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4 È Ö Ñ Ø Ó ÄÙ Ó Ù Ð ËÕÓÐ ËÑÙÖÒ Ì Ü Å Ñ Ø Â Ø Ì ÕÒÓÐÓ Ã Ø Ù ÙÒ Ë Ñ Û Â ÛÖ Ã Ø ÆºËº Å ÙÖÓ ÒÒ Ç Ñ ô ÙØ Ò ÕÓÐ ÕÖ º ÅÔÓÖÓ Ò Ò Ò Ô Ö Õ Ó Ò Ò Ò Ñ Ó Ò Ð Ö Ö¹ Ò Ñ Ò ÐÐ Ü ÑÓÖ ØÓÙº ØÓÒ Ô Ö ÓÖ Ñ ØÛÒ Ò Ô Ù ØÛÒ Ð ôò ÙÔ ÒØ ÙÒ Õ ÓÖ ô º Ò ÑÓÒØ Û ÕÓÙÒ Ó ÙÒØ Ø ØÓÙ Ò Ö Ñ Ù Ò ØÙÕ Ò ÔÖÓ Ð Ñ Ø ÔÓÙ Ò ÝÓÙÒ Ô Ø Ò ÕÖ ØÓÙº ÔÖ ÐÓÙ ¾¼½ ËØÓ Õ Ó Ø Ò Ñ ØÓ LATEXº

5 OdhgÐec Oi shmei seic autèc sunt qjhkan gia tic an gkec didaskalðac tou maj matoc Majhmatik Jetik c kai Teqnologik c KateÔjunshc} sto Prìtupo Peiramatikì Genikì LÔkeio. ApartÐzontai apì 4 mèrh: Kef laio 1 Perilamb nei upenjumðseic tôpwn apì thn Ôlh twn prohgoumènwn et n. K poioi lìgw twn suneq n ekpt sewn kai mei sewn thc Ôlhc den perilamb nontai plèon sthn didaktèa Ôlh. 'Eqoun sumperilhfjeð ed giatð eðnai qr simoi. Kef laio 2 Perièqei mia epilog erwt sewn anaforik me thn Ôlh thc jewrðac. Oi apant seic proèrqontai autoôsiec apì to sqolikì biblðo. Kef laio 3 Perièqei mia seir apì qr simec mikrèc jewrhtikèc epekt seic. Tuqìn qr sh touc stic exet seic prèpei na sunodeôetai apì apìdeixh. Kef laio 4 Perièqei mia seir erwt sewn tôpou Swstì-L joc. Oi majhtèc pou ja qrhsimopoi soun autèc tic shmei seic ja prèpei na gnwrðzoun ìti h melèth touc de mporeð na upokatast sei th melèth tou sqolikoô biblðou pou prèpei na gðnei polô prosektik. Kalì di basma kai kal epituqða. 31 MartÐou 2013, Majhmatikìc (MSc, PhD) i

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7 Perieqìmena OdhgÐec i 1 UpenjumÐseic Tautìthtec - Anisìthtec Dun meic -RÐzec-Log rijmoi Apìlutec Timèc OrÐzousec kai Grammik Sust mata Deuterob jmio Tri numo TrigwnometrÐa Merikèc exis seic Embad Suntetagmènec DianÔsmata EujeÐa-KÔkloc Kwnikèc Tomèc Grafikèc Parast seic Basik n Sunart sewn Erwt seic JewrÐac 19 3 Qr simec Prot seic 41 4 Erwt seic Swstì-L joc MigadikoÐ ArijmoÐ 'Oria-Sunèqeia Diaforikìc Logismìc Oloklhrwtikìc Logismìc iii

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9 Kef laio 1 UpenjumÐseic 'Oloi oi arijmoð an den anafèretai k ti llo jewroôntai pragmatikoð. O ν eðnai jetikìc akèraioc. 1.1 Tautìthtec - Anisìthtec 1. (α ± β) 2 = α 2 ± 2αβ + β 2 2. (α ± β) 3 = α 3 ± 3α 2 β + 3αβ 2 ± β 3 3. α 3 ± β 3 =(α ± β)(α 2 αβ + β 2 ) 4. (α + β + γ) 2 = α 2 + β 2 + γ 2 + 2αβ + 2βγ + 2γα 5. α ν β ν =(α β)(α ν 1 + α ν 2 β + +αβ ν 2 + β ν 1 ) 6. α 3 + β 3 + γ 3 3αβγ = =(α + β + γ)(α 2 + β 2 + γ 2 αβ βγ γα)= = 1 2 (α + β + γ) ((α β)2 +(β γ) 2 +(γ α) 2 ) 7. α 3 + β 3 + γ 3 = 3αβγ α = β = γ α + β + γ = 0 8. (x α) 2 +(y β) 2 +(z γ) 2 = 0 9. α 2 + β 2 ±2αβ 10. α 2 ± αβ + β 2 0 x = α y = β z = γ 1

10 Dun meic -RÐzec -Log rijmoi 11. (1 + α) ν > 1 + να, 1 < α 0, ν α 2ν+1 < β 2ν+1 α < β 1.2 Dun meic - RÐzec - Log rijmoi 1. x = ν α α 0 x 0 x ν = α 2. ν α ν = α, ν αν = α 3. An α, β 0 tìte ν αβ = ν α ν β 4. An α 0, β> 0 tìte ν α β = ν α ν β, 5. An α 0, β> 0 tìte (a ) (b ) (g ) ν α μ = ν α μ ν μ α = νμ α νλ α μλ = ν α μ 6. An α > 0 tìte α μ ν = ν α μ = ν α μ 7. An α > 0 tìte (a ) α x 1 α x 2 = α x 1 +x 2 (b ) αx 1 α x 2 = αx 1 x 2 (g ) (α x 1 ) x 2 = α x 1x 2 8. An α, β > 0 tìte (a ) (αβ) x = α x β x (b ) ( α β )x = αx β x 9. An α > 0, α 1 tìte α x 1 = α x 2 x1 = x An α > 1 tìte α x 1 < α x 2 x1 < x An α < 1 tìte α x 1 < α x 2 x1 > x log α x = y α > 0, α 1 x > 0 α y = x

11 Kef laio 1. UpenjumÐseic log x = log 10 x, ln x = log e x 14. log α x = ln x ln α 15. An α > 0, α 1 tìte x 1 = x 2 log α x 1 = log α x An α > 1 tìte x 1 < x 2 log α x 1 < log α x An 0 < α < 1 tìte x 1 < x 2 log α x 1 > log α x x 1 = x 2 ln x 1 = ln x x 1 < x 2 ln x 1 < ln x (a ) 0 < x < 1 ln x < 0 (b ) x > 1 ln x > 0 (g ) x = 1 ln x = An x 1,x 2,x> 0, α > 0,α 1 tìte (a ) log α (x 1 x 2 )=log α x 1 + log α x 2 (b ) log α ( x 1 x 2 ) = log α x 1 log α x 2 (g ) log α x k = k log α x 22. An x 1,x 2,x> 0 tìte (a ) ln (x 1 x 2 )=ln x 1 + ln x 2 (b ) ln x 1 x 2 = ln x 1 ln x 2 (g ) ln x k = k ln x 1.3 Apìlutec Timèc 1. α ={ α an α < 0 α an α 0 2. α =α α 0 3. α = α α 0 4. x = α x =±α 5. α α α

12 OrÐzousec kai Grammik Sust mata 6. αβ = α β 7. α α β = β 8. α ν = α ν 9. α 2ν = α 2ν 10. α β α ± β α + β 11. α + β = α + β αβ OrÐzousec kai Grammik Sust mata 1. α β γ δ = α γ =αδ βγ β δ 2. 'Estw to sôsthma { α 1x + β 1 y = γ 1 α 2 x + β 2 y = γ 2 }(Σ) ìpou k poioc apì touc α 1,β 1,α 2,β 2 eðnai di foroc tou 0. 'Estw D = α 1 β 1 α 2 β 2 Tìte: D x = γ 1 β 1 γ 2 β 2 D y = α 1 γ 1 α 2 γ 2 (a ) An D 0 to (Σ) èqei mða mìno lôsh (x, y) me x = D x D,y= D y D (b ) An D = 0 kai k poioc apì touc D x, D y eðnai di foroc tou mhdenìc to (Σ) eðnai adônato. (g ) An D = D x = D y = 0 tìte to (Σ) èqei peirec lôseic (x, y).

13 Kef laio 1. UpenjumÐseic Deuterob jmio Tri numo 'Estw h sun rthsh f (x)=αx 2 + βx + γ, α 0, Δ = β 2 4αγ. 1. Prìshmo-RÐzec (a ) An Δ > 0 tìte h f èqei dôo nisec rðzec ρ 1,2 = β± Δ 2α. 'Otan to x eðnai ektìc twn riz n h f(x) eðnai omìshmh tou α en ìtan eðnai metaxô twn riz n eðnai eterìshmh tou α. (b ) An Δ = 0 tìte h f èqei mða dipl rðza ρ = β 2α. 'Otan to x eðnai di foro thc dipl c rðzac h f(x) eðnai omìshmh tou α (g ) An Δ < 0 h f den èqei rðzec kai eðnai omìshmh tou α gia ìlec tic pragmatikèc timèc tou x 2. Mègista-El qista (a ) An α > 0 h f èqei el qisth tim thn Δ 4α gia x = β 2α. (b ) An α < 0 h f èqei mègisth tim thn Δ 4α gia x = β 2α. 3. Sqèseic tou Vieta (a ) An eðnai Δ 0 tìte to jroisma kai to ginìmeno twn riz n thc f eðnai S = ρ 1 + ρ 2 = β α, P = ρ 1ρ 2 = γ α (b ) An dôo arijmoð èqoun jroisma S kai ginìmeno P tìte eðnai rðzec thc exðswshc x 2 Sx + P = 0.

14 ½º º ÌÖ ÛÒÓÑ ØÖ ¼ ¼ ¼ «¼ «¾ ¾««¾ ¾«¾««¾ ¾«¾««¼ ¾««¾ ½º ÌÖ ÛÒÓÑ ØÖ ½º 0 π 6 1 ηµ 0 συν ½ εϕ ¼ σϕ π ½ π π ½ 1 2 ¼ 3 3 ½ 3 3 ¼ ¾º ηµx 2 +συνx 2 =1 º εφx σϕx=1 εφx= ηµx συνx σφx= συνx ηµx º συν 2 x= 1 1+εφ 2 x ηµx 2 = εφ2 x 1+εφ 2 x º συν( x)=συνx ηµ( x)= ηµx ÈÖ ØÙÔÓ È Ö Ñ Ø Ò Ä Ó Ù Ð ËÕÓÐ ËÑ ÖÒ ØØÔ»»ÐÝ ¹ Ú ¹Ò¹ ÑÝÖÒº Øغ º Ö ÆºËº Å ÙÖÓ ÒÒ ÛÛÛºÒ Ñ ÚÖÓ ÒÒ º Ö

15 Kef laio 1. UpenjumÐseic 7 εϕ ( x) = εϕx σϕ( x)= σϕx 6. συν (π x)= συνx ημ(π x)=ημx εϕ (π x)= εϕx σϕ(π x)= σϕx 7. συν (π + x)= συνx ημ(π + x)= ημx εϕ (π + x)=εϕx σϕ(π + x)=σϕx 8. συν ( π 2 x) = ημx ημ(π 2 x) = συνx εϕ ( π 2 x) = σϕx σϕ(π 2 x) = εϕx 9. συν (α ± β)=συνασυνβ ημαημβ ημ(α ± β)=ημασυνβ ± ημβσυνα εϕ (α ± β)= εϕα±εϕβ 1 εϕα εϕβ 10. ημ2α = 2ημασυνα συν2α = συν 2 α ημ 2 α = 2συν 2 α 1 = 1 2ημ 2 α 11. εϕ2α = 2εϕα 1 εϕ 2 α ημ2α = 2εϕα 1+εϕ 2 α συν2α = 1 εϕ2 α 1+εϕ 2 α 12. συν 2 α = 1+συν2α 2 ημ 2 α = 1 συν2α Se k je trðgwno ABG isqôoun α 2 = β 2 + γ 2 2βγσυνA (Nìmoc twn sunhmitìnwn) α ημa = 2R (Nìmoc twn hmitìnwn, R h aktðna tou perigegrammènou kôklou) 1.7 Merikèc exis seic 1. x ν = α 2. x =α ν rtioc ν perittìc α 0 x =± ν α x = ν α α < 0 adônath x = ν α

16 Embad 3. ημx = α α 0 α < 0 x =±α adônath α 1, α = ημθ x = θ + 2kπ,k Z, x = π θ + 2kπ,k Z α >1 adônath 4. συνx = α α 1, α = συνθ x = θ + 2kπ,k Z, x = θ + 2kπ,k Z α >1 adônath 5. εϕx = α, α = εϕθ x = θ + kπ,k Z 6. α x = β, α > 0 7. ln x = α α > 0 α 0 x = ln β ln α adônath x = e α 1.8 Embad 1. To embadìn E trig nou ABΓ eðnai E = 1 2 αυ α = 1 2 βγημa = τ (τ α)(τ β)(τ γ)= 1 2 D ìpou D = det ( AB, AΓ) kai τ = α+β+γ 2.

17 Kef laio 1. UpenjumÐseic 9 An to trðgwno eðnai isìpleuro pleur c α tìte E = α To embadìn parallhlogr mmou eðnai b sh Ôyoc tou tetrag nou pleur c α eðnai α 2 δ kai tou rìmbou me diagwnðouc δ 1,δ 2 eðnai 1 δ 2 2. To embadìn trapezðou me b seic B,β kai Ôyoc υ eðnai B+β 2 υ. 3. To embadìn kôklou aktðnac ρ eðnai πρ 2 (to m koc tou eðnai 2πρ). Gia to embadìn tomèa kai to m koc tìxou gwnðac ϕ èqoume: gwnða ϕ se aktðnia gwnða ϕ se moðrec m koc tìxou ρϕ πρϕ 180 embadìn tomèa ρ 2 ϕ 2 πρ 2 ϕ Suntetagmènec 'Estw ta shmeða A(x 1,y 1 ), B(x 2,y 2 ), Γ (x 3,y 3 ). 1. H apìstash twn A, B eðnai d = (x 1 x 2 ) 2 +(y 1 y 2 ) 2 2. To mèso tou tm matoc AB eðnai to M ( x 1+x 2 2, y 1+y 2 2 ) 3. O suntelest c dieujônsewc tou AB kaj c kai thc eujeðac AB (effloson x 1 x 2 )eðnai λ = y 2 y 1 x 2 x 1 4. 'Estw D = x 2 x 1 y 2 y 1 x 3 x 1 y 3 y 1 (a ) Ta A, B, Γ eðnai suneujeiak an kai mìno an D = 0. (b ) An D 0 tìte to embadìn tou trig nou ABΓ eðnai 1 2 D DianÔsmata An α =(x 1,y 1 ), β =(x 2,y 2 ) tìte: 1. To jroisma-diafor touc eðnai α ± β =(x 1 ± x 2,y 1 ± y 2 )

18 EujeÐa-KÔkloc 2. O grammikìc sunduasmìc touc κ α +λ β eðnai κ α +λ β =(κx 1 + λx 2,κy 1 + λy 2 ) 3. To eswterikì ginìmeno touc eðnai α β = x 1 x 2 + y 1 y 2 = α β συν ( α, β) 4. To mètro tou α eðnai α = α α = x y IsqÔei α β α ± β α + β 6. α + β = α + β α β = α β α β 7. α // β x 1 y 1 =0 (ef' ìson orðzontai oi suntelestèc dieujônsewc) λ α x 2 y 2 = λ β 8. α β α β = 0 x 1 x 2 +y 1 y 2 = 0 (effloson orðzontai oi suntelestèc dieujônsewc) λ α λ β = EujeÐa-KÔkloc 1. H genik exðswsh eujeðac eðnai h Ax + By + Γ = 0 me A + B 0. An B 0 h eujeða èqei suntelest dieujônsewc λ = A B = εϕω ìpou ω eðnai h gwnða pou sqhmatðzei o xonac x x me thn eujeða. x x x x

19 Kef laio 1. UpenjumÐseic H apìstash tou shmeðou M (x 0,y 0 ) apì thn eujeða Ax + By + Γ = 0 eðnai d = Ax 0+By 0 +Γ. A 2 +B 2 3. Mia eujeða me suntelest dieujônsewc α èqei exðswsh thc morf c y = αx + β. Oi y = α 1 x + β 1, y = α 2 x + β 2 tèmnontai an kai mìno an α 1 α 2 kai eðnai k jetec an α 1 α 2 = 1. An α 1 = α 2 = λ oi eujeðec èqoun thn Ðdia dieôjunsh kai apìstash β 1 β 2 kai an epiplèon β 1+λ 2 1 = β 2 tìte sumpðptoun. 4. O kôkloc me kèntro to K (x 0,y 0 ) kai aktðna (x x 0 ) 2 +(y y 0 ) 2 = ρ 2. An to K sumpðptei me thn arq twn axìnwn tìte h kôkloc gr fetai x 2 + y 2 = ρ 2 kai h efaptomènh tou se tuqìn shmeðo tou P (x 1,y 1 ) eðnai x 1 x + y 1 y = ρ HexÐswsh x 2 + y 2 + Ax + By + Γ = 0 eðnai exðswsh kôklou an kai mìno an A 2 + B 2 4Γ > 0. To kèntro tou eðnai to K ( A 2, B ) 2 kai h aktðna tou eðnai ρ = A 2 +B 2 4Γ 'Estw ènac kôkloc me kèntro K kai aktðna ρ kai d h apìstash tou K apì mða eujeða ε. 7. JewroÔme dôo kôklouc me kèntra K 1, K 2 kai aktðnec ρ 1 > ρ 2. 'Estw d h apìstash twn kèntrwn touc (di kentroc).

20 EujeÐa-KÔkloc 8. To sônolo twn shmeðwn M pou o lìgoc tw apost sewn apì dôo stajer shmeða A, B eðnai stajerìc kai Ðsoc me λ 1 eðnai kôkloc (KÔkloc tou ApollwnÐou) me di metro pou èqei kra ta shmeða ta opoða diairoôn to tm ma AB eswterik kai exwterik se lìgo λ.

21 Kef laio 1. UpenjumÐseic 13

22 Kwnikèc Tomèc 1.12 Kwnikèc Tomèc

23 Kef laio 1. UpenjumÐseic Grafikèc Parast seic Basik n Sunart sewn 1. y = αx + β 2. y = αx 2 3. y = α x 4. y = αx 3

24 Grafikèc Parast seic Basik n S unart sewn 5. y = α x, y = log α x 1 (a ) α > e 1 e (b ) α = e 1 e (g ) 1 < α < e 1 e 1 Για λεπτομέρειες: Μπάμπης Τουμάσης, Πόσο καλά έχουμε κατανοήσει την εκθετική και λογαριθμική συνάρτηση;, ΕΥΚΛΕΙΔΗΣ Β, 1994 τ.3, 52-55

25 Kef laio 1. UpenjumÐseic 17 (d ) e e < α < 1 (e ) α = e e, 6. y = ημx, y = συνx

26 Grafikèc Parast seic Basik n S unart sewn 7. y = εϕx

27 Kef laio 2 Erwt seic JewrÐac Erwthsh 1. Pìte dôo migadikoð arijmoð α + βi kai γ + δi eðnai Ðsoi? Apanthsh IsqÔei α + βi = γ + δi α = γ kai β = δ Erwthsh 2. Na apodeðxete ìti h dianusmatik aktðna tou ajroðsmatoc twn migadik n α + βi kai γ + δi eðnai to jroisma twn dianusmatik n aktðnwn touc. Apanthsh An M 1 (α, β) kai M 2 (γ, δ) eðnai oi eikìnec twn α +βi kai γ +δi antistoðqwc sto migadikì epðpedo, tìte to jroisma (α + βi)+(γ + δi) = (α + γ) +(β + δ)i parist netai me to shmeðo M(α + γ,β + δ). Epomènwc, OM = OM 1 + OM 2. Erwthsh 3. Na apodeðxete ìti h dianusmatik aktðna diafor c twn migadik n α + βi kai γ + δi eðnai h diafor twn dianusmatik n aktðnwn touc. 19

28 20 Apanthsh H diafor (α+βi) (γ +δi) =(α γ)+(β δ)i parist netai me to shmeðo N(α γ, β δ). Epomènwc, ON = OM 1 OM 2. Erwthsh 4. Na apodeðxete ìti (α + βi)(γ + δi)=(αγ βδ)+(αδ + βγ)i. Apanthsh 'Eqoume: (α + βi)(γ + δi) =α(γ + δi)+βi(γ + δi) =αγ + αδi + βγi+(βi)(δi) ==αγ+αδi+βγi+βδi 2 = αγ+αδi+βγi βδ =(αγ βδ)+(αδ+βγ)i Erwthsh 5. Ti onom zetai suzug c tou α + βi? Apanthsh O arijmìc α βi pou sumbolðzetai me α + βi. Erwthsh 6. Na ekfr sete to phlðko α+βi γ+δi κ + λi., ìpou γ + δi 0, sth morf Apanthsh Pollaplasi zoume touc ìrouc tou kl smatoc me to suzug tou paronomast kai èqoume: α + βi γ + δi = (α + βi)(γ δi) (γ + δi)(γ δi) = (αγ + βδ)+(βγ αδ)i γ 2 + δ 2 = αγ + βδ γ 2 + δ 2 βγ αδ + γ 2 + δ i 2 Dhlad α + βi γ + δi = αγ + βδ γ 2 + δ 2 βγ αδ + γ 2 + δ i 2

29 Kef laio 2. Erwt seic JewrÐac 21 Erwthsh 7. Poiec eðnai oi dunatèc dun meic tou i? Apanthsh 'Eqoume: i 0 = 1, i 1 = i, i 2 = 1, i 3 = i 2 i = i kai genik an ν = 4ρ + υ, ìpou ρ to phlðko kai υ to upìloipo thc EukleÐdeiac diaðreshc tou ν me to 4, tìte: i ν = i 4ρ+υ = i 4ρ i υ =(i 4 ) ρ i υ = 1 ρ i υ = i υ = 1, αν υ = 0 i, αν υ = 1 1, αν υ = 2 i, αν υ = 3 Erwthsh 8. Na apodeðxete ìti z 1 + z 2 = z 1 + z 2 Apanthsh z 1 + z 2 = (α + βi)+(γ + δi)=(α + γ)+(β + δ)i =(α +γ) (β + δ)i =(α βi)+(γ δi)= z 1 + z 2 Erwthsh 9. Na lôsete thnexðswsh αz 2 +βz+γ = 0,meα, β, γ R, α 0 kai Δ < 0 Apanthsh Ergazìmaste ìpwc sthn antðstoiqh perðptwsh sto R kai th metasqhmatðzoume, me th mèjodo sumpl rwshc tetrag nwn, sth morf : (z + β 2α ) 2 = Δ 4α 2 ìpou Δ = β 2 4αγ h diakrðnousa thc exðswshc. Epeid Δ i 2 ( Δ) 2 (2α) 2 =( ι Δ 2α )2, h exðswsh gr fetai: (z + β thc eðnai: z1, 2 = β±i Δ 2α 2α )2 =( i Δ 2α 4α 2 = ( 1)( Δ) 4α 2 = )2. 'Ara oi lôseic, oi opoðec eðnai suzugeðc migadikoð arijmoð. Erwthsh 10. Ti onom zetai mètro tou migadikoô z = x + yi? Apanthsh OrÐzoume wc mètro tou z thn apìstash tou M apì thn arq O, dhlad ton arijmì z = OM = x 2 + y 2 Erwthsh 11. Na apodeðxete ìti z 1 z 2 = z 1 z 2

30 22 Apanthsh 'Eqoume: z 1 z 2 = z 1 z 2 z 1 z 2 2 = z 1 2 z 2 2 (z 1 z 2 )(z 1 z 2 )=z 1 z 1 z 2 z 2 z 1 z 2 z 1 z 2 = z 1 z 1 z 2 z 2 Erwthsh 12. 'Estw Aènamhkenì uposônolo tou R. Ti onom zetai pragmatik sun rthsh f me pedðo orismoô to A kai ti tim thc f sto x A? Apanthsh Mia diadikasða (kanìna) me thn opoða k je stoiqeðo x A antistoiqðzetai se èna mìno pragmatikì arijmì y. To y onom zetai tim thc f sto x kai sumbolðzetai me f(x). Erwthsh 13. Ti onom zetai sônolo tim n mðac sun rthshc f A R? Apanthsh To sônolo f(a) ={y y = f(x) gia k poio x A} pou èqei gia stoiqeða tou tic timèc thc f se ìla ta x A. Erwthsh 14. Ti onom zetai grafik par stash mðac sun rthshc f A R? Apanthsh To sônolo C f twn shmeðwn M(x, y) gia ta opoða isqôei y = f(x), dhlad to sônolo twn shmeðwn M(x, f(x)), x A. Erwthsh 15. Pìte dôo sunart seic lègontai Ðsec? Apanthsh DÔo sunart seic f kai g lègontai Ðsec ìtan èqoun to ÐdiopedÐo orismoô A kai gia k je x A isqôei f(x)=g(x). Erwthsh 16. An f, g, eðnai dôo sunart seic na orðsete tic sunart seic f + g, f g, fg kai f g. Apanthsh OrÐzoume to jroisma f + g, diafor f g, ginìmeno fg kai phlðko f g twn f, g tic sunart seic metôpouc antistoðqwc touc (f + g)(x)=f(x)+g(x), (f g)(x)=f(x) g(x), (fg)(x)=f(x)g(x), ( f f(x) g )(x)= g(x) To pedðo orismoô twn f +g, f g kai fg eðnai h tom A B twn pedðwn orismoô A kai B twn sunart sewn f kai g antistoðqwc, en to pedðo orismoô thc f g eðnai to A B, exairoumènwn twn tim n tou x pou mhdenðzoun ton paronomast g(x), dhlad to sônolo

31 Kef laio 2. Erwt seic JewrÐac 23 {x x A kai x B, me g(x) 0} Erwthsh 17. An f, g, eðnai dôo sunart seic na orðsete th sônjesh g f thc f me thn g. Apanthsh EÐnai h sun rthsh me tôpo (gof)(x)=g(f(x)) kai pedðo orismoô to sônolo pou apoteleðtai apì ìla ta stoiqeða x tou pedðou orismoô thc f gia ta opoða to f(x) an kei sto pedðo orismoô thc g. Dhlad eðnai to sônolo A 1 ={x A f(x) B} Erwthsh 18. 'Estw f mða sun rthsh kai Δ èna di sthma tou pedðou orismoô thc. Pìte h f onom zetai gnhsðwc aôxousa, gnhsðwc fjðnousa, aôxousa, fjðnousa sto Δ? Apanthsh H f lègetai gnhsðwc aôxousa sto Δ ìtan gia opoiad pote x 1,x 2 Δ me x 1 < x 2 isqôei f(x 1 )<f(x 2 ) gnhsðwc fjðnousa sto Δ, ìtan gia opoiad pote x 1,x 2 Δ me x 1 < x 2 isqôei f(x 1 )>f(x 2 ) aôxousa sto Δ, ìtan gia opoiad pote x 1,x 2 Δ me x 1 < x 2 isqôei f(x 1 ) f(x 2 ) fjðnousa sto Δ, ìtan gia opoiad pote x 1,x 2 Δ me x 1 < x 2 f(x 1 ) f(x 2 ) isqôei Erwthsh 19. Pìte h sun rthsh f parousi zei mègisto el qisto sto shmeðo x 0 tou pedðou orismoô thc? Apanthsh Mia sun rthsh f me pedðo orismoô A ja lème ìti: Parousi zei sto x 0 A (olikì) mègisto, to f(x 0 ), ìtan f(x) f(x 0 ) gia k je x A Parousi zei sto x 0 A (olikì) el qisto, to f(x 0 ), ìtan f(x) f(x 0 ) gia k je x A Erwthsh 20. Ti eðnai ta olik akrìtata mðac sun rthshc f?

32 24 Apanthsh To (olikì) mègisto kai to (olikì) el qisto thc f (efìson up rqoun) lègontai (olik ) akrìtata thc f. Erwthsh 21. Pìte mða sun rthsh lègetai 1-1? Apanthsh Mia sun rthsh f A R lègetai sun rthsh 1 1, ìtan gia opoiad pote x 1,x 2 A isqôei h sunepagwg an x 1 x 2, tìte f(x 1 ) f(x 2 ) Erwthsh 22. Na apodeðxete ìti h sun rthsh f(x)=αx + β,meα 0 eðnai sun rthsh 1 1. Apanthsh An upojèsoume ìti f(x 1 )=f(x 2 ), tìte èqoume diadoqik : αx 1 + β = αx 2 + β αx 1 = αx 2 x 1 = x 2 Erwthsh 23. Pwc orðzetai h antðstrofh mðac 1-1 sun rthshc? Apanthsh 'Estw mia 1 1 sun rthsh f A R. Tìte gia k je stoiqeðo y tou sunìlou tim n, f(a), thc f up rqei monadikì stoiqeðo x tou pedðou orismoô thc A gia to opoðo isqôei f(x)=y kai epomènwc orðzetai mia sun rthsh g f(a) R me thn opoða k je y f(a) antistoiqðzetai sto monadikì x A gia to opoðo isqôei f(x) =y. H g lègetai antðstrofh sun rthsh thc f kai sumbolðzetai me f 1. Erwthsh 24. Na apodeðxete ìti gia k je polu numo P (x)=α ν x ν + α ν 1 x ν 1 + +α 1 x + α 0 kai k je x 0 R isqôei lim x x0 P (x)=p (x 0 ). Apanthsh Efarmìzontac tic idiìthtec twn orðwn èqoume: lim P (x)= lim (α ν x ν + α ν 1 x ν 1 + +α 0 ) x x 0 x x0 = lim x x0 (α ν x ν )+ lim x x0 (α ν 1 x ν 1 )+ + lim x x0 α 0 = α ν lim x x0 x ν + α ν 1 lim x x0 x ν 1 + +lim x x0 α 0 = α ν x ν 0 + α ν 1 x ν α 0 = P (x 0 )

33 Kef laio 2. Erwt seic JewrÐac 25 Erwthsh 25. Na apodeðxete ìti gia k je rht sun rthsh f(x) = P (x) kai Q(x) P (x) k je x 0 R me Q(x 0 ) 0 isqôei lim x x0 Q(x) = P (x 0) Q(x 0 ). Apanthsh 'Estw h rht sun rthsh f(x)= P (x) Q(x), ìpou P (x), Q(x) polu - P (x) numa tou x kai x 0 R me Q(x 0 ) 0. Tìte lim f(x) = lim x x0 x x0 P (x 0 ) Q(x 0 ) Q(x) = lim P (x) x x 0 lim Q(x) = x x 0 Erwthsh 26. Pìte mða sun rthsh f ja eðnai suneq c se èna shmeðo x 0 tou pedðou orismoô thc? Apanthsh 'Otan isqôei lim x x0 f(x)=f(x 0 ) Erwthsh 27. Pìte mða sun rthsh f den eðnai suneq c se èna shmeðo x 0 tou pedðou orismoô thc? Apanthsh 'Otan: a) Den up rqei to ìriì thc sto x 0 b) Up rqei to ìriì thc sto x 0, all eðnai diaforetikì apì thn tim thc, f(x 0 ), sto shmeðo x 0. Erwthsh 28. Pìte ja lème ìti mia sun rthsh f eðnai suneq c se èna anoiktì di sthma (α,β)? Apanthsh 'Otan eðnai suneq c se k je shmeðo tou (α, β) Erwthsh 29. Pìte ja lème ìti mia sun rthsh f eðnai suneq c se èna kleistì di sthma [α,β]? Apanthsh 'Otan eðnai suneq c se k je shmeðo tou (α, β) kai epiplèon lim f(x)=f(α) kai lim f(x)=f(β) x α + x β Erwthsh 30. Na diatup sete to je rhma tou Bolzano. Apanthsh 'Estw mia sun rthsh f, orismènh se èna kleistì di sthma [α, β]. An:

34 26 h f eðnai suneq c sto [α, β] kai, epiplèon, isqôei f(α) f(β)<0, tìte up rqei èna, toul qiston, x 0 (α, β) tètoio, ste f(x 0 )=0 Dhlad : Up rqei mia, toul qiston, rðza thc exðswshc f(x) =0 sto anoiktì di sthma (α, β). Erwthsh 31. Na diatup sete kai na apodeðxete to je rhma endi meswn tim n Apanthsh DiatÔpwsh: 'Estw mia sun rthsh f, h opoða eðnai orismènh se èna kleistì di sthma [α, β]. An: h f eðnai suneq c sto [α, β] kai f(α) f(β) tìte, gia k je arijmì hmetaxô twn f(α) kai f(β) up rqei ènac, toul qiston x 0 (α, β) tètoioc, ste f(x 0 )=η Apìdeixh: Ac upojèsoume ìti f(α) < f(β). Tìte ja isqôei f(α)<η < f(β). An jewr soume th sun rthsh g(x)= f(x) η, x [α, β], parathroôme ìti: h g eðnai suneq c sto [α, β] kai g(α)g(β)<0, afoô g(α)=f(α) η < 0 kai g(β)=f(β) η > 0 Epomènwc, sômfwna me to je rhma tou Bolzano, up rqei x 0 (α, β) tètoio, ste g(x 0 )=f(x 0 ) η = 0, opìte f(x 0 )=η.

35 Kef laio 2. Erwt seic JewrÐac 27 Erwthsh 32. Na diatup sete to je rhma mègisthc kai el qisthc tim c. Apanthsh An f eðnai suneq c sun rthsh sto [α, β], tìte h f paðrnei sto [α, β] mia mègisth tim M kai mia el qisth tim m. Dhlad, up rqoun x 1,x 2 [α, β] tètoia, ste, an m = f(x 1 ) kai M = f(x 2 ), na isqôei m f(x) M, gia k je x [α, β]. Erwthsh 33. Poio eðnai to sônolo tim n miac suneqoôc, ìqi stajer c, sun rthshc f me pedðo orismoô to [α, β]? Apanthsh To kleistì di sthma [m,m], ìpou m h el qisth tim kai M h mègisth tim thc. Erwthsh 34. Poio eðnai to sônolo tim n mðac gnhsðwc aôxousac (antistoðqwc fjðnousac) kai suneqoôc sun rthshc orismènhc se èna anoiktì di sthma (α, β)? Apanthsh To di sthma (A, B) (antistoðqwc (B, A)) ìpou A = lim x α + f(x) kai B = lim x β f(x). Erwthsh 35. Pwc orðzetai h efaptomènh thc C f sto shmeðo thca? Apanthsh 'Estw f mia sun rthsh kai A(x 0,f(x 0 )) èna shmeðo thc C f. An f(x) f(x up rqei to lim 0 ) x x0 x x 0 kai eðnai ènac pragmatikìc arijmìc λ, tìte orðzoume wc efaptomènh thc C f sto shmeðo thca, thn eujeða ε: y f(x 0 )=λ(x x 0 ) pou dièrqetai apì to A kai èqei suntelest dieôjunshc λ. Erwthsh 36. Pìte ìti mia sun rthsh f eðnai paragwgðsimh s' èna shmeðo x 0 tou pedðou orismoô thc? f(x) f(x Apanthsh An up rqei to lim 0 ) x x0 x x 0 kai eðnai pragmatikìc arijmìc. To ìrio autì onom zetai par gwgoc thc f sto x 0 kai sumbolðzetai me f (x 0 ). Dhlad : f f(x) f(x (x 0 )= lim 0 ) x x0 x x 0. Erwthsh 37. Ti onom zetai klðsh thc C f sto A(x 0,f(x 0 )) klðsh thc f sto x 0?

36 28 Apanthsh H klðsh f (x 0 ) thc efaptomènhc e sto A(x 0,f(x 0 )). Erwthsh 38. Na apodeðxete ìti an mia sun rthsh f eðnai paragwgðsimh s' èna shmeðo x 0, tìte eðnai kai suneq c sto shmeðo autì. Apanthsh Gia x x 0 èqoume opìte f(x) f(x 0 )= f(x) f(x 0) x x 0 (x x 0 ) lim [f(x) f(x 0 )] = lim [ f(x) f(x 0) (x x 0 )] x x 0 x x0 x x 0 f(x) f(x 0 ) = lim lim (x x 0 )=f (x 0 ) 0 = 0 x x0 x x 0 x x0 afoô h f eðnai paragwgðsimh sto x 0. Epomènwc, lim x x0 f(x)=f(x 0 ), dhlad h f eðnai suneq c sto x 0. Erwthsh 39. Na apodeðxete ìti h sun rthsh f(x)= x an kai suneq c sto x 0 = 0, den eðnai paragwgðsimh s' autì. Apanthsh 'Estw h sun rthsh f(x) = x. H f eðnai suneq c sto x 0 = 0, f(x) f(0) x all den eðnai paragwgðsimh s' autì, afoô x 0 = lim x 0 x = 1, en f(x) f(0) x lim x 0 x 0 = lim x 0 x = 1. lim x 0 + Erwthsh 40. Pìte lème gia mða sun rthsh f me pedðo orismoô èna sônolo A lème ìti: 1. H f eðnai paragwgðsimh sto A? 2. H f eðnai paragwgðsimh se èna anoiktì di sthma (α, β) tou pedðou orismoô thc? 3. H f eðnai paragwgðsimh se èna kleistì di sthma [α, β] tou pedðou orismoô thc? Apanthsh 1. H f eðnai paragwgðsimh sto A ìtan eðnai paragwgðsimh se k je shmeðo x 0 A.

37 Kef laio 2. Erwt seic JewrÐac H f eðnai paragwgðsimh se èna anoiktì di sthma (α, β) tou pedðou orismoô thc, ìtan eðnai paragwgðsimh se k je shmeðo x 0 (α, β). 3. H f eðnai paragwgðsimh se èna kleistì di sthma [α, β] tou pedðou orismoô thc, ìtan eðnai paragwgðsimh sto (α, β) kai epiplèonisqôei lim f(x) f(β) R kai lim x β x β R. x α + Erwthsh 41. Ti onom zetai par gwgoc miac sun rthshc f me pedðoorismoô A? Apanthsh 'Estw A 1 to sônolo twn shmeðwn tou A sta opoða aut eðnai paragwgðsimh. AntistoiqÐzontac k je x A 1 sto f (x), orðzoume th sun rthsh f(x) f(α) x α f A 1 R, x f (x), h opoða onom zetai pr th par gwgoc thc f apl par gwgoc thc f. Erwthsh 42. Na apodeðxete ìti h stajer sun rthsh f(x)=c, c R eðnai paragwgðsimh sto R kai isqôei f (x) =0. c c x x 0 Apanthsh An x 0 eðnai èna shmeðo tou R, tìte gia x x 0 isqôei: f(x) f(x 0) = 0. Epomènwc lim x x0 f(x) f(x 0 ) x x 0 = 0, dhlad (c) = 0. x x 0 = Erwthsh 43. Na apodeðxete ìti h sun rthsh f(x) =x eðnai paragwgðsimh sto R kai isqôei f (x)=1. x x 0 x x 0 Apanthsh An x 0 eðnai èna shmeðo tou R, tìte gia x x 0 isqôei: f(x) f(x 0) f(x) f(x = 1. Epomènwc lim 0 ) x x0 x x 0 = lim 1 = 1, dhlad (x) = 1. x x0 x x 0 = Erwthsh 44. Na apodeðxete ìti h sun rthsh f(x)=x ν eðnai paragwgðsimh sto R kai isqôei f (x)=νx ν 1. Apanthsh An x 0 eðnai èna shmeðo tou R, tìte gia x x 0 isqôei: f(x) f(x 0 ) x x 0 = xν x ν 0 x x 0 opìte = (x x 0)(x ν 1 + x ν 2 x 0 + +x ν 1 0 ) = x ν 1 +x ν 2 x 0 + +x ν 1 0 x x 0 f(x) f(x 0 ) lim = lim (x ν 1 +x ν 2 x 0 + +x ν 1 x x 0 x x 0 x x0 0 )=x ν 1 0 +x ν x ν 1 0 = νx ν 1 0 dhlad (x ν ) = νx ν 1.

38 30 Erwthsh 45. Na apodeðxeteìti h sun rthsh f(x)= x eðnai paragwgðsimh sto (0, + ) kai isqôei f (x) = 1 2. Akìmh na apodeðxete ìti an kai suneq c x sto 0 den eðnai paragwgðsimh s' autì. Apanthsh An x 0 eðnai èna shmeðo tou (0, + ),tìte gia x x 0 isqôei: f(x) f(x 0 ) x x0 = = ( x x0 )( x + x 0 ) x x 0 x x 0 (x x 0 ) ( x + x 0 ) f(x) f(x opìte lim 0 ) x x0 x x 0 Tèloc lim f(x) f(0) x 0 x 0 paragwgðzetai sto 0. = x x 0 (x x 0 ) ( x + x 0 ) = 1 x + x0 = lim 1 x x0 x+ x0 = 1 x x = lim x 0 2 x 0, dhlad ( x) = 1 2 x. = lim x 1 =+ kai epomènwc h sun rthsh den x 0 Erwthsh 46. Na apodeðxete ìti h sun rthsh f(x)=ημx eðnai paragwgðsimh sto R kai isqôei f (x) =συνx. Apanthsh Gia k je x R kai h 0 isqôei f(x + h) f(x) h = ημ(x + h) ημx h = ημx = (συνh 1) h ημx συνh + συνx ημh ημx h + συνx ημh h ημh συνh 1 f(x+h) f(x) Epeid lim h 0 h = 1 kai lim h 0 h = 0, èqoume lim h 0 h = ημx 0+συνx 1 = συνx. Dhlad, (ημx) = συνx. Erwthsh 47. Na apodeðxete ìti h sun rthsh f(x)=συνx eðnai paragwgðsimh sto R kai isqôei f (x)= ημx. Apanthsh Gia k je x R kai h 0 isqôei: f(x + h) f(x) συν(x + h) συνx συνx συνh ημx ημh συνx = = h h h = συνx συνh 1 ημx ημh h h f(x+h) f(x) opìte lim h 0 h = lim (συνx συνh 1 h 0 h ) lim (ημx ημh h 0 h )= συνx 0 ημx 1 = ημx. Dhlad, (συνx) = ημx.

39 Kef laio 2. Erwt seic JewrÐac 31 Erwthsh 48. Na apodeðxete ìti an oi sunart seic f,g eðnai paragwgðsimec sto x 0, tìte h sun rthsh f + g eðnai paragwgðsimh sto x 0 kai isqôei: (f + g) (x 0 )=f (x 0 )+g (x 0 ) Apanthsh Gia x x 0, isqôei: (f + g)(x) (f + g)(x 0 ) x x 0 = f(x)+g(x) f(x 0) g(x 0 ) x x 0 == f(x) f(x 0) x x 0 + g(x) g(x 0) x x 0 Epeid oi sunart seic f,g eðnai paragwgðsimec sto x 0, èqoume: (f + g)(x) (f + g)(x 0 ) lim = lim x x 0 x x 0 dhlad (f + g) (x 0 )=f (x 0 )+g (x 0 ). f(x) f(x 0 ) x x0 x x 0 + lim x x0 g(x) g(x 0 ) x x 0 = f (x 0 )+g (x 0 ) Erwthsh 49. Na apodeðxete ìti h sun rthsh f(x) =x ν, ν N paragwgðsimh sto R kai isqôei f (x)= νx ν 1. eðnai Apanthsh Gia k je ν N èqoume: (x ν ) = ( 1 x ν ) = (1) x ν 1(x ν ) (x ν ) 2 = νxν 1 x 2ν = νx ν 1 Erwthsh 50. Na apodeðxete ìti h sun rthsh f(x)=εφx eðnai paragwgðsimh sto R 1 = R {x συνx = 0} kai isqôei f (x) = 1 συν 2 x. Apanthsh Gia k je x R 1 èqoume: (εφx) =( ημx συνx ) = (ημx) συνx ημx(συνx) συν 2 x = συνxσυνx + ημxημx συν 2 x = συν2 x + ημ 2 x συν 2 x = = 1 συν 2 x Erwthsh 51. Na apodeðxete ìti sun rthsh f(x) =x α, α R Z eðnai paragwgðsimh sto (0, + ) kai isqôei f (x)=αx α 1. Apanthsh An y = x α = e α ln x kai jèsoume u = α ln x, tìte èqoume y = e u. Epomènwc, y =(e u ) = e u u = e α ln x α 1 x = xα α x = αxα 1. Erwthsh 52. Na apodeðxete ìti h sun rthsh f(x)=α x, α > 0 eðnai paragwgðsimh sto R kai isqôei f (x)=α x ln α.

40 32 Apanthsh Any = α x = e x ln α kai jèsoume u = x ln α, tìte èqoume y = e u. Epomènwc y =(e u ) = e u u = e x ln α ln α = α x ln α Erwthsh 53. Na apodeðxete ìti h sun rthsh f(x) =ln x, x R eðnai paragwgðsimh sto R kai isqôei (ln x ) = 1 x. Apanthsh Pr gmati. h an x > 0, tìte (ln x ) =(ln x) = 1 x, en h an x < 0,tìte ln x =ln( x),opìte, an jèsoume y = ln( x) kai u = x, èqoume y = ln u. Epomènwc, y =(ln u) = 1 u u = 1 x ( 1)= 1 x kai ra (ln x ) = 1 x. Erwthsh 54. Ti onom zetai rujmìc metabol c tou y = f(x) wc proc x? Apanthsh Rujmìc metabol c tou y wc proc to x sto shmeðo x 0 eðnai h par gwgoc f (x 0 ). Erwthsh 55. Na diatup sete to je rhma tou Rolle kai na d sete thn gewmetrik ermhneða tou. Apanthsh An mia sun rthsh f eðnai: suneq c sto kleistì di sthma [α, β] paragwgðsimh sto anoiktì di sthma (α, β) kai f(α)=f(β) tìte up rqei èna, toul qiston, ξ (α, β) tètoio, ste: f (ξ)=0 Gewmetrik, autì shmaðnei ìti up rqei èna, toul qiston, ξ (α, β) tètoio, - ste h efaptomènh thc C f sto M(ξ,f(ξ)) na eðnai par llhlh ston xona twn x. Erwthsh 56. Na diatup seteto je rhmamèshc tim c diaforikoô logismoô kai na d sete thn gewmetrik ermhneða tou. Apanthsh An mia sun rthsh f eðnai: suneq c sto kleistì di sthma [α, β] kai paragwgðsimh sto anoiktì di sthma (α, β)

41 Kef laio 2. Erwt seic JewrÐac 33 tìte up rqei èna, toul qiston, ξ (α, β) tètoio, ste: f (ξ)= f(β) f(α) β α. Gewmetrik, autì shmaðnei ìti up rqei èna, toul qiston, ξ (α, β) tètoio, - ste h efaptomènh thc grafik c par stashc thc f sto shmeðo M(ξ,f(ξ)) na eðnai par llhlh thc eujeðac AB. Erwthsh 57. Na apodeðxete ìti an f eðnai mia sun rthsh orismènh se èna di sthma Dkai h f eðnai suneq c sto Dkai f (x) =0 gia k je eswterikì shmeðo x tou D, tìte hf eðnai stajer se ìlo to di sthma D. Apanthsh ArkeÐ na apodeðxoume ìti gia opoiad pote x 1,x 2 Δ isqôei f(x 1 )=f(x 2 ). Pr gmati An x 1 = x 2,tìte profan c f(x 1 )=f(x 2 ). An x 1 < x 2, tìte sto di sthma [x 1,x 2 ] h f ikanopoieð tic upojèseic tou jewr matoc mèshc tim c. Epomènwc, up rqei ξ (x 1,x 2 ) tètoio, ste f (ξ)= f(x 2) f(x 1 ) x 2 x 1 (1) Epeid to xeðnai eswterikì shmeðo tou D, isqôei f (ξ)=0, opìte, lìgw thc (1), eðnai f(x 1 )=f(x 2 ). An x 2 < x 1, tìte omoðwc apodeiknôetai ìti f(x 1 )=f(x 2 ). Se ìlec, loipìn, tic peript seic eðnai f(x 1 )=f(x 2 ). Erwthsh 58. Na apodeðxete ìti an duo sunart seic f, g orismènec se èna di sthma Δ kai oi f,g eðnai suneqeðc sto Dkai f (x) =g (x) gia k je eswterikì shmeðo x tou D, tìte up rqei stajer c tètoia, ste gia k je x Δ na isqôei: f(x)=g(x)+c Apanthsh H sun rthsh f g eðnai suneq c sto D kai gia k je eswterikì shmeðo x Δ isqôei (f g) (x) =f (x) g (x) =0. Epomènwc, sômfwna me to parap nw je rhma, h sun rthsh f g eðnai stajer sto Δ. 'Ara, up rqei stajer c tètoia, ste gia k je x Δ na isqôei f(x) g(x) =c, opìte f(x)=g(x)+c.

42 34 Erwthsh 59. 'Estw mia sun rthsh f, h opoða eðnai suneq c se èna di sthma D.NaapodeÐxete ìti An f (x)>0 se k je eswterikì shmeðo x tou Δ, tìte hf eðnai gnhsðwc aôxousa se ìlo to Δ. An f (x)<0 se k je eswterikì shmeðo x tou Δ, tìte hf eðnai gnhsðwc fjðnousa se ìlo to Δ. Apanthsh ApodeiknÔoume to je rhma sthn perðptwsh pou eðnai f (x)>0. 'Estw x 1 x 2 Δ me x 1 < x 2. Ja deðxoume ìti f(x 1 )<f(x 2 ). Pr gmati, sto di sthma [x 1,x 2 ] h f ikanopoieð tic proôpojèseic tou jewr matoc mèshc tim c. Epomènwc, up rqei ξ (x 1,x 2 ) tètoio, ste f (ξ) = f(x 2) f(x 1 ) x 2 x 1 opìte èqoume f(x 2 ) f(x 1 )=f (ξ)(x 2 x 1 ). Epeid f (ξ) >0 kai x 2 x 1 > 0, èqoume f(x 2 ) f(x 1 )>0, opìte f(x 1 )<f(x 2 ). Sthn perðptwsh pou eðnai f (x)<0 ergazìmaste analìgwc. Erwthsh 60. Pwc orðzetai h jèsh topikoô megðstou kai topikoô elaqðstou mðac sun rthshc f? Apanthsh Mia sun rthsh f, me pedðo orismoô A, ja lème ìti parousi - zei sto x 0 A topikì mègisto (antistoðqwc: topikì el qisto), ìtan up rqei δ > 0, tètoio ste f(x) f(x 0 ) (antistoðqwc f(x) f(x 0 ) ) gia k je x A (x 0 δ, x 0 + δ). To x 0 lègetai jèsh shmeðo topikoô megðstou, en to f(x 0 ) topikì mègisto (antistoðqwc topikì el qisto), thc f. Erwthsh 61. Na apodeðxete to je rhma tou Fermat: 'Estw mia sun rthsh f orismènh s' èna di sthma D kai x 0 èna eswterikì shmeðo tou D. An h f parousi zei topikì akrìtato sto x 0 kai eðnai paragwgðsimh sto shmeðo autì, tìte f (x 0 )=0. Apanthsh Ac upojèsoume ìti h f parousi zei sto x 0 topikì mègisto. Epeid to x 0 eðnai eswterikì shmeðo tou D kai h f parousi zei s' autì topikì mègisto, up rqei δ > 0 tètoio, ste (x 0 δ, x 0 + δ) Δ kai f(x) f(x 0 ), gia k je x (x 0 δ, x 0 + δ) (1) Epeid, epiplèon, h f eðnai paragwgðsimh sto x 0, isqôei f f(x) f(x 0 ) f(x) f(x 0 ) (x 0 )= lim = lim x x 0 x x 0 x x + 0 x x 0 Epomènwc,

43 Kef laio 2. Erwt seic JewrÐac 35 an x (x 0 δ, x 0 ), tìte, lìgw thc (1), ja eðnai f(x) f(x 0) x x 0 0, opìte ja èqoume f (x 0 )= lim x x 0 f(x) f(x 0 ) x x 0 0 (2) an x (x 0,x 0 + δ),tìte, lìgw thc (1), ja eðnai f(x) f(x 0) x x 0 0,opìte ja èqoume f (x 0 )= lim x x + 0 f(x) f(x 0 ) x x 0 0 (3) 'Etsi, apì tic (2) kai (3) èqoume f (x 0 )=0. H apìdeixh gia topikì el qisto eðnai an logh. Erwthsh 62. 'Estwmia sun rthsh f paragwgðsimh s' èna di sthma (α, β), me exaðresh Ðswc èna shmeðo tou x 0, sto opoðo ìmwc h f eðnai suneq c. Na apodeðxete ìti: 1. An f (x) >0 sto (α, x 0 ) kai f (x) <0 sto (x 0,β), tìte to f(x 0 ) eðnai topikì mègisto thc f. 2. An h f (x) diathreð prìshmo sto (α, x 0 ) (x 0,β), tìte to f(x 0 ) den eðnai topikì akrìtato kai h f eðnai gnhsðwc monìtonh sto (α, β). Apanthsh 1. Epeid f (x) >0 gia k je x (α, x 0 ) kai h f eðnai suneq c sto x 0, h f eðnai gnhsðwc aôxousa sto (α, x 0 ]. 'Etsi èqoume f(x) f(x 0 ) gia k je x (α, x 0 ] (1) Epeid f (x) <0 gia k je x (x 0,β) kai h f eðnai suneq c sto x 0, h f eðnai gnhsðwc fjðnousa sto [x 0,β). 'Etsi èqoume: f(x) f(x 0 ) gia k je x [x 0,β) (2) Epomènwc, lìgw twn (1) kai (2), isqôei: f(x) f(x 0 ) gia k je x (α, β) pou shmaðnei ìti to f(x 0 ) eðnai mègisto thc f sto (α, β) kai ra topikì mègisto aut c.

44 36 2. 'Estw ìti f (x)>0, gia k je x (α, x 0 ) (x 0,β) Epeid h f eðnai suneq c sto x 0 ja eðnai gnhsðwc aôxousa se k je èna apì ta diast mata (α, x 0 ] kai [x 0,β). Epomènwc, gia x 1 < x 0 < x 2 isqôei f(x 1 )<f(x 0 )<f(x 2 ). 'Ara to f(x 0 ) den eðnai topikì akrìtato thc f. Ja deðxoume, t ra, ìti h f eðnai gnhsðwc aôxousa sto (α, β). Pr gmati, èstw x 1,x 2 (α, β) me x 1 < x 2. An x 1,x 2 (α, x 0 ], epeid h f eðnai gnhsðwc aôxousa sto (α, x 0 ], ja isqôei f(x 1 )<f(x 2 ). An x 1,x 2 [x 0,β), epeid h f eðnai gnhsðwc aôxousa sto [x 0,β), ja isqôei f(x 1 )<f(x 2 ). Tèloc, an x 1 < x 0 < x 2, tìte ìpwc eðdame f(x 1 )<f(x 0 )<f(x 2 ). Epomènwc, se ìlec tic peript seic isqôei f(x 1 )<f(x 2 ), opìte h f eðnai gnhsðwc aôxousa sto (α, β). OmoÐwc, an f (x)<0 gia k je x (α, x 0 ) (x 0,β). Erwthsh 63. Pìte mða sun rthsh f ja lègetai kurt (antistoðqwc koðlh) se èna di sthma D? Apanthsh An eðnai suneq c sto D kai paragwgðsimh sto eswterikì tou D kai h f eðnai gnhsðwc aôxousa (antistoðqwc: gnhsðwc fjðnousa ) sto eswterikì tou D. Erwthsh 64. 'Estw mia sun rthsh f paragwgðsimh s' èna di sthma (α, β), me exaðresh Ðswc èna shmeðo tou x 0. Pìte to shmeðo A(x 0,f(x 0 )) onom zetai shmeðo kamp c thc grafik c par stashc thc f? Apanthsh To shmeðo A(x 0,f(x 0 )) onom zetai shmeðo kamp c thc grafik c par stashc thc f an isqôei: h f eðnai kurt sto (α, x 0 ) kai koðlh sto (x 0,β), antistrìfwc, kai h C f èqei efaptomènh sto shmeðo A(x 0,f(x 0 )). Erwthsh 65. Pìte h eujeða x = x 0 lègetai katakìrufh asômptwth thc grafik c par stashc thc f? Apanthsh An èna toul qiston apì ta ìria lim f(x), lim f(x) eðnai + x x + 0 x x 0, tìte h eujeða x = x 0 lègetai katakìrufh asômptwth thc grafik c par stashc thc f.

45 Kef laio 2. Erwt seic JewrÐac 37 Erwthsh 66. Pìte h eujeða y = l lègetai orizìntia asômptwth thc grafik c par stashc thc f sto + (antistoðqwc sto )? Apanthsh An isqôei lim f(x)=l (antistoðqwc lim x + x f(x)=l) ) Erwthsh 67. Pìte h eujeða y = λx + β lègetai asômptwth thc grafik c par stashc thc f sto + (antistoðqwc sto )? Apanthsh An isqôei lim [f(x) (λx+β)] = 0,(antistoÐqwc lim [f(x) x + x (λx + β)] = 0 ) Erwthsh 68. An eujeða y = λx+ β eðnai asômptwth thc grafik c par stashc thc f sto +, antistoðqwc sto poiec sqèseic mac dðnoun ta λ, β? lim x + lim x Apanthsh f(x) x = λ kai lim [f(x) λx] =β antistoðqwc x + f(x) x = λ kai lim [f(x) λx] =β x Erwthsh 69. Na diatup sete touc kanìnec tou de l' Hospital. Apanthsh Morf 0 0 f to lim (x) x x0 g (x) Morf + + f up rqei to lim (x) x x0 g (x) An lim x x 0 f(x) =0, lim x x0 g(x)=0, x 0 R {, + } kai up rqei f(x) (peperasmèno peiro), tìte: lim x x 0 g(x) = lim f (x) x x 0 g (x). An lim f(x) =+, lim x x 0 g(x) =+, x 0 R {, + } kai x x0 f(x) (peperasmèno peiro), tìte: lim x x 0 g(x) = lim f (x) x x 0 g (x). Erwthsh 70. 'Estw f mia sun rthsh orismènh se èna di sthma D. Ti onom zetai par gousa thc f sto D? Apanthsh Onom zetai k je sun rthsh F pou eðnai paragwgðsimh sto D kai isqôei F (x)=f(x), gia k je x Δ. Erwthsh 71. 'Estw f mia sun rthsh orismènh se èna di sthma D. Na apodeðxete ìti an F eðnai mia par gousa thc f sto D, tìte ìlec oi sunart seic thc morf c G(x) =F (x)+c, c R eðnai par gousec thc f sto Dkai

46 38 k je llh par gousa G thc f sto DpaÐrnei th morf G(x)=F (x)+c, c R Apanthsh K je sun rthsh thc morf c G(x) = F (x) + c, ìpou c R, eðnai mia par gousa thc f sto D afoô G (x) =(F (x)+c) = F (x) =f(x) gia k je x Δ. 'Estw G eðnai mia llh par gousa thc f sto D. Tìte gia k je x Δ isqôoun F (x) =f(x) kai G (x) =f(x), opìte G (x) =F (x), gia k je x Δ. 'Ara up rqei stajer c tètoia, ste G(x) =F (x)+c, gia k je x Δ. Erwthsh 72. Ti onom zetai aìristo olokl rwma thc f sto D? Apanthsh Onom zetai to sônolo ìlwn twn paragous n thc sun rthshc f sto di sthma D kai sumbolðzetai f(x)dx. Erwthsh 73. 'Estw f mia suneq c sun rthsh s' èna di sthma [α, β] kai G mia par gousa thcf sto [α, β]. Na apodeðxete ìti: β α f(t)dt = G(β) G(α) Apanthsh H sun rthsh F (x) = x α f(t)dt eðnai mia par gousa thc f sto [α, β]. Epeid kai h G eðnai mia par gousa thc f sto [α, β], ja up rqei c R tètoio, ste G(x)=F (x)+c (1) Apì thn (1), gia x = α, èqoume G(α)=F (α)+c = α α f(t)dt + c = c opìte c = G(α). Epomènwc, G(x) =F (x)+g(α), opìte, gia x = β, èqoume G(β)=F (β)+g(α)= β α f(t)dt + G(α) kai ra α β f(t)dt = G(β) G(α) Erwthsh 74. 'Estw duo sunart seic f kai g, suneqeðc sto di sthma [α, β] me f(x) g(x) 0 gia k je x [α, β] kai W to qwrðo pou perikleðetai apì tic grafikèc parast seic twn f,g kai tic eujeðec x = α kai x = β. Na apodeðxete ìti gia to embadìn E(W) tou WisqÔei E(Ω) = β α (f(x) g(x))dx.

47 Kef laio 2. Erwt seic JewrÐac 39 Apanthsh ParathroÔme ìti E(Ω)=E(Ω 1 ) E(Ω 2 )= α β f(x)dx Epomènwc: E(Ω)= β α (f(x) g(x))dx α β g(x)dx = α β (f(x) g(x))dx Erwthsh 75. 'Estw duo sunart seic f kai g, suneqeðc sto di sthma [α, β] me f (x) g (x) gia k je x [α, β] kai W to qwrðo pou perikleðetai apì tic

48 40 grafikèc parast seic twn f,g kai tic eujeðec x = α kai x = β. Na apodeðxete ìti gia to embadìn E(W) tou WisqÔei E(Ω) = β α (f(x) g(x))dx. Apanthsh Pr gmati, epeid oi sunart seic f,g eðnai suneqeðc sto [α, β], ja up rqei arijmìc c R tètoioc ste f(x)+c g(x)+c 0, gia k je x [α, β]. EÐnai fanerì ìti to qwrðow(sq. a) èqei to Ðdio embadìn meto qwrðo Ω (Sq. b). Epomènwc, èqoume: E(Ω) =E(Ω )= β α [(f(x)+c) (g(x)+c)]dx = β α (f(x) g(x))dx. 'Ara E(Ω)= β α (f(x) g(x))dx.

49 Kef laio 3 Qr simec Prot seic Shmeiwsh An kapoia apo tic epomenec protaseic qrhsimopoihjei stic exetaseic qreiazetai apodeixh. Exairesh apoteloun oi (1),(3),(13), (21) Protash 1. 'Enac migadikìc eðnai pragmatikìc an kai mìno an eðnai Ðsoc me ton suzug tou. Apodeixh: An z = α + βi, α,β R tìte z z = 2βi kai epomènwc z R β = 0 z z = 0 z = z Protash 2. An mða suneq c sun rthsh orismènh se ènaanoiktì di sthma (σ 1,σ 2 ) èqei thn idiìthta lim f (x) =, lim f (x) =+ tìte to sônolo 1 2 tim n thc eðnai to R. Apodeixh: ArkeÐ na deðxoume ìti k je pragmatikìc arijmìc y eðnai tim thc f. AfoÔ lim f (x)= h f ja paðrnei kai timèc mikrìterec tou y dhlad 1 ja up rqei x 1 (σ 1,σ 2 ) ste f (x 1 )<y. AfoÔ lim f (x)=+ h f ja paðrnei 2 kai timèc megalôterec tou y dhlad ja up rqei x 2 (σ 1,σ 2 ) ste y < f (x 2 ). Profan c x 1 x 2 kai apì to je rhma endiamèswn tim n ja up rqei x sto di sthma me kra ta x 1,x 2 tètoio ste f (x) =y. Epomènwc o y eðnai tim thc f. Protash 3. Gia k je x > 0 eðnai kai to =} isqôei mìno gia x = 1. ln x x 1 Apodeixh: Efarmog tou sqolikoô biblðou. 41

50 42 Protash 4. Gia k je x eðnai e x x + 1 kai to =}isqôei mìno gia x = 0. Apodeixh: Gia ìlouc touc jetikoôc arijmoôc x isqôei ln x x 1 kai to =} isqôei mìno gia x = 1. Epomènwc kai gia ton jetikì e x isqôei ln e x e x 1 kai to to =} isqôei mìno gia e x = 1 dhlad x = 0. Epomènwc x e x 1 kai to =} isqôei mìno gia x = 0. 'Ara e x x + 1 kai to =} isqôei mìno gia x = 0. Protash 5. An oi sunart seic f,g eðnai orismènec sto di sthma Δ kai isqôei g (x) < m gia ìla ta x Δ kai lim f (x)=0 tìte lim f (x) g (x)=0. Apodeixh: EÐnai: 'Ara gia ìla ta x isqôei kai epomènwc f (x) g (x) = f (x) g (x) f (x) m f (x) g (x) f (x) m f (x) m f (x) g (x) f (x) m All afoô lim f (x)=0 eðnai kai lim f (x) = 0 epomènwc lim m f (x) = lim ( m f (x) ) = 0 (3.1) Apì thn (1) kai to krit rio thc parembol c sun goume ìti lim f (x) g (x)=0 Protash 6. H sun rthsh x èqei gia x 0 par gwgo x x = x x den paragwgðzetai. en sto 0 Apodeixh: To ìti den paragwgðzetai sto 0eÐnai gnwstì. EpÐshc gia x > 0 eðnai ( x ) =(x) = 1 = x x = x x. Akìmh gia x < 0 eðnai ( x ) =( x) = 1 = x x = x x. 'Ara gia x 0 eðnai ( x ) = x x x kai profan c isqôei x = x x diìti x2 = x 2.

51 Kef laio 3. Qr simec Prot seic 43 Protash 7. An f [α, β] R suneq c kai f (α) f (β) 0 tìte h f èqei mða toul qiston rðza sto [α, β]. Apodeixh: f (α) f (β)=0. AfoÔ isqôei f (α) f (β) 0 ja eðnai f (α) f (β) <0 eðte An f (α) f (β) <0 tìte apì to je rhma tou Bolzano h f èqei mða toul qiston rðza sto (α, β) kai epomènwc sto [α, β]. An f (α) f (β) =0 tìte f (α) =0 eðte f (β) =0. 'Ara h f èqei mða toul qiston rðza sto {α, β} kai epomènwc sto [α, β] Se k je perðptwsh h f èqei mða toul qiston rðza sto [α, β]. Protash 8. An h f eðnai gnhsðwc aôxousa tìte ta koin shmeða twn grafik n parast sewn thc f kai thc antðstrof c thc f 1, ef' ìson up rqoun, an koun sthn eujeða y = x. Apodeixh: 'Estw M (α, β) èna shmeðo pou an kei kai sthn C f kai C f 1. Ja isqôei f (α) =β kai f (β) =α. Ja deðxoume ìti to M an kei kai sthn y = x dhlad ìti α = β. An eðnai α β tìte ja eðnai α < β eðte β < α. Sthn pr th perðptwsh ja èqoume f (α) < f (β) dhlad β < α ( topo). Sth deôterh perðptwsh èqoume ìti f (β) <f (α) dhlad α < β ( topo). 'Ara apokleðetai na eðnai α β kai apomènei ìti α = β. Protash 9. 'Estw f [ α, α] R suneq c. 1. An h f eðnai rtia tìte α α f (x) dx = 2 α 0 f (x) dx 2. An h f eðnai peritt tìte α α f (x) dx = 0 Apodeixh: EÐnai α α f (x) dx = 0 α f (x) dx + α 0 f (x) dx = u= x 0 α f ( u) du + α 0 f (x) dx = α 0 f ( x) dx + α 0 f (x) dx 'Otan h f eðnai rtia tìte tìte f ( x)=f (x) kai α 0 f ( x) dx + α 0 f (x) dx = α 0 f (x) dx + α 0 f (x) dx = 2 α 0 f (x) dx. 'Otan h f eðnai peritt tìte f ( x) = f (x) kai α 0 f ( x) dx + α 0 f (x) dx = α 0 f (x) dx + α 0 f (x) dx = 0 Protash 10. H sun rthsh x ln x x eðnai mða par gousa thc ln x. Apodeixh: Profan c isqôei (x ln x x) =(x ln x) (x) =(x) ln x + x (ln x) (x) = ln x + x 1 x 1 = ln x Protash 11. (εϕx) = 1 + εϕ 2 x

52 44 Apodeixh: EÐnai (εϕx) 1 = 1 eðnai συν 2 x = 1 + εϕ2 x. συν 2 x kai apì gnwst sqèsh thc trigwnometrðac Protash 12. Me z C isqôei z 2 = z 2 an kai mìno an z R. Apodeixh: 'Estw z = α + βi. EÐnai z 2 = z 2 α 2 + β 2 =(α + βi) 2 α 2 + β 2 = α 2 β 2 + 2αβi (α 2 + β 2 = α 2 β 2 kai 2αβ = 0) (2β 2 = 0 kai αβ = 0) β = 0 z R Protash 13. 'Estw ìti isqôei f (x) g (x) kont sto σ. epìmena: lim f (x)=+ lim g (x)=+ lim g (x)= lim f (x)= IsqÔoun ta Aitiologhsh: Prìkeitai gia mesh sunèpeia tou orismoô tou orðou. IsqÔei kat' analogða me tic idiìthtec twn peperasmènwn orðwn 1 Protash 14. An gia tic sunart seic f, g pou eðnai orismènec kai suneqeðc sto di sthma [α, β] isqôei f (x) g (x) gia ìla ta x kai f g tìte β α f (x) dx > β α g (x) dx. Apodeixh: Gia thn sun rthsh h = f g isqôei h (x) 0gia ìla ta x kai h 0. Epomènwc β α h (x) dx > 0 apì thn opoða èqoume β α (f (x) g(x)) dx > 0 ra kai β α f (x) dx β α g (x) dx > 0 apì thn opoða prokôptei ìti β α f (x) dx > g (x) dx. β α Protash 15. AnmÐasun rthsh f eðnai paragwgðsimh sto di sthma Δ tìte metaxô dôo opoiwnd pote diaforetik n riz n thc f brðsketai mða toul qiston rðza thc parag gou thc f. Apodeixh: 'Estw ρ 1 < ρ 2 dôo rðzec thc f sto Δ. H f eðnai paragwgðsimh sto di sthma [ρ 1,ρ 2 ] kai isqôei f (ρ 1 )=f (ρ 2 )=0. IkanopoioÔntai epomènwc oi proôpojèseic tou jewr matoc tou Rolle ra ja up rqei ξ me ρ 1 < ξ < ρ 2 tètoio ste f (ξ)=0. Protash 16. An h f eðnai gnhsðwc aôxousa kai f (x 1 )<f (x 2 ) tìte eðnai x 1 < x 2. Apodeixh: Gia touc x 1,x 2 up rqoun ta endeqìmena: x 1 = x 2, x 1 > x 2 kai x 1 < x 2. To pr to mac odhgeð sto topo sumpèrasma f (x 1 )=f (x 2 ). To deôtero, se sunduasmì me to ìti h f eðnai gnhsðwc aôxousa mac odhgeð sto epðshc topo sumpèrasma f (x 1 )>f (x 2 ). 'Ara anagkastik ja isqôei x 1 < x 2. 1 Βλ. σχολικόβιβλίο αρχή της σελίδας 184

53 Kef laio 3. Qr simec Prot seic 45 Protash 17. MÐa gnhsðwc monìtonh sun rthsh èqei to polô mða rðza. Apodeixh: 'Estw f mða gnhsðwc monìtonh sun rthsh. Tìte h f eðnai gnhsðwc aôxousa gnhsðwc fjðnousa kai se k je perðptwsh eðnai 1-1. An ρ 1,ρ 2 eðnai rðzec thc f tìte f (ρ 1 )=f(ρ 2 )=0 kai apì thn sqèsh f (ρ 1 )= f (ρ 2 ) sun goume ìti ρ 1 = ρ 2. Epomènwc h f èqei to polô mða rðza. Protash 18. An h f eðnai gnhsðwc aôxousa tìte kai h f 1 eðnai gnhsðwc aôxousa. Apodeixh: 'Estw y 1,y 2 D f 1 tètoia ste y 1 < y 2. Ja deðxoume ìti f 1 (y 1 )<f 1 (y 2 ). Ja up rqoun x 1,x 2 D f ètoia ste f (x 1 )=y 1 kai f (x 2 )=y 2 ja eðnai de f 1 (y 1 )=x 1 kai f 1 (y 2 )=x 2. Xèroume ìti f (x 1 )< f (x 2 ) kai jèloume x 1 < x 2. H apìdeixh sumplhr netai epiqeirhmatolog ntac ìpwc akrib c sthn prìtash (16.). Protash 19. An z =ρ 0 tìte z = ρ2 z. Apodeixh: AfoÔ z 0 eðnai kai z 0. 'Eqoume t ra: z =ρ z 2 = ρ 2 z z = ρ 2 z = ρ2 z. Protash 20. An z C me z R tìte z 3 = 1 z 2 + z + 1 = 0 z = 1 2 ± i 3 2 Apodeixh: z 3 = 1 z 3 1 = 0 z = 0 (z 1)(z 2 + z + 1) = 0 z R z 2 + z + 1 = 0 (EΠIΛYOYME) z = 1 2 ± i 3 2 Protash 21. Oi paragwgðsimec sunart seic f R R me thn idiìthta f = f eðnai akrib c ekeðnec thc morf c f (x)=ce x ìpou c R stajer. Apodeixh: Efarmog tou sqolikoô biblðou. Protash 22. An lim f (x) = 0 tìte lim f (x)=0. Apodeixh: Apì thn anisìthta A A A èqoume ìti gia k je x isqôei f (x) f (x) f (x) EÐnai lim f (x) = lim ( f (x) ) = 0 kai apì to krit rio thc parembol c èqoume ìti lim f (x)=0. Protash 23. An giamða paragwgðsimh sun rthsh f isqôei f (x) 0 gia k je eswterikì shmeðo x tou Δ tìte hf eðnai aôxousa sto Δ

54 46 Apodeixh: EÐnai ìmoia me thn an logh apìdeixh tou sqolikoô biblðou gia thn perðptwsh ìpou h par gwgoc eðnai jetik. To mìno pou all zei eðnai h teleutaða gramm : Epeid f (ξ) 0 kai x 2 x 1 > 0, èqoume f (x 2 ) f (x 1 ) 0 opìte f (x 1 ) f (x 2 )} Protash 24. MÐa gnhsðwc monìtonh sun rthsh f orismènh seènaanoiktì di sthma Δ den èqei akrìtata. Apodeixh: Ac upojèsoume ìti h f eðnai gnhsðwc aôxousa (h perðptwsh ìpou h f eðnai gnhsðwc fjðnousa antimetwpðzetai analìgwc). An p roume èna opoiod pote shmeðo x 0 Δ. Gia k je δ > 0 to sônolo Δ (x 0 δ, x 0 + δ) perièqei èna toul qiston x 1 < x 0 kai èna toul qiston x 2 > x 0. Lìgw thc monotonðac ja eðnai f (x 1 )<f(x 0 )<f(x 2 ). 'Ara den up rqei δ > 0 ste gia ìla ta x Δ (x 0 δ, x 0 + δ) na isqôei f (x) f (x 0 ) eðte gia ìla ta x Δ (x 0 δ, x 0 + δ) na isqôei f (x) f (x 0 ). 'Ara kanèna x 0 de mporeð na eðnai jèsh topikoô akrot tou.

55 Kef laio 4 Erwt seic Swstì-L joc 'Otan sta epìmena up rqei h er thsh an ènac isqurismìc alhjeôei ennoeðtai ìti zhteðtai na apanthjeð an alhjeôei gia ìlec tic dunatèc peript seic. 'Etsi sthn er thsh an eðnai swstì l joc ìti: An f(1) =1 tìte f(x) >0 gia ìla ta x} ja apant soume ìti eðnai l joc giatð up rqoun sunart seic pou den epalhjeôoun ton isqurismì (l.q. f(x)= 4 x) par to gegonìc ìti up rqoun sunart seic pou ton epalhjeôoun (l.q. f(x)=x 2 + 1) 4.1 MigadikoÐ ArijmoÐ 1. An dôo migadikoð èqoun to Ðdio pragmatikì kai to Ðdio fantastikì mèroc tìte eðnai Ðsoi. 2. An to ginìmeno dôo migadik n arijm n eðnai mhdèn tìte k poioc eðnai mhdèn. 3. An to jroisma twn tetrag nwn dôo migadik n eðnai mhdèn tìte oi migadikoð eðnai Ðsoi me mhdèn. 4. An ènac migadikìc den eðnai pragmatikìc tìte eðnai fantastikìc. 5. 'Enac pragmatikìc arijmìc den eðnai migadikìc. 6. 'Enac pragmatikìc arijmìc den eðnai fantastikìc. 7. 'Enac migadikìc ja eðnai pragmatikìc eðte fantastikìc. 8. De mporeð ènac migadikìc arijmìc na eðnai kai pragmatikìc kai fantastikìc. 9. An z C kai μ, ν eðnai fusikoð tìte z 4μ+ν = z ν. 10. K je migadikìc èqei antðjeto. 11. K je migadikìc èqei antðstrofo. 47

56 MigadikoÐ ArijmoÐ 12. An oi z 1,z 2 eðnai suzugeðc tìte o z 1 + z 2 eðnai pragmatikìc kai o z 1 z 2 eðnai fantastikìc. 13. An o z 1 +z 2 eðnai pragmatikìc kai o z 1 z 2 eðnai fantastikìc tìte oi z 1,z 2 eðnai suzugeðc. 14. An dôo migadikoð arijmoð eðnai Ðsoi tìte oi suzugeðc touc eðnai Ðsoi. 15. An dôo migadikoð arijmoð eðnai Ðsoi tìte ta mètra touc eðnai Ðsa. 16. An dôo migadikoð eðnai antðjetoi tìte kai oi suzugeðc touc eðnai antðjetoi. 17. An dôo migadikoð eðnai antðjetoi tìte èqoun Ðsa mètra. 18. An z 1 = z 2 tìte z 1 = z An z 1 = z 2 tìte z 1 = z An z 1 = z 2 tìte z 1 = z z 1 + z 2 = z 1 + z z n = z n (n jetikìc akèraioc) 23. z n =z 2n (n jetikìc akèraioc) 24. Oi eikìnec dôo suzug n migadik n eðnai summetrikèc wc proc thn arq twn axìnwn. 25. An h eikìna enìc migadikoô arijmoô an kei ston monadiaðo kôklo tìte oi eikìnec tou antðstrofou kai tou suzugoôc tou sumpðptoun. 26. An oi eikìnec dôo migadik n an koun ston monadiaðo kôklo tìte h eikìna tou ginomènou touc an kei epðshc ston monadiaðo kôklo. 27. An dôo migadikoð èqoun mètro 1 tìte to jroisma touc èqei mètro z = z z z =1 30. H apìstash twn eikìnwn twn z 1,z 2 eðnai z 1 z z 2 z 32. An h exðswsh αx 2 +βx+γ = 0 (α, β, γ R) èqei arnhtik diakrðnousa tìte èqei fantastikèc rðzec. 33. An o n eðnai akèraioc tìte i n { 1, i, i, 1}.

57 Kef laio 4. Erwt seic Swstì-L joc An μ, ν eðnai jetikoð akèraioi kai i μ = i ν tìte μ = ν. 35. O i kai o antðstrofoc tou eðnai arijmoð antðjetoi. 36. An to jroisma kai to ginìmeno twn riz n thc exðswshc αx 2 + βx+ γ = 0 (α, β, γ R) eðnai arijmoð pragmatikoð tìte oi rðzec eðnai pragamtikèc. 37. An z 3 = 1 tìte z = Re (z 1 + z 2 )=Re (z 1 )+Re (z 2 ) 39. Re (2z)=2Re (z) 40. Re (z 1 z 2 )=Re (z 1 ) Re (z 2 ) 41. Re (z) R 42. Im (z) R 43. An h αx 2 + βx + γ = 0 (α, β, γ R) èqei rðza ton z tìte èqei rðza kai ton z. 44. O gewmetrikìc tìpoc twn eikìnwn twn migadik n arijm n z gia touc opoðouc isqôei z z 1 = z z 2 (z 1 z 2 stajeroð) eðnai eujeða. 45. O gewmetrikìc tìpoc twn eikìnwn twn migadik n arijm n z gia touc opoðouc isqôei z z 0 =ρ (ρ > 0 stajerìc) eðnai kôkloc. 46. O gewmetrikìc tìpoc twn eikìnwn twn migadik n arijm n z gia touc opoðouc isqôei z z 1 = z z 2 (z 1,z 2 stajeroð) eðnai eujeða. 47. O gewmetrikìc tìpoc twn eikìnwn twn z gia touc opoðouc eðnai Re (z)=a ìpou a > 0 stajerìc eðnai eujeða. 48. O gewmetrikìc tìpoc twn eikìnwn twn z gia touc opoðouc eðnai Re (z)>0 eðnai eujeða. 49. z 1 + z 2 z 1 + z z 1 z 2 z 1 z z 1 z 2 = z 1 z z 1 z 2 2 = z 1 2 z z 1 z 2 2 = z 2 1 z z 1 z 2 = z 1 z ( z 1 z 2 ) = z 1 z z 1 + z 2 2 = z z z 1z 2