = lim

x→cf(x) + lim

x→cg(x).

L mma 2.4. ('Oria Ðswn se di�sthma sunart sewn) 'Estw pwc c ∈ (a, b) kai f(x) =g(x) gia k�je x ∈ (a, b), ektìc Ðswc tou c. Tìte an up�rqei to ìrio lim

x→cf(x), ja up�rqei

kai to limx→c

g(x) kai m�lista limx→c

g(x) = limx→c

f(x).

Apìdeixh. ProkÔptei eÔkola apì ton orismì tou orÐou.

Par�deigma 2.17. Ja upologÐsoume to ìrio limx→−3

x2−x−12x2+x−6 .

limx→−3

x2 − x− 12

x2 + x− 6= lim

x→−3

(x− 4)(x+ 3)

(x+ 3)(x− 2)= lim

x→−3

x− 4

x− 2=

7

5.

Sth deÔterh isìthta, qrhsimopoi same to L mma 2.4.

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2.3. IDI�OTHTES OR�IOU 39

L mma 2.5. (Allagèc metablht c)

1. limx→0

f(x) = A ann limx→ b

a

f(ax− b) = A, ìpou a 6= 0.

2. An limx→0

f(x) = A, tìte limx→0

f(x2) = A.

Apìdeixh. ParaleÐpetai.

Par�deigma 2.18. B�sei tou l mmatoc, afoÔ limx→0|x| = 0, tìte lim

x→2

∣∣x2 − 1

∣∣ = 0.

Je¸rhma 2.3. (Je¸rhma Parembol c) 'Estw f , g, h sunart seic me

f(x) ≤ g(x) ≤ h(x)

kont� kai gÔrw apì k�poio c, all� ìqi aparaÐthta sto c. Dhlad , up�rqei k�poio δ1

tètoio ¸ste an 0 < |x − c| < δ1, tìte isqÔei h �nw dipl  anisìthta. An limx→c

f(x) =

limx→c

h(x) = L, tìte kai limx→c

g(x) = L.

Apìdeixh. 'Estw ε > 0. Ex upojèsewc, up�rqei k�poio δ2 > 0 tètoio ¸ste

0 < |x− c| < δ2 ⇒ |h(x)− L| < ε⇒ h(x) < L+ ε.

ParomoÐwc, up�rqei δ3 > 0 tètoio ¸ste

0 < |x− c| < δ3 ⇒ |f(x)− L| < ε⇒ f(x) > L− ε.'Estw δ = min{δ1, δ2, δ3} > 0. Sundu�zontac ta �nw me thn upìjesh, èqoume:

0 < |x− c| < δ ⇒ L− ε < f(x) ≤ g(x) ≤ h(x) < L+ ε⇒ |g(x)− L| < ε.

'Ara, apì ton orismì tou orÐou prokÔptei to zhtoÔmeno. H gewmetrik  ermhneÐa touJewr matoc thc Parembol c emfanÐzetai sto Sq ma 2.7.

Par�deigma 2.19. Ja deÐxoume, me qr sh tou Jewr matoc thc Parembol c, ìti

limx→0

x sin

(1

x

)= 0.

Parathr ste ìti gia k�je x èqoume | sinx| ≤ 1, �ra gia k�je x 6= 0 ja èqoume| sin 1

x| ≤ 1, �ra kai |x sin 1x| ≤ |x|, kai telik�

−|x| ≤ x sin

(1

x

)≤ |x|

gia k�je x 6= 0. Epeid , ìmwc, kai h |x| kai h −|x| èqoun ìrio sto 0 to 0, me qr shtou Jewr matoc thc Parembol c prokÔptei to zhtoÔmeno.

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40 KEF�ALAIO 2. �ORIO

f(x)

g(x)

h(x)

x

c

L

Sq ma 2.7: Gewmetrik  ermhneÐa tou Jewr matoc thc Parembol c. AfoÔ oi sunart seic f(x), h(x)fr�ssoun k�tw kai �nw, antÐstoiqa, thn g(x), kai teÐnoun sto Ðdio ìrio sto c, kai h f(x) ja moir�zetaito ìrio touc sto c. Parathr ste ìti arkeÐ oi f(x), h(x) na fr�ssoun thn g(x) se mia anoiqt  geitoni�tou c, kai ìqi opoud pote.

Je¸rhma 2.4. (Idiìthtec pleurik¸n orÐwn) 'Olec oi idiìthtec twn orÐwn pou dìjhkanse aut  thn par�grafo, isqÔoun kai gia ta pleurik� ìria, me kat�llhlec tropopoi seic.

Apìdeixh. An�logh twn apodeÐxewn twn antÐstoiqwn prot�sewn.

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2.4. �ORIA TRIGWNOMETRIK�WN SUNART�HSEWN 41

2.4 'Oria Trigwnometrik¸n Sunart sewn

Je¸rhma 2.5. (Apl� ìria trigwnometrik¸n sunart sewn) Gia k�je pragmatikì a-rijmì c mèsa sto pedÐo orismoÔ thc antÐstoiqhc sun�rthshc, èqoume

limx→c

sinx = sin c, limx→c

cosx = cos c, limx→c

tanx = tan c, limx→c

cotx = cot c.

Apìdeixh. Pr¸ta ja deÐxoume ìti limx→c

sinx = sin c. Ja xekin soume deÐqnontac ìti

limx→0

sinx = 0. 'Estw 0 < x < π/2. Apì to Sq ma 2.8 èqoume ìti

0 < |PR| < |arc(PA)| ⇒ 0 < sinx < x.

(Me arc(PA) sumbolÐzoume to tìxo PA, kai me |arc(PA)| to m koc tou tìxou PA.)OmoÐwc, gia −π/2 < x < 0, brÐskoume ìti

x < sinx < 0,

�ra apì to Je¸rhma thc Parembol c, èqoume limx→0

sinx = 0. (Er¸thsh: poiec eÐnai oi

sunart seic pou paremb�lloun thn sinx?)Epiplèon, isqÔei lim

x→0cosx = 1. Pr�gmati:

limx→0

cosx = limx→0

√1− sin2 x =

√1− lim

x→0sin2 x =

√1− 02 = 1.

H pr¸th exÐswsh isqÔei gia x kont� sto 0 (¸ste na mhn isqÔei cosx = −√

1− sin2 x),en¸ h deÔterh apì to Par�deigma 2.7.

Qrhsimopoi¸ntac ta dÔo aut� ìria kai mia apl  allag  metablht c èqoume:

limx→c

sinx = limh→0

sin(c+ h) = limh→0

(sin c cosh+ cos c sinh)

= (sin c)limh→0

cosh+ (cos c)limh→0

sinh = (sin c)× 1 + (cos c)× 0 = sin c.

Me an�logo trìpo:

limx→c

cosx = limh→0

cos(c+ h) = limh→0

(cos c cosh− sin c sinh)

= (cos c)limh→0

cosh− (sin c)limh→0

sinh = (cos c)× 1 + (sin c)× 0 = cos c.

Oi apodeÐxeic twn exis¸sewn gia thn efaptìmenh kai sunefaptomènh prokÔptoun�mesa me efarmog  tou Jewr matoc 2.1.

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42 KEF�ALAIO 2. �ORIO

P=(cosx,sinx)

O=(0,0) A=(1,0)

B=(0,1)

x

R=(cosx,0)

Q=(0,sinx)

S

Sq ma 2.8: Apìdeixh twn Jewrhm�twn 2.5, 2.6.

Je¸rhma 2.6. (Mh tetrimmèna trigwnometrik� ìria)

limx→0

sinx

x= 1, lim

x→0

1− cosx

x= 0.

Apìdeixh. 1. Katarq n, parathroÔme ìti h epif�neia kuklikoÔ tomèa aktÐnac r kaigwnÐac θ isoÔtai me 1

2θr2. (Pr�gmati, h epif�neia prèpei na eÐnai an�logh thc gw-

nÐac, an�logh tou tetrag¸nou thc aktÐnac, kai gia ton sugkekrimèno suntelest 12 paÐrnoume to swstì apotèlesma gia θ = 2π.)

'Estw x ∈ (0, π2 ). Sto Sq ma 2.8 parathroÔme ìti h epif�neia tou kuklikoÔ tomèaORS, pou isoÔtai me 1

2(cos2 x)x, eÐnai mikrìterh thc epif�neiac tou trig¸nouORP , pou eÐnai 1

2 cosx sinx, pou me th seir� thc eÐnai mikrìterh apì thn epif�neiatou kuklikoÔ tomèa OAP , pou eÐnai 1

2 × 12 × x. 'Ara, gia x ∈ (0, π2 ),

1

2(cos2 x)x <

1

2cosx sinx <

1

2× 12 × x⇒ cosx <

sinx

x<

1

cosx.

Me parìmoio trìpo mporoÔme na deÐxoume ìti h teleutaÐa sqèsh isqÔei kai gia x ∈(−π

2 , 0). ParathroÔme ìti to aristerì kai to dexÐ skèloc èqoun ìrio th mon�da.'Ara, qrhsimopoi¸ntac to Je¸rhma thc Parembol c, prokÔptei to zhtoÔmeno.

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2.4. �ORIA TRIGWNOMETRIK�WN SUNART�HSEWN 43

2.

limx→0

1− cosx

x= lim

x→0

(1− cosx

x× 1 + cos x

1 + cos x

)= lim

x→0

(1− cos2 x

x(1 + cos x)

)= lim

x→0

(sin2 x

x(1 + cos x)

)=

(limx→0

sinx

x

limx→0

sinx

limx→0

(1 + cos x)= 1× 0

2= 0.

Par�deigma 2.20. Ja upologÐsoume to

limx→0

x2(1 + sin x)

(2x+ sinx)2.

ParathroÔme pwc

limx→0

x2(1 + sin x)

(2x+ sinx)2= lim

x→0

x2 + x2 sinx

4x2 + sin2 x+ 4x sinx

= limx→0

1 + sin x

4 + sin2 xx2 + 4 sinx

x

=1

4 + 1 + 4=

1

9.

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44 KEF�ALAIO 2. �ORIO

2.5 'Oria sto 'Apeiro, ApeirÐzonta 'Oria

Orismìc 2.3. ('Oria sto ± �peiro)

1. 'Estw f orismènh sto [c,∞) gia k�poio c ∈ R. Lème ìti h f èqei ìrio sto ∞ toL, kai gr�foume lim

x→∞f(x) = L an gia k�je ε > 0 up�rqei X ∈ R ¸ste

x > X ⇒ |f(x)− L| < ε.

2. 'Estw f sun�rthsh orismènh sto (−∞, c] gia k�poio c ∈ R. Lème ìti h fèqei ìrio sto −∞ to L, kai gr�foume lim

x→−∞f(x) = L an gia k�je ε > 0 up�rqei

X ∈ R ¸stex < X ⇒ |f(x)− L| < ε.

Parat rhsh: Mia gewmetrik  perigraf  tou orÐou L kaj¸c to x p�ei sto ∞emfanÐzetai sto Sq ma 2.9. H ermhneÐa gia to ìrio sto −∞ eÐnai an�logh.

Par�deigma 2.21. Ja deÐxoume ìti limx→∞

x+1x+2 = 1. 'Estw ε > 0. Parathr ste pwc,

an x 6= −2, ∣∣∣∣x+ 1

x+ 2− 1

∣∣∣∣ < ε⇔∣∣∣∣x+ 1− x− 2

x+ 2

∣∣∣∣ < ε⇔ |x+ 2| > 1

ε.

B�sei thc �nw, jètoume X = 1ε − 2, kai parathroÔme pwc:

x > X ⇒ x >1

ε− 2⇒ x+ 2 >

1

ε⇒ |x+ 2| > 1

ε⇒∣∣∣∣x+ 1

x+ 2− 1

∣∣∣∣ < ε

Sthn teleutaÐa sunepagwg , qrhsimopoi same thn arqik  akoloujÐa isodunami¸n, lam-b�nontac upìyin pwc x 6= −2.

Orismìc 2.4. (ApeirÐzonta ìria)

1. H sun�rthsh f èqei ìrio sto c to ∞, kai gr�foume limx→c

f(x) = ∞, an gia k�je

M ∈ R mporoÔme na broÔme δ > 0 tètoio ¸ste

0 < |x− c| < δ ⇒ f(x) > M.

2. H sun�rthsh f èqei ìrio sto∞ to∞, kai gr�foume limx→∞

f(x) =∞, an gia k�je

M ∈ R mporoÔme na broÔme X ∈ R tètoio ¸ste

x > X ⇒ f(x) > M.

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2.5. �ORIA STO �APEIRO, APEIR�IZONTA �ORIA 45

x

f(x)

L

L+ε

L-ε

X

ΒΗΜΑ 1

ΒΗΜΑ 2

Sq ma 2.9: Gewmetrik  perigraf  tou orismoÔ tou orÐou sto �peiro. Gia k�je p�qoc 2ε orizìntiaclwrÐdac perÐ thn eujeÐa y = L (BHMA 1), up�rqei èna hmiepÐpedo x > X (BHMA 2) tètoio ¸ste anto (x, f(x)) eÐnai entìc tou hmiepipèdou, na eÐnai kai entìc thc orizìntiac lwrÐdac.

3. Parìmoia mporoÔme na orÐsoume ta akìlouja ìria:

limx→c+

f(x) =∞, limx→c−

f(x) =∞, limx→c

f(x) = −∞,limx→c+

f(x) = −∞, limx→c−

f(x) = −∞, limx→−∞

f(x) =∞,. . . . . . . . . .

Parathr seic

1. Mia gewmetrik  perigraf  tou orÐou ∞ kaj¸c to x teÐnei sto c emfanÐzetai stoSq ma 2.10.

2. Pìsec peript¸seic orÐwn up�rqoun sunolik�? MporeÐte na k�nete ta antÐstoiqasq mata?

Par�deigma 2.22. Ja deÐxoume ìti limx→a

1(x−a)2 = ∞, ìpou a ∈ R. Pr�gmati, èstw

k�poio aujaÐreto M . An to M ≤ 0, tìte gia k�je δ > 0 èqoume ìti 0 < |x − a| <δ ⇒ 1

(x−a)2 > M . Opìte, èstw pwc M > 0. ParathroÔme pwc, gia x 6= a,

1

(x− a)2> M ⇔ |x− a| < 1√

M.

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46 KEF�ALAIO 2. �ORIO

x

f(x)

c+δc-δ c

M

BHMA 1

BHMA 2

y

f(x)

Sq ma 2.10: Gewmetrik  perigraf  tou orismoÔ tou �peirou orÐou. Gia k�je hmiepÐpedo y > M(BHMA 1), up�rqei mia k�jeth lwrÐda 0 < |x− c| < δ, exairoÔmenhc thc eujeÐac x = c, (BHMA 2)tètoia ¸ste an to (x, f(x)) eÐnai entìc thc k�jethc lwrÐdac, na eÐnai kai entìc tou hmiepipèdou.

'Ara, an jèsoume δ = 1√M, tìte

0 < |x− a| < δ ⇒ 0 < |x− a| < 1√M⇒ 1

(x− a)2> M.

Par�deigma 2.23. IsqÔei limx→∞

xn = ∞ ìtan n ∈ N. Pr�gmati, èstw M . Jètw

X = |M | 1n ⇔ Xn = |M |, kai èqoume:x > X ⇒ xn > Xn ⇒ xn > |M | ≥M ⇒ xn > M.

Par�deigma 2.24. Ja deÐxoume ìti limx→a+

1(x−a) = ∞, gia a ∈ R. Se aut  thn

perÐptwsh, o orismìc tou orÐou eÐnai ìti gia k�je M ∈ R mporoÔme na broÔme ènaδ > 0 tètoio ¸ste a < x < a + δ ⇒ f(x) > M . Katarq n parathr ste ìti anM ≤ 0, tìte opoiod pote δ mac k�nei, afoÔ h sun�rthsh 1

x−a eÐnai jetik  gia x > a.'Estw loipìn pwc M > 0. Parathr ste pwc an x > a ja isqÔei ìti

1

x− a > M ⇔ x− a < 1

M⇔ x < a+

1

M.

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2.5. �ORIA STO �APEIRO, APEIR�IZONTA �ORIA 47

Opìte, jètontac δ = 1M , èqoume

a < x < a+ δ ⇒ a < x < a+1

M⇒ 1

x− a > M,

kai h apìdeixh oloklhr¸jhke.IsqÔei epÐshc ìti lim

x→a−1

(x−a) = −∞. MporeÐte na gr�yete thn apìdeixh?

Par�deigma 2.25. Ja deÐxoume ìti an limx→c+

f(x) = +∞, tìte limx→0

f(c+x2) = +∞,

qrhsimopoi¸ntac apokleistik� touc orismoÔc twn antÐstoiqwn orÐwn. Pr�gmati, èstwM > 0. Ex upojèsewc, ja up�rqei δ > 0 tètoio ¸ste

c < x < c+ δ ⇒ f(x) > M. (2.5)

Jètoume δ′ =√δ. ParathroÔme pwc

0 < |x− 0| < δ′ ⇒ 0 < x2 < (δ′)2 ⇒ c < x2 + c < c+ δ ⇒ f(x2 + c) > M.

H teleutaÐa sunepagwg  isqÔei lìgw thc (2.5).

L mma 2.6. ('Orio thc 1/f(x))

1. An limx→∞

f(x) =∞ tìte limx→∞

1f(x) = 0.

2. An limx→∞

f(x) = 0 kai f(x) > 0, tìte limx→∞

1f(x) =∞.

Apìdeixh. 1. 'Estw ε > 0. Jètw M = 1ε > 0. Up�rqei X ∈ R tètoio ¸ste

x > X ⇒ f(x) > M > 0⇒ 0 <1

f(x)<

1

M= ε⇒

∣∣∣∣ 1

f(x)− 0

∣∣∣∣ < ε.

'Ara, gia to ε pou epilèxame brèjhke èna X ∈ R tètoio ¸ste an x > X na isqÔei∣∣∣ 1f(x) − 0

∣∣∣ < ε, kai o orismìc tou orÐou ikanopoieÐtai.

2. 'Estw M > 0. 'Estw to ε = 1M > 0. Apì to ìrio pou mac dÐnetai, èqoume ìti

up�rqei X ∈ R tètoio ¸ste

x > X ⇒ |f(x)− 0| < ε⇒ 0 < f(x) < ε⇒ 1

f(x)>

1

ε= M.

Sthn deÔterh sunepagwg  qrhsimopoÐhsa kai thn upìjesh f(x) > 0. 'Ara telik�gia to M pou epilèxame, up�rqei X ∈ R tètoio ¸ste x > X ⇒ 1

f(x) > M .Sunep¸c apodeÐxame to zhtoÔmeno.

Ja Ðsque to zhtoÔmeno an den Ðsque h upìjesh f(x) > 0?

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48 KEF�ALAIO 2. �ORIO

Par�deigma 2.26. Me efarmog  tou �nw l mmatoc kai tou prohgoÔmenou para-deÐgmatoc �mesa èqoume ìti lim

x→∞x−n = 0, ìtan n ∈ N.

Je¸rhma 2.7. 'Olec oi idiìthtec twn orÐwn thc Paragr�fou 2.3 isqÔoun kai gia taìria aut c thc paragr�fou, me tic kat�llhlec tropopoi seic.

Apìdeixh. An�logh twn apodeÐxewn twn antÐstoiqwn prot�sewn.

Par�deigma 2.27. Me efarmog  tou Jewr matoc thc Parembol c, mporoÔme nadeÐxoume pwc lim

x→∞sinxx = 0. Pr�gmati, gia x 6= 0,

| sinx| ≤ 1⇒∣∣∣∣sinxx

∣∣∣∣ ≤ ∣∣∣∣1x∣∣∣∣⇒ ∣∣∣∣sinxx

∣∣∣∣ ≤ 1

x⇒ −1

x≤ sinx

x≤ 1

x,

kai epeid  tìso to aristerì ìso kai to dexÐ skèloc teÐnoun sto 0 kaj¸c x → ∞,prokÔptei ìti lim

x→∞sinxx = 0.

Par�deigma 2.28. Ja upologÐsoume to limx→∞

2x+sin9 x3x+4 . ParathroÔme pwc

2x− 1

3x+ 4≤ 2x+ sin9 x

3x+ 4≤ 2x+ 1

3x+ 4.

'Omwc èqoume

limx→∞

2x± 1

3x+ 4= lim

x→∞2± 1/x

3 + 4/x=

2± 0

3 + 0=

2

3.

Sth deÔterh isìthta qrhsimopoi same thn kat�llhlh ekdoq  tou Jewr matoc 2.2.

'Ara, apì to Je¸rhma thc Parembol c, èqoume telik� pwc limx→∞

2x+sin9 x3x+4 = 2

3 .

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2.6. �ORIO AKOLOUJ�IAS 49

2.6 'Orio AkoloujÐac

Orismìc 2.5. (AkoloujÐa) Mia sun�rthsh a : N → R me pedÐo orismoÔ toucfusikoÔc 1, 2, . . . kaleÐtai akoloujÐa. SumbolÐzetai wc a(n), {an}   an.

Parathr seic

1. Epeid  oi akoloujÐec eÐnai sunart seic me pedÐo orismoÔ to N, o katallhlìteroctrìpoc na tic apeikonÐsoume grafik� eÐnai autìc tou Sq matoc 2.11.

2. Merik� paradeÐgmata akolouji¸n eÐnai ta ex c:

(aþ) O mèsoc (  el�qistoc,   mègistoc) qrìnoc a(n) pou ja qreiasteÐ gia nalÔsoume èna prìblhma me par�metro n, gia par�deigma na antistrèyoume ènapÐnaka megèjouc n, na broÔme thn diadrom  el�qistou kìstouc se èna gr�fome n kìmbouc, k.o.k.

(bþ) O bajmìc enìc foitht  sthn n-ost  telik  exètash pou ja d¸sei sta Ma-jhmatik� I.

(gþ) To b�roc tou n-ostoÔ granazioÔ pou par�gei mia mhqan .

(dþ) H tim  kleisÐmatoc tou qrhmatisthrÐou thn n-ost  mèra.

(eþ) H axÐa tou sunìlou twn metoq¸n enìc qartofulakÐou thn n-ost  mèra.

(�þ) k.o.k.

Orismìc 2.6. ('Orio akoloujÐac) Mia akoloujÐa an èqei ìrio to L,   teÐnei sto L,  sugklÐnei sto L kai gr�foume lim

n→∞an = L, an gia k�je ε > 0 up�rqei fusikìc N

tètoioc ¸sten > N ⇒ |an − L| < ε.

Mia akoloujÐa pou èqei ìrio kaleÐtai sugklÐnousa, alli¸c kaleÐtai apoklÐnousa kailème pwc apoklÐnei.

Par�deigma 2.29. Ja deÐxoume ìti h an = 1n èqei ìrio to 0, qrhsimopoi¸ntac ton

orismì. ParathroÔme ìti

|an − 0| < ε⇔∣∣∣∣1n∣∣∣∣ < ε⇔ n >

1

ε.

'Ara, arkeÐ na epilèxw N = b1/εc+ 1, kai ja èqoume

n > N ⇒ n >1

ε⇒ 1

n< ε⇒ |an − 0| < ε.

H deÔterh sunepagwg  proèkuye me qr sh twn �nw isodunami¸n.

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50 KEF�ALAIO 2. �ORIO

n

1

a(n)

2 3 4 5 6 7 8 9

Sq ma 2.11: Grafik  apeikìnish miac akoloujÐac.

Par�deigma 2.30. Sthn perÐptwsh thc an = n+4n+5 , èqoume ìrio L = 1, ìpwc pro-

kÔptei me qr sh tou orismoÔ.Pr�gmati, gia èna dosmèno ε > 0, parathr ste katarq n pwc∣∣∣∣n+ 4

n+ 5− 1

∣∣∣∣ < ε⇔∣∣∣∣n+ 4− n− 5

n+ 5

∣∣∣∣ < ε⇔ 1

n+ 5< ε⇔ n >

1

ε− 5.

'Ara, an epilèxoume N = b1/εc− 4, pou eÐnai p�nta fusikìc, anex�rthta thc tim c touε, tìte

n > N ⇒ n >1

ε− 5⇒

∣∣∣∣n+ 4

n+ 5− 1

∣∣∣∣ < ε⇒ |an − 1| < ε.

H deÔterh sunepagwg  proèkuye me qr sh twn �nw isodunami¸n.

Je¸rhma 2.8. (Basikèc idiìthtec orÐou akoloujÐac) 'Ola ta jewr mata autoÔ toukefalaÐou gia to ìrio miac sun�rthshc kaj¸c to x → ∞ isqÔoun, me kat�llhlectropopoi seic, gia to ìrio akoloujÐac kaj¸c to n→∞.

Apìdeixh. Se k�je perÐptwsh, h apìdeixh eÐnai entel¸c an�logh thc apìdeixhc gia thnperÐptwsh tou orÐou sun�rthshc.

Par�deigma 2.31. Ja qrhsimopoi soume thn kat�llhlh morf  tou Jewr matocthc Parembol c gia na deÐxoume ìti h an = 1

n! sugklÐnei. UpenjumÐzoume ìti

n! = n(n− 1) . . . 1.

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2.6. �ORIO AKOLOUJ�IAS 51

x,n

1

f(x), f(n)

2 3 4 5 6 7 8 9 10 11 12

L

Sq ma 2.12: Gewmetrik  ermhneÐa tou Jewr matoc 2.9: An h f(x) eÐnai entìc thc lwrÐdac gia ìlata pragmatik� x > X, ja eÐnai kai gia èna uposÔnolo aut¸n, kai sugkekrimèna gia touc fusikoÔcn > X.

Pr�gmati,

0 ≤ 1

n!≤ 1

n.

'Omwc, h akoloujÐa 1n teÐnei sto mhdèn, ìpwc eÐdame sto Par�deigma 2.29. AfoÔ kai to

aristerì skèloc profan¸c teÐnei sto 0, apì to je¸rhma thc Parembol c, prokÔpteitelik� pwc h an = 1

n! sugklÐnei sto 0.

Je¸rhma 2.9. ('Orio sun�rthshc ⇒ ìrio akoloujÐac) An limx→∞

f(x) = L, tìte

limn→∞

f(n) = L.

Apìdeixh. DeÐte to Sq ma 2.12 gia mia gewmetrik  ermhneÐa tou jewr matoc. 'Estwε > 0. AfoÔ h f(x)→ L, up�rqei X tètoio ¸ste

x > X ⇒ |f(x)− L| < ε.

H sunepagwg  isqÔei gia ìla ta x ∈ R pou eÐnai megalÔtera apì to X, �ra kai giatouc fusikoÔc n > X. Jètoume loipìn N = max{bXc+ 1, 1}, kai parathroÔme pwc

n > N ⇒ n > X ⇒ |f(n)− L| < ε,

kai h apìdeixh oloklhr¸jhke.

Parat rhsh: IsqÔei to antÐstrofo?

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52 KEF�ALAIO 2. �ORIO

a2 Ma1 a3 ...a4 a

Sq ma 2.13: Mia monìtonh akoloujÐa eÐnai fragmènh an kai mìno an sugklÐnei.

Par�deigma 2.32. Sthn perÐptwsh thc an = n+4n+5 tou ParadeÐgmatoc 2.29, mporoÔ-

me na broÔme ìti èqei ìrio kaj¸c n→∞ to L = 1, qwrÐc qr sh tou orismoÔ, apl¸corÐzontac thn sun�rthsh

f(x) =x

x+ 1=

1

1 + 1x

,

lamb�nontac to ìrio thc sto ∞, kai efarmìzontac to Je¸rhma 2.9.

Orismìc 2.7. (MonotonÐa kai fr�gmata)

1. Mia akoloujÐa an kaleÐtai aÔxousa an an+1 ≥ an gia k�je n ∈ N, kaleÐtaignhsÐwc aÔxousa an an+1 > an gia k�je n ∈ N, kai kaleÐtai �nw fragmènh me�nw fr�gma M an an ≤M gia k�je n ∈ N.

2. AntÐstoiqa orÐzontai oi ènnoiec thc fjÐnousac kai gnhsÐwc fjÐnousac akoloujÐac,thc k�tw fragmènhc akoloujÐac kai tou k�tw fr�gmatoc akoloujÐac.

3. Oi (gnhsÐwc) aÔxousec kai fjÐnousec akoloujÐec kaloÔntai apì koinoÔ (gnhsÐwc)monìtonec.

4. Oi akoloujÐec pou eÐnai kai �nw kai k�tw fragmènec kaloÔntai fragmènec. Para-thr ste pwc mia akoloujÐa eÐnai fragmènh ann up�rqei M tètoio ¸ste |an| ≤M

gia k�je n ∈ N.

Je¸rhma 2.10. (MonotonÐa ⇒ (fr�gma ⇔ sÔgklish) )

1. 'Estw h an aÔxousa akoloujÐa. H an sugklÐnei ann eÐnai fragmènh �nw.

2. 'Estw h an fjÐnousa akoloujÐa. H an sugklÐnei ann eÐnai fragmènh k�tw.

Apìdeixh. DeÐte to Sq ma 2.13 gia mia apl  ex ghsh tou Jewr matoc.ArkeÐ na apodeÐxoume to pr¸to skèloc. To deÔtero mporeÐ na prokÔyei �mesa apì

to pr¸to,   anex�rthta me parìmoia apìdeixh.'Estw pwc h an eÐnai fragmènh �nw, èstw apì k�poio M , �ra to sÔnolo {an} èqei

supremum, èstw a. To a eÐnai to ìrio thc an. Pr�gmati, èstw opoiod pote ε > 0.

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2.6. �ORIO AKOLOUJ�IAS 53

Apì ton orismì tou supremum, up�rqei k�poio N tètoio ¸ste

a− ε < aN ≤ a⇒ ∀n > N, a− ε < an ≤ a⇒ ∀n > N, |an − a| < ε.

(H pr¸th sunepagwg  prokÔptei giatÐ h an eÐnai aÔxousa, kai fragmènh �nw apì toa.) 'Ara, apì ton orismì tou orÐou, lim

n→∞an = a.

Antistrìfwc, èstw pwc h an èqei ìrio, èstw to L. Tìte up�rqei N ètsi ¸ste giak�je n > N , na èqoume |an − L| < 1 ⇒ an < L + 1. 'Ara, èna �nw fr�gma eÐnai omegalÔteroc apì touc L+ 1, a1, a2, . . . , aN .

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54 KEF�ALAIO 2. �ORIO

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Kef�laio 3

Sunèqeia

3.1 Orismìc kai Basikèc Idiìthtec Sunèqeiac

Orismìc 3.1. (Sunèqeia se shmeÐo)

1. H sun�rthsh f eÐnai suneq c sto c an limx→c

f(x) = f(c). (Dhlad  orÐzetai to

f(c), to ìrio up�rqei, kai ta dÔo isoÔntai.)

2. An h sun�rthsh den eÐnai suneq c sto c, kaleÐtai asuneq c, kai lème ìti sto cup�rqei asunèqeia.

3. H sun�rthsh f eÐnai dexi� suneq c sto c an limx→c+

f(x) = f(c).

4. H sun�rthsh f eÐnai arister� suneq c sto c an limx→c−

f(x) = f(c).

5. An h f(x) eÐnai asuneq c sto c, all� mporoÔme na tropopoi soume thn f(x)orÐzontac thn f(c) ètsi ¸ste h f(x) na gÐnei suneq c sto c, h asunèqeia kaleÐtaiepousi¸dhc. Alli¸c, kaleÐtai ousi¸dhc.

Parat rhsh: Sto Sq ma 3.1 èqoume sqedi�sei mia sun�rthsh me dÔo ousi¸deic kaimia epousi¸dh asunèqeia. MporeÐte na skefteÐte �lla paradeÐgmata? Ti sumbaÐnei, giapar�deigma, me thn sin

(1x

)sto 0?

L mma 3.1. (Pleurik  sunèqeia ⇔ sunèqeia) H f eÐnai suneq c sto c ann eÐnaiarister� suneq c sto c kai dexi� suneq c sto c.

Apìdeixh. Profan c.

55

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56 KEF�ALAIO 3. SUN�EQEIA

x

y

Sq ma 3.1: Mia sun�rthsh me dÔo ousi¸deic kai mia epousi¸dh asunèqeia.

L mma 3.2. (Sunèqeia kat� Cauchy) 'Estw f : A→ R, kai èstw c ∈ A.

1. H f eÐnai suneq c sto c ann

∀ε > 0 ∃δ > 0 : |x− c| < δ ⇒ |f(x)− f(c)| < ε. (3.1)

2. H f eÐnai dexi� suneq c sto c ann

∀ε > 0 ∃δ > 0 : c ≤ x < c+ δ ⇒ |f(x)− f(c)| < ε. (3.2)

3. H f eÐnai arister� suneq c sto c ann

∀ε > 0 ∃δ > 0 : c− δ < x ≤ c⇒ |f(x)− f(c)| < ε. (3.3)

Apìdeixh. ProkÔptei �mesa apì ton orismì.

Parathr seic

1. Apì thn (3.1) uponoeÐtai ìti to δ eÐnai tètoio ¸ste (c− δ, c+ δ) ⊆ A. Pr�gmati,se diaforetik  perÐptwsh h (3.1) den ja eÐqe nìhma.

2. OmoÐwc, apì thn (3.2) uponoeÐtai ìti to δ eÐnai tètoio ¸ste [c, c+ δ) ⊆ A.

3. OmoÐwc, apì thn (3.3) uponoeÐtai ìti to δ eÐnai tètoio ¸ste (c− δ, c] ⊆ A.

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3.1. ORISM�OS KAI BASIK�ES IDI�OTHTES SUN�EQEIAS 57

Par�deigma 3.1. (H x2 eÐnai suneq c sto 0) Ja deÐxoume ìti h sun�rthsh f(x) =x2 eÐnai suneq c sto mhdèn, qrhsimopoi¸ntac ton orismì kat� Cauchy.

Prèpei na deÐxoume ìti gia k�je ε > 0, up�rqei δ > 0 tètoio ¸ste |x − 0| < δ ⇒|f(x)− f(0)| < ε. 'Estw loipìn ε > 0. OrÐzoume δ =

√ε > 0. 'Eqoume:

|x− 0| < δ ⇒ x2 < δ2 = ε⇒ |x2 − 02| < ε⇒ |f(x)− f(0)| < ε.

Je¸rhma 3.1. (Sunèqeia gnwst¸n sunart sewn)

1. Oi poluwnumikèc sunart seic eÐnai pantoÔ suneqeÐc.

2. Oi rhtèc sunart seic eÐnai suneqeÐc, ìpou orÐzontai.

3. Oi trigwnometrikèc sunart seic eÐnai suneqeÐc, ìpou orÐzontai.

4. H sun�rthsh f(x) = |x| eÐnai pantoÔ suneq c.

Apìdeixh. Ta ìria pou qreiazìmaste èqoun  dh upologisjeÐ, ektìc tou teleutaÐou poueÐnai aplì.

Je¸rhma 3.2. (Pr�xeic metaxÔ suneq¸n sunart sewn) 'Estw oi sunart seic f(x)kai g(x), kai èstw pwc eÐnai suneqeÐc (  arister� suneqeÐc,   dexi� suneqeÐc) sto c.'Estw k ∈ R, n ∈ N. Tìte eÐnai suneqeÐc (  arister� suneqeÐc,   dexi� suneqeÐc) stoc kai oi kf(x), f(x) + g(x), f(x)− g(x), f(x)g(x), f(x)/g(x) (efìson g(c) 6= 0), kaih (f(x))n.

Apìdeixh. Se ìlec tic peript¸seic, h apìdeixh eÐnai apl  efarmog  basik¸n idiot twntou orÐou.

Je¸rhma 3.3. (SÔnjesh suneq¸n sunart sewn) An limx→c

g(x) = L kai f(t) eÐnai

suneq c sto L, tìtelimx→c

f(g(x)) = f(limx→c

g(x)) = f(L).

Epiplèon, an h g(x) eÐnai suneq c sto c, dhlad  g(c) = L, tìte h f(g(x)) eÐnai suneq csto c:

limx→c

f(g(x)) = f(g(c)).

Apìdeixh. 'Estw ε > 0. AfoÔ h f(t) eÐnai suneq c sto L, mpor¸ na brw δ1 > 0 tètoio¸ste

|t− L| < δ1 ⇒ |f(t)− f(L)| < ε.

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58 KEF�ALAIO 3. SUN�EQEIA

Autì ja isqÔei kai an t = g(x) gia k�poio x:

|g(x)− L| < δ1 ⇒ |f(g(x))− f(L)| < ε.

'Omwc h g(x) èqei ìrio to L, �ra, gia to sugkekrimèno δ1 up�rqei δ2 tètoio ¸ste

0 < |x− c| < δ2 ⇒ |g(x)− L| < δ1.

'Ara, gia to ε pou epèlexa, br ka k�poio δ2 tètoio ¸ste

0 < |x− c| < δ2 ⇒ |f(g(x))− f(L)| < ε,

�ra telik� limx→c

f(g(x)) = f(L).

An t¸ra eÐnai kai h g(x) suneq c, �mesa prokÔptei, apì ton orismì thc sunèqeiackai thc sÔnjethc sun�rthshc, ìti h f(g(x)) eÐnai suneq c sto c.

Parat rhsh: Ta Jewr mata 3.2 kai 3.3, kai eidik� to Je¸rhma 3.3, mac epitrèpounna apodeiknÔoume th sunèqeia sunart sewn se peript¸seic pou h qr sh tou orismoÔja  tan apì kopiastik  èwc adÔnath. DeÐte ta akìlouja paradeÐgmata.

Par�deigma 3.2. Apì prohgoÔmena paradeÐgmata, prokÔptei ìti h f(x) =√x eÐnai

suneq c se k�je c ∈ (0,+∞). 'Ara, me efarmog  tou Jewr matoc 3.3 prokÔptei ìtian lim

x→cf(x) > 0, tìte

limx→c

√f(x) =

√limx→c

f(x).

Par�deigma 3.3. Ja deÐxoume ìti h

h(x) = cos

(x3 + 2x+ 7

x2 + x− 2

)eÐnai suneq c pantoÔ, ektìc apì ta 1 kai −2.

Katarq n parathroÔme ìti x2 + x− 2 = (x− 1)(x+ 2). 'Ara h rht  sun�rthsh

g(x) =x3 + 2x+ 7

x2 + x− 2

eÐnai pantoÔ suneq c ektìc apì tic peript¸seic x = 1,−2. Epiplèon, h cos(x) eÐnaisuneq c pantoÔ. 'Ara telik� h sÔnjeth sun�rthsh f(g(x)) eÐnai suneq c pantoÔ ektìcapì ta 1,−2. 'Ara, gia par�deigma,

limx→3

cos

(x3 + 2x+ 7

x2 + x− 2

)= cos

(33 + 2× 3 + 7

32 + 3− 2

)= cos 4.

Par�deigma 3.4. MporeÐ na efarmosteÐ to �nw je¸rhma gia g(x) = x2, c = L = 0,kai f(x) orismènh wc ex c?

f(x) =

{1, x > 0,

0, x ≤ 0.

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3.2. SUN�EPEIES SUN�EQEIAS SE KLEIST�O FRAGM�ENO DI�ASTHMA 59

3.2 Sunèpeiec Sunèqeiac se Kleistì Fragmèno Di�sth-

ma

Orismìc 3.2. (Sunèqeia se di�sthma)

1. H sun�rthsh f orÐzetai wc suneq c sto anoiqtì di�sthma (a, b) an eÐnai suneq cse k�je x ∈ (a, b).

2. H sun�rthsh f orÐzetai wc suneq c sto kleistì di�sthma [a, b] an eÐnai suneq cse k�je x ∈ (a, b), dexi� suneq c sto a kai arister� suneq c sto b.

3. AntÐstoiqa oi peript¸seic gia ta diast mata [a, b), [a,∞), ktl.

Je¸rhma 3.4. (Diat rhshc prìshmou)

1. 'Estw f : A → R suneq c sto c kai f(c) 6= 0. Up�rqei δ > 0 tètoio ¸ste(c− δ, c+ δ) ⊆ A kai gia k�je x ∈ (c− δ, c+ δ), to f(x) na èqei to Ðdio prìshmome to f(c).

2. 'Estw f : A → R dexi� suneq c sto c kai f(c) 6= 0. Up�rqei δ > 0 tètoio ¸ste[c, c + δ) ⊆ A kai gia k�je x ∈ [c, c + δ), to f(x) na èqei to Ðdio prìshmo me tof(c).

3. 'Estw f : A → R arister� suneq c sto c kai f(c) 6= 0. Up�rqei δ > 0 tètoio¸ste (c − δ, c] ⊆ A kai gia k�je x ∈ (c − δ, c], to f(x) na èqei to Ðdio prìshmome to f(c).

Apìdeixh. Katarq n parathr ste ìti diaisjhtik� to je¸rhma eÐnai profanèc: An hsun�rthsh eÐnai suneq c se k�poio c, autì shmaÐnei ìti arkoÔntwc kont� sto c, oitimèc ja eÐnai arkoÔntwc kont� sto f(c), �ra moiraÐa ja moir�zontai kai to prìshmìtou.

Ja apodeÐxoume mìno thn pr¸th perÐptwsh, kaj¸c oi �llec prokÔptoun an�loga.Gia na apodeÐxoume to zhtoÔmeno, ìpwc faÐnetai kai sto Sq ma 3.2, èstw ìti f(c) > 0.

(H perÐptwsh f(c) < 0 prokÔptei an�loga.) 'Estw ε = f(c)2 > 0. Lìgw thc sunèqeiac

thc f(x), up�rqei δ tètoio ¸ste

x ∈ (−δ + c, c+ δ)⇒ |f(x)− f(c)| < ε =f(c)

2

⇒ f(x)− f(c) > −f(c)

2⇒ f(x) >

f(c)

2> 0.

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60 KEF�ALAIO 3. SUN�EQEIA

c c+δc-δ

f(c)

f(c)/2

3f(c)/2

f(x)

x

y

Sq ma 3.2: Apìdeixh tou Jewr matoc Diat rhshc Prìshmou.

Je¸rhma 3.5. (Bolzano) 'Estw f(x) suneq c sto [a, b] kai èstw f(a)f(b) < 0.Tìte up�rqei c ∈ (a, b) tètoio ¸ste f(c) = 0.

Apìdeixh. QwrÐc bl�bh thc genikìthtac, èstw f(a) < 0, �ra f(b) > 0. 'Estw

S = {x ∈ [a, b] : f(x) ≤ 0},dhlad  to sÔnolo ìlwn twn x gia ta opoÐa h sun�rthsh lamb�nei arnhtikèc timèc  thn tim  0. To sÔnolo eÐnai mh kenì, afoÔ perièqei to a, en¸ eÐnai fragmèno �nw, apìto b (an kai to b den an kei sto sÔnolo). Apì to AxÐwma thc Plhrìthtac, to S èqeisupremum, èstw c. To c an kei sto [a, b] giatÐ (a) wc �nw fr�gma, eÐnai megalÔtero  Ðso tou a ∈ S kai (b) wc el�qisto �nw fr�gma, eÐnai mikrìtero   Ðso tou fr�gmatocb. DeÐte to Sq ma 3.3.

O skopìc thc upìloiphc apìdeixhc eÐnai na deÐxoume ìti f(c) = 0. DiaqwrÐzoumepeript¸seic:

1. 'Estw pwc f(c) < 0. Profan¸c c < b. MporeÐ na èqoume c = a   c 6= a.

(aþ) An c = a, afoÔ h f eÐnai suneq c sto a, apì to Je¸rhma 3.4 (Je¸rhma dia-t rhshc prìshmou) prokÔptei ìti up�rqei k�poio δ > 0 tètoio ¸ste f(x) < 0gia k�je x ∈ [c, c+ δ).

(bþ) An c 6= a, tìte apì to Je¸rhma 3.4 prokÔptei ìti up�rqei k�poio δ > 0tètoio ¸ste f(x) < 0 gia k�je x ∈ (−δ + c, c+ δ).

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3.2. SUN�EPEIES SUN�EQEIAS SE KLEIST�O FRAGM�ENO DI�ASTHMA 61

(a,f(a))

(b,f(b))

(c,f(c)=0)

x

y

f(x)

S S

Sq ma 3.3: Apìdeixh tou Jewr matoc tou Bolzano.

Epomènwc se k�je perÐptwsh to c den eÐnai supremum tou S, afoÔ up�rqei ènacarijmìc, ac poÔme to c + δ

2 , pou eÐnai megalÔteroc tou c kai an kei sto S, diìtif(c + δ

2) < 0. 'Ara, ft�noume se �topo, kai h perÐptwsh f(c) < 0 prèpei naapokleisteÐ.

2. 'Estw pwc f(c) > 0. Profan¸c c > a. MporeÐ na èqoume c = b   c 6= b.

(aþ) An c = b, afoÔ h f eÐnai suneq c sto b, apì to Je¸rhma 3.4 (Je¸rhma diat -rhshc prìshmou), prokÔptei ìti up�rqei k�poio δ > 0 tètoio ¸ste f(x) > 0gia k�je x ∈ (−δ + c, c].

(bþ) An c 6= b, tìte apì to Je¸rhma 3.4 prokÔptei ìti up�rqei k�poio δ > 0 tètoio¸ste f(x) > 0 gia k�je x ∈ (c− δ, c+ δ).

Epomènwc se k�je perÐptwsh to c den eÐnai supremum, giatÐ, apì to L mma 1.3,ja èprepe na up rqe arijmìc x entìc tou (c−δ, c] me f(x) ≤ 0, ¸ste x ∈ S. 'Ara,ft�noume p�li se �topo, kai h perÐptwsh f(c) > 0 epÐshc prèpei na apokleisteÐ.

3. 'Ara telik�, prèpei na èqoume f(c) = 0.

Parathr seic

1. Ja  tan adÔnato na apodeÐxoume to Je¸rhma Bolzano qwrÐc qr sh k�poiac mor-f c tou Axi¸matoc thc Plhrìthtac.

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62 KEF�ALAIO 3. SUN�EQEIA

2. To Je¸rhma Bolzano eÐnai exairetik� qr simo gia dÔo lìgouc:

(aþ) Se autì basÐzontai oi apodeÐxeic poll¸n �llwn shmantik¸n jewrhm�twn,ìpwc gia par�deigma tou Jewr matoc Endi�meshc Tim c pou ja doÔme sthsunèqeia, all� kai giatÐ

(bþ) apì mìno tou èqei pollèc efarmogèc. DeÐte ta akìlouja paradeÐgmata.

Par�deigma 3.5. (RÐza exÐswshc) Ja deÐxoume ìti h exÐswsh x2 + 4 sinx− 2 = 0èqei toul�qiston mia lÔsh. Pr�gmati, èstw f(x) = x2 + 4 sinx− 2. Ja efarmìsoumeto Je¸rhma Bolzano gia th sun�rthsh f(x) sto di�sthma [−π/2, π/2]. Pr�gmati,h f eÐnai suneq c sto [−π/2, π/2] wc grammikìc sunduasmìc suneq¸n sunart sewn,Epiplèon,

f(−π/2) = π2/4 + 4 sin(−π/2)− 2 = π2/4− 6 < 0,

f(π/2) = π2/4 + 4 sin(π/2)− 2 = π2/4 + 2 > 0.

Sunep¸c oi proôpojèseic tou Jewr matoc Bolzano ikanopoioÔntai kai up�rqei x0 ∈[−π/2, π/2] tètoio ¸ste f(x0) = 0, dhlad  x2

0 + 4 sinx0 − 2 = 0.

Par�deigma 3.6. (Sun�rthsh me stajerì shmeÐo) 'Estw sun�rthsh suneq c stokleistì di�sthma [a, b], me f(a) ≥ a kai f(b) ≤ b. Ja deÐxoume ìti h sun�rthsh èqeistajerì shmeÐo sto [a, b], dhlad  up�rqei c ∈ [a, b] tètoio ¸ste f(c) = c.

OrÐzoume th sun�rthsh g(x) = f(x)−x, pou eÐnai suneq c sto [a, b] wc grammikìcsunduasmìc suneq¸n sunart sewn. ParathroÔme pwc: g(a) = f(a) − a ≥ 0 kaig(b) = f(b)−b ≤ 0. Up�rqoun dÔo peript¸seic: eÐte èna apì ta g(a), g(b) eÐnai 0, eÐtekai ta dÔo eÐnai diaforetik� apì to 0. Sth deÔterh perÐptwsh, èqoume anagkastik�g(a) > 0 kai g(b) < 0. Efarmìzontac to Je¸rhma Bolzano, èqoume ìti up�rqeic ∈ (a, b) tètoio ¸ste g(c) = 0. 'Ara se k�je perÐptwsh, up�rqei c ∈ [a, b], tètoio¸ste g(c) = 0 ⇒ f(c) − c = 0 ⇒ f(c) = c, dhlad  se k�je perÐptwsh up�rqeistajerì shmeÐo thc f(c).

H gewmetrik  ermhneÐa tou apotelèsmatoc eÐnai h ex c: An xekin soume apì ènashmeÐo (a, f(a)) pou an kei sthn eujeÐa y = x   brÐsketai p�nw apì aut , kai prèpeina p�me se èna shmeÐo (b, f(b)) pou an kei sthn eujeÐa   brÐsketai k�tw apì aut ,me mia suneq  gramm , eÐnai bèbaio ìti se k�poio shmeÐo thc diadrom c ja prèpei nasunant soume thn eujeÐa y = x. 'Ena par�deigma faÐnetai sto Sq ma 3.4.

Par�deigma 3.7. (Antipar�deigma) Gia na efarmìsoume to Je¸rhma tou Bolzano,prèpei na elègxoume prosektik� ìti ikanopoioÔntai ìlec oi proôpojèseic. DeÐte giapar�deigma sto Sq ma 3.5 th sun�rthsh f(x) = 1

x sto di�sthma [−1, 1]. GiatÐ denisqÔei ed¸ to Je¸rhma tou Bolzano?

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3.2. SUN�EPEIES SUN�EQEIAS SE KLEIST�O FRAGM�ENO DI�ASTHMA 63

x=y

a b

f(a)

f(b)

c

f(c)=cf(x)

x

y

Sq ma 3.4: H sun�rthsh tou ParadeÐgmatoc 3.6.

−1 −0.5 0 0.5 1−10

−5

0

5

10

Sq ma 3.5: Par�deigma 3.7: Mia sun�rthsh gia thn opoÐa den mporeÐ na efarmosteÐ to Je¸rhma Bol-zano.

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64 KEF�ALAIO 3. SUN�EQEIA

x

y

a b

x1x2

f(x1)

f(x2)

Sq ma 3.6: Gewmetrik  ermhneÐa tou Jewr matoc Endi�meshc Tim c. Prokeimènou h sun�rthsh f(x)na p�ei apì thn tim  f(x1) sthn tim  f(x2), prèpei anagkastik� na p�rei ìlec tic endi�mesec timèc(f(x1), f(x2)) (endeqomènwc na p�rei kai �llec).

Je¸rhma 3.6. (Endi�meshc Tim c) 'Estw f(x) suneq c sto [a, b]. 'Estw x1, x2 ∈[a, b] me x1 < x2 kai f(x1) 6= f(x2). H f(x) paÐrnei ìlec tic timèc metaxÔ twn f(x1)kai f(x2) sto (x1, x2).

Apìdeixh. 'Estw f(x1) < f(x2). (H perÐptwsh f(x1) > f(x2) antimetwpÐzetai an�lo-ga.) 'Estw opoiad pote tim  k tètoia ¸ste f(x1) < k < f(x2). 'Estw h sun�rthshg(x) = f(x) − k. 'Olec oi sunj kec tou Jewr matoc Bolzano ikanopoioÔntai sto[x1, x2], �ra up�rqei c ∈ (x1, x2) tètoio ¸ste g(c) = f(c)− k = 0⇒ f(c) = k. DeÐteto Sq ma 3.6 gia mia gewmetrik  ermhneÐa tou jewr matoc.

Par�deigma 3.8. Efarmìzontac to je¸rhma gia thn sun�rthsh f(x) = sinx kaigia a = 0, b = 2π, x1 = π

2 , x2 = 3π2 , prokÔptei pwc h sun�rthsh lamb�nei ìlec tic

timèc apì to −1 mèqri to 1.

Par�deigma 3.9. Ja qrhsimopoi soume to Je¸rhma Endi�meshc Tim c gia na deÐ-xoume ìti ìtan mia suneq c sun�rthsh f(x) paÐrnei mìno rhtèc timèc se èna di�sthmaI, tìte eÐnai stajer . Pr�gmati, èstw pwc h sun�rthsh den eÐnai stajer , dhlad up�rqoun x1, x2 ∈ I me f(x1) < f(x2). Apì to Je¸rhma Endi�meshc Tim c, h f(x) japaÐrnei ìlec tic timèc sto (f(x1), f(x2)). Autì ìmwc eÐnai �topo, giatÐ autì to sÔnoloperièqei kai �rrhtouc, pou ex' upojèsewc èqoun apokleisteÐ.

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3.2. SUN�EPEIES SUN�EQEIAS SE KLEIST�O FRAGM�ENO DI�ASTHMA 65

Parathr ste ìti me ton Ðdio trìpo ja mporoÔsame na apodeÐxoume epÐshc pwc miasuneq c sun�rthsh pou paÐrnei mìno �rrhtec timèc se èna di�sthma eÐnai stajer .

Je¸rhma 3.7. (Fr�gma suneq¸n sunart sewn) 'Estw f(x) suneq c sto [a, b]. Hf(x) eÐnai fragmènh, dhlad , up�rqei M tètoio ¸ste |f(x)| < M gia k�je x ∈ [a, b].

Apìdeixh. Ja qrhsimopoi soume apagwg  sto �topo. 'Estw pwc h f(x) den eÐnaifragmènh sto [a, b], dhlad  gia k�je M up�rqei x tètoio ¸ste |f(x)| > M . 'Arah f(x) den eÐnai fragmènh eÐte sto [a, a+b

2 ] eÐte sto [a+b2 , b]. OrÐzw wc [a1, b1] to

antÐstoiqo di�sthma. SuneqÐzontac, mpor¸ na orÐzw diast mata [an, bn], fjÐnontocmegèjouc

|bn − an| =b− a

2n

sta opoÐa h f(x) den eÐnai fragmènh. An se k�poio b ma h f(x) den eÐnai fragmènh kaista dÔo upodiast mata, epilègw to aristerì.

'Estw x0 = sup{an}, pou up�rqei giatÐ to sÔnolo {an} eÐnai mh kenì kai fragmèno�nw (apì to b). Up�rqoun treic peript¸seic, a < x0 < b, x0 = a kai x0 = b:

1. 'Estw pwc a < x0 < b. 'Estw ε = 1. Lìgw sunèqeiac sto x0,

∃δ > 0 : x0 − δ < x < x0 + δ ⇒ |f(x)− f(x0)| < 1⇒ |f(x)| < 1 + |f(x0)|.'Omwc, gia èna arkoÔntwc meg�lo n, èqoume ìti [an, bn] ⊂ (x0 − δ, x0 + δ), �raèqoume �topo.

2. (Parathr ste pwc se aut  thn perÐptwsh epilègetai p�nta to aristerì upodi�-sthma.) An x0 = a, tìte èstw p�li ε = 1. Lìgw dexi�c sunèqeiac sto x0,èqoume:

∃δ > 0 : x0 ≤ x < x0 + δ ⇒ |f(x)− f(x0)| < 1⇒ |f(x)| < 1 + |f(x0)|.'Omwc, gia èna arkoÔntwc meg�lo n, èqoume ìti [an, bn] ⊂ [x0, x0 + δ), �ra kaip�li èqoume �topo.

3. (Parathr ste pwc se aut  thn perÐptwsh epilègetai p�nta to dexÐ upodi�sthma.)An x0 = b, tìte èstw p�li ε = 1. Lìgw arister c sunèqeiac sto x0, èqoume:

∃δ > 0 : x0 − δ < x ≤ x0 ⇒ |f(x)− f(x0)| < 1⇒ |f(x)| < 1 + |f(x0)|.'Omwc, gia èna arkoÔntwc meg�lo n, èqoume ìti [an, bn] ⊂ (x0 − δ, x0], �ra kaip�li èqoume �topo.

'Ara se k�je perÐptwsh èqoume �topo, �ra h f(x) eÐnai fragmènh.

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66 KEF�ALAIO 3. SUN�EQEIA

−1 −0.5 0 0.5 1−0.5

0

0.5

1

0 5 100

0.2

0.4

0.6

0.8

1

Sq ma 3.7: Par�deigma 3.10.

Je¸rhma 3.8. (Akrìtatwn) 'Estw f suneq c sto [a, b]. Tìte up�rqoun shmeÐa c,d entìc tou [a, b] gia ta opoÐa isqÔei

f(c) = supa≤x≤b

f, f(d) = infa≤x≤b

f.

Apìdeixh. Ja apodeÐxoume to skèloc gia to supremum. To skèloc gia to infimumprokÔptei tìte apl�, exet�zontac thn sun�rthsh −f kai efarmìzontac to skèloc giato supremum.

Ja qrhsimopoi soume kai p�li apagwg  se �topo. 'Estw ìti sup f = M , kai pwcf(x) < M gia k�je x ∈ [a, b], �ra h suneq c sun�rthsh g(x) = M − f(x) > 0 giak�je x ∈ [a, b]. H 1

g(x) eÐnai epÐshc suneq c sto [a, b], �ra kai fragmènh sto [a, b], �ra1

g(x) < C ⇒ M − f(x) > 1C ⇒ f(x) < M − 1

C gia ìla ta x ∈ [a, b]. Autì antibaÐneiton orismì tou supremum wc to mikrìtero �nw fr�gma, �ra ft�same se �topo.

Parathr seic

1. Parathr ste ìti an mia sun�rthsh eÐnai suneq c sto [a, b], apì to Je¸rhma 3.7ja eÐnai kai fragmènh, �ra ja èqei infimum kai supremum, �ra to �nw je¸rhmaèqei nìhma.

2. To je¸rhma lèei ìti to supremum eÐnai kai maximum, kai to infimum eÐnai kaiminimum. 'Opwc faÐnetai kai apì to Par�deigma 3.10, autì den isqÔei genik�.

Par�deigma 3.10. (AntiparadeÐgmata) Exet�ste giatÐ to Je¸rhma twn Akrìtatwnden mporeÐ na efarmosteÐ stic akìloujec peript¸seic tou Sq matoc 3.7.

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3.2. SUN�EPEIES SUN�EQEIAS SE KLEIST�O FRAGM�ENO DI�ASTHMA 67

1. a = −1, b = 1, f(x) =

0, x = 0,

x+ 1, −1 < x < 0,

1− x, 0 < x < 1.

2. a = 1, b = +∞, f(x) = 1x .

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68 KEF�ALAIO 3. SUN�EQEIA

3.3 Mèjodoc Diqotìmhshc

Orismìc 3.3. (RÐza) 'Estw f(x) aujaÐreth sun�rthsh, kai x0 tètoio ¸ste f(x0) =0. To x0 kaleÐtai rÐza thc f .

Parat rhsh: To Je¸rhma Bolzano mac epitrèpei na entopÐzoume arijmhtik� ticrÐzec suneq¸n sunart sewn ìtan den up�rqoun se kleist  morf  (  den brÐskoumekleist  morf ,   den jèloume na y�xoume gia kleist  morf ). Sugkekrimèna,

1. 'Estw f(x) sun�rthsh suneq c sto [a0, b0].

2. 'Estw f(a0)f(b0) < 0, �ra apì to je¸rhma tou Bolzano up�rqei toul�qiston miarÐza sto (a0, b0).

3. 'Estw E > 0 to sf�lma me to opoÐo jèloume na entopÐsoume thn jèsh miac rÐzac,dhlad  jèloume na broÔme k�poion arijmì x′ pou na apèqei apì k�poia rÐza x0

ligìtero apì E: |x0 − x′| < E.

MporoÔme na ektelèsoume thn akìloujh Mèjodo thc Diqotìmhshc.

/* BISECTION METHOD */

INPUT: a, b, E, f() /* f(a)f(b)<0 */

OUTPUT: m /* ESTIMATED LOCATION OF ROOT */

m=(a+b)/2;

DO

IF f(m)=0,

EXIT;

ELSEIF f(m)f(b)<0,

a=m;

ELSE, /* WE MUST HAVE f(m)f(a)<0 */

b=m;

END;

m=(a+b)/2;

IF b-a < E,

EXIT;

END;

END

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3.3. M�EJODOS DIQOT�OMHSHS 69

3 3.5 4 4.5 5−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

3

3.5

4

0

0

x

y

1

2

3

4 5

Sq ma 3.8: Par�deigma 3.11.

Parathr seic

1. H mèjodoc eÐnai arg , se sqèsh me �llec (kakì).

2. Den qrei�zetai o upologismìc parag¸gwn (kalì).

3. Met� apì 10 epanal yeic, h abebaiìthta èqei meiwjeÐ perÐpou 1000 forèc (giatÐ?).

4. H mèjodoc sugklÐnei p�nta se k�poia rÐza (kalì).

5. Perissìtera se maj mata Arijmhtik c An�lushc.

Par�deigma 3.11. 'Estw ìti y�qnoume mia rÐza thc sun�rthshc x − 2 + 2 cosxsto di�sthma [3, 5]. Sto Sq ma 3.8 èqoume sqedi�sei thn sun�rthsh, kai ta pr¸ta 5shmeÐa pou dhmiourgeÐ o algìrijmoc. An ektelèsoume thn epan�lhyh tou algìrijmou10 sunolik� forèc, ja èqoume:

n 0 1 2 2 4 5 6 7 8 9 10a(n) 3 3 3.5 3.5 3.625 3.6875 3.6875 3.6875 3.6953 3.6953 3.6973b(n) 5 4 4 3.75 3.75 3.75 3.7188 3.7031 3.7031 3.6992 3.6992

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70 KEF�ALAIO 3. SUN�EQEIA

3.4 AntÐstrofh SuneqoÔc Sun�rthshc

Orismìc 3.4. (AntÐstrofh sun�rthsh) 'Estw sun�rthsh f : A → R, pou eÐnai1-1. OrÐzw thn antÐstrofh sun�rthsh f−1 : f(A)→ R wc ex c:

f−1(y) = x⇔ f(x) = y.

Parathr seic

1. H apaÐthsh na eÐnai h f 1-1 eÐnai aparaÐthth prokeimènou na mporeÐ na oristeÐ hantÐstrofh thc.

2. 'Eqei h sun�rthsh tou Sq matoc 3.9, orismènh sto [a, c], antÐstrofh sun�rthsh?An nai, mporeÐte na th sqedi�sete?

3. 'Enac trìpoc na dhmiourg soume to gr�fhma thc antÐstrofhc sun�rthshc, eÐnaina parathr soume ìti ta graf mata thc f kai thc f−1 eÐnai summetrik� gÔrwapì thn eujeÐa x = y. (Nooumènou ìti k�je mia sun�rthsh èqei touc dikoÔc thc�xonec.)

4. 'Enac �lloc trìpoc na fantastoÔme pwc eÐnai to gr�fhma thc antÐstrofhc, eÐnaina fantastoÔme ìti koit�zoume to arqikì gr�fhma mèsa apì th selÐda, èqontacgurismèno to kef�li mac 90 moÐrec aristerìstrofa.

Je¸rhma 3.9. (AntÐstrofh suneqoÔc gnhsÐwc aÔxousac sun�rthshc) 'Estw su-n�rthsh f , me pedÐo orismoÔ to [a, b], suneq c kai gnhsÐwc aÔxousa. Tìte

1. H f eÐnai 1-1 me pedÐo tim¸n to f([a, b]) = [c, d], ìpou c = f(a), d = f(b).Sunep¸c èqei antÐstrofh sun�rthsh f−1 : [c, d]→ R me f−1([c, d]) = [a, b].

2. H f−1 eÐnai gnhsÐwc aÔxousa.

3. H f−1 eÐnai suneq c.

Apìdeixh. ParaleÐpetai. DeÐte to Sq ma 3.10 gia na katal�bete thn gewmetrik  sh-masÐa tou jewr matoc.

Je¸rhma 3.10. To Je¸rhma 3.9 isqÔei, me tic kat�llhlec tropopoi seic, gia thnperÐptwsh pou h f eÐnai gnhsÐwc fjÐnousa.

Apìdeixh. ParaleÐpetai.

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3.4. ANT�ISTROFH SUNEQO�US SUN�ARTHSHS 71

x

y

f(a)

ca b

f(b)

f(c)

f(x)

Sq ma 3.9: 'Eqei h sun�rthsh f(x) antÐstrofh sun�rthsh f−1(y)?

Par�deigma 3.12. ('Artiec rÐzec) 'Estw n �rtioc. Epeid  h sun�rthsh f(x) = xn

eÐnai suneq c kai gnhsÐwc aÔxousa sto [0, b], h sun�rthsh f−1(y), dhlad  h x = y1n ,

eÐnai suneq c kai gnhsÐwc aÔxousa sto di�sthma [0, bn]. Epeid  autì isqÔei gia k�jeb > 0, h f−1(y) ja eÐnai suneq c kai gnhsÐwc aÔxousa kai sto [0,+∞). (DeÐte toJe¸rhma 1.3.)

Par�deigma 3.13. (Perittèc rÐzec) 'Estw n perittìc. H sun�rthsh f(x) = xn eÐnaisuneq c kai gnhsÐwc aÔxousa se k�je di�sthma [a, b] ìpou a, b ∈ R. 'Ara h sun�rthshf−1(y), dhlad  h x = y

1n , eÐnai suneq c kai gnhsÐwc aÔxousa sto di�sthma [an, bn].

Epeid  autì isqÔei gia aujaÐreta a, b, h f−1(y) ja eÐnai suneq c kai gnhsÐwc aÔxousakai sto (−∞,+∞). (DeÐte to Je¸rhma 1.3.)

Par�deigma 3.14. (Rhtèc dun�meic) H sun�rthsh xr, r ∈ Q, r > 0, eÐnai gnhsÐwcaÔxousa kai suneq c sto [0,∞).

Pr�gmati, èstw r = pq , me p, q pr¸ta metaxÔ touc kai p, q > 0. H xr eÐnai suneq c

sto [0,∞) giatÐ eÐnai akèraia dÔnamh thc x1q pou eÐnai suneq c sto [0,∞), ìpwc pro-

kÔptei apì ta prohgoÔmena paradeÐgmata. EÐnai epÐshc kai gnhsÐwc aÔxousa giatÐ h x1q

eÐnai aÔxousa (wc antÐstrofh aÔxousac), �ra:

0 ≤ x1 < x2 ⇒ 0 ≤ x1q

1 < x1q

2 ⇒(x

1q

1

)p<

(x

1q

2

)p.

H pr¸th sunepagwg  prokÔptei apì to ìti h x1q eÐnai aÔxousa, en¸ h deÔterh apì

gnwst  idiìthta twn pragmatik¸n arijm¸n.

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72 KEF�ALAIO 3. SUN�EQEIA

x

y

a b

f(a)=c

f(b)=d

y=f(x), x=f -1(y)

Sq ma 3.10: Je¸rhma 3.9.

AfoÔ loipìn h sun�rthsh xr, r ∈ Q, r > 0, eÐnai gnhsÐwc aÔxousa kai suneq csto [0,∞), eÔkola prokÔptei kai pwc h sun�rthsh x−r = 1/xr, r ∈ Q, r > 0, eÐnaignhsÐwc fjÐnousa kai suneq c sto (0,∞).

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3.5. SUN�EQEIA LIPSCHITZ 73

3.5 Sunèqeia Lipschitz

Orismìc 3.5. (Sunèqeia Lipschitz) H f : A→ R kaleÐtai Lipschitz (suneq c) stodi�sthma I ⊆ A an up�rqei èna C > 0 tètoio ¸ste gia k�je x1, x2 ∈ I,

|f(x1)− f(x2)| ≤ C|x1 − x2|.

Parat rhsh: Gia na katal�bete thn ènnoia thc sunèqeiac Lipschitz se di�sthmaI, parathr ste ta ex c:

1. An x1 6= x2, tìte

|f(x1)− f(x2)| ≤ C|x1 − x2| ⇔|f(x1)− f(x2)||x1 − x2|

≤ C.

Sunep¸c, mia sun�rthsh eÐnai Lipschitz suneq c sto I an up�rqei k�poio C tè-toio ¸ste h klÐsh opoioud pote eujÔgrammou tm matoc sundèei dÔo shmeÐa tougraf matoc thc f eÐnai to polÔ C, kat' apìluth tim .

2. EpÐshc, parathr ste pwc ìlec oi grammikèc sunart seic thc morf c f(x) =±Cx + k oriak� ikanopoioÔn thn sunj kh. 'Ara, opoiad pote sun�rthsh me-tab�lletai ligìtero taqÔtera apì autèc, eÐnai Lipschitz suneq c sto I.

3. Tèloc, èstw x0 èna stajerì shmeÐo, kai èstw x > x0. Efarmìzontac ton orismìgia ta x0, x, èqoume

|f(x)− f(x0)| ≤ C(x− x0)⇔ −C(x− x0) ≤ f(x)− f(x0) ≤ C(x− x0)

⇔ −C(x− x0) + f(x0) ≤ f(x) ≤ C(x− x0) + f(x0)

An gia to Ðdio x0 l�boume èna x < x0, me parìmoio trìpo èqoume ìti

C(x− x0) + f(x0) ≤ f(x) ≤ −C(x− x0) + f(x0).

'Ara, gia k�je x0 ∈ I h f(x) prèpei na perioristeÐ sto uposÔnolo tou epipèdoupou brÐsketai metaxÔ thc eujeÐac me klÐsh C pou dièrqetai apì to (x0, f(x0)),dhlad  thc eujeÐac

y = C(x− x0) + f(x0),

kai thc eujeÐac me klÐsh −C pou dièrqetai apì to (x0, f(x0)), dhlad  thc eujeÐac

y = −C(x− x0) + f(x0).

To uposÔnolo autì emfanÐzetai skiasmèno sto Sq ma 3.11.

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74 KEF�ALAIO 3. SUN�EQEIA

x

a

f(x)

b

y

x

Sq ma 3.11: Gewmetrik  ermhneÐa thc sunèqeiac Lipschitz se èna di�sthma I = [a, b].

−2 −1 0 1 20

1

2

3

4

−2 −1 0 1 20

0.5

1

1.5

2

0 2 4 6

−1

−0.5

0

0.5

1

Sq ma 3.12: Oi sunart seic tou ParadeÐgmatoc 3.15.

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3.5. SUN�EQEIA LIPSCHITZ 75

Par�deigma 3.15. Ja exet�soume thn sunèqeia Lipschitz twn akìloujwn sunar-t sewn:

1. f(x) = cos x.

2. f(x) = x2.

3. f(x) =√|x|.

(DeÐte to Sq ma 3.12.)

1. ParathroÔme pwc gia k�je x1, x2 isqÔei

|f(x1)− f(x2)| = | cosx1 − cosx2|

=

∣∣∣∣2 sinx1 + x2

2sin

x2 − x1

2

∣∣∣∣ ≤ 2

∣∣∣∣sin x2 − x1

2

∣∣∣∣ ≤ 2|x2 − x1|

2= |x1 − x2|,

H teleutaÐa anisìthta prokÔptei apì th gnwst  sqèsh | sinx| ≤ |x|. B�sei thc�nw, h cosx eÐnai Lipschitz suneq c se k�je uposÔnolo tou R, me C = 1.

2. 'Estw katarq n èna opoiod pote di�sthma I fragmèno, dhlad  x ∈ I ⇒ |x| < M ,gia k�poioM ∈ R. (Gia par�deigma, to I = (0, 1) eÐnai aut c thc morf c.) 'EstwdÔo x1, x2 ∈ I. Tìte:

|x21 − x2

2| = |x1 − x2||x1 + x2| ≤ 2M |x1 − x2|.

'Ara, an to I eÐnai fragmèno, h f eÐnai Lipschitz sto I, me C = 2M , ìpou M ènafr�gma tou I.

'Estw t¸ra pwc to I den eÐnai fragmèno. Tìte toul�qiston èna apì ta �kra toueÐnai to ∞   to −∞, gia par�deigma I = (0,∞), k.o.k. Den mporeÐ na isqÔei

|x21 − x2

2| = |x1 − x2||x1 + x2| ≤M |x1 − x2|.

gia kanènaM , giatÐ mporoÔme na k�noume to |x1 +x2| ìso meg�lo jèloume. 'Ara,an to I den eÐnai fragmèno, h f den eÐnai Lipschitz sto I.

3. 'Estw pwc h√|x| èqei oristeÐ se èna I pou eÐte perièqei to 0 eÐte èqei to 0 wc

sÔnoro. Ac upojèsoume gia par�deigma ìti I = (0, a]. (Oi �llec peript¸seicantimetwpÐzontai an�loga.) An x1, x2 ∈ (0, a], tìte to

|f(x1)− f(x2)| = |√|x1| −

√|x2|| =

|x1 − x2|√x1 +

√x2,

mporeÐ na gÐnei aujaÐreta megalÔtero apì to C|x1−x2|, gia k�je epilog  tou C,arkeÐ na p�rw ta x1, x2 arkoÔntwc mikr�.

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76 KEF�ALAIO 3. SUN�EQEIA

'Estw t¸ra opoiod pote di�sthma I pou den èqei sto �kro tou to 0. 'Estw giapar�deigma pwc I = [a,∞), ìpou a > 0. (Oi �llec peript¸seic antimetwpÐzontaian�loga.) 'Eqoume, gia k�je x1, x2 ∈ I,

|√|x1| −

√|x2|| =

|x1 − x2|√x1 +

√x2≤ 1

2√a|x1 − x2|,

�ra se aut  thn perÐptwsh h f eÐnai Lipschitz suneq c me C = 12√a, kai genik� h

sun�rthsh eÐnai Lipschitz suneq c se diast mata pou oÔte perièqoun to 0, oÔteto èqoun �kro.

Je¸rhma 3.11. (Sunèqeia Lipschitz ⇒ sunèqeia) An h f : A→ R eÐnai Lipschitzsuneq c se èna di�sthma I, tìte eÐnai kai suneq c sto I.

Apìdeixh. 'Estw èna eswterikì shmeÐo c tou I. Lìgw thc sunèqeiac Lipschitz, jaèqoume, gia k�je x ∈ I, ìti

|f(x)− f(c)| ≤ C|x− c| ⇔ −C|x− c|+ f(c) ≤ f(x) ≤ C|x− c|+ f(c).

Efarmìzontac to Je¸rhma thc Parembol c, prokÔptei pwc h f(x) èqei ìrio limx→c

f(x) =

f(c), kai �ra eÐnai suneq c.Se perÐptwsh pou to c eÐnai oriakì shmeÐo tou I, dhlad  I = [c, b]   I[a, c], tìte h

�nw apìdeixh tropopoieÐtai an�loga.

Parathr seic

1. To antÐstrofo tou jewr matoc den isqÔei! SkefteÐte thn f(x) =√|x| sto

[−1, 1].

2. Sunep¸c, mporoÔme na skeftìmaste thn sunèqeia Lipschitz san èna isqurì/stenìeÐdoc sunèqeiac.

3. Antijètwc, h sunèqeia Lipschitz eÐnai pio eureÐa apì thn paragwgisimìthta. Dh-lad , up�rqoun sunart seic pou den eÐnai paragwgÐsimec se k�poio di�sthma, all�eÐnai Lipschitz suneq c. 'Ena par�deigma eÐnai h f(x) = |x| sto R.

4. H praktik  qrhsimìthta thc sunèqeiac Lipschitz eÐnai ìti pollèc teqnikèc thcArijmhtik c An�lushc leitourgoÔn ìtan oi emplekìmenec sunart seic eÐnai Lip-schitz suneqeÐc.

5. Perissìtera se maj mata Arijmhtik c An�lushc.

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Kef�laio 4

Par�gwgoc

4.1 Orismìc

Orismìc 4.1. (Par�gwgoc kai paragwgisimìthta)

1. H par�gwgoc f ′(a) orÐzetai wc to ìrio

f ′(a) = limh→0

f(a+ h)− f(a)

h,

efìson to ìrio up�rqei. Se aut  thn perÐptwsh, h f kaleÐtai paragwgÐsimh stoa.

2. H dexi� par�gwgoc f ′+(a) orÐzetai wc to ìrio

f ′+(a) = limh→0+

f(a+ h)− f(a)

h,

efìson to ìrio up�rqei. Se aut  thn perÐptwsh, h f kaleÐtai dexi� paragwgÐsimhsto a.

3. H arister  par�gwgoc f ′−(a) orÐzetai an�loga.

4. H arister  kai h dexi� par�gwgoc kaloÔntai apì koinoÔ pleurikèc par�gwgoi.

5. An h f eÐnai paragwgÐsimh gia k�je x se èna di�sthma I (arister�/dexi� paragw-gÐsimh sto antÐstoiqo �kro, an autì perièqetai sto I), tìte kaleÐtai paragwgÐsimhsto I, kai mèsw tou �nw orismoÔ dhmiourgeÐtai mia kainoÔrgia sun�rthsh, h f ′,orismènh pantoÔ sto I.

Parathr seic

1. Ektìc kai an anafèretai rht¸c, ìtan lème ìti up�rqei h par�gwgoc, ennooÔme kai

77

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78 KEF�ALAIO 4. PAR�AGWGOS

a a+h3

f(x)y

x

tanθ=f’(a)

θ

a+h2 a+h1

f(a+h3)

f(a+h1)

f(a+h2)

f(a)

Sq ma 4.1: Gewmetrik  ermhneÐa thc parag¸gou.

ìti eÐnai peperasmènh.

2. Lìgw thc sqetik c idiìthtac twn orÐwn, h par�gwgoc se èna shmeÐo up�rqei annup�rqoun kai oi dÔo pleurikèc par�gwgoi kai tautÐzontai.

3. H akoloujÐa twn phlÐkwn

f(a+ h1)− f(a)

h1,f(a+ h2)− f(a)

h2,f(a+ h3)− f(a)

h3

teÐnei sthn par�gwgo f ′(a) kaj¸c ta hi → 0. Epomènwc

(aþ) H par�gwgoc sth jèsh a isoÔtai kai me thn klÐsh thc efaptìmenhc eujeÐ-ac sto a, dhlad  thn efaptìmenh thc gwnÐac θ pou sqhmatÐzetai apì thnefaptìmenh eujeÐa sto a kai ton �xona x. DeÐte to Sq ma 4.1.

(bþ) H par�gwgoc eÐnai epÐshc o rujmìc metabol c thc f sto a.

An h par�gwgoc den up�rqei, den up�rqoun oÔte h efaptìmenh eujeÐa, oÔte orujmìc metabol c.

4. Sth Fusik  kai �llec epist mec apant�tai suqn� h ènnoia tou rujmoÔ metabol c(taqÔthta, epit�qunsh, rujmìc an�ptuxhc, k.o.k.) kai autìc eÐnai ènac apì toucbasikoÔc lìgouc pou h par�gwgoc mac eÐnai tìso qr simh.

5. H par�gwgoc eÐnai b�sh �llwn majhmatik¸n montèlwn, gia par�deigma twn me-rik¸n parag¸gwn. (Oi perissìteroi fusikoÐ nìmoi ekfr�zontai mèsw sqèsewnmetaxÔ merik¸n parag¸gwn fusik¸n posot twn.)

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4.1. ORISM�OS 79

Par�deigma 4.1. An f(x) = cx+ b, tìte f ′(x) = c. Pr�gmati, efarmìzontac tonorismì:

f ′(x) = limh→0

f(x+ h)− f(x)

h= lim

h→0

c(x+ h) + b− cx− bh

= c.

Par�deigma 4.2. An f(x) = xn, me n jetikì akèraioi, tìte f ′(x) = nxn−1. Pr�g-mati, efarmìzontac ton orismì:

f ′(x) = limh→0

f(x+ h)− f(x)

h= lim

h→0

(x+ h)n − xnh

= limh→0

h[(x+ h)n−1 + (x+ h)n−2x+ · · ·+ (x+ h)xn−2 + xn−1

]h

= nxn−1.

Sthn �nw, qrhsimopoi same th gnwst  tautìthta:

an − bn = (a− b)(an−1 + an−2b+ · · ·+ abn−2 + bn−1),

pou apodeiknÔetai eÔkola an anaptÔxoume to ginìmeno tou dexioÔ mèlouc.

Par�deigma 4.3. An f(x) = sin x, tìte f ′(x) = cos x. EpÐshc, an f(x) = cos x,tìte f ′(x) = − sinx. Pr�gmati, efarmìzontac ton orismì gia to sinx, èqoume:

f ′(x) = limh→0

sin(x+ h)− sinx

h= lim

h→0

sin(h2

)h2

cos

(x+

h

2

)= 1× cosx = cosx.

Sthn �nw, qrhsimopoi same th gnwst  idiìthta

sinx− sin y = 2 cos

(x+ y

2

)sin

(x− y

2

).

Me parìmoio trìpo, upologÐsoume pwc (cosx)′ = − sinx.

Par�deigma 4.4. An f(x) = |x|, tìte h f ′(x) = |x|x gia x 6= 0, en¸ den orÐzetai gia

x = 0. Pr�gmati, èstw x > 0. Tìte

f ′(x) = limh→0

f(x+ h)− f(x)

h= lim

h→0

x+ h− xh

= 1 =|x|x.

H deÔterh isìthta isqÔei ìtan to h eÐnai arket� mikrì ¸ste to x + h na eÐnai jetikì.Parìmoia upologÐzetai kai h par�gwgoc gia x < 0. AntÐjeta, an x = 0, èqoume:

limh→0+

f(x+ h)− f(x)

h= lim

h→0+

h

h= lim

h→0+1 = 1,

limh→0−

f(x+ h)− f(x)

h= lim

h→0−

−hh

= limh→0−

− 1 = −1.

'Ara up�rqoun mìno pleurikèc par�gwgoi, pou diafèroun, �ra to ìrio den up�rqei. Toapotèlesma eÐnai diaisjhtik� anamenìmeno.

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80 KEF�ALAIO 4. PAR�AGWGOS

Parathr seic

1. Suqn� gr�foume thn f ′(x) wc (f(x))′. 'Ara, mporoÔme na gr�youme:

(cx+ b)′ = c, (xn)′ = nxn−1, (sinx)′ = cosx, (cosx)′ = − sinx, k.o.k.

2. An gr�foume y = f(x), tìte mporoÔme na sumbolÐsoume thn par�gwgo wc y′.

3. O NeÔtwnac sumbìlize thn par�gwgo thc y = f(x) wc f   y.

4. SunhjÐzetai epÐshc na sumbolÐzoume thn f ′ wc Dxf .

5. EpÐshc gr�foume

f ′(x) =df(x)

dx, f ′(a) =

df(x)

dx(a) =

df(x)

dx

∣∣∣∣x=a

.

6. An gr�foume y = f(x), tìte mporoÔme na sumbolÐsoume thn par�gwgo wc dydx .

7. O sumbolismìc dydx proèrqetai apì to gegonìc ìti, an èqoume mia sun�rthsh y =

f(x), tìte mporoÔme na orÐsoume tic diaforèc:

∆x = h, ∆y = f(x+ h)− f(x),

kai tìte

f ′(x) = limh→0

f(x+ h)− f(x)

h= lim

∆x→0

∆y

∆x,dy

dx.

Parìmoia ermhneÔetai o sumbolismìc df(x)dx . Oi sumbolismoÐ autoÐ (kai ìsoi �lloi

basÐzontai parìmoia sto ddx) kaloÔntai sumbolismoÐ tou Leibniz.

8. Ta dx, dy kaloÔntai diaforik�, kai mporoÔme na ta fantazìmaste san apeirostècmetabolèc, tìso mikrèc ¸ste to phlÐko touc na isoÔtai akrib¸c me thn par�gwgo(kai ìqi perÐpou, pou ja Ðsque an apl¸c  tan polÔ mikrèc).

Par�deigma 4.5. Ja deÐxoume ìti

(√x)′ =

1

2√x

gia x > 0, en¸ gia x = 0 h dexi� par�gwgoc eÐnai ∞.'Estw katarq n x > 0. 'Eqoume:

limh→0

√x+ h−√x

h= lim

h→0

(√x+ h−√x)(

√x+ h+

√x)

h(√x+ h+

√x)

= limh→0

x+ h− xh(√x+ h+

√x)

= limh→0

1

(√x+ h+

√x)

=1

limh→0

(√x+ h+

√x) =

1

2√x.

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4.1. ORISM�OS 81

An t¸ra x = 0, ja èqoume:

limh→0+

√0 + h−

√0

h= lim

h→0+

1√h,

pou eÔkola mporoÔme na deÐxoume, apeujeÐac apì ton orismì, ìti eÐnai ∞.

Par�deigma 4.6. Ja deÐxoume ìti (√|x|)′ = |x|

x · 1

2√|x|

gia x 6= 0, en¸ gia x = 0

to antÐstoiqo ìrio den up�rqei.An x > 0, erqìmaste sto Par�deigma 4.5. An x < 0, tìte èqoume:

limh→0

√|x+ h| −

√|x|

h= lim

h→0

(√|x+ h| −

√|x|)(

√|x+ h|+

√|x|)

h(√|x+ h|+

√|x|)

= limh→0

−x− h+ x

h(√−x− h+

√−x)= lim

h→0− 1

(√−x− h+

√−x)

= − 1

limh→0

(√−x− h+

√−x) = − 1

2√−x =

|x|x· 1

2√|x|.

Gia thn perÐptwsh x = 0 parathroÔme ìti to ìrio apì ta dexi� èqei upologisteÐ stoPar�deigma 4.5 kai eÐnai ∞, en¸ gia to ìrio apì ta arister� èqoume:

limh→0−

√|0 + h| −

√|0|

h= lim

h→0−

√−hh

= − limh→0−

√−h−h = − lim

h→0−

1√−h

,

pou eÔkola mporoÔme na deÐxoume, apeujeÐac apì ton orismì tou pleurikoÔ orÐou, ìtieÐnai −∞. 'Ara, afoÔ ta pleurik� ìria diafèroun, to ìrio den up�rqei. An sqedi�setethn

√|x|, to apotèlesma ja gÐnei �mesa katanohtì.

Par�deigma 4.7. Ja deÐxoume ìti(1

x

)′= − 1

x2, x 6= 0.

Parìmoia me ta prohgoÔmena paradeÐgmata,

limh→0

1x+h − 1

x

h= lim

h→0

x− (x+ h)

xh(x+ h)= lim

h→0

−1

x(x+ h)= − 1

limh→0

(x(x+ h))= − 1

x2.

Par�deigma 4.8. Poia apì ta akìlouja ìria mporoÔn na qrhsimopoihjoÔn wc e-nallaktikìc orismìc thc parag¸gou?

1. limh→a

f(h)−f(a)h−a .

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82 KEF�ALAIO 4. PAR�AGWGOS

2. limh→0

f(a+10h)−f(a)10h .

3. limh→0

f(a+6h)−f(a+5h)h .

4. limh→0

f(a+h2)−f(a)h .

'Eqoume kat� perÐptwsh:

1. To ìrio up�rqei an kai mìno an up�rqei h par�gwgoc, kai isoÔtai me aut n. Pr�g-mati, gia na to apodeÐxoume, efarmìzoume ston orismì thc parag¸gou thn idiìthta

limh→0

f(h) = A⇔ limh→a

f(h− a) = A.

pou prokÔptei apì to L mma 2.5.

2. ProkÔptei to Ðdio apotèlesma, efarmìzontac ston orismì thc parag¸gou thnidiìthta

limh→0

f(h) = A⇔ limh→0

f(ah) = A, a 6= 0, (4.1)

gia a = 10, pou epÐshc prokÔptei apì to L mma 2.5.

3. Parathr ste pwc:

f(a+ 6h)− f(a+ 5h)

h= 6

f(a+ 6h)− f(a)

6h− 5

f(a+ 5h)− f(a)

5h.

An up�rqei h par�gwgoc, ta kl�smata pou emfanÐzontai sto dexÐ skèloc èqoun kaita dÔo ìrio, kaj¸c h → 0, thn par�gwgo f ′(a). Autì prokÔptei efarmìzontacthn (4.1) gia a = 6 kai a = 5 antÐstoiqa. 'Ara ja up�rqei kai to ìrio tougrammikoÔ touc sunduasmoÔ:

limh→0

f(a+ 6h)− f(a+ 5h)

h= 6lim

h→0

f(a+ 6h)− f(a)

6h− 5lim

h→0

f(a+ 5h)− f(a)

5h= 6f ′(a)− 5f ′(a) = f ′(a).

To antÐstrofo ìmwc den isqÔei: mporeÐ na up�rqei to �nw ìrio, ìqi ìmwc kai h pa-r�gwgoc. Gia antipar�deigma, mporoÔme na p�roume opoiad pote sun�rthsh eÐnaipantoÔ paragwgÐsimh ektìc apì to a ìpou eÐnai asuneq c. Mia tètoia sun�rthsheÐnai h f(x) = x2 + 1 gia x 6= 0 kai f(x) = 0 gia x = 0, gia thn opoÐa denup�rqei par�gwgoc sto 0, up�rqei ìmwc to �nw ìrio, kai isoÔtai me 0. (MporeÐtena sumplhr¸sete tic leptomèreiec?)

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4.1. ORISM�OS 83

4. Parathr ste ìti

f(a+ h2)− f(a)

h= h

f(a+ h2)− f(a)

h2.

An up�rqei h par�gwgoc, tìte to deÔtero kl�sma èqei ìrio thn f ′(a), ìpwcprokÔptei apì to L mma (2.5). 'Ara:

limh→0

f(a+ h2)− f(a)

h= lim

h→0h× lim

h→0

f(a+ h2)− f(a)

h2= 0× f ′(a) = 0,

�ra to �nw ìrio den mporeÐ na apotelèsei enallaktikì orismì thc parag¸gou.

Orismìc 4.2. (Par�gwgoi an¸terwn t�xewn) An mia sun�rthsh f èqei par�gwgof ′ pantoÔ se èna di�sthma, mporoÔme na exet�soume thn Ôparxh thc parag gou thcf ′. H (f ′)′ kaleÐtai deÔterh par�gwgoc, kai sumbolÐzetai wc f ′′. Parìmoia mporoÔmena orÐzoume thn trÐth par�gwgo, thn tètarth par�gwgo, kai genik¸c thn par�gwgo nt�xhc. Gia tic parag gouc an¸terhc t�xhc qrhsimopoioÔntai oi akìloujoi sumbolismoÐ:

Par�gwgoc Sumb. f ′ Sumb. NeÔtwna Sumb. y′ Sumb. D Sumb. Leibniz

Pr¸th f ′(x) y y′ Dxf(x) dy/dx

DeÔterh f ′′(x) y y′′ D2xf(x) d2y/dx2

TrÐth f ′′′(x)...y y′′′ D3

xf(x) d3y/dx3

n-ost  f (n)(x) y(n) y(n) Dnxf(x) d4y/dx4

Par�deigma 4.9. Me diadoqikèc paragwgÐseic, eÔkola prokÔptoun ta akìlouja:

(sinx)(4n+1) = cosx, (sinx)(4n+2) = − sinx,

(sinx)(4n+3) = − cosx, (sinx)(4n) = sinx,

(cosx)(4n+1) = − sinx, (cosx)(4n+2) = − cosx,

(sinx)(4n+3) = sinx, (sinx)(4n) = cosx,

ìpou n ∈ N.

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84 KEF�ALAIO 4. PAR�AGWGOS

4.2 Basikèc Idiìthtec Parag¸gwn

Je¸rhma 4.1. (Paragwgisimìthta ⇒ sunèqeia) An h f ′(a) up�rqei, tìte h f eÐnaisuneq c sto a. (EpÐshc, an up�rqei h dexi�   h arister  par�gwgoc sto a, h sun�rthsheÐnai antÐstoiqa dexi�   arister� suneq c.)

Apìdeixh. Parathr ste ìti

f(x) = f(a) +f(x)− f(a)

x− a (x− a), x 6= a.

'Ara

limx→a

f(x) = limx→a

[f(a) +

f(x)− f(a)

x− a (x− a)

]= lim

x→af(a) + lim

x→af(x)− f(a)

x− a × limx→a

(x− a)

= f(a) + f ′(a)× 0 = f(a).

H apìdeixh gia tic peript¸seic thc dexi�   arister� sunèqeiac paraleÐpetai.

Je¸rhma 4.2. (Par�gwgoi algebrik¸n sunduasm¸n sunart sewn) 'Estw stajerècc, c1, c2, . . . , cn ∈ R kai f, g, h1, h2, . . . , hn sunart seic.

1. (f + g)′ = f ′ + g′.

2. (cf)′ = cf ′.

3. (∑n

i=1 cihi)′=∑n

i=1 cih′i.

4. (fg)′ = f ′g + fg′.

5.(

1g

)′= − g′

g2 , efìson g(x) 6= 0.

6.(fg

)′= f ′g−fg′

g2 , efìson g(x) 6= 0.

Me ta �nw, ennooÔme ìti an up�rqoun ta dexi� mèlh, se èna shmeÐo x   se èna di�sthma,up�rqoun kai ta arister� mèlh, sto antÐstoiqo shmeÐo   di�sthma. Ta �nw isqÔoun kaisthn perÐptwsh pou oi par�gwgoi eÐnai pleurikèc.

Apìdeixh. Ja deÐxoume mìno thn isìthta se shmeÐa gia thn perÐptwsh twn parag¸gwn.H genÐkeush gia diast mata prokÔptei autìmata, en¸ oi apodeÐxeic sthn perÐptwshpleurik¸n parag¸gwn prokÔptoun ìlec entel¸c an�loga.

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4.2. BASIK�ES IDI�OTHTES PARAG�WGWN 85

1. 'Eqoume:

(f(x+ h) + g(x+ h))− (f(x) + g(x))

h=f(x+ h)− f(x)

h+g(x+ h)− g(x)

h.

Kaj¸c h → 0, to pr¸to kl�sma sugklÐnei sto f ′(x) kai to deÔtero kl�smasugklÐnei sto g′(x). 'Ara to aristerì skèloc sugklÐnei sto f ′(x) + g′(x).

2. Parìmoia.

3. Qrhsimopoi¸ntac ta pr¸ta dÔo skèlh, eÔkola mporoÔme na deÐxoume ìti h isìthtaisqÔei gia n = 2:

(c1h1 + c2h2)′ = (c1h1)

′ + (c2h2)′ = c1h

′1 + c2h

′2.

(Gia thn pr¸th isìthta qrhsimopoi same to pr¸to skèloc, kai gia thn deÔterhto deÔtero.) H genik  perÐptwsh prokÔptei me apl  qr sh epagwg c.

4. ParathroÔme pwc:

f(x+ h)g(x+ h)− f(x)g(x)

h

= g(x)f(x+ h)− f(x)

h+ f(x+ h)

g(x+ h)− g(x)

h.

PaÐrnontac ìria, prokÔptei to zhtoÔmeno.

5. ParathroÔme pwc:

limh→0

1g(x+h) − 1

g(x)

h= lim

h→0

[−g(x+ h)− g(x)

h× 1

g(x)× 1

g(x+ h)

]=

[limh→0− g(x+ h)− g(x)

h

]×[limh→0

1

g(x)

]×[limh→0

1

g(x+ h)

]= −g′(x)× 1

g(x)× 1

g(x)= − g

′(x)

g2(x).

6. ProkÔptei me efarmog  twn prohgoÔmenwn dÔo skel¸n.

Par�deigma 4.10. (xn)′ = nxn−1 kai gia n < 0, kai efìson x 6= 0. Pr�gmati,

(xn)′ =

(1

x|n|

)′= −|n|x

|n|−1

x2|n| = nxn−1.

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86 KEF�ALAIO 4. PAR�AGWGOS

Par�deigma 4.11.

(tanx)′ =

(sinx

cosx

)′=

(sinx)′ cosx− sinx(cosx)′

cos2 x=

cos2 x+ sin2 x

cos2 x=

1

cos2 x.

(cotx)′ =(cosx

sinx

)′=

(cosx)′ sinx− cosx(sinx)′

sin2 x=− sin2 x− cos2 x

sin2 x= − 1

sin2 x.

Je¸rhma 4.3. (Kanìnac thc AlusÐdac) 'Estw h sÔnjeth sun�rthsh

h(x) = f(g(x)) = f ◦ g(x).

'Estw ìti up�rqoun oi par�gwgoi f ′(g(x0)) kai g′(x0). Tìte up�rqei kai h par�gwgoch′(x0), kai m�lista isoÔtai me

h′(x0) = f ′(g(x0))g′(x0).

Parathr seic

1. O kanìnac eÐnai apolÔtwc eÔlogoc. 'Estw oi posìthtec x, g, f , gia tic opoÐecxèroume ìti h tim  thc g exart�tai apì thn tim  thc x, kai h tim  thc f exart�taiapì thn tim  thc g. 'Estw epÐshc ìti an all�xei kat� dx h tim  thc x, tìte h tim thc g all�zei kat� dg = g′(x)dx, kai ìti an all�xei kat� dg h tim  thc g, tìte htim  thc f all�zei kat� df = f ′(g)dg. Tìte prokÔptei ìti an all�xei h tim  thcx kat� dx, h tim  thc f ja all�xei kat� df = f ′(g)dg = f ′(g)g′(x)dx.

2. Me sumbolismì Leibniz, h exÐswsh paÐrnei th morf 

df

dx

∣∣∣∣x0

=df

dg

∣∣∣∣g(x0)

dg

dx

∣∣∣∣x0

,

 , pio apl�,df

dx=df

dg× dg

dx.

pou eÔkola mporoÔme na jumìmaste, kai epiplèon mac dÐnei kai th swst  fusik diaÐsjhsh (deÐte kai to par�deigma pou akoloujeÐ.) H idiìthta moi�zei me tetrim-mènh algebrik  idiìthta, all� den eÐnai!

Apìdeixh. Ja deÐxoume th basik  idèa thc apìdeixhc tou Kanìna thc AlusÐdac, giatÐh pl rhc apìdeixh eÐnai ekten c kai thn paraleÐpoume.

Xekin�me parathr¸ntac pwc:

f(g(x0 + h))− f(g(x0))

h=f(g(x0 + h))− f(g(x0))

g(x0 + h)− g(x0)× g(x0 + h)− g(x0)

h. (4.2)

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4.2. BASIK�ES IDI�OTHTES PARAG�WGWN 87

'Omwc

limh→0

g(x0 + h)− g(x0)

h= g′(x),

en¸ gia to pr¸to phlÐko parathroÔme ìti h g(x) eÐnai suneq c, wc paragwgÐsimh, �rakaj¸c h → 0 èqoume g(x0 + h) → g(x0) �ra to phlÐko teÐnei sto f ′(g(x0)). 'ArapaÐrnontac ìrio wc proc h, prokÔptei to zhtoÔmeno.

Sto sullogismì mèqri t¸ra up�rqei èna l�joc: mporeÐ g(x0 + h) = g(x0) gia timèctou h 6= 0 aujaÐreta kont� sto 0. Se aut  thn perÐptwsh, h sqèsh (4.2) den isqÔeigia k�poio di�sthma gÔrw apì to h = 0, exairoumènou tou h = 0, ìpwc apaiteÐtai apìton orismì tou orÐou. 'Enac aplìc trìpoc na katal�bete ti phgaÐnei strab� eÐnai naprospaj sete efarmìsete thn �nw mèjodo sthn eidik  perÐptwsh pou g(x) = c. Hantimet¸pish autoÔ tou l�jouc ja olokl rwne thn apìdeixh, all� paraleÐpetai.

Par�deigma 4.12. EÐnai shmantikì sthn efarmog  tou kanìna thc alusÐdac namhn sugqèoume th seir� me thn opoÐa oi sunart seic f , g dhmiourgoÔn thn sÔnjeshf(g(x)).

Gia par�deigma, èstw h f(x) = sin x, kai h g(x) = x2. Tìte

((f ◦ g)(x))′ = (sinx2)′ = 2x cos(x2), all� ((g ◦ f)(x))′ = (sin2 x)′ = 2 sin x cosx.

Par�deigma 4.13. O Kanìnac thc AlusÐdac mporeÐ na efarmosteÐ gia ep�llhlecsunjèseic. Gia par�deigma,

(sin sin sinx)′ = (cos sin sinx) (sin sinx)′ = (cos sin sinx) (cos sin x) cosx.

MporeÐte na gr�yete thn perÐptwsh pou èqoume n ep�llhla sunhmÐtona?

Par�deigma 4.14. O sunduasmìc twn Jewrhm�twn 4.2 kai 4.3 mac epitrèpei tonupologismì parag¸gwn sunart sewn arket� polÔplokhc morf c, me sqetik� mikrìkìpo. Gia par�deigma:(

cosx3 cos3 x

x2 + x

)′=

(cosx3 cos3 x

)′(x2 + x)− (2x+ 1) cosx3 cos3 x

(x2 + x)2

=

(−3x2 sinx3 cos3 x− 6 cosx3 cos2 x sinx

)(x2 + x)− (2x+ 1) cosx3 cos3 x

(x2 + x)2.

Gia poia x isqÔei h �nw isìthta?

Par�deigma 4.15. Me qr sh tou Kanìna thc AlusÐdac, mporoÔme se orismènecpeript¸seic na broÔme mia exÐswsh pou ikanopoieÐ h par�gwgoc miac sun�rthshc, qwrÐc

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88 KEF�ALAIO 4. PAR�AGWGOS

na gnwrÐzoume thn Ðdia thn sun�rthsh. Gia par�deigma, èstw ìti up�rqei sun�rthshy(x) orismènh se k�poio di�sthma a < x < b, kai paragwgÐsimh se autì, gia thn opoÐax2 + y2(x) = r2. Tìte, paÐrnontac parag¸gouc wc proc x, èqoume:

2x+ 2y(x)y′(x) = 0⇒ y′(x) = − x

y(x), y 6= 0.

Parathr ste pwc den gnwrÐzoume thn morf  thc y, all� par' ìla aut� mporoÔme napoÔme k�ti gia thn par�gwgì thc. To apotèlesma èqei nìhma mìno an h paragwgisi-mìthta thc y(x) mac èqei exasfalisjeÐ ek twn protèrwn.

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4.3. PAR�AGWGOS ANT�ISTROFHS SUN�ARTHSHS 89

4.3 Par�gwgoc AntÐstrofhc Sun�rthshc

Je¸rhma 4.4. (Par�gwgoc antÐstrofhc sun�rthshc) 'Estw f : [a, b]→ R gnhsÐwcmonìtonh kai suneq c. Gia k�je x0 ∈ (a, b) gia to opoÐo up�rqei h par�gwgoc f ′(x0)kai f ′(x0) 6= 0, up�rqei kai h par�gwgoc thc antÐstrofhc f−1(y) , g(y) sto shmeÐoy0 = f(x0), kai isqÔei

g′(y0) =1

f ′(x0)⇔(f−1(y0)

)′=

1

f ′ (f−1(y0))

Parathr seic

1. Me sumbolismì Leibniz, to �nw je¸rhma èqei thn akìloujh morf :

dx

dy=

1dydx

,

'Opwc kai sthn perÐptwsh tou kanìna thc alusÐdac, h �nw morf  moi�zei me te-trimmènh algebrik  idiìthta, all� den eÐnai!

2. Diaisjhtik�, to apotèlesma eÐnai anamenìmeno. 'Estw ìti ta x kai y sqetÐzontaimèsw miac sun�rthshc y = f(x)⇔ x = f−1(y). An, metab�llontac to x kat� ∆xto y metab�lletai kat�, ac poÔme, ∆y = 2∆x, tìte an metab�lloume to y kat�∆y, eÐnai anamenìmeno to x na all�xei perÐpou kat� 1

2∆y. Sto ìrio ∆x,∆y → 0,prokÔptei to je¸rhma.

3. MporeÐ mia gnhsÐwc monìtonh sun�rthsh na èqei k�pou mhdenik  par�gwgo?

4. To je¸rhma mporeÐ na genikeuteÐ kai ìtan to x0 = a   x0 = b, me ton profan trìpo.

Apìdeixh. Ja apodeÐxoume thn sqèsh gia thn perÐptwsh pou h f (�ra kai h antÐstro-f  thc) eÐnai gnhsÐwc aÔxousa. H perÐptwsh gnhsÐwc fjÐnousac f antimetwpÐzetaian�loga.

KaloÔmaste na upologÐsoume to ìrio

limh→0

f−1(y0 + h)− f−1(y0)

h.

Gia k�je y0 + h, ìpou h arkoÔntwc mikrì, mporoÔme na gr�youme

y0 + h = f(x0 + k(h))⇔ k(h) = f−1(y0 + h)− f−1(y0),

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90 KEF�ALAIO 4. PAR�AGWGOS

ìpou k(h) eÐnai mia suneq c sun�rthsh (afoÔ h f−1 eÐnai suneq c) me k(0) = 0.Epiplèon, k(h) = 0 mìno gia h = 0, alli¸c ft�noume se �topo qrhsimopoi¸ntac thngn sia monotonikìthta thc f . 'Ara to ìrio mporeÐ na grafeÐ

limh→0

f−1(y0 + h)− f−1(y0)

h= lim

h→0

f−1(f(x0 + k(h)))− x0

f(x0 + k(h))− y0

= limh→0

k(h)

f(x0 + k(h))− f(x0)= lim

h→0

1f(x0+k(h))−f(x0)

k(h)

.

Parathr ste ìti kaj¸c h→ 0, èqoume k(h)→ 0, �ra to teleutaÐo ìrio isoÔtai me(f ′(x0))

−1. Autì eÐnai diaisjhtik� profanèc. An jèloume na to deÐxoume kai apolÔtwcausthr�, mporoÔme na orÐsoume th sun�rthsh

l(k) =

1

f(x0+k)−f(x0)k

, k 6= 0,

1f ′(x0) , k = 0.

H l(k) eÐnai suneq c, �ra, apì gnwstì je¸rhma gia th sÔnjesh sunart sewn, jaèqoume:

limh→0

1f(x0+k(h))−f(x0)

k(h)

= limh→0

l(k(h)) = l(limh→0

k(h)) = l(0) =1

f ′(x0).

Par�deigma 4.16. Ja deÐxoume ìti an q ∈ N kai perittìc, tìte (x1q )′ = 1

qx1q−1 gia

k�je x 6= 0. An q ∈ N kai �rtioc, tìte (x1q )′ = 1

qx1q−1 gia k�je x > 0.

Pr�gmati, èstw q perittìc (h perÐptwsh �rtiou q eÐnai an�logh). H sun�rthshf(x) = xq eÐnai suneq c kai gnhsÐwc aÔxousa sto R, kai èqei wc antÐstrofh thn suneq kai gnhsÐwc aÔxousa f−1(y) = y

1q . Gia k�je x 6= 0 h par�gwgoc f ′(x) = qxq−1 6= 0.

'Ara, apì to �nw je¸rhma, h antÐstrofh sun�rthsh f−1(y) ja èqei par�gwgo, giay 6= 0,

(y1q )′ = (f−1)′(y) =

1

qxq−1=

1

qyq−1q

=1

qy

1q−1.

AkoloÔjwc, eÔkola prokÔptei pwc(x−

1q

)′=(−1q

)x−

1q−1.

Par�deigma 4.17. An p ∈ Z, p 6= 0, q ∈ N kai q perittìc, tìte (xpq )′ = p

qxpq−1

gia k�je x 6= 0. An p ∈ Z, p 6= 0, q ∈ N kai q �rtioc, tìte (xpq )′ = p

qxpq−1 gia k�je

x > 0.

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4.3. PAR�AGWGOS ANT�ISTROFHS SUN�ARTHSHS 91

Pr�gmati, èstw q perittìc (h perÐptwsh �rtiou q eÐnai an�logh). Me qr sh toukanìna alusÐdac èqoume:

f ′(x) =((x

1q

)p)′= p

(x

1q

)p−1 1

qx

1q−1 =

p

qx[pq− 1

q+ 1q−1] =

p

qx

pq−1.

Orismìc 4.3. (AntÐstrofec trigwnometrikèc sunart seic)

1. H sun�rthsh

f(x) = sinx, −π2≤ x ≤ π

2,

eÐnai gnhsÐwc aÔxousa kai suneq c, �ra, h antÐstrofh sun�rthsh

f−1(y) = arcsin y, −1 ≤ y ≤ 1,

eÐnai gnhsÐwc aÔxousa kai suneq c.

2. H sun�rthsh

f(x) = cos x, 0 ≤ x ≤ π,

eÐnai gnhsÐwc fjÐnousa kai suneq c, �ra, h antÐstrofh sun�rthsh

f−1(y) = arccos y, −1 ≤ y ≤ 1,

eÐnai gnhsÐwc fjÐnousa kai suneq c.

3. H sun�rthsh

f(x) = tan x, −π2< x <

π

2,

eÐnai gnhsÐwc aÔxousa kai suneq c, �ra, h antÐstrofh sun�rthsh

f−1(y) = arctan y, y ∈ R,

eÐnai gnhsÐwc aÔxousa kai suneq c.

DeÐte to Sq ma 4.2 gia tic grafikèc parast�seic ìlwn twn �nw sunart sewn.

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92 KEF�ALAIO 4. PAR�AGWGOS

Par�deigma 4.18. (Par�gwgoi twn antÐstrofwn trigwnometrik¸n sunart sewn)Ja deÐxoume ìti isqÔoun ta akìlouja:

(arcsin y)′ =1√

1− y2, −1 < y < 1,

(arccos y)′ = − 1√1− y2

, −1 < y < 1,

(arctan y)′ =1

1 + y2, y ∈ R.

'Eqoume:

1.

(arcsin y)′ =1

(sinx)′=

1

cosx=

1√1− sin2 x

=1√

1− y2.

Parathr ste ìti h par�gwgoc tou pr¸tou skèlouc eÐnai wc proc y, en¸ toudeÔterou wc proc x. EpÐshc, parathr ste ìti gia to sugkekrimèno pedÐo orismoÔ,

èqoume p�nta cosx =√

1− sin2 x. BebaiwjeÐte ìti èqete katal�bei ìla tab mata.

2. Me parìmoio trìpo èqoume:

(arccos y)′ =1

(cosx)′= − 1

sinx= − 1√

1− cos2 x= − 1√

1− y2.

3. Kat' arq n, parathroÔme pwc:

(tanx)′ =1

cos2 x= 1 + tan2 x.

'Ara èqoume:

(arctan y)′ =1

(tanx)′=

1

1 + tan2 x=

1

1 + y2.

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4.3. PAR�AGWGOS ANT�ISTROFHS SUN�ARTHSHS 93

−1 0 1−1

−0.5

0

0.5

1sin x

−1 0 1−1.5

−1

−0.5

0

0.5

1

1.5arcsin y

0 1 2 3−1

−0.5

0

0.5

1cosx

−1 0 10

0.5

1

1.5

2

2.5

3

arccosy

−1 0 1−2

−1

0

1

2tanx

−2 −1 0 1 2−1

0

1arctany

Sq ma 4.2: Oi grafikèc parast�seic twn sunart sewn tou OrismoÔ 4.3.

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94 KEF�ALAIO 4. PAR�AGWGOS

4.4 Je¸rhma Mèshc Tim c

Je¸rhma 4.5. (AnagkaÐo Krit rio Akrìtatwn Fermat) 'Estw f orismènh kai pa-ragwgÐsimh se èna di�sthma (a, b). An h f èqei topikì akrìtato sto c ∈ (a, b), tìtef ′(c) = 0.

Parat rhsh: BebaiwjeÐte ìti èqete katal�bei ti lèei to je¸rhma! To Sq ma 4.3mporeÐ na sac dieukolÔnei sthn katanìhsh. Melet ste ti sumbaÐnei sta shmeÐa A wcF .

Apìdeixh. 'Estw h sun�rthsh

Q(x) =

{f(x)−f(c)

x−c , x 6= c,

f ′(c), x = c,

pou eÐnai suneq c sto c lìgw Ôparxhc thc parag¸gou thc f sto c. DiakrÐnoumepeript¸seic:

1. 'Estw pwc f ′(c) = Q(c) > 0. Tìte apì to Je¸rhma Diat rhshc PrìshmouprokÔptei pwc up�rqei δ tètoio ¸ste

|x− c| < δ ⇒ Q(x) > 0.

'Ara an x ∈ (c, c + δ) apì ton orismì thc Q(x) prokÔptei ìti f(x) − f(c) > 0,en¸ an x ∈ (c−δ, c) kai p�li apì ton orismì thc Q(x) prokÔptei ìti f(x) < f(c).'Ara to c den eÐnai akrìtato, kai prokÔptei �topo.

2. 'Estw pwc f ′(c) = Q(c) < 0. Me parìmoio trìpo, kai p�li mporoÔme na deÐxoume�topo.

3. 'Ara anagkastik� f ′(c) = Q(c) = 0.

Je¸rhma 4.6. (Rolle) 'Estw f suneq c sto [a, b] kai me par�gwgo pantoÔ sto (a, b).An f(a) = f(b), tìte up�rqei c ∈ (a, b) tètoio ¸ste f ′(c) = 0.

Apìdeixh. H f eÐnai suneq c sto kleistì di�sthma [a, b], �ra èqei mègisto kai el�qistosto [a, b]. An èna apì ta dÔo eÐnai entìc tou (a, b), se k�poio shmeÐo c, tìte apì toJe¸rhma 4.5 kai thn paragwgisimìthta thc f sto (a, b) prokÔptei pwc f ′(c) = 0. Anto el�qisto kai to mègisto eÐnai kai ta dÔo sta a, b, tìte profan¸c, epeid  f(a) = f(b),h f eÐnai stajer , �ra f ′(c) = 0 gia ìla ta c ∈ (a, b). Mia gewmetrik  perigraf  toujewr matoc emfanÐzetai sto Sq ma 4.4.

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4.4. JE�WRHMA M�ESHS TIM�HS 95

x

y

B

C

D

E

A

F

Sq ma 4.3: Poia apì ta shmeÐa A èwc F eÐnai akrìtata? Pou up�rqei h par�gwgoc thc sqediasmènhcsun�rthshc?

Par�deigma 4.19. Ja qrhsimopoi soume to Je¸rhma Rolle gia na deÐxoume ìti topolu¸numo f(x) = x3 + 3x2 + 24x+ 7 èqei to polÔ mia rÐza, an kai trÐtou bajmoÔ.

Pr�gmati, èstw pwc h f(x) èqei dÔo rÐzec x1, x2, ìpou f(x1) = f(x2) = 0. Meefarmog  tou Jewr matoc tou Rolle, h par�gwgoc f ′(x) = 3x2 + 6x + 24 = 3(x2 +2x + 8) èqei mia rÐza, x3, dhlad  f ′(x3) = 0. Autì ìmwc eÐnai adÔnato, giatÐ hdiakrÐnousa tou tri¸numou x2 + 2x + 8 eÐnai arnhtik , kai h f ′(x) eÐnai p�nta jetik .'Ara, odhghj kame se �topo, kai h f(x) èqei to polÔ mia rÐza. M�lista, me qr sh touJewr matoc Bolzano prokÔptei pwc èqei akrib¸c mia rÐza. (MporeÐte na sumplhr¸setetic leptomèreiec?)

Par�deigma 4.20. Ja qrhsimopoi soume to Je¸rhma Rolle gia na deÐxoume ìti tapolu¸numa thc morf c f(x) = ax4 + bx+ c, ìpou a 6= 0, èqoun to polÔ dÔo rÐzec, ankai tet�rtou bajmoÔ.

Pr�gmati, èstw x1 < x2 < x3 tètoia ¸ste f(x1) = f(x2) = f(x3) = 0. Meefarmog  tou Jewr matoc Rolle, prokÔptei pwc up�rqoun y1, y2 tètoia ¸ste y1 ∈(x1, x2), y2 ∈ (x2, x3), kai f ′(y1) = f ′(y2) = 0. Autì ìmwc eÐnai �topo, giatÐ h

f ′(x) = 4ax3 + b

èqei mìno mia rÐza, thn(−b

4a

) 13 .

Je¸rhma 4.7. (Mèshc Tim c) 'Estw f suneq c sto [a, b] kai me par�gwgo sto(a, b). Tìte up�rqei c ∈ (a, b) tètoio ¸ste

f(b)− f(a) = f ′(c)(b− a).

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96 KEF�ALAIO 4. PAR�AGWGOS

x

y

a b

f(a)=f(b)

f’(c1)=0

f’(c2)=0

c1 c2

Sq ma 4.4: Gewmetrik  perigraf  tou Jewr matoc Rolle. Se autì to sq ma, up�rqoun dÔo shmeÐa,c1 kai c2, ìpou h par�gwgoc f eÐnai mhdenik . To Je¸rhma Rolle exasfalÐzei thn Ôparxh toul�qistonenìc tètoiou shmeÐou.

Apìdeixh. Mia gewmetrik  perigraf  tou Jewr matoc up�rqei sto Sq ma 4.5. 'Estwh sun�rthsh

h(x) = (b− a)f(x)− x [f(b)− f(a)] .

ParathroÔme pwc h h(x) eÐnai suneq c sto [a, b] kai paragwgÐsimh sto (a, b). Epiplèon,

h(a) = bf(a)− af(a)− af(b) + af(a) = bf(a)− af(b),

h(b) = bf(b)− af(b)− bf(b) + bf(a) = bf(a)− af(b) = h(a).

'Ara, ikanopoioÔntai oi apait seic tou Jewr matoc Rolle, �ra up�rqei k�poio c tètoio¸ste

h′(c) = 0⇒ f ′(c)(b− a)− [f(b)− f(a)] = 0⇒ f ′(c) =f(b)− f(a)

b− a .

Par�deigma 4.21. 'Estw sun�rthsh f : [a, b] → R suneq c sto [a, b], kai dipl�paragwgÐsimh sto (a, b). 'Estw epÐshc pwc to eujÔgrammo tm ma pou en¸nei ta shmeÐa(a, f(a)), (b, f(b)) tèmnei to gr�fhma thc sun�rthshc se èna shmeÐo (x0, f(x0)) ìpoux0 ∈ (a, b). Ja deÐxoume ìti up�rqei èna shmeÐo x1 ∈ (a, b) tètoio ¸ste f ′′(x1) = 0.

Pr�gmati, me efarmog  tou Jewr matoc Mèshc Tim c sto [a, x0] prokÔptei ìtiup�rqei k1 ∈ (a, x0) tètoio ¸ste

f ′(k1) =f(x0)− f(a)

x0 − a.

Me efarmog  sto [x0, b], prokÔptei ìti up�rqei k2 ∈ (x0, b) tètoio ¸ste

f ′(k2) =f(b)− f(x0)

b− x0.

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4.4. JE�WRHMA M�ESHS TIM�HS 97

x

y

a b

f(a)

c1 c2

f’(c1)=[f(b)-f(a)]/[b-a]

f(b)

f’(c2)=[f(b)-f(a)]/[b-a]

Sq ma 4.5: Gewmetrik  perigraf  tou Jewr matoc Mèshc Tim c. Se autì to sq ma up�rqoun dÔoshmeÐa c1, c2 ìpou h par�gwgoc eÐnai aut  pou prosdiorÐzei to je¸rhma.

'Omwc ta dexi� mèlh eÐnai Ðsa, afoÔ ekfr�zoun thn klÐsh tou Ðdiou eujÔgrammou tm -matoc, opìte kai f ′(k1) = f ′(k2). Apì to Je¸rhma Rolle, tou opoÐou ìlec oi pro-ôpojèseic ikanopoioÔntai, èqoume ìti up�rqei x1 tètoio ¸ste f ′′(x1) = 0. DeÐte toSq ma 4.6 gia èna par�deigma.

Je¸rhma 4.8. (Mhdenik c Parag¸gou) 'Estw f suneq c sto [a, b] kai me par�gwgof ′(x) = 0 pantoÔ sto (a, b). Tìte h f(x) eÐnai stajer  sto [a, b].

Apìdeixh. 'Estw dÔo opoiad pote x1 < x2 ∈ [a, b]. IsqÔoun oi proôpojèseic touJewr matoc Mèshc Tim c gia thn sun�rthsh orismènh sto di�sthma [x1, x2], �ra jaup�rqei k�poio c ∈ (x1, x2) tètoio ¸ste

f(x2)− f(x1) = f ′(c)(x2 − x1) = 0⇒ f(x1) = f(x2).

Je¸rhma 4.9. (Mèshc Tim c Cauchy) An oi f(x), g(x) suneqeÐc sto [a, b] kaiparagwgÐsimec sto (a, b), tìte up�rqei arijmìc c ∈ (a, b) tètoioc ¸ste

[f(b)− f(a)]g′(c) = [g(b)− g(a)]f ′(c).

Apìdeixh. 'Estw h sun�rthsh

h(x) = f(x)[g(b)− g(a)]− g(x)[f(b)− f(a)].

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98 KEF�ALAIO 4. PAR�AGWGOS

x

y

a b

f(a)

k1 k2

f ’(k1)=[f(b)-f(a)]/[b-a]

f(b)

f ’(k2)=[f(b)-f(a)]/[b-a]

x0

f ’’(x1)=0f(x0)

x1

Sq ma 4.6: Par�deigma 4.21.

H h(x) eÐnai suneq c sto [a, b], paragwgÐsimh sto (a, b), kai

h(a) = f(a)g(b)− g(a)f(b) = h(b).

'Ara, mporoÔme na efarmìsoume to Je¸rhma tou Rolle, apì to opoÐo prokÔptei ìtiup�rqei c ∈ (a, b) tètoio ¸ste

0 = f ′(c)[g(b)− g(a)]− g′(c)[f(b)− f(a)],

kai h apìdeixh oloklhr¸jhke.

Je¸rhma 4.10. (Fragmènh par�gwgoc ⇒ Lipschitz suneq c) 'Estw f : I → RparagwgÐsimh sto I me fragmènh par�gwgo, dhlad  up�rqei C ≥ 0 tètoio ¸ste giak�je x ∈ I na isqÔei |f ′(x)| ≤ C. Tìte h f(x) eÐnai Lipschitz suneq c.

Apìdeixh. Pr�gmati, èstw x1, x2 ∈ I. 'Estw pwc x1 < x2. Apì to Je¸rhma MèshcTim c, up�rqei x0 ∈ (x, y) tètoio ¸ste

f(x2)− f(x1) = f ′(x0)(x2 − x1)⇒ |f(x2)− f(x1)| = |f ′(x0)||x2 − x1|⇒ |f(x2)− f(x1)| ≤ C|x2 − x1|.

Me parìmoio trìpo mporoÔme na deÐxoume ìti h anisìthta pou br kame isqÔei kai ìtanx2 < x1, en¸ an x1 = x2 kai ta dÔo mèlh eÐnai Ðsa me to 0. 'Ara, h f(x) eÐnai Lipschitzsuneq c.

Parat rhsh: To antÐstrofo den isqÔei, afoÔ up�rqoun Lipschitz suneqeÐc sunar-t seic se èna di�sthma I pou den eÐnai kan paragwgÐsimec se autì to di�sthma.

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4.5. PAR�AGOUSES 99

4.5 Par�gousec

Orismìc 4.4. (Par�gousa) 'Estw sun�rthsh f(x) orismènh se èna di�sthma I.An gia k�poia F (x) orismènh sto I isqÔei ìti F ′(x) = f(x) gia k�je x ∈ I, tìte hF (x) kaleÐtai par�gousa (  antipar�gwgoc   arqik  sun�rthsh) thc f sto I.

Parathr seic

1. An to I perilamb�nei k�poio �kro, h par�gwgoc F ′(x) ekeÐ ennoeÐtai ìti eÐnai hpleurik .

2. Den èqoun ìlec oi sunart seic par�gousa se k�poio di�sthma I.

3. All� èqoun ìlec oi suneqeÐc, ìpwc ja doÔme sÔntoma.

4. Akìma kai tìte, mporeÐ na eÐnai adÔnato na broÔme tÔpo gia thn par�gousa.

5. An h F (x) eÐnai par�gousa thc f(x), tìte eÐnai kai h F (x) +C gia k�je C ∈ R.Antistrìfwc, an oi F1(x) kai F2(x) eÐnai par�gousec thc f(x), anagkastik�

(F1(x)− F2(x))′ = f(x)− f(x) = 0,

�ra apì to Je¸rhma thc Mhdenik c Parag¸gou,

F1(x)− F2(x) = C ⇒ F1(x) = F2(x) + C,

gia k�poio C ∈ R.

Orismìc 4.5. (Sumbolismìc Leibniz) An h f èqei k�poia par�gousa F sto I, tosÔnolo twn paragous¸n {F (x) + C : C ∈ R} gr�fetai pio sunoptik� wc F (x) + C,sumbolÐzetai me ∫

f(x) dx  

∫f

kai kaleÐtai aìristo olokl rwma (  genik  par�gousa) . 'Ara:∫f(x) dx = F (x) + C ⇔ F ′(x) = f(x).

Par�deigma 4.22. ∫cosx dx = sin x+ C,∫x4 dx =

x5

5+ C.

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100 KEF�ALAIO 4. PAR�AGWGOS

Parathr seic

1. Suqn� se exis¸seic ìpwc tic �nw, to di�sthma I den anafèretai all� uponoeÐtai.

2. UpenjumÐzoume ton orismì tou ajroÐsmatoc sunìlwn: an A, B sÔnola,

A+B , {a+ b : a ∈ A, b ∈ B}.EpÐshc, an c ∈ R, orÐzoume to ginìmeno arijmoÔ me sÔnolo

cA , {ca : a ∈ A}.Oi orismoÐ isqÔoun kai ìtan ta A, B, eÐnai sÔnola sunart sewn.

3. To aìristo olokl rwma eÐnai suntomografÐa gia oikogèneia sunart sewn, kaiekfr�seic pou perilamb�noun aìrista oloklhr¸mata prèpei na ermhneÔontai ana-lìgwc. 'Etsi, an F ′(x) = f(x) kai a 6= 0, tìte

a

∫f(x) dx = a{F (x) + C : C ∈ R}

= {aF (x) + aC : C ∈ R} = {aF (x) + C : C ∈ R},ìpou sthn teleutaÐa isìthta qrhsimopoi same to a 6= 0. An epiplèon G′(x) =g(x) kai èna apì ta a, b eÐnai di�foro tou mhdenìc,

a

∫f(x) dx+ b

∫g(x) dx

= a{F (x) + C1 : C1 ∈ R}+ b{G(x) + C2 : C2 ∈ R}= {aF (x) + aC1 + bG(x) + bC2 : C1, C2 ∈ R}= {aF (x) + bG(x) + C : C ∈ R},

ìpou sthn teleutaÐa isìthta qrhsimopoi same to ìti èna ek twn a, b eÐnai di�forotou mhdenìc.

Je¸rhma 4.11. (Grammikìthta par�gousac) An oi f, g èqoun par�gousec sto I,kai èna ek twn a, b eÐnai di�foro tou mhdenìc, tìte èqei par�gousa kai h af + bg kaim�lista ∫

[af(x) + bg(x)] dx = a

∫f(x) dx+ b

∫g(x) dx.

Apìdeixh. 'Estw F (x) mia par�gousa thc f(x) kai G(x) mia par�gousa thc g(x).Profan¸c h aF (x) + bG(x) eÐnai par�gousa thc [af(x) + bg(x)], �ra∫

[af(x) + bg(x)] dx = {aF (x) + bG(x) + C : C ∈ R},

to opoÐo eÐdame �nw ìti isoÔtai me to a∫f(x) dx+ b

∫g(x) dx.

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4.5. PAR�AGOUSES 101

Je¸rhma 4.12. (Kanìnac thc AlusÐdac) 'Estw g(x) paragwgÐsimh sto I, kai forismènh sto J me antipar�gwgo F . 'Estw pwc to pedÐo tim¸n g(I) ⊆ J . Tìte hf(g(x))g′(x) : I → R èqei par�gousa sto I thn F (g(x)) kai aìristo olokl rwma:∫

f(g(x))g′(x) dx = F (g(x)) + C. (4.3)

Apìdeixh. ProkÔptei apì ton kanìna thc alusÐdac, paragwgÐzontac thn F (g(x)).

Parat rhsh: H (4.3) suqn� gr�fetai wc∫f(u) du = F (u) + C,

ìpou u ennoeÐtai ìti eÐnai ìpoia sun�rthsh g(x) jèloume pou ikanopoieÐ tic sunj kectou jewr matoc kai du , g′(x)dx.

Par�deigma 4.23. Ja upologÐsoume ta

I1 =

∫x sinx2 dx, I2 =

∫x2√x3 + 1 dx, I3 =

∫sin√x√

xdx,

efarmìzontac ton Kanìna thc AlusÐdac. 'Eqoume, kat� perÐptwsh:

1. ∫x sinx2 dx =

∫1

2sin(x2)2x dx =

1

2

∫sinu du

= −1

2cosu+ C = −1

2cosx2 + C,

ìpou u = g(x) = x2 kai du = g′(x)dx = 2xdx.

2. ∫x2√x3 + 1 dx =

1

3

∫(x3 +1)

12 (3x2dx) =

1

3

∫u

12 du =

2

9u

32 =

2

9(x3 +1)

32 +C,

ìpou u = g(x) = x3 + 1.

3. ∫sin√x√

xdx = 2

∫sin√x

1

2√xdx = 2

∫sinu du

= −2 cosu+ C = −2 cos√x+ C,

ìpou u = g(x) =√x.

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102 KEF�ALAIO 4. PAR�AGWGOS

Je¸rhma 4.13. (Olokl rwsh kat� par�gontec) An oi f, g eÐnai paragwgÐsimec stoI kai oi f ′g, fg′ èqoun par�gousec sto I, tìte∫

f(x)g′(x) dx = f(x)g(x)−∫f ′(x)g(x) dx.

Apìdeixh. To je¸rhma ermhneÔetai wc ex c: to sÔnolo twn paragous¸n thc fg′ pro-kÔptei an afairèsoume apì thn fg to sÔnolo twn paragous¸n thc f ′g. (H afaÐreshenìc sunìlou sunart sewn apì mia sun�rthsh orÐzetai me ton profan  trìpo.) Hapìdeixh prokÔptei eÔkola parathr¸ntac ìti

(fg)′ = f ′g + fg′,

kai apì to Je¸rhma 4.11.

Par�deigma 4.24. Ja upologÐsoume ta

I1 =

∫x sinx dx, I2 =

∫x cosx dx, I3 =

∫x2 sinx dx,

qrhsimopoi¸ntac olokl rwsh kat� par�gontec.'Eqoume, kat� perÐptwsh:

1.

I1 =

∫x sinx dx =

∫x (− cosx)′ dx = −

∫x(cosx)′ dx

= −x cosx+

∫cosx(x)′ dx = −x cosx+

∫cosx dx = −x cosx+ sinx+C.

2.

I2 =

∫x cosx dx =

∫x(sinx)′ dx = x sinx−

∫x′ sinx dx

= x sinx−∫

sinx dx = x sinx+ cosx+ C.

3.

I3 =

∫x2 sinx dx = −

∫x2(cosx)′ dx = −x2 cosx+

∫cosx(x2)′ dx

= −x2 cosx+ 2

∫x cosx dx = −x2 cosx+ 2I2

= −x2 cosx+ 2x sinx+ 2 cosx+ C.

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Kef�laio 5

Efarmogèc Parag¸gou

5.1 ProseggÐseic

Parat rhsh: AfoÔ

limh→0

f(x0 + h)− f(x0)

h= f ′(x0),

eÐnai logikì na èqoume, ìtan to h eÐnai mikrì:

f(x0 + h)− f(x0)

h' f ′(x0).

An jèsoume ∆x , h kai ∆f , f(x0 + h)− f(x0), prokÔptei

∆f ' f ′(x0)∆x⇔ f(x) ' f(x0) + f ′(x0)∆x = f(x0) + f ′(x0)(x− x0). (5.1)

Gia par�deigma,

√4.1−

√4 ' 1

2√

4× 0.1⇔

√4.1 ' 2 +

1

2√

4× 0.1 = 2.025,

en¸ sthn pragmatikìthta√

4.1 = 2.0248.Mia grafik  ex ghsh faÐnetai sto Sq ma 5.1. Sugkekrimèna, antÐ na qrhsimopoi -

soume thn Ðdia th sun�rthsh gia ton upologismì miac tim c thc f(x), qrhsimopoioÔmemia grammik  prosèggis  thc, kai sugkekrimèna thn efaptìmenh thc eujeÐa sto shmeÐo(x0, f(x0)) pou dÐnetai apì thn exÐswsh

y = f(x0) + f ′(x0)(x− x0).

Sto sq ma, to sf�lma isoÔtai me to m koc thc k�jethc gramm c, kai eÐnai mikrì ìtanx ' x0. 'Ara, an eÐnai eÔkolo na upologÐsoume to f ′(x0), mporoÔme na upologÐsoumeproseggistik�, ìsec timèc jèloume, kont� sto x0, qrhsimopoi¸ntac thn apl , grammik exÐswsh (5.1).

103

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104 KEF�ALAIO 5. EFARMOG�ES PARAG�WGOU

y

f(x)

y=f(x0)+f’(x0)(x-x0)x

Sq ma 5.1: Gewmetrik  ex ghsh thc prosèggishc (5.1).

Par�deigma 5.1. Gia thn perÐptwsh f(x) =√x, f ′(x) = 1

2√x, x0 = 2, lamb�noume:

x f(x) =√x f(x0) + f ′(x0)(x− x0) Sf�lma

3 1.7321 1.75 0.01793.9 1.9748 1.975 1.5823× 10−4

3.99 1.9975 1.9975 1.5645× 10−6

4 2 2 04.01 2.0025 2.0025 1.5605× 10−6

4.1 2.0248 2.025 1.5433× 10−4

5 2.2361 2.25 0.0139

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5.2. H M�EJODOS TOU NE�UTWNA 105

5.2 H Mèjodoc tou NeÔtwna

'Estw pwc mac endiafèrei h arijmhtik  eÔresh thc rÐzac x∗ miac sun�rthshc f(x) mepar�gwgo f ′(x). Me dedomèno ìti diajètoume thn par�gwgo, mporoÔme na th qrhsi-mopoi soume gia na proseggÐsoume th rÐza, ìpwc faÐnetai sto Sq ma 5.2.

Sto sq ma faÐnetai pwc, xekin¸ntac apì mia arqik  ektÐmhsh x0 gia th rÐza x∗,mporoÔme na broÔme mia kalÔterh ektÐmhsh, x1, sugkekrimèna to shmeÐo x1 sto opoÐoh efaptìmenh eujeÐa tèmnei ton �xona twn x, dhlad  to shmeÐo x1 ìpou

0 = f(x0) + f ′(x0)(x1 − x0)⇔ x1 = x0 −f(x0)

f ′(x0).

O upologismìc mporeÐ na epanalhfjeÐ pollèc forèc, dÐnontac k�je for� èna shmeÐokalÔtero tou prohgoÔmenou (elpÐzoume), b�sei tou tÔpou

xn = xn−1 −f(xn−1)

f ′(xn−1), n ∈ N.

Prèpei, bèbaia, se k�je b ma f ′(xn) 6= 0. (Ti sumbaÐnei se aut  thn perÐptwsh?)Sunep¸c, mporoÔme na ektelèsoume ton akìloujo algìrijmo, pou eÐnai gnwstìc wc

h mèjodoc tou NeÔtwna   mèjodoc twn Newton-Raphson.

/* NEWTON-RAPHSON METHOD */

INPUT: x0, f(), f’(), E /* x0 IS INITIAL POINT, E>0 IS TOLERANCE */

OUTPUT: x’ /* ESTIMATED LOCATION OF ROOT */

DO

x’=x0-f(x0)/f’(x0);

IF |x’-x0| < E,

EXIT;

END;

x0=x’;

END

Parathr seic

1. Ti epÐdrash èqei h epilog  tou E sthn ektèlesh tou algìrijmou?

2. H mèjodoc eÐnai polÔ pio gr gorh apì th mèjodo thc diqotìmhshc (kalì).

3. An den epilèxoume arqikì shmeÐo arket� kont� sth rÐza, tìte mporeÐ na mhn sug-klÐnoume potè, akìma kai an faÐnetai arqik� ìti up�rqei sÔgklish (polÔ kakì).

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106 KEF�ALAIO 5. EFARMOG�ES PARAG�WGOU

y

x

y=f(x0)+f’(x0)(x-x0)

f(x)

x0

y0

x1=x0-f(x0)/f’(x0)

x*

Sq ma 5.2: H basik  idèa thc mejìdou tou NeÔtwna: an gnwrÐzoume thn tim  f(x0) miac sun�rthshcf se èna shmeÐo x0, kaj¸c kai thn tim  f ′(x0) thc parag¸gou f sto Ðdio shmeÐo x0, tìte mporoÔmena upologÐsoume mia prosèggish x1 gia mia kontin  rÐza x∗ thc f .

An ìmwc epilèxoume èna shmeÐo arkoÔntwc kont� sth rÐza, ja sugklÐnoume seaut . (DeÐte to epìmeno je¸rhma.)

4. Ti ja k�name an eÐqame sth di�jes  mac kai thn deÔterh par�gwgo?

Je¸rhma 5.1. (SÔgklish mejìdou Newton) An h f èqei suneq  deÔterh par�gwgo,kai up�rqei rÐza x∗ tètoia ¸ste f(x∗) = 0 all� f ′(x∗) 6= 0, tìte, an to x0 eÐnaiarkoÔntwc kont� sto x∗, o �nw algìrijmoc sugklÐnei sto x∗ ètsi ¸ste se k�je b mato sf�lma na eÐnai to tetr�gwno tou sf�lmatoc sto prohgoÔmeno b ma.

Apìdeixh. ParaleÐpetai.

Par�deigma 5.2. 'Estw h sun�rthsh f(x) = cos x− x, gia thn opoÐa xekin�me thnanaz thsh miac rÐzac apì dÔo diaforetik� shmeÐa, to x0 = 0 kai to x0 = −3. 'OpwcfaÐnetai kai sto Sq ma 5.3, sthn pr¸th perÐptwsh èqw sÔgklish, ìqi ìmwc kai sth

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5.2. H M�EJODOS TOU NE�UTWNA 107

deÔterh! Ta shmeÐa pou epistrèfei o algìrijmoc sthn pr¸th perÐptwsh eÐnai ta

0, 1, 0.750363867840244, 0.739112890911362, 0.739085133385284,

0.739085133215161, 0.739085133215161, 0.739085133215161,

en¸ sth deÔterh eÐnai ta

−3, −0.6597, 3.0858, −0.7829, 4.2797, −46.7126, 29.5906, −896.8049.

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108 KEF�ALAIO 5. EFARMOG�ES PARAG�WGOU

0 0.2 0.4 0.6 0.8 1 1.2

−0.5

0

0.5

1

x

0

1

2

3

4

cosx − x

−60 −40 −20 0 20 40 60−60

−40

−20

0

20

40

60

x

0 1

2

3

4

5

6

cosx − x

Sq ma 5.3: Par�deigma 5.2: H ektèlesh thc mejìdou tou NeÔtwna gia thn sun�rthsh cosx− x kaigia dÔo diaforetik� arqik� shmeÐa, to x0 = 0 kai to x0 = −3. H sun�rthsh èqei sqediasteÐ mediakekommènh gramm .

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5.3. MONOTON�IA KAI AKR�OTATA SUNART�HSEWN 109

5.3 MonotonÐa kai Akrìtata Sunart sewn

Je¸rhma 5.2. (Ikan� krit ria monotonÐac) 'Estw sun�rthsh f suneq c sto [a, b]kai paragwgÐsimh sto (a, b).

1. An f ′(x) = 0 gia k�je x ∈ (a, b), tìte h f eÐnai stajer  sto [a, b]. (Je¸rhmaMhdenik c Parag gou 4.8)

2. An f ′(x) > 0 gia k�je x ∈ (a, b), tìte h f eÐnai gnhsÐwc aÔxousa sto [a, b].

3. An f ′(x) ≥ 0 gia k�je x ∈ (a, b), tìte h f eÐnai aÔxousa sto [a, b].

4. An f ′(x) < 0 gia k�je x ∈ (a, b), tìte h f eÐnai gnhsÐwc fjÐnousa sto [a, b].

5. An f ′(x) ≤ 0 gia k�je x ∈ (a, b), tìte h f eÐnai fjÐnousa sto [a, b].

Apìdeixh. To pr¸to skèloc èqei  dh apodeiqjeÐ, wc to Je¸rhma Mhdenik c Parag¸-gou (Je¸rhma 4.8).

Gia to deÔtero skèloc, èstw x, y, me a ≤ x < y ≤ b. Apì to Je¸rhma Mèshc Tim c

∃c ∈ (a, b) tètoio ¸ste f ′(c) = f(y)−f(x)y−x . Epeid  f ′(c) > 0, anagkastik� f(x) < f(y),

kai h f eÐnai gnhsÐwc aÔxousa.Ta upìloipa skèlh apodeiknÔontai an�loga.

Je¸rhma 5.3. (KrÐsima shmeÐa)

1. 'Estw f orismènh kai paragwgÐsimh se di�sthma (a, b). An h f èqei topikìakrìtato sto c ∈ (a, b) prèpei f ′(c) = 0.

2. Ta akrìtata miac sun�rthshc f entìc enìc diast matoc I brÐskontai an�mesa staakìlouja krÐsima shmeÐa:

(aþ) Ta �kra tou I, an an koun sto I,

(bþ) ta shmeÐa ìpou h par�gwgoc f ′ up�rqei kai eÐnai mhdenik ,

(gþ) kai ta shmeÐa ìpou h par�gwgoc den up�rqei.

Apìdeixh. To pr¸to skèloc èqei  dh apodeiqjeÐ (Je¸rhma 4.5). To deÔtero skèlocprokÔptei �mesa apì to pr¸to.

Parathr seic

1. To �nw je¸rhma eÐnai to basikì ergaleÐo gia ton entopismì twn akrìtatwn miacsun�rthshc.

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110 KEF�ALAIO 5. EFARMOG�ES PARAG�WGOU

2. Den eÐnai ìla ta krÐsima shmeÐa akrìtata, ìmwc b�sei tou �nw ìla ta akrìtataeÐnai krÐsima shmeÐa.

Je¸rhma 5.4. (Ikan� krit ria akrìtatwn) 'Estw f suneq c sto [a, b] kai paragw-gÐsimh sto (a, b), ektìc Ðswc apì k�poio c ∈ (a, b). Tìte

1. An f ′(x) ≥ 0 gia k�je x < c kai f ′(x) ≤ 0 gia k�je x > c, tìte h f(x) èqeitopikì mègisto sto x = c.

2. An f ′(x) ≤ 0 gia k�je x < c kai f ′(x) ≥ 0 gia k�je x > c, tìte h f(x) èqeitopikì el�qisto sto x = c.

Apìdeixh. 1. An f ′(x) ≥ 0 gia k�je x ∈ (a, c), tìte, apì to Je¸rhma 5.2, h f(x)eÐnai aÔxousa sto [a, c], �ra

a ≤ x < c⇒ f(x) ≤ f(c).

Epiplèon, an f ′(x) ≤ 0 gia k�je x ∈ (c, b), tìte, apì to Je¸rhma 5.2, h f(x)eÐnai fjÐnousa sto [c, b], �ra

c < x ≤ b⇒ f(x) ≤ f(c).

Sundu�zontac kai ta dÔo, prokÔptei telik� pwc h f(x) èqei topikì mègisto stoc, to opoÐo m�lista eÐnai kai olikì mègisto sto [a, b].

2. An�loga.

Je¸rhma 5.5. (Ikan� krit ria akrìtatwn deÔterhc t�xhc) 'Estw sun�rthsh f para-gwgÐsimh sto (a, b), me thn par�gwgo epÐshc paragwgÐsimh sto (a, b). 'Estw c ∈ (a, b).

1. An f ′(c) = 0 kai f ′′(x) ≤ 0 sto (a, b), tìte h f(x) èqei topikì mègisto sto c.

2. An f ′(c) = 0 kai f ′′(x) ≥ 0 sto (a, b), tìte h f(x) èqei topikì el�qisto sto c.

Apìdeixh. 1. AfoÔ h f ′′(x) ≤ 0 sto (a, b), apì to Je¸rhma 5.2 h f ′(x) eÐnai fjÐnousasto (a, b), kai afoÔ f ′(c) = 0, ja èqoume f ′(x) ≤ 0 sto (c, b) kai f ′(x) ≥ 0 sto(a, c), �ra apì to Je¸rhma 5.4 h f(x) èqei topikì mègisto sto c.

2. An�loga.

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5.3. MONOTON�IA KAI AKR�OTATA SUNART�HSEWN 111

−5 0 5−10

−5

0

5

10

15

20f(x) = (x + 1)2/(x − 1)

x

Sq ma 5.4: H sun�rthsh tou ParadeÐgmatoc 5.3.

Par�deigma 5.3. DÐnetai h sun�rthsh

f(x) =(x+ 1)2

x− 1, x ∈ R− {1}.

Gia na broÔme ta akrìtata kai ta diast mata ìpou h sun�rthsh eÐnai monìtonh, topr¸to b ma eÐnai na upologÐsoume thn pr¸th par�gwgo thc f(x), h opoÐa parathroÔmeup�rqei pantoÔ ektìc tou x = 1. 'Eqoume:

f ′(x) =2(x+ 1)(x− 1)− (x+ 1)2

(x− 1)2=

(x+ 1)(2x− 2− x− 1)

(x− 1)2=

(x+ 1)(x− 3)

(x− 1)2.

H par�gwgoc eÐnai arnhtik  sta diast mata (−1, 1) kai (1, 3), �ra ekeÐ h sun�rthsheÐnai gnhsÐwc fjÐnousa. EpÐshc, h par�gwgoc eÐnai jetik  sta (−∞,−1) kai (3,∞),�ra ekeÐ h sun�rthsh eÐnai gnhsÐwc aÔxousa. 'Ara up�rqei èna topikì mègisto stox = −1, kai èna topikì el�qisto sto x = 3, me timèc, antÐstoiqa, f(−1) = 0, kaif(3) = 8

Epiplèon, isqÔoun ta ex c, ìpwc eÔkola prokÔptei apì basikèc isìthtec twn orÐwn:

limx→−∞

f(x) = −∞, limx→1−

f(x) = −∞, limx→1+

f(x) =∞, limx→∞

f(x) =∞.

'Ara, den up�rqei olikì mègisto kai olikì el�qisto.H sun�rthsh èqei sqediasteÐ sto Sq ma 5.4.

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112 KEF�ALAIO 5. EFARMOG�ES PARAG�WGOU

5.4 Kurtèc Sunart seic

Orismìc 5.1. (Kurtèc kai koÐlec sunart seic)

1. Mia sun�rthsh f : A→ R kaleÐtai kurt  se k�poio di�sthma I ⊆ A an

∀x0, x1 ∈ I, ∀θ ∈ [0, 1], f(θx0 + (1− θ)x1) ≤ θf(x0) + (1− θ)f(x1).

2. Mia sun�rthsh f : A → R orismènh sto I ⊆ A kaleÐtai koÐlh an h −f eÐnaikurt  sto I.

3. An stouc �nw orismoÔc den anafèretai rht¸c to I, ennoeÐtai pwc eÐnai to pedÐoorismoÔ A.

4. 'Ena shmeÐo c kaleÐtai shmeÐo kamp c thc f an h f eÐnai kurt  (koÐlh) se ènadi�sthma [a, c] kai koÐlh (kurt ) se èna di�sthma [c, b].

Parathr seic

1. Exet�ste to Sq ma 5.5. Gia na d¸soume mia ermhneÐa ston orismì, parathroÔmeìti h exÐswsh thc eujeÐac pou dièrqetai apì ta shmeÐa (x0, f(x0)) kai (x1, f(x1))eÐnai h akìloujh:

y =f(x1)− f(x0)

x1 − x0(x− x0) + f(x0).

Gia na peisteÐte, parathr ste ìti pr�gmati dièrqetai kai apì ta dÔo shmeÐa. E-nallaktik�, parathreÐste ìti dièrqetai apì to (x0, f(x0)) (autì eÐnai profanèc)kai èqei kai th swst  klÐsh.

An sthn �nw eujeÐa jèsoume x = θx0 + (1− θ)x1, lamb�noume

y =f(x1)− f(x0)

x1 − x0(θx0 + (1− θ)x1 − x0) + f(x0)

= (f(x1)− f(x0))(1− θ)(x1 − x0)

x1 − x0+ f(x0)

= θf(x0) + (1− θ)f(x1),

lamb�noume dhlad  to dexÐ skèloc thc anisìthtac tou orismoÔ thc kurtìthtac.Parathr ste ìti

(aþ) 'Otan θ = 0, x = x1.

(bþ) 'Otan θ = 1, x = x0.

(gþ) 'Otan 0 < θ < 1, x0 < x < x1.

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5.4. KURT�ES SUNART�HSEIS 113

x

x0

f(x0)

f(x1)

x1θx0+(1-θ)x1

f(θx0+(1-θ)x1)

θf(x0)+(1-θ)f(x1)

y

f(x)

y={[f(x1)-f(x0)]/[x1-x0]}(x-x0)+f(x0)

Sq ma 5.5: Gewmetrik  ermhneÐa thc kurtìthtac sun�rthshc.

Sunep¸c, ìtan to θ metab�lletai apì to θ = 0 sto θ = 1, to x diatrèqei todi�sthma [x0, x1] apì to x1 proc to x0.

'Ara, mia sun�rthsh eÐnai kurt  an k�je qord  thc den brÐsketai potè k�tw apìto gr�fhm� thc.

ParomoÐwc, mporoÔme na doÔme pwc h sun�rthsh eÐnai koÐlh an to gr�fhma denbrÐsketai potè k�tw apì th qord .

2. Oi �nw orismoÐ thc kurtìthtac kai thc koilìthtac isqÔoun wc èqoun kai sthnperÐptwsh sunart sewn f : Rn → R poll¸n metablht¸n.

3. H kurtìthta/koilìthta miac sun�rthshc (miac   poll¸n metablht¸n) mac endia-fèrei giatÐ o entopismìc elaqÐstwn (megÐstwn) me arijmhtikèc mejìdouc se kurtèc(koÐlec) sunart seic eÐnai polÔ aplìc, se sqèsh me th genik  perÐptwsh. Mpo-reÐte na fantasteÐte giatÐ? Perissìtera se maj mata Arijmhtik c An�lushc kaiBeltistopoÐhshc.

Par�deigma 5.4. EÐnai oi sunart seic tou Sq matoc 5.6 kurtèc   koÐlec, kai annai, se poia diast mata?

Par�deigma 5.5. Ja deÐxoume ìti an oi sunart seic f1, f2, . . . , fm eÐnai kurtèc seèna di�sthma, tìte eÐnai kurt  kai h f = a1f1 + a2f2 + · · ·+ amfm ìpou ai ≥ 0.

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114 KEF�ALAIO 5. EFARMOG�ES PARAG�WGOU

0 1 2 3 4−1

−0.5

0

0.5

1

x

y

−1 −0.5 0 0.5 10

0.2

0.4

0.6

0.8

1

x

y

−1 −0.5 0 0.5 1−1

0

1

2

3

x

y

−1 −0.5 0 0.5 10

0.2

0.4

0.6

0.8

1

x

y

Sq ma 5.6: EÐnai oi sunart seic tou sq matoc kurtèc, kai an nai, se poia diast mata?

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5.4. KURT�ES SUNART�HSEIS 115

Pr�gmati,

f(θx0 + (1− θ)x1)

= a1f1(θx0 + (1− θ)x1) + a2f2(θx0 + (1− θ)x1) + · · ·+ amfm(θx0 + (1− θ)x1)

≤ a1θf1(x0) + a1(1− θ)f1(x1) + a2θf2(x0) + a2(1− θ)f2(x1)

+ · · ·+ amθfm(x0) + am(1− θ)fm(x1)

= θ [a1f1(x0) + a2f2(x0) + · · ·+ amfm(x0)]

+(1− θ) [a1f1(x1) + a2f2(x1) + · · ·+ amfm(x1)]

= θf(x0) + (1− θ)f(x1).

Par�deigma 5.6. An oi f1, f2 eÐnai kurtèc se èna di�sthma, tìte eÐnai kurt  kaih max(f1, f2). (H sun�rthsh max(f1, f2) lamb�nei se k�je shmeÐo x to mègisto twntim¸n f1(x) kai f(x2).)

Gia na to apodeÐxoume, parathroÔme kat' arq n pwc isqÔei:

max{a+ b, c+ d} ≤ max{a, c}+ max{b, d}. (5.2)

Gia na apodeÐxoume thn �nw, arkeÐ na p�roume tic 4 peript¸seic:(a ≥ c, b ≥ d

),(a ≥ c, b < d

),(a < c, b ≥ d

),(a < c, b < d

).

kai na sugkrÐnoume se k�je perÐptwsh ta dÔo skèlh.'Estw loipìn [a, b] to pedÐo orismoÔ twn f1, f2. Jètw f = max{f1, f2}. 'Estw

x0, x1 ∈ [a, b], kai èstw θ ∈ [0, 1]. 'Eqoume:

f(θx0 + (1− θ)x1) = max{f1(θx0 + (1− θ)x1), f2(θx0 + (1− θ)x1)}≤ max{θf1(x0) + (1− θ)f1(x1), θf2(x0) + (1− θ)f2(x1)}≤ max{θf1(x0), θf2(x0)}+ max{(1− θ)f1(x1), (1− θ)f2(x1)}= θmax{f1(x0), f2(x0)}+ (1− θ) max{f1(x1), f2(x1)}= θf(x0) + (1− θ)f(x1).

Sthn pr¸th anisìthta qrhsimopoi same ton orismì thc kurtìthtac, en¸ sth deÔterhthn (5.2). Sthn deÔterh isìthta qrhsimopoi same to ìti θ, 1− θ ≥ 0.

EÐnai p�nta kurt  h min(f1, f2)? Me lÐgh skèyh mporoÔme na doÔme pwc ìqi! 'Enaantipar�deigma èqei sqediasteÐ sto Sq ma 5.7, ìpou f1(x) = x2 kai f2(x) = (x− 4)2.

Je¸rhma 5.6. (Krit ria kurtìthtac) 'Estw f suneq c se di�sthma I.

1. An h f ′ up�rqei sto I kai eÐnai aÔxousa (fjÐnousa), tìte h f eÐnai kurt  (koÐlh)sto I.

2. An h f ′′ up�rqei sto I kai eÐnai mh arnhtik  (jetik ), tìte h f eÐnai kurt  (koÐlh)sto I.

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116 KEF�ALAIO 5. EFARMOG�ES PARAG�WGOU

−2 0 2 4 6−5

0

5

10

15

20

25

30

35

40

x

f1(x) = x2 f2(x) = (x − 4)2

Sq ma 5.7: Par�deigma 5.6. Oi f1(x), f2(x) eÐnai kurtèc, �ra ja eÐnai kai h max{f1(x), f2(x)}. Hmin{f1(x), f2(x)} ìmwc den eÐnai kurt !

Apìdeixh. ja apodeÐxoume tic sunj kec kurtìthtac. Oi apodeÐxeic twn sunjhk¸n koi-lìthtac prokÔptoun an�loga,   me qr sh twn sunjhk¸n kurtìthtac.

1. 'Estw x0, x1 ∈ I, kai èstw opoiod pote x = θx0 + (1 − θ)x1 ∈ (x0, x1), meθ ∈ (0, 1). Apì to Je¸rhma Mèshc Tim c, up�rqoun c, d tètoia ¸ste

f ′(c) =f(x)− f(x0)

x− x0, c ∈ (x0, x),

f ′(d) =f(x1)− f(x)

x1 − x, d ∈ (x0, x).

'Omwc h f ′ eÐnai aÔxousa, �ra

f ′(c) ≤ f ′(d)⇒ f(x)− f(x0)

x− x0≤ f(x1)− f(x)

x1 − x.

EpÐshc,

x− x0 = θx0 + (1− θ)x1 − x0 = (1− θ)(x1 − x0),

x1 − x = x1 − θx0 − (1− θ)x1 = θ(x1 − x0).

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5.4. KURT�ES SUNART�HSEIS 117

kai me antikat�stash sthn �nw anisìthta èqoume

f(x)− f(x0)

1− θ ≤ f(x1)− f(x)

θ⇒ θf(x)−θf(x0) ≤ (1−θ)f(x1)− (1−θ)f(x)

⇒ f(x) ≤ (1− θ)f(x1) + θf(x0).

H �nw profan¸c isqÔei kai gia θ = 0, 1. 'Ara telik� h f eÐnai kurt .

2. An h f ′′ up�rqei sto (a, b) kai eÐnai mh arnhtik , tìte me efarmog  tou Jewr ma-toc 5.2 prokÔptei pwc h f ′ eÐnai aÔxousa, kai mporoÔme na qrhsimopoi soume toprohgoÔmeno skèloc.

Par�deigma 5.7. Sqetik� me thn kurtìthta thc sun�rthshc tou ParadeÐgmatoc 5.3:

f ′′(x)

=

[(x+ 1)(x− 3)

(x− 1)2

]′=

[(x− 3) + (x+ 1)](x− 1)2 − 2(x+ 1)(x− 3)(x− 1)

(x− 1)4

=(2x− 2)(x− 1)− 2(x+ 1)(x− 3)

(x− 1)3=

8

(x− 1)3.

'Ara, h f(x) eÐnai kurt  sto di�sthma (1,∞) (ìpou h deÔterh par�gwgoc eÐnai jetik )kai koÐlh sto di�sthma (−∞, 1) (ìpou h deÔterh par�gwgoc eÐnai arnhtik .)

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118 KEF�ALAIO 5. EFARMOG�ES PARAG�WGOU

5.5 Kanìnac tou L’Hopital

Je¸rhma 5.7. (Kanìnac tou L’Hopital) An isqÔoun ta ìria limx→A

f(x) = limx→A

g(x) =

B kai limx→A

f ′(x)/g′(x) = C, tìte limx→A

f(x)/g(x) = C, ìpou

1. to A eÐnai èna apì ta a, a−, a+,∞,−∞,

2. to B eÐnai èna apì ta 0,∞,−∞,

3. kai to C eÐnai èna apì ta l ∈ R,∞,−∞.

Apìdeixh. ParaleÐpetai.

Par�deigma 5.8. Katarq n, parathr ste pwc me qr sh tou kanìna èqoume

limx→0

sinx

x= lim

x→0

cosx

1= 1, lim

x→0

1− cosx

x= lim

x→0

sinx

1= 0.

Parathr ste, p�ntwc, pwc o upologismìc tou pr¸tou orÐou me autìn ton trìpo eÐnaiproblhmatikìc, afoÔ k�noume qr sh thc parag¸gou (sinx)′ = cosx pou èqei upolo-gisteÐ me qr sh autoÔ tou orÐou.

O kanìnac èqei �mesh efarmog  (qwrÐc logikèc antif�seic) se polÔ pio sÔnjetaìria pou emplèkoun trigwnometrikèc sunart seic. Gia par�deigma,

limx→0

x− sinx

x− tanx= lim

x→0

1− cosx

1− 1/ cos2 x= lim

x→0

(1− cosx) cos2 x

(cosx− 1)(cosx+ 1)

= −limx→0

cos2 x

1 + cos x= −1

2.

Par�deigma 5.9. Ja upologÐsoume to ìrio limx→0

(1

sinx − 1x

), to opoÐo parousi�zei

aprosdioristÐa thc morf c ∞ − ∞, kai to opoÐo ek pr¸thc ìyewc den faÐnetai naepidèqetai lÔsh me ton Kanìna tou L’Hopital. Parathr ste ìmwc pwc

limx→0

(1

sinx− 1

x

)= lim

x→0

x− sinx

x sinx= lim

x→0

1− cosx

sinx+ x cosx

= limx→0

sinx

cosx+ cosx− x sinx=

0

2= 0.

O Kanìnac tou L’Hopital efarmìsthke dÔo forèc, sthn deÔterh kai thn trÐth isìthta.

Par�deigma 5.10. DeÐte thn akìloujh dipl  efarmog  tou Kanìna tou L’Hopital.SumfwneÐte?

limx→2

(x2 − x− 2

x2 + x− 6

)= lim

x→2

(2x− 1

2x+ 1

)= lim

x→2

2

2= 1.

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Kef�laio 6

Olokl rwma

6.1 Orismìc

Orismìc 6.1. (DiamerÐseic)

1. 'Estw kleistì di�sthma [a, b]. KaloÔme diamèrish mia peperasmènh sullog  sh-meÐwn P = {t0, t1, . . . , tn} gia ta opoÐa isqÔei:

a = t0 < t1 < · · · < tn = b.

2. An P , Q diamerÐseic me P ⊂ Q, to Q kaleÐtai eklèptunsh tou P kai lème ìti toQ perièqei to P .

Orismìc 6.2. ('Anw kai K�tw AjroÐsmata Darboux) 'Estw f : [a, b]→ R, fragmè-nh, kai P = {t0, t1, . . . , tn} mia diamèrish tou [a, b]. 'Estw

mi , inf{f(x) : ti−1 ≤ x ≤ ti}, i = 1, 2, . . . , n,

Mi , sup{f(x) : ti−1 ≤ x ≤ ti}, i = 1, 2, . . . , n.

1. To k�tw �jroisma (Darboux) thc f gia th diamèrish P orÐzetai wc

L(f, P ) ,n∑i=1

mi(ti − ti−1).

2. To �nw �jroisma (Darboux) thc f gia th diamèrish P orÐzetai wc

U(f, P ) ,n∑i=1

Mi(ti − ti−1).

119

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120 KEF�ALAIO 6. OLOKL�HRWMA

0 2 4 6−1

−0.5

0

0.5

1

0 2 4 6−1

−0.5

0

0.5

1

0 2 4 6−1

−0.5

0

0.5

1

0 2 4 6−1

−0.5

0

0.5

1

0 2 4 6−1

−0.5

0

0.5

1

0 2 4 6−1

−0.5

0

0.5

1

Sq ma 6.1: To embadìn k�je miac skiasmènhc klimakwt c sun�rthshc ekfr�zei èna k�tw (gia ticparast�seic sta arister�)   �nw (gia tic parast�seic sta dexi�) �jroisma Darboux.

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6.1. ORISM�OS 121

Parathr seic

1. P = {partition}, L = {lower}, U = {upper}.

2. Ta mi, Mi Ðswc na mhn up rqan an h f den  tan fragmènh,   an aut� orÐzontanwc el�qista kai mègista.

3. 'Otan f ≥ 0, to k�tw (�nw) �jroisma eÐnai h epif�neia metaxÔ tou graf matoc kaitou �xona twn x miac mh arnhtik c klimakwt c sun�rthshc pou eÐnai k�tw (�nw)apì thn sun�rthsh f kai ef�ptetai se aut .

4. Sthn genik  perÐptwsh, ti ekfr�zoun ta �nw kai k�tw ajroÐsmata?

5. DeÐte to Sq ma 6.1 gia paradeÐgmata �nw kai k�tw ajroism�twn Darboux.

6. IsqÔei ìti L(f, P ) ≤ U(f, P ). Pr�gmati, genik� inf S ≤ supS gia k�je sÔ-nolo, �ra kai gia ta sÔnola thc morf c [tk−1, tk]. Epomènwc mi ≤ Mi giai = 1, 2, . . . , n, kai h anisìthta prokÔptei pollaplasi�zontac k�je mia apì ticanisìthtec autèc me ta m kh twn antÐstoiqwn diasthm�twn kai prosjètontac tec.

L mma 6.1. (AjroÐsmata ekleptÔnsewn) An h diamèrish Q perièqei th diamèrish P ,tìte

L(f, P ) ≤ L(f,Q), U(f, P ) ≥ U(f,Q).

Apìdeixh. An prosjèsoume èna shmeÐo se mia diamèrish, tìte

1. o ìroc tou k�tw ajroÐsmatoc pou antistoiqeÐ sto di�sthma pou diqotom jhke jamegal¸sei   ja meÐnei o Ðdioc, kai sunep¸c to k�tw �jroisma ja megal¸sei, en¸

2. o ìroc tou �nw ajroÐsmatoc pou antistoiqeÐ sto di�sthma pou diqotom jhke jamikrÔnei   ja meÐnei o Ðdioc, kai sunep¸c to �nw �jroisma ja mikrÔnei.

Tèloc, afoÔ oi anisìthtec isqÔoun gia thn prosj kh enìc shmeÐou, me qr sh epagw-g c ja isqÔoun kai gia thn prosj kh opoioud pote peperasmènou pl jouc shmeÐwn,prokeimènou h P na metasqhmatisjeÐ sthn Q.

L mma 6.2. (K�tw AjroÐsmata ≤ 'Anw AjroÐsmata) 'Estw P1, P2 dÔo opoiesd potediamerÐseic tou diast matoc [a, b]. Ja isqÔei:

L(f, P1) ≤ U(f, P2).

Apìdeixh. 'Estw P = P1 ∪ P2. Ja èqoume:

L(f, P1) ≤ L(f, P ) ≤ U(f, P ) ≤ U(f, P2).

H pr¸th kai h trÐth anisìthta prokÔptoun me qr sh tou L mmatoc 6.1.

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122 KEF�ALAIO 6. OLOKL�HRWMA

Je¸rhma 6.1. ('Anw kai K�tw Olokl rwma Darboux)To k�tw olokl rwma Darboux thc f sto [a, b] orÐzetai wc to supremum ìlwn twnk�tw ajroism�twn:

supP{L(f, P )}.

EpÐshc, to �nw olokl rwma Darboux thc f sto [a, b] orÐzetai wc to infimum ìlwntwn �nw ajroism�twn:

infP{U(f, P )}.

IsqÔoun oi akìloujec anisìthtec:

supP{L(f, P )} ≤ inf

P{U(f, P )}, (6.1)

L(f, P ′) ≤ supP{L(f, P )} ≤ U(f, P ′), ∀P ′, (6.2)

L(f, P ′) ≤ infP{U(f, P )} ≤ U(f, P ′), ∀P ′. (6.3)

Apìdeixh. Apì to L mma 6.2, prokÔptei ìti k�je mèloc tou sunìlou {L(f, P )} eÐnaimikrìtero   Ðso apì opoiod pote mèloc tou sunìlou {U(f, P )}. 'Ara, to supremumtou pr¸tou sunìlou kai to infimum tou deÔterou sunìlou up�rqoun kai epiplèonto supremum tou pr¸tou sunìlou eÐnai mikrìtero   Ðso tou infimum tou deÔterousunìlou (Par�deigma 1.9). 'Ara, apodeÐxame thn (6.1).

Epeid  ex orismoÔ infP{U(f, P )} ≤ U(f, P ′) gia k�je P ′, prokÔptei kai h deÔte-rh anisìthta thc (6.2), en¸ h pr¸th prokÔptei epeid  ex' orismoÔ supP{L(f, P )} ≥L(f, P ′) gia k�je P ′. Oi anisìthtec thc (6.3) prokÔptoun parìmoia.

Parathr seic

1. BebaiwjeÐte ìti èqete katal�bei ti ekfr�zoun to k�tw kai to �nw olokl rwma.

2. 'Eqoume dÔo dunatèc peript¸seic:

(aþ) supP{L(f, P )} = infP{U(f, P )}.(bþ) supP{L(f, P )} < infP{U(f, P )}.

Sthn pr¸th perÐptwsh

(aþ) Efìson f(x) ≥ 0, o koinìc arijmìc ekfr�zei to embadìn thc epif�neiacmetaxÔ tou graf matoc thc f , tou �xona twn x, kai twn eujei¸n x = a,x = b.

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6.1. ORISM�OS 123

(bþ) Sth genik  perÐptwsh, o koinìc arijmìc ekfr�zei to proshmasmèno embadìnthc epif�neiac metaxÔ tou graf matoc thc f , tou �xona twn x, kai twn eu-jei¸n x = a, x = b, dhlad  to embadìn tou uposÔnolou aut c thc epif�neiacpou brÐsketai �nw tou �xona x meÐon to embadìn tou uposÔnolou aut c thcepif�neiac pou brÐsketai k�tw tou �xona x.

3. H deÔterh perÐptwsh eÐnai pajologik , all� uparkt , kai ekfr�zei to gegonìc ìtih sun�rthsh èqei tìso perÐergh morf , pou den mporoÔme na orÐsoume to embadìnthc antÐstoiqhc epif�neiac! Ja doÔme sÔntoma èna par�deigma.

Orismìc 6.3. (Olokl rwma kai oloklhrwsimìthta kat� Darboux) Mia sun�rthshf fragmènh sto [a, b] kaleÐtai oloklhr¸simh (kat� Darboux) an

supP{L(f, P )} = inf

P{U(f, P )}

Se aut  thn perÐptwsh, to koinì autì fr�gma kaleÐtai (orismèno) olokl rwma (Dar-boux) kai sumbolÐzetai wc ∫ b

a

f,

∫ b

a

f(x) dx.

Ta a, b kaloÔntai to k�tw ìrio olokl rwshc kai to �nw ìrio olokl rwshc antÐstoiqa.

Je¸rhma 6.2. (Enallaktikìc orismìc tou oloklhr¸matoc)Mia sun�rthsh f frag-mènh sto [a, b] eÐnai oloklhr¸simh me olokl rwma I ann to I eÐnai o monadikìc arijmìcme thn idiìthta

L(f, P ) ≤ I ≤ U(f, P )

gia ìlec tic diamerÐseic P .

Apìdeixh. EÐnai apl  kai paraleÐpetai.

Par�deigma 6.1. (Stajer  sun�rthsh) 'Estw f(x) = c, gia x ∈ [a, b]. Na deiqjeÐìti ∫ b

a

f = c(b− a).

Pr�gmati, èstw mia opoiad pote diamèrish P = {t0, t1, . . . , tn}. Ja èqoume:

L(f, P ) =n∑i=1

mi(ti − ti−1) = cn∑i=1

(ti − ti−1) = c(b− a).

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124 KEF�ALAIO 6. OLOKL�HRWMA

Parathr ste pwc to �jroisma aplopoieÐtai sto b−a qrhsimopoi¸ntac thn parat rhshìti eÐnai thleskopikì, dhlad  thc morf c

(a1 − a0) + (a2 − a1) + (a3 − a2) + · · ·+ (an − an−1) = an − a0.

Enallaktik�, parathr ste ìti to �jroisma ekfr�zei to �jroisma twn mhk¸n ìlwn twndiasthm�twn thc diamèrishc, pou eÐnai b− a.

AfoÔ ìla ta k�tw ajroÐsmata èqoun thn Ðdia tim , c(b− a), kai to supremum toucja èqei aut  thn tim , dhlad  supP{L(f, P )} = c(b− a).

Me an�logo trìpo, prokÔptei pwc infP{U(f, P )} = c(b− a).'Ara, afoÔ to �nw kai to k�tw olokl rwma tautÐzontai, h sun�rthsh eÐnai oloklh-

r¸simh me∫ ba f = c(b− a).

Par�deigma 6.2. (Sun�rthsh Dirichlet) 'Estw h akìloujh sun�rthsh f , orismènhsto [0, 1]:

f(x) =

{1, x ∈ Q ∩ [0, 1],

0, x ∈ (R−Q) ∩ [0, 1].

EÔkola mporoÔme na doÔme ìti h sun�rthsh den eÐnai poujen� suneq c. Ja deÐxoumekai ìti den eÐnai oloklhr¸simh.

Pr�gmati, èstw opoiad pote diamèrish

P = {0 = t0, t1, . . . , tn = 1}.

Epeid  se k�je upodi�sthma [ti−1, ti] thc diamèrishc up�rqei k�poioc rhtìc, to supre-mum se ekeÐno to upodi�sthma ja eÐnai 1. 'Ara,

∀P, U(f, P ) =n∑i=1

1× (ti − ti−1) = 1⇒ infP{U(f, P )} = 1.

H sunepagwg  prokÔptei giatÐ an eÐnai ìla ta �nw ajroÐsmata Ðsa me th mon�da, jaeÐnai kai to infimum touc. Parìmoia, se k�je upodi�sthma thc diamèrishc [ti−1, ti] thcdiamèrishc up�rqei k�poioc �rrhtoc, �ra to infimum se ekeÐno to upodi�sthma ja eÐnai0. 'Ara,

∀P, L(f, P ) =n∑i=1

0× (ti − ti−1) = 0⇒ supP{L(f, P )} = 0.

'Ara h sun�rthsh den eÐnai oloklhr¸simh, kai m�lista èqei �nw kai k�tw oloklhr¸mataDarboux

infP{U(f, P )} = 1, sup

P{L(f, P )} = 0.

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6.2. KRIT�HRIA OLOKLHRWSIM�OTHTAS 125

6.2 Krit ria Oloklhrwsimìthtac

Je¸rhma 6.3. (Ikanì kai anagkaÐo Krit rio Darboux) 'Estw f fragmènh sto[a, b]. H f eÐnai oloklhr¸simh ann gia k�je ε > 0 up�rqei diamèrish P tou [a, b] tètoia¸ste

U(f, P )− L(f, P ) < ε. (6.4)

Apìdeixh. (⇐) Ac upojèsoume kat' arq n ìti gia k�je ε up�rqei diamèrish tètoia ¸stena isqÔei h (6.4). Epeid  gia opoiad pote P isqÔei

infP ′{U(f, P ′)} ≤ U(f, P ), sup

P ′{L(f, P ′)} ≥ L(f, P ),

ja prèpei na èqoume kai

infP ′{U(f, P ′)} − sup

P ′{L(f, P ′)} < ε.

Epeid  to �nw isqÔei gia opoiod pote ε > 0, ja prèpei na èqoume

supP ′{L(f, P ′)} = inf

P ′{U(f, P ′)}, (6.5)

alli¸c prokÔptei eÔkola antÐfash. 'Ara, h f eÐnai oloklhr¸simh.(⇒) 'Estw t¸ra ìti h f eÐnai oloklhr¸simh, �ra isqÔei h (6.5). 'Estw ε > 0. Ex'

orismoÔ twn infimum kai supremum, ja up�rqoun diamerÐseic P1, P2 tètoiec ¸ste:

U(f, P1) < infP ′{U(f, P ′)}+

ε

2, L(f, P2) > sup

P ′{L(f, P ′)} − ε

2,

�raU(f, P1)− L(f, P2) < ε.

OrÐzw thn diamèrish P = P1 ∪ P2, gia thn opoÐa isqÔei, apì to Je¸rhma 6.1, ìti

U(f, P ) ≤ U(f, P1), L(f, P ) ≥ L(f, P2),

�ra, sundu�zontac tic treic anisìthtec, ja isqÔei kai h (6.4).

Par�deigma 6.3. 'Estw f orismènh sto [0, 2] me

f(x) =

{0, x 6= 1,

1, x = 1.

Ja deÐxoume ìti h f eÐnai oloklhr¸simh sto [0, 2], me olokl rwma 0.

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126 KEF�ALAIO 6. OLOKL�HRWMA

'Estw ε > 0. Kataskeu�zoume thn ex c diamèrish:

P ={t0 = 0, t1 = 1− ε

3, t2 = 1 +

ε

3, t3 = 2

}.

Gia th sugkekrimènh diamèrish, profan¸c L(f, P ) = 0 kai eÔkola mporoÔme na deÐxou-me, apì ton orismì, oti U(f, P ) = 2ε/3, �ra

U(f, P )− L(f, P ) < ε.

Epeid  to ε  tan aujaÐreto, apì to Je¸rhma 6.3 prokÔptei ìti h sun�rthsh eÐnaioloklhr¸simh. Epiplèon, parathr ste ìti gia k�je diamèrish P , ja èqoume L(f, P ) =0, �ra kai supP{L(f, P )} = 0, �ra ∫ 2

0

f = 0.

Je¸rhma 6.4. (Ikanì kai anagkaÐo krit rio oloklhrwsimìthtac) 'Estw f fragmè-nh sto [a, b]. H f eÐnai oloklhr¸simh me olokl rwma I ann up�rqei akoloujÐa apìdiamerÐseic {Pn} tètoia ¸ste

L(f, Pn)→ I, U(f, Pn)→ I. (6.6)

Apìdeixh. (⇐) 'Estw kat' arq n ìti up�rqei h akoloujÐa diamerÐsewn me thn dosmènhidiìthta (6.6). 'Estw tuqaÐo ε > 0. ProkÔptei ìti up�rqoun diamerÐseic Pn1 kai Pn2tètoiec ¸ste

|L(f, Pn1)− I| <ε

2, |U(f, Pn2)− I| <

ε

2.

OrÐzoume thn diamèrish R = Pn1 ∪ Pn2, gia thn opoÐa parathroÔme pwc isqÔei:

U(f,R)− L(f,R) ≤ U(f, Pn2)− L(f, Pn1) = U(f, Pn2)− I + I − L(f, Pn1)

≤ |U(f, Pn2)− I|+ |L(f, Pn1)− I| ≤ε

2+ε

2= ε.

H pr¸th anisìthta prokÔptei lìgw tou L mmatoc 6.1. Epeid  to ε  tan aujaÐreto,apì to Je¸rhma 6.3 prokÔptei pwc h f eÐnai oloklhr¸simh.

Gia na deÐxoume ìti to olokl rwma eÐnai to ìrio I, qrhsimopoioÔme eic �topon apa-gwg . 'Estw pwc to olokl rwma∫ b

a

f = infP{L(f, P} = sup

P{U(f, P},

eÐnai diaforetikì tou I. Ac poÔme pwc eÐnai megalÔtero, kai ac upojèsoume pwc∫ b

a

f = I + ε.

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6.2. KRIT�HRIA OLOKLHRWSIM�OTHTAS 127

'Omwc apì thn (6.6) èqoume ìti up�rqei diamèrish Pn0 tètoia ¸ste

|U(f, Pn0)− I| < ε⇒ U(f, Pn0) < I + ε =

∫ b

a

f = infPU(f, P ),

pou bèbaia eÐnai �topo, giatÐ br kame diamèrish me �nw �jroisma mikrìtero apì toinfimum twn �nw ajroism�twn. Parìmoia, mpor¸ na apokleÐsw to endeqìmeno toolokl rwma na eÐnai mikrìtero tou I, �ra telik� ja eÐnai Ðso.

(⇒) DeÐqnoume tèloc kai to eujÔ mèroc. 'Estw pwc h f eÐnai oloklhr¸simh, meolokl rwma I. 'Estw h akoloujÐa 1

n > 0. Apì to Je¸rhma 6.3, gia k�je ènan ìrothc akoloujÐac, ja up�rqei diamèrish, èstw Pn, tètoia ¸ste

U(f, Pn)− L(f, Pn) <1

n.

Epiplèon,U(f, Pn) ≥ I,

kai afair¸ntac, èqoume

L(f, Pn) > I − 1

n.

GnwrÐzoume ìmwc ìti L(f, Pn) ≤ I, �ra apì to krit rio thc parembol c, prokÔptei topr¸to ìrio. To deÔtero prokÔptei me trìpo an�logo.

Par�deigma 6.4. 'Estw f orismènh sto [0, b] me f(x) = x. Ja deÐxoume ìti h feÐnai oloklhr¸simh, kai ja upologÐsoume to olokl rwm� thc.

'Estw h diamèrish

Pn =

{t0 = 0, t1 =

b

n, t2 =

2b

n, . . . , ti =

ib

n, . . . , tn =

nb

n= b

}.

Profan¸c, sto di�sthma [ti−1, ti], to infimum ja isoÔtai me ti−1. 'Ara, to k�tw �jroi-sma isoÔtai me:

L(f, Pn) =n∑i=1

ti−1 (ti − ti−1) =n∑i=1

[(i− 1)b

n

]× b

n

=

[n∑i=1

(i− 1)

]× b2

n2=

(n− 1)n

2× b2

n2=n− 1

n× b2

2.

Sthn tètarth isìthta, qrhsimopoi same thn gnwst  sqèsh

n∑i=1

=n(n+ 1)

2.

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128 KEF�ALAIO 6. OLOKL�HRWMA

0 0.5 10

0.2

0.4

0.6

0.8

1

0 0.5 10

0.2

0.4

0.6

0.8

1

0 0.5 10

0.2

0.4

0.6

0.8

1

0 0.5 10

0.2

0.4

0.6

0.8

1

0 0.5 10

0.2

0.4

0.6

0.8

1

0 0.5 10

0.2

0.4

0.6

0.8

1

Sq ma 6.2: Ta k�tw kai �nw ajroÐsmata tou ParadeÐgmatoc 6.4 wc epif�neiec klimakwt¸n sunart -sewn.

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6.2. KRIT�HRIA OLOKLHRWSIM�OTHTAS 129

Me an�logo trìpo, èqoume:

U(f, Pn) =n∑i=1

ti (ti − ti−1) =n∑i=1

[ib

n

]× b

n

=

[n∑i=1

i

]× b2

n2=

(n+ 1)n

2× b2

n2=n+ 1

n× b2

2.

ParathroÔme pwc

U(f, Pn)→b2

2, L(f, Pn)→

b2

2,

�ra apì to krit rio tou Jewr matoc 6.4 prokÔptei ìti∫ b

a

f =b2

2.

Parathr ste ìti ta k�tw kai �nw ajroÐsmata isoÔntai me ta oloklhr¸mata klima-kwt¸n sunart sewn, me diark¸c auxanìmeno arijmì epipèdwn, oi opoÐec proseggÐsounoloèna kai me megalÔterh akrÐbeia thn f(x) apì k�tw kai apì �nw antÐstoiqa.

Par�deigma 6.5. 'Estw f orismènh sto [0, b] me f(x) = x2. Ja deÐxoume ìti h feÐnai oloklhr¸simh kai ja upologÐsoume to olokl rwm� thc.

Kai p�li, qrhsimopoioÔme thn diamèrish

Pn =

{t0 = 0, t1 =

b

n, t2 =

2b

n, . . . , ti =

ib

n, . . . , tn =

nb

n= b

}gia thn opoÐa èqoume

mi = f(ti−1) = t2i−1, Mi = f(ti) = t2i .

To k�tw �jroisma isoÔtai me

L(f, Pn) =n∑i=1

t2i−1(ti − ti−1)

=n∑i=1

(i− 1)2 b2

n2× b

n=b3

n3

n∑i=1

(i− 1)2 =b3n(n− 1)(2n− 1)

6n3.

Sthn teleutaÐa isìthta, qrhsimopoi same thn tautìthta

n∑i=1

i2 =n(n+ 1)(2n+ 1)

6. (6.7)

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130 KEF�ALAIO 6. OLOKL�HRWMA

To �nw �jroisma isoÔtai me

U(f, Pn) =n∑i=1

t2i (ti − ti−1) =n∑i=1

i2b2

n2× b

n=b3

n3

n∑i=1

i2 =b3n(n+ 1)(2n+ 1)

6n3.

Sthn teleutaÐa isìthta, qrhsimopoi same thn isìthta (6.7).Parathr ste ìti

L(f, Pn)→b3

3, U(f, Pn)→

b3

3,

�ra apì to krit rio tou Jewr matoc 6.4 prokÔptei pwc h f eÐnai oloklhr¸simh meolokl rwma ∫ b

a

f =b3

3.

Je¸rhma 6.5. (Sunèqeia ⇒ oloklhrwsimìthta) An h f eÐnai suneq c sto [a, b],tìte eÐnai oloklhr¸simh sto [a, b].

Apìdeixh. ParaleÐpetai.

Je¸rhma 6.6. (MonotonÐa⇒ oloklhrwsimìthta) 'Estw f fragmènh sto [a, b]. Anh f eÐnai monìtonh sto [a, b], tìte eÐnai kai oloklhr¸simh sto [a, b].

Apìdeixh. Ac upojèsoume ìti h f eÐnai aÔxousa. (H apìdeixh gia thn perÐptwsh poueÐnai fjÐnousa eÐnai entel¸c an�logh). H basik  idèa brÐsketai sto Sq ma 6.3. 'Estwdiamèrish

P =

{t0 = a, t1 = a+

b− an

, . . . , ti = a+i(b− a)

n, . . . , tn = a+

n(b− a)

n= b

}.

'Eqoume:

U(f, P )− L(f, P )

=n∑i=1

Mi(ti − ti−1)−n∑i=1

mi(ti − ti−1)

=n∑i=1

(Mi −mi)(ti − ti−1) =b− an

n∑i=1

(Mi −mi)

=b− an

[f(x1)− f(x0) + f(x2)− f(x1) + · · ·+ f(xn)− f(xn−1)]

=(b− a) [f(b)− f(a)]

n.

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6.2. KRIT�HRIA OLOKLHRWSIM�OTHTAS 131

y

x

f(x)

a b

f(a)

f(b)

ti=a+i(b-a)/n

Sq ma 6.3: Apìdeixh tou Jewr matoc 6.6.

Sthn tètarth isìthta qrhsimopoi same to gegonìc ìti h f eÐnai aÔxousa, �ra se k�jedi�sthma thc diamèrishc, to supremum lamb�netai sto dexÐ �kro, kai to infimum stoaristerì. Sthn pèmpth isìthta qrhsimopoi same to gegonìc ìti to �jroisma eÐnaithleskopikì.

B�sei thc �nw, prokÔptei ìti gia opoiod pote ε > 0, mpor¸ na epilèxw opoiod pote

n ≥ (b− a) [f(b)− f(a)]

ε,

kai gia thn antÐstoiqh diamèrish ja isqÔei

U(f, P )− L(f, P ) < ε.

Apì to Krit rio tou Darboux (Je¸rhma 6.3) prokÔptei ìti h sun�rthsh eÐnai oloklh-r¸simh.

H idèa thc apìdeixhc, kai to Sq ma 6.3, an koun ston NeÔtwna. Parathr ste ìti taskiasmèna orjog¸nia tou sq matoc èqoun ìla thn Ðdia b�sh, kai wc ek toÔtou mporoÔnna kalÔyoun akrib¸c èna orjog¸nio thc Ðdiac b�shc kai Ôyouc f(b)− f(a).

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132 KEF�ALAIO 6. OLOKL�HRWMA

6.3 Idiìthtec Oloklhr¸matoc

Je¸rhma 6.7. (Olokl rwma sthn ènwsh diasthm�twn) 'Estw a < c < b. An h feÐnai oloklhr¸simh sto [a, b], tìte eÐnai oloklhr¸simh kai sta [a, c], [c, b]. Antistrì-fwc, an h f eÐnai oloklhr¸simh sta [a, c], [c, b], tìte eÐnai oloklhr¸simh kai sto [a, b].EpÐshc, an h f eÐnai oloklhr¸simh sto [a, b], tìte∫ b

a

f =

∫ c

a

f +

∫ b

c

f. (6.8)

Apìdeixh. ParaleÐpetai.

Orismìc 6.4. (AujaÐreta ìria olokl rwshc)∫ a

a

f , 0,

∫ b

a

f , −∫ a

b

f, an a > b.

L mma 6.3. (Olokl rwma sthn {ènwsh} diasthm�twn) Gia k�je a, b, c ∈ R, isqÔei∫ b

a

f =

∫ c

a

f +

∫ b

c

f,

efìson up�rqoun ìla ta �nw oloklhr¸mata.

Apìdeixh. PaÐrnoume peript¸seic sqetik� me th merik  di�taxh twn a, b, c, kai efarmì-zoume to Je¸rhma 6.7. Gia par�deigma, èstw a < b < c. Tìte:∫ b

a

f +

∫ c

b

f =

∫ c

a

f ⇒∫ b

a

f =

∫ c

a

f −∫ c

b

f =

∫ c

a

f +

∫ b

c

f

⇒∫ b

a

f =

∫ c

a

f +

∫ b

c

f.

Je¸rhma 6.8. ('Ajroisma sunart sewn) An oi f , g eÐnai oloklhr¸simec sto [a, b],tìte h f + g eÐnai oloklhr¸simh sto [a, b] kai∫ b

a

(f + g) =

∫ b

a

f +

∫ b

a

g.

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6.3. IDI�OTHTES OLOKLHR�WMATOS 133

Apìdeixh. ParaleÐpetai.

Je¸rhma 6.9. (Olokl rwma thc cf) An h f eÐnai oloklhr¸simh sto [a, b], tìte giak�je arijmì c h sun�rthsh cf eÐnai oloklhr¸simh, kai m�lista∫ b

a

cf = c

∫ b

a

f.

Apìdeixh. ParaleÐpetai.

L mma 6.4. An oi f , g eÐnai oloklhr¸simec sto [a, b], kai c1, c2 ∈ R, tìte h c1f+c2geÐnai oloklhr¸simh sto [a, b] kai∫ b

a

(c1f + c2g) = c1

∫ b

a

f + c2

∫ b

a

g.

Apìdeixh. ProkÔptei �mesa apì ta Jewr mata 6.8 kai 6.9.

Je¸rhma 6.10. (Fr�gma sth sun�rthsh ⇒ fr�gma sto olokl rwma) 'Estw h foloklhr¸simh sto [a, b] kai m ≤ f(x) ≤M gia k�je x ∈ [a, b]. Tìte

m(b− a) ≤∫ b

a

f ≤M(b− a).

Apìdeixh. 'Estw mia opoiad pote diamèrish P . Ja èqoume:

L(f, P ) =n∑i=1

mi(ti − ti−1) ≥n∑i=1

m(ti − ti−1) = m(b− a),

apì to opoÐo prokÔptei ìti∫ b

a

f = supPL(f, P ) ≥ m(b− a).

OmoÐwc, èqoume:

U(f, P ) ≤M(b− a)⇒∫ b

a

f = infPU(f, P ) ≤M(b− a).

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134 KEF�ALAIO 6. OLOKL�HRWMA

Par�deigma 6.6. To Je¸rhma 6.10 mac epitrèpei na prosdiorÐzoume fr�gmata o-loklhrwm�twn pou den mporoÔme na upologÐsoume se kleist  morf . Gia par�deigma,ja deÐxoume pwc

π

9≤∫ π/2

π/6

dx

sinx+ sin2 x+ sin3 x≤ 8π

21.

Pr�gmati, parathroÔme pwc h sun�rthsh f(x) =[sinx+ sin2 x+ sin3 x

]−1eÐnai fjÐ-

nousa, �ra h mègisth kai h el�qisth tim  thc sto di�sthma[π6 ,

π2

]eÐnai oi

f(π

6

)=

1

sin(π6

)+ sin2

(π6

)+ sin3

(π6

) =8

7,

f(π

2

)=

1

sin(π2

)+ sin2

(π2

)+ sin3

(π6

) =1

3,

antÐstoiqa. Me efarmog  tou Jewr matoc 6.10 prokÔptei to zhtoÔmeno.

Je¸rhma 6.11. (Je¸rhma Mèshc Tim c gia oloklhr¸mata) An h f(x) suneq c sto[a, b], tìte, gia k�poio ξ ∈ [a, b],∫ b

a

f(x) dx = (b− a)f(ξ).

Apìdeixh. 'Estw m h el�qisth tim  thc sun�rthshc, kai M h mègisth tim  thc sun�r-thshc sto di�sthma [a, b]. (Ta m,M up�rqoun, epeid  h sun�rthsh eÐnai suneq c.)Apì to Je¸rhma 6.10, to olokl rwma an kei sto [m(b− a),M(b− a)], �ra ja isoÔ-tai me k(b − a) gia k�poio k ∈ [m,M ]. Epeid  h f eÐnai suneq c, apì to Je¸rhmaEndi�meshc Tim c (Je¸rhma 3.6) ja up�rqei ξ tètoio ¸ste f(ξ) = k, kai prokÔptei tozhtoÔmeno.

Par�deigma 6.7. 'Ena endiafèron apotèlesma tou �nw jewr matoc eÐnai ìti an mia

sun�rthsh f : [a, b] → R eÐnai suneq c sto [a, b] kai èqei∫ ba f = 0, tìte up�rqei

k�poio ξ ìpou f(ξ) = 0. Pr�gmati, apì to �nw je¸rhma èqoume ìti up�rqei k�poio ξtètoio ¸ste ∫ b

a

f(x) dx = (b− a)f(ξ)⇒ 0 = (b− a)f(ξ)⇒ f(ξ) = 0.

L mma 6.5. (Anisìthtec)

1. 'Estw f oloklhr¸simh sto [a, b] me f(x) ≥ 0 ∀x ∈ [a, b]. IsqÔei∫ b

a

f ≥ 0.

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6.3. IDI�OTHTES OLOKLHR�WMATOS 135

2. 'Estw f, g oloklhr¸simec sto [a, b] me f(x) ≥ g(x) ∀x ∈ [a, b]. IsqÔei∫ b

a

f ≥∫ b

a

g.

Apìdeixh. To pr¸to skèloc prokÔptei �mesa apì to Je¸rhma 6.10. To deÔtero skè-loc prokÔptei me qr sh tou pr¸tou skèlouc kai tou L mmatoc 6.4. (MporeÐte nasumplhr¸sete tic leptomèreiec thc apìdeixhc?)

L mma 6.6. (Olokl rwma thc |f |) An h f eÐnai oloklhr¸simh sto [a, b], tìte eÐnaikai h |f |, kai m�lista ∣∣∣∣∫ b

a

f

∣∣∣∣ ≤ ∫ b

a

|f |.

Apìdeixh. ParaleÐpetai.

Parat rhsh: An h |f | eÐnai oloklhr¸simh, eÐnai kai h f ?

Je¸rhma 6.12. (To olokl rwma eÐnai suneqèc) An h f eÐnai oloklhr¸simh sto[a, b], kai x0 ∈ [a, b], tìte h

F (x) =

∫ x

x0

f(t) dt (6.9)

eÐnai suneq c sto [a, b].

Apìdeixh. Katarq n parathroÔme ìti wc oloklhr¸simh h f eÐnai fragmènh, kai èstw

|f(x)| ≤ C ∀x ∈ [a, b].

'Estw dÔo shmeÐa x1, x2 ∈ [a, b]. ParathroÔme pwc

|F (x1)− F (x2)| =

∣∣∣∣∫ x1

x0

f −∫ x2

x0

f

∣∣∣∣ =

∣∣∣∣∫ x2

x1

f

∣∣∣∣ =

∣∣∣∣∣∫ max{x1,x2}

min{x1,x2}f

∣∣∣∣∣≤∫ max{x1,x2}

min{x1,x2}|f | ≤ C|x2 − x1|.

Sthn pr¸th anisìthta efarmìsame to L mma 6.6. Sthn deÔterh, to Je¸rhma 6.10.'Ara telik� h f eÐnai Lipschitz suneq c sto [a, b], kai epomènwc suneq c sto [a, b].

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136 KEF�ALAIO 6. OLOKL�HRWMA

0 0.5 10

0.2

0.4

0.6

0.8

1

0 0.5 10

0.2

0.4

0.6

0.8

1

0 0.5 1

−0.4

−0.2

0

0.2

0.4

0.6

0 0.5 10

0.2

0.4

0.6

0.8

Sq ma 6.4: Par�deigma 6.8.

Par�deigma 6.8. Poioi sunduasmoÐ twn sunart sewn twn Sqhm�twn 6.4, 6.5 ika-nopoioÔn th sqèsh (6.9) gia to di�sthma [a, b] = [0, 1]?

Je¸rhma 6.13. (Metatìpish sun�rthshc) 'Estw f(x) oloklhr¸simh sto [a, b].Tìte h f(x−c), dhlad  h f(x) metatopismènh proc ta dexi� kat� c, eÐnai oloklhr¸simhsto [a+ c, b+ c], kai m�lista∫ b

a

f(x) dx =

∫ b+c

a+c

f(x− c) dx.

Apìdeixh. ParaleÐpetai.

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6.3. IDI�OTHTES OLOKLHR�WMATOS 137

0 0.5 10

0.2

0.4

0.6

0.8

1

0 0.5 10

0.2

0.4

0.6

0.8

1

0 0.5 10

0.2

0.4

0.6

0.8

1

0 0.5 10

0.10.2

0 0.5 10

0.2

0.4

0.6

Sq ma 6.5: Par�deigma 6.8.

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138 KEF�ALAIO 6. OLOKL�HRWMA

−2 −1 0 1 2

0

0.5

1f(x)

−2 −1 0 1 2

0

0.5

1f(2x)

−2 −1 0 1 2

0

0.5

1f(− 1

2x)

Sq ma 6.6: Par�deigma 6.9.

Je¸rhma 6.14. (Ginìmeno sunart sewn) An oi f, g eÐnai oloklhr¸simec sto [a, b],tìte kai h fg eÐnai oloklhr¸simh.

Apìdeixh. ParaleÐpetai.

Je¸rhma 6.15. (SmÐkrunsh/epim kunsh sunart sewn) 'Estw f : [ca, cb] → Rme a, b, c ∈ R. To

∫ cbca f(t) dt up�rqei ann up�rqei to

∫ ba f(ct) dt, kai se aut  thn

perÐptwsh ∫ cb

ca

f(t) dt = c

∫ b

a

f(ct) dt.

Apìdeixh. ParaleÐpetai.

Par�deigma 6.9. 'Estw th sun�rthsh

f(x) =

{1− x, x ∈ [0, 1],

0, x 6∈ [0, 1],

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6.3. IDI�OTHTES OLOKLHR�WMATOS 139

gia thn opoÐa eÔkola prokÔptei pwc∫ 1

0 f(t) dt = 12 . Efarmìzontac to Je¸rhma 6.15

gia a = 0, b = 12 , c = 2, èqoume∫ 1

0

f(t) dt = 2

∫ 1/2

0

f(2t) dt,

en¸ efarmìzontac to Je¸rhma 6.15 gia a = 0, b = −12 , c = −1

2 , èqoume∫ 1

0

f(t) dt = −1

2

∫ −2

0

f

(− t

2

)dt.

Sto Sq ma 6.6 èqoun sqediasteÐ oi f(x), f(2x), kai f(−x

2

).

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140 KEF�ALAIO 6. OLOKL�HRWMA

6.4 Olokl rwma Riemann

Orismìc 6.5. (AjroÐsmata Riemann kai Olokl rwma Riemann)

1. 'Estw sun�rthsh f : [a, b]→ R. K�je �jroisma thc morf c

n∑i=1

f(xi)(ti − ti−1)

ìpou P = {t0, . . . , tn} diamèrish tou [a, b] kai gia ta {xi} isqÔei ìti ti−1 ≤ xi ≤ tikaleÐtai �jroisma Riemann.

2. OrÐzoume thn nìrma ‖P‖ thc diamèrishc P = {t0, t1, . . . , tn} wc:

‖P‖ = maxi=1,...,n

{|ti − ti−1|}.

3. Lème ìti h f eÐnai oloklhr¸simh kat� Riemann kai èqei olokl rwma RiemannI an gia k�je ε > 0 up�rqei δ > 0 tètoio ¸ste gia opoiad pote P me ‖P‖ < δ

kai gia opoiod pote �jroisma Riemann pou proèrqetai apì thn P isqÔei∣∣∣∣∣n∑i=1

f(xi)(ti − ti−1)− I∣∣∣∣∣ < ε.

Perilhptik�, h �nw sunj kh gr�fetai:

lim‖P‖→0

n∑i=1

f(xi)(ti − ti−1) = I.

Je¸rhma 6.16. (Je¸rhma Darboux: Olokl rwma Darboux = Olokl rwma Rie-mann) H f eÐnai oloklhr¸simh kat� Darboux ann eÐnai oloklhr¸simh kat� Riemann.'Otan up�rqoun, ta dÔo oloklhr¸mata isoÔntai.

Apìdeixh. ParaleÐpetai. BebaiwjeÐte p�ntwc ìti katalabaÐnetai giatÐ isqÔei.

Parathr seic

1. To ìrio lim‖P‖→0

pou emfanÐzetai ston orismì eÐnai polÔ diaforetikì apì ta sun jh

ìria!

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6.4. OLOKL�HRWMA RIEMANN 141

2. To Je¸rhma tou Darboux mac lèei ìti ìtan h f eÐnai oloklhr¸simh,

lim‖P‖→0

n∑i=1

f(xi)(ti − ti−1) =

∫ b

a

f.

'Ara, ìtan h diamèrish eÐnai pukn ,   isodÔnama to ‖P‖ mikrì, tìten∑i=1

f(xi)(ti − ti−1) '∫ b

a

f,

kai sto ìrio ‖P‖ → 0 èqoume isìthta. Aut  eÐnai Ðswc h pio diaisjhtik  ermhneÐatou oloklhr¸matoc.

L mma 6.7. 'Estw f : [a, b] → R oloklhr¸simh sto [a, b]. IsqÔoun ta akìloujaìria, kaj¸c n→∞:

n∑i=1

b− an

f

(a+

i(b− a)

n

)→

∫ b

a

f,

n∑i=1

b− an

f

(a+

(i− 1)(b− a)

n

)→

∫ b

a

f.

Apìdeixh. ParaleÐpetai.

Par�deigma 6.10. To �nw l mma eÐnai idiaÐtera qr simo ston upologismì orismè-nwn orÐwn pou èqoun thn morf  ajroism�twn, kai ta opoÐa polÔ dÔskola na upolo-gÐzontan me opoiond pote �llo trìpo. Gia par�deigma, ìpwc ja doÔme sto epìmenokef�laio, ∫ π

0

sinx dx = 2,

�ra, me qr sh tou �nw l mmatoc, kaj¸c to n→∞,

n∑i=1

π

nsin

(πi

n

)→ 2.

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142 KEF�ALAIO 6. OLOKL�HRWMA

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Kef�laio 7

Olokl rwsh

7.1 Jemeli¸dh Jewr mata LogismoÔ

Je¸rhma 7.1. (Pr¸to Jemeli¸dec Je¸rhma LogismoÔ) 'Estw f oloklhr¸simh sto[a, b] kai èstw h F : [a, b]→ R pou orÐzetai wc ex c:

F (x) =

∫ x

a

f.

An h f eÐnai suneq c sto c ∈ [a, b], tìte h F eÐnai paragwgÐsimh sto c kai

F ′(c) = f(c).

EnnoeÐtai ìti an c = a   c = b, h �nw sunèqeia eÐnai apl¸c pleurik , kai h �nwpar�gwgoc eÐnai epÐshc apl¸c pleurik  (antistoÐqwc dexi�   arister ).

Apìdeixh. Ja epikentrwjoÔme sthn perÐptwsh c ∈ (a, b). Oi peript¸seic c = a kaic = b antimetwpÐzontai parìmoia.

KaloÔmaste na deÐxoume ìti

limh→0

F (c+ h)− F (c)

h= f(c)⇔ lim

h→0

[F (c+ h)− F (c)

h− f(c)

]= 0

⇔ limh→0

[1

h

∫ c+h

c

f − f(c)

]= 0⇔ lim

h→0

1

h

[∫ c+h

c

[f(x)− f(c)] dx

]= 0

⇔ ∀ε > 0 ∃δ > 0 : 0 < |h| < δ ⇒∣∣∣∣1h∫ c+h

c

[f(x)− f(c)] dx

∣∣∣∣ < ε.

'Estw aujaÐreto ε > 0. 'Ara kai ε/2 > 0. Epeid  h f(x) eÐnai suneq c sto c, up�rqeiδ > 0 tètoio ¸ste

|x− c| < δ ⇒ |f(x)− f(c)| < ε/2.

143

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144 KEF�ALAIO 7. OLOKL�HRWSH

'Estw t¸ra k�poio h me 0 < h < δ. ParathroÔme pwc gia k�je x tètoio ¸stec ≤ x ≤ c+ h, èqoume ìti |x− c| < δ ⇒ |f(x)− f(c)| < ε/2, �ra∣∣∣∣1h

∫ c+h

c

(f(x)− f(c)) dx

∣∣∣∣ ≤ 1

h

∫ c+h

c

|f(x)− f(c)| dx ≤ 1

h× (ε/2)× h = ε/2

⇒∣∣∣∣1h∫ c+h

c

(f(x)− f(c)) dx

∣∣∣∣ < ε.

H pr¸th anisìthta proèkuye me qr sh tou L mmatoc 6.6, kai h deÔterh apì to Je¸-rhma 6.10. H Ðdia anisìthta mporeÐ na deiqjeÐ kai ìtan −δ < h < 0, me an�logo trìpo,kai h apìdeixh oloklhr¸jhke.

Parathr seic

1. An h f eÐnai pantoÔ suneq c sto [a, b], tìte F ′ = f sto [a, b], kai �ra h F eÐnaipar�gousa thc f sto [a, b]. 'Ara, ìlec oi suneqeÐc sunart seic èqoun par�gousa,kai èqoume kai trìpo na thn upologÐsoume. (Praktik� bèbaia to olokl rwmamporeÐ na mhn mporeÐ na upologisteÐ se kleist  morf .)

2. BebaiwjeÐte ìti èqete katal�bei giatÐ isqÔei to je¸rhma: an, gia par�deigma, hf(x) èqei meg�lh tim , eÐnai logikì to olokl rwma na aux�netai gr gora kaj¸cperilamb�nontai oi timèc gÔrw apì to x sthn olokl rwsh.

3. To je¸rhma apokalÔptei th sqèsh an�mesa sthn par�gwgo kai to olokl rwma:qondrik� mil¸ntac, {h par�gwgoc tou oloklhr¸matoc miac sun�rthshc eÐnai hÐdia h sun�rthsh}.

4. SÔmfwna me to je¸rhma, an h f(x) eÐnai oloklhr¸simh sto [a, b] kai suneq c stox ∈ [a, b], tìte (∫ x

a

f

)′= f(x).

To epìmeno l mma genikeÔei to �nw.

L mma 7.1. (Pr¸to Jemeli¸dec Je¸rhma LogismoÔ - praktik  morf ) An h f eÐnaioloklhr¸simh sto [a, b], kai c, x ∈ [a, b], tìte:(∫ x

c

f

)′= f(x), (7.1)(∫ c

x

f

)′= −f(x), (7.2)

ìpou oi par�gwgoi eÐnai wc proc x.

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7.1. JEMELI�WDH JEWR�HMATA LOGISMO�U 145

Apìdeixh. (Parathr ste ìti to c mporeÐ na eÐnai megalÔtero tou x.) H (7.1) isqÔeigiatÐ(∫ x

c

f

)′=

(∫ a

c

f +

∫ x

a

f

)′=

(∫ a

c

f

)′+

(∫ x

a

f

)′= 0 + f(x) = f(x).

Sqetik� me thn (7.2), parathroÔme apl¸c pwc(∫ c

x

f

)′= −

(∫ x

c

f

)′= −f(x),

ìpou sthn teleutaÐa isìthta qrhsimopoi same thn (7.1).

Par�deigma 7.1. Ja upologÐsoume tic parag¸gouc wc proc x twn akìloujwn:∫ x

a

cos3 t dt,

∫ sinx

a

cos3 t dt,

∫ a

x

t+ cos t

1 + t2dt,

∫ a

x

x+ cosx

1 + t2dt.

Se ìla ta �nw, h par�metroc a ∈ R. 'Eqoume, kat� perÐptwsh:

1. H sun�rthsh cos3 t eÐnai suneq c kai �ra oloklhr¸simh se k�je kleistì di�sthma,�ra apì to Pr¸to Jemeli¸dec Je¸rhma tou LogismoÔ, ja èqoume:(∫ x

a

cos3 t dt

)′= cos3 x.

2. Parathr ste ìti h sun�rthsh eÐnai h sÔnjesh F (g(x)) thc sun�rthshc F (x) =∫ xa cos3 t dt me th sun�rthsh g(x) = sinx. Parathr ste pwc h F (x) èqei oristeÐswst�, afoÔ h cos3 t eÐnai oloklhr¸simh, wc suneq c. Epiplèon, h F (x) eÐnaiparagwgÐsimh, afoÔ h cos3 t eÐnai suneq c. 'Ara, mporoÔme na efarmìsoume tonkanìna thc alusÐdac:(∫ sinx

a

cos3 t dt

)′= (F (g(x)))′ = F ′(g(x))g′(x) = (cos3 sinx)× cosx,

ìpou h F ′ proèkuye apì to Pr¸to Jemeli¸dec Je¸rhma tou LogismoÔ.

3. H oloklhrwtèa sun�rthsh eÐnai oloklhr¸simh kai suneq c pantoÔ (eÔkola prokÔ-ptei ìti o paronomast c den mhdenÐzetai gia k�poio t). Apì to Pr¸to Jemeli¸decJe¸rhma tou LogismoÔ sth morf  tou L mmatoc 7.1, prokÔptei:(∫ a

x

t+ cos t

1 + t2dt

)′= −x+ cosx

1 + x2.

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146 KEF�ALAIO 7. OLOKL�HRWSH

4. Parathr ste pwc entìc tou oloklhr¸matoc, h èkfrash (x+cos x) eÐnai stajer�.(∫ a

x

x+ cosx

1 + t2dt

)′=

((x+ cosx)

∫ a

x

dt

1 + t2

)′= −x+ cosx

1 + x2+ (1− sinx)

∫ a

x

dt

1 + t2.

Je¸rhma 7.2. (DeÔtero Jemeli¸dec Je¸rhma LogismoÔ)An h f eÐnai oloklhr¸simh sto [a, b] kai f = g′ pantoÔ sto [a, b] gia k�poia g, tìte:∫ b

a

f = g(b)− g(a).

Apìdeixh. 'Estw P = {t0, . . . , tn} opoiad pote diamèrish tou [a, b]. Apì to Je¸rhmaMèshc Tim c (Je¸rhma 4.7) ja èqoume gia thn g ìti

g(ti)− g(ti−1) = g′(xi)(ti − ti−1) = f(xi)(ti − ti−1),

gia k�poio xi ∈ (ti−1, ti). 'Omwc, an orÐsoume, kat� ta gnwst�,

mi , inf{f(x) : ti−1 ≤ x ≤ ti}, Mi , sup{f(x) : ti−1 ≤ x ≤ ti},ja èqoume:

mi ≤ f(xi) ≤Mi ⇔ mi(ti − ti−1) ≤ f(xi)(ti − ti−1) ≤Mi(ti − ti−1)

⇔ mi(ti − ti−1) ≤ g(ti)− g(ti−1) ≤Mi(ti − ti−1),

kai an prosjèsoume kat� mèlh tic �nw gia i = 1, 2, . . . , n, telik�, epeid  to mesaÐo�jroisma eÐnai thleskopikì, ja èqoume:

n∑i=1

mi(ti − ti−1) ≤ g(b)− g(a) ≤n∑i=1

Mi(ti − ti−1),

dhlad L(f, P ) ≤ g(b)− g(a) ≤ U(f, P ).

H �nw isqÔei gia opoiad pote diamèrish P . Kat� ta gnwst� apì th jewrÐa (Je¸-rhma 6.2), o mìnoc arijmìc o opoÐoc fr�ssetai �nw kai k�tw me autì ton trìpo giaaujaÐretec diamerÐseic eÐnai to olokl rwma thc f , dhlad ∫ b

a

f = g(b)− g(a).

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7.1. JEMELI�WDH JEWR�HMATA LOGISMO�U 147

Parathr seic

1. To DeÔtero Jemeli¸dhc Je¸rhma oloklhr¸nei th sqèsh metaxÔ parag¸gou kaioloklhr¸matoc pou xekÐnhse to Pr¸to. Qondrik�, mac lèei ìti {to olokl rwmathc parag¸gou miac sun�rthshc eÐnai h Ðdia h sun�rthsh}.

2. An h f eÐnai suneq c (kai ìqi apl¸c oloklhr¸simh), to DeÔtero Jemeli¸decJe¸rhma prokÔptei �mesa apì to Pr¸to (mporeÐte na to deÐxete?).

3. To DeÔtero Jemeli¸dhc Je¸rhma èqei meg�lh praktik  shmasÐa apì mìno tou,giatÐ mac epitrèpei na upologÐzoume to olokl rwma miac sun�rthshc, arkeÐ nagnwrÐzoume mia antipar�gwgo.

Par�deigma 7.2. Epeid , gia n 6= −1,[xn+1

n+ 1

]′= xn,

prokÔptei pwc: ∫ b

a

xn =bn+1

n+ 1− an+1

n+ 1.

An n ≥ 0, tìte a, b ∈ R, alli¸c,an n < 0, prèpei ta a, b na eÐnai kai ta dÔo arnhtik�  kai ta dÔo jetik�, ¸ste na mhn mhdenÐzetai o paronomast c sto 0 kai na ikanopoioÔntaioi proôpojèseic tou jewr matoc.

Epiplèon, èqoume ta akìlouja:

(sinx)′ = cosx ⇒∫ b

a

cosx dx = sin b− sin a,

(cosx)′ = − sinx ⇒∫ b

a

sinx dx = cos a− cos b,

(tanx)′ =1

cos2 x⇒

∫ b

a

1

cos2 xdx = tan b− tan a,

(cotx)′ = − 1

sin2 x⇒

∫ b

a

1

sin2 xx dx = cot a− cot b.

Poioi eÐnai oi periorismoÐ stic timèc twn a, b stic �nw exis¸seic?

Parat rhsh: Epiplèon, to DeÔtero Jemeli¸dec Je¸rhma eÐnai qr simo kai èmme-sa, giatÐ odhgeÐ �mesa se dÔo teqnikèc me meg�lh praktik  efarmog  ston upologismìoloklhrwm�twn: thn olokl rwsh kat� par�gontec, kai thn olokl rwsh me antikat�-stash. Kai oi dÔo mèjodoi eÐnai an�logec twn antÐstoiqwn mejìdwn me efarmog  staaìrista oloklhr¸mata pou eÐdame sthn Par�grafo 4.5.

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148 KEF�ALAIO 7. OLOKL�HRWSH

Je¸rhma 7.3. (Olokl rwsh kat� par�gontec) 'Estw sunart seic f, g : [a, b]→ R,me suneqeÐc parag gouc f ′, g′ sto [a, b]. Tìte∫ b

a

f(x)g′(x) dx = f(x)g(x)|ba −∫ b

a

f ′(x)g(x) dx.

Apìdeixh. Katarq n parathr ste ìti ta �nw oloklhr¸mata up�rqoun, giatÐ oi fg′ kaif ′g eÐnai suneqeÐc, �ra kai oloklhr¸simec.

Gia ton Ðdio lìgo ja up�rqoun kai oi:

F1(x) =

∫ x

a

f(t)g′(t) dt, F2(x) =

∫ x

a

f ′(t)g(t) dt,

gia tic opoÐec parathroÔme pwc

F1(x) + F2(x) =

∫ x

a

[f ′(t)g(t) + f(t)g′(t)] dt

=

∫ x

a

[f(t)g(t)]′ dt = f(x)g(x)− f(a)g(a).

Efarmìzontac thn �nw gia x = b, prokÔptei to zhtoÔmeno.

Par�deigma 7.3.∫ π/2

0

x2 cosx dx =

∫ π/2

0

x2 (sinx)′ dx = x2 sinx∣∣π/20−∫ π/2

0

2x sinx dx

=π2

4− 0 +

∫ π/2

0

2x (cosx)′ dx

=π2

4+ 2x cosx|π/20 − 2

∫ π/2

0

cosx dx

=π2

4+ 0− 0− 2

∫ π/2

0

(sinx)′ dx =π2

4− 2 sinx|π/20 =

π2

4− 2.

Par�deigma 7.4.∫ 1

0

x arctanx dx =

∫ 1

0

(x2 + 1

2

)′arctanx dx

=x2 + 1

2arctanx

∣∣∣∣10

−∫ 1

0

x2 + 1

2(arctanx)′ dx

= arctan 1− 1

2arctan 0−

∫ 1

0

x2 + 1

2(x2 + 1)dx =

π

4− 1

2.

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7.1. JEMELI�WDH JEWR�HMATA LOGISMO�U 149

Je¸rhma 7.4. (Olokl rwsh me antikat�stash) 'Estw sun�rthsh g : [a, b] → Rme suneq  par�gwgo g′, kai èstw f : J → R suneq c, me g([a, b]) ⊆ J . Tìte:∫ g(b)

g(a)

f(u) du =

∫ b

a

f(g(x))g′(x) dx. (7.3)

Apìdeixh. Kat' arq�c parathr ste ìti kai ta dÔo oloklhr¸mata up�rqoun, b�sei twnupojèsewn pou k�name.

H f , wc suneq c, ja èqei par�gousa, èstw F . Dhlad  F ′ = f sto J . 'Ara, apìto DeÔtero Jemeli¸dec Je¸rhma,∫ g(b)

g(a)

f(u) du = F |g(b)g(a) = F (g(b))− F (g(a)).

Parathr ste ìmwc ìti (F (g(x)))′ = F ′(g(x))g′(x) = f(g(x))g′(x). 'Ara, h F (g(x))eÐnai par�gousa thc f(g(x))g′(x), �ra, p�li apì to DeÔtero Jemeli¸dec Je¸rhma,∫ b

a

f(g(x))g′(x) dx = F (g(x))|ba = F (g(b))− F (g(a)).

Sundu�zontac tic dÔo exis¸seic, prokÔptei to apotèlesma.

Parathr seic

1. Stic perissìterec peript¸seic olokl rwshc me antikat�stash, prospajoÔme nametatrèyoume èna arqikì olokl rwma thc morf c tou dexioÔ mèlouc thc (7.3) (wcproc x) se èna aploÔstero, thc morf c tou aristeroÔ mèlouc (wc proc u). HdiadikasÐa eÐnai h ex c:

(aþ) Epilègoume/manteÔoume mia sun�rthsh u = g(x) pou ikanopoieÐ tic proôpo-jèseic tou jewr matoc.

(bþ) All�zoume ta ìria olokl rwshc, a→ g(a), kai b→ g(b).

(gþ) AntikajistoÔme to g′(x)dx me to du (  to dx me to du/g′(x), an to g′(x) denemfanÐzetai autoÔsio).

(dþ) Eis�goume to u sthn oloklhrwtèa sun�rthsh (pou t¸ra endeqomènwc peri-lamb�nei to g′(x) pou eis�game), endeqomènwc me kat�llhlouc algebrikoÔcmetasqhmatismoÔc.

(eþ) An ìla p�ne kal�, to olokl rwm� mac èqei èrjei sth morf ∫ g(b)

g(a) f(u) du thnopoÐa ja eÐmaste se jèsh na oloklhr¸soume.

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150 KEF�ALAIO 7. OLOKL�HRWSH

2. Se arketèc peript¸seic, h diadikasÐa ekteleÐtai an�poda: prospajoÔme na me-tatrèyoume èna arqikì olokl rwma thc morf c tou aristeroÔ mèlouc thc (7.3)(wc proc u) se èna aploÔstero, thc morf c tou dexioÔ mèlouc (wc proc x). HdiadikasÐa eÐnai an�logh thc prohgoÔmenhc perÐptwshc.

Par�deigma 7.5. 'Estw to olokl rwma:∫ 1

0

x+ 1

(x2 + 2x+ 6)2dx.

Parathr ste ìti o arijmht c eÐnai (sqedìn) h par�gwgoc thc posìthtac pou èqeiuywjeÐ sto tetr�gwno ston paronomast . Jètoume g(x) = u = x2 + 2x + 6. 'Arax = 0⇒ u = 6 kai x = 1⇒ u = 9. Epiplèon, dudx = 2x+ 2. 'Ara:∫ 1

0

x+ 1

(x2 + 2x+ 6)2dx =

∫ 9

6

du

2u2=

1

2u

∣∣∣∣69

=1

12− 1

18=

1

36.

Par�deigma 7.6. 'Estw to olokl rwma∫ π/3

0

sinx

cos2 xdx.

'Estw u = cosx. Tìte dudx = − sinx, kai x = 0⇒ u = 1, x = π

3 ⇒ u = 12 . 'Ara:∫ π/3

0

sinx

cos2 xdx = −

∫ 1/2

1

du

u2=

∫ 1/2

1

(1

u

)′du =

1

u

∣∣∣∣1/21

= 1.

Par�deigma 7.7. 'Estw to olokl rwma∫ 1

0

√1− x2 dx.

Jètoume x = sin t, me to t ∈ (0, π/2). 'Ara, dx = cos tdt, kai epiplèon x = 0⇒ t = 0,en¸ x = 1⇒ t = π/2. Sunep¸c:∫ 1

0

√1− x2 dx =

∫ π/2

0

√1− sin2 t cos t dt =

∫ π/2

0

cos2 t dt =

∫ π/2

0

1 + cos 2t

2dt

=

∫ π/2

0

1

2dt+

∫ π/2

0

cos 2t

2dt =

π

4+

∫ π/2

0

(sin 2t)′

4dt =

π

4+

sin 2t

4

∣∣∣∣π/20

4.

Sth deÔterh isìthta qrhsimopoi same to gegonìc ìti metaxÔ twn orÐwn olokl rwshcèqoume cos t ≥ 0. To apotèlesma anamenìtan, kaj¸c to olokl rwma ekfr�zei toembadìn tetartokuklÐou aktÐnac 1 kai kèntro to shmeÐo (0, 0).

Page 161: ΣΗΜΕΙΩΣΕΙΣ-ΜΑΘΗΜΑΤΙΚΑ 1 ΠΛΗΡΟΦΟΡΙΚΗ ΑΣΟΕΕ 1 ΕΞΑΜΗΝΟ

7.2. LOGARIJMIK�H KAI EKJETIK�H SUN�ARTHSH 151

7.2 Logarijmik  kai Ekjetik  Sun�rthsh

Orismìc 7.1. (Logarijmik  sun�rthsh)An x > 0, orÐzw ton (fusikì   nepèrio) log�rijmo tou x wc

log x ,∫ x

1

1

tdt.

Je¸rhma 7.5. (Basikèc idiìthtec tou logarÐjmou)

1. O log�rijmoc eÐnai suneq c sun�rthsh sto (0,+∞) me pr¸th kai deÔterh par�-gwgo

log′ x =1

x, log′′ x = − 1

x2.

2. O log�rijmoc eÐnai gnhsÐwc aÔxousa kai koÐlh sun�rthsh sto (0,+∞).

3. An x < 1, tìte log x < 0, an x > 1, tìte log x > 0, kai log 1 = 0.

4. An x, y > 0,

log xy = log x+ log y, (7.4)

logx

y= log x− log y. (7.5)

5. An x > 0,log x−1 = − log x. (7.6)

6. An x > 0, n ∈ Z,log xn = n log x. (7.7)

7.limx→0+

log x = −∞, limx→+∞

log x = +∞. (7.8)

Apìdeixh. 1. O log�rijmoc eÐnai suneq c sun�rthsh wc olokl rwma. H pr¸th pa-r�gwgoc prokÔptei apì to Pr¸to Jemeli¸dec Je¸rhma tou LogismoÔ, en¸ hexÐswsh gia th deÔterh par�gwgo eÐnai profan c.

2. Profan c, exet�zontac thn pr¸th kai th deÔterh par�gwgo.

3. Profan c, exet�zontac to olokl rwma tou orismoÔ.

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152 KEF�ALAIO 7. OLOKL�HRWSH

4. Parathr ste pwc:

log xy =

∫ xy

1

1

tdt =

∫ x

1

1

tdt+

∫ xy

x

1

tdt = log x+

∫ xy

x

1

tdt,

kai to teleutaÐo olokl rwma eÔkola prokÔptei pwc eÐnai Ðso me log y, me allag metablht c u = t/x.

H (7.5) prokÔptei parathr¸ntac pwc:

log x = log

(x

yy

)= log

x

y+ log y.

5. ProkÔptei �mesa me efarmog  thc (7.5) gia x = 1.

6. ProkÔptei �mesa me qr sh epagwg c kai efarmog  twn (7.4), (7.6).

7. Parathr ste ìti, me efarmog  thc (7.7), èqoume:

log 2n = n log 2, log

(1

2

)n= n log

1

2= −n log 2.

Epeid  log 2 > 0 eÐnai eÔkolo na prokÔyoun ta �nw ìria me efarmog  twn orism¸ntou k�je orÐou. (MporeÐte na sumplhr¸sete tic leptomèreiec?)

Parathr seic

1. Oi �nw idiìthtec tou logarÐjmou antikatoptrÐzontai ìlec sthn grafik  tou pa-r�stash, pou emfanÐzetai sto Sq ma 7.1.

2. AfoÔ o log�rijmoc eÐnai gnhsÐwc aÔxousa, suneq c sun�rthsh, ja èqei antÐstro-fh, h opoÐa ja orÐzetai gia k�je x ∈ R, afoÔ o log�rijmoc paÐrnei ìlec tic timècsto R, kai ja paÐrnei jetikèc timèc, afoÔ o log�rijmoc orÐzetai mìno gia jetikèctimèc.

Orismìc 7.2. (Ekjetik  sun�rthsh)

1. H ekjetik  sun�rthsh expx orÐzetai wc h antÐstrofh thc log x:

y = expx⇔ x = log y.

2. OrÐzoume th stajer� e wc

e , exp 1⇔ 1 = log e =

∫ e

1

1

tdt.

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7.2. LOGARIJMIK�H KAI EKJETIK�H SUN�ARTHSH 153

0 2 4 6 8−5

−4

−3

−2

−1

0

1

2

3

x

ylogx

−4 −2 0 2 4−1

0

1

2

3

4

5

6

7

x

yexp x

Sq ma 7.1: H logarijmik  sun�rthsh kai h ekjetik  sun�rthsh.

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154 KEF�ALAIO 7. OLOKL�HRWSH

Je¸rhma 7.6. (Basikèc idiìthtec thc ekjetik c sun�rthshc)

1.

∀x ∈ R, log expx = x,

∀x ∈ (0,+∞), exp log x = x.

2. H ekjetik  sun�rthsh eÐnai jetik , suneq c, me par�gwgo

exp′ x = expx,

kai genik¸c ìlec tic parag gouc an¸terhc t�xhc Ðsec me expx.

3. H ekjetik  sun�rthsh eÐnai gnhsÐwc aÔxousa kai kurt .

4. An x < 0, tìte expx < 1, an x > 0, tìte expx > 1, kai exp 0 = 1.

5. An x, y ∈ R,exp(x+ y) = exp(x) exp(y). (7.9)

6.exp(−x) = (exp x)−1 . (7.10)

7.limx→+∞

expx = +∞, limx→−∞

expx = 0. (7.11)

8. An to x ∈ Q, tìteexpx = ex. (7.12)

9. An y ∈ R, x ∈ Q, tìteexp(xy) = (exp y)x, (7.13)

en¸ an a > 0, x ∈ Q,ax = exp(x log a).

Apìdeixh. 1. ProkÔptei apì to ìti h ekjetik  eÐnai h antÐstrofh thc logarijmik c.

2. H ekjetik  sun�rthsh eÐnai jetik  giatÐ to pedÐo orismoÔ thc antÐstrof c thceÐnai to (0,+∞). EpÐshc, eÐnai suneq c wc antÐstrofh suneqoÔc. Epiplèon, kat�ta gnwst� apì th jewrÐa, an mia sun�rthsh y = f(x) me par�gwgo f ′(x) èqeiantÐstrofh sun�rthsh x = f−1(y), me par�gwgo (f−1)′(y), tìte

(f−1)′(y) =1

f ′(f−1(y)).

Page 165: ΣΗΜΕΙΩΣΕΙΣ-ΜΑΘΗΜΑΤΙΚΑ 1 ΠΛΗΡΟΦΟΡΙΚΗ ΑΣΟΕΕ 1 ΕΞΑΜΗΝΟ

7.2. LOGARIJMIK�H KAI EKJETIK�H SUN�ARTHSH 155

An efarmìsoume thn �nw gia thn perÐptwsh

y = f(x) = log x⇔ x = f−1(y) = exp y,

kai lamb�nontac upìyin ìti f ′(x) = log′ x = 1/x, èqoume

(exp y)′ = (f−1)′(y) =1

f ′(f−1(y))=

1

1/ exp y= exp y,

�ra èqoume (all�zontac to x me to y), (expx)′ = expx.

3. ProkÔptei �mesa apì th morf  thc parag¸gou.

4. ProkÔptei �mesa apì to ìti h expx eÐnai gnhsÐwc aÔxousa, kai to ìti log 1 =0⇔ exp 0 = 1.

5. Parathr ste pwc

exp(x+ y) = exp x exp y ⇔ log exp(x+ y) = log (expx exp y)

⇔ log exp(x+ y) = log expx+ log exp y ⇔ x+ y = x+ y.

Sthn pr¸th isodunamÐa qrhsimopoi same to ìti h logarijmik  sun�rthsh eÐnai ènaproc èna. Sthn deÔterh, qrhsimopoi same thn (7.4), en¸ sthn trÐth to gegonìc ìtih ekjetik  sun�rthsh eÐnai antÐstrofh thc logarijmik c. AfoÔ isqÔei h teleutaÐaisìthta, lìgw twn isodunami¸n ja isqÔei kai h pr¸th.

6. ProkÔptei efarmìzontac to prohgoÔmeno skèloc gia y = −x.

7. Gia na apodeÐxoume to pr¸to ìrio, prèpei na deÐxoume ìti gia k�je M , up�rqeik�poio x0 tètoio ¸ste

x > x0 ⇒ expx > M.

'Estw èna opoiod pote M ∈ R. An M ≤ 0, profan¸c h �nw isqÔei gia opoiad -pote epilog  tou x0. 'Estw loipìn M > 0. Tìte up�rqei to logM , kai jètoumex0 = logM . ParathroÔme pwc

x > x0 ⇒ expx > exp logM ⇒ expx > M,

kai h apìdeixh oloklhr¸jhke. (Sthn pr¸th sunepagwg , qrhsimopoi same togegonìc ìti h exp eÐnai aÔxousa.)

Gia na apodeÐxoume to deÔtero ìrio, prèpei na deÐxoume ìti gia k�je ε > 0 up�rqeix0 tètoio ¸ste

x < x0 ⇒ | expx| < ε.

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156 KEF�ALAIO 7. OLOKL�HRWSH

'Estw loipìn ε > 0, kai èstw x0 = log ε. Tìte

x < x0 ⇒ expx < exp log ε⇒ | expx| < ε,

kai h apìdeixh oloklhr¸jhke. Sthn pr¸th sunepagwg  qrhsimopoi same to ge-gonìc ìti h exp eÐnai gnhsÐwc aÔxousa, en¸ sth deÔterh ìti eÐnai p�nta jetik .

8. Katarq n, qrhsimopoi¸ntac thn (7.9), prokÔptei ìti

exp 2 = exp(1 + 1) = exp 1 exp 1 = (exp 1)2 = e2,

kai genik� me epagwg  èqoume, gia n ∈ N:

exp(n) = en.

Epiplèon apì thn (7.10) prokÔptei pwc

exp(−n) = (expn)−1 = (en)−1 = e−n.

(Sthn teleutaÐa isìthta qrhsimopoi same gnwst  idiìthta gia akèraiouc ekjètec.)'Ara, mèqri t¸ra apodeÐxame thn (7.12) gia x akèraio. Epiplèon ìmwc:

exp(1) = exp

(1

n+ · · ·+ 1

n

)︸ ︷︷ ︸

n ìroi

=

[exp

1

n

]n⇒ exp

1

n= (exp(1))

1n = e

1n .

(H sunepagwg  prokÔptei lamb�nontac th n-ost  rÐza.) Sth sunèqeia,

exp(mn

)= exp

(1

n+ · · ·+ 1

n

)︸ ︷︷ ︸

m ìroi

= exp

(1

n

). . . exp

(1

n

)

=

[exp

1

n

]m=(e

1n

)m= e

mn ,

�ra apodeÐxame thn (7.12) gia jetikoÔc rhtoÔc. Tèloc, èqoume:

exp(−mn

)=[exp

(mn

)]−1

=(e

mn

)−1= e−

mn ,

kai ètsi apodeÐxame thn (7.12) gia ìlouc touc rhtoÔc. (Parathr ste ìti h teleu-taÐa isìthta prokÔptei apì gnwst  idiìthta twn rht¸n dun�mewn.)

9. H (7.13) prokÔptei polÔ parìmoia me to prohgoÔmeno skèloc, en¸ gia th deÔterhapl� èqoume:

ax = [exp log a]x = exp(x log a).

H deÔterh isìthta prokÔptei lìgw thc (7.13).

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7.2. LOGARIJMIK�H KAI EKJETIK�H SUN�ARTHSH 157

Parathr seic

1. ApodeÐxame ìti ìtan x ∈ Q,expx = ex.

To dexÐ skèloc èqei oristeÐ wc t¸ra mìno gia x ∈ Q. To aristerì ìmwc, orÐzetaigia x ∈ R.

2. OmoÐwc, apodeÐxame ìti ìtan x ∈ Q,

exp(x log a) = ax,

ìpou a > 0, a ∈ R. OmoÐwc, en¸ to aristerì skèloc orÐzetai gia x ∈ R, to dexÐèqei oristeÐ wc t¸ra gia x ∈ Q.

Oi �nw parathr seic odhgoÔn stouc akìloujouc orismoÔc:

Orismìc 7.3. (Pragmatikèc dun�meic)

1. OrÐzoume, gia k�je x ∈ R,ex , expx.

2. An a > 0, orÐzoume, gia k�je x ∈ R,

ax , exp(x log a).

Parathr seic

1. Apì dw kai pèra, to ex kai to expx ja shmaÐnoun to Ðdio pr�gma, kai ja taqrhsimopoioÔme kai ta dÔo, an�loga thn perÐstash.

2. Sto Sq ma 7.2 èqoume sqedi�sei tic sunart seic 0.5x, 1x, 1.5x, ex.

Je¸rhma 7.7. (Idiìthtec pragmatik¸n dun�mewn) 'Estw a ∈ R, a > 0.

1. H ax èqei pr¸th par�gwgo(ax)′ = (log a)ax,

kai genik¸c gia thn n-ost  par�gwgo isqÔei:

(ax)(n) = (log a)nax.

2. H ax eÐnai jetik  kai kurt .

3. An a > 1, h ax eÐnai gnhsÐwc aÔxousa, limx→+∞

ax = +∞, kai limx→−∞

ax = 0.

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158 KEF�ALAIO 7. OLOKL�HRWSH

4. An 0 < a < 1, h ax eÐnai gnhsÐwc fjÐnousa, limx→+∞

ax = 0, kai limx→−∞

ax = +∞.

5. An a = 1, ax = 1.

6.a1 = a.

7. Gia k�je x, y ∈ R,(ax)y = axy.

8. Gia k�je x, y ∈ R,axay = ax+y.

9. Gia k�je x ∈ R,axbx = (ab)x.

10. An a 6= 1, a > 0, tìte h antÐstrofh thc y = ax eÐnai h log ylog a , dhlad :

y = ax ⇔ x =log y

log a.

Apìdeixh. 1. Apì ton orismì thc ax kai ton kanìna thc alusÐdac èqoume:

(ax)′ =(ex log a

)′= (log a)ex log a = (log a)ax.

Oi par�gwgoi an¸terhc t�xewc prokÔptoun me parìmoio trìpo.

2. -6. ApodeiknÔontai eÔkola, lamb�nontac upìyh ton orismì thc ax kai tic idiìthtecthc ekjetik c sun�rthshc.

7. 'Eqoume:

(ax)y = ey log(ax) (Orismìc ax)

= ey log(ex log a) (Orismìc ax)

= eyx log a (exp, log antÐstrofec)

= axy. (Orismìc ax)

8. ParomoÐwc:

axay = ex log aey log b (Orismìc ax)

= ex log a+y log a (Idiìthta (7.9))

= e(x+y) log a

= ax+y. (Orismìc ax)

Page 169: ΣΗΜΕΙΩΣΕΙΣ-ΜΑΘΗΜΑΤΙΚΑ 1 ΠΛΗΡΟΦΟΡΙΚΗ ΑΣΟΕΕ 1 ΕΞΑΜΗΝΟ

7.2. LOGARIJMIK�H KAI EKJETIK�H SUN�ARTHSH 159

9. ParomoÐwc:

axbx = ex log aex log b (Orismìc ax)

= ex log a+x log b (Idiìthta (7.9))

= ex log ab (Idiìthta (7.4))

= (ab)x. (Orismìc ax)

10. Parathr ste kat' arq n ìti an a > 1 h ax eÐnai suneq c kai gnhsÐwc aÔxousa, en¸an 0 < a < 1 h ax eÐnai suneq c kai gnhsÐwc fjÐnousa. 'Ara se k�je perÐptwsh,h antÐstrofh up�rqei. Parathr ste pwc:

y = ax ⇔ log y = log ax ⇔ log y = x log a⇔ x =log y

log a.

Orismìc 7.4. (Logarijmik  sun�rthsh me b�sh 6= e) An a > 0, a 6= 1, orÐzw tonlog�rijmo me b�sh a wc

loga y =log y

log a, y ∈ (0,+∞),

dhlad  thn antÐstrofh thc y = ax.

Parat rhsh: Sto Sq ma 7.2 èqoume tic sunart seic log1.5 x, log x, log0.5.

Par�deigma 7.8. (H sun�rthsh f(x)h(x)) 'Estw paragwgÐsimh f(x) : I → R mef(x) > 0 kai paragwgÐsimh g(x) : I → R. Ja deÐxoume ìti h f(x)g(x) eÐnai paragw-gÐsimh pantoÔ sto I, kai ja upologÐsoume thn par�gwgì thc. Katarq n parathroÔmeìti, ex' orismoÔ,

f(x)g(x) = exp(g(x) log f(x)).

Epeid  h f(x) eÐnai paragwgÐsimh, ja eÐnai kai h sÔnjesh log f(x), �ra kai toginìmeno g(x) log f(x), �ra kai h sÔnjesh exp(g(x) log f(x)). 'Ara h f(x)g(x) eÐnaiparagwgÐsimh. Gia na upologÐsoume thn par�gwgo, arkeÐ na qrhsimopoi soume tonkanìna thc alusÐdac:[

f(x)g(x)]′

= [exp(g(x) log f(x))]′

= exp(g(x) log f(x)) [g′(x) log f(x) + g(x)(log f(x))′]

= f(x)g(x)

[g′(x) log f(x) + g(x)

f ′(x)

f(x)

].

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160 KEF�ALAIO 7. OLOKL�HRWSH

−4 −2 0 2 4−1

0

1

2

3

4

5

6

7

x

y

0.5x

1x

1.5x

ex

0 2 4 6 8−5

−4

−3

−2

−1

0

1

2

3

x

y

log0.5 x

log1.5 x

logx

Sq ma 7.2: Ekjetikèc kai logarijmikèc sunart seic gia di�forec b�seic.

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7.2. LOGARIJMIK�H KAI EKJETIK�H SUN�ARTHSH 161

Par�deigma 7.9. (Idiìthtec thc xa) Sto prohgoÔmeno par�deigma, sthn eidik  pe-rÐptwsh pou I = (0,+∞), g(x) = x, h(x) = a, èqoume:

(xa)′ = (exp(a log x))′ = exp(a log x)

(a

1

x

)= xa

a

x= axa−1,

dhlad  isqÔei o gnwstìc tÔpoc

(xa)′ = axa−1, x > 0, a 6= 0,

akìma kai an to a eÐnai �rrhtoc! AfoÔ h xa eÐnai paragwgÐsimh, eÐnai kai suneq c.Epiplèon, me deÔterh parag¸gish, èqoume pwc

(xa)′′ = a(a− 1)xa−2, x > 0, a 6= 0, 1.

Sunep¸c, h xa eÐnai aÔxousa kai kurt  gia a > 1, aÔxousa kai koÐlh gia 0 < a < 1,kai fjÐnousa kai kurt  gia a < 0.

Tèloc, èqoume ta akìlouja ìria, pou dÐnontai qwrÐc apìdeixh:

limx→∞

xa = ∞, a > 0,

limx→∞

xa = 0, a < 0,

limx→0+

xa = 0, a > 0,

limx→0+

xa = ∞, a < 0.

(MporeÐte na gr�yete tic apodeÐxeic?)

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162 KEF�ALAIO 7. OLOKL�HRWSH

7.3 Ask seic me thn Logarijmik  kai thn Ekjetik  Su-

n�rthsh

Par�deigma 7.10. (Sun jh ìria me thn logarijmik  kai thn ekjetik  sun�rthsh)Sto par�deigma autì ja upologÐsoume merik� ìria pou emfanÐzoun thn ekjetik  kai thlogarijmik  sun�rthsh kai prokÔptoun suqn� se probl mata kai efarmogèc.

1. limx→+∞

ex/xn =∞, ìpou n ∈ N.

2. limx→+∞

log xxa = 0, ìpou a ∈ R, a > 0.

3. limx→+∞

(log x)n

xa = 0, ìpou a ∈ R, a > 0, n ∈ N.

4. limx→0+

x(log x)n = 0, ìpou n ∈ N.

5. limx→0+

xx = 1.

6. limx→0

log(1+ax)x = a, ìpou a ∈ R.

7. limx→+∞

x log(1 + ax) = a, ìpou a ∈ R.

8. limx→+∞

(1 + ax)x = ea, ìpou a ∈ R.

'Eqoume, kat� perÐptwsh:

1. Arijmht c kai paronomast c teÐnoun sto �peiro, sunep¸c èqoume aprosdioristÐa,mporoÔme ìmwc na qrhsimopoi soume ton Kanìna tou L’Hopital:

limx→+∞

ex

xn= lim

x→+∞ex

nxn−1= · · · = lim

x→+∞ex

n!= +∞.

Parathr ste pwc efarmìsame ton kanìna n forèc mèqri na afairejeÐ h aprosdio-ristÐa. 'Ara, h ekjetik  sun�rthsh aux�netai taqÔtera apì opoiad pote dÔnamhtou x.

2. Parathr ste ìti kai p�li èqoume aprosdioristÐa, all� me mia mìno qr sh toukanìna tou L’Hopital èqoume:

limx→+∞

log x

xa= lim

x→+∞1/x

axa−1= lim

x→+∞1

axa,

pou teÐnei sto 0 (DeÐte to Par�deigma 7.9). Epeid  to a mporeÐ na eÐnai osod potemikrì (all� bèbaia jetikì), prokÔptei pwc o log�rijmoc aux�nei men, all� polÔarg�.

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7.3. ASK�HSEIS ME THN LOGARIJMIK�H KAI THN EKJETIK�H SUN�ARTHSH 163

3. Gia n = 1, prokÔptei to prohgoÔmeno skèloc. An n ≥ 2, me efarmog  pr¸ta toukanìna L’Hopital kai met� tou kanìna thc alusÐdac, sunolik� n forèc, èqoume:

limx→+∞

(log x)n

xa= lim

x→+∞n(log x)n−1 1

x

axa−1

= limx→+∞

n(n− 1)(log x)n−2

a2xa= · · · = lim

x→+∞n!

an1

xa= 0.

'Ara, o log�rijmoc aux�nei akìma pio arg� apì ìti mac deÐqnei to prohgoÔmenoskèloc.

4. Ja qrhsimopoi soume thn idiìthta

limx→0+

f(x) = limt→∞

f(1/t), (7.14)

h opoÐa eÔkola prokÔptei b�sei twn orism¸n twn antÐstoiqwn orÐwn. H �nwshmaÐnei ìti an up�rqei to èna ìrio, ja up�rqei kai to �llo, kai ja eÐnai Ðsa.'Eqoume loipìn,

limx→0+

x(log x)n = limt→+∞

1

t

(log

1

t

)n= lim

t→+∞(−1)n

(log t)n

t= 0.

H teleutaÐa exÐswsh prokÔptei me qr sh tou prohgoÔmenou skèlouc.

5. Ed¸ èqoume thn aprosdioristÐa 00. Parathr ste pwc

limx→0+

xx = limx→0+

exp(x log x) = exp( limx→0+

x log x) = 1.

H deÔterh isìthta prokÔptei epeid  h ex eÐnai suneq c. H teleutaÐa isìthtaprokÔptei efarmìzontac to prohgoÔmeno skèloc gia n = 1.

6. ParathroÔme pwc èqoume aprosdioristÐa 0/0, kai mporoÔme na qrhsimopoi soumeton kanìna L’Hopital:

limx→0

log(1 + ax)

x= lim

x→0

a/(1 + ax)

1= a.

7. 'Eqoume aprosdioristÐa 0×∞, kai qrhsimopoioÔme ton kanìna tou L’Hopital wcex c:

limx→+∞

x log(

1 +a

x

)= lim

x→+∞log(1 + a

x

)1x

= limt→0+

log (1 + at)

t= a.

Sthn deÔterh isìthta efarmìsame thn idiìthta (7.14), en¸ sthn trÐth isìthta toprohgoÔmeno skèloc.

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164 KEF�ALAIO 7. OLOKL�HRWSH

8. Parathr ste pwc èqoume thn k�pwc asun jisth aprosdioristÐa 1+∞, thn opoÐaìmwc mporoÔme na antimetwpÐsoume wc ex c:

limx→+∞

(1 +

a

x

)x= lim

x→+∞exp

[x log

(1 +

a

x

)]= exp

[limx→+∞

x log(

1 +a

x

)]= exp a.

Sthn deÔterh isìthta qrhsimopoi same thn sunèqeia thc ex, en¸ sthn teleutaÐaqrhsimopoi same to prohgoÔmeno skèloc. Orismènoi suggrafeÐc orÐzoun to emèsw autoÔ tou orÐou!

Par�deigma 7.11. (Aìrista oloklhr¸mata me thn logarijmik  kai thn ekjetik sun�rthsh) Ja upologÐsoume ta akìlouja:

1.∫xex dx.

2.∫

log x dx, x > 0.

3.∫

log xx2 dx, x > 0.

'Eqoume, kat� perÐptwsh:

1. ∫xex dx =

∫x(ex)′ dx = xex −

∫(x)′ex dx

= xex −∫ex dx = xex − ex + C.

2. ∫log x dx =

∫(x)′ log x dx = x log x−

∫x(log x)′ dx

= x log x−∫x

1

xdx = x log x−

∫dx = x log x− x+ C.

3. ∫log x

x2=

∫(−x−1)′ log x dx = − log x

x+

∫x−1x−1 dx

= − log x

x+

∫x−2 dx = − log x

x− x−1 + C = − log x+ 1

x+ C, C ∈ R.

Par�deigma 7.12. (Orismèna oloklhr¸mata) Ja upologÐsoume ta akìlouja:

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7.3. ASK�HSEIS ME THN LOGARIJMIK�H KAI THN EKJETIK�H SUN�ARTHSH 165

1.∫ e

1 x2 log x dx.

2.∫ e3e log2 x dx.

3.∫ a

0 xe−2x dx.

4.∫ ab

x1+x2 dx.

'Eqoume, kat� perÐptwsh,

1. Ja qrhsimopoi soume paragontik  olokl rwsh:∫ e

1

x2 log x dx =

∫ e

1

(x3

3

)′log x dx =

[x3

3log x

]e1

−∫ e

1

x3

3(log x)′ dx

=e3

3−∫ e

1

x3

3× 1

xdx =

e3

3−[x3

9

]e1

=2e3 + 1

9.

2. Ja qrhsimopoi soume paragontik  olokl rwsh:∫ e3

e

log2 x dx =

∫ e3

e

(x)′ log2 x dx = x log2 x∣∣e3e−∫ e3

e

2x log x1

xdx

= e3 log2 e3 − e log2 e−∫ e3

e

2 log x dx = 9e3 − e− 2

∫ e3

e

log x dx.

Me mia akìma paragontik  olokl rwsh èqoume:∫ e3

e

log x dx =

∫ e3

e

(x)′ log x dx = x log x|e3e −∫ e3

e

dx = 3e3−e−(e3−e) = 2e3,

kai sundu�zontac ta �nw prokÔptei telik�:∫ e3

e

(log x)2 dx = 5e3 − e.

3. Ja qrhsimopoi soume paragontik  olokl rwsh:∫ a

0

xe−2x dx = −1

2

∫ a

0

x(e−2x

)′dx = − xe−2x

2

∣∣∣∣a0

+1

2

∫ a

0

e−2x dx =

− ae−2a

2− 1

4

∫ a

0

(e−2x

)′dx = −ae

−2a

2− e−2a

4+

1

4.

4. ∫ a

b

x

1 + x2dx =

∫ a

b

1

2

[log(1 + x2)

]′dx

=1

2log(1 + a2)− 1

2log(1 + b2) =

1

2log

1 + a2

1 + b2.

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166 KEF�ALAIO 7. OLOKL�HRWSH

7.4 Kataqrhstik� Oloklhr¸mata

Parat rhsh: 'Estw pwc kat� m koc tou jetikoÔ hmi�xona {t : 0 ≤ t ≤ ∞} kata-skeu�zoume toÐqo metablhtoÔ Ôyouc 1/t2. Pìsh mpogi� qreiazìmaste gia na b�youmeto tm ma tou toÐqou (1,∞)? Pìsh mpogi� qreiazìmaste gia na b�youme to tm ma toutoÐqou (0, 1)? Probl mata autoÔ tou tÔpou (orismèna ek twn opoÐwn eÐnai pio peistik�apì to sugkekrimèno) mac odhgoÔn ston akìloujo orismì.

Orismìc 7.5. (Kataqrhstik� oloklhr¸mata pr¸tou tÔpou)OrÐzoume wc kataqrhstik� oloklhr¸mata pr¸tou tÔpou ta akìlouja ìria:

1. ∫ +∞

a

f , limx→∞

∫ x

a

f,

efìson up�rqoun to olokl rwma (gia k�je x > a) kai to ìrio.

2. ∫ b

−∞f , lim

x→−∞

∫ b

x

f,

efìson up�rqoun to olokl rwma (gia k�je x < b) kai to ìrio.

3. ∫ +∞

−∞f ,

∫ c

−∞f +

∫ +∞

c

f,

gia k�poio c ∈ R, efìson up�rqoun ta kataqrhstik� oloklhr¸mata sto dexÐ skè-loc. (H epilog  tou c den èqei shmasÐa: an up�rqei to kataqrhstikì olokl rwmagia k�poio c, ja up�rqei gia ìla ta c, kai h tim  tou ja eÐnai h Ðdia gia ìla ta c.)

Par�deigma 7.13. TrÐa apl� paradeÐgmata eÐnai ta akìlouja:

1. ∫ +∞

1

1

t2dt = lim

x→∞

∫ x

1

1

t2dt = lim

x→∞

∫ x

1

(−1

t

)′dt = lim

x→∞

(1− 1

x

)= 1.

2. ∫ +∞

1

1√tdt = lim

x→∞

∫ x

1

1√tdt = lim

x→∞

∫ x

1

(2√t)′dt = 2lim

x→∞

(√x− 1

)=∞.

3. ∫ ∞0

sin t dt = limx→∞

∫ x

0

sin t dt = limx→∞

∫ x

0

(− cos t)′ dt = limx→∞

(1− cosx),

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7.4. KATAQRHSTIK�A OLOKLHR�WMATA 167

pou den up�rqei. 'Ara to antÐstoiqo kataqrhstikì olokl rwma den up�rqei.

Parathr ste ìti en¸ kai h 1/t2 kai h 1/√t oloklhr¸jhkan kat� m koc apeÐrwn dia-

sthm�twn, h pr¸th fjÐnei arkoÔntwc gr gora ¸ste to olokl rwm� thc na eÐnai pepe-rasmèno, en¸ h �llh tìso arg�, ¸ste to olokl rwm� thc na eÐnai �peiro.

Orismìc 7.6. (Kataqrhstik� oloklhr¸mata deÔterou tÔpou)OrÐzoume wc kataqrhstik� oloklhr¸mata deÔterou tÔpou ta akìlouja ìria:

1. Efìson h f den eÐnai oloklhr¸simh sto [a, b],∫ b

a+f , lim

x→a+

∫ b

x

f,

efìson up�rqei to olokl rwma entìc tou orÐou gia k�je x ∈ (a, b], kai efìsonup�rqei kai to ìrio.

2. Efìson h f den eÐnai oloklhr¸simh sto [a, b],∫ b−

a

f , limx→b−

∫ x

a

f,

efìson up�rqei to olokl rwma entìc tou orÐou gia k�je x ∈ [a, b), kai efìsonup�rqei kai to ìrio.

Par�deigma 7.14. DÔo apl� paradeÐgmata eÐnai ta akìlouja:

1. ∫ 1

0+

1

t2dt = lim

x→0+

∫ 1

x

1

t2dt = lim

x→0+

∫ 1

x

(−1

t

)′dt = lim

x→0+

[1

x− 1

]=∞.

2. ∫ 1

0+

1√tdt = lim

x→0+

∫ 1

x

1√tdt = lim

x→0+

∫ 1

x

(2√t)′dt = lim

x→0+

[2− 2

√x]

= 2.

'Ara, parìti kai h 1/t2 kai h 1/√t apeirÐzontai sto 0, to olokl rwma thc pr¸thc eÐnai

�peiro, en¸ to olokl rwma thc deÔterhc eÐnai peperasmèno!

Parathr seic

1. Se orismènec peript¸seic kataqrhstik¸n oloklhrwm�twn deÔterou tÔpou gr�-

foume∫ ba f antÐ gia

∫ ba+  

∫ b−a , me to ± na uponoeÐtai apì to gegonìc ìti h f den

eÐnai oloklhr¸simh sto [a, b].

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168 KEF�ALAIO 7. OLOKL�HRWSH

2. H basik  idèa eÐnai saf c: an to èna �kro eÐnai {problhmatikì}, upologÐzoumepr¸ta to olokl rwma gia ìrio olokl rwshc x, kai met� paÐrnoume kat�llhloìrio wc proc x.

(aþ) Sta kataqrhstik� oloklhr¸mata pr¸tou tÔpou to problhmatikì �kro eÐnaito ±∞.

(bþ) Sta kataqrhstik� oloklhr¸mata deÔterou tÔpou sto problhmatikì �kro holoklhrwtèa sun�rthsh eÐnai pajologik , kai praktik� stic perissìterecpeript¸seic pou ja doÔme teÐnei sto ±∞.

3. Oi �nw orismoÐ mporoÔn na sunduastoÔn metaxÔ touc, me ton profan  trìpo. Giapar�deigma ∫ ∞

a+f = lim

x→a+

∫ c

x

f + limx→∞

∫ x

c

f,

ìpou c > a, kai h epilog  tou c den èqei shmasÐa. (Dhlad , efìson ta ìriaup�rqoun gia èna c > a ja up�rqoun gia ìla ta c > a, kai h tim  tou ajroÐsmatocden all�zei.)

4. Sthn perÐptwsh pou prèpei na upologÐsoume �jroisma kataqrhstik¸n oloklh-rwm�twn, mporeÐ na prokÔyei aprosdioristÐa thc morf c ∞ − ∞, kai tìte to�jroisma den orÐzetai.

Par�deigma 7.15. 'Estw pwc kaloÔmaste na upologÐsoume to olokl rwma∫ ∞0

t−s dt

ìpou s ∈ R, s > 0.Se aut  thn perÐptwsh, eÐnai problhmatik� kai ta dÔo �kra, kai èqoume∫ ∞

0

t−s dt =

∫ 1

0

t−s dt+

∫ ∞1

t−s dt, (7.15)

ìpou to mesaÐo ìrio (dhlad  to 1) epilèqjhke aujaÐreta. 'Estw katarq n s 6= 1, opìteèqoume: ∫ b

a

t−s dt =

∫ b

a

[t−s+1

−s+ 1

]′dt =

1

1− s[b1−s − a1−s] .

'Ara gia to pr¸to olokl rwma èqoume∫ 1

0

t−s dt = limx→0+

∫ 1

x

t−s dt = limx→0+

1

1− s[11−s − x1−s]

=1

1− s

[1− lim

x→0+x1−s

]=

{1

1−s , s < 1,

∞, s > 1.

Page 179: ΣΗΜΕΙΩΣΕΙΣ-ΜΑΘΗΜΑΤΙΚΑ 1 ΠΛΗΡΟΦΟΡΙΚΗ ΑΣΟΕΕ 1 ΕΞΑΜΗΝΟ

7.4. KATAQRHSTIK�A OLOKLHR�WMATA 169

Gia to deÔtero olokl rwma èqoume:∫ ∞1

t−s dt = limx→∞

∫ x

1

t−s dt = limx→∞

1

1− s[x1−s − 11−s] =

{∞, s < 1,

1s−1 , s > 1.

Gia thn perÐptwsh s = 1 èqoume∫ ∞0

dt

t=

∫ 1

0

dt

t+

∫ ∞1

dt

t= lim

x→0+

∫ 1

x

dt

t+ lim

x→∞

∫ x

1

dt

t

= limx→0+

∫ 1

x

(log t)′ dt+ limx→∞

∫ x

1

(log t)′ dt = limx→0+

log t

∣∣∣∣1x

+ limx→∞

log t|x1= 0− lim

x→0+log x+ lim

x→∞log x− 0 =∞+∞ =∞.

'Ara telik� to �jroisma twn dÔo kataqrhstik¸n oloklhrwm�twn thc (7.15) ja eÐnai�peiro, eÐte lìgw apeirismoÔ tou pr¸tou ìrou tou, gia s > 1, eÐte lìgw apeirismoÔtou deÔterou ìrou tou, gia s < 1, eÐte lìgw apeirismoÔ kai twn dÔo ìrwn, gia s = 1.

Parat rhsh: Mia teqnik  pou qrhsimopoieÐtai suqn� se probl mata me kataqrh-stik� oloklhr¸mata eÐnai na qeirizìmaste to problhmatikì �kro thc olokl rwshc sanna  tan èna sunhjismèno �kro, kai thn olokl rwsh san mia sunhjismènh olokl rwsh,kai na emfanÐzoume to ìrio afoÔ upologÐsoume to olokl rwma. H mèjodoc mporeÐ naodhg sei se sÔgqush, kai ja prèpei na eÐmaste polÔ prosektikoÐ. DeÐte to akìloujopar�deigma.

Par�deigma 7.16. Ja upologÐsoume merik� apì ta kataqrhstik� oloklhr¸matapou emfanÐzontai sto Par�deigma 7.15 me thn �nw sunoptik  diadikasÐa.

Gia ton deÔtero ìro thc (7.15), an s 6= 1, èqoume∫ ∞1

t−s dt =

∫ ∞1

(t1−s

1− s

)′dt =

t1−s

1− s

∣∣∣∣∞1

= limt→∞

t1−s

1− s −11−s

1− s =

{∞, s < 1,

1s−1 , s > 1.

Gia ton pr¸to ìro thc (7.15), èqoume, ìtan s > 1,∫ 1

0

t−s dt =

∫ 1

0

(t1−s

1− s

)′dt =

t1−s

1− s

∣∣∣∣10

=11−s

1− s − limt→0+

t1−s

1− s =∞,

en¸ ìtan s < 1,∫ 1

0

t−s dt =

∫ 1

0

(t1−s

1− s

)′dt =

t1−s

1− s

∣∣∣∣10

=11−s

1− s −01−s

1− s =1

1− s.

Parathr ste ìti sthn teleutaÐa perÐptwsh, epeid  h sun�rthsh pou prokÔptei apì thnolokl rwsh eÐnai suneq c sto problhmatikì ìrio olokl rwshc, telik� den qrei�sthkena upologÐsoume kanèna ìrio!

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170 KEF�ALAIO 7. OLOKL�HRWSH

Par�deigma 7.17. (Sun jh kataqrhstik� oloklhr¸mata me thn logarijmik  kaithn ekjetik  sun�rthsh) Sto par�deigma autì ja upologÐsoume merik� kataqrhstik�oloklhr¸mata pou emfanÐzoun thn ekjetik  kai th logarijmik  sun�rthsh.∫ ∞−∞

e−ax dx,

∫ ∞0

xe−2x dx,

∫ ∞−∞

e−a|x| dx,

∫ ∞10

x

1 + x2dx,

∫ ∞1+

1

x log xdx.

'Eqoume, kat� perÐptwsh:

1. ∫ ∞−∞

e−ax dx =

∫ 0

−∞e−ax dx+

∫ ∞0

e−ax dx

= limb→−∞

∫ 0

b

e−ax dx+ limb→∞

∫ b

0

e−ax dx = limb→−∞

[e−ab − 1

a

]+ lim

b→∞

[1− e−ab

a

].

Parathr ste ìti qrhsimopoi same to ìti (−e−ax/a)′ = e−ax kai ìti upojèsameìti a 6= 0. An a > 0, to pr¸to ìrio eÐnai ∞ kai to deÔtero peperasmèno. Ana < 0, tìte to deÔtero ìrio eÐnai ∞ kai to pr¸to peperasmèno. Sthn perÐptwsha = 0, eÔkola prokÔptei ìti to olokl rwma eÐnai ∞. 'Ara, se k�je perÐptwshto olokl rwma eÐnai∞. To apotèlesma mporeÐ na exhghjeÐ apl� an sqedi�soumethn e−ax, kai parathr soume ìti gia opoiad pote tim  thc a, h epif�neia k�tw apìto gr�fhm� thc prèpei na eÐnai ∞.

2. Sto Par�deigma 7.12 br kame ìti∫ a

0

xe−2x dx = −ae−2a

2− e−2a

4+

1

4.

Sunep¸c,∫ ∞0

xe−2x dx = lima→∞

∫ a

0

xe−2x dx = lima→∞

[−ae

−2a

2− e−2a

4+

1

4

]=

1

4.

To ìrio lima→∞

ae−2a = 0 upologÐzetai me ton Kanìna tou L’Hopital.

3. ∫ ∞−∞

e−a|x| dx

=

∫ 0

−∞e−a|x| dx+

∫ ∞0

e−a|x| dx = limb→−∞

∫ 0

b

e−a|x| dx+ limb→∞

∫ b

0

e−a|x| dx

= limb→−∞

∫ 0

b

eax dx+ limb→∞

∫ b

0

e−ax dx = 2limb→∞

∫ b

0

e−ax dx

= −2

alimb→∞

∫ b

0

[e−ax

]′dx = −2

alimb→∞

[e−ab − 1

]=

{2/a, a > 0,

∞, a < 0.

Page 181: ΣΗΜΕΙΩΣΕΙΣ-ΜΑΘΗΜΑΤΙΚΑ 1 ΠΛΗΡΟΦΟΡΙΚΗ ΑΣΟΕΕ 1 ΕΞΑΜΗΝΟ

7.4. KATAQRHSTIK�A OLOKLHR�WMATA 171

H tètarth isìthta prokÔptei me thn allag  metablht c sto pr¸to olokl rwmax → −x, kai thn epakìloujh tropopoÐhsh tou orÐou apì b → −∞ se b → ∞.Parathr ste ìti sta �nw upojèsame a 6= 0. An a = 0, eÔkola prokÔptei ìti toolokl rwma eÐnai ∞.

4. Sto Par�deigma 7.12 br kame ìti∫ a

b

x

1 + x2dx =

1

2log

1 + a2

1 + b2.

AkoloÔjwc, èqoume∫ ∞10

x

1 + x2dx = lim

a→∞

∫ a

10

x

1 + x2dx = lim

a→∞

(1

2log

1 + a2

101

)=∞,

kai telik� to kataqrhstikì olokl rwma eÐnai �peiro.

5. Katarq n parathroÔme ìti to olokl rwma eÐnai kataqrhstikì giatÐ èqei èna ìrioÐso me∞. Epiplèon, parathroÔme ìti gia x→ 1, h oloklhrwtèa sun�rthsh apei-rÐzetai. 'Ara, èqoume èna kataqrhstikì olokl rwma meiktoÔ pr¸tou kai deÔteroutÔpou. MporoÔme na upologÐzoume to olokl rwma eÔkola, an parathr soume ìti

(log(log x))′ =1

log x(log x)′ =

1

x log x.

'Ara èqoume:∫ ∞1+

1

x log xdx =

∫ K

1+

1

x log xdx+

∫ ∞K

1

x log xdx

= limm→1+

∫ K

m

1

x log xdx+ lim

M→∞

∫ M

K

1

x log xdx

= limm→1+

∫ K

m

(log log x)′ dx+ limM→∞

∫ M

K

(log log x)′ dx

= limm→1+

log log x|Km + limM→∞

log log x|MK= lim

m→1+[log logK − log logm] + lim

M→∞[log logM − log logK]

= log logK − (−∞) +∞− log logK =∞.Sta �nw, to K eÐnai mia aujaÐreth stajer� megalÔterh thc mon�dac, pou ìpwcanamènetai den emfanÐzetai sto telikì apotèlesma.

Par�deigma 7.18. Ja upologÐsoume to

I =

∫ 1

−1

|x|√1− x2

dx.

Page 182: ΣΗΜΕΙΩΣΕΙΣ-ΜΑΘΗΜΑΤΙΚΑ 1 ΠΛΗΡΟΦΟΡΙΚΗ ΑΣΟΕΕ 1 ΕΞΑΜΗΝΟ

172 KEF�ALAIO 7. OLOKL�HRWSH

Katarq n parathr ste pwc to olokl rwma eÐnai kataqrhstikì olokl rwma lìgw twnapeirism¸n thc oloklhrwtèac sun�rthshc kai sta dÔo �kra. 'Eqoume:

I =

∫ 1

−1

|x|√1− x2

dx = lima→−1+

∫ 0

a

|x|√1− x2

dx+ lima→1−

∫ a

0

|x|√1− x2

dx

= lima→−1+

∫ 0

a

−x√1− x2

dx+ lima→1−

∫ a

0

x√1− x2

dx

= lima→1−

∫ 0

−a

−x√1− x2

dx+ lima→1−

∫ a

0

x√1− x2

dx

= lima→1−

∫ a

0

x√1− x2

dx+ lima→1−

∫ a

0

x√1− x2

dx = 2 lima→1−

∫ a

0

x√1− x2

dx.

H tètarth isìthta prokÔptei me efarmog  thc idiìthtac limx→c−

f(x) = limx→−c+

f(−x).

(MporeÐte na thn apodeÐxete?) H pèmpth me ton metasqhmatismì x→ −x. Parathr -ste, t¸ra, pwc ∫

x√1− x2

dx = −√

1− x2.

Sunep¸c,

I = 2 lima→1−

∫ a

0

x√1− x2

dx = 2 lima→1−

∫ a

0

(−√

1− x2)′dx

= 2 lima→1−

[−√

1− x2]a

0= 2(−0 + 1) = 2.

Par�deigma 7.19. Ja upologÐsoume to∫ 1

−1

1

t2dt.

Katarq n, eÐnai l�joc na antimetwpÐsoume autì to olokl rwma wc sunhjismèno epeid èqei peperasmèna ìria olokl rwshc kai den apeirÐzetai h sun�rthsh se aut�. H su-n�rthsh apeirÐzetai entìc tou diast matoc olokl rwshc! Epomènwc, qrhsimopoi¸ntacton sunoptikì trìpo upologismoÔ kataqrhstik¸n oloklhrwm�twn,∫ 1

−1

1

t2dt =

∫ 0

−1

1

t2dt+

∫ 1

0

1

t2dt =

∫ 0

−1

(−1

t

)′dt+

∫ 1

0

(−1

t

)′dt

=

[1

t

]−1

0

+

[1

t

]0

1

= −1− limt→0−

1

t+ lim

t→0+

1

t− 1

= −1 +∞+∞− 1 =∞.

Page 183: ΣΗΜΕΙΩΣΕΙΣ-ΜΑΘΗΜΑΤΙΚΑ 1 ΠΛΗΡΟΦΟΡΙΚΗ ΑΣΟΕΕ 1 ΕΞΑΜΗΝΟ

Kef�laio 8

Efarmogèc Oloklhrwm�twn

8.1 Upologismìc EmbadoÔ se Kartesianèc Suntetag-

mènec

Je¸rhma 8.1. (Olokl rwma = embadìn) 'Estw sun�rthsh f mh arnhtik  kai olo-klhr¸simh sto di�sthma [a, b]. H epif�neia A(R) tou qwrÐou R pou brÐsketai an�mesasto gr�fhma thc sun�rthshc, ton �xona x, thn eujeÐa x = a, kai thn eujeÐa x = b,dhlad  tou sunìlou

R = {(x, y) : a ≤ x ≤ b, 0 ≤ y ≤ f(x)},

isoÔtai me

A(R) =

∫ b

a

f.

Apìdeixh. 'Estw mia opoiad pote diamèrish P = {t0, t1, . . . , tn} me

a = t0 < t1 < · · · < tn = b.

'Estw epÐshc, kat� ta gnwst�,

mi = inf{f(x) : ti−1 ≤ x ≤ ti},Mi = sup{f(x) : ti−1 ≤ x ≤ ti}.

ParathroÔme pwc to k�tw �jroisma

L(f, P ) =n∑i=1

mi(ti − ti−1) ≤ A(R),

giatÐ to k�tw �jroisma isoÔtai me to embadìn enìc qwrÐou pou brÐsketai pl rwc entìctou qwrÐou R. To en lìgw qwrÐo eÐnai to deÔtero skiasmèno qwrÐo tou Sq matoc 8.1,

173

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174 KEF�ALAIO 8. EFARMOG�ES OLOKLHRWM�ATWN

a b

f(x)

R

x a b

f(x)

x

xa b

f(x)

a b x

f(x)

Sq ma 8.1: To qwrÐo R an�mesa sto gr�fhma miac mh arnhtik c sun�rthshc f kai ton �xona twn x,kai treic klimakwtèc proseggÐseic thc sun�rthshc f .

kai apoteleÐtai apì orjog¸nia, me embadìn to k�je èna Ðso me mi(ti − ti−1). Meparìmoio trìpo, èqoume

U(f, P ) =n∑i=1

Mi(ti − ti−1) ≥ A(R),

giatÐ to qwrÐo R eÐnai uposÔnolo enìc qwrÐou me embadìn Ðso me to �nw �jroisma (deÐteto trÐto skiasmèno qwrÐo tou Sq matoc 8.1). 'Ara, gia opoiad pote diamèrish, isqÔei

L(f, P ) ≤ A(R) ≤ U(f, P ).

'Omwc, epeid  h f eÐnai oloklhr¸simh, up�rqei mìno ènac arijmìc pou ikanopoieÐ aut 

th dipl  anisìthta gia opoiad pote diamèrish, kai autìc eÐnai to∫ ba f (Je¸rhma 6.2).

'Ara, prokÔptei to zhtoÔmeno.

Parathr seic

1. R=region, A=area.

2. O pr¸toc pou èkane ton �nw sullogismì (ìqi akrib¸c se aut  th morf )  tan oArqim dhc, gia thn perÐptwsh f(x) = x2, a = 0, b = 1.

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8.1. UPOLOGISM�OS EMBADO�U SE KARTESIAN�ES SUNTETAGM�ENES 175

3. Den èqoume orÐsei axiwmatik� thn epif�neia, kai basizìmaste se idiìthtec poudiaisjhtik� faÐnetai ìti ikanopoioÔntai, �ra h apìdeix  mac den eÐnai entel¸c au-sthr .

4. Mia pio diaisjhtik , ligìtero austhr  apìdeixh, mèsw ajroism�twn Riemann,eÐnai h ex c: 'Estw mia diamèrish

P = {a = t0, t1, . . . , tn = b},

kai èstw n timèc x1, x2, . . . , xn, mia gia k�je di�sthma [ti−1, ti], pou epilègontaiaujaÐreta, dhlad :

ti−1 ≤ xi ≤ ti.

Tìte to �jroisma Riemann

n∑i=1

f(xi)(ti − ti−1),

apì to Je¸rhma Darboux (Je¸rhma 6.16), sugklÐnei sto olokl rwma∫ ba f , ka-

j¸c h diamèrish pukn¸nei. Diaisjhtik� ìmwc, koit¸ntac to tètarto skiasmènoqwrÐo tou Sq matoc 8.1), eÐnai profanèc ìti to �jroisma sugklÐnei kai sthn e-pif�neia A(R), kaj¸c h diamèrish pukn¸nei. 'Ara, ta dÔo ìria (olokl rwma kaiepif�neia) eÐnai Ðsa.

Je¸rhma 8.2. (Epif�neia metaxÔ grafhm�twn) 'Estw sunart seic f, g oloklhr¸-simec sto [a, b]. H epif�neia A(R) tou qwrÐou R pou brÐsketai an�mesa sta graf matatwn sunart sewn, thn eujeÐa x = a, kai thn eujeÐa x = b, dhlad  tou sunìlou

R = {(x, y) : a ≤ x ≤ b, min(f(x), g(x)) ≤ y ≤ max(f(x), g(x))},

dÐnetai apì to olokl rwma:

A(R) =

∫ b

a

|f − g|.

Apìdeixh. H apìdeixh basÐzetai sth l yh peript¸sewn kai sth qr sh profan¸n idiot -twn thc epif�neiac, kai paraleÐpetai. MporeÐ epÐshc na dojeÐ mia diaisjhtik  ex ghshparìmoia me aut  tou Jewr matoc 8.1.

Par�deigma 8.1. (Upologismìc epifanei¸n me oloklhr¸mata) Ja sqedi�soume toqwrÐo pou perikleÐetai metaxÔ twn grafhm�twn twn sunart sewn f(x) kai g(x) giaa ≤ x ≤ b, kai ja upologÐsoume thn epif�nei� tou, gia tic akìloujec peript¸seic:

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176 KEF�ALAIO 8. EFARMOG�ES OLOKLHRWM�ATWN

a b

f(x)

g(x)

xR

−1 −0.5 0 0.5 1

0

0.5

1

1.5

2

x

y

x3 + x2

x3 + 1

−1 0 1

−2

−1

0

1

2

x

y

cosx

sin x

π2− π

2

Sq ma 8.2: ('Anw) Par�deigma qwrÐou me embadìn pou upologÐzetai apì to Je¸rhma 8.2, kai (k�tw)ta qwrÐa tou ParadeÐgmatoc 8.1.

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8.1. UPOLOGISM�OS EMBADO�U SE KARTESIAN�ES SUNTETAGM�ENES 177

1. f(x) = x3 + 1, g(x) = x3 + x2, a = −1, b = 1.

2. f(x) = 2 cosx, g(x) = 2 sinx, a = −π/2, b = π/2.

Katarq n, kai gia tic dÔo peript¸seic, ta graf mata twn f(x), g(x) emfanÐzontaisto Sq ma 8.2, kai to qwrÐo tou opoÐou thn epif�neia kaloÔmaste na broÔme emfanÐ-zetai skiasmèno.

Sqetik� me ta embad�, èqoume kat� perÐptwsh:

1. ParathroÔme ìti x ∈ [−1, 1]⇒ x2 ≤ 1⇒ x3 +x2 ≤ x3 + 1⇒ g(x) ≤ f(x), �ra,apì to Je¸rhma 8.2, èqoume:

A =

∫ 1

−1

[f(x)− g(x)] dx =

∫ 1

−1

(1− x2) dx = 2−∫ 1

−1

x2 dx

= 2−[x3

3

]1

−1

= 2− 2

3=

4

3.

2. ParathroÔme pwc an x ∈ (−π/2, π/2), ¸ste cosx > 0, tìte

2 sinx Q 2 cosx⇔ tanx Q 1⇔ x Q π/4.

Epomènwc,

A(R) =

∫ π/2

−π/2|2 sinx− 2 cosx| dx

= 2

∫ π/4

−π/2(cosx− sinx) dx+ 2

∫ π/2

π/4

(sinx− cosx) dx

= 2 [sinx+ cosx]π/4−π/2 + 2 [− cosx− sinx]

π/2π/4

= 2[√

2/2 +√

2/2 + 1− 0]

+ 2[0− 1 +

√2/2 +

√2/2]

= 2√

2.

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178 KEF�ALAIO 8. EFARMOG�ES OLOKLHRWM�ATWN

8.2 Upologismìc EmbadoÔ se Polikèc Suntetagmènec

Orismìc 8.1. (Polikèc suntetagmènec) To shmeÐo (x, y) èqei polikèc suntetagmènec(r, θ), ìpou r ≥ 0, θ ∈ R, an{

x = r cos θ,y = r sin θ

}⇔{

r =√x2 + y2,

cos θ = x√x2+y2

, sin θ = y√x2+y2

}

Parathr seic

1. H sqèsh metaxÔ twn x, y, r, θ emfanÐzetai sto Sq ma 8.3.

2. Merik� paradeÐgmata faÐnontai sto Sq ma 8.4.

3. Se antÐjesh me tic kartesianèc suntetagmènec, èna shmeÐo èqei pollèc polikècsuntetagmènec:

(aþ) An èna shmeÐo di�foro tou (x, y) = (0, 0) èqei polikèc suntetagmènec (r, θ),èqei kai (r, θ + 2kπ) gia k�je k ∈ Z.

(bþ) Eidik� to (x, y) = (0, 0) èqei polikèc suntetagmènec r = 0 kai θ opoiod potesto R.

4. Pwc mporoÔme na skeftìmaste tic polikèc suntetagmènec:

(aþ) Me tic kartesianèc suntetagmènec, gia na broÔme to shmeÐo (x, y) proqw-r�me x kat� m koc tou �xona x kai met� y par�llhla ston �xona y. (OikateujÔnseic exart¸ntai apì ta antÐstoiqa prìshma.)

(bþ) Me tic polikèc suntetagmènec, pr¸ta koit�me proc thn kateÔjunsh θ, kaimet� proqwr�me apìstash r (pou eÐnai p�nta mh arnhtik ) epÐ aut c.

5. P�nta r ≥ 0. θ ∈ R, all� gia k�je shmeÐo tou epipèdou up�rqei k�poio θ ∈[0, 2π).

6. Gia èna shmeÐo (x, y) up�rqoun dÔo gwnÐec θ1, θ2 ∈ [0, 2π) tètoiec ¸ste

cos θ1 = cos θ2 =x√

x2 + y2.

(Ti sqèsh èqoun metaxÔ touc autèc oi gwnÐec?) ParomoÐwc, up�rqoun dÔo diafo-retikèc gwnÐec θ1, θ2 ∈ [0, 2π) tètoiec ¸ste

sin θ1 = sin θ2 =y√

x2 + y2.

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8.2. UPOLOGISM�OS EMBADO�U SE POLIK�ES SUNTETAGM�ENES 179

x

y

P=(x,y)=(r,θ)

y=r sinθ

x=r cosθ

0

r

θ

Sq ma 8.3: Sqèsh metaxÔ kartesian¸n kai polik¸n suntetagmènwn.

−3 −2 −1 0 1 2 3−3

−2

−1

0

1

2

3

x

y

r = 1, θ = π/4

r = 2, θ = π/4

r = 0, θ = ∗

r = 1, θ = 3π/2

r = 2, θ = π

Sq ma 8.4: ParadeÐgmata polik¸n suntetagmènwn.

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180 KEF�ALAIO 8. EFARMOG�ES OLOKLHRWM�ATWN

(Ti sqèsh èqoun metaxÔ touc autèc oi gwnÐec?) Gi autì, ìtan k�noume ton meta-sqhmatismì (x, y)→ (r, θ), apaitoÔme

cos θ =x√

x2 + y2kai sin θ =

y√x2 + y2

.

Oi dÔo autèc exis¸seic ikanopoioÔntai p�nta gia mìno mia gwnÐa θ ∈ [0, 2π).

7. Par� tic epiplokèc pou èqoun (gia par�deigma, den up�rqei 1 − 1 antistoÐqishmetaxÔ shmeÐwn sto epÐpedo kai zeug¸n (r, θ), ìpwc me tic kartesianèc sunte-tagmènec), oi polikèc suntetagmènec eÐnai polÔ qr simec. Mac epitrèpoun, giapar�deigma, na perigr�foume polÔ apl� orismènec kampÔlec.

Orismìc 8.2. (KampÔlh se polikèc suntetagmènec) An h f : [θ1, θ2] → R+ eÐnaisuneq c, tìte h exÐswsh r = f(θ) perigr�fei mia kampÔlh se polikèc suntetagmènec.

Par�deigma 8.2. AntistoiqÐsete tic kampÔlec twn Sqhm�twn 8.5, 8.6, me ta

1. r = θ, θ ∈ [0, 2π]. (SpeÐra tou Arqim dh)

2. r = 1, θ ∈ [0, 2π].

3. r = log θ, θ ∈ [1, 50].

4. r = | cos θ|, θ ∈ [0, 2π].

5. r = 1 + cos θ, θ ∈ [0, 2π]. (Kardioeidèc)

6. r = | sin 2θ|, θ ∈ [0, 2π].

7. r = 2 cos θ, θ ∈ [−π2 ,

π2 ].

8. r = 2 + cos θ, θ ∈ [0, 2π].

Parat rhsh: EÐnai polÔ shmantikì na mhn sugqèoume th morf  pou èqei mia sun�r-thsh r = f(θ) se kartesianèc suntetagmènec (r, θ) me th morf  thc kampÔlhc r = f(θ)ìtan ta r, θ eÐnai polikèc suntetagmènec. DeÐte to Sq ma 8.7 ìpou h r = | cos 2θ|, meθ ∈ [0, 2π], èqei sqediasteÐ se kartesianèc suntetagmènec (r, θ), kai ek twn ustèrwnèqei qrhsimopoihjeÐ gia na orÐsei mia kampÔlh r = | cos 2θ| se polikèc suntetagmènec.

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8.2. UPOLOGISM�OS EMBADO�U SE POLIK�ES SUNTETAGM�ENES 181

−1 −0.5 0 0.5 1−0.5

0

0.5

−5 0 5 10−6

−4

−2

0

2

4

−1 −0.5 0 0.5 1−1

−0.5

0

0.5

1

−2 0 2 4−3

−2

−1

0

1

2

3

−5 0 5−5

0

5

−1 −0.5 0 0.5 1

−1

−0.5

0

0.5

1

Sq ma 8.5: KampÔlec tou ParadeÐgmatoc 8.2.

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182 KEF�ALAIO 8. EFARMOG�ES OLOKLHRWM�ATWN

0 1 2−2

−1

0

1

2

0 1 2−2

−1

0

1

2

Sq ma 8.6: KampÔlec tou ParadeÐgmatoc 8.2.

Je¸rhma 8.3. (Embadìn qwrÐou se polikèc suntetagmènec) 'Estw gwnÐec a, b ∈[0, 2π] me a < b kai èstw f : [a, b] → R mh arnhtik  oloklhr¸simh sun�rthsh. Toembadìn A(R) tou qwrÐou

R = {(r, θ) : a ≤ θ ≤ b, 0 ≤ r ≤ f(θ)} (8.1)

dÐnetai apì to olokl rwma

A(R) =1

2

∫ b

a

f 2(θ) dθ.

Apìdeixh. ParaleÐpetai.

Parat rhsh: Mia diaisjhtik  ex ghsh, mèsw ajroism�twn Riemann, eÐnai h ex c:'Estw mia diamèrish

P = {a = θ0, θ1, . . . , θn = b},

kai èstw n timèc x1, x2, . . . , xn, mia gia k�je di�sthma [θi−1, θi], pou epilègontai aujaÐ-reta, dhlad :

θi−1 ≤ xi ≤ θi.

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8.2. UPOLOGISM�OS EMBADO�U SE POLIK�ES SUNTETAGM�ENES 183

0 1 2 3 4 5 6−0.5

0

0.5

1

1.5

θ

| cos 2θ|

−1 −0.5 0 0.5 1

−1

−0.5

0

0.5

1

x

y

Sq ma 8.7: H sun�rthsh r = | cos 2θ|, me θ ∈ [0, 2π], sqediasmènh se kartesianèc suntetagmènec(r, θ), kai h kampÔlh r = | cos 2θ|, se polikèc suntetagmènec.

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184 KEF�ALAIO 8. EFARMOG�ES OLOKLHRWM�ATWN

r = f(θ) θ = a

θ = b

x

y

R

Sq ma 8.8: Diaisjhtik  ermhneÐa tou Jewr matoc 8.3.

'Estw to �jroisma Riemann

n∑i=1

1

2f 2(xi)(θi − θi−1),

pou isoÔtai me to embadìn enìc qwrÐou pou proseggÐzei to R. (DeÐte to deÔteroskiasmèno qwrÐo tou Sq matoc 8.8.) Kat� ta gnwst� apì th jewrÐa, to �jroisma

Riemann sugklÐnei sto olokl rwma∫ ba

12f

2 kaj¸c h diamèrish pukn¸nei. Diaisjhtik�ìmwc, koit¸ntac sto sq ma, eÐnai profanèc ìti, kaj¸c h diamèrish pukn¸nei, to Ðdio�jroisma sugklÐnei kai sthn epif�neia A(R). (UpenjumÐzoume ìti to embadìn kukli-koÔ tomèa gwnÐac θ kai aktÐnac r eÐnai 1

2θr2.) 'Ara, ta dÔo ìria, h epif�neia kai to

olokl rwma, eÐnai Ðsa.

Par�deigma 8.3. Ja upologÐsoume to embadìn twn qwrÐwn pou perigr�fontai wcex c:

1. R = {0 ≤ θ ≤ 2π, 0 ≤ r ≤ f(θ) = θ} (Parat rhsh: h exÐswsh r = θ orÐzeithn SpeÐra tou Arqim dh).

2. R = {−π2 ≤ θ ≤ π

2 , 0 ≤ r ≤ f(θ) = 2 cos θ}. (Parat rhsh: h exÐswshr = 2 cos θ orÐzei èna kÔklo, me kèntro to (1, 0) kai aktÐna 1. MporeÐte na todeÐxete?).

3. R = {−π ≤ θ ≤ π, 0 ≤ r ≤ min{1 + cos θ, 1− cos θ}}. Parathr ste pwc se

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8.2. UPOLOGISM�OS EMBADO�U SE POLIK�ES SUNTETAGM�ENES 185

aut  thn perÐptwsh, to R eÐnai h tom  twn dÔo qwrÐwn

R1 = {−π ≤ θ ≤ π, 0 ≤ r ≤ 1 + cos θ},R2 = {−π ≤ θ ≤ π, 0 ≤ r ≤ 1− cos θ}.

Ta qwrÐa emfanÐzontai skiasmèna sto Sq ma 8.9. 'Eqoume, kat� perÐptwsh:

1.

A(R) =1

2

∫ 2π

0

θ2 dθ =1

2

[θ3

3

]2π

0

=4

3π3.

2.

A(R) =1

2

∫ π/2

−π/24 cos2 θ dθ = 2

∫ π/2

−π/2

1 + cos 2θ

2dθ

= 2

∫ π/2

−π/2

1

2dθ +

∫ π/2

−π/2cos 2θ dθ = π +

1

2[sin 2θ]

π/2−π/2 = π.

3. O pio aplìc trìpoc na upologÐsoume to embadìn eÐnai na parathr soume ìti autìeÐnai 4 forèc to embadìn tou qwrÐou

R′ = {(r, θ) : 0 ≤ θ ≤ π/2, 0 ≤ r ≤ 1− cos θ},

dhlad  tou tm matoc tou zhtoÔmenou qwrÐou pou brÐsketai sto pr¸to tetarthmì-rio. Sunep¸c,

A(R) = 4A(R′) = 2

∫ π/2

0

(1− cos θ)2 dθ = 2

∫ π/2

0

(1 + cos2 θ − 2 cos θ) dθ

= 2

∫ π/2

0

dθ + 2

∫ π/2

0

1 + cos 2θ

2dθ − 4

∫ π/2

0

cos θ dθ

= π + π/2 +

∫ π/2

0

cos 2θ dθ − 4

∫ π/2

0

cos θ dθ

=3π

2+

1

2[sin 2θ]

π/20 − 4 [sin θ]

π/20 =

2− 4.

Parat rhsh: Se ask seic autoÔ tou tÔpou, kai genikìtera ìsec emplèkoun kam-pÔlec se polikèc suntetagmènec, to pr¸to b ma ja prèpei na eÐnai o sqediasmìc twnkampul¸n thc �skhshc, me kat� to dunatìn kalÔterh akrÐbeia. Autì mporeÐ na gÐnei methn eÔresh diafìrwn shmeÐwn p�nw sthn kampÔlh, me th qr sh thc exÐswshc r = f(θ)gia di�fora θ, kai me th qr sh tuqìn summetri¸n.

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186 KEF�ALAIO 8. EFARMOG�ES OLOKLHRWM�ATWN

−5 0 5 10−6

−4

−2

0

2

4

x

y

r = θ, 0 ≤ θ ≤ 2π

0 0.5 1 1.5 2

−1

−0.5

0

0.5

1

x

y

r = 2 cos θ, − π/2 ≤ θ ≤ π/2

−2 −1 0 1 2

−1.5

−1

−0.5

0

0.5

1

1.5

x

y

r(θ) = 1 + cos θr(θ) = 1 − cos θ

−π ≤ θ ≤ π−π ≤ θ ≤ π

Sq ma 8.9: Ta qwrÐa tou ParadeÐgmatoc 8.3.

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8.3. UPOLOGISM�OS �OGKOU ME TH M�EJODO TWN D�ISKWN 187

8.3 Upologismìc 'Ogkou me th Mèjodo twn DÐskwn

Je¸rhma 8.4. (Mèjodoc dÐskwn) 'Estw oloklhr¸simh f : [a, b] → R+. 'Estw Rto qwrÐo pou perikleÐetai apì to gr�fhma thc f , ton �xona twn x, kai tic eujeÐec x = akai x = b. 'Estw S to stereì pou dhmiourgeÐtai apì thn peristrof  tou R perÐ ton�xona twn x. O ìgkoc V (S) tou stereoÔ S dÐnetai apì thn exÐswsh

V (S) = π

∫ b

a

f 2.

Apìdeixh. ParaleÐpetai.

Parathr seic

1. DeÐte to Sq ma 8.10 gia èna par�deigma stereoÔ thc morf c tou jewr matoc.

2. S =solid, V =volume.

3. Mia diaisjhtik  ex ghsh tou jewr matoc, mèsw ajroism�twn Riemann, eÐnai hex c: 'Estw mia diamèrish

P = {a = t0, t1, . . . , tn = b},

kai èstw n timèc x1, x2, . . . , xn, mia gia k�je di�sthma [ti−1, ti], pou epilègontaiaujaÐreta, dhlad :

ti−1 ≤ xi ≤ ti.

'Estw to �jroisma Riemann

n∑i=1

πf 2(xi)(ti − ti−1),

to opoÐo isoÔtai me to �jroisma twn ìgkwn n kulÐndrwn i = 1, . . . , n, aktÐnacf(xi) kai Ôyouc ti− ti−1. (Sqetik� me ton ìgko kulÐndrwn, deÐte to Sq ma 8.11.)Apì to Je¸rhma Darboux (Je¸rhma 6.16), to �jroisma autì sugklÐnei sto o-

lokl rwma∫ ba πf

2 kaj¸c h diamèrish pukn¸nei. Diaisjhtik� ìmwc, koit¸ntac toSq ma 8.12, eÐnai profanèc ìti, kaj¸c h diamèrish pukn¸nei, to Ðdio �jroismasugklÐnei kai ston ìgko V (S). 'Ara, ta dÔo, olokl rwma kai ìgkoc, eÐnai Ðsa.

4. Epeid  to Ôyoc twn kulÐndrwn, kaj¸c h diamèrish P pukn¸nei, gÐnetai ìlo kai piomikrì, oi kÔlindroi moi�zoun me dÐskouc, ex ou kai to ìnoma thc mejìdou.

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188 KEF�ALAIO 8. EFARMOG�ES OLOKLHRWM�ATWN

x

y

a b

f(x)

R

Sq ma 8.10: Mia sun�rthsh, to qwrÐo R metaxÔ tou graf matìc thc kai tou �xona x, kai o ìgkocpou dhmiourgeÐtai me thn peristrof  tou R perÐ ton �xona twn x.

h

r

h

r1

r2

Sq ma 8.11: 'Enac kÔlindroc me Ôyoc h, aktÐna b�shc r, kai ìgko V = πr2h, kai èna kèlufoc me Ôyoch, meg�lh aktÐna b�shc r2, mikr  aktÐna b�shc r1, kai ìgko V = h(r22 − r21) = 2πh(r1 − r2) r1+r2

2.

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8.3. UPOLOGISM�OS �OGKOU ME TH M�EJODO TWN D�ISKWN 189

x

f(x)

y

z

Sq ma 8.12: ProseggÐseic tou ìgkou tou Sq matoc 8.10 wc ènwsh dÐskwn.

Sq ma 8.13: Ta stere� ek peristrof c tou ParadeÐgmatoc 8.4.

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190 KEF�ALAIO 8. EFARMOG�ES OLOKLHRWM�ATWN

Par�deigma 8.4. Ja upologÐsoume ton ìgko tou stereoÔ pou dhmiourgeÐtai me thnperistrof , perÐ ton �xona twn x, tou qwrÐou pou perikleÐetai apì to gr�fhma thc f ,ton �xona twn x, kai tic eujeÐec x = a, x = b, gia tic akìloujec peript¸seic:

1. f(x) =√x, a = 0, b = 1.

2. f(x) = sinx, a = 0, b = π.

'Eqoume, kat� perÐptwsh,

1.

V (S) =

∫ 1

0

π(√x)2 dx = π

∫ 1

0

x dx = π

[x2

2

]1

0

2.

2.

V (S) =

∫ π

0

π sin2 x dx =

∫ π

0

π1− cos 2x

2dx

=π2

2− π

2

∫ π

0

cos 2x dx =π2

2− π

4[sin 2x]2π0 =

π2

2.

Parat rhsh: Se ask seic aut c thc morf c, eÐnai shmantikì na mporoÔme na sqe-di�soume proseggistik� ta stere� twn opoÐwn ton ìgko kaloÔmaste na upologÐsoume.

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8.4. UPOLOGISM�OS �OGKOU ME TH M�EJODOS TWN KEL�UFWN 191

8.4 Upologismìc 'Ogkou me th Mèjodoc twn KelÔfwn

Je¸rhma 8.5. (Mèjodoc kelÔfwn) 'Estw oloklhr¸simh f : [a, b] → R+. 'EstwR to qwrÐo metaxÔ tou graf matìc thc, tou �xona twn x, kai twn eujei¸n x = a,x = b, me 0 < a < b. 'Estw S to stereì ek peristrof c pou dhmiourgeÐtai apì thnperistrof  tou R perÐ ton �xona twn y. O ìgkoc V (S) tou stereoÔ S dÐnetai apì thnexÐswsh

V (S) = 2π

∫ b

a

xf(x) dx.

Apìdeixh. ParaleÐpetai.

Parathr seic

1. DeÐte to Sq ma 8.14 gia èna par�deigma stereoÔ ek peristrof c thc morf c touJewr matoc 8.5.

2. Sto Sq ma 8.15 èqei sqediasteÐ h probol  tou S epÐ tou epipèdou xz.

3. Mia diaisjhtik  ex ghsh tou jewr matoc, mèsw ajroism�twn Riemann, eÐnai hakìloujh.

Katarq n, parathr ste ìti èna kulindrikì kèlufoc me mikr  aktÐna b�shc r1,meg�lh aktÐna b�shc r2, kai Ôyoc h (deÐte to Sq ma 8.11), èqei ìgko

V = hπ(r22 − r2

1) = 2πr2 + r1

2h(r2 − r1).

(Dhlad , o ìgkoc tugq�nei na isoÔtai me ton ìgko enìc orjog¸niou parallhlepÐ-pedou me diast�seic 2π r2+r1

2 , h, kai (r2−r1). Proseggistik�, to parallhlepÐpedoautì ja to p�roume an k�noume mia diam kh tom  sto kèlufoc kai to apl¸soumesto epÐpedo!)

'Estw t¸ra mia diamèrish

P = {a = t0, t1, . . . , tn = b},

kai èstw oi n timèc x1, x2, . . . , xn, mia gia k�je di�sthma [ti−1, ti], pou epilègontaina eÐnai sth mèsh tou k�je diast matoc, dhlad :

xi =ti−1 + ti

2.

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192 KEF�ALAIO 8. EFARMOG�ES OLOKLHRWM�ATWN

x

y

a b

f(x)

R

Sq ma 8.14: Mia sun�rthsh f ≥ 0 kai o ìgkoc pou dhmiourgeÐtai me thn peristrof  tou qwrÐou Rpou perikleÐetai apì to gr�fhm� thc f , ton �xona twn x, kai tic eujeÐec x = a, x = b perÐ ton �xonatwn y.

x

z

a b

a2≤ x2+z2≤ b2

Sq ma 8.15: Probol  tou stereoÔ tou Sq matoc 8.14 sto epÐpedo xz. O �xonac y eÐnai proc ta �nw.

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8.4. UPOLOGISM�OS �OGKOU ME TH M�EJODOS TWN KEL�UFWN 193

Sq ma 8.16: Prosèggish tou ìgkou enìc stereoÔ ek peristrof c me ton ìgko stereoÔ pou apoteleÐtaiapì diadoqik� kelÔfh.

(Parathr ste ìti mèqri t¸ra se autì to kef�laio, ìpote anèkupte �jroismaRiemann, epilègame aujaÐreta ta shmeÐa xi.) 'Estw to �jroisma Riemann

n∑i=1

2πxif(xi)(ti − ti−1) =n∑i=1

2π(ti + ti−1)

2f(xi)(ti − ti−1)

=n∑i=1

πf(xi)(t2i − t2i−1).

Parathr ste ìti oi ekfr�seic sta dexi� ekfr�zoun (me diaforetikì trìpo h k�jemia) to �jroisma twn ìgkwn twn ep�llhlwn keluf¸n tou Sq matoc 8.16. Diai-sjhtik�, sto ìrio pou h diamèrish gÐnetai �peira pukn , autèc oi ekfr�seic jasugklÐnoun ston ìgko tou S, V (S). Epeid  ìmwc h èkfrash sto pr¸to skèloc

ja teÐnei sto∫ ba xf(x) dx, prokÔptei telik� h isìthta.

Par�deigma 8.5. Ja upologÐsoume touc ìgkouc twn stere¸n pou dhmiourgoÔn-tai apì thn peristrof  perÐ ton �xona twn y tou qwrÐou pou brÐsketai metaxÔ tougraf matoc thc f , tou �xona twn x, kai twn eujei¸n x = a, x = b, ìtan:

1. f(x) = x−1, a = 12 , b = 5

2 . (HfaÐsteio)

2. f(x) = x− 2, a = 2, b = 5. (Jèatro)

3. f(x) = 2[1− (x− 1)2], a = 0, b = 2. (Poluwnumikì kèhk)

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194 KEF�ALAIO 8. EFARMOG�ES OLOKLHRWM�ATWN

−20

2 −2

0

20

1

2

zx

y

−5

0

5 −5

0

50

2

zx

y

−2

0

2 −2

0

20

1

2

zx

y

Sq ma 8.17: Par�deigma 8.5.

V (S1) = 2π

∫ 52

12

x× x−1 dx =

∫ 52

12

dx = 4π,

V (S2) = 2π

∫ 5

2

x(x− 2) dx = 2π

∫ 5

2

(x2 − 2x) dx = 2π

∫ 5

2

(x3

3− x2

)′dx

= 2π

[53

3− 52 − 23

3+ 4

]= 36π,

V (S3) = 2π

∫ 2

0

2x(1− (x− 1)2) dx = 4π

∫ 2

0

x(1− x2 − 1 + 2x) dx

= 4π

∫ 2

0

x(2x− x2) dx = 4π

∫ 2

0

(2x2 − x3) dx

= 4π

∫ 2

0

(2

3x3 − x4

4

)′dx = 4π

(24

3− 24

4

)=

16

3π.

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8.4. UPOLOGISM�OS �OGKOU ME TH M�EJODOS TWN KEL�UFWN 195

−2

0

2 −2

0

20

1

2

3

zx

y

Sq ma 8.18: Par�deigma 8.5. Peristrof  kai twn dÔo skiasmènwn sunìlwn sta arister� perÐ toaristerì touc sÔnoro dhmiourgeÐ to stereì sta dexi�.

Par�deigma 8.6. Ja upologÐsoume ton ìgko tou stereoÔ pou dhmiourgeÐtai apìthn peristrof , perÐ ton �xona x = 3 tou graf matoc thc f(x) = x(x+ 1) metaxÔ toux = 3 kai tou x = 5. ParathroÔme pwc to stereì pou sqhmatÐzetai eÐnai Ðdio me autìpou ja sqhmatisteÐ an peristrèyoume thn f(x) perÐ ton �xona twn y, apì to x = 0mèqri to x = 2, afoÔ pr¸ta metafèroume thn f(x) proc ta arister� kat� 3. H nèasun�rthsh eÐnai h g(x) = f(x + 3) = (x + 3)(x + 4). Gia na peisteÐte ìti pr�gmati hg(x) eÐnai mia metatopismèna arister� kat� 3 èkdosh thc f(x), mporeÐte na dokim�setena tic sqedi�sete. DeÐte to Sq ma 8.18. Sqetik� me ton ìgko, kat� ta sunhjismènaplèon, èqoume

V (S) = 2π

∫ 2

0

x(x+ 3)(x+ 4) dx = 2π

∫ 2

0

(x3 + 7x2 + 12x) dx

= 2π

∫ 2

0

(x4

4+

7x3

3+ 6x2

)′dx = 280π/3.

Parat rhsh: Se ask seic aut c thc morf c, eÐnai shmantikì na mporoÔme na sqe-di�soume proseggistik� ta stere� twn opoÐwn ton ìgko kaloÔmaste na upologÐsoume.

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196 KEF�ALAIO 8. EFARMOG�ES OLOKLHRWM�ATWN

8.5 Upologismìc M kouc KampÔlhc

Parat rhsh: Pwc pisteÔete ìti apojhkeÔoun oi suskeuèc GPS thn diadrom  pouakoloujeÐte? 'Enac trìpoc eÐnai mèsw tou akìloujou orismoÔ.

Orismìc 8.3. (KampÔlec) OrÐzoume wc kampÔlh sto epÐpedo k�je suneq  apeikìnish

apì to R sto R2 thc morf c:

x = f(t), y = g(t), a ≤ t ≤ b, (8.2)

ìpou oi f, g suneqeÐc sunart seic. Oi �nw exis¸seic kaloÔntai parametrikèc exis¸seicthc kampÔlhc, kai to t par�metroc. OrÐzoume wc Ðqnoc to sÔnolo twn shmeÐwn apì taopoÐa dièrqetai h kampÔlh, dhlad  to sÔnolo

R = {(x, y) : x = f(t1), t = g(t1), gia k�poio t1 ∈ [a, b]}.

Par�deigma 8.7. H kampÔlh tou Sq matoc 8.19 perigr�fetai apì tic sunart seic

x = t cos t, y = t sin t, 0 ≤ t ≤ 10.

Parathr seic

1. MporoÔme na fantazìmaste thn kampÔlh wc thn troqi� enìc antikeimènou, thnpar�metro t wc qrìno, kai tic sunart seic f(t), g(t) wc thn x kai y sunist¸satou antikeimènou sunart sei tou qrìnou.

2. Oi f, g prèpei na eÐnai suneqeÐc, alli¸c h kampÔlh ja eÐnai diakekommènh.

Par�deigma 8.8. AntistoiqÐste ta Ðqnh tou Sq matoc 8.20 me tic akìloujec para-metrikèc exis¸seic:

1. f(t) = cos t, g(t) = sin t, 0 ≤ t ≤ 2π.

2. f(t) = cos t, g(t) = sin 2t, 0 ≤ t ≤ 2π.

3. f(t) = cos 3t, g(t) = sin 2t, 0 ≤ t < 2π.

4. f(t) = t− sin t, g(t) = 1− cos t, 0 ≤ t < 4π.

5. f(t) = t, g(t) = |t|, −1 ≤ t ≤ 1.

6. f(t) = t, g(t) = sin t, −π ≤ t ≤ π.

7. f(t) = cos 2t, g(t) = sin 2t, 0 ≤ t ≤ π.

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8.5. UPOLOGISM�OS M�HKOUS KAMP�ULHS 197

−10 −5 0 5 10−10

−5

0

5

10

t = 0

t = π/2

t = π

t = 3π/2

t = 2π

t = 5π/2

t = 3π

t = 10

Sq ma 8.19: H kampÔlh tou ParadeÐgmatoc 8.7.

8. f(t) = t3, g(t) = |t3|, −1 ≤ t ≤ 1.

Parathr seic

1. 'Opwc faÐnetai kai sto prohgoÔmeno par�deigma, mporeÐ mia kampÔlh na èqei pollècperigrafèc thc morf c (8.2).

2. An up�rqoun oi f ′, g′, ti ekfr�zoun? Ti ekfr�zei h√f ′2(t) + g′2(t)?

Orismìc 8.4. (LeÐa kampÔlh) Mia kampÔlh kaleÐtai leÐa an èqei perigraf  thcmorf c (8.2) kai oi f ′, g′ up�rqoun kai eÐnai suneqeÐc sto [a, b], kai ìqi tautìqronamhdenikèc.

Parathr seic

1. Diaisjhtik�, mia leÐa kampÔlh den èqei gwnÐec.

2. H apaÐthsh oi f ′, g′ na mhn eÐnai tautìqrona 0 den eÐnai profan c, all� eÐnaiaparaÐthth gia na mhn up�rqoun gwnÐec. (DeÐte thn teleutaÐa perigraf  touParadeÐgmatoc 8.8.)

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198 KEF�ALAIO 8. EFARMOG�ES OLOKLHRWM�ATWN

−1 −0.5 0 0.5 1

−1

−0.5

0

0.5

1

−1 −0.5 0 0.5 1

−1

−0.5

0

0.5

1

−1 −0.5 0 0.5 1

−1

−0.5

0

0.5

1

−1 −0.5 0 0.5 1

−1

−0.5

0

0.5

1

0 5 10−2

0

2

−2 0 2−1

0

1

Sq ma 8.20: Oi kampÔlec tou ParadeÐgmatoc 8.8.

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8.5. UPOLOGISM�OS M�HKOUS KAMP�ULHS 199

Orismìc 8.5. (M koc kampÔlhc) 'Estw kampÔlh C me perigraf 

x = f(t), y = g(t), a ≤ t ≤ b,

kai èstw diamèrish P = {a = t0, t1, . . . , tn = b}. 'Estw

l(C,P ) =n∑i=1

√[f(ti)− f(ti−1)]2 + [g(ti)− g(ti−1)]2

to m koc thc tmhmatik� grammik c prosèggishc thc kampÔlhc.OrÐzoume to m koc thc kampÔlhc l(C) wc to

l(C) = supPl(C,P ),

efìson eÐnai peperasmèno.

Parathr seic

1. Sto Sq ma 8.21 faÐnetai mia kampÔlh gia thn opoÐa eÐnai diaisjhtik� profanèc ìtiup�rqei to supremum, kai to plhsi�zoume kaj¸c h nìrma thc diamèrishc teÐneisto 0.

2. Up�rqoun kampÔlec me �peiro m koc, gia par�deigma orismèna fractal, ìpwc hnif�da tou Koch, all� kai h akìloujh kampÔlh me thn apl  perigraf 

x = t, y =

{t sin 1

t , t 6= 0,

0, t = 0,0 ≤ t ≤ 1.

3. 'Opwc orÐsthke, to m koc thc kampÔlhc den exart�tai apì thn parametropoÐhs thc. (Eutuq¸c!) Autì den eÐnai profanèc, all� h apìdeixh paraleÐpetai.

Je¸rhma 8.6. (M koc kampÔlhc) 'Estw kampÔlh

x = f(t), y = g(t), a ≤ t ≤ b.

An oi f ′, g′ up�rqoun sto [a, b] kai eÐnai suneqeÐc, tìte

l(C) =

∫ b

a

√(f ′(t))2 + (g′(t))2 dt. (8.3)

Apìdeixh. H apìdeixh eÐnai meg�lh kai paraleÐpetai, dÐnoume ìmwc thn akìloujh ex -ghsh: 'Estw diamèrish P , kai èstw l(C,P ) to m koc thc tmhmatik� suneqoÔc kampÔlhc

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200 KEF�ALAIO 8. EFARMOG�ES OLOKLHRWM�ATWN

Sq ma 8.21: Kaj¸c h diamèrish tou [a, b] gÐnetai oloèna kai pio pukn , to m koc thc tmhmatik�suneqoÔc kampÔlhc teÐnei sto m koc thc kampÔlhc C.

pou dhmiourgeÐ:

l(C,P ) =n∑i=1

√[f(ti)− f(ti−1)]2 + [g(ti)− g(ti−1)]2.

Parathr ste ìti h �nw èkfrash ekfr�zei to m koc thc tmhmatik� suneqoÔc kampÔ-lhc wc to �jroisma twn mhk¸n twn epimèrouc eujÔgrammwn tmhm�twn.

Parathr ste ìti

l(C,P ) =n∑i=1

√[f(ti)− f(ti−1)

ti − ti−1

]2

+

[g(ti)− g(ti−1)

ti − ti−1

]2

(ti − ti−1).

An fantastoÔme ìti to t ekfr�zei qrìno kai h kampÔlh thn troqi� enìc antikeimènou,tìte sto �nw �jroisma k�je tetragwnik  rÐza ekfr�zei thn mèsh taqÔthta me thn opoÐakinoÔtan epÐ thc kampÔlhc to antikeÐmeno sto antÐstoiqo eujÔgrammo tm ma.

Epeid  oi f ′, g′ up�rqoun, apì to Je¸rhma Mèshc Tim c, ja up�rqoun t′i ∈ [ti−1, ti]kai t′′i ∈ [ti−1, ti] tètoia ¸ste

l(C,P ) =n∑i=1

√[f ′(t′i)]

2 + [g′(t′′i )]2(ti − ti−1).

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8.5. UPOLOGISM�OS M�HKOUS KAMP�ULHS 201

Se autì to shmeÐo ja k�noume mia prosèggish pou den ja dikaiolog soume austhr�,all� eÐnai diaisjhtik� logik : Epeid  oi f ′, g′ eÐnai suneqeÐc, ìtan h diamèrish eÐnaiarket� pukn , h tim  tou m kouc den ja all�zei polÔ an, antÐ gia ta t′i, t

′′i−1 ∈ [ti−1, ti],

qrhsimopoi soume k�poio τi ∈ [ti−1, ti]. 'Ara:

l(C,P ) 'n∑i=1

√[f ′(τi)]2 + [g′(τi)]2(ti − ti−1). (8.4)

'Otan h diamèrish gÐnetai �peira pukn , to men aristerì skèloc teÐnei sto m koc thckampÔlhc l(C), to de dexÐ skèloc, pou èqei th morf  ajroÐsmatoc Riemann, teÐnei

sto olokl rwma l(C) =∫ ba

√(f ′)2 + (g′)2, en¸ h prosèggish (8.4) gÐnetai ìlo kai

kalÔterh. ProkÔptei ètsi ìti telik� l(C) =∫ ba

√(f ′)2 + (g′)2.

'Ara, telik� o tÔpoc (8.3) ekfr�zei to sunolikì m koc thc kampÔlhc wc to �jroi-sma, sto ìrio, twn mhk¸n pou dianÔontai se apeiroel�qistouc qrìnouc, ìpou to k�jeapeirostì m koc dÐnetai wc to ginìmeno thc stigmiaÐac taqÔthtac me ton antÐstoiqoapeiroel�qisto qrìno.

Par�deigma 8.9. Ja upologÐsoume to m koc twn kampul¸n

x = f(t), y = g(t), a ≤ t ≤ b,

gia tic akìloujec peript¸seic:

1. f(t) = R cosωt, g(t) = R sinωt, t0 ≤ t ≤ t1. (Tìxo kÔklou)

2. f(t) = x0 + (x1−x0)t, g(t) = y0 + (y1− y0)t, 0 ≤ t ≤ 1. (EujÔgrammo tm ma)

3. f(t) = a(t − sin t), g(t) = a(1 − cos t), 0 ≤ t ≤ 2π, me a ∈ R, a > 0.(Kukloeidèc)

4. f(t) = a cos3 t, g(t) = a sin3 t, 0 ≤ t ≤ 2π. (Upokukloeidèc)

Oi kampÔlec emfanÐzontai sto Sq ma 8.22. Sqetik� me to m koc touc, èqoume, kat�perÐptwsh

1.

l(C) =

∫ t1

t0

√(−Rω sinωt)2 + (Rω cosωt)2 dt

=

∫ t1

t0

Rω√

sin2 ωt+ cos2 ωt dt =

∫ t1

t0

Rω dt = Rω(t1 − t0).

Sthn eidik  perÐptwsh ω = 1 paÐrnoume to m koc tìxou metaxÔ twn gwni¸n t0, t1.Gia par�deigma, gia t0 = 0, t1 = 2π, paÐrnoume to m koc thc perifèreiac kÔklou,l(C) = 2πR.

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202 KEF�ALAIO 8. EFARMOG�ES OLOKLHRWM�ATWN

x

y

t = t0, θ0 = ωt0

t = t1, θ1 = ωt1

R

x

y

(x0, y0), t = 0

(x1, y1), t = 1

(0, 0), t = 0 (2πr, 0), t = 2π

2r

x

y

t = 0, 2π

t = π/2

t = π

t = 3π/2

(a, 0)

(0, a)

(−a, 0)

(0,−a)

x

y

Sq ma 8.22: Par�deigma 8.9.

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8.5. UPOLOGISM�OS M�HKOUS KAMP�ULHS 203

2. Parathr ste ìti h kampÔlh eÐnai to eujÔgrammo tm ma pou sundèei ta shmeÐa(x0, y0), (x1, y1).

l(C) =

∫ 1

0

√(x1 − x0)2 + (y1 − y0)2 dt =

√(x1 − x0)2 + (y1 − y0)2.

AnaktoÔme dhlad  th gnwst  exÐswsh gia thn apìstash metaxÔ dÔo shmeÐwn.

3.

l(C) =

∫ 2π

0

√(a− a cos t)2 + (a sin t)2 dt

=

∫ 2π

0

√a2 + a2 cos t− 2a2 cos t+ a2 sin2 t dt

=

∫ π

0

a√

2− 2 cos t dt = 2a

∫ 2π

0

√1− cos t

2= 2a

∫ 2π

0

sint

2dt

= 4a

∫ 2π

0

(− cos

t

2

)′dt = 4a cos

t

2

∣∣∣∣02π

= 4a[1− (−1)] = 8a.

Sthn pèmpth isìthta qrhsimopoi same thn tautìthta (1− cos 2θ)/2 = sin2 θ, kaito gegonìc ìti t ∈ [0, 2π]⇒ sin t

2 ≥ 0.

4.

l(C) =

∫ π

0

√(3a cos2 t sin t)2 + (3a sin2 t cos t)2 dt

= 3a

∫ 2π

0

√sin2 t cos2 t(sin2 t+ cos2 t) dt = 3a

∫ 2π

0

√sin2 t cos2 t dt

= 12a

∫ π/2

0

√sin2 t cos2 t dt = 12a

∫ π/2

0

sin t cos t dt

= 6a

∫ π/2

0

sin 2t dt = 3a

∫ π/2

0

(− cos 2t)′ dt = 3a cos 2t|0π/2= 3a[1− (−1)] = 6a.

Sthn tètarth isìthta qrhsimopoi same to gegonìc ìti h oloklhrwtèa eÐnai pe-riodik  me perÐodo π/2.

Parat rhsh: Se ask seic aut c thc morf c, eÐnai shmantikì na mporoÔme na sqe-di�soume, èstw proseggistik�, thn dosmènh kampÔlh. Autì mporeÐ na gÐnei apl¸cupologÐzontac tic suntetagmènec arket¸n shmeÐwn p�nw sthn kampÔlh, gia di�forat ∈ [a, b], me qr sh twn sunart sewn f(t), g(t), kai endeqomènwc ekmetalleuìmenoituqìn summetrÐec.

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204 KEF�ALAIO 8. EFARMOG�ES OLOKLHRWM�ATWN

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Kef�laio 9

Diaforikèc Exis¸seic

9.1 Diaforikèc Exis¸seic Pr¸thc T�xewc

Orismìc 9.1. (Diaforikèc exis¸seic pr¸thc t�xewc kai eÐdh lÔsewn)

1. KaloÔme diaforik  exÐswsh (DE) pr¸thc t�xewc mia exÐswsh thc morf c

dy

dx= f(x, y), (9.1)

ìpou h exarthmènh metablht  y = y(x) eÐnai mia paragwgÐsimh sun�rthsh thcanex�rththc metablht c x, kai f(x, y) eÐnai mia sun�rthsh dÔo metablht¸n. HexÐswsh isqÔei gia ìla ta x se k�poio di�sthma I.

2. KaloÔme lÔsh thc DE (9.1) k�je sun�rthsh y(x) : I → R pou ikanopoieÐ thn(9.1).

3. KaloÔme genik  lÔsh thc DE (9.1) to sÔnolo twn sunart sewn pou thn ikano-poioÔn.

4. An, ektìc thc (9.1), èqei dojeÐ kai h arqik  sunj kh y(x0) = y0, ìpou x0 ∈ I,tìte kaloÔme eidik  (  merik ) lÔsh thc (9.1) mia sun�rthsh pou ikanopoieÐ kaithn DE, kai thn arqik  sunj kh.

Parathr seic

1. Sthn pr�xh, to I eÐte anafèretai rht¸c, eÐte uponoeÐtai apì ta sumfrazìmena.

2. Den èqoun ìlec oi DE genikèc kai/  eidikèc lÔseic. Ta probl mata Ôparxhc kaimonadikìthtac lÔsewn diaforik¸n exis¸sewn eÐnai en gènei polÔ dÔskola.

Par�deigma 9.1. (y′(x) = g(x)) Parathr ste ìti to na broÔme ìlec tic lÔseic thcDE y′(x) = g(x) isodunameÐ me to na broÔme to sÔnolo twn paragous¸n thc g(x).

205

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206 KEF�ALAIO 9. DIAFORIK�ES EXIS�WSEIS

1. SÔmfwna me to Je¸rhma thc Mhdenik c Parag¸gou (Je¸rhma 4.8), h exÐswshdydx = 0 èqei wc genik  lÔsh to sÔnolo twn stajer¸n sunart sewn.

2. H exÐswsh dydx = xn, ìpou n ∈ N, èqei wc genik  lÔsh to sÔnolo twn sunart sewn

thc morf c

y(x) =1

n+ 1xn+1 + C, C ∈ R, (9.2)

ìpou C ∈ R. Pr�gmati, h sun�rthsh g(x) = 1n+1x

n+1 eÐnai lÔsh thc DE, en¸ìlec oi �llec lÔseic eÐnai oi upìloipec par�gousec thc xn pou, kat� ta gnwst�gia tic par�gousec, èqoun thn morf  thc (9.2).

3. Pio genik�, an h g(x) eÐnai suneq c se k�poio di�sthma I, tìte h DE

dy

dx= g(x)

èqei wc genik  lÔsh to sÔnolo twn sunart sewn

y(x) =

∫ x

a

g(t) dt+ C. (9.3)

Pr�gmati, me qr sh tou Pr¸tou Jemeli¸douc Jewr matoc tou LogismoÔ prokÔ-ptei ìti h

∫ xa g(t) dt ikanopoieÐ thn DE, en¸ ìlec oi �llec pou thn ikanopoioÔn,

wc par�gousec thc g(x), prèpei na eÐnai thc morf c thc exÐswshc (9.3).

Par�deigma 9.2. (y′(x) = ky(x)) H exÐswsh

dy

dx= ky

èqei genik  lÔsh thny(x) = Cekx, C ∈ R.

Pr�gmati, parathr ste katarq n ìti gia aut  thn y(x) èqoume

y′(x) = (Cekx)′ = Ckekx = ky(x).

Antistrìfwc, èstw mia y(x) pou ikanopoieÐ thn DE, dhlad  y′(x) = ky(x). Parath-r ste ìti

(y(x)e−kx)′ = y′(x)e−kx − ky(x)e−kx = ky(x)e−kx − ky(x)e−kx = 0

⇒ y(x)e−kx = C ⇒ y(x) = Cekx.

H deÔterh exÐswsh proèkuye giatÐ h y(x) ex' upojèsewc ikanopoieÐ thn DE, kai hpr¸th sunepagwg  kai p�li apì to Je¸rhma Mhdenik c Parag¸gou.

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9.1. DIAFORIK�ES EXIS�WSEIS PR�WTHS T�AXEWS 207

Parathr seic

1. H DE y′(x) = ky(x) eÐnai exairetik� shmantik  giatÐ emfanÐzetai suqnìtata sthfÔsh. Sugkekrimèna, emfanÐzetai ìpote o rujmìc aÔxhshc y′(t) = dy

dt me to qrìnot, kat� th qronik  stigm  t, miac posìthtac y(t) eÐnai an�logoc thc Ðdiac thcposìthtac y(t), dhlad 

y′(t) = ky(t),

ìpou h stajer� analogÐac k mporeÐ na eÐnai jetik    arnhtik . Merik� paradeÐg-mata tètoiwn posot twn eÐnai:

(aþ) O plhjusmìc anjr¸pwn/bakthrÐwn/lag¸n/monom�qwn/k.o.k.

(bþ) H diafor� thc jermokrasÐac tou faghtoÔ apì th jermokrasÐa tou perib�l-lonta q¸ro, ìtan to bg�loume apì to foÔrno.

(gþ) To posì miac radienergoÔc posìthtac kaj¸c aut  diasp�tai.

(dþ) H taqÔthta tou autokin tou mac, an pat soume to sumplèkth.

(eþ) (Proseggistik�) Ta qr mat� mac sthn tr�peza, an stamat soume na k�noumeanal yeic   katajèseic.

(Sta �nw paradeÐgmata, h stajer� eÐnai jetik /arnhtik ? Meg�lh/mikr ?)

2. Ta prohgoÔmena dÔo paradeÐgmata mac deÐqnoun ti sumbaÐnei se dÔo eidikèc pe-ript¸seic thc sun�rthshc f(x, y), dhlad  tic f(x, y) = f(x) kai f(x, y) = y.Gia thn genik  perÐptwsh, den mporoÔme na broÔme eÔkola th morf  twn lÔsewn.Stic epìmenec dÔo paragr�fouc, ja exet�soume akìma dÔo eidikèc peript¸seic.

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208 KEF�ALAIO 9. DIAFORIK�ES EXIS�WSEIS

9.2 Grammikèc Diaforikèc Exis¸seic Pr¸thc T�xewc

Orismìc 9.2. (Grammikèc DE pr¸thc t�xewc)KaloÔme grammik  DE pr¸thc t�xewc k�je DE thc morf c

y′ + P (x)y = Q(x), (9.4)

me to x na an kei se k�poio di�sthma I.

Parat rhsh: To aristerì mèloc eÐnai sqedìn thc morf c f ′g + gf ′ = (fg)′. An tan akrib¸c thc morf c aut c, ja mporoÔsame na broÔme eÔkola th genik  lÔsh thcDE. Me th bo jeia ìmwc thc JeÐac EpifoÐthshc, parathroÔme pwc an pollaplasi�-soume to aristerì skèloc me thn posìthta exp[R(x)], ìpou R′(x) = P (x), dhlad  hR eÐnai mia opoiad pote par�gousa thc P , èqoume:

exp[R(x)](y(x)′ + P (x)y(x)) = exp[R(x)]y′(x) + exp[R(x)]P (x)y(x)

= y′(x) exp[R(x)] + y(x) exp[R(x)]R′(x) =(

exp[R(x)]y(x))′.

B�sei thc �nw parat rhshc, prokÔptei to akìloujo je¸rhma:

Je¸rhma 9.1. (Genik  kai merik  lÔsh grammik¸n DE pr¸thc t�xewc) 'Estw h DE

y′(x) + P (x)y(x) = Q(x), (9.5)

se èna di�sthma I, ìpou oi P (x), Q(x) eÐnai suneqeÐc sunart seic. 'Estw R(x) miaopoiad pote par�gousa thc P (x), dhlad  R′(x) = P (x), kai S(x) mia opoiad potepar�gousa thc Q(x) exp[R(x)], dhlad  S ′(x) = Q(x) exp[R(x)].

1. H genik  lÔsh thc DE eÐnai to sÔnolo twn sunart sewn thc morf c

y(x) = [S(x) + C] exp[−R(x)], C ∈ R. (9.6)

2. Up�rqei akrib¸c mia sun�rthsh y(x) pou ikanopoieÐ thn DE mazÐ me thn oriak sunj kh y(x0) = y0, kai aut  eÐnai h

y(x) =

{y0 +

∫ x

x0

Q(u) exp

[∫ u

x0

P (t) dt

]du

}exp

[−∫ x

x0

P (t) dt

].

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9.2. GRAMMIK�ES DIAFORIK�ES EXIS�WSEIS PR�WTHS T�AXEWS 209

Apìdeixh. 1. 'Estw y(x) paragwgÐsimh. Parathr ste pwc:

y′(x) + P (x)y(x) = Q(x)

⇔ y′(x) exp[R(x)] + P (x)y(x) exp[R(x)] = Q(x) exp[R(x)]

⇔ [y(x) exp[R(x)]]′ = S ′(x)

⇔ y(x) = [S(x) + C] exp[−R(x)], C ∈ R.

'Ara, an h y(x) ikanopoieÐ thn DE, ja eÐnai thc morf c (9.6) gia k�poio C ∈ R,en¸, antistrìfwc, an eÐnai thc morf c (9.6), tìte ja ikanopoieÐ thn DE.

2. H merik  lÔsh ja èqei th morf  (9.6) gia k�poio C, kai gia aujaÐretec par�gousecR(x) (thc P (x)) kai S(x) (thc Q(x) exp[R(x)]). QwrÐc bl�bh thc genikìthtac,paÐrnoume

R(x) =

∫ x

x0

P (t) dt, S(x) =

∫ x

x0

Q(u) exp

[∫ u

x0

P (t) dt

]du.

'Ara, h genik  lÔsh apokt� th morf 

y(x) =

{∫ x

x0

Q(u) exp

[∫ u

x0

P (t) dt

]du+ C

}exp

[−∫ x

x0

P (t) dt

].

Me antikat�stash x = x0, y(x0) = y0, prokÔptei pwc C = y0. Jètontac C = y0

sthn �nw, prokÔptei to zhtoÔmeno.

Parathr seic

1. H sunèqeia twn P (x), Q(x) qrei�zetai gia na exasfalÐsoume thn Ôparxh twnparagous¸n R(x), S(x).

2. Gia na efarmìsoume to je¸rhma, mporoÔme na epilèxoume opoiesd pote par�gou-sec S(x), R(x), jèloume.

3. Den èqoun ìlec oi DE akrib¸c mia merik  lÔsh gia k�je arqik  sunj kh. (Sanèna genikìtero sqìlio, h grammikìthta thc DE k�nei tic lÔseic na èqoun di�foraqr sima qarakthristik�, èna ek twn opoÐwn eÐnai kai autì).

4. To je¸rhma èqei thn idiaiterìthta ìti eÐnai pio eÔkolo na efarmìzoume thn apì-deixh tou autoÔsia stic ask seic, par� na jumìmaste to apotèlesma. Pr�gmati,gia thn eÔresh thc genik c lÔshc (kai endeqomènwc zhtoÔmenwn merik¸n lÔsewn),to mìno pou èqoume na k�noume, eÐnai na efarmìsoume thn akìloujh mejodologÐa,ta b mata thc opoÐac akoloujoÔn thn apìdeixh tou jewr matoc:

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210 KEF�ALAIO 9. DIAFORIK�ES EXIS�WSEIS

(aþ) An h dosmènh DE den èqei th morf  thc exÐswshc (9.4), pou suqn� kaleÐtaikanonik , thn fèrnoume se aut  th morf , pollaplasi�zontac kai ta dÔomèlh me thn kat�llhlh sun�rthsh ¸ste h y′(x) na emfanÐzetai me suntelest mon�da.

(bþ) UpologÐzoume mia par�gousa R(x) thc P (x) kai pollaplasi�zoume thn exÐ-swsh me thn exp(R(x)).

(gþ) Fèrnoume thn exÐswsh sth morf  [y(x) exp[R(x)]]′ = Q(x) exp[R(x)].

(dþ) UpologÐzoume mia par�gousa S(x) thc Q(x) exp[R(x)].

(eþ) Oloklhr¸nontac kai ta dÔo skèlh thc exÐswshc, brÐskoume th genik  lÔshy(x) = [S(x) + C] exp[−R(x)], C ∈ R.

(�þ) An mac zhteÐtai mia merik  lÔsh, antikajistoÔme sthn genik  lÔsh tic dosmè-nec timèc x0, y0, kai upologÐzoume to �gnwsto C.

DeÐte to akìloujo par�deigma gia merikèc qarakthristikèc efarmogèc aut c thcmejodologÐac.

Par�deigma 9.3. Ja broÔme tic genikèc lÔseic twn akìloujwn diaforik¸n exis¸-sewn sta diast mata pou dÐnontai, kai akoloÔjwc ja prosdiorÐsoume thn merik  touclÔsh pou ikanopoieÐ thn dosmènh arqik  sunj kh.

1. y′ + 4y = 2, ìpou x ∈ R, me arqik  sunj kh x0 = 1, y0 = 1.

2. y′ + 1xy = cosx, ìpou x > 0, me arqik  sunj kh x0 = π, y0 = 0.

3. (x + 3)y′ − 2(x2 + 3x)y = ex2

/(x + 3), ìpou x ∈ (−3,∞), me arqik  sunj khx0 = 0, y0 = 0.

LÔsh:

1. Mia antipar�gwgoc tou 4 eÐnai profan¸c h 4x, kai akolouj¸ntac thn �nw mejo-dologÐa èqoume:

y′ + 4y = 2⇔ e4xy′ + 4ye4x = 2e4x ⇔(e4xy

)′=

1

2

(e4x)′

⇔ e4xy − 1

2e4x = C ⇔ y(x) =

1

2+ Ce−4x.

An h y(x) dièrqetai apì to shmeÐo x0 = 1, y0 = 1, tìte èqoume

1 =1

2+ Ce−4 ⇒ C =

1

2e4,

kai me antikat�stash prokÔptei:

y(x) =1

2

(1 + e4(1−x)

).

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9.2. GRAMMIK�ES DIAFORIK�ES EXIS�WSEIS PR�WTHS T�AXEWS 211

2. Mia antipar�gwgoc tou 1x eÐnai h log x. (Prosoq , an eÐqame x < 0 tìte ja  tan

h log(−x).). 'Ara, prèpei na pollaplasi�soume thn exÐswsh me ton par�gontaelog x = x, dhlad :

y′ +1

xy = cosx⇔ xy′ + y = x cosx⇔ (xy)′ = x cosx

⇔ (xy)′ = [x sinx+ cosx]′ ⇔ xy = x sinx+ cosx+ C

⇔ y(x) = sin x+cosx

x+C

x.

Thn par�gousa tou x cosx thn br kame wc ex c:∫x cosx =

∫x(sinx)′ = x sinx−

∫x′ sinx

= x sinx−∫

sinx = x sinx+ cosx+ C.

Gia na broÔme th merik  lÔsh, antikajistoÔme ta x0 = π, y0 = 0 sthn genik lÔsh, kai lamb�noume

0 = sinπ +cos π

π+C

π⇒ C = 1,

�ra h merik  lÔsh eÐnai h

y(x) = sin x+cosx+ 1

x.

3. AfoÔ x > −3, mporoÔme na diairèsoume me to x+ 3, kai h exÐswsh gÐnetai h

y′ − 2xy =ex

2

(x+ 3)2,

pou eÐnai h kanonik  morf  miac grammik c DE pr¸thc t�xhc. Mia par�gousa tousuntelest  −2x eÐnai h −x2, �ra pollaplasi�zoume me thn e−x

2

, kai èqoume:

y′ − 2xy =ex

2

(x+ 3)2⇔ y′e−x

2 − 2xe−x2

y =1

(x+ 3)2

⇔ (ye−x2

)′ =

(− 1

x+ 3

)′⇔ y(x) = ex

2

(C − 1

x+ 3

).

Sqetik� me thn zhtoÔmenh merik  lÔsh, antikajist¸ntac ta x0 = 0 kai y0 = 0,lamb�noume C = 1/3, �ra telik� h merik  lÔsh eÐnai h

y(x) = ex2

(1

3− 1

x+ 3

).

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212 KEF�ALAIO 9. DIAFORIK�ES EXIS�WSEIS

9.3 DiaqwrÐsimec Diaforikèc Exis¸seic Pr¸thc T�xe-

wc

Orismìc 9.3. (DiaqwrÐsimh DE pr¸thc t�xewc) Mia DE pr¸thc t�xewc kaleÐtaidiaqwrÐsimh (  qwrizomènwn metablht¸n) ìtan eÐnai thc morf c

a(y)y′ = g(x),

me to x na an kei se k�poio di�sthma I.

Parat rhsh: Parathr ste ìti an A′ = a, tìte, apì ton kanìna thc alusÐdac,to aristerì skèloc thc exÐswshc eÐnai h par�gwgoc thc A(y(x)), opìte, an mporoÔmena broÔme kai mia antipar�gwgo thc g(x), tìte praktik� èqoume lÔsei thn DE. Pioausthr�, èqoume to akìloujo je¸rhma:

Je¸rhma 9.2. (Genik  lÔsh diaqwrÐsimwn DE pr¸thc t�xewc) 'Estw h diaqwrÐsimhDE

a(y)y′ = g(x) (9.7)

se èna di�sthma I. 'Estw pwc h g(x) èqei par�gousa G sto I, kai pwc h a èqeipar�gousa A sto dikì thc pedÐo orismoÔ. H y(x) eÐnai lÔsh thc DE ann ikanopoieÐ thnexÐswsh

A(y(x)) = G(x) + C (9.8)

sto I gia k�poia stajer� C ∈ R.

Apìdeixh. 'Estw katarq n pwc h y(x) eÐnai lÔsh. Ja èqoume pantoÔ sto I:

a(y(x))y′(x) = g(x)⇒ A′(y(x))y′(x) = G′(x)

⇒ (A(y(x)))′ = G′(x)⇒ A(y(x)) = G(x) + C,

gia k�poio C ∈ R. H deÔterh sunepagwg  prokÔptei me qr sh tou kanìna thc alusÐ-dac, en¸ h trÐth me qr sh tou Jewr matoc Mhdenik c Parag¸gou (Je¸rhma 4.8).

Antistrìfwc, an isqÔei h (9.8), tìte paragwgÐzontac kai ta dÔo mèlh prokÔptei h(9.7).

Parathr seic

1. Praktik�, an mac dÐnetai h DE a(y)y′ = g(x) kai gnwrÐzoume par�gousec A, G,

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9.3. DIAQWR�ISIMES DIAFORIK�ES EXIS�WSEIS PR�WTHS T�AXEWS 213

gia tic a, g antÐstoiqa, tìte to je¸rhma lèei pwc mporoÔme na gr�youme:

a(y)dy

dx= g(x)⇔ a(y)dy = g(x)dx

⇔∫a(y)dy =

∫g(x)dx⇔ A(y) = G(x) + C.

Parathr ste ìti to je¸rhma exasfalÐzei thn isodunamÐa thc pr¸thc exÐswshcme thn teleutaÐa. Oi endi�mesec dÔo isqÔoun kataqrhstik�, afoÔ ta dx, dy eÐnaisumbolismìc kai den mporoÔme na ta qwrÐsoume, en¸ h y eÐnai sun�rthsh kai ìqimetablht  olokl rwshc. 'Omwc, h �nw akoloujÐa twn isodunami¸n eÐnai ènacaplìc mnhmonikìc kanìnac pou mporoÔme na qrhsimopoi soume gia na jumìmastepwc leitourgeÐ to je¸rhma, kai p�nw se autìn basÐzetai h akìloujh mejodologÐa:

(aþ) Fèrnoume thn exÐswsh sth morf  a(y)dy = g(x)dx.

(bþ) Oloklhr¸noume ta dÔo mèlh, san na  tan h y mia apl  metablht  olokl rw-shc. ProkÔptei ètsi mia exÐswsh thc morf c A(y(x)) = G(x) + C.

(gþ) Efìson eÐnai dunatìn (kai den eÐnai p�nta), lÔnoume thn �nw exÐswsh wc procthn y(x).

(dþ) Se k�je perÐptwsh, h stajer� C mporeÐ na apaleifjeÐ me qr sh miac arqik csunj khc, ìpwc kai sthn perÐptwsh twn grammik¸n DE.

2. 'Otan oi DE qwrizomènwn metablht¸n den eÐnai kai grammikèc, h epÐlus  toucmporeÐ na emfanÐzei di�forec epiplokèc. Gia par�deigma:

(aþ) Endeqomènwc, h A(y(x)) = G(x) + C na mhn lÔnetai wc proc y(x).

(bþ) PolÔ suqn�, h genik  lÔsh den eÐnai ìlec oi sunart seic pou ikanopoioÔnthn A(y(x)) = G(x) + C gia k�je C ∈ R, giatÐ gia orismèna C ∈ R aut  hexÐswsh mporeÐ apl¸c na mhn ikanopoieÐtai gia k�poia x ∈ I.

(gþ) EÐnai epÐshc dunatì na mhn up�rqei C pou na exasfalÐzei thn dièleush k�poiacmerik c lÔshc apì èna shmeÐo (x0, y0), en¸ na up�rqei C pou na exasfalÐzeithn dièleush k�poiac merik c lÔshc apì èna shmeÐo (x0, y1).

Gia touc �nw lìgouc, akìma kai aplèc DE mporoÔn na èqoun sobar� probl matasthn antimet¸pis  touc. Sta plaÐsia autoÔ tou maj matoc, mac arkeÐ na mporoÔmena upologÐsoume tic sunart seic A, G kai na ft�noume sth morf  (9.8).

Par�deigma 9.4. Ja broÔme tic genikèc lÔseic twn akìloujwn diaforik¸n exi-s¸sewn sta diast mata kai me touc periorismoÔc pou dÐnontai, kai akoloÔjwc japrosdiorÐsoume thn merik  touc lÔsh pou ikanopoieÐ thn dosmènh arqik  sunj kh.

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214 KEF�ALAIO 9. DIAFORIK�ES EXIS�WSEIS

1. dydx = ex−y, ìpou x ∈ R, me arqik  sunj kh x0 = 1, y0 = 2.

2. dydx = x4√y, ìpou x, y(x) > 0, me arqik  sunj kh x0 = 2, y0 = 100, kai katìpinme arqik  sunj kh x0 = 2, y0 = 4.

3. dydx = cos2 y, ìpou x ∈ R, y(x) ∈ (−π/2, π/2), me arqik  sunj kh x0 = 0, y0 =π/4.

4. y′(x) = ex

y√y2+1

, ìpou x ∈ R, y(x) 6= 0, me arqik  sunj kh x0 = 0, y0 =√

8.

'Eqoume, kat� perÐptwsh:

1. H exÐswsh eÐnai diaqwrÐsimh kai

dy

dx= ex−y ⇔ ey dy = ex dx⇔

∫ey dy =

∫ex dx

⇔ ey = ex + C ⇔ y(x) = log(ex + C),

gia C ∈ R. AfoÔ jèloume x ∈ R, prèpei C ≥ 0. 'Ara, telik�, h genik  lÔsheÐnai h

y(x) = log(ex + C), C ≥ 0.

Sqetik� me thn merik  lÔsh, antikajist¸ntac tic timèc x0 = 1, y0 = 2, prokÔpteitelik� pwc C = e2 − e, �ra h merik  lÔsh eÐnai h

y(x) = log(ex + e2 − e).

2. H exÐswsh eÐnai diaqwrÐsimh kai

dy

dx= x4√y ⇔ y−1/2dy = x4dx.

SÔmfwna me thn �nw mejodologÐa, sto shmeÐo autì ja oloklhr¸soume kai ta dÔomèlh thc exÐswshc, k�je èna wc proc th dik  tou metablht :

y−1/2 dy = x4 dx⇔∫y−1/2dy =

∫x4 dx⇔ 2y1/2 =

x5

5+ C

⇔ 2y1/2 − x5

5= C ⇔ y(x) =

1

4(C + x5/5)2.

Kat� ta gnwst� apì th jewrÐa, h y eÐnai lÔsh, an kai mìno an ikanopoieÐ thn �nwexÐswsh gia k�poio C. Kaj¸c jèloume y(x) > 0 sto di�sthma x > 0, prèpeiC ≥ 0. 'Ara, h genik  lÔsh eÐnai h

y(x) =1

4(C + x5/5)2, C ≥ 0.

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9.3. DIAQWR�ISIMES DIAFORIK�ES EXIS�WSEIS PR�WTHS T�AXEWS 215

Sqetik� me thn pr¸th arqik  sunj kh, me antikat�stash twn shmeÐwn x0 = 2,y = 100, prokÔptei pwc C = 68/5, �ra telik� h merik  lÔsh eÐnai h

y(x) =1

4(68/5 + x5/5)2.

Sqetik� me thn deÔterh arqik  sunj kh, me antikat�stash twn shmeÐwn x0 = 2,y = 4, prokÔptei pwc C = −12/5, gia to opoÐo ìmwc den up�rqei pantoÔ stox > 0 h lÔsh! 'Ara den up�rqei merik  lÔsh pou na dièrqetai apì autì to shmeÐotou epipèdou.

3. H exÐswsh eÐnai diaqwrÐsimh kai sÔmfwna me thn �nw mejodologÐa,

dy

dx= cos2 y ⇔ dy

cos2 y= dx⇔

∫dy

cos2 y=

∫dx⇔ tan y = x+ C.

gia k�poio C ∈ R. AfoÔ den proèkuye k�poioc periorismìc sto C, telik� hgenik  lÔsh eÐnai ìlec oi y(x) oi opoÐec ikanopoioÔn thn

tan y(x) = x+ C

gia k�poio C,   isodÔnama, ìlec oi

y(x) = arctan(x+ C), C ∈ R.

Gia na broÔme thn pr¸th merik  lÔsh, k�noume antikat�stash sthn genik  lÔshtwn tim¸n x0 = 0, y0 = π/4, kai brÐskoume pwc C = 1, �ra h merik  lÔsh eÐnai h

y(x) = arctan(x+ 1).

4. Parathr ste ìti h exÐswsh eÐnai diaqwrÐsimh, kai mporeÐ na grafeÐ wc

y√y2 + 1 dy = ex dx.

Parathr ste epÐshc ìti (1

3

[1 + y2

] 32

)′= y√

1 + y2.

(EnnoeÐtai ìti h par�gwgoc eÐnai wc proc y.) SÔmfwna me thn �nw mejodologÐa,èqoume:

y√y2 + 1 dy = ex dx⇔

∫y√y2 + 1 dy =

∫ex dx⇔ 1

3

[1 + y2

] 32 = ex + C.

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216 KEF�ALAIO 9. DIAFORIK�ES EXIS�WSEIS

Epomènwc, mia y(x) eÐnai lÔsh thc DE gia x ∈ R ann ikanopoieÐ thn

1

3

[1 + y2

] 32 = ex + C

gia k�poio C ∈ R. Parathr ste ìmwc, ìti prèpei C ≥ 13 , alli¸c h �nw exÐswsh

den ikanopoieÐtai gia polÔ arnhtik� x (giatÐ to aristerì skèloc eÐnai p�nta mega-lÔtero tou 1/3.) LÔnontac pwc proc y(x), èqoume telik� pwc h genik  lÔsh thcDE eÐnai h

y(x) = ±√

[3(ex + C)]23 − 1, C ≥ 1

3.

Gia na prosdiorÐsoume th zhtoÔmenh merik  lÔsh, qrhsimopoioÔme to gegonìc ìtito gr�fhma thc y(x) dièrqetai apì to shmeÐo (0,

√8), �ra

1

3[1 + 8]

32 = e0 + C ⇒ C = 8.

'Ara telik� h zhtoÔmenh merik  lÔsh eÐnai h

y(x) =

√[3(ex + 8)]

23 − 1.

(Profan¸c prèpei na krat soume mìno to jetikì mèloc.)

Parathr seic

1. An kai se ìla ta �nw paradeÐgmata d¸same mia exÐswsh gia thn y(x), gia tican�gkec tou maj matoc, arkeÐ na mporoÔme na d¸soume thn y(x) sthn peplegmènhmorf  A(y(x)) = G(x) + C.

2. An kai den to k�name sta �nw paradeÐgmata, se ìlec tic peript¸seic, gia epal -jeush, mporoÔme na bebaiwjoÔme ìti h genik  lÔsh pou br kame ikanopoieÐ thnarqik  DE, me apl  antikat�stas  thc se aut .

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9.4. H M�EJODOS TOU EULER 217

9.4 H Mèjodoc tou Euler

Parat rhsh: Me th bo jeia twn pedÐwn dieujÔnsewn, mporoÔme na èqoume miapolÔ kal  diaisjhtik  katanìhsh opoiasd pote DE pr¸thc t�xewc, akìma kai an denlÔnetai se kleist  morf . Sugkekrimèna, an gnwrÐzoume ìti to gr�fhma thc lÔshc y(x)dièrqetai apì to shmeÐo (x0, y0), gnwrÐzoume ìti ja èqei klÐsh ekeÐ Ðsh me f(x0, y0). An,gia èna plègma shmeÐwn (x0, y0), parast soume thn klÐsh me èna di�nusma (1, f ′(x0, y0))pou deÐqnei thn dieÔjunsh pou {kineÐtai} to gr�fhma, dhmiourgoÔme èna pedÐo dieÔjun-shc me th bo jeia tou opoÐou mporoÔme na apokt soume mia eikìna gia th sumperifor�thc sun�rthshc.

Par�deigma 9.5. Sta Sq mata 9.1, 9.2, èqoume sqedi�sei (me tuqaÐa seir�) pedÐadieujÔnsewn gia tic akìloujec DE:

y′ = y, y′ = x, y′ = x− y, y′ = (1 + y2)ex.

'Eqoume epÐshc sqedi�sei merikèc endeiktikèc lÔseic. MporeÐte na breÐte to pedÐo pouantistoiqeÐ se k�je DE?

Parathr seic

1. Pwc mporoÔme na fti�xoume èna pedÐo dieÔjunshc qrhsimopoi¸ntac N koub�decme mpogi�, K puxÐdec, kai N +K anjr¸pouc?

2. Pollèc DE, akìma kai pr¸thc t�xewc, den mporoÔn an lujoÔn se kleist  morf .Pollèc �llec DE mporoÔn na lujoÔn, all� den mac endiafèrei h lÔsh touc sekleist  morf , par� mìno mia grafik  apeikìnish. Gia touc �nw lìgouc, h arij-mhtik  epÐlush diaforik¸n exis¸sewn eÐnai ter�stia perioq  èreunac. Ed¸ jadoÔme thn aploÔsterh mèjodo epÐlushc, th mèjodo tou Euler.

3. H basik  idèa thc mejìdou proèrqetai apì ta pedÐa dieÔjunshc:

(aþ) 'Estw h diaforik  exÐswsh y′ = f(x, y). 'Estw pwc jèloume na broÔmeproseggistik� mia lÔsh y(x) pou brÐsketai dexi� apì to shmeÐo (x0, y0).

(bþ) MporoÔme na upologÐsoume thn par�gwgo thc y(x) sto (x0, y0): eÐnai hy′(x0) = f(x0, y0).

(gþ) 'Estw mikrì ∆x. Tìte, kat� prosèggish, h tim  thc y(x0 + ∆x) ja isoÔtaime

y(x0 + ∆x) ' y0 + ∆x× y′(x0, y0).

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218 KEF�ALAIO 9. DIAFORIK�ES EXIS�WSEIS

−1 −0.5 0 0.5 1−1

−0.5

0

0.5

1

−1 −0.5 0 0.5 1−1

−0.5

0

0.5

1

Sq ma 9.1: PedÐa dieujÔnsewn tou ParadeÐgmatoc 9.5.

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9.4. H M�EJODOS TOU EULER 219

−1 −0.5 0 0.5 1−1

−0.5

0

0.5

1

−1 −0.5 0 0.5 1−1

−0.5

0

0.5

1

Sq ma 9.2: PedÐa dieujÔnsewn tou ParadeÐgmatoc 9.5.

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220 KEF�ALAIO 9. DIAFORIK�ES EXIS�WSEIS

(dþ) 'Ara, proseggistik�, br kame �llo èna shmeÐo, to

(x0 + ∆x, y0 + ∆x× y′(x0, y0)),

kinoÔmenoi ìqi epÐ tou graf matoc, all� epÐ thc efaptìmenhc eujeÐac se autì.

(eþ) H diadikasÐa mporeÐ na epanalhfjeÐ apì to nèo shmeÐo pou br kame.

O algìrijmoc, se yeudok¸dika, eÐnai o akìloujoc.

/* EULER METHOD */

INPUT: f(.),

Dx, /* STEP */

x(0), y(0), /* INITIAL POINT */

N /* ITERATIONS */

OUTPUT: (x(1),y(1)),(x(2),y(2)),...,(x(N),y(N))

FOR i=1:N,

x(i)=x(i-1)+Dx;

y(i)=y(i-1)+Dx*f(x(i-1),y(i-1));

END

Parathr seic

1. Me th mèjodo aut , brÐskoume èna sÔnolo shmeÐwn pou proseggÐzoun to gr�fhmathc �gnwsthc sun�rthshc, ìqi mia èkfrash gia mia sun�rthsh.

2. MporoÔme epÐshc na metakinhjoÔme proc ta arister� apì to shmeÐo (x0, y0), antÐgia ta dexi�, jètontac arnhtikì ∆x.

3. 'Ena par�deigma thc mejìdou faÐnetai sto Sq ma 9.3, gia thn exÐswsh y′ = y, mearqikèc sunj kec x0 = 1, y0 = e. 'Eqoume epÐshc sqedi�sei thn lÔsh, y(x) = ex.Epeid  èqoume epilèxei arket� meg�lo b ma, to sf�lma, met� apì mìno pènteepanal yeic tou algìrijmou, eÐnai upologÐsimo.

4. An èqoume epilèxei arket� mikrì b ma ∆x, elpÐzoume ìti to sf�lma ja eÐnaiarket� mikrì akìma kai met� apì arketì arijmì bhm�twn. MporoÔme na deÐxoumejewrhtik� ìti to ajroistikì sf�lma, dhlad  to sf�lma sthn teleutaÐa epan�lhyhtou algìrijmou eÐnai an�logo tou m kouc tou b matoc.

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9.4. H M�EJODOS TOU EULER 221

5. 'Ena �llo par�deigma faÐnetai sto Sq ma 9.4, ìpou èqei lujeÐ arijmhtik� hy′ = cosx sto di�sthma [0, 10], gia arqik  sunj kh x0 = 0, y0 = 0, kai giameioÔmeno mègejoc b matoc. 'Eqoume epÐshc sqedi�sei thn akrib  lÔsh, pou mpo-reÐ na upologisteÐ se kleist  morf . (Poia eÐnai?).

6. H mèjodoc genikeÔetai kai gia �lla eÐdh DE, gia par�deigma DE pou perilamb�-noun parag¸gouc p�nw apì pr¸thc t�xhc.

7. H mèjodoc eÐnai polÔ apl  all� me mètria epÐdosh, kai up�rqoun polÔ kalÔterec.Gia par�deigma, h mèjodoc Runge-Kutta tètarthc t�xewc èqei ajroistikì sf�lmaan�logo thc tètarthc dÔnamhc tou b matoc, kai ìqi thc pr¸thc dÔnamhc, ìpwc hmèjodoc pou deÐxame.

8. Perissìtera se maj mata Arijmhtik c An�lushc.

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222 KEF�ALAIO 9. DIAFORIK�ES EXIS�WSEIS

0 0.5 1 1.5 2

1

2

3

4

5

6

7

54

32

1

0

1

2

3

4

5

x

y

y(x) = ex

Sq ma 9.3: Arijmhtik  epÐlush thc DE y′(x) = y(x), me arqikèc sunj kec x0 = 1, y0 = e, stodi�sthma [0, 2]. 'Eqoume ektelèsei 5 epanal yeic proc ta arister�, kai 5 proc ta dexi�. Parathr steìti kaj¸c aux�noun oi epanal yeic, aux�nei kai to sf�lma an�mesa sthn sun�rthsh pou upologÐzoumearijmhtik� kai thn pragmatik  lÔsh y(x) = ex.

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9.4. H M�EJODOS TOU EULER 223

0 2 4 6 8 10

−1

0

1 Δx = 1

0 2 4 6 8 10

−1

0

1 Δx = 0.5

0 2 4 6 8 10

−1

0

1 Δx = 0.2

0 2 4 6 8 10

−1

0

1Δx = 0.1

Sq ma 9.4: Arijmhtik  epÐlush thc DE y′ = cosx, me arqikèc sunj kec x0 = 0, y0 = 0, kai giadi�fora megèjh bhm�twn, sto di�sthma [0, 10]. Parathr ste ìti kaj¸c mikraÐnei to mègejoc toub matoc, mei¸netai kai to sf�lma an�mesa sthn sun�rthsh pou upologÐzoume arijmhtik� (kai pouèqoume sqedi�sei me diakekommènh gramm ) kai thn pragmatik  lÔsh y = sinx (pou èqoume sqedi�seime suneqìmenh gramm ).

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224 KEF�ALAIO 9. DIAFORIK�ES EXIS�WSEIS

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Kef�laio 10

Polu¸numo Taylor

10.1 Orismìc kai Basikèc Idiìthtec

Orismìc 10.1. (Polu¸numo Taylor) 'Estw f gia thn opoÐa up�rqoun oi par�gwgoimèqri kai n t�xhc sto shmeÐo a. OrÐzoume to polu¸numo Taylor bajmoÔ n sth jèsh awc to

Pn,a(x) , f(a) + f ′(a)(x− a) +f ′′(a)

2(x− a)2 + · · ·+ f (n)(a)

n!(x− a)n

, a0 + a1(x− a) + a2(x− a)2 + · · ·+ an(x− a)n.

ìpou

ak ,f (k)(a)

k!, 0 ≤ k ≤ n.

Parathr seic

1. Ston orismì tou suntelest  a0, ennoeÐtai pwc 0! , 1, f (0) , f .

2. Sthn perÐptwsh a = 0, to polu¸numo kaleÐtai kai polu¸numo Maclaurin.

3. Merikèc forèc paraleÐpoume to a kai gr�foume Pn(x).

4. 'Otan jèloume na anafèroume rht¸c thn sun�rthsh f apì thn opoÐa to polu¸numoproèrqetai, to gr�foume wc Pn,a,f(x).

5. Apì thn kataskeu  tou, to polu¸numo Taylor tautÐzetai me th sun�rthsh sthjèsh a, kai epiplèon èqei parag¸gouc mèqri kai n bajmoÔ Ðsec me tic antÐstoiqec

225

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226 KEF�ALAIO 10. POLU�WNUMO TAYLOR

parag¸gouc thc sun�rthshc f sto shmeÐo a. Pr�gmati,

P (0)n,a(a) =

[f(a) + f ′(a)(x− a) + · · ·+ f (n)(a)

n!(x− a)n

]x=a

= f(a),

P (1)n,a(a) =

[f ′(a) + f ′′(a)(x− a) +

f ′′′(a)

2(x− a)2 + . . .

· · ·+ f (n)(a)

(n− 1)!(x− a)n−1

]x=a

= f ′(a),

P (2)n,a(a) =

[f ′′(a) + f ′′′(a)(x− a) +

f (4)(a)

2(x− a)2 + . . .

· · ·+ f (n)(a)

(n− 2)!(x− a)n−2

]x=a

= f ′′(a),

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

P (n)n,a (a) = f (n)(a).

6. EÐnai eÔkolo na deÐxoume ìti dÔo polu¸numa bajmoÔ ≤ n pou èqoun Ðsec parag¸-gouc t�xhc 0 èwc n se èna shmeÐo a tautÐzontai,dhlad  ìloi oi suntelestèc touceÐnai Ðsoi. (MporeÐte na deÐxete tic leptomèreiec?) 'Ara, to polu¸numo Taylormiac sun�rthshc f se èna shmeÐo a eÐnai to mìno polu¸numo t�xhc ≤ n me thnidiìthta na tautÐzontai oi par�gwgoÐ tou apì 0 mèqri n t�xhc me tic antÐstoiqecthc sun�rthshc se ekeÐno to shmeÐo.

7. Parathr ste ìti gia n = 1 to polu¸numo Taylor tautÐzetai me thn efaptìmenheujeÐa thc f sto shmeÐo a.

8. Epeid  to polu¸numo Taylor bajmoÔ n thc sun�rthshc f kai h sun�rthsh fèqoun Ðsec parag¸gouc, eÐnai anamenìmeno ìti to polu¸numo ja proseggÐzei thnsun�rthsh kont� sto a, kai m�lista ìso megalÔtero eÐnai to n, tìso kalÔterhja eÐnai h prosèggish. Pr�gmati, deÐte to Sq ma 10.1, ìpou èqoume sqedi�seitic expx, sinx kai ta antÐstoiqa polu¸numa Taylor, sto shmeÐo a = 0, diafìrwnbajm¸n. Parathr ste ìti ìso megal¸nei to n, tìso megal¸nei h akrÐbeia thcprosèggishc. Sthn epìmenh par�grafo ja asqolhjoÔme me to pìso kal  eÐnai hprosèggish.

Par�deigma 10.1. Ja deÐxoume ìti ta polu¸numa Taylor twn sunart sewn expx,

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10.1. ORISM�OS KAI BASIK�ES IDI�OTHTES 227

0 0.5 1 1.5 2 2.5 30

2

4

6

8

10

12

14

16

18

20

x

n = 1

n = 2

n = 3

n = 4

n = 5

n = 0

ex

0 1 2 3 4 5 6−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

3

x

n = 3, 4

n = 5, 6

n = 7, 8

n = 9, 10

sin x

n = 1, 2

Sq ma 10.1: Polu¸numa Taylor di�forwn bajm¸n gia tic sunart seic expx, sinx.

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228 KEF�ALAIO 10. POLU�WNUMO TAYLOR

log(x+ 1), 11−x , sinx, kai cosx, sth jèsh a = 0, eÐnai:

Pn,0,expx(x) = 1 + x+x2

2!+x3

3!+ · · ·+ xn

n!,

Pn,0,log(x+1)(x) = x− x2

2+x3

3− x4

4+ · · ·+ (−1)n+1x

n

n,

Pn,0,(1−x)−1(x) = 1 + x+ x2 + x3 + · · ·+ xn,

P2n+1,0,sinx(x) = x− x3

3!+x5

5!− x7

7!+ · · ·+ (−1)n

x2n+1

(2n+ 1)!,

P2n+2,0,sinx(x) = P2n+1,0,sinx(x),

P2n,0,cosx(x) = 1− x2

2!+x4

4!− x6

6!+ · · ·+ (−1)n

x2n

(2n)!,

P2n+1,0,cosx(x) = P2n,0,cosx(x).

'Eqoume, kat� perÐptwsh:

1. Gia thn f(x) = expx profan¸c èqoume f (n)(x) = expx ⇒ f (n)(0) = 1, �raeÔkola prokÔptei to zhtoÔmeno.

2. Gia na upologÐsoume to polu¸numo thc f(x) = log(x + 1) parathroÔme ìti gian ≥ 1,

f (n)(x) =(−1)n+1(n− 1)!

(1 + x)n⇒ f (n)(0) = (−1)n+1(n− 1)!

H �nw sqèsh mporeÐ na apodeiqjeÐ me epagwg . Pr�gmati, isqÔei gia n = 1,afoÔ qrhsimopoioÔme thn sÔmbash 0! = 1, en¸ an upojèsoume ìti isqÔei gia n,me parag¸gish prokÔptei �mesa ìti ja isqÔei gia n + 1. Efarmìzontac thn �nwston orismì tou polu¸numou Taylor, prokÔptei telik� to zhtoÔmeno.

3. Gia na upologÐsoume to polu¸numo Taylor thc f(x) = 11−x parathroÔme ìti gia

n ≥ 0,

f (n)(x) =n!

(1− x)n+1⇒ f (n)(0) = n!.

H �nw sqèsh mporeÐ na apodeiqjeÐ me epagwg . Pr�gmati, isqÔei gia n = 0, en¸an upojèsoume ìti isqÔei gia n, me parag¸gish prokÔptei �mesa ìti ja isqÔei gian+ 1. Efarmìzontac thn �nw ston orismì tou polu¸numou Taylor prokÔptei tozhtoÔmeno.

4. Sthn perÐptwsh tou hmitìnou, me epagwg  mporoÔme na deÐxoume ìti isqÔei, giak ≥ 0,

sin(2k) x = (−1)k sinx⇒ sin(2k) 0 = 0,

sin(2k+1) x = (−1)k cosx⇒ sin(2k+1) 0 = (−1)k.

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10.1. ORISM�OS KAI BASIK�ES IDI�OTHTES 229

Efarmìzontac tic �nw exis¸seic ston orismì tou polu¸numou Taylor, prokÔpteito zhtoÔmeno.

5. Gia na upologÐsoume to polu¸numo thc cosx parathroÔme ìti, me epagwg , mpo-roÔme na deÐxoume ìti isqÔei, gia k ≥ 0,

cos(2k) x = (−1)k cosx⇒ cos(2k) 0 = 1,

cos(2k+1) x = (−1)k+1 sinx⇒ cos(2k+1) 0 = 0,

�ra eÔkola prokÔptei to zhtoÔmeno.

Je¸rhma 10.1. (Logismìc me polu¸numa Taylor) Ta polu¸numa Taylor èqoun ticakìloujec idiìthtec:

1. To polu¸numo Taylor tou grammikoÔ sunduasmoÔ dÔo sunart sewn isoÔtai meton grammikì sunduasmì twn poluwnÔmwn Taylor twn sunart sewn: an oi c1, c2

stajerèc, tìte

Pn,a,c1f1+c2f2(x) = c1Pn,a,f1(x) + c2Pn,a,f2(x).

2. H par�gwgoc enìc polu¸numou Taylor eÐnai èna polu¸numo Taylor thc parag gou:

(Pn,a,f(x))′ = Pn−1,a,f ′(x).

3. To olokl rwma enìc polu¸numou Taylor eÐnai èna polu¸numo Taylor tou oloklh-r¸matoc: 'Estw g(x) =

∫ xa f . Tìte:∫ x

a

Pn,a,f(x) = Pn+1,a,g(x).

4.Pn,a,f(cx)(x) = Pn,ca,f(x)(cx),

dhlad  to polu¸numo Taylor thc f(cx) sth jèsh a prokÔptei apì to polu¸numoTaylor thc f(x) sth jèsh ca.

Apìdeixh. H apìdeixh eÐnai apl  kai paraleÐpetai. Se ìlec tic peript¸seic arkeÐ nasugkrÐnoume to polu¸numo tou aristeroÔ mèlouc gia na doÔme pwc eÐnai Ðdio me topolu¸numo tou dexioÔ mèlouc.

Par�deigma 10.2. To �nw je¸rhma mac epitrèpei na brÐskoume kainoÔrgia polu¸-numa Taylor, qrhsimopoi¸ntac aut� pou gnwrÐzoume  dh. Gia par�deigma, gnwrÐzoume

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230 KEF�ALAIO 10. POLU�WNUMO TAYLOR

ìti

Pn,0,exp(x)(x) = 1 + x+x2

2!+x3

3!+ · · ·+ xn

n!,

�ra, efarmìzontac to teleutaÐo skèloc tou jewr matoc gia a = 0, c = −1, èqoume:

Pn,0,exp(−x)(x) = Pn,0,exp(x)(−x) = 1− x+x2

2!− x3

3!+ · · ·+ (−x)n

n!.

San èna �llo par�deigma, èstw h sun�rthsh uperbolikoÔ sunhmitìnou (pou emfa-nÐzetai suqn�, se lÔseic diaforik¸n exis¸sewn, ektìc twn �llwn):

cosh(x) ,ex + e−x

2.

Tìte, apì to pr¸to skèloc tou jewr matoc, èqoume

Pn,0,cosh(x)(x) =1

2Pn,0,exp(x)(x) +

1

2Pn,0,exp(−x)(x)

= 1 +x2

2!+x4

4!+ · · ·+ x2n

(2n)!.

Parìmoia orÐzetai kai to uperbolikì hmÐtono:

sinh(x) ,ex − e−x

2,

gia to opoÐo, entel¸c an�loga, prokÔptei

Pn,0,sinh(x)(x) =1

2Pn,0,exp(x)(x)− 1

2Pn,0,exp(−x)(x)

= 1 + x+x3

3!+x5

5!+ · · ·+ x2n+1

(2n+ 1)!.

Tèloc, paragwgÐzontac thn

Pn,0,log(x+1)(x) = x− x2

2+x3

3− x4

4+ · · ·+ (−1)n+1x

n

n,

prokÔptei pwc

Pn−1,a,1/(x+1)(x) =(Pn,a,log(x+1)(x)

)′= 1− x+ x2 − x3 + · · ·+ (−1)n−1xn−1.

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10.2. UP�OLOIPO 231

10.2 Upìloipo

Je¸rhma 10.2. (AkrÐbeia thc prosèggishc sto ìrio)An f eÐnai mia sun�rthsh me parag gouc n t�xhc sto a, kai Pn,a(x) to antÐstoiqopolu¸numo Taylor n bajmoÔ gÔrw apì to a, tìte:

limx→a

f(x)− Pn,a(x)

(x− a)n= 0. (10.1)

Epiplèon, to Pn,a(x) eÐnai to monadikì polu¸numo bajmoÔ k ≤ n me aut  thn idiìthta,dhlad  an

limx→a

f(x)−Q(x)

(x− a)n= 0 (10.2)

me Q(x) polu¸numo bajmoÔ mikrìterou   Ðsou tou n, tìte

Q(x) = Pn,a(x).

Apìdeixh. Pr¸ta ja apodeÐxoume thn (10.1), kai xekin�me parathr¸ntac pwc to ìrioparousi�zei aprosdioristÐa thc morf c 0/0, afoÔ profan¸c lim

x→a(x − a)n = 0, en¸

epiplèon

limx→a

(f(x)− Pn,a(x)) =

limx→a

f(x)− limx→a

[f(a) + f ′(a)(x− a) + · · ·+ f (n)(a)

n!(x− a)n

]= f(a)− f(a) = 0.

(Sth deÔterh isìthta, qrhsimopoi same th sunèqeia thc f(x) pou exasfalÐzetai a-pì thn Ôparxh thc parag¸gou f ′(a).) 'Ara, mporoÔme na efarmìzoume ton KanìnaL’Hopital. ParathroÔme ìmwc pwc an i < n, tìte

[(x− a)n](i) = n(n− 1) . . . (n− i+ 1)(x− a)n−i → 0, kaj¸c x→ a.

Sqetik� me ton arijmht , eÔkola prokÔptei ìti gia i < n,

[f(x)− Pn,a(x)](i) =

[f(x)−

[f(a) + f ′(a)(x− a) + · · ·+ f (n)(a)

n!(x− a)n

]](i)

= f (i)(x)

−[f (i)(a) + f (i+1)(a)(x− a) + · · ·+ f (n)(a)

(x− a)n−i

(n− i)!

]→ f (i)(a)− f (i)(a) = 0, kaj¸c x→ a.

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232 KEF�ALAIO 10. POLU�WNUMO TAYLOR

Parathr ste pwc kai p�li qrhsimopoi same th sunèqeia thc f i sto a, pou exasfalÐze-tai apì thn Ôparxh thc f i+1. 'Ara, o Kanìnac L’Hopital mporeÐ na efarmosteÐ n− 1forèc, katal gontac sto

limx→a

f(x)− Pn,a(x)

(x− a)n= lim

x→af (n−1)(x)− f (n−1)(a)− f (n)(a)(x− a)

n!(x− a)

= limx→a

f (n−1)(x)− f (n−1)(a)

n!(x− a)− lim

x→af (n)(a)

n!

=f (n)(a)

n!− f (n)(a)

n!= 0.

H trÐth isìthta prokÔptei apì ton orismì thc f (n) sto x = a.Ja deÐxoume t¸ra ìti to Pn,a(x) eÐnai to monadikì polu¸numo bajmoÔ ≤ n me aut 

thn idiìthta. 'Estw pwc to Pn,a(x) den eÐnai monadikì, isqÔei dhlad  h (10.2) gia k�poioQ me bajmì ≤ n. Apì tic (10.1) kai (10.2) prokÔptei pwc

limx→a

Pn,a(x)−Q(x)

(x− a)n= 0⇒ lim

x→aR(x)

(x− a)n= 0,

ìpou R(x) = Pn,a(x) − Q(x) polu¸numo bajmoÔ ≤ n di�foro tou mhdenikoÔ. 'Estwpwc

R(x) = rk(x− a)k + · · ·+ rl(x− a)l,

ìpou 0 ≤ k ≤ l ≤ n, kai rk, rl 6= 0. Dhlad , o ìroc mikrìterhc t�xhc eÐnai o rk(x−a)k,kai o ìroc megalÔterhc t�xhc eÐnai o rl(x− a)l. (Endeqomènwc ta k, l na tautÐzontai.)'Ara ja prèpei

limx→a

rk(x− a)k + · · ·+ rl(x− a)l

(x− a)n= 0.

Epeid  limx→a

(x− a)n−k = 0 (ìtan k < n)   limx→a

(x− a)n−k = 1 (ìtan k = n), mporoÔme

na pollaplasi�soume to phlÐko entìc tou �nw orÐou me (x− a)n−k, gia na l�boume

limx→a

rk(x− a)n + rk+1(x− a)n+1 + · · ·+ rl(x− a)n−k+l

(x− a)n= 0⇒

limx→a

[rk + rk+1(x− a) + · · ·+ rl(x− a)l−k

]= 0⇒ rk = 0,

pou eÐnai �topo. 'Ara, prèpei tautotik� R(x) = 0, dhlad  to Pn,a(x) eÐnai monadikì.

Parathr seic

1. To �nw je¸rhma epibebai¸nei ìti pr�gmati ta polu¸numa Taylor proseggÐzounmia sun�rthsh kaj¸c x → a, kai prosdiorÐzei kai to pìso kal  eÐnai aut  hprosèggish, gia sugkekrimèno n.

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10.2. UP�OLOIPO 233

2. Epiplèon, deÐqnei ìti h prosèggish eÐnai h kalÔterh dunat , an perioristoÔme sepolu¸numa tou Ðdiou bajmoÔ n.

3. 'Oso pio meg�lo eÐnai to n, tìso pio gr gora p�ei to n sto 0, �ra kai tìsokalÔterh prèpei na eÐnai h prosèggish an�mesa sth sun�rthsh kai to polu¸numoTaylor.

Orismìc 10.2. (Upìloipo) Wc upìloipo,   sf�lma, tou polu¸numou Taylor orÐzou-me thn posìthta

Rn(x) , f(x)− Pn,a,f(x)⇔ f(x) =n∑k=0

f (k)(a)

k!(x− a)k +Rn(x).

Je¸rhma 10.3. (Je¸rhma Taylor)

1. 'Estw pwc h sun�rthsh f èqei suneqeÐc parag gouc mèqri kai n+1 t�xewc pantoÔsto kleistì di�sthma I me �kra x, a, ìpou x 6= a. (Dhlad , I = [a, x] an a < x,  I = [x, a], an a > x.) IsqÔei o akìloujoc tÔpoc gia to sf�lma, sth jèsh x,tou polu¸numou Taylor Pn,a(x) gÔrw apì to shmeÐo a:

Rn(x) =1

n!

∫ x

a

(x− t)nf (n+1)(t) dt.

2. An isqÔei to fr�gma|f (n+1)(t)| ≤M, ∀t ∈ I,

tìte isqÔei kai to fr�gma

|Rn(x)| ≤M|x− a|n+1

(n+ 1)!.

Apìdeixh. 1. Ja qrhsimopoi soume epagwg  sto n, sunep¸c to pr¸to b ma eÐnai oupologismìc tou Rn(x) gia n = 1. Pr�gmati:

R1(x) = f(x)− f(a)− (x− a)f ′(a) =

∫ x

a

f ′(t) dt− f ′(a)

∫ x

a

dt

=

∫ x

a

[f ′(t)− f ′(a)] dt =

∫ x

a

(t− x)′[f ′(t)− f ′(a)] dt

= [(t− x)[f ′(t)− f ′(a)]]xa −

∫ x

a

(t− x)f ′′(t) dt

=

∫ x

a

(x− t)f ′′(t) dt.

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234 KEF�ALAIO 10. POLU�WNUMO TAYLOR

Sthn 5h isìthta qrhsimopoi same paragontik  olokl rwsh. 'Estw t¸ra pwcisqÔei to je¸rhma gia n. Ja apodeÐxoume ìti isqÔei kai gia n+ 1. Pr�gmati:

Rn+1(x) = Rn(x)− f (n+1)(a)

(n+ 1)!(x− a)n+1

=1

n!

∫ x

a

(x− t)nf (n+1)(t) dt− f (n+1)(a)

n!

∫ x

a

(x− t)n dt

=1

n!

∫ x

a

(x− t)n[f (n+1)(t)− f (n+1)(a)

]dt

=1

n!

∫ x

a

[−(x− t)n+1

n+ 1

]′ [f (n+1)(t)− f (n+1)(a)

]dt

=

[−(x− t)n+1

(n+ 1)!

[f (n+1)(t)− f (n+1)(a)

]]xa

+1

(n+ 1)!

∫ x

a

(x− t)n+1f (n+2)(t) dt

=1

(n+ 1)!

∫ x

a

(x− t)n+1f (n+2)(t) dt.

Sthn deÔterh isìthta qrhsimopoi same thn epagwgik  upìjesh, kai ton tÔpo∫ x

a

(x− t)n dt =(x− a)n+1

n+ 1.

Sthn pèmpth isìthta qrhsimopoi same kai p�li paragontik  olokl rwsh.

2. H apìdeixh autoÔ tou skèlouc eÐnai apl  kai paraleÐpetai.

Parat rhsh: To je¸rhma eÐnai polÔ shmantikì giatÐ mac epitrèpei na prosdiorÐ-soume to sf�lma pou k�noume ston upologismì thc tim c miac sun�rthshc qrhsimo-poi¸ntac to polu¸numo Taylor thc. DeÐte to parak�tw par�deigma:

Par�deigma 10.3. Ja upologÐsoume ta akìlouja, qrhsimopoi¸ntac apokleistik�tic tèsseric basikèc pr�xeic metaxÔ akeraÐwn:

1. To√e, me sf�lma mikrìtero tou 10−4.

2. To cos 12 , me sf�lma mikrìtero tou 10−5.

'Eqoume, kat� perÐptwsh:

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10.2. UP�OLOIPO 235

1. Efarmìzoume to Je¸rhma Taylor me f(x) = ex, a = 0 kai x = 12 . JewroÔme

gnwstì oti e < 4. Me dedomèno ìti ìlec oi par�gwgoi thc f(x) isoÔntai mef (n)(x) = ex, kai pwc sto

[0, 1

2

]ex ≤ e

12 < 2,

prokÔptei pwc

|Rn(t)| ≤ 2

(1

2

)n+1/(n+ 1)!.

Me antikatast�seic gia di�fora n, prokÔptei pwc to �nw fr�gma sto sf�lmagÐnetai mikrìtero tou 10−4 gia n = 5. 'Ara, prokÔptei pwc√e ' f(0) + f ′(0)x+

1

2f ′′(0)x2 +

1

3!f ′′′(0)x3 +

1

4!f (4)(x) +

1

5!f (5)(x)

= 1 +1

2+

1

2

(1

2

)2

+1

3!

(1

2

)3

+1

4!

(1

2

)4

+1

5!

(1

2

)5

= 887/538 = 1.648697916666666 . . .

An k�noume ton upologismì me hlektronikì upologist  kai akrÐbeia 15 dekadik�,√e ' 1.648721270700128,

epomènwc to sf�lma eÐnai perÐpou 2× 10−5.

2. Se aut  thn perÐptwsh, efarmìzoume to Je¸rhma Taylor me f(x) = cos x, a = 0kai x = 1

2 . Me dedomèno ìti ìlec oi par�gwgoi thc f(x) eÐnai fragmènec apì thmon�da, prokÔptei

|Rn(t)| ≤ 1×(

1

2

)n+1

/(n+ 1)!.

Me antikatast�seic gia di�fora n, prokÔptei pwc to sf�lma gÐnetai mikrìterotou 10−5 gia n = 6. 'Ara, prokÔptei pwc

cos

(1

2

)' f(0) +

1

2f ′′(0)x2 +

1

4!f (4)(x)x4 +

1

6!f (6)(x)x6

= 1− 1

2

(1

2

)2

+1

4!

(1

2

)4

− 1

6!

(1

2

)6

= 1147/1307 = 0.87758246527777 . . .

An k�noume ton upologismì me H/U kai akrÐbeia 15 dekadik�,

cos

(1

2

)' 0.877582561890373,

epomènwc to sf�lma eÐnai thc t�xhc tou 2 × 10−7, arket� mikrìtero apì to zh-toÔmeno. (MporeÐte na d¸sete mia ex ghsh?)

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236 KEF�ALAIO 10. POLU�WNUMO TAYLOR

Page 247: ΣΗΜΕΙΩΣΕΙΣ-ΜΑΘΗΜΑΤΙΚΑ 1 ΠΛΗΡΟΦΟΡΙΚΗ ΑΣΟΕΕ 1 ΕΞΑΜΗΝΟ

Kef�laio 11

Seirèc

11.1 Seirèc

Orismìc 11.1. (Seirèc) 'Estw {an} akoloujÐa. H nèa akoloujÐa sn =∑n

i=1 ai twnmerik¸n ajroism�twn kaleÐtai (�peirh) seir�, kai sumbolÐzetai enallaktik� wc:

a1 + a2 + . . . ,

∞∑n=1

an,∑

an

An mia seir�∑an sugklÐnei se k�poio ìrio, tìte kaleÐtai sugklÐnousa, h antÐstoiqh

akoloujÐa an kaleÐtai ajroÐsimh, en¸ kaloÔme to ìrio limn→∞

∑nk=1 ak �jroisma thc an,

kai to sumbolÐzoume wc∞∑k=1

ak , limn→∞

n∑k=1

ak. (11.1)

An h seir� den sugklÐnei, kaleÐtai apoklÐnousa.

Parathr seic

1. Parathr ste ìti me∑∞

n=1 an sumbolÐzoume, kataqrhstik�, kai th seir� kai toìriì thc, efìson autì up�rqei.

2. Poia apì tic akìloujec seirèc sugklÐnei?

1− 1 + 1− 1 + 1− 1 + 1− . . .1 +

1

2+

1

22+

1

23+

1

24+

1

25+ . . .

1 +1

2+

1

3+

1

4+

1

5+

1

6+ . . .

Sth sunèqeia ja melet soume trìpouc me touc opoÐouc mporoÔme na diapist¸sou-me thn sÔgklish   ìqi seir¸n.

237

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238 KEF�ALAIO 11. SEIR�ES

3. Mia apoklÐnousa seir� endeqomènwc na teÐnei sto ∞   sto −∞. Enallaktik�,parousi�zei {talant¸seic} ep' �peiron.

Par�deigma 11.1. (H gewmetrik  seir�) 'Estw h gewmetrik  prìodoc

an = rn

kai h gewmetrik  seir�∞∑n=1

rn = r + r2 + r3 + . . .

ParathroÔme pwc to merikì �jroisma

sn = r + r2 + r3 + · · ·+ rn =

{r−rn+1

1−r , r 6= 1,

n, r = 1.

H isìthta gia thn perÐptwsh r 6= 1 prokÔptei qrhsimopoi¸ntac thn gnwst  tautìthta

an − bn = (a− b)(an−1 + an−2b+ · · ·+ abn−2 + bn−1),

gia a = 1 kai b = r, kai pollaplasi�zontac me r. DiakrÐnoume peript¸seic:

1. 'Otan |r| < 1, limn→∞

rn+1 = 0, �ra h gewmetrik  seir� sugklÐnei sto

∞∑n=1

rn =r

1− r .

2. 'Otan r > 1 to rn+1 teÐnei sto ∞, �ra h seir� teÐnei sto ∞.

3. 'Otan r < −1, to rn+1 apoklÐnei (gia thn akrÐbeia oi perittoÐ ìroi p�ne sto ∞kai oi �rtioi sto −∞), �ra apoklÐnei kai h seir�.

4. 'Otan r = 1, èqoume sn = n, pou teÐnei sto ∞.

5. 'Otan r = −1, èqoume

sn =n∑i=1

(−1)n+1 =

{1, n = 2k + 1,

0, n = 2k,

pou apoklÐnei.

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11.1. SEIR�ES 239

Par�deigma 11.2. (H armonik  seir�) 'Estw h armonik  seir�

∞∑n=1

1

n= 1 +

1

2+

1

3+

1

4+ . . .

Ja deÐxoume ìti h armonik  seir� apoklÐnei, kai m�lista me mia stoiqeiwdèstath apì-deixh.

ParathroÔme pwc:∞∑n=1

1

n= 1

+1

2

+1

3+

1

4

+1

5+

1

6+

1

7+

1

8

+1

9+

1

10+

1

11+

1

12+

1

13+

1

14+

1

15+

1

16+ . . .

Parathr ste ìti k�je gramm  èqei �jroisma toul�qiston 12 , kai mporoÔme na dhmiour-

g soume �peirec tètoiec grammèc, diplasi�zontac se k�je epan�lhyh ton arijmì twnìrwn pou qrhsimopoioÔme. 'Ara, profan¸c to �jroisma eÐnai ∞.

Par�deigma 11.3. (Thleskopikèc seirèc) 'Estw h akoloujÐa bn, h akoloujÐa an =bn − bn+1, kai h seir�

∑an =

∑(bn − bn+1). ParathroÔme pwc (upojètontac ìti ta

ìria pou gr�foume sth sunèqeia up�rqoun):∞∑k=1

ak = limn→∞

n∑k=1

ak = limn→∞

n∑k=1

(bk − bk+1)

= limn→∞

[b1 − b2 + b2 − b3 + · · ·+ bn − bn+1] = limn→∞

[b1 − bn+1]

= b1 − limn→∞

bn+1 = b1 − limn→∞

bn.

'Ara, h seir�∑

(bn − bn+1) sugklÐnei ann sugklÐnei h bn, kai an sugklÐnoun isqÔei∞∑n=1

an = b1 − limn→∞

bn.

San par�deigma, efarmìzontac ta �nw gia an = 1n(n+1) kai bn = 1

n , èqoume

∞∑n=1

1

n(n+ 1)=

∞∑n=1

(1

n− 1

n+ 1

)=

1

1− lim

n→∞1

n= 1.

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240 KEF�ALAIO 11. SEIR�ES

Je¸rhma 11.1. (Apokop  peperasmènou arijmoÔ ìrwn) 'Estw k ∈ N. H∑∞

n=1 ansugklÐnei ann h

∑∞n=1 an+k sugklÐnei.

Apìdeixh. ParaleÐpetai. ProkÔptei �mesa me qr sh tou orismoÔ tou orÐou akoloujÐac.

Je¸rhma 11.2. (GrammikoÐ sunduasmoÐ seir¸n) 'Estw dÔo seirèc∑an,

∑bn kai

λ1, λ2 stajerèc. An sugklÐnoun kai oi dÔo seirèc, tìte sugklÐnei kai h∑(λ1an + λ2bn) = (λ1a1 + λ2b1) + (λ1a2 + λ2b2) + . . .

kai èqoume thn akìloujh isìthta orÐwn:

∞∑i=1

(λ1ai + λ2bi) = λ1

∞∑i=1

ai + λ2

∞∑i=1

bi.

Apìdeixh. ParathroÔme pwc:

n∑i=1

(λ1ai + λ2bi) = λ1

n∑i=1

ai + λ2

n∑i=1

bi.

PaÐrnontac to ìrio n→∞, prokÔptei to zhtoÔmeno.

Je¸rhma 11.3. (AnagkaÐo krit rio Cauchy) An mia seir�∑an sugklÐnei tìte

limn→∞

an = 0.

Apìdeixh. An h∑an sugklÐnei sto L, tìte ja sugklÐnei kai h

∑an+1, profan¸c sto

L− a1. 'Ara, ja èqoume kai∑(an − an+1) =

∑an −

∑an+1 → L− (L− a1) = a1.

to ìrio prokÔptei apì to Je¸rhma 11.2. Parathr ste ìmwc pwc∑

(an − an+1) =a1 − an+1, �ra telik� an+1 → 0, kai telik� an → 0.

Par�deigma 11.4. H seir�∑

nn+2 apoklÐnei, afoÔ h akoloujÐa an = n

n+2 profan¸cden teÐnei sto 0.

Par�deigma 11.5. H armonik  seir� ikanopoieÐ to krit rio all� apoklÐnei. 'Ara tokrit rio eÐnai mìno anagkaÐo, kai ìqi ikanì.

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11.2. SEIR�ES MH ARNHTIK�WN �ORWN 241

11.2 Seirèc mh Arnhtik¸n 'Orwn

Je¸rhma 11.4. (Krit rio sÔgkrishc) 'Estw dÔo akoloujÐec an, bn ≥ 0. An up�r-qoun c ∈ R, c > 0, kai n0 ∈ N, tètoia ¸ste

∀n > n0, an ≤ cbn, (11.2)

tìte an sugklÐnei h∑bn, sugklÐnei kai h

∑an.

Apìdeixh. Katarq n parathr ste ìti, apì to Je¸rhma 2.10, mia seir� mh arnhtik¸nìrwn (kai �ra me aÔxousa akoloujÐa merik¸n ajroism�twn), sugklÐnei an kai mìno anh akoloujÐa twn merik¸n ajroism�twn thc eÐnai fragmènh.

AkoloÔjwc, parathr ste ìti an sugklÐnei h∑bn, ja sugklÐnei kai h

∑bn+n0

(Je¸rhma 11.1), �ra ja eÐnai fragmèna ta merik� ajroÐsmat� thc. 'Omwc, apì thn(11.2) èqoume:

n∑i=1

an0+i ≤ c

n∑i=1

bn0+i ≤ cM,

ìpouM to fr�gma twn merik¸n ajroism�twn thc∑bn+n0. 'Ara, ja eÐnai fragmèna kai

ta merik� ajroÐsmata thc∑an+n0, �ra, apì thn arqik  mac parat rhsh, ja sugklÐnei

kai h∑an+n0, �ra ja sugklÐnei kai h

∑an (Je¸rhma 11.1).

Par�deigma 11.6. H∑

1n! sugklÐnei, giatÐ

1

n!≤ 1

2× 1

2n

gia k�je n > n0 = 4, en¸ h∑

12n sugklÐnei wc gewmetrik  (r = 1

2).

Je¸rhma 11.5. (Krit rio sÔgkrishc sto ìrio) 'Estw dÔo akoloujÐec an, bn > 0 kai

limn→∞

anbn

= c.

Tìte:

1. An c 6= 0,∞, tìte h∑an sugklÐnei ann sugklÐnei kai h

∑bn.

2. An c = 0, tìte an sugklÐnei h∑bn sugklÐnei kai h

∑an, en¸ an apoklÐnei h∑

an apoklÐnei kai h∑bn.

3. An c = ∞, tìte an sugklÐnei h∑an sugklÐnei kai h

∑bn, en¸ an apoklÐnei h∑

bn apoklÐnei kai h∑an.

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242 KEF�ALAIO 11. SEIR�ES

Apìdeixh. Katarq n parathr ste ìti ìla ta skèlh eÐnai diaisjhtik� profan .Ja apodeÐxoume to pr¸to skèloc, kaj¸c ta �lla prokÔptoun me parìmoio trìpo.

Ex' orismoÔ tou orÐou, gia ε = c2 > 0 ja up�rqei n0 ∈ N tètoio ¸ste

∀n > n0,c

2≤ anbn≤ 3c

2⇔ c

2bn ≤ an ≤

3c

2bn.

To zhtoÔmeno prokÔptei efarmìzontac to krit rio sÔgkrishc dÔo forèc, mia gia thnanisìthta (c/2)bn ≤ an ⇔ bn ≤ (2/c)an kai mia for� gia thn anisìthta an ≤ (3c/2)bn.

Par�deigma 11.7. AfoÔ h∑

1n apoklÐnei, ja apoklÐnei kai h

∑1√

n(n+1), afoÔ

limn→∞

1/n

1/√n(n+ 1)

= limn→∞

√n(n+ 1)

n2= lim

n→∞

√1 +

1

n= 1.

EpÐshc, afoÔ h∑(

12

)nsugklÐnei (wc gewmetrik ), kai

limn→∞

1log n

(12

)n(12

)n = limn→∞

1

log n= 0,

ja sugklÐnei kai h∑

1log n

(12

)n.

Je¸rhma 11.6. (Krit rio lìgou tou d’ Alembert) 'Estw akoloujÐa an > 0 kai

limn→∞

an+1

an= r.

1. An r > 1, h seir�∑an apoklÐnei (kai gia thn akrÐbeia teÐnei sto ∞).

2. An r < 1, h seir�∑an sugklÐnei.

Parathr seic

1. An r = 1, den mporoÔme na apofanjoÔme. Dhlad  kai ta dÔo �nw endeqìmena(sÔgklish, apìklish) eÐnai dunat�.

2. BebaiwjeÐte ìti èqete katal�bei to krit rio. Sthn pr¸th perÐptwsh, h seir�sumperifèretai wc apoklÐnousa gewmetrik , en¸ sth deÔterh perÐptwsh sumpe-rifèretai san sugklÐnousa gewmetrik . (Parathr ste ìti gia thn gewmetrik prìodo an = rn isqÔei an+1

an= r p�nta, kai ìqi mìno sto ìrio.)

3. To krit rio tou lìgou eÐnai eÔkolo na efarmosteÐ ìtan h morf  thc an eÐnai tètoia¸ste o lìgoc an+1

anna èqei apl  morf .

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11.2. SEIR�ES MH ARNHTIK�WN �ORWN 243

Apìdeixh. Ja exet�soume pr¸ta to endeqìmeno r < 1. Parathr ste ìti, apì tonorismì tou orÐou, prokÔptei pwc up�rqoun n0 ∈ N kai s ∈ R me r < s < 1 tètoia¸ste gia k�je n > n0,

an+1

an< s⇔ an+1

sn+1<ansn,

�ra h an/sn eÐnai fjÐnousa gia n > n0, kai sunep¸c up�rqei K tètoio ¸ste

∀n > n0,ansn≤ K ⇔ an ≤ Ksn,

kai apì to krit rio thc sÔgkrishc, epeid  sugklÐnei h∑sn, ja sugklÐnei kai h

∑an.

Exet�zoume t¸ra to endeqìmeno r > 1. Parathr ste ìti, apì ton orismì tou orÐou,prokÔptei pwc up�rqoun n0 ∈ N kai s ∈ R me r > s > 1 tètoia ¸ste gia k�je n > n0,

an+1

an> s⇒ an+1 > an,

opìte gia ìla ta n > n0, an > an0, opìte den èqw sÔgklish thc an sto 0, �ra kai hseir� den mporeÐ na sugklÐnei.

Par�deigma 11.8. Ja apofanjoÔme gia thn sÔgklish twn parak�tw seir¸n qrh-simopoi¸ntac to krit rio tou lìgou:∑ (n!)2

n(2n)!,∑ 2nn!

nn,∑ 3nn!

nn

'Eqoume, kat� perÐptwsh:

1. Parathr ste ìti ìloi oi ìroi an > 0, �ra mpor¸ na qrhsimopoi sw to krit riotou lìgou. 'Eqoume:

an+1

an=

[[(n+ 1)!]2

(n+ 1)(2(n+ 1))!

] [[n!]2

n(2n)!

]−1

=n(n+ 1)!(n+ 1)!(2n)!

(n+ 1)[2(n+ 1)]!n!n!=

n(n+ 1)(n+ 1)

(n+ 1)(2n+ 1)(2n+ 2)→ 1

4,

�ra b�sei tou krithrÐou tou lìgou h seir� sugklÐnei.

2. 'Oloi oi ìroi eÐnai jetikoÐ, �ra

an+1

an=

[2n+1(n+ 1)!

(n+ 1)n+1

] [2nn!

nn

]−1

=2n+1(n+ 1)!nn

(n+ 1)n+12nn!=

2(1 + 1

n

)n → 2

e< 1.

Epeid  to ìrio eÐnai mikrìtero thc mon�dac, h seir� sugklÐnei.

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244 KEF�ALAIO 11. SEIR�ES

3. Se aut  thn perÐptwsh, epanalamb�nontac thn Ðdia diadikasÐa, prokÔptei:

an+1

an=

3

e> 1,

�ra me aut  th mikr  allag  sthn morf  twn ìrwn, h seir� apoklÐnei.

Par�deigma 11.9. Efarmìzontac to krit rio tou lìgou gia th seir�∑

1/np, è-qoume:

an+1

an=

[1

(n+ 1)p

] [1

np

]−1

=

[n

n+ 1

]p=

[1

1 + 1n

]p→ 1.

'Ara briskìmaste sthn perÐptwsh pou den mporoÔme na apofanjoÔme.

Je¸rhma 11.7. (Krit rio thc rÐzac tou Cauchy) 'Estw akoloujÐa an > 0 kai

limn→∞

(an)1n = r.

1. An r > 1, h seir�∑an apoklÐnei (kai gia thn akrÐbeia teÐnei sto ∞).

2. An r < 1, h seir�∑an sugklÐnei.

Apìdeixh. H apìdeixh paraleÐpetai, giatÐ moi�zei arket� me thn apìdeixh tou krithrÐoutou lìgou.

Parathr seic

1. An r = 1, den mporoÔme na apofanjoÔme. Dhlad  kai ta dÔo �nw endeqìmenaeÐnai dunat�.

2. BebaiwjeÐte ìti èqete katal�bei to krit rio. Sthn pr¸th perÐptwsh, h seir�sumperifèretai wc sugklÐnousa gewmetrik  seir�, en¸ sth deÔterh perÐptwshsumperifèretai san apoklÐnousa gewmetrik  seir�. (Parathr ste ìti gia thngewmetrik  prìodo an = rn isqÔei (an)

1n = r akrib¸c, kai ìqi mìno sto ìrio.)

3. To krit rio thc rÐzac eÐnai eÔkolo na efarmosteÐ ìtan h morf  thc an eÐnai tètoia¸ste h rÐza (an)

1n na èqei apl  morf .

Par�deigma 11.10. Ja broÔme an sugklÐnoun oi akìloujec seirèc, efarmìzontacto krit rio tou lìgou: ∑(

n1n − 1

2

)n,∑

e−n2

'Eqoume, kat� perÐptwsh:

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11.2. SEIR�ES MH ARNHTIK�WN �ORWN 245

1. Parathr ste pwc

(an)1n = n

1n − 1

2.

H f(x) = x1x teÐnei sto 1 gia x→∞ (deÐte to Par�deigma 7.10), opìte ja èqoume

ìti h f(n) = n1n − 1

2 teÐnei sto 12 gia n→∞, kai h seir� sugklÐnei.

2. Lìgw thc morf c twn ìrwn thc seir�c, eÐnai kai p�li profanèc ìti prèpei naqrhsimopoi soume to krit rio thc rÐzac. Pr�gmati:

a1nn =

[e−n

2] 1

n

= e−n → 0,

kai h seir� sugklÐnei.

Par�deigma 11.11. Efarmìzontac to krit rio thc rÐzac gia th seir�∑

1/np,èqoume: [

1

np

] 1n

=1(n

1n

)p → 1,

�ra kai p�li den mporoÔme na apofanjoÔme. (Qrhsimopoi same to ìrio limn→∞

n1n = 1

tou prohgoÔmenou paradeÐgmatoc.)

Je¸rhma 11.8. (Krit rio tou oloklhr¸matoc) 'Estw f : [1,∞) → R jetik  fjÐ-nousa sun�rthsh. 'Estw oi akoloujÐec

sn =n∑k=1

f(k), tn =

∫ n

1

f(x) dx.

Oi akoloujÐec sn kai tn sugklÐnoun   apoklÐnoun mazÐ.

Parathr seic

1. Genik¸c eÐnai pio eÔkolo na upologÐzoume oloklhr¸mata apì ajroÐsmata, gi' autìkai to krit rio eÐnai polÔ qr simo.

2. H sn eÐnai akoloujÐa kai seir�, en¸ h tn eÐnai akoloujÐa kai olokl rwma.

Apìdeixh. Kat' arq n parathr ste ìti, epeid  h f eÐnai fjÐnousa, isqÔei ìti

n∑k=2

f(k) ≤∫ n

1

f(x) dx ≤n−1∑k=1

f(k). (11.3)

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246 KEF�ALAIO 11. SEIR�ES

x

y

1 2 n...

y

1 2 n...

x

f(x)f(x)

Sq ma 11.1: Gewmetrik  ex ghsh thc sqèshc (11.3).

H �nw dipl  anisìthta mporeÐ na ermhneuteÐ me th bo jeia tou Sq matoc 11.1. Toaristerì mèloc isoÔtai me to embadìn thc arister c skiasmènhc epif�neiac, en¸ to dexÐmèloc isoÔtai me to embadìn thc dexi� skiasmènhc epif�neiac.

An h tn =∫ n

1 f(x) dx sugklÐnei, wc aÔxousa (h f eÐnai jetik ) eÐnai kai fragmènh(Je¸rhma 2.10), �ra apì thn pr¸th anisìthta thc (11.3) ja eÐnai fragmènh kai h∑n

k=2 f(k), �ra kai h sn =∑n

k=1 f(k), kai wc aÔxousa (h f eÐnai jetik ) ja sugklÐnei.Parìmoia, an sugklÐnei h sn =

∑nk=1 f(k), ja eÐnai fragmènh, �ra apì thn deÔterh

anisìthta thc (11.3) ja eÐnai fragmènh kai h tn =∫ n

1 f(x) dx, kai wc aÔxousa (h feÐnai jetik ) ja sugklÐnei.

'Ara h sn sugklÐnei ann h tn sugklÐnei, kai profan¸c h sn apoklÐnei ann h tn apo-klÐnei.

Par�deigma 11.12. Ja deÐxoume ìti h seir�∑

1np , p ∈ R, sugklÐnei ann p > 1.

Ja efarmìsoume to krit rio tou oloklhr¸matoc gia th sun�rthsh f(x) = x−p. OiakoloujÐec gÐnontai:

sn =n∑k=1

1

kp, tn =

∫ n

1

1

xpdx =

{n1−p−1

1−p , p 6= 1,

log n, p = 1.

'Ara, an p > 1, h n1−p teÐnei sto 0, �ra h tn sugklÐnei, kai apì to krit rio sugklÐneikai h sn. An p = 1, log n → ∞, �ra h tn apoklÐnei, kai mazÐ thc apoklÐnei kai hsn. Tèloc, an p < 1, h n1−p teÐnei sto ∞, �ra kai p�li h tn apoklÐnei, kai mazÐ thcapoklÐnei kai h sn.

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11.2. SEIR�ES MH ARNHTIK�WN �ORWN 247

Parathr seic

1. Sun jwc h qr sh enìc krithrÐou eÐnai polÔ aploÔsterh apì th qr sh twn upì-loipwn, an kai den eÐnai p�nta profanèc poio eÐnai autì.

2. Me ta krit ria mporoÔme na diapist¸soume an mia seir� sugklÐnei   ìqi. DenmporoÔme ìmwc na diapist¸soume se pio arijmì sugklÐnei. Sun jwc autì eÐnaipolÔ duskolìtero, ektìc orismènwn exairèsewn (p.q. thleskopik¸n seir¸n.)

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248 KEF�ALAIO 11. SEIR�ES

11.3 ParadeÐgmata

Par�deigma 11.13. Ja exet�soume thn sÔgklish thc∑[(

12

)n+(

13

)n]. To pio

aplì eÐnai na parathr soume pwc h seir� eÐnai to �jroisma dÔo seir¸n pou sugklÐnoun,wc gewmetrikèc, kai epomènwc me qr sh tou Jewr matoc 11.2 prokÔptei ìti sugklÐneikai h dosmènh. (Se poio ìrio sugklÐnei?)

Par�deigma 11.14. Ja exet�soume thn sÔgklish thc∑n tan (1/n). ParathroÔme

pwc

limn→∞

n tan

(1

n

)= lim

n→∞sin(

1n

)1n

[cos

(1

n

)]−1

= limn→∞

sin(

1n

)1n

[limn→∞

cos

(1

n

)]−1

= limx→0+

sinx

x

[limx→0+

cos (x)

]−1

= 1.

Sthn trÐth isìthta qrhsimopoi same to gegonìc ìti an limx→0+

f(x) = L, tìte kai

limn→∞

f(1/n) = L. 'Ara, afoÔ oi ìroi thc seir�c den sugklÐnoun sto 0, apì to anagkaÐo

krit rio tou Cauchy, h seir� apoklÐnei.

Par�deigma 11.15. Ja exet�soume thn sÔgklish thc∑

(n +√n)(n + 2)/n!.

ParathroÔme pwc:

an+1

an=

[(n+ 1 +

√n+ 1)(n+ 3)

(n+ 1)!

] [(n+

√n)(n+ 2)

n!

]−1

=(n+ 1 +

√n+ 1)(n+ 3)

(n+ 1)(n+√n)(n+ 2)

=1

n

[(1 + 1/n+

√1/n+ 1/n2)(1 + 3/n)

(1 + 1/n)(1 + 1/√n)(1 + 2/n)

],

pou profan¸c teÐnei sto 0, opìte apì to krit rio tou lìgou prokÔptei ìti h seir�sugklÐnei.

Par�deigma 11.16. Ja exet�soume thn sÔgklish thc∑(

13

)n | cos πn2

11888|. Parath-roÔme pwc ìloi oi ìroi thc eÐnai jetikoÐ, kai epÐshc pwc eÐnai, ìro proc ìro, mikrìterhapì thn gewmetrik  seir�

∑(13

)nh opoÐa eÐnai gnwstì pwc sugklÐnei. 'Ara, ja sug-

klÐnei kai h∑(

13

)n | cos πn2

11888|.Par�deigma 11.17. Ja exet�soume thn sÔgklish thc

∑n10000e−n. Qrhsimopoi¸n-

tac to krit rio tou lìgou, prokÔptei pwc

an+1

an=

(n+ 1)10000e−n−1

n10000e−n=

(n+ 1

n

)10000

e−1 → e−1,

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11.3. PARADE�IGMATA 249

�ra h seir� sugklÐnei.

Par�deigma 11.18. Ja exet�soume thn sÔgklish thc∑

n!10000n . Qrhsimopoi¸ntac

to krit rio tou lìgou, prokÔptei pwc

an+1

an=

[(n+ 1)!

10000n+1

]/[ n!

10000n

]=n+ 1

10000→∞,

�ra h seir� apoklÐnei.

Par�deigma 11.19. Ja exet�soume thn sÔgklish thc∑

(log n)n/en2+n+1. Lìgw

twn dun�mewn tou n, h profan c prosèggish eÐnai na efarmìsoume to krit rio thcrÐzac. Pr�gmati, èstw an = (log n)n/en

2+n+1. 'Eqoume:

limn→∞

a1nn = lim

n→∞log n

en+1+ 1n

= limn→∞

1/n

en+1+1/n (1− 1/n2)= 0.

Sthn deÔterh isìthta, efarmìsame ton kanìna tou L’Hopital. 'Ara, apì to krit riothc rÐzac, prokÔptei pwc h seir� sugklÐnei.

Par�deigma 11.20. Ja exet�soume thn sÔgklish thc∑

(n+√n)/(2n3 − 1). An

dokim�soume to krit rio tou lìgou den ja broÔme th sÔgklish, kaj¸c o sqetikìclìgoc ja sugklÐnei sto 1. (Dokim�ste to!) To krit rio thc rÐzac den faÐnetai na eÐnaieÔkolo sthn efarmog , kai epiplèon h sqetik  rÐza ja sugklÐnei sto 1. 'Omwc, anan = (n+

√n)/(2n3 − 1) oi ìroi thc seir�c, parathroÔme pwc

an1/n2

=n3 + n2

√n

2n3 − 1=

1 + n−1/2

2− n−3→ 1

2> 0,

�ra, apì to krit rio thc sÔgkrishc sto ìrio, afoÔ sugklÐnei h∑(

1/n2)ja sugklÐnei

kai h dosmènh.

Par�deigma 11.21. Ja exet�soume thn sÔgklish thc∑∞

n=1 n2/(n3 + 4n − 5).

An dokim�soume to krit rio tou lìgou, ìpwc kai sto prohgoÔmeno par�deigma denja broÔme th sÔgklish, kaj¸c o sqetikìc lìgoc ja sugklÐnei sto 1. EpÐshc, tokrit rio thc rÐzac den faÐnetai na eÐnai kat�llhlo. ParathroÔme ìmwc ìti oi ìroi anthc sugkekrimènhc akoloujÐac, gia meg�la n, eÐnai perÐpou Ðsoi me 1

n . H antÐstoiqhseir� eÐnai h armonik , pou apoklÐnei. 'Ara h seir� anamènoume na apoklÐnei. Gia nato deÐxoume austhr�, qrhsimopoioÔme to krit rio sÔgkrishc sto ìrio. 'Estw bn = 1

n .Tìte

anbn

=n2

n3+4n−51n

=n3

n3 + 4n− 5→ 1,

kai me dedomèno ìti apoklÐnei h∑∞

n=1 bn, apì to krit rio sÔgklishc sto ìrio apoklÐneikai h

∑∞n=1

nn2+2n+3 .

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250 KEF�ALAIO 11. SEIR�ES

Par�deigma 11.22. Ja exet�soume thn sÔgklish thc∑

(n3 [2 + sinn]n)/4n. 'E-stw an = (n3 [2 + sinn]n)/4n oi ìroi thc seir�c. Lìgw thc morf c touc, eÐnai logikìna prospaj soume na efarmìsoume to krit rio thc rÐzac. An to k�noume, ja diapist¸-soume ìti den up�rqei ìrio, lìgw tou ìrou sinn. (MporeÐte na to epibebai¸sete?) AutìmporeÐ na antimetwpisteÐ an orÐsoume thn akoloujÐa bn = (n3 [2 + 1]n)/4n = n3(3/4)n.ParathroÔme pwc

0 ≤ an ≤ bn.

SÔmfwna me to krit rio thc sÔgkrishc, h∑an ja sugklÐnei an sugklÐnei kai h

∑bn.

H∑bn ìmwc sugklÐnei, giatÐ apì to krit rio thc rÐzac èqoume:

(bn)1n = n

3n

(3

4

)→(

3

4

)< 1.

Parathr ste pwc qrhsimopoi same to ìrio limn→∞

n1n = 1.

Par�deigma 11.23. Ja exet�soume thn sÔgklish thc∑(

1n − n10000e−n

). H seir�

apoteleÐtai apì dÔo seirèc, ek twn opoÐwn h pr¸th eÐnai h armonik , kai apoklÐnei, en¸h deÔterh sugklÐnei, sÔmfwna me to Par�deigma 11.17. Sunep¸c, h seir� prèpei naapoklÐnei. Pr�gmati, èstw pwc sunèkline. Tìte, afoÔ sugklÐnei kai h

∑n10000e−n,

apì gnwstì je¸rhma ja prèpei na sugklÐnei kai to �jroisma touc, pou eÐnai h∑

1n ,

kai èqoume �topo.

Par�deigma 11.24. Ja exet�soume thn sÔgklish thc seir�c∑∞

n=21

n(log n)2 . EÐnaieÔkolo na doÔme ìti oÔte to krit rio thc rÐzac, oÔte to krit rio tou lìgou mporeÐ nad¸sei k�poia lÔsh, kai ja efarmìsoume to krit rio tou oloklhr¸matoc. Parathr steìti oi ìroi thc seir�c xekinoÔn apì to n = 2, kai epomènwc to krit rio ja efarmosteÐan�loga.

Prèpei na doÔme an sugklÐnei to∫∞

2dx

x(log x)2 . K�nontac thn allag  metablht c

y = log x⇒ dy = dxx èqoume:∫

dx

x(log x)2=

∫dy

y2= −y−1 + C = − 1

log x+ C.

Sqetik� me th seir�, sÔmfwna me to krit rio tou oloklhr¸matoc, ta∞∑n=2

1

n(log n)2,

∫ ∞2

1

x(log x)2dx

sugklÐnoun   apoklÐnoun tautìqrona. ParathroÔme ìmwc pwc∫ ∞2

1

x(log x)2dx =

∫ ∞2

(− 1

log x

)′dx = −0 + (log 2)−1,

kai epomènwc prokÔptei pwc h seir� sugklÐnei.

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Kef�laio 12

DianÔsmata

12.1 Rn

Orismìc 12.1. (DianÔsmata)

1. OrÐzoume wc Rn to sÔnolo twn diatetagmènwn n-�dwn

Rn = {(x1, x2, . . . , xn) : xi ∈ R}.

Ta stoiqeÐa tou Rn kaloÔntai dianÔsmata. Ta xi kaloÔntai sunist¸sec, stoiqeÐa,  suntetagmènec tou (x1, x2, . . . , xn).

2. OrÐzoume thn isìthta dianusm�twn wc isìthta kat� stoiqeÐo:

(x1, x2, . . . , xn) = (y1, y2, . . . , yn)⇔ x1 = y1, x2 = y2, . . . , xn = yn.

3. OrÐzoume thn prìsjesh dianusm�twn wc:

(x1, x2, . . . , xn) + (y1, y2, . . . , yn) , (x1 + y1, x2 + y2, . . . , xn + yn).

4. OrÐzoume to antÐjeto enìc dianÔsmatoc wc:

−(x1, x2, . . . , xn) , (−x1,−x2, . . . ,−xn).

5. OrÐzoume thn afaÐresh dianusm�twn wc:

(x1, x2, . . . , xn)− (y1, y2, . . . , yn) , (x1, x2, . . . , xn) + [−(y1, y2, . . . , yn)]

= (x1 − y1, x2 − y2, . . . , xn − yn).

6. OrÐzoume thn pollaplasiasmì dianÔsmatoc me pragmatikì c wc:

c(x1, x2, . . . , xn) , (cx1, cx2, . . . , cxn).

251

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252 KEF�ALAIO 12. DIAN�USMATA

X

Y

X+Y

X-Y

-X

2X

0

Sq ma 12.1: Gewmetrik  apeikìnish pr�xewn metaxÔ dianusm�twn.

7. OrÐzoume to mhdenikì di�nusma wc:

0 , (0, 0, . . . , 0).

8. DÔo mh mhdenik� dianÔsmata kaloÔntai par�llhla an

(x1, x2, . . . , xn) = λ(y1, y2, . . . , yn),

gia k�poio λ 6= 0. An λ > 0, ta dianÔsmata kaloÔntai omìrropa. An λ < 0, tìteta dianÔsmata lègontai antÐrropa.

Parathr seic

1. Sqetik� me touc sumbolismoÔc, parathroÔme ta ex c:

(aþ) Ja sumbolÐzoume sun jwc ta dianÔsmata me kefalaÐouc latinikoÔc qarakt -rec. Gia par�deigma:

X = (x1, x2, . . . , xn), Y = (y1, y2, . . . , yn), . . .

(bþ) An den dieukrinÐzoume ti sumbolÐzei akrib¸c to 0 (dhlad  to mhdenikì di�nu-sma   to mhdèn), autì ja mporeÐ na gÐnei katanohtì apì ta sumfrazìmena.

2. Sqetik� me thn tim  tou n, parathroÔme ta ex c:

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12.1. RN 253

(aþ) Ta dianÔsmata tou R1 antistoiqoÔn 1− 1 sta shmeÐa pou brÐskontai epÐ miaceujeÐac.

(bþ) Ta dianÔsmata tou R2 antistoiqoÔn 1−1 sta shmeÐa epÐ enìc epipèdou (efìsonèqoume orÐsei èna kartesianì sÔsthma suntetagmènwn.)

(gþ) Ta dianÔsmata tou R3 antistoiqoÔn 1 − 1 sta shmeÐa tou q¸rou (efìsonèqoume orÐsei èna kartesianì sÔsthma suntetagmènwn.)

(dþ) H genik  perÐptwsh tou aujaÐretou n eÐnai polÔ qr simh gia pollèc efarmo-gèc. To n mporeÐ na eÐnai:

i. O arijmìc k�poiwn metr sewn.

ii. O arijmìc twn paramètrwn proc tic opoÐec jèloume na k�noume k�poiabeltistopoÐhsh.

iii. O arijmìc twn pixels se mia eikìna.

iv. O arijmìc twn stoiqeÐwn se èna hlektrikì kÔklwma.

v. O arijmìc twn zeÔxewn se èna dÐktuo epikoinwni¸n.

vi. k.t.l.

(eþ) Pollèc apì tic idiìthtec twn dianusm�twn sto epÐpedo (n = 2) kai sto q¸ro(n = 3) prokÔptoun apì idiìthtec pou mporoÔme na deÐxoume mia kai kal  giaaujaÐreto n.

3. Gia n = 1, 2, 3, h gewmetrik  ermhneÐa twn ennoi¸n tou OrismoÔ 12.1 eÐnai:

(aþ) To di�nusma X antistoiqeÐ se èna shmeÐo tou q¸rou.

(bþ) To 0 antistoiqeÐ sthn arq  twn axìnwn.

(gþ) To di�nusma X +Y antistoiqeÐ sthn tètarth koruf  tou parallhlogr�mmoutou opoÐou oi �llec korufèc eÐnai ta X, Y , kai 0.

(dþ) To di�nusma −X eÐnai to shmeÐo tou q¸rou pou eÐnai summetrikì tou X wcproc thn arq  twn axìnwn.

(eþ) Ti ekfr�zei to di�nusma cX?

(�þ) Ti shmaÐnei dÔo dianÔsmata na eÐnai par�llhla?

DeÐte to Sq ma 12.1. Oi �nw gewmetrikèc ermhneÐec eÐnai exairetik� qr simec kaisth genik  perÐptwsh gia n 6= 1, 2, 3.

4. Par� thn ter�stia shmasÐa pou èqei h ermhneÐa twn dianusm�twn wc shmeÐa stoq¸ro, austhr¸c ta èqoume orÐsei wc diatetagmènec n-�dec, kai h jewrÐa pou jaanaptÔxoume basÐzetai se autì ton orismì, kai ìqi sth gewmetrik  mac diaÐsjhsh.

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254 KEF�ALAIO 12. DIAN�USMATA

Je¸rhma 12.1. (Idiìthtec dianusm�twn) 'Estw X, Y, Z aujaÐreta dianÔsmata kaia, b ∈ R. IsqÔoun oi akìloujec idiìthtec:

1. (Antimetajetik  Idiìthta) X + Y = Y +X.

2. (Prosetairistik  Idiìthta Prìsjeshc Dianusm�twn)X+(Y +Z) = (X+Y )+Z.

3. (Oudètero StoiqeÐo) 0 +X = X.

4. (AntÐjeto StoiqeÐo) Gia k�jeX up�rqei stoiqeÐo Y = −X tètoio ¸steX+Y = 0.

5. (Epimeristik  Idiìthta Prìsjeshc Dianusm�twn) a(X + Y ) = aX + aY .

6. (Epimeristik  Idiìthta Prìsjeshc Pragmatik¸n) (a+ b)X = aX + bX.

7. (Prosetairistik  Idiìthta PollaplasiasmoÔ Pragmatik¸n) a(bX) = (ab)X.

8. (Oudètero StoiqeÐo PollaplasiasmoÔ) 1×X = X.

9. (H ParallhlÐa eÐnai Metabatik ) An ta X, Y eÐnai par�llhla kai ta Y, Z eÐnaipar�llhla, tìte eÐnai par�llhla kai ta X,Z.

Apìdeixh. Profan c, kai basÐzetai sth qr sh twn antÐstoiqwn idiot twn twn pragma-tik¸n.

Orismìc 12.2. (M koc ston Rn) Gia k�je di�nusma (x1, x2, . . . , xn) ston Rn, orÐ-zoume to m koc   nìrma

‖(x1, x2, . . . , xn)‖ ,√x2

1 + x22 + · · ·+ x2

n.

Parat rhsh: Gia n ≤ 3 to m koc tou dianÔsmatoc tautÐzetai me thn apìstash toushmeÐou X apì thn arq  twn axìnwn. To m koc ìmwc eÐnai exairetik� qr simh ènnoiakai gia n > 3.

Je¸rhma 12.2. (Idiìthtec m kouc) 'Estw X, Y ∈ Rn kai a ∈ R.

1. ‖X‖ ≥ 0.

2. ‖X‖ = 0⇔ X = 0.

3. (Idiìthta Omogèneiac) ‖aX‖ = |a|‖X‖.

Apìdeixh. Profan c.

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12.1. RN 255

Orismìc 12.3. (Eswterikì ginìmeno) OrÐzoume to eswterikì ginìmeno dÔo dianu-sm�twn wc ex c:

(x1, x2, . . . , xn) · (y1, y2, . . . , yn) = x1y1 + x2y2 + · · ·+ xnyn =n∑i=1

xiyi.

Je¸rhma 12.3. (Basikèc idiìthtec eswterikoÔ ginomènou) 'Estw X, Y, Z ∈ Rn, kaia ∈ R. To eswterikì ginìmeno èqei tic akìloujec idiìthtec:

1. (Antimetajetik ) X · Y = Y ·X.

2. (Prosetairistik ) (aX) · Y = a(X · Y ).

3. (Epimeristik ) (X + Y ) · Z = X · Z + Y · Z.

4. ‖X‖ =√X ·X ⇔ ‖X‖2 = X ·X.

5. X ·X = 0⇔ X = 0.

Apìdeixh. Profan c.

Je¸rhma 12.4. (Anisìthta Cauchy-Schwarz)

1. Gia k�je X = (x1, x2, . . . , xn) Y = (y1, y2, . . . , yn) ∈ Rn,

|X · Y | ≤ ‖X‖‖Y ‖ ⇔∣∣∣∣∣n∑i=1

xiyi

∣∣∣∣∣ ≤(

n∑i=1

x2i

)1/2( n∑i=1

y2i

)1/2

. (12.1)

H �nw anisìthta emfanÐzetai me pollèc morfèc se di�forouc kl�douc twn majh-matik¸n, kai eÐnai gnwst  wc Anisìthta Cauchy-Schwarz.

2. An X, Y 6= 0 isqÔei ìti

X · Y = ‖X‖‖Y ‖ ⇔ X, Y omìrropa,

3. An X, Y 6= 0 isqÔei ìti

X · Y = −‖X‖‖Y ‖ ⇔ X, Y antÐrropa,

Apìdeixh. 1. ParathroÔme pwc gia opoiod pote λ ∈ R èqoume:

‖X − λY ‖2 ≥ 0⇔ (X − λY ) · (X − λY ) ≥ 0

⇔ X ·X + λ2Y · Y − 2λX · Y ≥ 0⇔ ‖Y ‖2λ2 + (−2X · Y )λ+ ‖X‖2 ≥ 0.

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256 KEF�ALAIO 12. DIAN�USMATA

Epeid  to �nw tri¸numo den eÐnai potè arnhtikì, den mporeÐ h diakrÐnousa na eÐnaijetik . 'Ara:

4(X · Y )2 − 4‖X‖2 ≤ 0⇔ |X · Y | ≤ ‖X‖ ‖Y ‖.

2. 'Estw t¸ra ìti ta X, Y 6= 0 eÐnai omìrropa, dhlad  X = λY gia k�poio λ > 0.Tìte:

X · Y = X · (λX) = λ(X ·X) = λ‖X‖2 = ‖X‖‖λX‖ = ‖X‖‖Y ‖.

Antistrìfwc, èstw pwc X · Y = ‖X‖‖Y ‖. ParathroÔme pwc

‖X − ‖X‖‖Y ‖Y ‖2 = X ·X +

‖X‖2

‖Y ‖2Y · Y − 2

‖X‖‖Y ‖X · Y

= ‖X‖2 + ‖X‖2 − 2‖X‖2 = 0⇒ X =‖X‖‖Y ‖Y,

dhlad  ta X, Y eÐnai omìrropa, afoÔ ‖X‖‖Y ‖ > 0.

3. 'Estw t¸ra ìti ta X, Y eÐnai antÐrropa, dhlad  X = λY gia k�poio λ < 0. Tìte:

X · Y = X · (λX) = λ(X ·X) = λ‖X‖2 = −‖X‖‖λX‖ = −‖X‖‖Y ‖.

Antistrìfwc, èstw pwc X · Y = −‖X‖‖Y ‖. ParathroÔme pwc

‖X +‖X‖‖Y ‖Y ‖

2 = X ·X +‖X‖2

‖Y ‖2Y · Y + 2

‖X‖‖Y ‖X · Y

= ‖X‖2 + ‖X‖2 − 2‖X‖2 = 0⇒ X = −‖X‖‖Y ‖Y,

dhlad  ta X, Y eÐnai antÐrropa.

Je¸rhma 12.5. (Trigwnik  anisìthta kai nìmoc tou parallhlogr�mmou)

1.‖X + Y ‖ ≤ ‖X‖+ ‖Y ‖. (12.2)

(H �nw eÐnai gnwst  wc trigwnik  anisìthta, kaj¸c ekfr�zei to gegonìc ìti to�jroisma ‖X‖+‖Y ‖ twn mhk¸n twn dÔo pleur¸n enìc trig¸nou eÐnai megalÔtero  Ðso tou m kouc ‖X+Y ‖ thc trÐthc. DeÐte to Sq ma 12.2. Kai aut  h anisìthtaemfanÐzetai se polloÔc kl�douc twn majhmatik¸n upì di�forec morfèc.)

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12.1. RN 257

||X||||Y||

||X+Y||

||X||

||Y||

Y

X

X+Y

0

X-Y

||X-Y||

Sq ma 12.2: SÔmfwna me thn trigwnometrik  anisìthta, to �jroisma ‖X‖+‖Y ‖ twn mhk¸n twn dÔopleur¸n enìc trig¸nou eÐnai megalÔtero   Ðso tou m kouc ‖X + Y ‖ thc �llhc pleur�c. SÔmfwna meton nìmo tou parallhlogr�mmou, to �jroisma ‖X + Y ‖2 + ‖X − Y ‖2 twn tetrag¸nwn twn mhk¸ntwn dÔo diagwnÐwn enìc parallhlogr�mmou isoÔtai me to �jroisma 2(‖X‖2 + ‖Y ‖2) twn tetrag¸nwntwn mhk¸n twn tess�rwn pleur¸n tou.

2. 'Otan X, Y 6= 0, h trigwnik  anisìthta isqÔei me isìthta ann ta X, Y eÐnaiomìrropa.

3.‖X + Y ‖2 + ‖X − Y ‖2 = 2(‖X‖2 + ‖Y ‖2). (12.3)

(H �nw isìthta eÐnai gnwst  wc nìmoc tou parallhlogr�mmou, kaj¸c ekfr�zei

to gegonìc ìti to �jroisma 2(‖X‖2 + ‖Y ‖2) twn tetrag¸nwn twn mhk¸n twntess�rwn pleur¸n enìc parallhlogr�mmou isoÔtai me to �jroisma ‖X + Y ‖2 +‖X−Y ‖2 twn tetrag¸nwn twn mhk¸n twn diagwnÐwn. DeÐte to Sq ma 12.2. Kaiaut  emfanÐzetai se polloÔc kl�douc twn majhmatik¸n upì di�forec morfèc.)

Apìdeixh. 1. ParathroÔme pwc

‖X + Y ‖2 = (X + Y ) · (X + Y ) = X ·X + Y · Y + 2X · Y= ‖X‖2 + ‖Y ‖2 + 2X · Y ≤ ‖X‖2 + ‖Y ‖2 + 2‖X‖‖Y ‖= (‖X‖+ ‖Y ‖)2.

H anisìthta prokÔptei apì thn anisìthta Cauchy-Schwarz.

2. 'Estw pwc ta X, Y 6= 0 eÐnai omìrropa, dhlad  up�rqei λ > 0 tètoio ¸ste

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258 KEF�ALAIO 12. DIAN�USMATA

Y = λX. 'Eqoume:

‖X + Y ‖ = ‖X + λX‖ = |1 + λ|‖X‖= ‖X‖+ λ‖X‖ = ‖X‖+ ‖λX‖ = ‖X‖+ ‖Y ‖.

Qrhsimopoi same gnwstèc idiìthtec tou m kouc. Epiplèon, sth deÔterh isìthtaqrhsimopoi same to gegonìc ìti λ > 0.

'Estw t¸ra pwc isqÔei h trigwnik  anisìthta me isìthta. Parathr¸ntac thnapìdeixh thc anisìthtac, blèpoume pwc autì sunep�getai ìti X · Y = ‖X‖‖Y ‖,dhlad , apì to Je¸rhma 12.4, ìti ta X, Y eÐnai omìrropa.

3. Parathr ste pwc:

‖X + Y ‖2 + ‖X − Y ‖2

= (X + Y ) · (X + Y ) + (X − Y ) · (X − Y )

= X ·X + Y · Y + 2X · Y +X ·X + Y · Y − 2X · Y= 2X ·X + 2Y · Y = 2‖X‖2 + 2‖Y ‖2.

Orismìc 12.4. (MonadiaÐa kai basik� dianÔsmata)

1. 'Ena di�nusma P me monadiaÐo m koc ‖P‖ = 1 kaleÐtai monadiaÐo.

2. Opoiod pote �jroisma thc morf c V =∑n

i=1 xiVi ìpou Vi aujaÐreta n-di�statadianÔsmata kai xi ∈ R, kaleÐtai grammikìc sunduasmìc twn xi.

3. OrÐzoume ta basik� dianÔsmata

E1 = (1, 0, . . . , 0),

E2 = (0, 1, . . . , 0),

. . .

Ei = (0, 0, . . . , 1, . . . , 0), (To mh mhdenikì stoiqeÐo eÐnai sth jèsh i.)

. . .

En = (0, 0, . . . , 1).

K�je di�nusma (x1, x2, . . . , xn) mporeÐ na grafeÐ wc grammikìc sunduasmìc twnEi wc ex c:

(x1, x2, . . . , xn) =n∑i=1

xiEi.

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12.1. RN 259

4. An A ∈ Rn, A 6= 0, orÐzw wc to monadiaÐo di�nusma sthn kateÔjunsh tou A to

A =A

‖A‖ .

Parat rhsh: Parathr ste ìti

‖X+Y ‖2 = (X+Y ) · (X+Y ) = X ·X+Y ·Y + 2X ·Y = ‖X‖2 +‖Y ‖2 + 2(X ·Y ).

'Estw n = 2   n = 3, X, Y 6= 0, kai θ ∈ [0, π] h gwnÐa an�mesa sta dianÔsmata. Apìthn �nw exÐswsh, prokÔptoun ta akìlouja:

X · Y = 0⇔ ‖X + Y ‖2 = ‖X‖2 + ‖Y ‖2 ⇔ θ =π

2, (12.4)

X · Y > 0⇔ ‖X + Y ‖2 > ‖X‖2 + ‖Y ‖2 ⇔ θ ∈[0,π

2

), (12.5)

X · Y < 0⇔ ‖X + Y ‖2 < ‖X‖2 + ‖Y ‖2 ⇔ θ ∈(π

2, π]. (12.6)

Parathr ste ìti h exÐswsh ‖X + Y ‖2 = ‖X‖2 + ‖Y ‖2 eÐnai to Pujagìreio Je¸rhma.Qrhsimopoi¸ntac thn (12.4), dikaiologoÔmaste na k�noume ton akìloujo orismì, pouisqÔei gia k�je n.

Orismìc 12.5. (Orjogwniìthta ston Rn) Duo dianÔsmata X, Y ∈ Rn kaloÔntaik�jeta   orjog¸nia metaxÔ touc an X · Y = 0.

Parathr seic

1. O �nw orismìc isqÔei akìma kai an X = 0   Y = 0. Bèbaia se autèc ticpeript¸seic den eÐnai idiaÐtera qr simoc.

2. O �nw orismìc isqÔei akìma kai gia n 6= 2, 3!

3. Er¸thsh: mporeÐ èna di�nusma na eÐnai k�jeto me ton eautì tou, sÔmfwna me ton�nw orismì?

Je¸rhma 12.6. (An�lush dianÔsmatoc se sunist¸sec) 'Estw dÔo dianÔsmataA 6= 0kai B 6= 0. Mpor¸ na analÔsw to A se dÔo sunist¸sec:

A‖ =

[A ·B‖B‖2

]B, A⊥ = A− A‖.

H pr¸th eÐnai par�llhlh sto B (ektìc an eÐnai mhdenik , pou sumbaÐnei ìtan ta A,BeÐnai k�jeta, dhlad  A · B = 0), kai h deÔterh eÐnai k�jeth sto B (kai endeqomènwcmhdenik ). DeÐte to Sq ma 12.3.

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260 KEF�ALAIO 12. DIAN�USMATA

A

B

A^

A||

θ

A

BA||

θ

A^

00

Sq ma 12.3: An�lush enìc dianÔsmatoc A se se mia par�llhlh kai mia k�jeth sunist¸sa sto B.

Apìdeixh. Kat' arq n parathr ste pwc to[A·B‖B‖2

]∈ R. Profan¸c to �jroisma twn

A‖, A⊥ mac k�nei to A, kai epiplèon to A‖ eÐnai par�llhlo sto B (ektìc an eÐnaimhdenikì). Gia na deÐxoume ìti to A⊥ eÐnai k�jeto sto B, apl¸c parathroÔme pwc:

A⊥ ·B = (A− A‖) ·B = A ·B − A ·B‖B‖2

B ·B = A ·B − A ·B‖B‖2

‖B‖2 = 0.

Parat rhsh: Gia n = 2 kai n = 3, gia th gwnÐa θ an�mesa sta A,B, èqoume ìti:

0 ≤ θ ≤ π

2⇒ cos θ =

‖A‖‖‖A‖ =

|A·B|‖B‖‖B‖2

‖A‖ =A ·B‖A‖‖B‖ ,

π

2≤ θ ≤ π ⇒ cos θ = −‖A‖‖‖A‖ = −

|A·B|‖B‖‖B‖2

‖A‖ =A ·B‖A‖‖B‖ .

Sthn teleutaÐa isìthta thc pr¸thc gramm c qrhsimopoi same tic (12.4), (12.5), kaisthn teleutaÐa isìthta thc deÔterhc gramm c qrhsimopoi same tic (12.4), (12.6). 'Ara,se k�je perÐptwsh,

cos θ =A ·B‖A‖‖B‖ .

Parathr ste ìti, apì thn anisìthta Cauchy-Schwarz, h posìthta sta dexi� èqei tim p�ntote mèsa sto [−1, 1]. 'Ara, mporoÔme na genikeÔsoume thn ènnoia thc gwnÐac seopoiad pote di�stash n wc ex c:

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12.1. RN 261

Orismìc 12.6. (GwnÐa metaxÔ dianusm�twn) 'Estw A,B ∈ Rn di�fora tou mhdeni-koÔ dianÔsmatoc. OrÐzw th gwnÐa an�mes� touc wc

θ , arccos

(A ·B‖A‖‖B‖

),

apì thn opoÐa sunep�getai pwc

cos θ =A ·B‖A‖‖B‖ , A ·B = ‖A‖‖B‖ cos θ.

Parat rhsh: Apì ton orismì tou tìxou sunhmitìnou,

arccosx ∈ [0, π] ∀x ∈ [−1, 1],

�ra ìpwc orÐsthke, h gwnÐa metaxÔ dÔo dianusm�twn eÐnai p�nta jetik .

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262 KEF�ALAIO 12. DIAN�USMATA

12.2 R2

Orismìc 12.7. (EujeÐa)

1. KaloÔme eujeÐa k�je sÔnolo shmeÐwn thc morf c

L = {(x, y) : (x, y) = (x0, y0)+λ(x1, y1), λ ∈ R} = {(x0, y0)+λ(x1, y1), λ ∈ R},

gia k�poia (x0, y0) kai (x1, y1) 6= 0.

2. H dianusmatik  exÐswsh

(x, y) = (x0, y0) + λ(x1, y1)⇔ {x = x0 + λx1, y = y0 + λy1}

kaleÐtai parametrik  exÐswsh eujeÐac, me par�metro λ.

3. H eujeÐa kaleÐtai par�llhlh (k�jeth) se k�je (x2, y2) pou eÐnai par�llhlo (k�-jeto) sto (x1, y1).

Parathr seic

1. DeÐte to Sq ma 12.4 gia mia gewmetrik  ermhneÐa thc parametrik c exÐswshc thceujeÐac.

2. Profan¸c, mia eujeÐa mporeÐ na grafeÐ me thn �nw morf  gia poll� diafore-tik� (x0, y0) kai (x1, y1). MporeÐ na deiqjeÐ ìti ìla ta epitrept� (x1, y1) eÐnaipar�llhla metaxÔ touc.

Orismìc 12.8. (Par�llhlec eujeÐec) 'Estw oi eujeÐec

La = {(xa0, ya0) + λ(xa1, ya1), λ ∈ R}, (12.7)

Lb = {(xb0, yb0) + λ(xb1, yb1), λ ∈ R}. (12.8)

An oi eujeÐec den èqoun kanèna koinì shmeÐo, kaloÔntai par�llhlec, kai tìte mporoÔmena deÐxoume ìti ta (xa1, ya1) kai (xb1, yb1) eÐnai par�llhla. Antistrìfwc, an ta (xa1, ya1)kai (xb1, yb1) eÐnai par�llhla tìte oi La kai Lb eÐte tautÐzontai eÐte eÐnai par�llhlec.MporeÐ epÐshc na deiqteÐ ìti dÔo eujeÐec eÐte tautÐzontai, eÐte eÐnai par�llhlec, eÐteèqoun èna akrib¸c koinì shmeÐo.

Orismìc 12.9. (K�jetec eujeÐec) DÔo eujeÐec kaloÔntai k�jetec an mporoÔn nagrafoÔn sth morf  twn exis¸sewn (12.7) kai (12.8) me ta (xa1, ya1), (xb1, yb1) k�jetametaxÔ touc. MporeÐ na deiqjeÐ ìti an oi La, Lb eÐnai k�jetec kai dÐnontai apì tic

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12.2. R2 263

y

x

(x0,y0)

(x,y)=(x0,y0)+λ(x1,y1)

L

(x1,y1)

(x1,y1)

Sq ma 12.4: Gewmetrik  ermhneÐa thc parametrik c exÐswshc eujeÐac.

exis¸seic (12.7) kai (12.8), tìte ta (xa1, ya1), (xb1, yb1) eÐnai anagkastik� k�jeta metaxÔtouc.

Je¸rhma 12.7. (Exis¸seic eujeÐac)

1. (Genik  exÐswsh eujeÐac) 'Olec oi eujeÐec mporoÔn na perigrafoÔn apì exÐswshthc morf c

ax+ by = c, (12.9)

ìpou a 6= 0   b 6= 0. Dhlad  èna shmeÐo (x, y) ∈ L ann ikanopoieÐ thn (12.9) giak�poia a, b, c pou antistoiqoÔn sthn L.

(aþ) An a = 0, h �nw exÐswsh gÐnetai y = cb .

(bþ) An a 6= 0, h �nw exÐswsh gÐnetai x = ca − b

ay.

(gþ) An b = 0, h �nw exÐswsh gÐnetai x = ca .

(dþ) An b 6= 0, h �nw exÐswsh gÐnetai y = cb − a

bx.

Ta a, b, c profan¸c den eÐnai monadik� gia k�je eujeÐa L.

2. H eujeÐa L dierqìmenh apì èna shmeÐo (x0, y0) kai par�llhlh se di�nusma (λ, µ)mporeÐ na perigrafeÐ me thn exÐswsh

µ(x− x0)− λ(y − y0) = 0. (12.10)

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264 KEF�ALAIO 12. DIAN�USMATA

y

x

(λ,μ)(x0,y0)

(λ,μ)

(x,y)

L

Sq ma 12.5: EujeÐa L dierqìmenh apì shmeÐo (x0, y0) kai par�llhlh se di�nusma (λ, µ).

(Parathr ste pwc an, gia par�deigma, λ 6= 0 kai x 6= x0, h �nw exÐswsh gÐnetai:

y − y0

x− x0=µ

λ.

DeÐte to Sq ma 12.5.)

3. H eujeÐa L dierqìmenh apì to shmeÐo (x0, y0) kai k�jeth sto di�nusma (A,B) 6= 0mporeÐ na perigrafeÐ apì thn exÐswsh

Ax+By = Ax0 +By0. (12.11)

(Parathr ste pwc an to (x, y) ∈ L, tìte to (x − x0, y − y0) eÐnai k�jeto sto(A,B), dhlad 

(x− x0, y − y0) · (A,B) = 0⇒ Ax+By = Ax0 +By0.

DeÐte to Sq ma 12.6.)

4. H eujeÐa L dierqìmenh apì ta shmeÐa (x0, y0) kai (x1, y1) mporeÐ na perigrafeÐapì thn exÐswsh

(y − y0)(x1 − x0) = (x− x0)(y1 − y0). (12.12)

(Parathr ste ìti h exÐswsh eÐnai grammik , dhlad  thc morf c thc (12.9), kaiikanopoieÐtai kai sta dÔo shmeÐa (x0, y0), (x1, y1). DeÐte to Sq ma 12.7.)

5. H eujeÐa efaptìmenh sth paragwgÐsimh sun�rthsh f(x) sto shmeÐo (x0, y0) mpo-reÐ na perigrafeÐ apì thn exÐswsh:

y = y0 + f ′(x0)(x− x0).

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12.2. R2 265

y

x

(x0,y0)

(A,B)

L

(A,B)

(x,y)

Sq ma 12.6: EujeÐa L dierqìmenh apì shmeÐo (x0, y0) kai k�jeth se di�nusma (A,B).

Apìdeixh. ParaleÐpetai.

Par�deigma 12.1. 'Estw oi eujeÐec

1. x = 1.

2. x+ 2y = 3.

3. 2x− 5y = 0.

Parathr ste ìti èqoun dojeÐ sth morf  ax + by = c. Ja d¸soume perigrafèc toucstic morfèc (12.10), (12.11), (12.12), prosdiorÐzontac se k�je perÐptwsh ta sqetik�dianÔsmata. 'Eqoume, kat� perÐptwsh:

1.x = 1⇔ 1(x− 1)− 0(y − k) = 0

'Ara, h eujeÐa eÐnai par�llhlh sto (0, 1), kai dièrqetai apì to (1, k) gia opoiod -pote k.

EpÐshc,x = 1⇔ 1x+ 0y = 1× 1 + 0× k,

�ra h eujeÐa eÐnai k�jeth sto (1, 0), kai dièrqetai apì to (1, k) gia opoiod potek.

Tèloc,x = 1⇔ (y − k)(1− 1) = (x− 1)× (l − k),

ìpou k, l ∈ R, kai k 6= l. 'Ara, mporeÐ na perigrafeÐ wc h eujeÐa dièrqetai apì tashmeÐa (1, k), (1, l), gia opoiad pote k, l di�fora metaxÔ touc.

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266 KEF�ALAIO 12. DIAN�USMATA

y

x(x0,y0)

(x,y)

L

(x1,y1)

Sq ma 12.7: EujeÐa L dierqìmenh apì dÔo shmeÐa (x0, y0), (x1, y1).

2.x+ 2y = 3⇔ 1(x− 1) + 2(y − 1) = 0.

'Ara, h eujeÐa eÐnai par�llhlh sto (−2, 1), kai dièrqetai apì to (1, 1).

EpÐshc,x+ 2y = 3⇔ x+ 2y = 1× 1 + 2× 1,

�ra h eujeÐa eÐnai k�jeth sto (1, 2), kai dièrqetai apì to (1, 1).

Tèloc,x+ 2y = 3⇔ (y − 1)(3− 1) = (x− 1)(0− 1).

'Ara, h eujeÐa mporeÐ na perigrafeÐ wc h eujeÐa dièrqetai apì ta shmeÐa (1, 1),(3, 0). (AntÐ na prospaj soume apeujeÐac na fèroume thn exÐswsh sth deÔterhmorf , ja mporoÔsame apl¸c na broÔme dÔo shmeÐa pou an koun se aut , kai nata antikatast soume sth genik  morf  (12.12).)

3.2x− 5y = 0⇔ 2(x− 0)− 5(y − 0) = 0.

'Ara, h eujeÐa eÐnai par�llhlh sto (5, 2), kai dièrqetai apì to (0, 0).

EpÐshc,2x− 5y = 0⇔ 2x− 5y = 2× 0− 5× 0.

�ra h eujeÐa eÐnai k�jeth sto (2,−5), kai dièrqetai apì to (0, 0).

Tèloc,2x− 5y = 0⇔ (y − 0)(10− 0) = (x− 0)(4− 0).

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12.2. R2 267

'Ara, h eujeÐa mporeÐ na perigrafeÐ wc h eujeÐa dièrqetai apì ta shmeÐa (0, 0),(10, 4).

Parathr ste ìti kamÐa apì thc �nw perigrafèc den eÐnai monadik .

Par�deigma 12.2. Ja apant soume tic akìloujec erwt seic:

1. Poio eÐnai to shmeÐo tom c twn eujei¸n x− 2y + 3 = 0, x+ y − 3 = 0?

2. Poio eÐnai to shmeÐo tom c twn eujei¸n x− 2y + 5 = 0, −2x+ 4y + 6 = 0?

3. Poio eÐnai to shmeÐo tom c twn eujei¸n x− 2y + 5 = 0, −2x+ 4y − 10 = 0?

4. Poia eÐnai h sqèsh an�mesa stic eujeÐec 5x+ 2y = 1 kai 2x− 5y = 3?

5. Poia eÐnai h sqèsh an�mesa stic eujeÐec 7x − 2y = 1 kai −14x + 4y = k, ìpouk par�metroc?

'Eqoume, gia k�je mia perÐptwsh:

1. To shmeÐo tom c prèpei na ikanopoieÐ tautoqrìnwc kai tic dÔo exis¸seic. LÔnontacloipìn to sÔsthma 2 × 2, prokÔptei telik� pwc to shmeÐo tom c eÐnai to x = 1,y = 2.

2. To shmeÐo tom c prèpei na ikanopoieÐ tautoqrìnwc kai tic dÔo exis¸seic. Tosugkekrimèno sÔsthma, ìmwc, den èqei lÔsh. Pr�gmati, an pollaplasi�soume thnpr¸th exÐswsh me 2 kai thn prosjèsoume sth deÔterh, prokÔptei �topo. O lìgoceÐnai ìti oi dÔo eujeÐec pou antistoiqoÔn se autèc tic exis¸seic eÐnai par�llhlec.

3. To shmeÐo tom c prèpei na ikanopoieÐ tautoqrìnwc kai tic dÔo exis¸seic. Tosugkekrimèno sÔsthma, ìmwc, èqei �peirec lÔseic. Pr�gmati, h mÐa exÐswsh eÐnaipollapl�sio thc �llhc. Se aut  thn perÐptwsh, oi dÔo perigrafèc pou èqoumeaforoÔn thn Ðdia eujeÐa.

4. Parathr ste ìti h pr¸th eujeÐa èqei k�jeto di�nusma to (5, 2) kai h deÔterh to(2,−5). Epeid  to eswterikì ginìmeno (5, 2) · (2,−5) = 0, prokÔptei telik� ìtioi eujeÐec eÐnai k�jetec.

5. Parathr ste ìti kai oi dÔo eujeÐec eÐnai k�jetec sto (7,−2). 'Ara, eÐte eÐnaipar�llhlec (an k 6= 1), eÐte tautÐzontai (an k = 1).

Orismìc 12.10. (HmiepÐpedo) OrÐzoume wc hmiepÐpedo k�je sÔnolo thc morf c

C = {(x, y) : (A,B)(x, y) = Ax+By ≥ c},

gia k�poio di�nusma (A,B) 6= 0 kai k�poio c ∈ R.

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268 KEF�ALAIO 12. DIAN�USMATA

y

x(x0,y0)

(x0,y0)+(a,b)=(x1,y1)

L

(a,b)

(A,B)

C

(A,B)

Sq ma 12.8: HmiepÐpedo C me sÔnoro eujeÐa k�jeth se di�nusma (A,B) kai dierqìmenh apì shmeÐo(x0, y0).

Parathr seic

1. To hmiepÐpedo C = {(x, y) : (A,B)(x, y) = Ax + By ≥ c} eÐnai to sÔnolo twnshmeÐwn pou brÐsketai eÐte epÐ thc eujeÐac L me exÐswsh Ax + By = c, eÐte sthmeri� thc eujeÐac proc thn opoÐa deÐqnei to k�jeto se aut  di�nusma (A,B).

Pr�gmati, èstw èna shmeÐo (x1, y1) ∈ R2, kai èstw (x1, y1) = (x0, y0) + (a, b)ìpou (x0, y0) ∈ L, dhlad  Ax0 +By0 = c. Tìte

(x1, y1) ∈ C ⇔ A(x0 + a) +B(y0 + b) ≥ c⇔ (Ax0 +By0) + (A,B) · (a, b) ≥ c

⇔ (A,B) · (a, b) ≥ 0⇔ ‖(A,B)‖‖(a, b)‖ cos θ ≥ 0⇔ cos θ ≥ 0⇔ θ ∈ [0,π

2].

ìpou θ h gwnÐa pou sqhmatÐzoun ta (a, b), (A,B). AfoÔ θ ∈ [0, π2 ], to (a, b)deÐqnei proc thn Ðdia meri� me to (A,B),   to polÔ eÐnai k�jeto se autì.

2. Gia na prosdiorÐsoume th jèsh enìc hmiepipèdou Ax + By ≥ c, mporoÔme naprosdiorÐsoume thn eujeÐa Ax + By = c, kai katìpin na broÔme apì poia meri�thc ikanopoieÐtai h anisìthta, exet�zontac èna shmeÐo ektìc thc eujeÐac.

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12.2. R2 269

0 2 4 6−1

0

1

2

3

4

5

x

y

y ≥ 0

x+y ≥ 1

x+2y ≤ 6

x−4y ≥ −8x+y ≤ 6

y ≤ 4

Sq ma 12.9: To sÔnolo S tou ParadeÐgmatoc 12.3.

Par�deigma 12.3. (Perittèc anisìthtec) 'Estw to uposÔnolo

S = {(x, y) : y ≥ 0, x+ y ≥ 1, x+ y ≤ 6, y ≤ 4, x− 4y ≥ −8, x+ 2y ≤ 6}

tou R2 pou perigr�fetai mèsw 6 anis¸sewn, dhlad  isodÔnama wc tom  6 hmiepipèdwn.Pìsec apì tic �nw anisìthtec mporoÔn na afairejoÔn apì thn �nw perigraf  qwrÐcna all�xei to sÔnolo?

Sto Sq ma 12.9 emfanÐzetai skiasmèno to sÔnolo S, kaj¸c kai k�je èna apì ta 6hmiepÐpeda. 'Opwc faÐnetai apì to sq ma, ta hmiepÐpeda pou perigr�fontai mèsw twnanis¸sewn x+ y ≤ 6, y ≤ 4, mporoÔn na afairejoÔn apì thn perigraf  tou S, qwrÐcto S na all�xei.

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270 KEF�ALAIO 12. DIAN�USMATA

12.3 R3

Orismìc 12.11. (EpÐpedo)

1. OrÐzoume wc epÐpedo k�je sÔnolo shmeÐwn

P = {(x, y, z) : (x, y, z) = (x0, y0, z0)+λ1(x1, y1, z1)+λ2(x2, y2, z2), λ1, λ2 ∈ R},(12.13)

ìpou to di�nusma (x0, y0, z0) ∈ R3, kai ta dianÔsmata (x1, y1, z1), (x2, y2, z2) ∈ R3

eÐnai mh mhdenik�, kai ìqi par�llhla.

2. H dianusmatik  exÐswsh

(x, y, z) = (x0, y0, z0) + λ1(x1, y1, z1) + λ2(x2, y2, z2)

kaleÐtai parametrik  exÐswsh epipèdou, me paramètrouc λ1, λ2 ∈ R.

3. 'Ena di�nusma V lègetai par�llhlo sto P an up�rqoun λ1, λ2 ¸ste

V = λ1(x1, y1, z1) + λ2(x2, y2, z2).

4. 'Ena di�nusma V lègetai k�jeto sto P an to V eÐnai k�jeto se ìla ta dianÔsmatathc morf c λ1(x1, y1, z1) + λ2(x2, y2, z2).

Parathr seic

1. Sthn perigraf  (12.13) to shmeÐo (x0, y0, y0) an kei sto epÐpedo, kai ta dianÔsmata(x1, y1, z1), (x2, y2, z2) eÐnai par�llhla se autì. DeÐte to Sq ma 12.10.

2. P = {plane}.

3. GiatÐ ta (x1, y1, z1) kai (x2, y2, z2) prèpei na eÐnai mh mhdenik� kai ìqi par�llhla?

4. Ti gÐnetai an (x0, y0, z0) = 0?

Je¸rhma 12.8. (Tomèc epipèdou) DÔo epÐpeda eÐte tautÐzontai, eÐte tèmnontai kat�m koc miac eujeÐac, eÐte den èqoun kanèna koinì shmeÐo, opìte kai lègontai par�llhla.

Apìdeixh. ParaleÐpetai.

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12.3. R3 271

y

x

(x0,y0,z0)

P

z

(x1,y1,z1)

(x2,y2,z2)

Sq ma 12.10: Orismìc enìc epipèdou P ston R3 b�sei enìc dianÔsmatoc (x0, y0, z0) pou an kei seautì kai dÔo dianusm�twn (x1, y1, z1), (x2, y2, z2) par�llhlwn se autì.

Je¸rhma 12.9. (Exis¸seic epipèdou)

1. (Genik  exÐswsh epipèdou) K�je epÐpedo P mporeÐ na perigrafeÐ apì mia exÐswshthc morf c

ax+ by + cz = d, (12.14)

ìpou ta a, b, c den eÐnai ìla mhdèn.

2. To epÐpedo pou perigr�fetai apì thn (12.13), dhlad  to epÐpedo sto opoÐo an kei to(x0, y0, z0) kai sto opoÐo eÐnai par�llhla ta (x1, y1, z1), (x2, y2, z2), perigr�fetaiapì thn exÐswsh ∣∣∣∣∣∣

x− x0 y − y0 z − z0

x1 y1 z1

x2 y2 z2

∣∣∣∣∣∣ = 0, (12.15)

 , isodÔnama,

(x − x0)(y1z2 − y2z1) − (y − y0)(x1z2 − x2z1) + (z − z0)(x1y2 − x2y1) = 0.

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272 KEF�ALAIO 12. DIAN�USMATA

(Parathr ste pwc to (x, y, z) an kei sto epÐpedo ann to (x − x0, y − y0, z − z0)eÐnai grammikìc sunduasmìc twn (x1, y1, z1), (x2, y2, z2), dhlad 

(x− x0, y − y0, z − z0) = λ1(x1, y1, z1) + λ2(x2, y2, z2),

gia k�poia λ1, λ2 ∈ R, to opoÐo isqÔei ann h �nw orÐzousa eÐnai 0. DeÐte toSq ma 12.10.)

3. To epÐpedo pou dièrqetai apì ta mh suneujeiak� shmeÐa

(x0, y0, z0), (x1, y1, z1), (x2, y2, z2),

mporeÐ na perigrafeÐ apì thn exÐswsh∣∣∣∣∣∣x− x0 y − y0 z − z0

x1 − x0 y1 − y0 z1 − z0

x2 − x0 y2 − y0 z2 − z0

∣∣∣∣∣∣ = 0, (12.16)

  isodÔnama

(x− x0) [(y1 − y0)(z2 − z0)− (y2 − y0)(z1 − z0)]

− (y − y0) [(x1 − x0)(z2 − z0)− (x2 − x0)(z1 − z0)]

+ (z − z0) [(x1 − x0)(y2 − y0)− (x2 − x0)(y1 − y0)] = 0.

(Parathr ste ìti h exÐswsh mporeÐ na èrjei sth morf  (12.14), kai epiplèonikanopoieÐtai kai gia ta trÐa shmeÐa. DeÐte to Sq ma 12.11.)

4. To epÐpedo pou dièrqetai apì to shmeÐo (x0, y0, z0) kai eÐnai k�jeto sto (mh mhde-nikì) di�nusma (A,B,C) mporeÐ na perigrafeÐ apì thn exÐswsh

(A,B,C) · (x− x0, y − y0, z − z0) = 0

⇔ A(x− x0) +B(y − y0) + C(z − z0) = 0

⇔ Ax+By + Cz = Ax0 +By0 + Cz0. (12.17)

(Parathr ste pwc an to (x, y, z) an kei sto epÐpedo, to (x − x0, y − y0, z − z0)eÐnai k�jeto sto (A,B,C), dhlad  (x−x0, y− y0, z− z0) · (A,B,C) = 0.) DeÐteto Sq ma 12.12.)

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12.3. R3 273

y

x(x0,y0,z0)

P

z

(x1,y1,z1)

(x2,y2,z2)

Sq ma 12.11: Orismìc enìc epipèdou P ston R3 b�sei tri¸n dianusm�twn (x0, y0, z0), (x1, y1, z1),(x2, y2, z2) pou an koun se autì.

Par�deigma 12.4. 'Estw to epÐpedo pou dièrqetai apì ta shmeÐa x0 = (1, 0, 0),x1 = (0, 1, 0), x2 = (0, 0, 1). Ja d¸soume gia to epÐpedo autì perigrafèc stic morfèc(12.14), (12.15), (12.16), (12.17) tou Jewr matoc 12.9.

Katarq n, afoÔ mac dÐnontai trÐa shmeÐa tou epipèdou, h perigraf  thc morf c(12.16) eÐnai h ∣∣∣∣∣∣

x− 1 y − 0 z − 0−1 1 0−1 0 1

∣∣∣∣∣∣ = 0,

  isodÔnama, ektel¸ntac tic pr�xeic sthn orÐzousa, sth morf  thc (12.14),

x+ y + z = 1.

ParathroÔme pwc to epÐpedo eÐnai par�llhlo sta x1−x0 = (−1, 1, 0) kai x2−x0 =(−1, 0, 1). 'Ara to epÐpedo èqei thn akìloujh perigraf  thc morf c (12.15):∣∣∣∣∣∣

x− 1 y − 0 z − 0−1 1 0−1 0 1

∣∣∣∣∣∣ = 0.

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274 KEF�ALAIO 12. DIAN�USMATA

y

x

(x0,y0,z0)

P

z

(A,B,C)

φ=π/2

(A,B,C)

Sq ma 12.12: Orismìc enìc epipèdou P ston R3 b�sei enìc dianÔsmatoc (x0, y0, z0) pou an kei seautì kai enìc dianÔsmatoc (A,B,C) k�jetou se autì.

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12.3. R3 275

H perigraf  tautÐzetai me thn perigraf  thc morf c thc morf c (12.16) pou èqoume dh brei.

Tèloc, parathroÔme pwc h

x+ y + z = 1⇔ 1× (x− 1) + 1× (y − 0) + 1× (z − 0) = 0

⇔ 1× x+ 1× y + 1× z = 1× 1 + 1× 0 + 1× 0,

pou eÐnai thc morf c (12.17). 'Ara, to epÐpedo dièrqetai apì to (1, 0, 0) kai eÐnaik�jeto sto (1, 1, 1). (Parathr ste pwc to (1, 1, 1) eÐnai k�jeto kai sta dÔo dianÔsmata(−1, 1, 0) kai (−1, 0, 1) pou xèroume pwc eÐnai par�llhla sto epÐpedo.)

Par�deigma 12.5. Ja upologÐsoume thn gwnÐa an�mesa sta epÐpeda 7x+6y+2 = 7kai 3x− 2y + 4 = 2.

Kat� ta gnwst� apì th jewrÐa, ta dÔo epÐpeda eÐnai k�jeta sta dÔo dianÔsmata(7, 6, 2), (3,−2, 4), antÐstoiqa. 'Ara, h gwnÐa pou ja sqhmatÐzoun ta dÔo epÐpeda jaeÐnai Ðsh me θ, ìpou

(7, 6, 2) · (3,−2, 4) = |(7, 6, 2)||(3,−2, 4)| cos θ

⇒ θ = arccos(7, 6, 2) · (3,−2, 4)

‖(7, 6, 2)‖‖(3,−2, 4)‖ ' 1.2296.

Parathr ste ìti dÔo epÐpeda pou tèmnontai sqhmatÐzoun mia oxeÐa kai mia ambleÐagwnÐa, me �jroisma metaxÔ touc π. H �nw mèjodoc mac èdwse thn oxeÐa, an eÐqameepilèxei ta dianÔsmata (7, 6, 2), −(3,−2, 4) = (−3, 2,−4), ja eÐqame l�bei thn ambleÐa.

Par�deigma 12.6. (Apìstash an�mesa se shmeÐo kai epÐpedo) Ja upologÐsoumethn apìstash an�mesa sto shmeÐo (x0, y0, z0) kai to epÐpedo Ax+By +Cz = D. JabroÔme epÐshc èkfrash gia thn probol  tou shmeÐou (x0, y0, z0) sto epÐpedo, dhlad to shmeÐo (x1, y1, z1) p�nw sto epÐpedo pou apèqei el�qista apì to (x0, y0, z0).

Pr�gmati, an to (x0, y0, z0) an kei sto epÐpedo, tìte h zhtoÔmenh apìstash eÐnai0, kai to (x0, y0, z0) eÐnai h probol  tou eautoÔ tou. 'Estw t¸ra pwc den an kei stoepÐpedo. Parathr ste pwc to (x0 − x1, y0 − y1, z0 − z1) eÐnai k�jeto sto epÐpedo, �rapar�llhlo sto (A,B,C), �ra up�rqei mh mhdenikì λ tètoio ¸ste

(x1, y1, z1)− (x0, y0, z0) = λ(A,B,C)

⇒ A(x1 − x0) +B(y1 − y0) + C(z1 − z0) = λ(A2 +B2 + C2)

⇒ λ =D − (Ax0 +By0 + Cz0)

A2 +B2 + C2.

H pr¸th sunepagwg  proèkuye paÐrnontac to eswterikì ginìmeno me to (A,B,C). HdeÔterh proèkuye parathr¸ntac pwc to (x1, y1, z1) an kei sto epÐpedo. 'Ara telik� to

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276 KEF�ALAIO 12. DIAN�USMATA

zhtoÔmeno shmeÐo eÐnai to

(x1, y1, z1) = (x0, y0, z0) +(A,B,C)

‖(A,B,C)‖2(D − Ax0 +By0 + Cz0),

en¸ h apìstash eÐnai Ðsh me

‖(x1 − x0, y1 − y0, z1 − z0)‖ =|D − Ax0 −By0 − Cz0|

‖(A,B,C)‖ .

Parathr ste ìti oi tÔpoi autoÐ isqÔoun kai an an kei to shmeÐo sto epÐpedo, sunep¸ceÐnai genikoÐ.

Orismìc 12.12. (EujeÐa sto q¸ro)

1. OrÐzoume wc eujeÐa sto R3 k�je sÔnolo thc morf c

L = {(x, y, z) : (x, y, z) = (x0, y0, z0) + λ(x1, y1, z1), λ ∈ R}, (12.18)

ìpou (x0, y0, y0), (x1, y1, z1) ∈ R3 kai to (x1, y1, z1) eÐnai mh mhdenikì.

2. H dianusmatik  exÐswsh

(x, y, z) = (x0, y0, z0) + λ(x1, y1, z1)

⇔ {x = x0 + λx1, y = y0 + λy1, z = z0 + λz1}

kaleÐtai parametrik  exÐswsh eujeÐac, me par�metro λ.

3. H eujeÐa kaleÐtai par�llhlh (k�jeth) se k�je (x2, y2, z2) pou eÐnai par�llhlo(k�jeto) sto (x1, y1, z1).

Parathr seic

1. Profan¸c, mia eujeÐa mporeÐ na grafeÐ me thn �nw morf  gia poll� diaforetik�(x0, y0, z0) kai (x1, y1, z1). MporeÐ na deiqjeÐ ìti ìla ta epitrept� (x1, y1, z1) eÐnaipar�llhla metaxÔ touc.

2. H eujeÐa pou perigr�fetai apì thn (12.18) dièrqetai apì to (x0, y0, z0) kai eÐnaipar�llhlh sto (x1, y1, z1).

Orismìc 12.13. (Par�llhlec eujeÐec) DÔo eujeÐec

La = {(xa0, ya0, za0) + λ(xa1, ya1, za1), λ ∈ R},Lb = {(xb0, yb0, zb0) + λ(xb1, yb1, zb1), λ ∈ R}.

kaloÔntai par�llhlec an eÐnai par�llhla ta (xa1, ya1, za1) kai (xb1, yb1, za1).

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12.3. R3 277

y

x

(x,y,z)=(x0,y0,z0)+λ(x1,y1,z1)

L

(x1,y1,z1)

(x1,y1,z1)

(x0,y0,z0)

z

Sq ma 12.13: Gewmetrik  ermhneÐa thc parametrik c exÐswshc eujeÐac sto q¸ro

Je¸rhma 12.10. (Exis¸seic eujeÐac)

1. H eujeÐa pou dhmiourgeÐtai apì thn tom  dÔo epipèdwn ikanopoieÐ kai tic dÔo exi-s¸seic touc. Gia par�deigma, h eujeÐa pou dièrqetai apì to shmeÐo (x0, y0, z0) kaieÐnai h tom  epipèdwn k�jetwn sta (a1, b1, c1), (a2, b2, c2) dÐnetai apì tic exis¸seic

a1x+ b1y + c1z = d1, a2x+ b2y + c2z = d2.

2. H eujeÐa pou perigr�fetai apì thn (12.18), dhlad  h eujeÐa pou dièrqetai apì to(x0, y0, z0) kai eÐnai par�llhlh sto (x1, y1, z1), perigr�fetai apì tic exis¸seic

x− x0

x1=y − y0

y1=z − z0

z1, an x1, y1, z1 6= 0,

x− x0

x1=y − y0

y1, z = z0, an x1, y1 6= 0, z1 = 0,

y = y0, z = z0, an x1 6= 0, y1 = z1 = 0.

(Oi peript¸seic pou paraleÐpontai eÐnai an�logec me k�poiec ek twn �nw.)

Apìdeixh. ParaleÐpetai.

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278 KEF�ALAIO 12. DIAN�USMATA

Orismìc 12.14. (EujeÐa par�llhlh se epÐpedo) Mia eujeÐa eÐte perièqetai ex' o-lokl rou se èna epÐpedo, eÐte to tèmnei se akrib¸c èna shmeÐo, eÐte den to tèmnei sekanèna shmeÐo, opìte kai kaleÐtai par�llhlh se autì.

Orismìc 12.15. (KateujÔnonta sunhmÐtona) 'Estw P = (x, y, z) mh mhdenikì di�-nusma, kai èstw θ1, θ2, θ3 oi gwnÐec pou sqhmatÐzei me ta monadiaÐa dianÔsmata, epÐ twnaxìnwn x, y, z antÐstoiqa. Ta sunhmÐton� touc kaloÔntai kateujÔnonta sunhmÐtona.ParathroÔme pwc

P · (1, 0, 0) = ‖P‖‖(1, 0, 0)‖ cos θ1 ⇒ cos θ1 =x√

x2 + y2 + z2,

kai omoÐwc èqoume:

cos θ2 =y√

x2 + y2 + z2, cos θ3 =

z√x2 + y2 + z2

.

Par�deigma 12.7. Ja deÐxoume ìti h eujeÐa x = 1 + t, y = 3 + t, z = 4− t eÐnaipar�llhlh sto epÐpedo x+ 2y + 3z = 4.

Pr�gmati, èstw ìti h eujeÐa den eÐnai par�llhlh. 'Ara up�rqei k�poio t tètoio¸ste to antÐstoiqo shmeÐo thc eujeÐac na an kei sto epÐpedo. Sunep¸c, ja prèpei naikanopoieÐ thn exÐswsh, kai me antikat�stash prokÔptei ìti

(1 + t) + 2(3 + t) + 3(4− t) = 4⇔ 1 + 6 + 12 + (1 + 2− 3)t = 4⇒ 19 = 4.

Ft�noume se �topo, �ra den up�rqei t gia to opoÐo h eujeÐa na pern� apì to epÐpedo,kai telik� h eujeÐa eÐnai par�llhlh.

'Enac �lloc trìpoc eÐnai o ex c: Katarq n, parathroÔme ìti h eujeÐa eÐnai par�l-lhlh sto di�nusma (1, 1,−1). EpÐshc, to shmeÐo (1, 0, 1) an kei sto epÐpedo, en¸ heujeÐa pou dièrqetai apì ekeÐno to shmeÐo kai prokÔptei apì thn arqik  me mia apl metatìpish eÐnai h x = 1 + t, y = 0 + t, z = 1− t. Me antikat�stash ston tÔpo touepipèdou, prokÔptei ìti h sugkekrimènh eujeÐa an kei sto epÐpedo gia k�je t. 'Ara, harqik  eujeÐa, apì thn opoÐa h nèa eujeÐa èqei prokÔyei me metatìpish, eÐnai par�llhlhsto epÐpedo.

'Alloc trìpoc epÐlushc eÐnai na upologÐsoume thn apìstash k�je shmeÐou thc eu-jeÐac apì to epÐpedo kai na doÔme ìti h apìstash paramènei stajer  kaj¸c kinoÔmasteepÐ thc eujeÐac. DeÐte, sqetik�, to epìmeno par�deigma.

Par�deigma 12.8. Ja upologÐsoume to shmeÐo tom c twn eujei¸n L1, L2, pou

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12.3. R3 279

perigr�fontai apì tic parametrikèc exis¸seic

L1 : x = 5 + t, y = 1− 2t, z = 7t,

L2 : x = 6− t, y = −1 + t, z = 7 + 3t,

kai akoloÔjwc to epÐpedo sto opoÐo an koun kai oi dÔo.Katarq n parathr ste pwc den eÐnai aparaÐthto dÔo eujeÐec sto q¸ro na tèmnontai.

Gia na tèmnontai oi sugkekrimènec, eÐnai aparaÐthto na up�rqoun dÔo par�metroi, t1,t2, tètoiec ¸ste na isqÔei

(5 + t1, 1− 2t1, 7t1) = (6− t2,−1 + t2, 7 + 3t2)

⇔ {5 + t1 = 6− t2, 1− 2t1 = −1 + t2, 7t1 = 7 + 3t2}

To �nw sÔsthma tri¸n exis¸sewn me dÔo agn¸stouc, eÔkola prokÔptei pwc èqei lÔsh,ta t1 = 1, t2 = 0, �ra oi exis¸seic èqoun shmeÐo tom c to (6,−1, 7). Kat� ta gnwst�apì th jewrÐa, to epÐpedo pou dièrqetai apì to shmeÐo (x0, y0, z0) = (6,−1, 7) kai eÐnaipar�llhlo sta (1,−2, 7) kai (−1, 1, 3) perigr�fetai apì thn exÐswsh∣∣∣∣∣∣

x− 6 y + 1 z − 71 −2 7−1 1 3

∣∣∣∣∣∣ = 0⇔ (x− 6)(−6− 7)− (y + 1)(3 + 7) + (z − 7)(1− 2)

⇔ 13x+ 10y + z = 75.

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280 KEF�ALAIO 12. DIAN�USMATA

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Kef�laio 13

Kwnikèc Tomèc

13.1 Kwnikèc Tomèc

{Orismìc} KaloÔme kwnik  tom  thn tom  enìc (diploÔ orjoÔ kuklikoÔ) k¸nou me ènaepÐpedo.

Parathr seic

1. Sto Sq ma 13.1 èqoume sqedi�sei tic treic mh tetrimmènec peript¸seic kwnik¸ntom¸n, dhlad  tic

(aþ) 'Elleiyh.

(bþ) Parabol .

(gþ) Uperbol .

2. Sto Sq ma 13.2 èqoume sqedi�sei tic tèsseric tetrimmènec peript¸seic kwnik¸ntom¸n:

(aþ) KÔkloc.

(bþ) DÔo eujeÐec.

(gþ) MÐa eujeÐa.

(dþ) 'Ena shmeÐo.

3. BebaiwjeÐte ìti èqete katal�bei pwc dhmiourgeÐtai h k�je perÐptwsh.

4. O �nw orismìc eÐnai o arqaiìteroc, all� ìqi o praktikìteroc gia tic an�gkec mac.Stic epìmenec paragr�fouc ja doÔme k�poiouc pio praktikoÔc orismoÔc.

281

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282 KEF�ALAIO 13. KWNIK�ES TOM�ES

Sq ma 13.1: Oi treic mh tetrimmènec peript¸seic kwnik¸n tom¸n: èlleiyh, parabol , kai uperbol antÐstoiqa.

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13.1. KWNIK�ES TOM�ES 283

Sq ma 13.2: Oi tèsseric tetrimmènec peript¸seic kwnik¸n tom¸n: kÔkloc, dÔo eujeÐec, mia eujeÐa,kai èna shmeÐo antÐstoiqa.

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284 KEF�ALAIO 13. KWNIK�ES TOM�ES

13.2 Allag  Suntetagmènwn

Par�deigma 13.1. 'Estw èna qwriì ìpou zoun qristianoÐ kai mousoulm�noi. Oimen qristianoÐ kanonÐzoun tic sunant seic touc orÐzontac tic suntetagmènec sun�nth-s c touc b�sei enìc kartesianoÔ sust matoc suntetagmènwn me arq  twn axìnwn thnekklhsÐa tou qwrioÔ kai me ton �xona twn x proc thn anatol . Oi de mousoulm�noikanonÐzoun tic sunant seic touc orÐzontac tic suntetagmènec sun�nths c touc b�seienìc kartesianoÔ sust matoc suntetagmènwn me arq  twn axìnwn to tzamÐ tou qwrioÔkai me ton �xona twn x proc th Mèkka. An ènac qristianìc kai èna mousoulm�nocjèloun na sunanthjoÔn, kai o qristianìc d¸sei tic suntetagmènec thc sun�nthshcsto dikì tou sÔsthma suntetagmènwn ston mousoulm�no, pwc o mousoulm�noc ja breito shmeÐo sun�nthshc sto dikì tou sÔsthma suntetagmènwn? To akìloujo je¸rhmaapant� autì to er¸thma (kai arket� �lla, praktikìtera).

Je¸rhma 13.1. (Allag  suntetagmènwn) 'Estw ta dÔo sust mata suntetagmènwnxy kai uv tou Sq matoc 13.3. H arq  twn axìnwn tou sust matoc uv brÐsketai sthjèsh (x0, y0) (wc proc to sÔsthma xy) kai to sÔsthma uv eÐnai peristrammèno kat�gwnÐa θ wc proc to xy. (H gwnÐa θ eÐnai proshmasmènh: an eÐnai jetik  (arnhtik )eÐnai antÐjeth (sÔmfwnh) me th for� tou rologioÔ.)

1. An èna shmeÐo P èqei suntetagmènec (x, y) sto sÔsthma suntetagmènwn xy, tìteèqei suntetagmènec (u, v) sto sÔsthma suntetagmènwn uv pou dÐnontai apì tonmetasqhmatismì{

u = (x− x0) cos θ + (y − y0) sin θ,v = −(x− x0) sin θ + (y − y0) cos θ

}⇔(uv

)=

(cos θ sin θ− sin θ cos θ

)(x− x0

y − y0

). (13.1)

2. An èna shmeÐo P èqei suntetagmènec (u, v) sto sÔsthma uv, tìte èqei suntetag-mènec (x, y) sto sÔsthma xy pou dÐnontai apì ton antÐstrofo metasqhmatismì{

x = x0 + u cos θ − v sin θ,y = y0 + u sin θ + v cos θ

}⇔(x

y

)=

(x0

y0

)+

(cos θ − sin θsin θ cos θ

)(u

v

). (13.2)

3. Sthn eidik  perÐptwsh pou θ = 0, oi metasqhmatismoÐ mporoÔn na grafoÔn wc(uv

)=

(x− x0

y − y0

)⇔(xy

)=

(x0 + uy0 + v

). (13.3)

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13.2. ALLAG�H SUNTETAGM�ENWN 285

y

x

vu

θ

x0

y0

P=(x,y)=(u,v)

Sq ma 13.3: Ta sust mata suntetagmènwn xy kai uv.

Apìdeixh. DeÐte to Sq ma 13.4. Parathr ste pwc

x = x0 + r cos(θ + φ) = x0 + r cosφ cos θ − r sinφ sin θ

⇒ x = x0 + u cos θ − v sin θ.

Epiplèon,

y = y0 + r sin(θ + φ) = y0 + r cosφ sin θ + r sinφ cos θ

⇒ y = y0 + u sin θ + v cos θ,

�ra br kame ton metasqhmatismì (u, v)→ (x, y).Gia na broÔme ton antÐstrofo metasqhmatismì (x, y)→ (u, v), arkeÐ na lÔsoume to

sÔsthma 2× 2 wc proc u, v.

Par�deigma 13.2. Ja apant soume ta akìlouja erwt mata:

1. Na brejoÔn oi suntetagmènec twn shmeÐwn (0, 0), (1, 1), (1,−1), (−1,−1), kai(−1, 1) (wc proc to arqikì sÔsthma suntetagmènwn xy) se èna nèo sÔsthmasuntetagmènwn, uv, me kèntro to shmeÐo (5, 2), kai peristrammèno se sqèsh meto arqikì, xy, kat� gwnÐa π/3. Ta shmeÐa èqoun sqediasteÐ sto Sq ma 13.5(arister�).

2. Na brejoÔn oi suntetagmènec, sto arqikì sÔsthma suntetagmènwn, xy twn sh-meÐwn pou sto �nw, nèo sÔsthma suntetagmènwn uv èqoun suntetagmènec (0, 0),(1, 1), (1,−1), (−1,−1), (−1, 1). Ta shmeÐa emfanÐzontai sto Sq ma 13.5 (ari-ster�).

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286 KEF�ALAIO 13. KWNIK�ES TOM�ES

y

x

v

u

θ

x0

y0

P=(x,y)=(u,v)

θ

φ

rv=r sinφ

u=r cosφ

y

x

Sq ma 13.4: Apìdeixh twn exis¸sewn allag c suntetagmènwn.

3. Na brejeÐ èna �llo sÔsthma suntetagmènwn uv b�sei tou opoÐou ta shmeÐa (0, 0)kai (1, 0) (wc proc to xy) apeikonÐzontai sta (0, 2

√2) kai (

√2/2, 3

√2/2) (wc

proc to uv).

'Eqoume, kat� perÐptwsh:

1. Ja efarmìsoume ton metasqhmatismì (13.1) ìpou x0 = 5, y0 = 2, kai θ = π/3,kai x, y tic dosmènec suntetagmènec, kat� perÐptwsh. Me apl  antikat�stash,prokÔptei telik� pwc ta shmeÐa, se suntetagmènec uv, me akrÐbeia dÔo dekadik¸nyhfÐwn, eÐnai ta (−4.23, 3.33), (−2.86, 2.96), (−4.59, 1.96), (−3.86, 4.69), kai(−5.59, 3.69).

2. Ja efarmìsoume ton antÐstrofo metasqhmatismì (13.2) me ta x0, y0, θ wc �nw,kai u, v tic dosmènec suntetagmènec, kat� perÐptwsh. Me apl  antikat�stash,prokÔptei telik� pwc oi arqikèc suntetagmènec, sto sÔsthma xy, eÐnai oi (5, 2),(4.63, 3.36), (6.36, 2.36), (5.36, 0.63), (3.63, 1.63).

3. Se aut  thn perÐptwsh, mac eÐnai �gnwsta ta x0, y0, kai θ, kai gia na ta upologÐ-soume ja qrhsimopoi soume tic (13.1) gia ta dÔo shmeÐa pou dÐnontai. Pr�gmati,apì to pr¸to zeÔgoc suntetagmènwn mac dÐnetai:

0 = (0− x0) cos θ + (0− y0) sin θ ⇒ x0 cos θ + y0 sin θ = 0 (13.4)

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13.2. ALLAG�H SUNTETAGM�ENWN 287

Sq ma 13.5: Oi �xonec suntetagmènwn thc 'Askhshc 13.2.

kai

2√

2 = −(0− x0) sin θ + (0− y0) cos θ ⇒ x0 sin θ − y0 cos θ = 2√

2, (13.5)

en¸ apì to deÔtero zeÔgoc suntetagmènwn mac dÐnetai:

√2/2 = (1− x0) cos θ + (0− y0) sin θ

⇒ x0 cos θ + y0 sin θ = cos θ −√

2/2 (13.6)

kai

3√

2/2 = −(1− x0) sin θ + (0− y0) cos θ

⇒ x0 sin θ − y0 cos θ = sin θ + 3√

2/2. (13.7)

Apì tic (13.4) kai (13.6) prokÔptei ìti cos θ =√

2/2, en¸ apì tic (13.5), (13.7)prokÔptei ìti sin θ =

√2/2, �ra telik� θ = π/4. Me dedomènh thn tim  tou θ,

apì thn (13.4) prokÔptei pwc x0 = −y0, kai th (13.5) pwc x0 = 2, �ra y0 = −2.Ta shmeÐa, oi arqikoÐ �xonec, kai oi nèoi �xonec emfanÐzontai sto Sq ma 13.5(dexi�).

Parathr seic

1. An ènac gewmetrikìc tìpoc perigr�fetai apì thn exÐswsh f(x, y) = 0 sto sÔ-sthma suntetagmènwn (x, y), tìte antikajist¸ntac ta x, y me qr sh thc (13.2)

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288 KEF�ALAIO 13. KWNIK�ES TOM�ES

mporoÔme na katal xoume se mia �llh exÐswsh g(u, v) = 0 pou perigr�fei tongewmetrikì tìpo sto sÔsthma suntetagmènwn uv.

Eidik� ìtan ta dÔo sust mata eÐnai par�llhla metaxÔ touc, dhlad  θ = 0, tìteisqÔei h eidik  perÐptwsh (13.3) kai h perigraf  tou gewmetrikoÔ tìpou f(x, y) =0 sto sÔsthma suntetagmènwn uv eÐnai h f(x, y) = 0⇔ f(u+ x0, v + y0) = 0.

Gia par�deigma, afoÔ o kÔkloc me kèntro thn arq  twn axìnwn kai aktÐna Rperigr�fetai apì thn exÐswsh

x2 + y2 = R2,

o Ðdioc kÔkloc sto sÔsthma suntetagmènwn uv me kèntro (x0, y0) = (1,−2) kai�xonec par�llhlouc stouc arqikoÔc perigr�fetai apì thn exÐswsh

(u+ 1)2 + (v − 2)2 = R2.

Me parìmoio trìpo, mporoÔme na xekin soume apì mia exÐswsh gewmetrikoÔ tìpousto sÔsthma uv kai na katal xoume se mia exÐswsh sto sÔsthma xy gia ton Ðdiotìpo.

2. 'Estw pwc mac dÐnetai ènac gewmetrikìc tìpoc f(x, y) = 0 kai èstw pwc jèloumena ton metatopÐsoume kat� x0 epÐ tou �xona twn x, kat� y0 epÐ tou �xona twn y,kai na ton peristrèyoume kat� gwnÐa θ. Parathr ste ìti o nèoc (metatopismènoc)gewmetrikìc tìpoc perigr�fetai apì thn exÐswsh f(u, v) = 0 ìpou ta u, v dÐnontaiapì thn (13.1).

Eidik� ìtan o gewmetrikìc tìpoc f(x, y) = 0 metafèretai qwrÐc peristrof ,dhlad  θ = 0, tìte isqÔei h eidik  perÐptwsh (13.3) kai o nèoc gewmetrikìc tìpoceÐnai o f(u, v) = 0⇔ f(x− x0, y − y0) = 0.

Gia par�deigma, afoÔ o kÔkloc me kèntro thn arq  twn axìnwn kai aktÐna Rperigr�fetai apì thn exÐswsh

x2 + y2 = R2,

o kÔkloc me aktÐna R kai kèntro to (x0, y0) = (1,−2) eÐnai o

(x− 1)2 + (y + 2)2 = R2.

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13.3. PARABOL�H 289

13.3 Parabol 

Orismìc 13.1. (Parabol ) O gewmetrikìc tìpoc P twn shmeÐwn pou isapèqoun apìmia eujeÐa d (pou kaleÐtai dieujetoÔsa) kai èna shmeÐo F (pou kaleÐtai estÐa) kaleÐtaiparabol . To shmeÐo V pou brÐsketai an�mesa sthn dieujetoÔsa kai thn estÐa kaleÐtaikoruf  thc parabol c. H eujeÐa pou eÐnai k�jeth sth dieujetoÔsa kai dièrqetai apìthn estÐa kaleÐtai �xonac thc parabol c.

Je¸rhma 13.2. (Kanonikèc morfèc parabol c) 'Estw p > 0.

1. H parabol  me estÐa to shmeÐo F = (p, 0) kai dieujetoÔsa thn eujeÐa x = −pèqei koruf  to shmeÐo (0, 0), �xona thn eujeÐa y = 0, kai exÐswsh

y2 = 4px.

2. H parabol  me estÐa to shmeÐo F = (−p, 0) kai dieujetoÔsa thn eujeÐa x = p

èqei koruf  to shmeÐo (0, 0), �xona thn eujeÐa y = 0, kai exÐswsh

y2 = −4px.

3. H parabol  me estÐa to shmeÐo F = (0, p) kai dieujetoÔsa thn eujeÐa y = −pèqei koruf  to shmeÐo (0, 0), �xona thn eujeÐa x = 0, kai exÐswsh

x2 = 4py.

4. H parabol  me estÐa to shmeÐo F = (0,−p) kai dieujetoÔsa thn eujeÐa y = pèqei koruf  to shmeÐo (0, 0), �xona thn eujeÐa x = 0, kai exÐswsh

x2 = −4py.

Oi �nw parabolèc (kai idiaÐtera h pr¸th) kaloÔntai suqn� kanonikèc morfèc.

Apìdeixh. Ja apodeÐxoume thn pr¸th perÐptwsh. Oi �llec prokÔptoun parìmoia. EÐnaiprofanèc ìti h koruf  kai o �xonac eÐnai autoÐ pou dÐnontai. Sqetik� me thn exÐsw-sh pou dÐnetai, èstw èna shmeÐo (x, y) thc parabol c. H apìstash tou d1 apì thdieujetoÔsa eÐnai Ðsh me thn apìstash tou d2 apì thn estÐa, �ra:

d1 = d2 ⇔√

(x+ p)2 + 02 =√

(p− x)2 + y2

⇔ x2 + 2px+ p2 = p2 − 2xp+ y2 + x2 ⇔ y2 = 4px.

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290 KEF�ALAIO 13. KWNIK�ES TOM�ES

Εστία FΚορυφή V

Παραβολή P

Άξονας

Διε

υθετ

ούσ

α d

d1

d2

d1=d2

Sq ma 13.6: Parabol c kai basik� qarakthristik�.

Parat rhsh: Akìma kai oi parabolèc pou den eÐnai se kanonik  morf , mporoÔnna èrjoun se kanonik  morf  me mia allag  suntetagmènwn se èna sÔsthma suntetag-mènwn uv me kèntro thn koruf  thc parabol c kai ton �xona u   v na tautÐzetai meton �xona thc parabol c. GnwrÐzontac th sqèsh pou èqoun ta sust mata uv kai xy,mporoÔme na broÔme thn exÐswsh thc sto sÔsthma xy. DeÐte to akìloujo par�deigma.

Par�deigma 13.3. Ja broÔme thn exÐswsh parabol c me estÐa sto (4, 3) kai dieu-jetoÔsa thn y + 1 = 0. ParathroÔme katarq n ìti h koruf  thc parabol c brÐsketaisto misì thc apìstashc thc estÐac apì thn dieujetoÔsa, dhlad  sto shmeÐo (4, 1).AfoÔ h dieujetoÔsa eÐnai par�llhlh ston �xona twn x, o �xon�c thc eÐnai par�llhlocston �xona twn y. 'Estw loipìn to sÔsthma uv me kèntro sth jèsh (4, 1) kai me �xonecx, u par�llhlouc kai y, v par�llhlouc. Oi metasqhmatismoÐ suntetagmènwn eÐnai oi

{x = 4 + u, y = 1 + v} ⇔ {u = x− 4, v = y − 1}.Sto sÔsthma uv h parabol  èqei kanonik  morf  me exÐswsh 4pv = u2. Mènei oprosdiorismìc tou p. ParathroÔme pwc p eÐnai h apìstash an�mesa sthn koruf  kaithn estÐa, �ra p = 2 kai telik� h exÐswsh eÐnai h

8v = u2 ⇔ 8(y − 1) = (x− 4)2.

H parabol  èqei sqediasteÐ sto Sq ma 13.8.

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13.3. PARABOL�H 291

x=

−p

F = (p, 0)(0, 0)

y2 = 4px

x

y

F = (−p, 0) (0, 0)

x

y

y2 = −4px

y = −p

F = (0, p)

(0, 0)

x2 = 4py

y

x

y = p

F = (0,−p)

(0, 0)

x2 = −4py

y

x

Sq ma 13.7: Oi tèsseric kanonikèc morfèc thc parabol c kai oi exis¸seic touc.

−2 0 2 4 6 8 10−2

0

2

4

(4,1)

(4,3)

y + 1 = 0

y

x

Sq ma 13.8: H parabol  tou ParadeÐgmatoc 13.3.

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292 KEF�ALAIO 13. KWNIK�ES TOM�ES

13.4 'Elleiyh

Orismìc 13.2. ('Elleiyh) O gewmetrikìc tìpoc P twn shmeÐwn gia ta opoÐa to�jroisma d1 +d2 twn apost�sewn touc d1 kai d2 apì dÔo estÐec F1 kai F2 eÐnai stajerìkaleÐtai èlleiyh. H eujeÐa pou dièrqetai apì tic dÔo estÐec kaleÐtai meg�loc �xonac.To shmeÐo O pou brÐsketai sto mèso tou eujugr�mmou tm matoc F1F2 kaleÐtai kèntrothc èlleiyhc. H eujeÐa pou dièrqetai apì to kèntro kai eÐnai k�jeth ston meg�lo �xonakaleÐtai mikrìc �xonac. Ta shmeÐa A1 kai A2 ìpou tèmnetai o meg�loc �xonac me thnèlleiyh kai B1 kai B2 ìpou tèmnetai o mikrìc �xonac me thn èlleiyh kaloÔntai korufèc.

'Estw 2a , |A1A2| = d1 + d2. OrÐzoume thn ekkentrìthta thc èlleiyhc wc ton ε giaton opoÐo h apìstash an�mesa stic estÐec F1, F2, isoÔtai me |F1F2| = 2aε. To |A1A2|kaleÐtai m koc tou meg�lou �xona kai to |B1B2| kaleÐtai m koc tou mikroÔ �xona.

Parathr seic

1. PerÐptwsh ε = 0: Oi estÐec tautÐzontai, kai lamb�noume kÔklo.

2. PerÐptwsh ε = 1: Oi estÐec tautÐzontai me tic korufèc, kai h èlleiyh ekfulÐzetaieujÔgrammo tm ma an�mesa stic estÐec.

3. PerÐptwsh ε > 1: O gewmetrikìc tìpoc eÐnai to kenì sÔnolo (giatÐ?).

4. PerÐptwsh 0 < ε < 1: H mình mh tetrimmènh.

Je¸rhma 13.3. (Exis¸seic èlleiyhc)

1. H èlleiyh me korufèc tou meg�lou �xona ta A1 = (−a, 0) kai A2 = (a, 0) kaiestÐec ta F1 = (−aε, 0), F2 = (aε, 0), ìpou 0 < ε < 1, èqei korufèc tou mikroÔ�xona ta B1 = (0, b), B2 = (0,−b), ìpou b = a

√1− ε2, kèntro to (0, 0), m kh

meg�lou kai mikroÔ �xona |A1A2| = 2a kai |B1B2| = 2b antÐstoiqa, kai exÐswsh

x2

a2+

y2

a2(1− ε2) = 1⇔ x2

a2+y2

b2= 1.

2. H èlleiyh me korufèc tou meg�lou �xona ta A1 = (0, a) kai A2 = (0,−a) kai meestÐec ta F1 = (0, aε), F2 = (0,−aε), ìpou 0 < ε < 1, èqei korufèc tou mikroÔ�xona ta B1 = (−b, 0), B2 = (b, 0), ìpou b = a

√1− ε2, kèntro to (0, 0), m kh

meg�lou kai mikroÔ �xona |A1A2| = 2a kai |B1B2| = 2b antÐstoiqa, kai exÐswsh

x2

a2(1− ε2) +y2

a2= 1⇔ x2

b2+y2

a2= 1.

Oi �nw elleÐyeic (kai eidik� h pr¸th) kaloÔntai suqn� kanonikèc morfèc.

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13.4. �ELLEIYH 293

Κέντρο Ο

Εστία F1 Εστία F2Κορυφή Α1 Κορυφή Α2

Κορυφή Β1

Κορυφή Β2

Έλλειψη P

Μεγάλος

Άξονας

Μικρός

Άξονας

d1 d2

d1+d2=2a, |F1F2|=2aε

Sq ma 13.9: 'Elleiyh kai basik� qarakthristik�.

Apìdeixh. Ja apodeÐxoume thn pr¸th perÐptwsh. H deÔterh prokÔptei an�loga. 'EnashmeÐo (x, y) ja an kei sthn èlleiyh ann

d1 + d2 = 2a⇔√

(x+ aε)2 + y2 +√

(x− aε)2 + y2 = 2a

⇔ (x+ aε)2 + y2 + (x− aε)2 + y2 + 2√

[(x+ aε)2 + y2][(x− aε)2 + y2] = 4a2

⇔[(x+ aε)2 + y2 + (x− aε)2 + y2 − 4a2

]2= 4[(x+ aε)2 + y2][(x− aε)2 + y2]

⇔ (x+ aε)4 + (x− aε)4 + 4y4 + 16a4 + 2(x+ aε)2(x− aε)2 + 4(x+ aε)2y2

− 8(x+ aε)2a2 + 4(x− aε)2y2 − 8(x− aε)2a2 − 16a2y2

= 4(x+ aε)2(x− aε)2 + 4y4 + 4y2(x− aε)2 + 4y2(x+ aε)2

⇔ (x+aε)4+(x−aε)4+16a4−2(x+aε)2(x−aε)2−8(x+aε)2a2−8(x−aε)2a2−16a2y2 = 0

⇔[(x+ aε)2 − (x− aε)2

]2 − 8a2[−2a2 + (x+ aε)2 + (x− aε)2 + 2y2

]= 0

⇔ 16a2x2ε2+16a4−16a2x2−16a4ε2−16a2y2 = 0⇔ x2(a2ε2−a2)−a2y2 = a4(ε2−1)

⇔ x2(1− ε2) + y2 = a2(1− ε2)⇔ x2

a2+

y2

a2(1− ε2) = 1.

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294 KEF�ALAIO 13. KWNIK�ES TOM�ES

x2

a2 + y2

b2 = 1

A1 = (−a, 0) A2 = (a, 0)

B1 = (0, b)

B2 = (0,−b)

F1 = (−aε, 0) F2 = (aε, 0)

O = (0, 0)

y

x

x2

b2 + y2

a2 = 1A1 = (0, a)

A2 = (0,−a)

B1 = (−b, 0) B2 = (b, 0)

F1 = (0, aε)

F2 = (0,−aε)

O = (0, 0)

y

x

Sq ma 13.10: Oi dÔo kanonikèc morfèc thc èlleiyhc kai oi exis¸seic touc.

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13.4. �ELLEIYH 295

1 2 3 4 5 6 7−2

−1

0

1

2

3

4

(4, 1)

(6.1,−1.1)

(1.9, 3.1)

(5.4, 2.4)

(2.6,−0.4)(5.6,−0.6)

(2.4, 2.6)

x

uv

Sq ma 13.11: H èlleiyh tou ParadeÐgmatoc 13.4.

Parat rhsh: Oi elleÐyeic pou den eÐnai se kanonik  morf , mporoÔn na èrjounse kanonik  morf  me mia allag  suntetagmènwn se èna kat�llhla epilegmèno sÔ-sthma suntetagmènwn uv. GnwrÐzontac th sqèsh pou èqoun ta sust mata uv kai xy,mporoÔme na broÔme thn exÐswsh touc sto sÔsthma xy. DeÐte to akìloujo par�deigma.

Par�deigma 13.4. Ja broÔme thn exÐswsh pou perigr�fei thn èlleiyh me kèntro toshmeÐo (4, 1), me m kh axìnwn 2a = 6 kai 2b = 4, kai me ton mikrì �xona na brÐsketaiupì gwnÐa π/4 se sqèsh me ton �xona twn x. H èlleiyh perigr�fetai apì thn exÐswsh

u2

b2+v2

a2= 1⇔ u2

4+v2

9= 1,

ìpou to sÔsthma suntetagmènwn brÐsketai me kèntro to shmeÐo (x0, y0) = (4, 1), kaiupì gwnÐa, se sqèsh me ton �xona twn x, Ðsh me θ = π/4. 'Ara

u = (x− x0) cos θ + (y − y0) sin θ = (x− 4)√

2/2 + (y − 1)√

2/2,

v = −(x− x0) sin θ + (y − y0) cos θ = −(x− 4)√

2/2 + (y − 1)√

2/2.

Me antikat�stash tou �nw metasqhmatismoÔ sthn arqik  exÐswsh, prokÔptei

9(

(x− 4)√

2/2 + (y − 1)√

2/2)2

+ 4(−(x− 4)

√2/2 + (y − 1)

√2/2)2

= 36

⇔ 13x2 + 13y2 + 10xy − 114x− 66y + 189 = 0.

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296 KEF�ALAIO 13. KWNIK�ES TOM�ES

13.5 Uperbol 

Orismìc 13.3. (Uperbol ) O gewmetrikìc tìpoc P twn shmeÐwn gia ta opoÐah apìluth diafor� |d1 − d2| twn apost�sewn touc d1 kai d2 apì dÔo estÐec F1 kai F2

(ennoeÐtai me F1 6= F2) eÐnai stajer  (kai ennoeÐtai jetik ) kaleÐtai uperbol . H eujeÐapou dièrqetai apì tic dÔo estÐec kaleÐtai �xonac. To shmeÐo O pou brÐsketai sto mèsotou eujugr�mmou tm matoc F1F2 kaleÐtai kèntro. Ta shmeÐa A1 kai A2 ìpou tèmnetai o

�xonac me thn uperbol  kaloÔntai korufèc. 'Estw 2a , |d1− d2| = |A1A2|. OrÐzoumethn ekkentrìthta thc uperbol c wc ton arijmì ε gia ton opoÐo h apìstash an�mesastic estÐec F1, F2, isoÔtai me |F1F2| = 2aε.

Parathr seic

1. PerÐptwsh ε = 0: AdÔnato diìti ex upojèsewc F1 6= F2.

2. PerÐptwsh ε < 1: Apì thn trigwnik  anisìthta, prokÔptei pwc o gewmetrikìctìpoc eÐnai to kenì sÔnolo. (Pwc akrib¸c?)

3. PerÐptwsh ε = 1: Oi estÐec tautÐzontai me tic korufèc, kai h èlleiyh ekfulÐzetaise dÔo hmieujeÐec.

4. PerÐptwsh ε > 1: H mình mh tetrimmènh.

Je¸rhma 13.4. (Exis¸seic uperbol c)

1. H uperbol  me korufèc ta shmeÐa A1 = (−a, 0) kai A2 = (a, 0) kai estÐec tashmeÐa F1 = (−aε, 0), F2 = (aε, 0), ìpou ε > 1, èqei exÐswsh

x2

a2− y2

a2(ε2 − 1)= 1⇔ x2

a2− y2

b2= 1.

2. H uperbol  me korufèc ta shmeÐa A1 = (0, a) kai A2 = (0,−a) kai estÐec tashmeÐa F1 = (0, aε), F2 = (0,−aε), ìpou ε > 1, èqei exÐswsh

y2

a2− x2

a2(ε2 − 1)= 1⇔ y2

a2− x2

b2= 1.

Stic �nw, b , a√ε2 − 1. Oi �nw uperbolèc (kai idiaÐtera h pr¸th) kaloÔntai suqn�

kanonikèc morfèc.

Apìdeixh. 'Opwc kai h antÐstoiqh gia thn perÐptwsh thc èlleiyhc.

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13.5. UPERBOL�H 297

Κορυφή A1

Άξονας

Υπερβολή P

Κορυφή A2

Κέντρο ΟΕστία F1 Εστία F2

d1d2

|d1-d2|=2a, |F1F2|=2aε

Sq ma 13.12: Uperbol  kai basik� qarakthristik�.

Parat rhsh: 'Opwc kai stic prohgoÔmenec peript¸seic, oi uperbolèc pou den eÐnaise kanonik  morf , mporoÔn na èrjoun se kanonik  morf  me mia allag  suntetagmè-nwn se èna kat�llhla epilegmèno sÔsthma suntetagmènwn uv. GnwrÐzontac th sqèshpou èqoun ta sust mata uv kai xy, mporoÔme na broÔme thn exÐswsh touc sto sÔsthmaxy. DeÐte to akìloujo par�deigma.

Par�deigma 13.5. Ja broÔme thn exÐswsh uperbol c me kèntro to (−2,−4) meton �xona na brÐsketai upì gwnÐa π

6 se sqèsh me ton �xona twn x, kai paramètrouca = 3, b = 2.

ParathroÔme pwc h uperbol  perigr�fetai apì thn exÐswsh

u2

9− v2

4= 1,

ìpou to sÔsthma axìnwn uv brÐsketai upì gwnÐa θ = π/6 se sqèsh me to sÔsth-ma axìnwn xy kai me kèntro to (x0, y0) = (−2,−4). 'Ara mpor¸ na efarmìsw tonmetasqhmatismì

u = (x− x0) cos θ + (y − y0) sin θ = (x+ 2)√

3/2 + (y + 4)/2,

v = −(x− x0) sin θ + (y − y0) cos θ = −(x+ 2)/2 + (y + 4)√

3/2.

Page 308: ΣΗΜΕΙΩΣΕΙΣ-ΜΑΘΗΜΑΤΙΚΑ 1 ΠΛΗΡΟΦΟΡΙΚΗ ΑΣΟΕΕ 1 ΕΞΑΜΗΝΟ

298 KEF�ALAIO 13. KWNIK�ES TOM�ES

x2

a2 − y2

b2 = 1

A1 = (−a, 0) A2 = (a, 0)

F1 = (−aε, 0) F2 = (aε, 0)O = (0, 0)

y

x

y2

a2 − x2

b2 = 1

A1 = (0, a)

A2 = (0,−a)

F1 = (0, aε)

F2 = (0,−aε)

O = (0, 0)

y

x

Sq ma 13.13: Oi dÔo kanonikèc morfèc thc uperbol c kai oi exis¸seic touc.

Page 309: ΣΗΜΕΙΩΣΕΙΣ-ΜΑΘΗΜΑΤΙΚΑ 1 ΠΛΗΡΟΦΟΡΙΚΗ ΑΣΟΕΕ 1 ΕΞΑΜΗΝΟ

13.5. UPERBOL�H 299

−6 −4 −2 0 2 4−10

−8

−6

−4

−2

0

(−2,−4)

(0.6,−2.5)

(−4.6,−5.5)

(1.1,−2.2)

(−5.1,−5.8)

xy

u

v

Sq ma 13.14: H uperbol  tou ParadeÐgmatoc 13.5.

Me antikat�stash tou �nw metasqhmatismoÔ sthn arqik  exÐswsh, prokÔptei

4((x+ 2)√

3/2 + (y + 4)/2)2 − 9(−(x+ 2)/2 + (y + 4)√

3/2)2 = 36

⇔ 3x2 − 23y2 + 26√

3xy + (12 + 104√

3)x+ (−184 + 52√

3)y = −208√

3.

H uperbol  èqei sqediasteÐ sto Sq ma 13.14.

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300 KEF�ALAIO 13. KWNIK�ES TOM�ES

13.6 Ax2 + Cy2 +Dx + Ey + F = 0

Parat rhsh: 'Olec oi exis¸seic kwnik¸n tom¸n me touc �xonèc touc par�llhloucproc touc �xonec x  /kai y eÐnai thc genik c morf c

Ax2 + Cy2 +Dx+ Ey + F = 0.

(MporeÐte na exhg sete giatÐ?) IsqÔei to antÐstrofo? Dhlad , h �nw exÐswsh peri-gr�fei p�nta kwnik  tom  tètoiac morf c? To parak�tw je¸rhma apant� jetik�.

Je¸rhma 13.5. (Kwnikèc tomèc me �xonec par�llhlouc stouc x, y) H exÐswsh

Ax2 + Cy2 +Dx+ Ey + F = 0 (13.8)

perigr�fei eÐte kwnik  tom  me �xona(ec) par�llhlo(ouc) proc touc �xonec twn x   y,eÐte orismènec tetrimmènec peript¸seic.

Apìdeixh. DiakrÐnoume peript¸seic:

1. A = 0, C 6= 0, D 6= 0. Parathr ste pwc

Cy2 +Dx+ Ey + F = 0⇔ y2 + 2

(E

2C

)y +

D

Cx+

F

C= 0

⇔ y2 + 2

(E

2C

)y +

E2

4C2=

E2

4C2− D

Cx− F

C

⇔(y +

E

2C

)2

=

(−DC

)[x− E2

4CD+F

D

].

Sunep¸c, h exÐswsh mporeÐ na èrjei sth morf  (y − y0)2 = ±4p(x − x0). Me

allag  suntetagmènwn u = x − x0, v = y − y0, prokÔptei kanonik  morf parabol c. ProkÔptei telik� pwc èqoume parabol  me kèntro to (x0, y0), kai�xona par�llhlo ston �xona twn x.

2. A = 0, C 6= 0, D = 0. Akolouj¸ntac th mèjodo thc pr¸thc perÐptwshc,prokÔptei h exÐswsh mporeÐ na èrjei sth morf  (y − y0)

2 = k, �ra èqoume dÔopar�llhlec eujeÐec par�llhlec ston �xona twn x, tic y = y0 ±

√k, an

k > 0, mia eujeÐa par�llhlh ston �xona twn x, thn y = y0, an k = 0,   tokenì sÔnolo, an k < 0.

3. A 6= 0, C = 0, E 6= 0. AnalÔetai ìpwc h pr¸th perÐptwsh, kai prokÔpteiparabol  me �xona par�llhlo ston �xona twn y.

Page 311: ΣΗΜΕΙΩΣΕΙΣ-ΜΑΘΗΜΑΤΙΚΑ 1 ΠΛΗΡΟΦΟΡΙΚΗ ΑΣΟΕΕ 1 ΕΞΑΜΗΝΟ

13.6. AX2 + CY 2 +DX + EY + F = 0 301

4. A 6= 0, C = 0, E = 0. AnalÔetai ìpwc h deÔterh perÐptwsh, kai prokÔptoundÔo par�llhlec eujeÐec, par�llhlec ston �xona twn y,   mia eujeÐapar�llhlh ston �xona twn y,   to kenì sÔnolo.

5. A = 0, C = 0. H exÐswsh gÐnetai Dx+Ey+F = 0, dhlad  perigr�fei eujeÐa.

6. AC 6= 0. Se aut  thn perÐptwsh, parathr ste ìti

Ax2 + Cy2 +Dx+ Ey + F = 0⇔ A

[x2 + 2

(D

2A

)x+

(D

2A

)2]

+

C

[y2 + 2

(E

2C

)y +

(E

2C

)2]

=D2

4A+E2

4C− F

⇔(x+ D

2A

)2

C+

(y + E

2C

)2

A=

D2

4A2C+

E2

4AC2− F

AC.

'Eqoume dÔo upopeript¸seic:

(aþ) AC > 0. Tìte h exÐswsh mporeÐ na èrjei sth morf 

(x− x0)2

a2+

(y − y0)2

b2= k,

me k ∈ R. Me allag  suntetagmènwn u = x − x0, v = y − y0, prokÔpteikanonik  morf  èlleiyhc. 'Ara èqoume èlleiyh me kèntro to (x0, y0), ank > 0, èna shmeÐo, an k = 0,   to kenì sÔnolo, an k < 0.

(bþ) AC < 0. Tìte h exÐswsh mporeÐ na èrjei sth morf 

(x− x0)2

a2− (y − y0)

2

b2= k,

me k ∈ R. Me allag  suntetagmènwn u = x − x0, v = y − y0, prokÔpteikanonik  morf  uperbol c. 'Ara èqoume uperbol  me kèntro to (x0, y0), ank 6= 0,   dÔo diastauroÔmenec eujeÐec pou dièrqontai apì to (x0, y0),kai exis¸seic (x− x0) = ± b

a(y − y0), an k = 0.

(Se ìlec tic peript¸seic, bebaiwjeÐte ìti mporeÐte na breÐte tic paramètrouc twn kw-nik¸n tom¸n x0, y0, p, a, b sunart sei twn A,C,D,E, F .)

Parat rhsh: 'Opote mac dÐnetai mia exÐswsh thc morf c (13.8), mporoÔme na epa-nal�boume ta b mata thc apìdeixhc kai na broÔme ètsi th morf  thc kwnik c tom cpou perigr�fei h exÐswsh. DeÐte to akìloujo par�deigma.

Page 312: ΣΗΜΕΙΩΣΕΙΣ-ΜΑΘΗΜΑΤΙΚΑ 1 ΠΛΗΡΟΦΟΡΙΚΗ ΑΣΟΕΕ 1 ΕΞΑΜΗΝΟ

302 KEF�ALAIO 13. KWNIK�ES TOM�ES

−1 0 1 2 3 4−1

0

1

2

3

4

5

6

(2, 3) (3, 3)

(1, 3)

(2, 5)

(2, 1)

y=

λ1x

y = λ2x

y

x

−3 −2 −1 0 1−5

−4

−3

−2

−1

0

1

y=

√ 2x−

2+

√ 2

y=− √

2x −2 − √

2(0, 0)

(−1,−2)

y

x

Sq ma 13.15: Oi gewmetrikoÐ tìpoi tou ParadeÐgmatoc 13.6.

Par�deigma 13.6. Ja perigr�youme touc gewmetrikoÔc tìpouc pou dÐnontai apìtic exis¸seic:

1. 4x2 − 16x + y2 − 6y + 21 = 0. AkoloÔjwc, ja brejoÔn ìlec oi eujeÐec poudièrqontai apì thn arq  twn axìnwn kai eÐnai efaptomenikèc ston �nw gewmetrikìtìpo.

2. 2x2 − y2 + 4x− 4y − 2 = 0.

'Eqoume, kat� perÐptwsh:

1. Sumplhr¸nontac ta tetr�gwna, eÔkola paÐrnoume

4x2− 16x+ y2− 6y+ 21 = 0⇔ 4(x2− 4x+ 4) + (y2− 6y+ 9) = 16 + 9− 21

⇔ (x− 2)2 +(y − 3)2

4= 1, (13.9)

�ra o gewmetrikìc tìpoc eÐnai èlleiyh me kèntro to (2, 3), mikrì �xona kat� m koctou �xona twn x Ðso me 2a = 2 kai meg�lo �xona kat� m koc tou �xona twn ym kouc 2b = 4.

Page 313: ΣΗΜΕΙΩΣΕΙΣ-ΜΑΘΗΜΑΤΙΚΑ 1 ΠΛΗΡΟΦΟΡΙΚΗ ΑΣΟΕΕ 1 ΕΞΑΜΗΝΟ

13.6. AX2 + CY 2 +DX + EY + F = 0 303

Sqetik� me thn eÔresh efaptomenik¸n eujei¸n, mia eujeÐa pou dièrqetai apì thnarq  twn axìnwn èqei exÐswsh

y = λx. (13.10)

'Ena shmeÐo (x0, y0) pou an kei kai sthn èlleiyh kai sthn eujeÐa ja prèpei naikanopoieÐ tic (13.9), (13.10), kai sunep¸c,

(x0 − 2)2 +(λx0 − 3)2

4= 1⇔ (4 + λ2)x2

0 + (−6λ− 16)x0 + 21 = 0.

An to �nw tri¸numo èqei dÔo rÐzec, profan¸c h eujeÐa tèmnei thn èlleiyh se dÔoshmeÐa. An den èqei rÐzec, tìte h èlleiyh kai h eujeÐa den tèmnontai. An èqeiakrib¸c mia rÐza, tìte profan¸c h eujeÐa eÐnai efaptomenik  sthn èlleiyh. Giana gÐnei autì, prèpei h diakrÐnousa na eÐnai 0. 'Ara:

∆ = 0⇔ (16 + 6λ)2 − 4× 21× (4 + λ2) = 0⇔ 3λ2 − 12λ+ 5 = 0,

pou èqei dÔo rÐzec, tic

λ1 = 2 +

√21

3, λ2 = 2−

√21

3.

'Ara up�rqoun dÔo efaptìmenec eujeÐec, oi

y = λ1x =

(2 +

√21

3

)x, y = λ2x =

(2−√

21

3

)x

H èlleiyh kai oi dÔo efaptìmenec èqoun sqediasteÐ sto Sq ma 13.15.

2. Sumplhr¸noume ta tetr�gwna:

2x2 − y2 + 4x− 4y − 2 = 0

⇔ 2(x2 + 2x+ 1)− (y2 + 4y + 4) = 2 + 2− 4 = 0⇔ 2(x+ 1)2 = (y + 2)2.

An k�noume ton metasqhmatismì suntetagmènwn u = x+1, v = y+2, h exÐswshgÐnetai

2u2 = v2 ⇔√

2u = ±v,sunep¸c o gewmetrikìc tìpoc apoteleÐtai apì dÔo eujeÐec, tic

√2u = v ⇔

√2(x+ 1) = y + 2⇔ y =

√2x− 2 +

√2,√

2u = −v ⇔√

2(x+ 1) = −(y + 2)⇔ y = −√

2x− 2−√

2.

Oi eujeÐec èqoun sqediasteÐ sto Sq ma 13.15.

Page 314: ΣΗΜΕΙΩΣΕΙΣ-ΜΑΘΗΜΑΤΙΚΑ 1 ΠΛΗΡΟΦΟΡΙΚΗ ΑΣΟΕΕ 1 ΕΞΑΜΗΝΟ

304 KEF�ALAIO 13. KWNIK�ES TOM�ES

13.7 Ax2 +Bxy + Cy2 +Dx + Ey + F = 0

Je¸rhma 13.6. (Peristrammènec kwnikèc tomèc) H exÐswsh

Ax2 +Bxy + Cy2 +Dx+ Ey + F = 0. (13.11)

perigr�fei eÐte kwnik  tom  (me aujaÐreth dieÔjunsh axìnwn), eÐte orismènec eidikècpeript¸seic.

Apìdeixh. Eis�goume tic suntetagmènec (u, v) ìpou{x = u cos θ − v sin θ,y = u sin θ + v cos θ

}⇔{

u = x cos θ + y sin θ,v = −x sin θ + y cos θ

}(13.12)

Sunep¸c, to kainoÔrgio sÔsthma suntetagmènwn uv eÐnai peristrammèno kat� gwnÐa θme to arqikì, en¸ oi arqèc twn axìnwn twn dÔo susthm�twn tautÐzontai. H (13.11)gÐnetai

A(u2 cos2 θ + v2 sin2 θ − 2uv cos θ sin θ)

+B(uv cos2 θ − uv sin2 θ + u2 cos θ sin θ − v2 sin θ cos θ)

+ C(u2 sin2 θ + v2 cos2 θ + 2uv sin θ cos θ)+

D(u cos θ − v sin θ) + E(u sin θ + v cos θ) + F = 0

⇔ u2[A cos2 θ +B cos θ sin θ + C sin2 θ

]+ uv

[−2A cos θ sin θ +B(cos2 θ − sin2 θ) + 2C sin θ cos θ

]+ v2

[A sin2 θ −B sin θ cos θ + C cos2 θ

]+ u [D cos θ + E sin θ] + v [−D sin θ + E cos θ] + F = 0.

An up�rqei gwnÐa θ gia thn opoÐa o suntelest c tou uv mhdenÐzetai, tìte èqou-me fèrei thn exÐswsh sthn gnwst  morf  (13.8), all� apl¸c se èna �llo sÔsthmasuntetagmènwn. H gwnÐa θ pou qreiazìmaste mporeÐ na upologisjeÐ wc ex c:

− 2A cos θ sin θ +B(cos2 θ − sin2 θ) + 2C sin θ cos θ = 0

⇔ (A− C) sin 2θ = B cos 2θ ⇔ cos 2θ

sin 2θ=A− CB

⇔ cot 2θ =A− CB

⇔ θ =1

2arccot

(A− CB

). (13.13)

Parathr ste ìti h sunefaptìmenh paÐrnei timèc apì to −∞ èwc to ∞, �ra p�ntotempor¸ na brw èna θ pou na mhdenÐzei ton suntelest  tou uv. (Gia thn akrÐbeia,up�rqoun p�nta akrib¸c dÔo θ ∈ (0, π) pou na mhdenÐzoun ton suntelest  tou uv.)

Page 315: ΣΗΜΕΙΩΣΕΙΣ-ΜΑΘΗΜΑΤΙΚΑ 1 ΠΛΗΡΟΦΟΡΙΚΗ ΑΣΟΕΕ 1 ΕΞΑΜΗΝΟ

13.7. AX2 +BXY + CY 2 +DX + EY + F = 0 305

H kainoÔrgia exÐswsh eÐnai thc morf c

A′u2 +B′v2 +D′u+ E ′v + F ′ = 0,

ìpou u, v eÐnai oi suntetagmènec tou nèou sust matoc suntetagmènwn. 'Omwc, apì taprohgoÔmena xèroume ìti aut  h exÐswsh ekfr�zei kwnik  tom  metatopismènh wc procton arq  twn axìnwn   k�poia tetrimmènh perÐptwsh.

Parat rhsh: H apìdeixh tou jewr matoc mac parèqei thn basik  mejodologÐa poumac epitrèpei na prosdiorÐzoume to eÐdoc kai ta basik� qarakthristik� opoiasd potekwnik c tom c mac èqei dojeÐ sth morf  thc exÐswshc (13.11). Sugkekrimèna:

1. ProsdiorÐzoume th gwnÐa θ me qr sh thc (13.13).

2. Eis�goume to nèo sÔsthma suntetagmènwn uv me qr sh thc (13.12). (Parathr -ste ìti autì to b ma perilamb�nei peristrof  twn axìnwn, kai ìqi metatìpish thcarq c touc.)

3. Sthn nèa exÐswsh o suntelest c tou uv eÐnai mhdenikìc (an den èqoume k�neil�joc). 'Ara, mporoÔme na sumplhr¸soume ta tetr�gwna, ìpwc k�name sthnPar�grafo 13.6.

4. 'Opwc kai sthn Par�grafo 13.6, me mia akìma allag  suntetagmènwn (pou aut th for� eÐnai apl  metatìpish thc arq c twn axìnwn, qwrÐc peristrof  touc),fèrnoume thn kwnik  tom  se kanonik  morf .

Par�deigma 13.7. 'Estw oi kwnikèc tomèc pou ikanopoioÔn tic exis¸seic:

1. x2 + y2 + (−6− 4√

2)x+ (6− 4√

2)y − 2xy + (9 + 4√

2) = 0.

2. (7/4)x2 + (13/4)y2− (6√

3/4)xy+ (−7 + 3√

3)x+ (−13 + 3√

3)y = (6√

3− 4).

Ja broÔme to eÐdoc touc, ja prosdiorÐsoume ta basik� qarakthristik� touc, kaitelik� ja tic sqedi�soume. 'Eqoume, kat� perÐptwsh:

1. H kwnik  tom  eÐnai thc morf c Ax2 + Bxy + Cy2 + Dx + Ey + F = 0. 'Ara,gia na apofanjoÔme gia th morf  thc, prèpei kat' arq n na apaleÐyoume ton ìroBxy. Autì ja gÐnei, kat� ta gnwst� apì th jewrÐa, eis�gontac mia peristrof twn axìnwn kat� mia gwnÐa θ ìpou

cot 2θ =A− CB

=1− 1

−2= 0.

Page 316: ΣΗΜΕΙΩΣΕΙΣ-ΜΑΘΗΜΑΤΙΚΑ 1 ΠΛΗΡΟΦΟΡΙΚΗ ΑΣΟΕΕ 1 ΕΞΑΜΗΝΟ

306 KEF�ALAIO 13. KWNIK�ES TOM�ES

Mia gwnÐa pou ikanopoieÐ thn �nw eÐnai h θ = π/4. 'Ara qrhsimopoioÔme tonmetasqhmatismì:{

x = u cos θ − v sin θ =√

22 (u− v)

y = u sin θ + v cos θ =√

22 (u+ v)

}

⇔{

u = x cos θ + y sin θ =√

22 (x+ y)

v = −x sin θ + y cos θ =√

22 (−x+ y)

}(13.14)

kai me antikat�stash tou pr¸tou zeÔgouc sthn arqik  exÐswsh èqoume:

1

2(u− v)2 +

1

2(u+ v)2 + (−6− 4

√2)

√2

2(u− v)

+ (6− 4√

2)

√2

2(u+ v)− u2 + v2 + 9 + 4

√2 = 0

⇔ 2v2 − 8u+ 6√

2v + 9 + 4√

2 = 0⇔(v +

3√

2

2

)2

= 4

(u−√

2

2

).

Sunep¸c, sto nèo sÔsthma axìnwn, èqoume mia parabol , me koruf  to shmeÐo(√

2/2,−3√

2/2), p = 1, dieujetoÔsa eujeÐa thn u =√

2/2 − 1, kai estÐa thn(√

2/2 + 1,−3√

2/2). Sto arqikì sÔsthma axìnwn brÐskoume, qrhsimopoi¸ntacton metasqhmatismì (13.14), ìti h koruf  èqei suntetagmènec (2,−1), h dieuje-toÔsa eÐnai h x+ y = 1−

√2 kai h estÐa h (2 +

√2/2,√

2/2− 1) ' (2.7,−0.3).H parabol  èqei sqediasteÐ sto Sq ma 13.16.

2. Kat� ta gnwst� apì th jewrÐa, eis�goume mia peristrof  twn axìnwn kat� miagwnÐa θ ìpou

cot 2θ =A− CB

=−6/4

−6√

3/4=

1√3.

Mia gwnÐa pou ikanopoieÐ thn �nw eÐnai h θ = π/6. 'Ara qrhsimopoioÔme tonmetasqhmatismì:{

x = u cos θ − v sin θ =√

32 u− 1

2v

y = u sin θ + v cos θ = 12u+

√3

2 v

}

⇔{

u = x cos θ + y sin θ =√

32 x+ 1

2y

v = −x sin θ + y cos θ = −12x+

√3

2 y

}(13.15)

Page 317: ΣΗΜΕΙΩΣΕΙΣ-ΜΑΘΗΜΑΤΙΚΑ 1 ΠΛΗΡΟΦΟΡΙΚΗ ΑΣΟΕΕ 1 ΕΞΑΜΗΝΟ

13.7. AX2 +BXY + CY 2 +DX + EY + F = 0 307

kai me antikat�stash tou pr¸tou zeÔgouc sthn arqik  exÐswsh èqoume:

7

(√3

2u− 1

2v

)2

+ 13

(1

2u+

√3

2v

)2

− 6√

3

(√3

2u− 1

2v

)(1

2u+

√3

2v

)

+ 4(−7 + 3√

3)

(√3

2u− 1

2v

)+ 4(−13 + 3

√3)

(1

2u+

√3

2v

)= 24

√3− 16

⇔ 16u2 + 64v2 + (−32− 32√

3)u+ (128− 128√

3)v = 16(6√

3− 4)

⇔ 4(u− 1−√

3)2 + 16(v + 1−√

3)2 = 64

⇔ (u− 1−√

3)2

16+

(v + 1−√

3)2

4= 1.

Sunep¸c, sto nèo sÔsthma axìnwn èqoume mia èlleiyh me m kh axìnwn 2a = 8kai 2b = 4 kai

(aþ) kèntro to (1 +√

3,√

3− 1),

(bþ) korufèc sta shmeÐa (5 +√

3,√

3− 1), (−3 +√

3,√

3− 1), (1 +√

3,√

3− 3),kai (1 +

√3,√

3 + 1), kai

(gþ) �xonec tic eujeÐec u = 1 +√

3, v =√

3− 1.

An efarmìsoume ton metasqhmatismì (13.15) prokÔptei pwc, sto sÔsthma xy,

(aþ) to kèntro eÐnai to (2, 2)

(bþ) oi korufèc eÐnai sta shmeÐa, antÐstoiqa, (2+2√

3, 4), (2−2√

3, 0), (3, 2−√

3),kai (1, 2 +

√3).

(gþ) kai �xonec eÐnai oi eujeÐec√

3x/2+y/2 = 1+√

3 kai−x/2+√

3y/2 =√

3−1.

H èlleiyh èqei sqediasteÐ sto Sq ma 13.16.

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308 KEF�ALAIO 13. KWNIK�ES TOM�ES

0 1 2 3 4−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

(2,−1)

(2.7,−0.3)x+

y=

1 − √2

x

−2 0 2 4 6−1

0

1

2

3

4

5

(2, 2)

(5.5, 4.0)

(−1.5, 0.0)

(1.0, 3.7)

(3.0, 0.3)

(5.0, 3.7)

(−1.0, 0.3)

x

y

( √3/2)x

+y/2

=1+ √

3

−x/2+

(√ 3/

2)y=

√ 3−1

Sq ma 13.16: Oi kwnikèc tomèc tou ParadeÐgmatoc 13.7.

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Kef�laio 14

Par�rthma

14.1 BibliografÐa

Ellhnik  BibliografÐa

1. An¸tera Majhmatik�, HlÐa Flutz�nh, Ekdìseic A. StamoÔlhc, Aj na-Peirai�c 1993.

2. Diaforikìc kai Oloklhrwtikìc Logismìc, tìmoi I-II, Tom M. Apostol, Ekdìseic AtlantÐc, 1hèkdosh, 1962.

3. Diaforikìc kai Oloklhrwtikìc Logismìc, Michael Spivak, 2h èkdosh, 1980.

4. Apeirostikìc Logismìc, R. L. Finney, F. R. Giordano, Panepisthmiakèc Ekdìseic Kr thc.

5. Majhmatik  An�lush, tìmoi I-II, G. N. PantelÐdh, Ekdìseic Z th, 1994.

Agglik  BibliografÐa

1. Calculus, Tom M. Apostol, John Willey and Sons, 2nd edition, 1969.

2. Calculus, Michael Spivak, Cambridge University Press, 3rd edition, 1993.

3. Calculus, D. Varberg, E. J. Purcell, S. E. Rigdon, Pearson International Edition, 9th edition,2007.

4. Calculus, James Stewart, Brooks Cole, 6th edition, 2007.

5. Calculus, G. B. Thomas, R. L. Finney, M. D. Weir, F. R. Giordano, Addison Wesley, 10thedition, 2002.

309

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310 KEF�ALAIO 14. PAR�ARTHMA

14.2 SumbolismoÐ

SuntomografÐec

• ∀ shmaÐnei {gia k�je}.

• ∃ shmaÐnei {up�rqei}.

• To : kai to | shmaÐnoun {tètoio ¸ste}.

• To , kai to ≡ sumbolÐzoun {orÐzetai wc}.

• {ann}= {an kai mìno an}.

• Stouc orismoÔc, to {an} ennoeÐtai wc {ann}.

• ∨ shmaÐnei { } kai ∧ shmaÐnei {kai}.

SÔnola

• R eÐnai oi pragmatikoÐ, Q eÐnai oi rhtoÐ, N eÐnai oi fusikoÐ 1, 2, 3, . . . , kai Z eÐnai oi akèraioiarijmoÐ.

• R+ eÐnai oi jetikoÐ pragmatikoÐ arijmoÐ.

• Q+ eÐnai oi jetikoÐ rhtoÐ arijmoÐ.

• To kenì sÔnolo sumbolÐzetai wc ∅.

• ⊆ shmaÐnei {eÐnai uposÔnolo} kai to ⊂ shmaÐnei {eÐnai gn sio uposÔnolo}.

• A ∩B = {x : x ∈ A ∧ x ∈ B} eÐnai h tom  dÔo sunìlwn.

• A ∪B = {x : x ∈ A ∨ x ∈ B} eÐnai h ènwsh dÔo sunìlwn.

• Ac = {x : x 6∈ A} eÐnai to sumpl rwma enìc sunìlou.

• A−B = A ∩Bc eÐnai h diafor� dÔo sunìlwn.

• ∈ shmaÐnei {an kei} kai 6∈ shmaÐnei {den an kei}.

Sunart seic

• domf eÐnai to pedÐo orismoÔ miac sun�rthshc.

• sinx eÐnai to hmÐtono tou x ( {sine} ), cosx eÐnai to sunhmÐtono tou x ( {cosine} ), tanx eÐnaih efaptomènh tou x ( {tangent} ) kai cotx eÐnai h sunefaptomènh tou x ( {cotangent} ).

• bxc eÐnai o megalÔteroc akèraioc pou eÐnai mikrìteroc   Ðsoc me ton x.ParadeÐgmata: b2.3c = 2, b2c = 2, b1.999999c = 1.

• dxe eÐnai o mikrìteroc akèraioc pou eÐnai megalÔteroc   Ðsoc me ton x.ParadeÐgmata: d2.3e = 3, d2e = 2, d1.999999e = 2.

• f |x shmaÐnei h f upologismènh sto x.

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14.3. LEXIK�O 311

14.3 Lexikì

Akèraioc: IntegerAnagkaÐo: Necessary'Axonac: Axis'Anw: Upper'Ajroisma: Sum, summationAkoloujÐa: Sequence'Akro (sunìlou   diast matoc): BoundAkrìtato(a): Extremum (extrema)Algìrijmoc: AlgorithmAnisìthta: InequalityAnoiktì (di�sthma): OpenAntÐjetoc: OppositeAntikat�stash: SubstitutionAntipar�gwgoc: AntiderivativeAntÐstrofoc: InverseAntimetajetikìthta: CommutativityAntipar�deigma: Counterexample'Anw fragmèno/h: Upper boundedAxÐwma: AxiomAìristo olokl rwma: Indefinite integral'Apeiro: Infinite, infinityApìdeixh: ProofApoklÐnw: DivergeApokop : TruncationApìluth tim : Absolute valueArmonikìc: Harmonic'Arrhtoc: Irrational'Artioc: EvenArq : PrincipleAÔxwn: IncreasingB�sh: BaseGeitoni�: NeighborhoodGenik  antipar�gwgoc: General antiderivativeGewmetrikìc: GeometricGewmetrikìc tìpoc: LocusGn sio uposÔnolo: Strict (proper) subsetGnhsÐwc aÔxwn: Strictly increasingGnhsÐwc fjÐnwn: Strictly decreasingGnhsÐwc monìtonoc: Strictly monotonousGrammikìc: LinearGrammik  'Algebra: Linear AlgebraGrammikìc sunduasmìc: Linear combinationGr�fhma: GraphGwnÐa: AngleDiamèrish: Partition

Di�nusma: VectorDianusmatikìc q¸roc: Vector space, linear spaceDi�sthma: IntervalDi�taxh: OrderingDiafor�: DifferenceDiaqwrÐsimoc: SeparableDiaforikì: DifferentialDieujetoÔsa (parabol c): DirectrixDiqotìmhsh: BisectionEkjetikìc: ExponentialEkkentrìthta (kwnik c tom c): EccentricityEklèptunsh: RefinementEl�qisto(a): Minimum (minima)'Elleiyh: Ellipse'Ena proc èna: One on oneEnall�ssousa: AlternatingEndi�mesh (tim ): Intermediate'Enwsh: UnionExÐswsh: EquationEstÐa(ec) (kwnik c tom c): Focus (foci)Eswterikì ginìmeno: Inner product, dot productEpimeristik  idiìthta: Distributive lawEpÐpedo: PlaneEpif�neia: Surface, areaEswterikì (sunìlou): InteriorEujeÐa: LineEfaptìmenh: TangentHmiepÐpedo: Half-planeHmÐtono: SineJemeli¸dec Je¸rhma: Fundamental TheoremJe¸rhma: TheoremIkanì: SufficientK�jetoc: VerticalKampÔlh: CurveKanìnac: Rule, lawKanìnac thc alusÐdac: Chain ruleKartesianìc: CartesianKataqrhstikì Olokl rwma: Improper IntegralK�tw: LowerK�tw fragmènoc: Lower boundedKèlufoc: ShellKenì SÔnolo: Empty setKèntro: CenterKleistì di�sthma: Closed intervalKlimakwt  sun�rthsh: Step functionKoÐloc: ConcaveKoruf  (kwnik c tom c): Vertex

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312 KEF�ALAIO 14. PAR�ARTHMA

Krit rio: CriterionKÔkloc: Circle (ìqi cycleKÔlindroc: CylinderKumatik  ExÐswsh: Wave EquationKurtìthta: ConvexityKurtìc: ConvexKwnik  tom : Conic sectionLeÐoc: SmoothLog�rijmoc: LogarithmL mma: LemmaLogismìc: CalculusMègisto(a): Maximum (maxima)Mèjodoc: MethodMerik : PartialMetasqhmatismìc: TransformationMetatìpish: TranslationMèsoc: MeanM koc: LengthMigadikìc: ComplexMonadiaÐo di�nusma: Unit vectorMonìtonoc: Monotonous, monotoneNìrma: Norm'Ogkoc: VolumeOlikìc: TotalOlokl rwma: IntegralOlokl rwsh: IntegrationOloklhr¸simoc/h/o: IntegrableOloklhrwsimìthta: IntegrabilityOmogen c: HomogeneousOmoiìmorfoc: UniformOrjog¸nioc: Orthogonal'Orio: LimitOrismènoc: Defined'Oroc: TermOudètero stoiqeÐo: Identity elementParabol : ParabolaPar�gousa: AntiderivativePar�gontac: FactorPar�gwgoc: DerivativePar�gwgoc an¸terhc t�xhc: Higher order deri-vativeParagwgÐsimoc/h/o: DifferentiableParagwgisimìthta: DifferentiabilityPar�llhloc: ParallelParametrikìc: ParametricPar�metroc: ParameterPedÐo: Field

PedÐo DieÔjunshc: Slope FieldPedÐo orismoÔ: DomainPedÐo tim¸n: Range, imagePeriodikìc: PeriodicPeristrof : RotationPerittìc: OddPÐnakac: MatrixPleurikìc: SidePolikìc: PolarPolu¸numo: PolynomialProbol : ProjectionProsèggish: ApproximationProsetairistik  idiìthta: Associative lawPrìshmo: SignPìrisma: CorollaryPrìodoc: ProgressionPr¸thc T�xhc: First OrderPuknì (sÔnolo): DenseOrjogwniìthta: OrthogonalityRhtìc: RationalRÐza: RootRujmìc: RateSeir�: SeriesShmeÐo Kamp c: Inflection PointStajerìc: ConstantStereì: SolidStoiqeÐo: ElementSugklÐnw: ConvergeSÔgklish: ConvergenceSumpl rwma: ComplementSun�rthsh: FunctionSunefaptìmenh: CotangentSunhmÐtono: CosineSun jhc: OrdinarySunist¸sa: ComponentSuntelest c: CoefficientSuntetagmènh: CoordinateSÔnolo: SetSunèqeia: ContinuityTeÐnw: TendTetragwnik  rÐza: Square rootThleskopikìc: TelescopicTom : IntersectionTìxo: ArcTopikìc: LocalTrigwnik  anisìthta: Triangle identityTrigwnometrikìc: Trigonometric

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14.3. LEXIK�O 313

UpakoloujÐa: SubsequenceUperbol : HyperbolaUperbolikì: HyperbolicUpì sunj kh: ConditionalUpìloipo: RemainderUposÔnolo: SubsetFjÐnwn: DecreasingFr�gma: BoundFragmènoc: BoundedFusikìc: NaturalQord : ChordQwrÐo: RegionQ¸roc: Space

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Euret rio

R2, 262R3, 270Rn, 251GF(5), 1'Ajroisma Darboux, 119

'Anw, 119K�tw, 119

'Ajroisma Riemann, 140'Apeiro, 44'Axonac, 289, 292, 296'Elleiyh, 292

Kanonik  morf , 292'Ogkoc

Ek peristrof c, 187, 191Mèjodoc dÐskwn, 187Mèjodoc kelÔfwn, 191

'Orio, 30AkoloujÐac, 49Algebrikèc pr�xeic orÐwn, 37Idiìthtec, 35Monadikìthta, 35Orismìc, 30, 33, 44Pleurikì, 33Poluwnumik¸n/rht¸n sunart sewn, 38Sto �peiro, 44Sunj kec mh Ôparxhc, 36Trigwnometrik¸n sunart sewn, 41

Infimum, 9Supremum, 9

11888, 248

Akèraio mèroc, 26AkoloujÐa, 49

ApoklÐnousa, 49SugklÐnousa, 49

Allag  Suntetagmènwn, 284Anisìthta Cauchy-Schwarz, 255AntÐjetoc, 1AntÐstrofoc, 1

Antipar�gwgoc, 99Apìluth tim , 6Arijmìc, 1

Arnhtikìc, 6Jetikìc, 6

Asunèqeia, 55Epousi¸dhc, 55Ousi¸dhc, 55

AxÐwma plhrìthtac, 11Axi¸mata di�taxhc, 6Axi¸mata pedÐou, 1

Di�nusma, 251AfaÐresh, 251AntÐjeto, 251AntÐrropo, 252Basikì, 258Idiìthtec, 254Isìthta, 251K�jeto, 259M koc, 254Mhdenikì, 251MonadiaÐo, 258Nìrma, 254Omìrropo, 252Orjog¸nio, 259Par�llhlo, 252Pollaplasiasmìc me pragmatikì, 251Prìsjesh, 251StoiqeÐo, 251Sunist¸sa, 251Suntetagmènh, 251

Diaforik�, 80Diaforikèc exis¸seic

Arqik  sunj kh, 205DiaqwrÐsimec pr¸thc t�xewc, 212Grammikèc pr¸thc t�xewc, 208LÔseic, 205Pr¸thc t�xewc, 205

Diamèrish, 119

314

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EURET�HRIO 315

Diast mata, 16DieujetoÔsa, 289

Efaptomènh, 20Ekkentrìthta, 292, 296Eklèptunsh, 119Embadìn

Kartesianèc suntetagmènec, 173Polikèc suntetagmènec, 182

EpÐpedo, 270Exis¸seic, 271Par�llhlo, 270

EstÐa, 289Eswterikì ginìmeno, 255

Idiìthtec, 255EujeÐa

Exis¸seic, 263, 277K�jeth, 262, 276Par�llhlh, 262, 276Par�llhlh se epÐpedo, 278Parametrik  exÐswsh, 262Ston R2, 262Ston R3, 276

fr�gma, 9

Geitoni�, 16Gewmetrik  prìodoc, 238

HmÐtono, 20HmiepÐpedo, 267

IdiìthtaAntimetajetik , 1Arqim deia, 14Epimeristik , 1Prosetairistik , 1

JeÐa EpifoÐthsh, 208Je¸rhma

Bolzano, 60Darboux, 140Rolle, 94Taylor, 233Akrìtatwn, 66DeÔtero Jemeli¸dec Je¸rhma LogismoÔ, 146Diat rhshc Prìshmou, 59Endi�meshc Tim c, 64Mèshc Tim c, 95

Mèshc Tim c Cauchy, 97Mèshc Tim c gia oloklhr¸mata, 134Mhdenik c Parag¸gou, 97Parembol c, 39Pr¸to Jemeli¸dec Je¸rhma LogismoÔ, 143,

144

Kèntro, 292, 296KampÔlh

'Iqnoc kampÔlhc, 196LeÐa kampÔlh, 197M koc kampÔlhc, 199Par�metroc kampÔlhc, 196Parametrikèc exis¸seic kampÔlhc, 196Parametrik  morf , 196Polikèc suntetagmènec, 180

Kanìnac thc AlusÐdac, 86, 101Kanìnac tou L’Hopital, 118KateujÔnonta sunhmÐtona, 278Koruf , 289, 292, 296KrÐsima ShmeÐa, 109Krit ria

Akrìtatwn, 109, 110Kurtìthtac, 115MonotonÐac, 109Oloklhr¸matoc, 245RÐzac, 244SÔgklishc seir¸n, 240SÔgkrishc, 241SÔgkrishc sto ìrio, 241

Krit rioDarboux, 125Oloklhrwsimìthtac, 125, 126, 130

Kwnik  Tom Genik  exÐswsh, 300, 304

Mèjodoc Euler, 217Mèjodoc Diqotìmhshc, 68Mèjodoc tou NeÔtwna, 105Merikì �jroisma, 237

Nìmoc tou parallhlogr�mmou, 256

Olokl rwma'Anw Olokl rwma Darboux, 122Darboux, 123Riemann, 140Aìristo, 99Idiìthtec, 132

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316 EURET�HRIO

K�tw Olokl rwma Darboux, 122Kataqrhstikì deÔterou tÔpou, 167Kataqrhstikì pr¸tou tÔpou, 166Orismèno, 123

Olokl rwshKat� par�gontec, 102, 148Me antikat�stash, 149

Oudètero stoiqeÐo, 1

Par�gousa, 99Genik , 99

Par�gwgoc, 77An¸terhc t�xhc, 83AntÐstrofhc sun�rthshc, 89Arister , 77Dexi�, 77Idiìthtec, 84Pleurik , 77

Parabol , 289Kanonik  morf , 289

Paragwgisimìthta, 77PedÐa dieujÔnsewn, 217PedÐo, 1Polikèc suntetagmènec, 178Polu¸numo Maclaurin, 225Polu¸numo Taylor, 225

Sf�lma, 233Upìloipo, 233

Pragmatikèc Dun�meic, 157Pukn� sÔnola, 14

RÐza, 14, 68Rhtèc dun�meic, 15

SÔnolo'Akro, 16Anoiktì, 16Eswterikì shmeÐo, 16Kleistì, 16Sumpl rwma, 16Sunoriakì shmeÐo, 16

S¸ma, 1Seir�, 237

ApoklÐnousa, 237Armonik , 239Gewmetrik , 238SugklÐnousa, 237

Sumbolismìc Leibniz, 80, 83, 99

Sun�rthsh, 17'Anw fragmènh, 17'Artia, 17'Ena proc èna, 17Dirichlet, 124Infimum, 18Maximum, 18Minimum, 18Supremum, 18AÔxousa, 17Akrìtata, 18AntÐstrofh, 70AntÐstrofh trigwnometrik , 91Arister� suneq c, 55Arqik , 99Asuneq c, 55Dexi� suneq c, 55Ekjetik , 152El�qisto, 18FjÐnousa, 17Fragmènh, 17GnhsÐwc aÔxousa, 17GnhsÐwc fjÐnousa, 17GnhsÐwc monìtonh, 17Jèsh topikoÔ akrìtatou, 18Jèsh topikoÔ el�qistou, 18Jèsh topikoÔ mègistou, 18K�tw fragmènh, 17KoÐlh, 112Kurt , 112Logarijmik , 151, 159Mègisto, 18Metatìpish, 136Monìtonh, 17Olik� akrìtata, 18Oloklhr¸simh, 123ParagwgÐsimh, 77PedÐo orismoÔ, 17PedÐo tim¸n, 17Periodik , 17Peritt , 17SÔnjeth, 19, 57SmÐkrunsh/epim kunsh, 138Suneq c, 55Topik� akrìtata, 18Topikì el�qisto, 18Topikì mègisto, 18

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EURET�HRIO 317

Sunèqeia, 55Lipschitz, 73AntÐstrofhc sun�rthshc, 70Apì arister�, 55Apì dexi�, 55Kat� Cauchy, 56Orismìc, 55Pleurik , 55Poluwnumik¸n sunart sewn, 57Rht¸n dun�mewn, 71Rht¸n sunart sewn, 57Riz¸n, 71SÔnjethc sun�rthshc, 57Se di�sthma, 59Se shmeÐo, 55Trigwnometrik¸n sunart sewn, 57

Sunefaptomènh, 20SunhmÐtono, 20

Trigwnik  anisìthta, 256Trigwnometrikèc sunart seic, 20

Uperbol Kanonik  morf , 296

Uperbolikì hmÐtono, 230Uperbolikì sunhmÐtono, 230UposÔnolo, 6

Gn sio, 6