PANEPISTHMIO DUTIKHS ATTIKHS SQOLH MHQANIKWN. Ask seic kai Jèmata sthn Pragmatik Anˆlush I TMHMA POLITIKWN MHQANIKWN

Σχετικά έγγραφα
Anaplhrwt c Kajhght c : Dr. Pappˆc G. Alèxandroc PANEPISTHMIO DUTIKHS ATTIKHS SQOLH MHQANIKWN TMHMA POLITIKWN MHQANIKWN ANWTERA MAJHMATIKA I

SUNARTHSEIS POLLWN METABLHTWN. 5h Seirˆ Ask sewn. Allag metablht n sto diplì olokl rwma

GENIKEUMENA OLOKLHRWMATA

PANEPISTHMIO DUTIKHS ATTIKHS SQOLH MHQANIKWN TMHMA POLITIKWN MHQANIKWN ANWTERA MAJHMATIKA II SUNARTHSEIS POLLWN METABLHTWN.

9. α 2 + β 2 ±2αβ. 10. α 2 ± αβ + β (1 + α) ν > 1+να, 1 <α 0, ν 2. log α. 14. log α x = ln x. 19. x 1 <x 2 ln x 1 < ln x 2

PANEPISTHMIO DUTIKHS ATTIKHS SQOLH MHQANIKWN TMHMA POLITIKWN MHQANIKWN ANWTERA MAJHMATIKA II DIAFORIKES EXISWSEIS.

6h Seirˆ Ask sewn. EpikampÔlia oloklhr mata

Τίτλος Μαθήματος: Γραμμική Άλγεβρα Ι

JEMATA EXETASEWN Pragmatik Anˆlush I

PANEPISTHMIO DUTIKHS ATTIKHS SQOLH MHQANIKWN TMHMA POLITIKWN MHQANIKWN ANWTERA MAJHMATIKA II DIAFORIKES EXISWSEIS DEUTERHS KAI ANWTERHS TAXHS

11 OktwbrÐou S. Malefˆkh Genikì Tm ma Majhmatikˆ gia QhmikoÔc

PANEPISTHMIO DUTIKHS ATTIKHS SQOLH MHQANIKWN TMHMA POLITIKWN MHQANIKWN ANWTERA MAJHMATIKA II DIAFORIKES EXISWSEIS.

AM = 1 ( ) AB + AΓ BΓ+ AE = AΔ+ BE. + γ =2 β + γ β + γ tìte α// β. OΓ+ OA + OB MA+ MB + M Γ+ MΔ =4 MO. OM =(1 λ) OA + λ OB

Pragmatik Anˆlush ( ) TopologÐa metrik n q rwn Ask seic

25 OktwbrÐou 2012 (5 h ebdomˆda) S. Malefˆkh Genikì Tm ma Majhmatikˆ gia QhmikoÔc

Τίτλος Μαθήματος: Γραμμική Άλγεβρα ΙΙ

Τίτλος Μαθήματος: Γραμμική Άλγεβρα ΙΙ

5. (12 i)(3+4i) 6. (1 + i)(2+i) 7. (4 + 6i)(7 3i) 8. (1 i)(2 i)(3 i)

Εφαρμοσμένα Μαθηματικά για Μηχανικούς

PANEPISTHMIO DUTIKHS ATTIKHS SQOLH MHQANIKWN TMHMA POLITIKWN MHQANIKWN ANWTERA MAJHMATIKA II SUNARTHSEIS POLLWN METABLHTWN EPIKAMPULIA OLOKLHRWMATA

Ανάλυση ις. συστήματα

Εφαρμοσμένα Μαθηματικά για Μηχανικούς

Εφαρμοσμένα Μαθηματικά για Μηχανικούς

Diˆsthma empistosônhc thc mèshc tim c µ. Statistik gia Hlektrolìgouc MhqanikoÔc EKTIMHSH EKTIMHSH PARAMETRWN - 2. Dhm trhc Kougioumtz c.

APEIROSTIKOS LOGISMOS I

ΕΛΛΗΝΙΚΗ ΔΗΜΟΚΡΑΤΙΑ ΠΑΝΕΠΙΣΤΗΜΙΟ ΚΡΗΤΗΣ. Μηχανική Μάθηση. Ενότητα 10: Θεωρία Βελτιστοποίησης. Ιωάννης Τσαμαρδίνος Τμήμα Επιστήμης Υπολογιστών

1 η Σειρά Ασκήσεων Θεόδωρος Αλεξόπουλος. Αναγνώριση Προτύπων και Νευρωνικά Δίκτυα


SofÐa ZafeirÐdou: GewmetrÐec

Diakritˆ Majhmatikˆ I. Leutèrhc KuroÔshc (EÔh Papaðwˆnnou)

Ask seic me ton Metasqhmatismì Laplace

Jerinì SqoleÐo Fusik c sthn EkpaÐdeush 28 IounÐou - 1 IoulÐou 2010 EstÐa Episthm n Pˆtrac

Anagn rish ProtÔpwn & Neurwnikˆ DÐktua Probl mata 2

Mègisth ro - elˆqisth tom

Eisagwg sthn KosmologÐa

Shmei seic sto mˆjhma Analutik GewmetrÐa

Κλασσική Ηλεκτροδυναμική II

ΜΑΘΗΜΑΤΙΚΑ ΤΗΛΕΠΙΚΟΙΝΩΝΙΩΝ

ΕΛΛΗΝΙΚΗ ΔΗΜΟΚΡΑΤΙΑ ΠΑΝΕΠΙΣΤΗΜΙΟ ΚΡΗΤΗΣ

Ανάλυση ασκήσεις. συστήματα

Shmei seic sto mˆjhma Analutik GewmetrÐa

Θεωρία Πιθανοτήτων και Στατιστική

ΠΑΝΕΠΙΣΤΗΜΙΟ ΠΑΤΡΩΝ ΤΜΗΜΑ ΕΠΙΣΤΗΜΗΣ ΤΩΝ ΥΛΙΚΩΝ ΠΛΗΡΟΦΟΡΙΚΗ ΙΙ Εξετάσεις Ιουνίου 2002

Statistik gia PolitikoÔc MhqanikoÔc ELEGQOS UPOJ

Statistik gia PolitikoÔc MhqanikoÔc EKTIMHSH PAR

HU215 - Frontist rio : Seirèc Fourier

ISTORIKH KATASKEUH PRAGMATIKWN ARIJMWN BIBLIOGRAFIA

Hmiomˆdec telest n sônjeshc kai pðnakec Hausdorff se q rouc analutik n sunart sewn

στο Αριστοτέλειο υλικού.

Στατιστική για Χημικούς Μηχανικούς

SUNOLA BIRKHOFF JAMES ϵ ORJOGWNIOTHTAS KAI ARIJMHTIKA PEDIA

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

Στατιστική για Χημικούς Μηχανικούς

στο Αριστοτέλειο υλικού.

1, 3, 5, 7, 9,... 2, 4, 6, 8, 10,... 1, 4, 7, 10, 13,... 2, 5, 8, 11, 14,... 3, 6, 9, 12, 15,...

Eukleideiec Gewmetriec

Statistik gia QhmikoÔc MhqanikoÔc EKTIMHSH PARA

Εφαρμοσμένα Μαθηματικά για Μηχανικούς

Anaz thsh eustaj n troqi n se triplˆ sust mata swmˆtwn

KBANTOMHQANIKH II (Tm ma A. Laqanˆ) 28 AugoÔstou m Upìdeixh: Na qrhsimopoihjeð to je rhma virial 2 T = r V.

Shmei seic Sunarthsiak c Anˆlushc

Upologistik Fusik Exetastik PerÐodoc IanouarÐou 2013

f(x) =x x 2 = x x 2 x =0 x(x 1) = 0,


ΑΛΓΕΒΡΑ Β ΛΥΚΕΙΟΥ ΕΙΣΑΓΩΓΗ ΑΠΑΙΤΟΥΜΕΝΕΣ ΓΝΩΣΕΙΣ. ΕΠΙΛΥΣΗ ΕΞΙΣΩΣΗΣ 2ου ΒΑΘΜΟΥ ΠΡΟΣΗΜΟ ΤΡΙΩΝΥΜΟΥ

Ασκήσεις Γενικά Μαθηµατικά Ι Λύσεις ασκήσεων Οµάδας

Σήματα Συστήματα Ανάλυση Fourier για σήματα και συστήματα συνεχούς χρόνου Περιοδικά Σήματα (Σειρά Fourier)

Ανάλυση ΙΙ Σεπτέµβριος 2012 (Λύσεις)

Farkas. αx+(1 α)y C. λx+(1 λ)y i I A i. λ 1,...,λ m 0 me λ 1 + +λ m = m. i=1 λ i = 1. i=1 λ ia i A. j=1 λ ja j A. An µ := λ λ k = 0 a λ k

ΜΕΤΑΒΟΛΙΚΕΣ ΑΝΙΣΟΤΗΤΕΣ ΚΑΙ ΠΡΟΒΛΗΜΑΤΑ ΕΛΕΥΘΕΡΩΝ ΣΥΝΟΡΩΝ ΣΤΗ ΜΑΘΗΜΑΤΙΚΗ ΧΡΗΜΑΤΟΟΙΚΟΝΟΜΙΑ ΜΕΤΑΠΤΥΧΙΑΚΗ ΕΡΓΑΣΙΑ ΝΙΠΥΡΑΚΗ ΜΑΡΙΑ ΠΑΝΕΠΙΣΤΗΜΙΟ ΚΡΗΤΗΣ

Θεωρία Πιθανοτήτων και Στατιστική

Ανάλυση. σήματα και συστήματα

Statistik gia QhmikoÔc MhqanikoÔc EKTIMHSH PARA

MELETH TWN RIZWN TWN ASSOCIATED ORJOGWNIWN

Φυλλο 3, 9 Απριλιου Ροδόλφος Μπόρης

Ergasthriak 'Askhsh 2

N.Σ. Μαυρογιάννης 2010

Å Ó Ó ÐÅÉÑÁÌÁÔÉÊÏ ËÕÊÅÉÏ. ÁóêÞóåéò. ôçò ÅÕÁÃÃÅËÉÊÇÓ Ó ÏËÇÓ ÓÌÕÑÍÇÓ Å ÅÔÏÓ É ÉÄÑÕÓÇÓ

Ο μαθητής που έχει μελετήσει το κεφάλαιο αυτό θα πρέπει:

Σχήμα 1.1: Διάφορες ισόχρονες καμπύλες με διαφορετικές μεταλλικότητες Ζ, και περιεκτικότητα σε ήλιο Υ.


Upologistikˆ Zht mata se Sumbibastikèc YhfoforÐec

Didaktorikèc spoudèc stic HPA, sta Majhmatikˆ. 20 MartÐou 2015

spin triplet S =1,M S =0 = ( + ) 2 S =1,M S = 1 = spin singlet S =0,M S =0 = ( )

ΓΕΝΙΚΕΣ ΑΣΚΗΣΕΙΣ 3 ου ΚΕΦΑΛΑΙΟΥ (Γ ΟΜΑ ΑΣ) Ασκήσεις σχολικού βιβλίου σελίδας

+#!, - ),,) " ) (!! + Henri Poincar e./ ', / $, 050.

2

Γιάνναρος Μιχάλης. 9x 2 t 2 7dx 3) 1 x 3. x 4 1 x 2 dx. 10x. x 2 x dx. 1 + x 2. cos 2 xdx. 1) tan xdx 2) cot xdx 3) cos 3 xdx.

ΑΛΓΕΒΡΑ Β Λυκείου ΑΣΚΗΣΕΙΣ. 2. Να υπολογίσετε την τιµή των παραστάσεων : α) συν π 18 συνπ 9 - ηµ π. 18 ηµπ 9. β) συν18 ο συν27 ο - ηµ18 ο ηµ27 ο

Νίκος Ζανταρίδης. Χρήσιμες γνώσεις Τριγωνομετρίας. Λυμένες Ασκήσεις. Προτεινόμενες Ασκήσεις

thlèfwno: , H YHFIAKH TAXH A' GumnasÐou Miqˆlhc TzoÔmac Sq. Sumb. kl.

1.06 Δίνεται ένα σύστημα (Σ) 2 γραμμικών

Τριγωνομετρία ΓΙΩΡΓΟΣ ΚΑΡΙΠΙΔΗΣ 2 ΑΝΘΟΥΛΑ ΣΟΦΙΑΝΟΠΟΥΛΟΥ

Σχόλια για το Μάθημα. Λουκάς Βλάχος

Tm ma Fusik c Mˆjhma: Pijanìthtec -Sfˆlmata-Statistik PerÐodoc: Febrouˆrioc 2008

Ημερομηνία: Σάββατο 29 Δεκεμβρίου 2018 Διάρκεια Εξέτασης: 3 ώρες ΕΚΦΩΝΗΣΕΙΣ

Γ ΩΝΙΕΣ Π ΟΥ Σ ΥΝΔΕΟΝΤΑΙ Μ ΕΤΑΞΥ Τ ΟΥΣ

EUSTAJEIA DUNAMIKWN SUSTHMATWN 1 Eisagwg O skop c tou par ntoc kefala ou e nai na parousi sei th basik jewr a gia th mel th thc eust jeiac en c mh gra

Ta Jewr mata Alexander kai Markov thc JewrÐac Kìmbwn

ΣΤΡΑΤΗΣ ΑΝΤΩΝΕΑΣ ΣΠΑΡΤΗ 2008

Transcript:

PANEPISTHMIO DUTIKHS ATTIKHS SQOLH MHQANIKWN Ask seic kai Jèmata sthn Pragmatik Anˆlush I TMHMA POLITIKWN MHQANIKWN Anaplhrwt c Kajhght c: Dr. Pappˆc G. Alèandroc

Perieqìmena. Sumbolismìc kai OrologÐa.. PÐnakac Basik n Trigwnometrik n Sunart sewn. PÐnakac Basik n Aìristwn Oloklhrwmˆtwn.3 PÐnakac Basik n Anaptugmˆtwn se Dunamoseirˆ 3. Ask seic sthn Pragmatik Anˆlush I 4. Jèmata Eetˆsewn. Akadhmaïkì ètoc 8 9. Akadhmaïkì ètoc 9 3. Akadhmaïkì ètoc 4. Akadhmaïkì ètoc 5. Akadhmaïkì ètoc 3 6. Akadhmaïkì ètoc 3... anart ntai sthn hlektronik dieôjunsh: http : // vplace.teipir.gr/pde math 5. Lumènec Ask seic 6. BibliografÐa 3

Sumbolismìc kai OrologÐa R R + to sônolo twn pragmatik n arijm n to sônolo twn jetik n pragmatik n arijm n to epektamèno sônolo twn pragmatik n arijm n. EÐnai to sônolo twn pragmatik n arijm n R R sto opoðo èqoume prosjèsei dôo stoiqeða, to ( + ) kai to. Dhlad R R {, },, ìpwc sun jwc grˆfetai, R [, ]. Z N Q (a, b) [a, b] [a, b) (a, b] to sônolo twn akeraðwn to sônolo twn jetik n akeraðwn to sônolo twn rht n anoiktì kai fragmèno diˆsthma kleistì kai fragmèno diˆsthma hmianoiktì diˆsthma (kleistì apì aristerˆ kai anoiktì apì deiˆ) hmianoiktì diˆsthma (anoiktì apì aristerˆ kai kleistì apì deiˆ) An n N, n! 3 n, (n)!! 4 6 (n ) (n) kai (n + )!! 3 5 (n ) (n + ). H akoloujða (a n ) pragmatik n arijm n lègetai aôousa (fjðnousa) an a n+ a n gia kˆje n N (a n+ a n gia kˆje n N). To akèraio mèroc tou R, sumbolðzetai me [], eðnai o monadikìc akèraioc k Z tètoioc ste k < k +. H sunˆrthsh f orismènh sto A R, A, eðnai ˆrtia (antðstoiqa peritt ), ìtan gia kˆje A to A kai f ( ) f () (antðstoiqa f ( ) f ()). H sunˆrthsh f orismènh sto diˆsthma I eðnai aôousa (antðstoiqa fjðnousa), an gia kˆje, I, me <, eðnai f ( ) f ( ) (antðstoiqa f ( ) f ( ). H sunˆrthsh f eðnai gn sia aôousa (antðstoiqa gn sia fjðnousa) sto diˆsthma I, an gia kˆje, I, me <, eðnai f ( ) < f ( ) (antðstoiqa f ( ) > f ( ). f (n) h n-ost parˆgwgoc miac sunˆrthshc f. y τoξηµ y arcsin, y [ π, ] π h antðstrofh thc sunˆrthshc ηµy sin y. y τoξσυν y arccos, y [, π] h antðstrofh thc sunˆrthshc συνy cos y. y τoξεϕ y arctan, y ( π, ) π h antðstrofh thc sunˆrthshc εϕy tan y. y τoξσϕ y arccot, y (, π) h antðstrofh thc sunˆrthshc σϕy cot y. 4

sinh e e, R uperbolikì hmðtono. cosh e +e, R uperbolikì sunhmðtono. tanh e e e +e, R cosh sinh uperbolik efaptomènh. h basik tautìthta twn uperbolik n sunart sewn. n a n, a n R seirˆ pragmatik n arijm n arijmhtik seirˆ apl c seirˆ. n a n n, a n R dunamoseirˆ seirˆ dunˆmewn tou. n a n ( ) n, a n R dunamoseirˆ tou dunamoseirˆ me kèntro to. n z n, z n C migadik seirˆ. Oi tôpoi e jθ cos θ + j sin θ kai e jθ cos θ j sin θ lègontai tôpoi tou Euler kai prosdiorðzoun thn ekjetik morf tou migadikoô arijmoô. M ij elˆsswn orðzousa tou stoiqeðou a ij. A ij ( ) i+j M ij algebrikì sumpl rwma tou stoiqeðou a ij. I monadiaðoc pðnakac. A t anˆstrofoc tou pðnaka A. A antðstrofoc tou pðnaka A. r(a) bajmìc tˆh tou pðnaka A. A λi qarakthristik sunˆrthsh tou pðnaka A. qarakthristik eðswsh tou pðnaka A. Oi rðzec thc kaloôntai qarakthristikèc rðzec idiotimèc tou pðnaka A. A λi α β α β συνω me ω π α β ewterikì ginìmeno dôo dianusmˆtwn. ( α ) (, β, γ α ) β γ miktì ginìmeno dianusmˆtwn. eswterikì ginìmeno dôo dianusmˆtwn. ( α) + (y β) R analutik eðswsh tou kôklou [kèntro K(α, β) kai aktðna R]. +y +A+By+Γ genik eðswsh tou kôklou [ kèntro K(α, β) ( A, B ) kai aktðna R A +B 4Γ ]. { α + R συνϑ y β + R ηµϑ parametrikèc eis seic tou kôklou [kèntro K(α, β) kai aktðna R]. α + y { α συνϑ y β ηµϑ β h eðswsh thc elleðyewc. parametrikèc eis seic thc elleðyewc. 5

PÐnakac Basik n Aìristwn Oloklhrwmˆtwn.. 3. 4. 5. 6. 7. 8. 9.... 3. 4. 5. 6. α d α + α+ + c, α, > d ln + c, a d ln a a + c, < a e d e + c sin d cos + c cos d sin + c (kπ cos d tan + c, π, kπ + π ), k Z sin d cot + c, { a d arcsin ( a ) + c, arccos ( (kπ, (k + ) π), k Z a ) + c, { a + d a arctan ( ) a + c, a arccot ( ) a + c, a >, ( a, a) a ( a + d ln + ) a + + c, a a d ln + a + c, > a > sinh d cosh + c cosh d sinh + c cosh d tanh + c sinh d coth + c, 6

PÐnakac Basik n Trigwnometrik n Sunart sewn. TrigwnometrikoÐ arijmoð oeðac gwnðac 'Estw èna orjog nio trðgwno ABΓ (A 9 ). ηµb β ( ) apènanti kˆjeth α upoteðnousa συνb γ ( ) proskeðmenh kˆjeth α upoteðnousa εϕb β ( ) apènanti kˆjeth γ proskeðmenh kˆjeth σϕb γ ( ) proskeðmenh kˆjeth β apènanti kˆjeth. Basikèc trigwnometrikèc tautìthtec ηµ ω + συν ω εϕω ηµω συνω σϕω συνω ηµω εϕω σϕω συν ω + εϕ ω ηµ ω εϕ ω + εϕ ω 3. GwnÐec antðjetec συν( ω) συνω ηµ( ω) ηµω εϕ( ω) εϕω σϕ( ω) σϕω 4. GwnÐec me ˆjroisma 8 ηµ(8 ω) ηµω συν(8 ω) συνω εϕ(8 ω) εϕω σϕ(8 ω) σϕω 5. GwnÐec pou diafèroun katˆ 8 ηµ(8 + ω) ηµω συν(8 + ω) συνω εϕ(8 + ω) εϕω σϕ(8 + ω) σϕω 7

6. GwnÐec me ˆjroisma 9 ηµ(9 ω) συνω συν(9 ω) ηµω εϕ(9 ω) σϕω σϕ(9 ω) εϕω 7. SunhmÐtono ajroðsmatoc kai diaforˆc gwni n συν(α β) συνασυνβ + ηµαηµβ συν(α + β) συνασυνβ ηµαηµβ 8. HmÐtono ajroðsmatoc kai diaforˆc gwni n ηµ(α + β) ηµασυνβ + συναηµβ ηµ(α β) ηµασυνβ συναηµβ 9. Efaptomènh ajroðsmatoc kai diaforˆc gwni n συν(α + β), συνα kai συνβ εϕα + εϕβ εϕ(α + β) εϕαεϕβ εϕα εϕβ εϕ(α β) + εϕαεϕβ. Sunefaptomènh ajroðsmatoc kai diaforˆc gwni n ηµ(α + β), ηµα kai ηµβ σϕ(α + β) σϕασϕβ σϕβ + σϕα σϕ(α β) σϕασϕβ + σϕβ σϕα. TrigwnometrikoÐ arijmoð thc gwnðac a ηµα ηµασυνα συνα συν α ηµ α συν α ηµ α εϕα εϕα εϕ α. TrigwnometrikoÐ arijmoð thc gwnðac a, an gnwrðzoume to συνα. ηµ α συνα συν α + συνα εϕ α συνα + συνα 8

3. P c ekfrˆzetai to ηµ wc sunˆrthsh thc εϕ, an συν. ηµ ηµ ηµ ηµ συν συν + ηµ diairoôme touc ìrouc tou klˆsmatoc me to συν ηµ συν συν συν +ηµ συν ηµ εϕ + εϕ ηµ συν συν συν + ηµ συν εϕ + εϕ 4. P c ekfrˆzetai to συν wc sunˆrthsh thc εϕ, an συν. συν συν συν συν ηµ συν + ηµ diairoôme touc ìrouc tou klˆsmatoc me to συν συν συν ηµ συν συν συν + ηµ συν συν εϕ + εϕ εϕ + εϕ 9

PÐnakac Basik n Anaptugmˆtwn se Dunamoseirˆ e + +! + sin 3 3! + 5 n 5! n cos! + 4 4! n n n!, R, ( ) n n+ (n + )!, R, ( ) n n (n)!, R, sinh + 3 3! + 5 5! + n+ (n + )!, R, cosh +! + 4 4! + n (n)!, R, + + + + + ln ( + ) + 3 ln ( ) ( + n n n, <, (gewmetrik seirˆ) n ( ) n n, <, (gewmetrik seirˆ) n 3 ( ) n n + 3 3 + ) n n n, <, n n, <, ( ) + ln + 3 3 + 5 5 + n+ n +, <, n arctan 3 3 + 5 5 ( ) n n+ n +, <, ( + ) α + α + ìpou α R, α (α ) + ( ) α kai n n ( ) α n ( ) α n, <, (diwnumik seirˆ) n α (α ) (α ) (α n + ) n!.

PANEPISTHMIO DUTIKHS ATTIKHS SQOLH MHQANIKWN TMHMA POLITIKWN MHQANIKWN ANWTERA MAJHMATIKA I Ask seic sthn Pragmatik Anˆlush I Jèma. (a) Na brejeð h eðswsh enìc kôklou pou pernˆ apì ta shmeða A(, ), B(5, 4), Γ(, 5) pou den keðntai sthn Ðdia eujeða. (mon.) (b) Na grammoskiasjeð h perioq R, pou eurðsketai pˆnw apì ton -ˆona kai kˆtw apì thn eujeða y kai thn kampôlh y. (.5mon.) Jèma. (a) Na prosdioristoôn oi diastˆseic tou orjogwnðou me to mègisto embadìn pou eggrˆfetai se kôklo aktðnac R. (mon.) R O R Y X (b) Na brejeð to èrgo W pou parˆgetai apì mia dônamh F (,, 3) ìtan metafèrei to shmeðo efarmog c thc apì to A(,, ) sto B(4, 3, ). (.5mon.)

Jèma 3. (a) Na eetasteð wc proc th sôgklish h seirˆ n ( ) n n ln n. (.8mon.) (b) Na brejoôn oi qarakthristikèc rðzec ( idiotimèc) tou pðnaka: A 3 4 7. (.7mon.) Jèma 4. (a) Eetˆsete an sugklðnei to: I + d ln. (.5mon.) (b) Na brejeð to embadìn tou trig nou me korufèc A(, ), B(4, 6), Γ(3, 5). (mon.) Jèma 5. (a) Na upologisteð to olokl rwma: I e + e d. (.5mon.) (b) Na brejeð h sunj kh twn k, λ, ν ste ta dianôsmata u λ i + λ j + k, v µ i + µ j + k kai w ν i + ν j + k na eðnai sunepðpeda. (mon.) Jèma 6. (a) 'Estw to olokl rwma: Eetˆsete eˆn sugklðnei. I d 4. (mon.) (b) 'Ena s ma mˆzac m 5gr. kineðtai sto q ro me taqôthta U (,, 3). Na brejeð h orm tou. (.5mon.)

Jèma 7. (a) Na brejeð to kèntro bˆrouc thc epifˆneiac pou periorðzetai apì tic grammèc y,, y. (mon.) y / (, ) y C O y (b) Eetˆste wc proc th sôgklish th seirˆ ( ) n n n. (.5mon.) Jèma 8. (a) Na upologisjeð h mèsh tim thc olik c antðdrashc mèqri th qronik stigm t 4, {R(t)} R(t) t R(t)dt t enìc organismoô s> èna fˆrmako, ìtan h antðdrash R(t) èqei rujmì metabol c R(t) kai h arqik antðdrash eðnai Ðsh me monˆdec. (mon.) (b) UpologÐste to ìrio : lim. (.5mon.) Jèma 9. (a) AfoÔ melet sete wc proc th sunèqeia sto, na eetˆsete an h sunˆrthsh eðnai paragwgðsimh sto shmeðo autì. f() +, an. (mon.) (b) H èntash tou magnhtikoô pedðou pou dhmiourgeðtai apì to reôma pou diarrèei metallik speðrac aktðnac a, se èna shmeðo tou ˆona thc, dðnetai apì th sqèsh C F (a + ) 3 ìpou h apìstash tou shmeðou apì to kèntro thc speðrac ( < < ) C stajer. Na prosdioristeð h apìstash gia thn opoða h tim thc F gðnetai mègisth. (.5mon.) 3

Jèma. (a) Na upologisjeð to olokl rwma: I d 9. (mon.) y 8 f() 9 6 3 3 4 4 3 3 4 (b) Na brejeð to ìrio: lim ( +3 ) +4. (.5mon.) Jèma. (a) Na upologisteð to embadìn thc èlleiyhc qrhsimopoi ntac tic parametrikèc eis seic thc a cos t, y b sin t, < t π. (.8mon.) y b a O a b (b) Na brejeð h parˆgwgoc thc y τoξεϕ, y ( π, π ). (.7mon.) 4

Jèma. (a) Oi grafikèc parastˆseic C f kai C g twn sunart sewn pou orðzontai me f() sin kai g() cos tèmnontai ˆpeirec forèc kai orðzoun qwrða Ðswn embad n. Na upologðsete to embadìn E enìc apì ta qwrða autˆ. (.8mon.) y.5 f() sin.5.5.5 g() cos π 4 5 π 4 3 3 4 5 6 (b) Na brejeð h eðswsh thc efaptomènhc kai thc kˆjethc sth grafik parˆstash thc sto shmeðo. f() e ηµ, (.7mon.) Jèma 3. (a) Na upologisjeð to olokl rwma I ln 4 ln ln d. (b) Na upologðsete to olokl rwma ( ) I ln + d. (.3mon.) (.mon.) Jèma 4. (a) Na upologisjeð to olokl rwma I (b) Na upologðsete to olokl rwma I e ( + ηµ) + συν τoξηµ + d. d (.mon.) (.3mon.) 5

Jèma 5. (a) Na brejeð to embadìn E tou qwrðou to opoðo perikleðetai apì tic grafikèc parastˆseic twn y gia [, + ), kai y. (.3mon.) (b) Na prosdioristeð h aktðna enìc kuklikoô tomèa tou opoðou h perðmetroc eðnai m, ètsi ste to embadìn tou na eðnai mègisto. (.mon.) Jèma 6. (a) Na brejeð to embadìn tou trig nou pou èqei korufèc ta shmeða A(,, ), B(3, 4, 5), Γ(, 5, 6). (b) Na brejeð h eðswsh thc efaptomènhc thc grafik c parˆstashc thc f me sto shmeðo me tetagmènh y /. f() 3 4 (.8mon.) (.7mon.) Jèma 7. (a) Na brejeð to kèntro bˆrouc thc epifˆneiac pou èqei sq ma hmikôkliou aktðnac R. (mon.) y y R O R R y R (b) Na upologisteð to olokl rwma: I ln d. (.5mon.) Jèma 8. (a) Na lujeð me ton algìrijmo tou Gauss to sôsthma: + y + ω y + ω 3 y + 3ω (b) Na upologisteð to olokl rwma: 3 3 + 5 + 6 d. (.3mon.) (.mon.) 6

Jèma 9. (a) Na brejeð to m koc tou tìou thc hmikubik c parabol c y 3 metaô twn shmeðwn (, ) kai (4, 8). (mon.) (b) Na brejeð to kèntro kai h aktðna tou kôklou: +y +4 6y. (.5mon.) Jèma. (a) Na brejeð to m koc tou tìou thc kampôlhc apì t èwc t 4. t, y t 3 (mon.) (b) Na brejeð h eðswsh thc elleðyewc me F ( 4, ), F (4, ) kai koruf A(6, ). (.5mon.) Jèma. (a) Gia poiˆ tim tou µ, oi eujeðec: µ + y 5, µy, y + µ pernoôn apì to Ðdio shmeðo. (.9mon.) (b) Na lujeð me th mèjodo Crammer to sôsthma: + y + z 6 y + z 3 3 + y z (.9mon.) (g) Na upologisteð to ìrio: τoξσυν lim. (.7mon.) Jèma. Na apodeiqjeð ìti oi parakˆtw seirèc sugklðnoun (a) ( ) n sin n n (.mon.) (b) ( ) ( ) n n n n. (.3mon.) 7

Jèma 3. (a) Na lujeð me ton algìrijmo tou Gauss to sôsthma: + 3ω 3 + y + 6ω 6 y + 3ω 3. (.3mon.) (b) H enèrgeia pou katanal nei ènac mikroorganismìc kinoômenoc sto aðma enìc asjenoôc me taqôthta v, eðnai : E v [ (v 35) + 75 ]. Me poiˆ taqôthta prèpei na kinhjeð gia na katanal sei thn mikrìterh enèrgeia; (.mon.) Jèma 4. Na upologistoôn ta oloklhr mata: (a) (b) 3 d. 5 3 4 d. (.mon.) (.3mon.) Jèma 5. (a) BreÐte tic parag gouc twn sunart sewn: f() ηµ 3, f() ηµ 3, (b) Na brejeð to ìrio: f() (συν) ηµ, f() συν, f() ln ( + 3y), f(y) ln ( + 3y). (mon.) lim ( ) +4 + 5. (.5mon.) Jèma 6. (a) Ekfrˆste tic sunart seic f() a kai f() e se anˆptugma poluwnômou, wc proc tic dunˆmeic tou. (mon.) (b) Na brejeð to ìrio: ( lim + ). (.5mon.) 8

Jèma 7. (a) Na upologisteð to embadìn tou qwrðou pou perikleðetai metaô twn diagrammˆtwn twn sunart sewn f() ηµ, g() συν kai tou ˆona y. (.3mon.) y f() sin O π 4 g() cos (b) Na brejeð to P.O thc sunˆrthshc: y π τoξσυν 6 τoξσυν. (.mon.) ShmeÐwsh. H sunˆrthsh y τoξσυν èqei pedðo orismoô to [, ] kai pedðo tim n to [, π]. y π Γ y y τoξσυν M (, π/) y A O M (π/, ) y B y συν π Jèma 8. (a) Na eetasjeð wc proc thn sôgklish h seirˆ: ( ) ( ) n n + n n (b) Na eetasjeð wc proc thn sôgklish h seirˆ: ( ) n+ n + 5 n(n + ). n. (.3mon.) (.mon.) 9

Jèma 9. (a) Na prosdiorðsete ta α, β R ste h sunˆrthsh f me tôpo: na eðnai suneq c sto R. ++α, < f() β + α,, (.mon.) (b) 'Estw trðgwno ABΓ kai AM h diˆmesoc tou. An isqôei: na brejoôn oi λ, µ R. AB + 3 AΓ λ BΓ + µ AM (.3mon.) Jèma 3. (a) Na brejeð h eðswsh enìc kôklou pou pernˆ apì ta shmeða A(3, ), B( 5, 3 ), Γ(, ) pou den keðntai sthn Ðdia eujeða. (.7mon.) (b) An gia ta mh sunepðpeda dianôsmata α, β, γ isqôei: ( k α ) ( + β + 3 γ + λ α ) ( β γ + µ α ) β γ 6 ( α + γ ), na brejoôn ta k, λ, µ R. (.8mon.) Jèma 3. (a) Na lujeð h eðswsh t t dt π, >. (.mon.).5 y f(t) t, t R \ [, ] t.5 f(t)dt.5.5 3 3 4 5 6 (b) Na upologisteð to 8 + 3 7 + 3 6 + 5 5 + 3 4 + 3 3 I ( + ) 3 ( + ) d. (.3mon.)

Jèma 3. Na eetastoôn wc proc th sôgklish ta genikeumèna oloklhr mata (a) ShmeÐwsh. To + + sin d. (mon.) d p, p R sugklðnei gia p > kai apoklðnei gia p y y sin y y (b) DikaiologeÐste gewmetrikˆ giatð to + sin d +. (.5mon.) Jèma 33. (a) Na apodeiqjeð ìti to genikeumèno olokl rwma ln d. ln d sugklðnei kai ìti (.6mon.) (b) Na eetˆsete wc proc th sôgklish to genikeumèno olokl rwma ln d. (.3mon.) (g) Na apodeiqjeð ìti ta genikeumèna oloklhr mata sugklðnoun kai ìti I π/ ln (sin ) d, J π/ ln (cos ) d I J π ln. (.6mon.)

Jèma 34. (a) DÐdontai ta monadiaða kai sunepðpeda dianôsmata α, β, γ, ste: ) ) ( ( β ) α, β ϑ,, γ ϕ, ( γ, α ω. DeÐte ìti: συνϑ συνω συνϑ συνϕ συνω συνϕ. (.3mon.) (b) Na brejeð to P.O thc sunˆrthshc: y τoξηµ(log ). (.mon.) ShmeÐwsh. H sunˆrthsh y τoξηµ èqei pedðo orismoô to [, ] kai pedðo tim n to [ π, π ]. y π/ y y τoξηµ y ηµ y y π/ A O π/ B y π/ Γ Jèma 35. (a) Na breðte thn eðswsh tou epipèdou pou dièrqetai apì ta shmeða P (,, ), P (3, 4, ), P 3 (, 3, 5). (b) 'Estw ta monadiaða dianôsmata δ, δ, ( δ δ, ste, ) δ π 3. DeÐte ìti isqôei: 4 δ δ > ( δ δ + ) δ (.3mon.). (.mon.)

Jèma 36. (a) Na upologisteð to aìristo olokl rwma I συν d, nπ + π, n Z. (mon.) 5 4 sec cos, nπ + π, n Z y 3 3π + π π + π π + π π + π π + π 3π + π 3 4 5 5 5 Apˆnthsh. συν d ln συν + εϕ + C ln + ηµ ηµ + C ln + εϕ (/) εϕ (/) + C, nπ + π/, n Z. (b) Na brejeð to m koc tìou thc kampôlhc y ln (συν) me [ π 6, π ] 4. (.5mon.) y π π y ln cos, [ π 6, π 4 ] 3 4 5.5.5.5.5 3

Jèma 37. (a) Na upologisteð to aìristo olokl rwma I d, nπ, n Z. ηµ (.mon.) 5 4 y csc sin, nπ, n Z 3 3π π π π π 3π 3 4 5 5 5 Apˆnthsh. ηµ d ln ηµ + σϕ + C ln συν εϕ συν + + C ln + C, nπ + π/, n Z. (b) Na brejeð to m koc tìou thc kampôlhc y ln (ηµ) me [ ] π 3, π. (.3mon.) 3 y π y ln sin, [ π 3, π 3 ] 3 f() ln sin, nπ nπ + π, n Z 4 5.5.5.5.5 3 3.5 4 4

Jèma 38. (a) Na apodeðete ìti o ìgkoc k nou me Ôyoc h kai aktðna bˆshc r eðnai V 3 πr h. (.3mon.) y r O h (b) Na upologðsete ton ìgko tou stereoô pou gennˆtai apì thn peristrof thc kleist c kampôlhc me eðswsh 6 + y 9 perð ton -ˆona. (.mon.) Jèma 39. (a) Na brejeð h rop adrˆneiac wc proc ton -ˆona thc hmiperifèreiac me eðswsh + y R. (.3mon.) y y R O R R y R (b) An < α <, na apodeðete ìti h sunˆrthsh f() 3 3 + α èqei akrib c mða lôsh sto diˆsthma (, ). (.mon.) 5

Jèma 4. (a) Na upologðsete to olokl rwma + d p, ìpou p R. (.mon.) Apˆnthsh. To + (b) Na apodeiqjeð ìti h armonik seirˆ p-tˆhc d, p R sugklðnei gia p > kai apoklðnei gia p. p + n n p sugklðnei sto R gia p > kai apoklðnei gia p. (.3mon.) Jèma 4. (a) 'Estw h akoloujða (a n ), me a n n!/n n. i. Na apodeiqjeð ìti lim n a n+ /a n lim n n a n /e. (.6mon.) ii. Na apodeiqjeð ìti lim n n n!. (.4mon.) (b) Na eetasteð wc proc th sôgklish h seirˆ n ( ) n n n!. (.5mon.) Jèma 4. 'Estw P {,, 3,..., n} mia diamèrish tou diast matoc I [, n], n. (a) UpologÐzontac to kat tero kai an tero ˆjroisma thc f() ln pou antistoiqeð sthn diamèrish P, kaj c epðshc kai to olokl rwma : na apodeðete ìti : n ln d, n n e n+ < n! < (n + ) n+ e n, n, 3,.... (.3mon.) (b) Na apodeiqjeð ìti sugklðnei kai na brejeð to ìriì thc. a n n n! n (.8mon.) (g) Na apodeiqjeð ìti n lim n!. n (.4mon.) 6

Jèma 43. (a) ApodeÐte ìti gia kˆje n N isqôei : ( n ) n ( n ) n e < n! < en (.4mon.) e e Upìdeih. ApodeÐte ìti h akoloujða a n ( n e ) n n! eðnai gnhsðwc fjðnousa kai h β n n a n eðnai gnhsðwc aôousa. SugkrÐnate me thn a kai β antðstoiqa. (b) Na apodeiqjeð ìti n lim n!. n (.mon.) Jèma 44. (a) Na brejeð o bajmìc tou pðnaka: A 4. (.7mon.) (b) Na upologisteð to olokl rwma: 4 ( ) d. (.8mon.) Jèma 45. Na eetasteð an to genikeumèno olokl rwma I + dt t t +, >, sugklðnei kai an nai na upologisteð. (.5mon.) ShmeÐwsh. To + ( ++ ) Apˆnthsh. I ln +. d, p R sugklðnei gia p > kai apoklðnei gia p. p Jèma 46. An n N, n, na apodeiqjoôn oi parakˆtw anagwgikoð tôpoi (aþ) sin n d n sinn cos + n sin n d n (bþ) cos n d n cosn sin + n cos n d. n (.5mon.) 7

Jèma 47. (a) Na apodeðete ìti to embadìn tou orjog niou trig nou pou sqhmatðzetai apì touc ˆonec kai thn efaptìmenh eujeða (ε) thc grafik c parˆstashc C f thc sunˆrthshc f : (, + ) R pou orðzetai me f(), an > sto tuqaðo shmeðo thc C f diathreð stajer tim (eðnai stajerì). (.6mon.) y y A (, y ) O B (b) Na apodeðete ìti ( lim + e. (.9mon.) + ) Jèma 48. (a) Na brejeð to kèntro bˆrouc tou epipèdou qwrðou pou orðzetai apì tic kampôlec y kai y. (.3mon.) (b) Na brejeð o ìgkoc thc sfaðrac. (.mon.) y y R O R R y R 8

Jèma 49. (a) Na upologisteð to olokl rwma I π + 3 cos d. (.5mon.) (b) Qrhsimopoi ntac paragontik olokl rwsh na apodeiqjeð ìti to genikeumèno olokl rwma + sin a d sugklðnei gia a >. (.5mon.) ShmeÐwsh. To + d, p R sugklðnei gia p > kai apoklðnei gia p. p Jèma 5. (a) Na upologisteð to olokl rwma I d. (.6mon.) 3 + 5ηµ.8 y.6.4.. f() 3+5 sin.4.6.8 8 6 4 4 6 8 Apˆnthsh. I 4 ln 3εϕ( )+ εϕ( )+3 + c. (b) Na apodeðete ìti ( lim + k e + ) k. (.9mon.) 9

Jèma 5. (a) Na eetasteð wc proc th sôgklish h seirˆ n Upìdeih. Na apodeiqjeð ìti ( ) πn a n sin n + ( ) πn sin n +. (.mon.) ( ) π ( ) n sin n + (b) Na eetˆsete wc proc th sôgklish to genikeumèno olokl rwma. (g) Na apodeiqjeð ìti I τoξεϕ + τoξεϕ ln( + ) d. (.6mon.) ( ) π, >. (.8mon.) Jèma 5. (a) Na apodeiqjeð ìti τoξηµ, gia kˆje [, ]. (.mon.) (b) 'Estw Na apodeiqjeð ìti h seirˆ sugklðnei an kai mìno an a >. a n n (τoξηµ) a d, a >. n a n (.3mon.) ShmeÐwsh. H seirˆ + n n p, p R sugklðnei gia p > kai apoklðnei gia p. Jèma 53. (a) Na apodeiqjeð ìti h sunˆrthsh f() tan, (, π ), eðnai gn sia aôousa. (b) Na eetasteð wc proc th sôgklish h seirˆ ( ) n ( ) n tan n n (.mon.). (.3mon.) 3

Jèma 54. (a) 'Estw f : [a, b] [c, d] mia kai epð sunˆrthsh sto kleistì diˆsthma [a, b] kai c, d mh arnhtikoð pragmatikoð arijmoð. An h f eðnai aôousa sunˆrthsh sto [a, b], na apodeðete ìti b f(t)dt + d a c f (t)dt bd ac. (.9mon.) y y f(), [a, b] d S c S O a b ShmeÐwsh. Epeid h sunˆrthsh f eðnai, epð kai aôousa, sunepˆgetai ìti h f eðnai suneq c sunˆrthsh. (b) Na upologðsete ton ìgko tou stereoô ek peristrof c gôrw apì ton -ˆona tou qwrðou pou perikleðetai apì thn kampôlh y ln kai tic eujeðec, y. (.6mon.) 3

Jèma 55. (a) Na upologðsete to embadìn E tou qwrðou pou perikleðetai apì th grafik parˆstash thc kai twn asumpt twn thc. y (.3mon.) y y y (b) Na eetˆsete wc proc th sôgklish th seirˆ cos(nπ) n. (.mon.) n Jèma 56. (a) Na upologisteð to olokl rwma I d. (.3mon.) ηµ + συν 5 4 y 3 f() sin +cos 3 4 5 8 6 4 4 6 8 Apˆnthsh. I ln εϕ ( ) + c. (b) Na upologisteð to olokl rwma I + 4 d. (.mon.) 3

Jèma 57. (a) Na brejeð h eðswsh thc eujeðac pou pernˆei apì to shmeðo (, 3) kai sqhmatðzei me touc ˆonec suntetagmènwn trðgwno embadoô t.m. (.mon.) y B 3 (, 3) y f() O A Apˆnthsh. y 3 3 ( ). (b) Na brejeð h eðswsh kôklou o opoðoc efˆptetai thc eujeðac y + sto shmeðo aut c (, 5) kai to kèntro tou keðtai epð thc eujeðac + y 9. (.3mon.) y M(, 5) K O + y 9 y + Apˆnthsh. ( 6) + (y 3). Jèma 58. (a) Na apodeðete ìti to m koc thc kampôlhc me eðswsh y + ln(cos ), ìpou π 4, eðnai (b) Na upologðsete to olokl rwma Apˆnthsh. + e d. ( ln + ). (.3mon.) + e d. (.mon.) 33

Jèma 59. (a) Na deiqjeð ìti an ta dianôsmata α, β, γ eðnai grammik c aneˆrthta tìte kai ta dianôsmata: u α + β γ, v α β + γ, ω α + 3 β γ, eðnai grammikˆ aneˆrthta. (.9mon.) (b) Na brejeð to embadìn thc perioq c pou sqhmatðzetai apì tic kampôlec + y 4, + y 4. (.6mon.) 3 y + 4 y A y + 4 O M N B 3 3 3 4 5 6 Apˆnthsh. E 8π 6 3 3. Jèma 6. (a) An f, g : [α, β] R eðnai oloklhr simec sunart seic sto kleistì diˆsthma [α, β], na apodeðete ìti: ( ( β ) ( β ) β f() g() d) f () d g () d. (.7mon.) α α α Anisìthta Cauchy Schwarz ShmeÐwsh. Gia kˆje [α, β] kai gia kˆje λ R alhjeôei (λf() + g()). Epomènwc β α (λf() + g()) d. (b) 'Estw f : [, ] R suneq c sunˆrthsh sto kleistì diˆsthma [, ] kai f() > gia kˆje [, ]. Na apodeðete ìti: ( ) ( ) f() d f() d. (.8mon.) 34

Jèma 6. (a) Na brejeð o ìgkoc tou stereoô pou parˆgetai apì thn peristrof thc kampôlhc y perð ton -ˆona kai tic eujeðec,. (.3mon.) y y, [, ] O Apˆnthsh. V 3π. (b) Na eetˆsete wc proc th sôgklish to genikeumèno olokl rwma I + e d. (.mon.) Jèma 6. (a) Na upologisteð to olokl rwma I 7 5 d. (.mon.) (b) Na upologisteð to olokl rwma I ηµ + συν + 3 d. (.3mon.) Jèma 63. (a) Na brejeð to m koc tìou thc kampôlhc ( ) y ln συν, an π 4. (.mon.) (b) H tim p lhshc enìc farmˆkou èqei kajorisjeð se eur. To kìstoc tou sunart sh tou qrìnou dðnetai apì th sqèsh : y t + 5t. Pìte pragmatopoi jhke to mègisto kèrdoc kai poiì tan autì ; 35 (.3mon.)

Jèma 64. (a) 'Estw f : [ a, a] R mia oloklhr simh sunˆrthsh sto kleistì diˆsthma [ a, a], a R. An f eðnai ˆrtia sunˆrthsh, tìte a a a An f eðnai peritt sunˆrthsh, tìte a f()d a f()d. f()d. (.6mon.) (b) Na upologisteð to olokl rwma I cos ln ( ) + d. (.9mon.) Jèma 65. (a) Na upologisteð to embadìn tou qwrðou pou perikleðetai apì tic kampôlec y, y kai tic eujeðec,. ShmeÐwsh. H koruf thc kampôlhc y eðnai to shmeðo ( β ( α, f β )) (, ). α (.9mon.) (b) Se poio shmeðo tou autokinhtìdromou (ε) prèpei na topojethjeð h stˆsh K tou lewforeðou ètsi ste oi kˆtoikoi tou qwrioô B na fjˆnoun sthn pìlh A ston elˆqisto dunatì qrìno. H taqôthta tou lewforeðou eðnai V kai twn pez n v, (v < V ). H apìstash tou qwrioô B apì ton autokinhtìdromo eðnai BM a kai h apìstash MA. (.6mon.) ShmeÐwsh. An kalèsoume th gwnða MBK, ja prosdiorðsoume th gwnða, ètsi ste h diadrom BKA na gðnetai ston elˆqisto qrìno. 36

Jèma 66. (a) Na upologisteð to olokl rwma I π + συν d. (.3mon.) y.5 +cos f().5.5 π 3 4 5 6 (b) Na upologisteð to olokl rwma 5 + 3 I 3 + d. (.mon.) Jèma 67. (a) 'Estw h sunˆrthsh f : R R eðnai suneq c kai periodik me perðodo T >, dhlad Na apodeiqjeð ìti gia kˆje a R eðnai f ( + T ) f (), gia kˆje R. (b) Na upologisteð to olokl rwma a+t a π ϕ ϕ T f () d f () d. sin θ dθ. (.6mon.) (.9mon.) 37

Jèma 68. (a) Na brejeð to embadìn E tou qwrðou pou perikleðetai apì tic grafikèc parastˆseic twn y sin, y cos, kai π. (.3mon.) y.5 f() sin.5.5 O.5 g() cos 3 3 4 5 6 7 8 π (b) Na brejeð to summetrikì tou shmeðou M(, 4) wc proc thn eujeða (ε) : y. (.mon.) Jèma 69. (a) Na apodeiqjeð ìti 3 e sin + d π e. (.mon.) (b) Na upologðsete ta ìria: i) lim + sin t sin t dt sin t dt (.3mon.) ii) lim 3 Jèma 7. (a) Na upologðsete to embadì tou qwrðou pou perikleðetai apì ton kôklo + y 8 kai thn parabol y. (.4mon.) (b) Na apodeiqjeð ìti h sunˆrthsh f : (, + ) R pou orðzetai me tôpo f() ηµ τoξηµ t dt + συν τoξσυν t dt lambˆnei thn tim f() π 4 gia kˆje (, π ). (.mon.) (g) 'Estw n N {}. Na brejeð o megalôteroc apì touc arijmoôc, 3, 3,..., n n. (.4mon.) 38

Jèma 7. (a) JewroÔme th sunˆrthsh f() α ηµ + α ηµ + α 3 ηµ3 +... + α ν ηµν ìpou α, α, α 3,..., α ν R kai ν N. DÐnetai ìti Na deiqteð ìti f() ηµ gia kˆje R. α + α + 3α 3 +... + να ν. (.mon.) (b) An (, y ) eðnai h lôsh tou sust matoc (ηµϑ) (συνϑ)y na upologisteð to olokl rwma y (συνϑ) + (ηµϑ)y, ( + 3) e + d. (.3mon.) Jèma 7. (a) Na apodeiqjeð ìti to ˆjroisma twn apostˆsewn enìc eswterikoô shmeðou M isopleôrou trig nou pleurˆc a apì tic pleurèc isoôtai me to Ôyoc tou isopleôrou trig nou. (.3mon.) ShmeÐwsh. PaÐrnoume thn pleurˆ BΓ tou trig nou pˆnw ston ˆona twn kai thn koruf B na sumpðptei me thn arq twn aìnwn. UpenjumÐzetai ìti to Ôyoc isopleôrou trig nou eðnai υ a 3 B. y A( a, υ) E Z M O B a Γ(a, ) (b) Na upologisteð to olokl rwma: I π + ηµ + συν e d. (.mon.) 39

Jèma 73. (a) An eðnai α (,, ), β (,, ), γ (,, ) na upologisteð to diˆnusma δ ( α β ) ( β γ ) α. (.3mon.) (b) Na apodeiqjeð ìti α β α ( β α ) β. (.mon.) Jèma 74. (a) Na breðte ton antðstrofo pðnaka tou : A [ 4 5 3 ]. (.9mon.) (b) Na breðte ton antðstrofo pðnaka tou : A 3 4. (.6mon.) Jèma 75. (a) DÐnontai oi pðnakec A [ 3 4 5 ], B 3 4. Na brejeð o A B. (.7mon.) (b) DÐnonta oi pðnakec A [ 3 6 8 ], B 4 6 8. Na brejoôn ta ginìmena A B kai B A. (.7mon.) (g) An A na brejeð o R ste na isqôei, B, A B B A I. (.mon.) 4

Jèma 76. DÐnetai èna trðgwno ABΓ, me pleurèc α, β, γ. An isqôoun oi isìthtec: β συνγ + γ συνb α γ συνa + α συνγ β () α συνb + β συνa γ kai α ηµa β ηµb + γ ηµγ β ηµa α ηµb γ ηµa α ηµγ () na deiqjeð ìti: (a) Oi () mac odhgoôn ston nìmo twn sunhmitìnwn. (b) Oi () deðqnoun ìti to trðgwno ABΓ eðnai orjog nio sto A. (.3mon.) (.mon.) Jèma 77. (a) Na upologðsete to olokl rwma: I συν d. (b) DÐnetai h sunˆrthsh f pou eðnai dôo forèc paragwgðsimh sto diˆsthma [α, β]. f (α) kai f () >, gia kˆje (α, β), na deiqjeð ìti o pðnakac A f(β) f(α) antistrèfetai. (.mon.) An isqôei (.4mon.) Jèma 78. Gia kˆje n N upologðsete to ìrio kai sth sunèqeia apodeðete ìti τoξσυν n+ lim τoξσυν n τoξσυν n lim τoξσυν n. (.9mon.) (.6mon.) ShmeÐwsh... τoξσυν n n τoξσυν k τoξσυν k+ τoξσυν k. a n b n (a b) ( a n + a n b + a n 3 b +... + ab n + b n ) ìpou n N. 4

Jèma 79. (a) ProsdiorÐste tic plèon oikonomikèc diastˆseic miac anoiqt c pisðnac 3 m 3 me tetragwnik bˆsh ètsi ste h epifˆneia twn eswterik n toðqwn kai tou pujmèna na eðnai elˆqisth. (.mon.) (b) Na brejeð to m koc tìou thc kampôlhc y τoξηµe, an ln. (.3mon.) y.5 y arcsine, [ ln, ].5 f() arcsine, R.5 5 4 3 ShmeÐwsh. ηµ d ln ηµ + σϕ + C ln συν εϕ συν + + C ln + C, nπ + π/, n Z. Jèma 8. i. DeÐte thn anisìthta tou Bernoulli (+) n > +n gia n, 3,..., an >,. ii. DeÐte ìti lim n n an <. iii. DeÐte ìti gia opoiand pote stajer a. a n lim n n! ShmeÐwsh. An lim n u n, tìte lim n u n. iv. DeÐte ìti (.9mon.) (.9mon.) (.8mon.) e + +! + 3 3! +.... (.mon.) v. DeÐte ìti o e eðnai ˆrrhtoc arijmìc. (.9mon.) vi. Na upologisteð to e d proseggistikˆ me to je rhma thc mèshc tim c tou Taylor kai na ektimhjeð to mègisto sfˆlma. (.mon.) 4

Jèma 8. BreÐte to m koc L AB thc makrôterhc skˆlac pou mporeð na perˆsei apì èna gwniakì diˆdromo, tou opoðou oi diastˆseic shmei nontai sto sq ma, an upotejeð ìti h skˆla metakineðtai parˆllhla proc to pˆtwma. (.5mon.) ShmeÐwsh. i. To m koc thc makrôterhc skˆlac eðnai to mikrìtero eujôgrammo tm ma AB pou akoumpˆ kai stouc dôo toðqouc kai sth gwnða pou sqhmatðzetai apì touc toðqouc. ii. ηµ εϕ, συν, +εϕ +εϕ ( ) iii. L τoξεϕ 3 b a > Apˆnthsh. L AB ( ) 3 a 3 + b 3. Jèma 8. (a) Gia kˆje R na apodeiqjeð ìti Gia kˆje > na apodeiqjeð ìti e + kai h isìthta isqôei an kai mìno an. ln kai h isìthta isqôei an kai mìno an. (b) Poioc arijmìc eðnai megalôteroc e π π e ; (g) An A n a + a + + a n, G n n n n a a a n kai H n a + a + + a n eðnai o arijmhtikìc mèsoc, o gewmetrikìc mèsoc kai o armonikìc mèsoc twn jetik n pragmatik n arijm n a, a,..., a n, antðstoiqa, na apodeiqjeð ìti n H n G n A n a + a + + n a a a n a + a + + a n. a n n Oi isìthtec kai stic dôo parapˆnw anisìthtec isqôoun an kai mìno an a a a n. Jèma 8 (5mon.) 43

JEMATA EXETASEWN. Akadhmaïkì ètoc 8-9. Akadhmaïkì ètoc 9-3. Akadhmaïkì ètoc - 4. Akadhmaïkì ètoc - 5. Akadhmaïkì ètoc -3 44

A.T.E.I. PEIRAIA SQOLH TEQNOLOGIKWN EFARMOGWN GENIKO TMHMA MAJHMATIKWN Kajhghtèc: Elènh JeofÐlh Alèandroc Pappˆc ONOMATEPWNUMO:... EXAMHNO:... A.M.:... JEMATA MAJHMATIKA I TMHMA POLITIKWN DOMIKWN ERGWN 4 FEBROUARIOU 9 Jèma. (a) Na upologðsete to embadìn E tou epipèdou qwrðou pou orðzetai apì tic grammèc (b) Na brejeð to m koc tìou thc kampôlhc y ln, y ln 3, e, e. (.mon.) y τoξηµe, an ln. (.4mon.) y.5 y arcsine, [ ln, ].5 f() arcsine, R.5 5 4 3 ShmeÐwsh. ηµ d ln ηµ + σϕ + C ln συν εϕ συν + + C ln + C, nπ + π/, n Z. 45

Jèma. (a) Na brejeð to embadì thc perioq c pou sqhmatðzetai apì tic kampôlec + y 4, + y 4. (.mon.) 3 y + 4 y A y + 4 O M N B 3 3 3 4 5 6 Apˆnthsh. E 8π 6 3 3. (b) Na apodeiqjeð ìti to genikeumèno olokl rwma sugklðnei gia k >. ShmeÐwsh. To + + sin d k+ d, p R sugklðnei gia p > kai apoklðnei gia p. p (.3mon.) Jèma 3. (a) Na brejeð to kèntro bˆrouc tou epipèdou qwrðou pou orðzetai apì tic kampôlec y kai y. (.mon.) (b) ProsdiorÐste tic plèon oikonomikèc diastˆseic miac anoiqt c pisðnac 3 m 3 me tetragwnik bˆsh ètsi ste h epifˆneia twn eswterik n toðqwn kai tou pujmèna na eðnai elˆqisth. (.3mon.) Jèma 4. (a) To embadìn tou trig nou pou èqei korufèc ta shmeða eðnai t.m.. Na brejeð to a. A(a,, ), B(4,, ), Γ(,, 3) (b) Na eetasteð wc proc th sôgklish h seirˆ ( ) n ( ) n εϕ n n (.mon.). (.4mon.) Diˆrkeia eètashc:.5 rec KALH EPITUQIA 46

A.T.E.I. PEIRAIA SQOLH TEQNOLOGIKWN EFARMOGWN GENIKO TMHMA MAJHMATIKWN Kajhghtèc: Elènh JeofÐlh Alèandroc Pappˆc ONOMATEPWNUMO:... EXAMHNO:... A.M.:... JEMATA MAJHMATIKA I TMHMA POLITIKWN DOMIKWN ERGWN 8 FEBROUARIOU 9 Jèma. (a) Na apodeðete ìti (b) Na upologisteð to ( lim + e + ). (.mon.) τoξσυν lim. (.mon.) (g) Na upologisjeð to olokl rwma: I d 6. (.4mon.) (d) Na eetˆsete wc proc th sôgklish to olokl rwma + ShmeÐwsh. To + sin d. (.4mon.) d, p R sugklðnei gia p > kai apoklðnei gia p. p Jèma. (a) Na brejeð to embadìn tou trig nou me korufèc A(, ), B(, 4), Γ(3, 5). (.7mon.) (b) Na brejeð to embadìn tou trig nou me korufèc A(, 3, ), B(3,, ), Γ(3, 4, 6). (.4mon.) (g) Na brejeð h eðswsh enìc kôklou pou pernˆ apì ta shmeða A(3, ), B(, ), Γ(, ) pou den keðntai sthn Ðdia eujeða. (.4mon.) (d) Na brejeð to kèntro bˆrouc tou epipèdou qwrðou pou orðzetai apì tic kampôlec y kai y 4. (.5mon.) 47

Jèma 3. (a) Na brejeð to embadìn E tou qwrðou pou perikleðetai apì tic grafikèc parastˆseic twn y sin, y cos, kai π. (b) Na upologisteð to aìristo olokl rwma I συν d, nπ + π, n Z. (.3mon.) (.mon.) 5 4 sec cos, nπ + π, n Z y 3 3π + π π + π π + π π + π π + π 3π + π 3 4 5 5 5 Apˆnthsh. συν d ln συν + εϕ + C ln + ηµ ηµ + C ln + εϕ (/) εϕ (/) + C, nπ + π/, n Z. (g) Na apodeðete ìti to embadìn tou orjog niou trig nou pou sqhmatðzetai apì touc ˆonec kai thn efaptìmenh eujeða (ε) thc grafik c parˆstashc C f thc sunˆrthshc f : (, + ) R pou orðzetai me f(), an > sto tuqaðo shmeðo thc C f diathreð stajer tim (eðnai stajerì). y (.5mon.) y A (, y ) O B Diˆrkeia eètashc:.5 rec KALH EPITUQIA 48

A.T.E.I. PEIRAIA SQOLH TEQNOLOGIKWN EFARMOGWN GENIKO TMHMA MAJHMATIKWN Kajhghtèc: Elènh JeofÐlh Alèandroc Pappˆc ONOMATEPWNUMO:... EXAMHNO:... A.M.:... JEMATA MAJHMATIKA I TMHMA POLITIKWN DOMIKWN ERGWN 4 IOUNIOU 9 Jèma. (a) Na apodeðete ìti ( lim + 3 e + ) 3. (.8mon.) (b) Na eetˆsete wc proc th sôgklish th seirˆ n cos(nπ) sin n. (.mon.) Jèma. (a) Na apodeðete ìti o ìgkoc k nou me Ôyoc h kai aktðna bˆshc r eðnai V 3 πr h. (.9mon.) y r O h (b) Na upologðsete to embadìn E tou epipèdou qwrðou pou orðzetai apì tic grammèc y ln, y ln 3, e, e. (.mon.) 49

Jèma 3. (a) Na brejeð to embadìn tou trig nou me korufèc A(, 3), B(, 5), Γ(, 4). (.4mon.) (b) Na brejeð to embadìn tou trig nou me korufèc A(, 3, ), B(, 3, ), Γ(3, 4, ). (.9mon.) (g) Na brejeð h eðswsh enìc kôklou pou pernˆ apì ta shmeða A(3, ), B(, ), Γ(, ) pou den keðntai sthn Ðdia eujeða. (.mon.) Jèma 4. (a) Na upologisteð to olokl rwma I d. (.4mon.) ηµ συν (b) Na brejeð to embadì thc perioq c pou sqhmatðzetai apì tic kampôlec + y 6, + y 8. (.6mon.) 3 y + 4 y A y + 4 O M N B 3 3 3 4 5 6 Jèma 5. (a) Na eetˆsete wc proc th sôgklish to olokl rwma + ShmeÐwsh. To + cos d. (.3mon.) d, p R sugklðnei gia p > kai apoklðnei gia p. p (b) Na brejeð to kèntro bˆrouc tou epipèdou qwrðou pou orðzetai apì tic kampôlec y kai y. (.mon.) Diˆrkeia eètashc:.5 rec KALH EPITUQIA 5

A.T.E.I. PEIRAIA SQOLH TEQNOLOGIKWN EFARMOGWN GENIKO TMHMA MAJHMATIKWN Kajhghtèc: Elènh JeofÐlh Alèandroc Pappˆc ONOMATEPWNUMO:... EXAMHNO:... A.M.:... JEMATA MAJHMATIKA I TMHMA POLITIKWN DOMIKWN ERGWN SEPTEMBRIOU 9 Jèma. (a) Na apodeðete ìti ( lim + e + ). (.9mon.) (b) Na eetˆsete wc proc th sôgklish to olokl rwma + y sin d. (.6mon.) y sin y y ShmeÐwsh. To + d, p R sugklðnei gia p > kai apoklðnei gia p. p Jèma. (a) Na brejeð to embadìn tou trig nou me korufèc A(, ), B(3, 4), Γ(6, 5). (.5mon.) (b) Na brejeð to embadìn tou trig nou me korufèc A(, 3, ), B(3,, ), Γ(3, 4, 6). (.mon.) (g) Na brejeð h eðswsh enìc kôklou pou pernˆ apì ta shmeða A(3, ), B(, ), Γ(, ) pou den keðntai sthn Ðdia eujeða. (.3mon.) 5

Jèma 3. (a) Na upologisteð to aìristo olokl rwma I συν d, nπ + π, n Z. (.mon.) (b) Na brejeð to m koc tìou thc kampôlhc y ln (συν) me [ π 6, π 4 ]. (.8mon.) Jèma 4. (a) Na apodeiqjeð ìti to genikeumèno olokl rwma ln d sugklðnei kai ìti ln d. (b) Na eetasteð wc proc th sôgklish h seirˆ n ( ) n n ln n. (.9mon.) (.mon.) Jèma 5. (a) Na brejeð o ìgkoc thc sfaðrac. (.mon.) y y R O R R y R (b) Na brejeð to kèntro bˆrouc tou epipèdou qwrðou pou orðzetai apì tic kampôlec y kai y. (.3mon.) Diˆrkeia eètashc:.5 rec KALH EPITUQIA 5

A.T.E.I. PEIRAIA SQOLH TEQNOLOGIKWN EFARMOGWN GENIKO TMHMA MAJHMATIKWN Kajhghtèc: Elènh JeofÐlh Alèandroc Pappˆc ONOMATEPWNUMO:... EXAMHNO:... A.M.:... JEMATA MAJHMATIKA I TMHMA POLITIKWN DOMIKWN ERGWN IANOUARIOU Jèma. (a) Na apodeðete ìti ( lim + 7 e + ) 7. (.8mon.) (b) Na eetˆsete wc proc th sôgklish th seirˆ n cos(nπ) sin n. (.7mon.) Jèma. (a) Na brejeð to embadìn thc perioq c pou periorðzetai apì thn èlleiyh a + y b. (.7mon.) y b a O a b (b) ProsdiorÐste tic plèon oikonomikèc diastˆseic miac anoiqt c pisðnac 3 m 3 me tetragwnik bˆsh ètsi ste h epifˆneia twn eswterik n toðqwn kai tou pujmèna na eðnai elˆqisth. (.3mon.) 53

Jèma 3. (a) Na brejeð to embadìn tou trig nou me korufèc A(, 5), B(, 3), Γ(4, ). (.4mon.) (b) To embadìn tou trig nou pou èqei korufèc ta shmeða A(a,, ), B(4,, ), Γ(,, 3) eðnai t.m.. Na brejeð to a. (.9mon.) (g) Na brejeð h eðswsh enìc kôklou pou pernˆ apì ta shmeða A(3, ), B(, ), Γ(, ) pou den keðntai sthn Ðdia eujeða. (.mon.) Jèma 4. (a) Na apodeiqjeð ìti to genikeumèno olokl rwma + sin d k+ sugklðnei gia k >. (.7mon.) ShmeÐwsh. To + d, p R sugklðnei gia p > kai apoklðnei gia p. p (b) Na brejeð to kèntro bˆrouc tou epipèdou qwrðou pou orðzetai apì tic kampôlec y kai y. (.3mon.) Diˆrkeia eètashc:.5 rec KALH EPITUQIA 54

A.T.E.I. PEIRAIA SQOLH TEQNOLOGIKWN EFARMOGWN GENIKO TMHMA MAJHMATIKWN Kajhghtèc: Elènh JeofÐlh Alèandroc Pappˆc ONOMATEPWNUMO:... EXAMHNO:... A.M.:... JEMATA MAJHMATIKA I TMHMA POLITIKWN DOMIKWN ERGWN FEBROUARIOU Jèma. (a) Oi grafikèc parastˆseic C f kai C g twn sunart sewn pou orðzontai me f() sin kai g() cos tèmnontai ˆpeirec forèc kai orðzoun qwrða Ðswn embad n. Na upologðsete to embadìn E enìc apì ta qwrða autˆ. (.9mon.) (b) Na eetˆsete wc proc th sôgklish to olokl rwma + y sin d. (.6mon.) y sin y y ShmeÐwsh. To + d, p R sugklðnei gia p > kai apoklðnei gia p. p Jèma. (a) Na brejeð to embadìn tou trig nou me korufèc A(, 3), B(, 5), Γ(6, 4). (.5mon.) (b) Na brejeð to embadìn tou trig nou me korufèc A(4,, ), B(,, ), Γ(3, 5, 6). (.mon.) (g) Na brejeð h eðswsh enìc kôklou pou pernˆ apì ta shmeða A(3, ), B(, ), Γ(, ) pou den keðntai sthn Ðdia eujeða. (.3mon.) 55

Jèma 3. (a) Na upologisteð to aìristo olokl rwma I συν d, nπ + π, n Z. (b) Na brejeð to m koc tìou thc kampôlhc [ π y ln (συν) me 6, π ] 4 (.4mon.). (.mon.) ( Jèma 4. (a) Na apodeðete ìti lim + + ) e. (.9mon.) (b) 'Estw h akoloujða (a n ), me a n n!/n n. i. Na apodeiqjeð ìti lim n a n+ /a n lim n n a n /e. (.8mon.) ii. Na apodeiqjeð ìti lim n n n!. (.5mon.) (g) Na eetasteð wc proc th sôgklish h seirˆ n ( ) n n n!. (.3mon.) Jèma 5. (a) Na brejeð o ìgkoc thc sfaðrac. (.mon.) y y R O R R y R (b) Na brejeð to kèntro bˆrouc tou epipèdou qwrðou pou orðzetai apì tic kampôlec y kai y 4. (.4mon.) (g) Na apodeiqjeð ìti τoξεϕ + τoξεϕ ( ) π, >. (.9mon.) Diˆrkeia eètashc:.5 rec KALH EPITUQIA 56

A.T.E.I. PEIRAIA SQOLH TEQNOLOGIKWN EFARMOGWN GENIKO TMHMA MAJHMATIKWN Kajhghtèc: Elènh JeofÐlh Alèandroc Pappˆc ONOMATEPWNUMO:... EXAMHNO:... A.M.:... JEMATA MAJHMATIKA I TMHMA POLITIKWN DOMIKWN ERGWN 3 IOUNIOU Jèma. (a) Na apodeðete ìti lim + ( + ) e. (.7mon.) (b) 'Estw h akoloujða (a n ), me a n n!/n n. i. Na apodeiqjeð ìti lim n a n+ /a n lim n n a n /e. (.7mon.) ii. Na apodeiqjeð ìti lim n n n!. (.3mon.) (g) Na eetasteð wc proc th sôgklish h seirˆ n ( ) n n n!. (.8mon.) Jèma. (a) Na brejeð to embadìn tou trig nou me korufèc A(, 5), B(, 3), Γ(4, ). (.4mon.) (b) To embadìn tou trig nou pou èqei korufèc ta shmeða A(a,, ), B(4,, ), Γ(,, 3) eðnai t.m.. Na brejeð to a. (.9mon.) (g) Na brejeð h eðswsh enìc kôklou pou pernˆ apì ta shmeða A(3, ), B(, ), Γ(, ) pou den keðntai sthn Ðdia eujeða. (.mon.) 57

Jèma 3. (a) Na brejeð to embadìn thc perioq c pou periorðzetai apì thn èlleiyh a + y b. (.3mon.) y b a O a b (b) ProsdiorÐste tic plèon oikonomikèc diastˆseic miac anoiqt c pisðnac 3 m 3 me tetragwnik bˆsh ètsi ste h epifˆneia twn eswterik n toðqwn kai tou pujmèna na eðnai elˆqisth. (.mon.) Jèma 4. (a) Na apodeiqjeð ìti to genikeumèno olokl rwma + sin d k+ sugklðnei gia k >. (.4mon.) ShmeÐwsh. To + d, p R sugklðnei gia p > kai apoklðnei gia p. p (b) Na brejeð to kèntro bˆrouc tou epipèdou qwrðou pou orðzetai apì tic kampôlec y kai y. (.mon.) Diˆrkeia eètashc:.5 rec KALH EPITUQIA 58

A.T.E.I. PEIRAIA SQOLH TEQNOLOGIKWN EFARMOGWN GENIKO TMHMA MAJHMATIKWN Kajhghtèc: Elènh JeofÐlh Alèandroc Pappˆc ONOMATEPWNUMO:... EXAMHNO:... A.M.:... JEMATA MAJHMATIKA I TMHMA POLITIKWN DOMIKWN ERGWN SEPTEMBRIOU Jèma. (a) Na apodeðete ìti ( lim + 4 e + ) 4. (.9mon.) (b) Na eetˆsete wc proc th sôgklish to olokl rwma + sin d. (.6mon.) y y sin y y ShmeÐwsh. To + (g) Na upologisjeð to olokl rwma: I (d) Na apodeiqjeð ìti d, p R sugklðnei gia p > kai apoklðnei gia p. p τoξεϕ + τoξεϕ d 5. (.mon.) 59 ( ) π, >. (.9mon.)

Jèma. (a) Na brejeð to embadìn tou trig nou me korufèc A(, 3), B(, 4), Γ(, 5). (.6mon.) (b) Na brejeð to embadìn tou trig nou me korufèc A(, 3, ), B(3,, ), Γ(3,, ). (.5mon.) (g) Na brejeð h eðswsh enìc kôklou pou pernˆ apì ta shmeða A(3, ), B(, ), Γ(, ) pou den keðntai sthn Ðdia eujeða. (.4mon.) Jèma 3. (a) Na upologisteð to aìristo olokl rwma I συν d, nπ + π, n Z. Apˆnthsh. συν (.mon.) d ln συν + εϕ + C ln + ηµ ηµ + C ln + εϕ (/) εϕ (/) + C, nπ + π/, n Z. (b) Na brejeð to m koc tìou thc kampôlhc [ π y ln (συν) me 6, π ] 4. (.8mon.) Jèma 4. (a) Na apodeiqjeð ìti to genikeumèno olokl rwma ln d. (b) Na eetasteð wc proc th sôgklish h seirˆ ( ) n n ln n. n ln d sugklðnei kai ìti (.mon.) (.4mon.) Jèma 5. (a) Na apodeðete ìti o ìgkoc k nou me Ôyoc h kai aktðna bˆshc r eðnai V 3 πr h. (.mon.) (b) Na brejeð to kèntro bˆrouc tou epipèdou qwrðou pou orðzetai apì tic kampôlec y kai y 3. (.3mon.) Diˆrkeia eètashc:.5 rec KALH EPITUQIA 6

A.T.E.I. PEIRAIA SQOLH TEQNOLOGIKWN EFARMOGWN GENIKO TMHMA MAJHMATIKWN Kajhghtèc: Elènh JeofÐlh Alèandroc Pappˆc ONOMATEPWNUMO:... EXAMHNO:... A.M.:... JEMATA MAJHMATIKA I TMHMA POLITIKWN DOMIKWN ERGWN 8 IANOUARIOU Jèma. (a) Na upologisteð to lim sin t dt 3. (.8mon.) (b) Na upologðsete to embadì tou qwrðou pou perikleðetai apì ton kôklo + y 8 kai thn parabol y. (.mon.) Jèma. (a) 'Estw n N {}. Na brejeð o megalôteroc apì touc arijmoôc,, (b) Na eetˆsete wc proc th sôgklish th seirˆ n 3 3,..., n n. ) cos(nπ) (n n (.mon.). (.9mon.) Jèma 3. (a) Na apodeiqjeð ìti h sunˆrthsh f : (, + ) R pou orðzetai me tôpo f() ηµ τoξηµ t dt + συν τoξσυν t dt lambˆnei thn tim f() π 4 gia kˆje (, π ). (.4mon.) (b) Na upologisteð to olokl rwma I (g) Na lujeð h eðswsh ηµ + συν + 3 d. t t dt π, >. (.mon.) (.4mon.) 6

Jèma 4. (a) To embadìn tou trig nou pou èqei korufèc ta shmeða A(a,, ), B(4,, ), Γ(,, 3) eðnai t.m.. Na brejeð to a. (mon.) (g) Na brejeð h eðswsh enìc kôklou pou pernˆ apì ta shmeða A(, ), B(, 3), Γ(, 4) pou den keðntai sthn Ðdia eujeða. (mon.) Jèma 5. (a) Na eetˆsete wc proc th sôgklish to olokl rwma + ShmeÐwsh. To + cos d. (.mon.) d, p R sugklðnei gia p > kai apoklðnei gia p. p y y cos y y (b) Na brejeð to kèntro bˆrouc tou epipèdou qwrðou pou orðzetai apì tic kampôlec y kai 4 y. (.8mon.) Diˆrkeia eètashc:.5 rec KALH EPITUQIA 6

A.T.E.I. PEIRAIA SQOLH TEQNOLOGIKWN EFARMOGWN GENIKO TMHMA MAJHMATIKWN Kajhghtèc: Elènh JeofÐlh Alèandroc Pappˆc ONOMATEPWNUMO:... EXAMHNO:... A.M.:... JEMATA MAJHMATIKA I TMHMA POLITIKWN DOMIKWN ERGWN 8 FEBROUARIOU ( Jèma. (a) Na apodeðete ìti lim + + ) e. (.9mon.) (b) 'Estw h akoloujða (a n ), me a n n!/n n. i. Na apodeiqjeð ìti lim n a n+ /a n lim n n a n /e. (.8mon.) ii. Na apodeiqjeð ìti lim n n n!. (.4mon.) (g) Na eetasteð wc proc th sôgklish h seirˆ n ( ) n n n!. (.9mon.) Jèma. (a) Na brejeð to embadìn thc perioq c pou periorðzetai apì thn èlleiyh a + y b. (.7mon.) y b a O a b (b) ProsdiorÐste tic plèon oikonomikèc diastˆseic miac anoiqt c pisðnac 3 m 3 me tetragwnik bˆsh ètsi ste h epifˆneia twn eswterik n toðqwn kai tou pujmèna na eðnai elˆqisth. (.3mon.) 63

Jèma 3. (a) Na brejeð to embadìn tou trig nou me korufèc A(, 4), B(5, ), Γ(3, 6). (.4mon.) (b) Na brejeð to embadìn tou trig nou me korufèc A(, 3, ), B(3,, ), Γ(3,, ). (.mon.) (g) Na brejeð h eðswsh enìc kôklou pou pernˆ apì ta shmeða A(3, ), B(, ), Γ(, ) pou den keðntai sthn Ðdia eujeða. (.mon.) (d) DÐdontai ta monadiaða kai sunepðpeda dianôsmata α, β, γ, ste: ) ) ( ( β ) α, β ϑ,, γ ϑ, ( γ, α ϑ 3. DeÐte ìti: συνϑ συνϑ 3 συνϑ συνϑ συνϑ 3 συνϑ. (.mon.) ShmeÐwsh. α, β, γ sunepðpeda k α + λ β + µ γ, me (k, λ, µ) (,, ). Jèma 4. (a) Na upologðsete to embadìn E tou qwrðou pou perikleðetai apì th grafik parˆstash thc kai twn asumpt twn thc. y (.8mon.) y y y ShmeÐwsh. To + d, p R sugklðnei gia p > kai apoklðnei gia p. p (b) Na upologðsete ton ìgko tou stereoô ek peristrof c gôrw apì ton -ˆona tou qwrðou pou perikleðetai apì thn kampôlh y ln kai tic eujeðec, y. (.4mon.) (b) Na brejeð to m koc tìou thc kampôlhc [ π y ln (ηµ) me 4, π ] 3. (.8mon.) Diˆrkeia eètashc:.5 rec KALH EPITUQIA 64

A.T.E.I. PEIRAIA SQOLH TEQNOLOGIKWN EFARMOGWN GENIKO TMHMA MAJHMATIKWN Kajhghtèc: Elènh JeofÐlh Alèandroc Pappˆc ONOMATEPWNUMO:... EXAMHNO:... A.M.:... JEMATA MAJHMATIKA I TMHMA POLITIKWN DOMIKWN ERGWN OKTWBRIOU Jèma. (a) Na upologisteð to lim sin t dt 3. (.mon.) (b) Oi grafikèc parastˆseic C f kai C g twn sunart sewn pou orðzontai me f() sin kai g() cos tèmnontai ˆpeirec forèc kai orðzoun qwrða Ðswn embad n. Na upologðsete to embadìn E enìc apì ta qwrða autˆ. (.9mon.) y.5 f() sin.5.5.5 g() cos π 4 5 π 4 3 3 4 5 6 (g) Na upologisjeð to olokl rwma: I d 6. (.3mon.) (d) ProsdiorÐste tic plèon oikonomikèc diastˆseic miac anoiqt c pisðnac 3 m 3 me tetragwnik bˆsh ètsi ste h epifˆneia twn eswterik n toðqwn kai tou pujmèna na eðnai elˆqisth. (.mon.) 65

Jèma. (a) Na brejeð to embadìn tou trig nou me korufèc A(, 4), B(, 3), Γ(4, 6). (.7mon.) (b) Na brejeð to embadìn tou trig nou me korufèc A(,, 4), B(, 3, ), Γ(5,, ). (.4mon.) (g) Na brejeð h eðswsh enìc kôklou pou pernˆ apì ta shmeða A(, ), B(, 3), Γ(, 4) pou den keðntai sthn Ðdia eujeða. (.4mon.) Jèma 3. (a) Na upologisteð to aìristo olokl rwma I d, nπ, n Z. ηµ (.mon.) 5 4 y csc sin, nπ, n Z 3 3π π π π π 3π 3 4 5 5 5 Apˆnthsh. ηµ d ln ηµ + σϕ + C ln συν εϕ συν + + C ln + C, nπ + π/, n Z. (b) Na brejeð to m koc tìou thc kampôlhc y ln (ηµ) me [ π 3, π ] 3. (.3mon.) ( Jèma 4. (a) Na apodeðete ìti lim + + ) e. (.9mon.) (b) 'Estw h akoloujða (a n ), me a n n!/n n. i. Na apodeiqjeð ìti lim n a n+ /a n lim n n a n /e. (.8mon.) ii. Na apodeiqjeð ìti lim n n n!. (.5mon.) (g) Na eetasteð wc proc th sôgklish h seirˆ Diˆrkeia eètashc:.5 rec n ( ) n n n!. (.3mon.) KALH EPITUQIA 66

A.T.E.I. PEIRAIA SQOLH TEQNOLOGIKWN EFARMOGWN GENIKO TMHMA MAJHMATIKWN Kajhghtèc: Elènh JeofÐlh Alèandroc Pappˆc ONOMATEPWNUMO:... EXAMHNO:... A.M.:... JEMATA MAJHMATIKA I TMHMA POLITIKWN DOMIKWN ERGWN 7 FEBROUARIOU Jèma. (a) Na Na apodeiqjeð ìti τoξεϕ + τoξεϕ ( ) π, >. (.7mon.) (b) Na brejeð to m koc tìou thc kampôlhc y τoξηµe, an ln. (.8mon.) y.5 y arcsine, [ ln, ].5 f() arcsine, R.5 5 4 3 Jèma. (a) 'Estw n N {}. Na brejeð o megalôteroc apì touc arijmoôc,, 3 3,..., n n. (.4mon.) (b) Na eetˆsete wc proc th sôgklish th seirˆ n cos(nπ) sin n. (.mon.) 67

Jèma 3. (a) Na brejeð to embadì thc perioq c pou sqhmatðzetai apì tic kampôlec + y 6, + y 8. (.4mon.) 5 y 4 y + 6 y + 8 3 3 4 5 5 5 (b) Na upologisteð to olokl rwma I 3+5ηµ d. (.mon.) Jèma 4. Jèma 5. Jèma 6. (a) Na brejeð to kèntro bˆrouc tou epipèdou qwrðou pou orðzetai apì tic kampôlec y kai y. (.3mon.) (b) Na upologisteð to olokl rwma I τoξεϕ d. (.mon.) (a) To embadìn tou trig nou pou èqei korufèc ta shmeða A(a,, ), B(4,, ), Γ(,, 3) eðnai t.m.. Na brejeð to a. (.4mon.) (b) Na apodeiqjeð ìti 3 (a) Na apodeiqjeð ìti to genikeumèno olokl rwma e sin + d π e. (.mon.) sugklðnei gia k >. ShmeÐwsh. To + + sin d k+ d, p R sugklðnei gia p > kai apoklðnei gia p. p (.6mon.) (b) Na brejeð h eðswsh enìc kôklou pou pernˆ apì ta shmeða A(, ), B(, 3), Γ(, 4) pou den keðntai sthn Ðdia eujeða. (.9mon.) Na epilèete tèssera (4) apì ta èi (6) jèmata Diˆrkeia eètashc:.5 rec KALH EPITUQIA 68

A.T.E.I. PEIRAIA SQOLH TEQNOLOGIKWN EFARMOGWN GENIKO TMHMA MAJHMATIKWN Kajhghtèc: Elènh JeofÐlh Alèandroc Pappˆc ONOMATEPWNUMO:... EXAMHNO:... A.M.:... JEMATA MAJHMATIKA I TMHMA POLITIKWN DOMIKWN ERGWN 5 SEPTEMBRIOU Jèma. (a) Na upologisteð to lim sin t dt 3. (.mon.) (b) Oi grafikèc parastˆseic C f kai C g twn sunart sewn pou orðzontai me f() sin kai g() cos tèmnontai ˆpeirec forèc kai orðzoun qwrða Ðswn embad n. Na upologðsete to embadìn E enìc apì ta qwrða autˆ. (.9mon.) y.5 f() sin.5.5.5 g() cos π 4 5 π 4 3 3 4 5 6 (g) Na upologisjeð to olokl rwma: I d 6. (.3mon.) (d) ProsdiorÐste tic plèon oikonomikèc diastˆseic miac anoiqt c pisðnac 3 m 3 me tetragwnik bˆsh ètsi ste h epifˆneia twn eswterik n toðqwn kai tou pujmèna na eðnai elˆqisth. (.mon.) 69

Jèma. (a) Na brejeð to embadìn tou trig nou me korufèc A(, 4), B(, 3), Γ(4, 6). (.7mon.) (b) Na brejeð to embadìn tou trig nou me korufèc A(,, 4), B(, 3, ), Γ(5,, ). (.4mon.) (g) Na brejeð h eðswsh enìc kôklou pou pernˆ apì ta shmeða A(, ), B(, 3), Γ(, 4) pou den keðntai sthn Ðdia eujeða. (.4mon.) ( Jèma 3. (a) Na apodeðete ìti lim + + ) e. (.9mon.) (b) 'Estw h akoloujða (a n ), me a n n!/n n. i. Na apodeiqjeð ìti lim n a n+ /a n lim n n a n /e. (.8mon.) ii. Na apodeiqjeð ìti lim n n n!. (.5mon.) (g) Na eetasteð wc proc th sôgklish h seirˆ n ( ) n n n!. (.3mon.) Jèma 4. (a) Na upologðsete ton ìgko tou stereoô ek peristrof c gôrw apì ton -ˆona tou qwrðou pou perikleðetai apì thn kampôlh y ln kai tic eujeðec, y. (.3mon.) (b) Na brejeð to kèntro bˆrouc thc epifˆneiac pou èqei sq ma hmikôkliou aktðnac R. (.mon.) y y R O R R y R (g) Na upologisteð to olokl rwma I 3+5ηµ d. (.3mon.) (d) Na eetˆsete wc proc th sôgklish to olokl rwma + ShmeÐwsh. To + Diˆrkeia eètashc:.5 rec cos d. (.8mon.) d, p R sugklðnei gia p > kai apoklðnei gia p. p KALH EPITUQIA 7

T.E.I. PEIRAIA SQOLH TEQNOLOGIKWN EFARMOGWN GENIKO TMHMA MAJHMATIKWN Kajhghtèc: Elènh JeofÐlh Alèandroc Pappˆc ONOMATEPWNUMO:... EXAMHNO:... A.M.:... JEMATA MAJHMATIKA I TMHMA POLITIKWN DOMIKWN ERGWN 3 FEBROUARIOU 3 Jèma. (a) Na upologðsete to orismèno olokl rwma: e e ( ln + ) d. (.9mon.) (b) Na brejeð to m koc tìou thc kampôlhc y ln (συν) me [ π 4, π 4 ]. (.6mon.) π y π y ln cos, [ π 4, π 4 ] 3 f() ln cos, n π n π, n Z 4 5.5.5.5.5 Jèma. (a) Na brejeð to embadìn tou trig nou me korufèc A(5,, ), B(4, 3, ), Γ(3, 7, 6). (.mon.) (b) Na brejeð to embadìn thc perioq c pou periorðzetai apì thn èlleiyh a + y b. (.4mon.) y b a O a b 7

Jèma 3. (a) 'Estw n N {}. Na brejeð o megalôteroc apì touc arijmoôc, 3, 3,..., n n. (.4mon.) (b) Na eetˆsete wc proc th sôgklish th seirˆ n cos(nπ) sin n. (.mon.) Jèma 4. (a) Na lujeð h eðswsh t t dt π, >. (.4mon.).5 y f(t) t, t R \ [, ] t.5 f(t)dt.5.5 3 3 4 5 6 (b) Na brejeð h eðswsh enìc kôklou pou pernˆ apì ta shmeða A(, ), B(, 3), Γ(, 4) pou den keðntai sthn Ðdia eujeða. (.mon.) Jèma 5. (a) Na eetˆsete wc proc th sôgklish to olokl rwma + y cos d. (.4mon.) y cos y y ShmeÐwsh. To + d, p R sugklðnei gia p > kai apoklðnei gia p. p (b) Na brejeð to kèntro bˆrouc tou epipèdou qwrðou pou orðzetai apì tic kampôlec y kai y. (.mon.) Na epilèete tèssera (4) apì ta pènte (5) jèmata Diˆrkeia eètashc:.5 rec KALH EPITUQIA 7

. Lumènec Ask seic 73

Jèma. (a) Na brejeð h eðswsh enìc kôklou pou pernˆ apì ta shmeða A(, ), B(5, 4), Γ(, 5) pou den keðntai sthn Ðdia eujeða. (mon.) (b) Na grammoskiasjeð h perioq R, pou eurðsketai pˆnw apì ton -ˆona kai kˆtw apì thn eujeða y kai thn kampôlh y. (.5mon.) LUSH (a) EÐswsh kôklou pou pernˆ apì trða shmeða EÐnai gnwstì, ìti trða mh suneujeiakˆ shmeða, orðzoun th jèsh enìc monadikoô kôklou. 'Estw trða shmeða M (, y ), M (, y ), M 3 ( 3, y 3 ) mh suneujeiakˆ, dhlad kai y y 3 y 3 () + y + A + By + Γ () h eðswsh tou kôklou pou pernˆei apì autˆ. Tìte ta M (, y ), M (, y ), M 3 ( 3, y 3 ) ja thn epalhjeôoun, opìte ja èqoume: + y + A + By + Γ + y + A + By + Γ (3) 3 + y3 + A 3 + By 3 + Γ To sôsthma (3) me agn stouc ta A, B, Γ, èqei monadik lôsh lìgw thc (). Tic timèc twn agn stwn an tic jèsoume sthn () brðskoume thn eðswsh tou kôklou. 74

H eðswsh tou kôklou dðnetai apì ton tôpo: + y + A + By + Γ kai epeid jèloume na pernˆ apì trða shmeða oi suntetagmènec touc ja thn epalhjeôoun, opìte èqoume to sôsthma: A B + Γ 5 H lôsh tou sust matoc mac dðnei opìte h eðswsh tou kôklou grˆfetai: 5A + 4B + Γ 4 A + 5B + Γ 5 A 8, B 6, Γ 5 + y 8 + 6y + 5 pou èqei kèntro ( K A ) (, B K 8 ), 6 K(9, 3) kai aktðna A + B R 4Γ ( 8) + 6 4 5 34 + 36 6 4 65 TROPOS 65 65. H eðswsh tou kôklou grˆfetai kai upì thn morf orðzousac 4 ης tˆhc: + y y + y y + y y 3 + y3 3 y 3. + y y + y y + y y 3 + y3 3 y 3 + y y 5 4 5 4 5 5 75

( + y ) 5 4 5 5 4 5 4 5 5 5 4 4 5 5 + y 5 4 5 5 ( + y ) (4 + 5 4 5 + ) ( + 5 5 5 5 + 8) + + y(5 + 4 + 5 65 5 4) (5 8 + 5 + 5 5) ( + y ) (9 35) ( 5 443) + y(56 76) ( 95 + 845) 6 ( + y ) + 468 56y 65 + y 8 + 6y + 5 A 8, B 6, Γ 5. H eðswsh tou kôklou + y 8 + 6y + 5 èqei kèntro K ( A, B ) K ( 8, 6 ) K(9, 3) kai aktðna A + B R 4Γ ( 8) + 6 4 5 34 + 36 6 4 65 65 65. y A(, ) y O y B(, ) (b) 76

Jèma. (a) Na prosdioristoôn oi diastˆseic tou orjogwnðou me to mègisto embadìn pou eggrˆfetai se kôklo aktðnac R. (mon.) R O R Y X (b) Na brejeð to èrgo W pou parˆgetai apì mia dônamh F (,, 3) ìtan metafèrei to shmeðo efarmog c thc apì to A(,, ) sto B(4, 3, ). (.5mon.) LUSH (a) 'Estw ìti oi diastˆseic tou orjogwnðou eðnai kai y. Tìte to embadìn tou eðnai : Epeid EÐnai Opìte E y. + y 4R. y 4R. E 4R, < < R. Gia ton prosdiorismì thc mègisthc tim c eetˆzoume thn pr th parˆgwgo thc sunˆrthshc : E 4R 4R. 'Eqei rðzec R kai R. 77