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

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2 Peiramatiko Lukeio Euaggelikhc Sqolhc Smurnhc Taxh G, Majhmatika Jetikhc kai Teqnologikhc Kateujunshc, Askhseic Kajhght c: Oi shmei seic autèc eðnai gia sqolik qr sh. MporoÔn na anaparaqjoôn kai na dianemhjoôn eleôjera arkeð na mhn all xei h morf touc. Gia ton periorismì, twn anapìfeuktwn, laj n upìkeintai se suneqeðc diorj seic. Dianèmontai wc èqoun kai o sunt kthc touc den fèrei kamða eujônh gia tuqìn probl mata pou anakôyoun apì thn qr sh touc. 6 SeptembrÐou Stoiqeiojet jhkan me to LATEX.

3 Kef laio MigadikoÐ ArijmoÐ. Isìthta,pr xeic migadik n arijm n.. A' OMADA. Na brejeð o λ R ste na isqôoun oi isìthtec:. +3i =+(λ +)i. λ ++5i =+ ( λ 3λ +7 ) i 3. λ ( + i) =+i. Na gðnoun oi pr xeic:. ( + 3i)+(6+i). (4 + 5i)+(6 i) 3. (3 9i) (6 8i) 4. (3 i)(+i) 5. ( i)(3+4i) 6. ( + i)(+i) 7. (4 + 6i)(7 3i) 8. ( i)( i)(3 i) 3. Na grafoôn sth morf α + βi,α,β R oi arijmoð:. +i i i 3i 5. i + +i. i +3i 4. ( i) (+i) 3 6. ( +3i + 4+i ) 4. 'Estw z = 3i, z =+i. Na upologðsete ta:. z + z. z (z z ) 3. z + z ( ) z +i z z z + z z z + z z z + z +z z z

4 .. Isìthta,pr xeic migadik n arijm n 5. Na epalhjeôsete thn isìthta: 3+i (+i)( i) = i 6. An z =+3i kai z = i na upologisjeð ì z ((z +z )) z 7. DeÐxte ìti (x i)(x +i)(x ++i)(x + i) =x DeÐxte ìti: ( t i )( t +i )( t +i )( 3 t i ) 3 = t 4 6t 3 +8t 6t+ 9. Gia poièc timèc twn pragmatik n arijm n x, y eðnai oi migadikoð eðnai suzugeðc? z =9y 4 xi 5, z =8y +i. To pragmatikì kai to fantastikì mèroc tou z eðnai antistoðqwc x kai y. Poiì eðnai to pragmatikì kai to fantastikì mèroc tou z 7?. An z 6 = i kai z 7 = 78 9i poiìc eðnai o z?. Na gr yete sth morf α + βi to migadikì: ( + i) 8 ( + i) 8 i Me z =3 4i na upologðsete thn par stash z + z +(z z) 4. An z =+5i na upologisjeð h par stash Re (z +)+Im(iz). 5. Gia poia tim tou λ R isqôei ( + i) 3 = λ +(5λ +)i? 6. Apì tic exet seic tou 954. Ejnikì Metsìbio PoluteqneÐo. UpologÐsate thn par stasin ( +i ) 3 + ( i ) Na breðte touc migadikoôc arijmoôc z, z an eðnai gnwstì ìti z +3z =8 kai 4z 3iz =7+9i. 8. Na paragontopoi sete sto C tic parast seic:. x + y. 4x + y 9. Pìte isqôei z + z i = z z i?. Na brejoôn oi migadikoð arijmoð z,w an eðnai gnwstì ìti z + w =7+i kai z = w +i 3. Na breðte touc pragmatikoôc arijmoô x, y an eðnai gnwstì ìti: x i 3+i + y i 3 i = i. Na brejoôn oi pragmatikoð κ, λ ètsi ste oi migadikoð arijmoð κ+3λ+3i kai +(κ λ) i na eðnai Ðsoi. 3. Na breðte ton z an eðnai gnwstì ìti z +3 z =5+7i. 4. Na lôsete tic exis seic:

5 Kef laio. MigadikoÐ ArijmoÐ 3. x 4x +3=. x x +5= 3. 3t t += 4. x (συνθ) x += 5. An z = a + bi kai z + z = 3+5i na brejoôn ta a, b. 6. An α, β, γ, δ, ε, ζ eðnai pragmatikoð arijmoð deðxte ìti oi z = α (β + γi) + δ (ε + ζi)+η, w = α (β γi) + δ (ε ζi)+η eðnai suzugeðc. 7. 'Estw z = + i 3 kai κ, λ pragmatikoð. Na apodeðxete ìti o arijmìc ( κz + λz )( κz + λz ) eðnai mh arnhtikìc pragmatikìc. 8. Gia ton migadikì arijmì z eðnai gnwstì ìti z = z 5. Na epalhjeôsete tic isìthtec:. z 3 = 4z 5. z 4 = 9z + 3. z 5 =z Na aplopoi sete thn par stash: Re(Re(z)) + Im(Re(z)) + Re(Im(z)) + Im(Im(z)) 3. Me z = + i 3 na upologðsete ta z, z 3, z 6 3. Na brejeð o x ìtan x = x+3i + x 3i 3. O Girolamo Cardano, ènac apì touc prwterg tec thc 'Algebrac sto èrgo tou Ars Magna (Meg lh Tèqnh) pou exedìjh to 545, gr fei: An k poioc sac pei na diairèsete to se dôo mèrh, pou an pollaplasiastoôn metaxô touc ja d soun 3 4, eðnai fanerì ìti prìkeitai gia èna adônato prìblhma. Girolamo Cardano Na exhg sete giatð o Cardano onom zei to prìblhma adônato.. Poia tim, antð tou 3 tou 4, ja mporoôse na k nei to prìblhma dunatì}? 33. 'Estw ìti oi α, β eðnai pragmatikoð arijmoð kai α+βi = z z+. Na apodeðxete ìti α = z z 4 z z+z+ z+4 kai β = i( z z) z z+z+ z Na apodeðxete ìti ( ) ( ) +z z z + z = z 35. Na lôsete thn exðswsh x+ + x+4 = x MporeÐ rage na isqôei z z =? z z =?

6 4.. Isìthta,pr xeic migadik n arijm n.. B' OMADA 37. DeÐxte ìti an a, b, x, y R me a, b kai isqôei tìte x = b kai y = a. (a + bi)(x + yi) = ( a + b ) i 38. 'Estw z = x + yi, x, y R kai w = ( x y + x ) +(xy + y) i. Na ekfr sete ton w sunart sei tou z. 39. Na breðte thn tim thc par stashc A = iν fusikoô arijmoô ν. i gia tic di forec timèc tou 4. Na gr yete sth morf α + βi ton arijmì z = p+qi p qi + p qi p+qi ìpou p, q R. 4. 'Estw z = x + yi. Na brejeð o z an eðnai gnwstì ìti o arijmìc w = (z ) ( z + i) eðnai pragmatikìc. 4. 'Estw ω ènac mh pragmatikìc migadikìc arijmìc. Na apodeðxete ìti k je migadikìc arijmìc z mporeð na grafeð kat monadikì trìpo sth morf : z = α + βω, α,β R 43. Na breðte touc pragmatikoôc arijmoôc κ, λ ste na isqôei κ + i + λ +i = i 44. DÐnontai oi migadikoð arijmoð z = 3i, z =4+i kai z = i. Na breðte pragmatikoôc arijmoôc κ, λ ètsi ste z = κz + λz. 45. Na apodeðxete ìti an α, β, γ R tìte Re ((α + β + γi)(β + γ + αi)(γ + α + βi)) = αβγ 46. Na apodeðxete ìti an z 3 =kai z/ R tìte ( z + z ) 5 + ( +z z ) 5 =3 47. Omigadikìc arijmìc z = x+yi pollaplasiazìmenoc epð ton arijmì +3i èqei ginìmeno fantastikì arijmì. Poia sqèsh sundèei ta x, y? 48. 'Estw h sun rthsh f (z) = αz+β z, z. An eðnai f (i) = 5 i kai f ( + i) = 5i breðte to f ( + i). 49. Na lôsete thn exðswsh ( i) 8 x =(+i) 'Estw z = +i i. BreÐte to pragmatikì kai to fantastikì mèroc tou z + z.

7 Kef laio. MigadikoÐ ArijmoÐ 5 5. Na breðte ton w an ( 3w + w + ) + ( w +w + ) =. 5. 'Estw u ènac mh mhdenikìc migadikìc arijmìc. Pwc prèpei na epilèxoume to migadikì arijmì z ste o arijmìc (z u)(z u) na eðnai pragmatikìc? 53. Gia to migadikì arijmì z eðnai gnwstì ìti Re (z) =kai Im (z) =. Na breðte ta Re ( z ) kai Im ( z ). 54. 'Estw hexðswsh x mx+m =me rðzec x x. Na breðte tic x,x an eðnai gnwstì ìti x 3 + x3 = x + x. 55. BreÐte ton z an eðnai gnwstì ìti z = 5+i. 56. Na apodeðxete ìti an o jetikìc akèraioc ν den eðnai pollapl sio tou 4 tìte ( +i ν) ( + i ν )=. 57. Apì tic exet seic tou 98. Na breðte to jroisma twn n-ìrwn: Σ=i +(+3i)+(4+5i)+... +[(ν ) + (ν ) i] 58. Na apodeðxete ìti o migadikìc arijmìc ( z z )( z z ) eðnai pragmatikìc. 59. Apì tic exet seic tou 983, Dèsmh I. 'Estw ω me ω 3 =,ìpou ω C kai ω. Na upologðsete thn tim thc par stashc 6. Na apodeðxete ìti y =( ω) ( ω )( ω 4)( ω 5) +ημθ + iσυνθ +ημθ iσυνθ = ημθ + iσυνθ 6. To jroisma kai to ginìmeno dôo migadik n arijm n eðnai arijmoð pragmatikoð. EÐnai rage oi dôo migadikoð suzugeðc? 6. Na lôsete thn exðswsh αz z + β (z z) = γ + δi ìpou α, β, γ, δ mh mhdenikoð pragmatikoð. 63. 'Estw ìti z + z =. Poièc timèc mporeð na p rei h par stash A ν = z ν + z ν? 64. O Bombelli se k poio prìblhma qrei sjhke na breð migadikoôc arijmoôc α + βi ste (α + βi) 3 =+i. Na epalhjeôsete ìti o migadikìc +i eðnai lôsh tou probl matoc.. Na breðte touc llouc migadikoôc arijmoôc pou eðnai epðshc lôseic tou probl matoc. Rafael Bombelli 56-57

8 6.. Isìthta,pr xeic migadik n arijm n 65. Na apodeðxete ìti an gia touc jetikoôc akèraiouc κ, λ, μ, ν isqôei i κ + i λ + i μ + i ν =tìte ja isqôei i κ+4λ + i λ+4μ + i μ+4ν + i ν+4κ =. 66. 'Estw w = p 3 3p i, w = pq i, w 3 = p + pqi ìpou oi p, q eðnai di foroi mh mhdenikoð pragmatikoð. Na apodeðxete ìti an o arijmìc w +iw w 3 eðnai fantastikìc tìte isqôei p + q =. 67. 'Estw F (z) =(+z) 4 + +z. An F ( + i) = F ( i)? 68. An z + z = na breðte to z + z i poio eðnai to 69. Gia poièc timèc twn pragmatik n arijm n x, y o arijmìc (x + yi) 3 eðnai pragmatikìc kai megalôteroc tou 8? 7. Na apodeðxete ìti ( ) xi y 4n x + yi ( ) x + yi 4n+ =+i xi y 7. 'Estw ìti (X + Yi) 3 = x + yi. Na apodeðxete ìti x X + y Y =4( X Y ) 7. 'Estw α, β, γ treic ana dôo di foroi migadikoð arijmoð. Na apodeðxete ìti an oi α, β, γ ikanopoioôn k poia apì tic parak tw sunj kec ikanopoiooôn kai tic upìloipec:. β γ + γ α + α β =. (β γ)(γ α)+(γ α)(α β)+(α β)(β γ) = 3. (β γ) =(γ α)(α β) 4. (γ α) =(α β)(β γ) 5. (α β) =(β γ)(γ α) 6. (β γ) +(γ α) +(α β) = 7. α + β + γ = αβ + βγ + γα 73. 'Estw ìti ζ 5 =. Na apodeiqjeð ìti: ζ +ζ + ζ +ζ 4 + ζ3 +ζ + ζ4 +ζ 3 =

9 Kef laio. MigadikoÐ ArijmoÐ 7..3 G' OMADA 74. Na apodeðxete ìti gia opoiad pote tri da jetik n akeraðwn κ, λ, μ eðnai i κ + i λ + i μ. 75. O arijmìc ( + i) n +( i) n eðnai pragmatikìc (n N). Pìte eðnai arnhtikìc? 76. Na apodeðxete ìti an w 3 =, w kai o n eðnai jetikìc rtioc tìte ( w + w )( w + w 4)( w 4 + w 8)... ( w n + w n) = n 77. 'Estw z,z,z 3 treic an dôo di foroi migadikoð arijmoð ste z z z z 3 / R. Na apodeðxete ìti k je migadikìc arijmìc z mporeð na grafeð kat monadikì trìpo wc z = αz + βz + γz 3 ìpou α, β, γ pragmatikoð me jroisma.. Mètro migadikwn arijm n.. A' OMADA 78. Na breðte to mètro tou migadikoô arijmoô z ìtan:. z =+i. z =4+3i 3. z = i 79. An z =poio eðnai to mètro tou z? Tou z? Tou z 3? 8. An o migadikìc arijmìc x+yi èqei mètro 3 poiì eðnai to mètro tou y+xi? 8. Apì tic exet seic tou. An gia to migadikì arijmì z isqôei z =na deðxete ìti z = z. 8. Na apodeðxete ìti z + z = z z + z z. 83. Gia poia tim tou λ eðnai z =3ìtan:. z =+λi. z = λ +3(λ +)i 84. Na breðte ta mètra twn migadik n: z = συνϕ + iημϕ, z = ημϕ + iσυνϕ, z 3 = 85. Na breðte ta mètra twn migadik n arijm n z =(+i)(+i)(3+i), z = t + i t +t +t +i ( + i)(3+i) 86. An z = kai z =3poio eðnai to mètro tou z3 z z 4? 87. 'Estw z = 4i. Na upologðsete ta

10 8.. Mètro migadikwn arijm n. z ++i 3. z z. z i 4. z+i 88. Gia poiouc migadikoôc arijmoôc isqôei z = z? 89. Na apodeðxete ìti an z >, z < tìte z + z > +z z. 9. Gia ton z èinai gnwstì ìti èqei mètro kai den eðnai pragmatikìc. Na apodeðxete ìti mporeð na grafeð sth morf z = α+i α i,α R. 9. Na apodeðxete ìti z + z z ( z + z ). z + z z 9. Na lôsete th exðswsh 4z +8 z = Na breðte ton z an eðnai gnwstì ìti z z 8i = 5 3 kai z 4 =. 94. Na breðte ìlouc touc migadikoôc arijmoôc z gia touc opoðouc isqôei z 3 = z. 95. Na apodeðxete ìti an isqôei z + αi = z + βi kai oi α, β eðnai di foroi pragmatikoð arijmoð tìte isqôei z z = (α + β) i. 96. Na apodeðxete ìti z z z z z An z =na breðte to mètro tou z (z i) +zi. 98. Na upologðsete thn par stash ( + i) 34 i Na breðte to mètro tou z = ( κ λ ) +κλi, κ, λ R.. Na lôsete thn exðswsh z + z =7+7i.. An z =, w =3na upologðsete thn par stash. An eðnai w A = z ( z + w) w (z w) 6 w =poiì eðnai to mètro tou w? 3. DeÐxte ìti Re (z) z kai Im (z) z 4. BreÐte ton z an eðnai gnwstì ìti z =kai z + =. 5. Gia ton migadikì arijmì z = x + yi isqôei 3x =4y>. Na apodeðxete ìti z = 5 4 x. 6. Gia k je migadikì z kai k je jetikì pragmatikì r isqôei rz = r z. GiatÐ?

11 Kef laio. MigadikoÐ ArijmoÐ 9 7. Gia touc z,z eðnai gnwstì ìti z kai z 3. Na apodeðxete ìti:. z +3z. 5z 6z z 5z 4. z z 3 8. 'Estw ìti u + v = u = v. DeÐxte ìti u v = + i 3 eðte u v = i 3 Upodeixh: DeÐte kai thn skhsh 4 9. To mètro tou ginomènou dôo migadik n arijm n eðnai 3 kai to mètro tou phlðkou touc eðnai 4. Poia eðnai ta mètra touc?.. B' OMADA. Gia touc migadikoôc arijmoôc z kai z isqôei z =5, z =3, z z = 8. Na breðte to mètro tou ajroðsmatoc touc.. An α, β, γ eðnai pragmatikoð kai z γi = α+βi α+γi na breðte to z.. Na apodeðxete ìti an gia ton migadikì arijmì z isqôei z + = z + tìte o z eðnai mh arnhtikìc pragmatikìc. 3. Na apodeðxete ìti an x + yi = a+bi c+di tìte x + y = a +b c +d. 4. Na breðte ìlouc touc migadikoôc z gia touc opoðouc isqôei z + z =. 5. Apì tic exet seic tou 98. An z +6 =4 z +, na deðxete ìti z =4. 6. Na apodeðxete ìti an u, v eðnai mh mhdenikoð mhgadikoð me u + v tìte isqôei u + v = u + v an kai mìno an o arijmìc u v eðnai fantastikìc. 7. Estw ìti gia touc mh mhdenikoôc migadikoôc z,z,z 3 isqôei z + z + z 3 =kai z = z = z 3 Na apodeðxete ìti oi z z, z 3 z eðnai rðzec thc exðswshc z + z +=. 8. Apì tic exet seic tou 979. An z +i kai z +i deðxte ìti:

12 .. Mètro migadikwn arijm n. 'Otan z =,tìte o arijmìc z z+. Otan o arijmìc z z+ eðnai kajarìc fantastikìc. eðnai kajarìc fantastikìc tìte z =. 9. Apì tic exet seic tou 99, Dèsmh I. An ω = z+αi iz+α kai z αi tìte na apodeiqjeð ìti: me α R. O ω eðnai fantastikìc arijmìc an kai mìno an o z eðnai fantastikìc arijmìc.. IsqÔei ω =an kai mìno an o z eðnai pragmatikìc arijmìc.. An z 9 =3na breðte to z. z. Na apodeðxete ìti an isqôei z z = z z tìte isqôei kai ( z 3 z + ( ) 3 z = z 3 z + ) 3 z. Na breðte to migadikì arijmì z an eðnai gnwstì ìti z +3i = z +5 i kai z 4i = z +i. 3. Na apodeðxete ìti an (α + β i)(α + β i)... (α ν + β ν i)=a + Bi tìte ( α + β )( α + β )... ( α ν + β ν) = A + B 4. Apì tic exet seic tou 98. Na deðxete ìti h exðswsh ( + iz) ν = +i 3 3+i me ν N kai z C den èqei pragmatik lôsh. 5. Gia poiouc z isqôei z i + z +3 =? 6. Gia ton migadikì arijmì z isqôei z = z. DeÐxte ìti z = z = z 7. Gia poiouc z isqôei z = z 3 = z i? 8. Na apodeðxete ìti an z +z + z z =tìte ènac toul qiston apìtouc z,z èqei mètro. 9. 'Estw hexðswsh z + az + b =ìpou a, b pragmatikoð me a 4b <. Na apodeðxete ìti oi rðzec thc exðswshc èqoun mètro mikrìtero tou an kai mìno an b<. 3. 'Estw ìti z + = z +3. BreÐte to Re (z +3). EnnoeÐ: fantastikìc EÐnai h skhsh B4 selðda tou sqolikoô biblðou

13 Kef laio. MigadikoÐ ArijmoÐ 3. Gia to migadikì arijmì u eðnai gnwstì ìti u iu < u i ìtioarijmìc +Im ( u ) eðnai arnhtikìc pragmatikìc. 3. Na apodeðxete ìti o arijmìc z z z+ z eðnai fantastikìc.. Na apodeðxete 33. Na apodeðxete ìti den up rqei migadikìc z ètsi ste z + i = kai z = 'Estw p, q R kai z = +pi pi, z = +qi qi. Na apodeðxete ìti an d = = p q z z tìte d 4 d +pq. 35. 'Estw ìti z + z = z = z. Na apodeðxete ìti z z = 3 z. 36. Na apodeðxete ìti an isqôei z =3 z tìte isqôei z = Na apodeðxete ìti. z + z = z + z +Re(z z ). z z = z + z Re (z z ) 38. Na apodeðxete ìti an oi migadikoð arijmoð z,z,z 3 èqoun mètro kai ginìmeno di foro tou - tìte o migadikìc eðnai pragmatikìc. 39. Na apodeðxete ìti z + z + z 3 + z z + z z 3 + z 3 z +z z z 3 z + z + z z = z z R 4. Na apodeðxete ìti z + z z z + z z. 4. Na apodeðxete ìti an z tìte z H tautìthta tou Lagrange. Na apodeðxete ìti: ( z + z )( w + w ) = z w + z w + z w z w 43. Apì tic exet seic tou 979 (Akur jhkan).. Na deðxete ìti k je migadikìc arijmìc mporeð na grafeð san tetr gwno migadikoô arijmoô. 3 3 DeÐte pr ta thn skhsh 55. To er thma autì tan, tìte, er thma jewrðac. Gia na to apant sete jewreðste migadikì arijmì α + βi kai deðxte ìti up rqei p nta migadikìc x + yi ètsi ste (x + yi) = α + βi. Sthn ousða prèpei na deðxete ìti to sôsthma x y = α, xy = β èqei lôsh.

14 .. Mètro migadikwn arijm n. An z,z C, na deðxete ìti 4 z + z + z z = ( z + z ) 3. An α, β C tìte sômfwna me thn er thsh (a') up rqei èna z C tètoio ste: αβ = z. Na deðxete thn isìthta: α + β = α + β + z + α + β z Joseph Louis Lagrange Apì tic exet seic tou 979. An Π(x) =x + z z x + (+ z )( + z ) ìpou z,z dojèntec migadikoð arijmoð na deðxete ìti Π(x) gia k je pragmatikì arijmì x. Pìte isqôei to Ðson? 45. Apì tic exet seic tou 7. DÐnetai o migadikìc arijmìc z = +αi α +i, me α R. Na apodeðxete ìti h eikìna tou migadikoô z an keo ston kôklo me kèntro to O(, ) kai aktðna ρ =.. 'Estw z,z oi migadikoð arijmoð pou prokôptoun apì ton tôpo gia α =kai α =antðstoiqa. z = +αi α +i (a ) Na brejeð h apìstash twn eikìnwn twn migadik n arijmwn z kai z. (b ) Na apodeiqjeð ìti isqôei (z ) ν =( z ) ν gia k je fusikì arijmì ν. 4 EÐnai h skhsh A9 selðda tou sqolikoô biblðou.

15 Kef laio. MigadikoÐ ArijmoÐ 3..3 G' OMADA 46. Na apodeðxete ìti Katìpin na apodeðxete ìti: z + z + z + z z + z + z + z z + z z ν + z + z z ν z + z + z + z z ν + z ν 47. Apo to diagwnismì Putnam, 989. Na apodeiqjeð ìti an z +iz 9 +iz = tìte z = 48. 'Estw S to sônolo twn z C gia touc opoðouc isqôei z + z = α. Na breðte pwc prèpei na epilegeð o α> ste h el qisth tim twn mètrwn twn stoiqeðwn tou S na eðnai. 49. Na apodeðxete ìti gia k je zeôgoc migadik n arijm n α, β kai gia k je jetikì akèraio r isqôei: α + β r r ( α r + β r ) Gia touc mh mhdenikoôc migadikoôc arijmoôc z,z,..., z ν isqôei ìti z + z z ν = z + z z ν Na apodeðxete ìti up rqoun jetikoð arijmoð λ,..., λ ν ètsi ste.3 To Migadikì EpÐpedo.3. A' OMADA z = λ z,..., z ν = λ ν z 5. Sto parak tw sq ma to shmeðo A eðnai h eikìna tou z. Na sqedi sete thn eikìna tou z.

16 4.3. To Migadikì EpÐpedo 5. Na apodeðxete ìti oi eikìnec twn migadik n arijm n x+yi,, x+yi eðnai shmeða suneujeiak. 5. Na sqedi sete to qwrðo tou epipèdou sto opoðo an koun oi eikìnec twn migadik n arijm n z gia touc opoðouc kat perðptwsh isqôei:. Im (z) > Re (z). Im (z) > Im ( z ) 53. Na pollaplasi sete ton 3+i me to i. Met to apotèlesma pou ja breðte na to pollaplasi sete xan me i. Na epanal bete llh mða for. Na k nete èna sq ma. 54. Heikìna tou z an kei sthn eujeða me exðswsh x 3y =. Na ekfr sete ton z sunart sei tou z. 55. Se k je èna apì ta parak tw sq mata emfanðzontai dôo omìkentroi kôkloi me kèntro thn eikìna tou z kai aktðnec ρ <ρ. Na perigr yete se k je perðptwsh to grammoskiasmèno qwrðo. ρ ρ ρ ρ ρ ρ z z z ) ) 3) ρ ρ ρ ρ ρ ρ z z z 4) 5) 56. DeÐxte ìti me z isqôei h isodunamða: z 5 = z 6 z = 6) 57. Oi eikìnec twn z,z apèqoun apìstash d. Na apodeðxete ìti d = z z z z z z + z z 58. Na apodeðxete ìti to trðgwno tou opoðou oi korufèc eðnai oi eikìnec twn eðnai isoskelèc. Upodeixh: EÐnai z z = z z 3 =5 z =+i, z =4 i, z 3 = 6i 59. Se ìlec tic parak tw peript seic o gewmetrikìc tìpoc thc eikìnac tou z eðnai kôkloc. Na breðte to kèntro tou K kai thn aktðna tou ρ.

17 Kef laio. MigadikoÐ ArijmoÐ 5. z =4. z i = 3. z 3i = 4. z ++3i = 5. z + 3i = 6. z +4i = 6. Na apodeðxete ìti oi eikìnec twn migadik n eðnai korufèc tetrag nou. +i, 3+i, +3i, +i 6. An h eikìna tou z an kei ston kôklo me exðswsh (x ) +(y 3) =4 tìteheikìna tou z an kei se èna kôklo. Se poiìn? 6. Na apodeðxete ìti an h eikìna tou z an kei sthn eujeða me exðswsh x + y =tìte isqôei z = z + i. 63. Gia poiec timèc tou x h eikìna tou migadikoô arijmoô z = x +(x ) i an kei ston kôklo me kèntro A(, 3) kai aktðna 4? 64. Na apodeðxete ìti oi eikìnec twn migadik n z, z, z, z, eðnai shmaða suneujeiak. 65. Apì tic exet seic tou 999, Dèsmh I. Estw o migadikìc arijmìc z = x + yi, x, y R. Na apodeðxete ìti sto migadikì epðpedo o gewmetrikìc tìpoc twn shmeðwn M(x, y) pou eðnai tètoia ste z + z 3 i =6eÐnai kôkloc. Na breðte to kèntro kai thn aktðna tou kôklou autoô. 66. Se k je èna apì ta epìmena sq mata apeikonðzontai dôo temnìmenoi kôkloi me kèntra tic eikìnec twn z,z kai aktðnec ρ ρ. Na perigr yete se k je perðptwsh to grammoskiasmèno qwrðo. ρ ρ ρ ρ z z z z ) ) ρ ρ ρ ρ z z z z 3) 4) 67. Na breðte to gewmetrikì tìpo twn eikìnwn twn migadik n arijm n z gia touc opoðouc isqôei 4(z z) =9z z 5.

18 6.3. To Migadikì EpÐpedo 68. Poiìc eðnai o gewmetrikìc tìpoc twn eikìnwn twn w gia touc opoðouc isqôei =? w 69. JewroÔme touc migadikoôc z =+i kai z =3+4i. Poiì eðnai to embadìn tou trig nou pou sqhmatðzoun oi eikìnec twn z, z,z + z? 7. Na breðte ton gewmetrikì tìpo twn eikìnwn twn z gia touc opoðouc isqôei z +3i =5 z + i. 7. Poioc eðnai o gewmetrikìc tìpoc twn eikìnwn twn migadik n arijm n z gia touc opoðouc isqôei Re ( z ) =? 7. Apì tic exet seic tou 986, Dèsmh I. 'Estw ìti z =(x 3) + (y )i me x, y R. Na deðxete ìti sto migadikì epðpedo o gewmetrikìc tìpoc twn shmeðwn (x, y) pou eðnai tètoia ste z +3i =3 eðnai kôkloc. Sth sunèqeia na breðte tic suntetagmènec tou kôklou autoô kai thn aktðna tou. 73. 'Estw z,z dôo migadikoð arijmoð kai u, u ta antðstoiqa dianôsmata. Na apodeðxete ìti isqôei: u u z z + z z = Katìpin na apodeðxete ìti to eujôgrammo tm ma me kra tic eikìnec twn u, v eðnai k jeto sto eujôgrammo tm ma me kra tic eikìnec twn w, z an kai mìno an o arijmìc u v w z eðnai fantastikìc 74. DÐnetai o migadikìc arijmìc z =+i. Na breðte to m koc thc tejlasmènhc grammhc ABGD ìpoutaa,b,g,deðnaioieikìnec twn z, z, z 3, z Na breðte ton gewmetrikì tìtpo twn eikìnwn twn migadik n z gia touc opoðouc isqôei ( ) z i Re = z ++i.3. B' OMADA 76. Oi eikìnec twn migadik n arijm n 3+i kai 4+i eðnai diadoqikèc korufèc enìc tetrag nou. TÐnwn migadik n mporeð na eðnai eikìnec oi llec dôo korufèc tou tetrag nou? 77. An α R, kai w C stajeroð na apodeðxete ìti ìloi oi migadikoð arijmoð z pou ikanopoioôn thn sqèsh Re(wz) =α an koun se mða eujeða.

19 Kef laio. MigadikoÐ ArijmoÐ Na apodeðxete ìti an oi eikìnec twn migadik n arijm n, z,z eðnai korufèc trig nou tìte kai oi eikìnec twn migadik n,, z z eðnai korufèc trig nou ìmoiou proc to arqikì. 79. 'Estw ìti eikìna tou migadikoô arijmoô z an kei sthn kampôlh Γ. 'Estw a ènac mh mhdenikìc migadikìc kai S, M oi eikìnec twn u = a + z kai w = az antistoðqwc. Na sumplhr sete ton pðnaka: η Γ είναι Το S ανήκει: Το M ανήκει: Ηευθεία z z = z z Ο κύκλος z z = ρ 'Estw z, z,z 3 treic an dôo di foroi migadikoð arijmoð. Na apodeðxete ìti oi eikìnec touc eðnai shmeða suneujeiak an kai mìno an 8. Na apodeðxete ìti: z z z z 3 R. To eswterikì ginìmeno twn dianusm twn pou ekfr zoun oi migadikoð arijmoð z,z eðnai Ðso me (z z + z z ). Ta embadìn tou trig nou pou èqei wc korufèc tic eikìnec twn migadik n arijm n z,z,z 3 eðnai Ðso me 4 z z + z z 3 + z 3 z z z z z 3 z 3 z 8. 'Estw z,z dôo migadikoð arijmoð kai u, u ta antðstoiqa dianôsmata. Na apodeðxete ìti isqôei: u u z z + z z = Katìpin na apodeðxete ìti to eujôgrammo tm ma me kra tic eikìnec twn u, v eðnai k jeto sto eujôgrammo tm ma me kra tic eikìnec twn w, z an kai mìno an o arijmìc u v w z eðnai fantastikìc 83. Oi eikìnec twn migadik n arijm n z kai z eðnai ta shmeða A kai B. Na apodeðxete ìti to trðgwno OAB eðnai orjog nio sto O kai isoskelèc an kai mìno an z + z =. 84. Na brejeð o gewmetrikìc ( tìpoc ) twn eikìnwn ) twn migadik n arijm n z gia touc opoðouc isqôei Re z z+ =Im( z z Na breðte ton gewmetrikì tìpo twn eikìnwn twn migadik n arijm n z gia touc opoðouc isqôei z z = z + z +8z +8 z. 86. Gia to migadikì arijmì z eðnai gnwstì ìti +z + z = c ìpou c>. Na breðte ton gewmetrikì tìpo thc eikìnac tou z.

20 8.3. To Migadikì EpÐpedo 87. Se k je migadikì arijmì z tou opoðou h eikìna anhkei sthn eujeða y = αx + β antistoiqoôme ton migadikì w = z. Na breðte ton gewmetrikì tìpo thc eikìnac tou w. 88. Na brejeð o gewmetrikìc tìpoc thc eikìnac tou w = iz +3ìtan, kat perðptwsh, isqôei:. H eikìna tou z an kei sthn eujeða x + y =.. z z = 3. z +3i = 4. Re(z) = 5. Im(z) =3 89. Na brejeð o gewmetrikìc tìpoc thc eikìnac tou w = z+i perðptwsh, isqôei: z+ ìtan, kat. (i +3)z +(i 3) z =i. z +3+i = 9. 'Estw ìti w = ( 3 +i) z 5 zi.. An z = α + βi na breðte ta Re (w), Im (w).. Na apodeðxete ìti oi eikìnec sto migadikì epðpedo twn arijm n w an koun sthn eujeða x +3y =. 9. Na lôsete thn skhsh 4 qrhsimopoi ntac gewmetrik epiqeir mata. 9. Oi arijmoð z,z èqoun thn idiìthta z + z z z. Poiìc eðnai o gewmetrikìc tìpoc thc eikìnac tou ginomènou touc? 93. Apì tic exet seic tou 993, Dèsmh I. DÐnetai h sun rthsh f (z) = (z ) ( z +) z + z z C kai Re (z).. Na apodeðxete ìti f ( z ) = f (z). Na breðte to eðdoc thc kampôlhc sthn opoða an koun ta shmeða M (x, y) gia ta opoða oi migadikoð arijmoð z = αx + βyi me α, β, x, y R kai αβx ikanopoioôn th sqèsh Re [f (z)] =

21 Kef laio. MigadikoÐ ArijmoÐ BreÐte poiìc apì touc migadikoôc pou ikanopoioôn thn sqèsh z +i = èqei to mègisto kai poiìc to el qisto mètro? jz +ij = 95. Apì tic exet seic tou 994, Dèsmh I. JewroÔme touc migadikoôc arijmoôc z,w kai w tètoiouc ste w = z zi kai w = α +αi, α R. Na deðxetec ìti an to α metab lletai sto R kai isqôei w = w, tìte h eikìna P tou z sto migadikì epðpedo kineðtai se mða uperbol. 96. 'Estw P, Q oi eikìnec twn migadik n z,z. Na apodeðxete ìti an to trðgwno OPQ eðnai isìpleuro tìte ja isqôei z + z = z z. 97. Na apodeðxete ìti an α, β R me α >βtìte oi eikìnec twn migadikwn z pou epalhjeôoun thn exðswsh z z = az + a z + b an koun se kôklo. 98. Apì tic exet seic tou 8. An gia touc migadikoôc arijmoôc z kai w isqôoun ( i + ) z =6 kai w ( i) = w (3 3i) tìte na breðte:. to gewmetrikì topo twn eikìnwn twn migadik n arijm n z.. to gewmetrikì topo twn eikìnwn twn migadik n arijm n w. 3. thn el qisth tim tou w. 4. thn el qisth tim tou z w. 99. Apo to diagwnismì Putnam, 959. Na apodeðxete ìti an oi eikìnec twn, di forwn, migadik n arijm n z kai z eðnai korufèc isopleôrou trig nou tìte h trðth koruf autoô tou trig nou ja eðnai h eikìna tou migadikoô arijmoô ωz ω z ìpou ω eðnai mða mh pragmatik rðza thc exðswshc x 3 =.

22 .3. To Migadikì EpÐpedo. 'Estw dôo stajeroð migadikoð p, q me Ðsa mètra kai h sun rthsh f (z) = pz + q z, z C. Na apodeðxete ìti oi eikìnec twn f (z) an koun se mða eujeða.. O kôkloc tou ApollwnÐou. 'Estw z,z dôo di foroi arijmoð. Na apodeðxete ìti an z z z z = c ìpou c jetikìc pragmatikìc arijmìc di foroc toutìte o gewmetrikìc tìpoc thc eikìnac tou z eðnai kôkloc me kèntro thn eikìna tou z c c z c kai aktðna c z z c Apoll nioc o PergaÐoc 6-7 p.q.. Apì ta Mathematical Tripos tou Cambridge, 998. 'Estw mða eujeða L kai èstw α o migadikìc pou antistoiqeð sto Ðqnoc thc k jethc pou getai apì thn arq twn axìnwn O sthn L. Na apodeðxete ìti h exðswsh thc L gr fetai z = α + λαi kai epomènwc sth morf zᾱ + α z =αᾱ Sth sunèqeia na breðte thn exðswsh thc kampôlhc pou prokôptei an sta shmeða thc L efarmosjeð o metasqhmatismìc z z. Na d sete gewmetrik ermhneða. 3. Apì ta Mathematical Tripos tou Cambridge,. Gia k je jetikì pragmatikì arijmì k orðzoume to sônolo S k = { z C : } z z + = k. Na apodeðxete ìti gia k je k oi eikìnec twn migadik n tou sunìlou S k parist noun kôklo tou opoðou na upologðsete to kèntro kai thn aktðna.. Na d sete akìmh mða pl rh perigraf tou sunìlou twn eikìnwn tou S. 3. 'Estw h sun rthsh f : C C me f (z) = z. Na d sete mða gewmetrik perigraf thc f kai na apodeðxete ìti f (S k )=S k 4. 'Estw α C me α <. Gia k je shmeðo tou migadikoô epipèdou to opoðo eðnai eikìna tou migadikoô arijmoô z èna apì ta parak tw jaisqôei: z α < z α = z α > ᾱz ᾱz ᾱz Epomènwc to migadikì epðpedo qwrðzetai se trða qwrða an loga me to poi sqèsh alhjeôei gia ta shmeða touc. Na perigr yete aut ta qwrða. Upodeixh: IsqÔei z α ᾱz = z α ᾱz = ᾱ z α. L bete up' ìyin thn skhsh z ᾱ

23 Kef laio. MigadikoÐ ArijmoÐ.3.3 G' OMADA 5. Oi eikìnec twn migadik n arijm n z, z+, z eðnai shmeða suneujeiak. Na breðte ton gewmetrikì tìpo thc eikìnac tou z. 6. Na apodeðxete ìtihexðswsh tou kôklou z z = ρ eðnai isodônamh me thn z z =Re(z z )+ρ z 7. JewroÔme touc ana dôo di forouc migadikoôc arijmoôc z,z,z 3,z 4 pou oi eikìnec touc an treic den an koun sthn Ðdia eujeða. Na apodeðxete ìti oi eikìnec touc eðnai shmeða omokuklik an kai mìno an isqôei KlaÔdioc PtolemaÐoc Upodeixh: Sundu ste tic ask seic (8), (87) (z z )(z 3 z 4 ) (z z 4 )(z 3 z ) R 8. To Je rhma PtolemaÐou-Euler. JewroÔme touc migadikoôc z, z, z 3, z 4.. Na apodeðxete ìti Leonhard Euler (z z )(z 3 z 4 )+(z z 4 )(z z 3 )=(z z 3 )(z z 4 ). Na apodeðxete ìti (z z )(z 3 z 4 ) + (z z 4 )(z z 3 ) (z z 3 )(z z 4 ) 3. Na apodeðxete ìti h isìthta thn prohgoômenh anisìthta isqôei an kai mìno an oi eikìnec twn z,z,z 3,z 4 eðnai shmeða omokuklik. August Ferdinard Möbius Upodeixh: DeÐte kai thn skhsh O Metasqhmatismìc tou Möbius. H apeikìnish z αz+β γz+δ onom zetai metasqhmatismìc tou Möbius. Na apodeðxete ìti an αδ βγ tìte o metasqhmatismìc Möbius apeikonðzei eujeðec - kôklouc se eujeðec kôklouc.. 'Estw ìti eikìna tou migadikoô arijmoô z an kei ston kôklo me exðswsh x + y + Ax + By + C. 'Estw m = z.. Na apodeðxete ìti m = (ARe (z)+b Im (z)+c). Na apodeðxete ìti m B + ( B C A B B 4) m + B C 3. Na apodeðxete ìti h mègisth kai h el qisth tim tou z eðnai oi (A + B ) C ± (A + B )(A + B 4C)

24 .4. Ask seic se ìlo to kef laio.4 Ask seic se ìlo to kef laio. Up rqei rage akèraioc n ètsi ste ( + i) n =(+i) n+?. Gia touc migadikoôc arijmoôc z,z eðnai gnwstì ìti z = z =. Na apodeðxete ìti an z z + z z = α me α R tìte isqôei z z + αz z = 3. Apì tic exet seic tou 3. DÐnontai oi migadikoð arijmoð z = α + βi, ìpouα, β R kai w =3z i z +4, ìpou z eðnai o suzug c tou z.. Na apodeðxete ìti Re (w) =3α β +4 Im (w) =3β α. Na apodeðxete ìti, an oi eikìnec tou w sto migadikì epðpedo kinoôntai sthn eujeða me exðswsh y = x, tìte oi eikìnec tou z kinoôntai sthn eujeða me exðswsh y = x. 3. Na breðte poioc apì touc migadikoôc arijmoôc z, oi eikìnec twn opoðwn kinoôntai sthn eujeða me exðswsh y = x, èqei to el qisto dunatì mètro. 4. ν migadikoð arijmoð an koun se stajerì kôklo me kèntro thn arq twn axìnwn kai èqoun stajerì jroisma. Na apodeðxete ìti kai h eikìna tou ajroðsmatoc twn antistrìfwn touc an kei epðshc se stajerì kôklo. 5. Na apodeðxete ìti an gia touc migadikoôc arijmoôc z,z,..., z ν isqôei z i z + i <, z i z + i <,..., z ν i z ν + i < tìte isqôei kai z + z z ν i z + z z ν + i < 6. Na perigr yete to sônolo twn migadik n z gia touc opoðouc isqôei < z < + z 7. Na perigr yete to sônolo twn migadik n z gia touc opoðouc to pragmatikì mèroc tou arijmoô ( + i) z eðnai jetikì.

25 Kef laio. MigadikoÐ ArijmoÐ 3 8. Na apodeiqjeð ìti an gia èna mhgadikì arijmì z up rqoun migadikoð arijmoð u, w me u = w ( ) ste z = u + w ( ) tìte z ( ) all kai antistrìfwc an gia ton z isqôei h (***) tìte up rqoun migadikoð arijmoð u, w ste na isqôoun oi (*), (**). 9. Apì tic exet seic tou 984, Dèsmh I. 'Estw z o migadikìc arijmìc x + yi me y (x, y pragmatikoð arijmoð). Jètoume ω = z z, ìpou z o suzughc tou z. Na apodeðxete ìti o ω eðnai pragmatikìc arijmìc, e n kai mìno an to shmeðo (x, y), wc proc èna orjokanonikì sôsthma anafor c xoy, an kei se mða uperbol apì thn opoða èqoun exairejeð oi korufèc thc.. Na breðte ìlouc touc migadikoôc z gia touc opoðouc isqôei z + z =. Na apodeðxete ìti an z <, z < tìte z z < z z. Gia touc z,z C isqôei z + z z + z =. DeÐxte ìti z = z. 3. Gia touc z,z C isqôei z + z z +3z =. Na upologðsete to z z. 4. 'Estw oi migadikoð arijmoð z = + 3i, z = +i. Na breðte pragmatikoôc arijmoôc κ, λ tètoiouc ste z + z = κz + λz. 5. Na apodeðxete ìti an α R kai z = z =tìte isqôei z + z + αz z = z + z z z + α 6. Na breðte touc migadikoôc w gia touc opoðouc isqôei w = 5 i. 7. JewroÔme ìlouc touc migadikoôc arijmoôc z gia touc opoðouc up rqei akèraioc m ste z m =(z +) m. Na apodeðxete ìti oi eikìnec ìlwn aut n twn arijm n an koun sthn Ðdia eujeða. 8. Na apodeðxete ìti an h eikìna tou migadikoô arijmoô an kei ston kôklo me kèntro to K (, ) kai aktðna tìte oi migadikoð arijmoð z ν+ z ν, z ν ìpou ν fusikìc, èqoun Ðsa mètra.

26 4.4. Ask seic se ìlo to kef laio 9. 'Estw oi an dôo di foroi migadikoð z,z,z 3,z 4. Na apodeðxete ìti an dôo apì touc arijmoôc z z 3 z z 4, z 3 z z z 4, z z z 3 z 4 eðnai fantastikoð tìte ja eðnai kai o trðtoc. 3. Na apodeðxete ìti an z + z + z 3 = kai z + z + z 3 = tìte z = z = z Gia poia tim tou p R to pragmatikì mèroc tou migadikoô arijmoô z = +i +pi + +3i 3+i eðnai Ðso me to fantastikì tou mèroc? 3. Na apodeðxete ìti an gia touc an dôo di forouc migadikoôc arijmoôc z,z,z 3 isqôei z + z + z 3 = z z + z z 3 + z 3 z tìte oi eikìnec touc eðnai korufèc isopleôrou trig nou. 33. Me th bo jeia thc skhshc.4 na apodeðxete ìti an α, β, γ eðnai treðc ana dôo di foroi migadikoð arijmoð oi opoðoi ikanopoioôn mða opoiad pote apì tic sunj kec thc skhshc 7 tìte oi eikìnec touc sto migadikì epðpedo eðnai korufèc isopleôrou trig nou. 34. Na upologðsete to embadìn tou trig nou tou opoðou korufèc eðnai oi eikìnec twn migadik n z =+3i, z = +i kai z z. 35. Na apodeðxete ìti an λ eðnai jetikìc pragmatikìc kai z,z migadikoð tìte isqôei ( z + z ( + λ) z + + ) z λ 36. Na breðte ton gewmetrikì tìpo twn eikìnwn twn migadik n arijm n z gia touc opoðouc isqôei z + z + =4 37. Na apodeðxete ìti gia k je migadikì arijmì z isqôei: ( z + z ) + (z + z)(z z) 38. Na epalhjeôsete tic isìthtec Re (iz) = Im (z) Im (iz) =Re(z) ( ) z z = z 39. 'Estw ìti gia to migadikì arijmì z isqôei z z <αkai z z <β. Na apodeðxete ìti z z +z < α+β. i

27 Kef laio. MigadikoÐ ArijmoÐ 5 4. Na apodeðxete ìti gia k je tri da migadik n arijm n z,z,z 3 isqôei z + z + z 3 + z + z + z 3 = z + z + z + z 3 + z 3 + z 4. Na breðte to gewmetrikì tìpo twn eikìnwn twn arijm n z gia touc opoðouc isqôei ( z i)( z +i) =(Re(z)+) 4. 'Estw ìti w = a+z a z, a R. Na apodeðxete ìti an to z diatrèqei ton xona twn fantastik n tìte to w diatrèqei ton kôklox + y =. 43. Na apodeðxete ìti an h eikìna tou migadikoô arijmoô z gr fei kôklo kèntrou O kai aktðnac α tìte h eikìna tou w = z + z gr fei thn èlleiyh me exðswsh x (+α ) + y ( α ) =. 44. 'Estw oimigadikoð arijmoð z,z,z 3 me z 3 =( λ) z +λz me λ [, ]. Na apodeðxete ìti z z 3 + z 3 z = z z. 45. Na apodeðxete ìti an o arijmìc 3 (z + z) + i (z z) eðnai fantastikìc tìte z i JewroÔme thn sun rthsh f : C C me f (z) = z + z. PoioÔc migadikoôc apeikonðzei h f se pragmatikoôc? 47. Gia touc migadikoôc arijmoôc z,z,z 3 me z z z 3 = z + z / R eðnai gnwstì ìti Na apodeðxete ìti oi eikìnec touc kai h arq twn axìnwn eðnai shmeða omokuklik. Upodeixh: DeÐte thn skhsh (87) 48. Na apodeðxete ìti wz w z = ( w ) ( z ). 49. An z 953 =tìte o arijmìc z 97 + z 97 eðnai pragmatikìc. GiatÐ? 5. Na apodeðxete ìti an (u +) ν + u ν = (ν jetikìc akèraioc) tìte Re (u) =. 5. Na breðte ton gewmetrikì tìpo thc eikìnac tou z an eðnai gnwstì ìti z + ki + z ki =k, k > 5. Na apodeðxete ìti an z + z =... = z ν tìte o arijmìc z + z z ν + z + z z ν effloson orðzetai eðnai pragmatikìc. 53. Na apodeðxete ìti an isqôei

28 6.4. Ask seic se ìlo to kef laio z + z z ν = z = z =... z ν = tìte isqôei: z z + z z z z ν ν 54. Na apodeðxete ìti an Oi λ,λ,..., λ ν eðnai mh arnhtikoð pragmatikoð me λ + λ λ ν = z <, z <,... z ν < tìte isqôei: λ z + λ z λ ν z ν < 55.. Na apodeðxete ìti z z + z z 3 + z 3 z + z + z + z 3 =3 ( z + z + z 3 ). Na apodeðxete ìti an z = z = z 3 =kai z + z + z 3 =3tìte z = z = z Apì touc migadikoôc z pou ikanopoioôn thn sqèsh z+ iz+ =poioc èqei to el qisto mètro? 57. Na apodeðxete ìti an oi eikìnec twn z,w isapèqoun apì thn arq twn ( ),ef'ìson axìnwn tìte o arijmìc z+w z w orðzetai, eðnai pragmatikìc. 58. Omigadikìc arijmìc z den eðnai pragmatikìc oôte fantastikìc kai isqôei R. DeÐxte ìti z =. 6z 4 +5z +6 3z 4 +z Na apodeðxete ìti an h eikìna tou migadikoô arijmoô z = x + yi an kei sto eswterikì tou kôklou me kèntro kai aktðna tìte h eikìna tou w = x + y i an kei sto eswterikì tou trig nou me korufèc ta shmeða O (, ), A (, ), B (, ). 6.. Na apodeðxete ìti an oi arijmoð z, z, z 3 èqoun pragmatikì mèroc mikrìtero Ðso tou tìte to pragmatikì mèroc tou z eðnai.. Na apodeðxete ìti an z + kai z + kai z 3 + tìte z =. 6. Na apodeðxete ìti oi eikìnec twn migadik n z,z,z 3 eðnai shmeða suneujeiak an kai mìno an isqôei z z + z z 3 + z 3 z R. Upodeixh: DeÐte kai thn skhsh (8).

29 Kef laio. MigadikoÐ ArijmoÐ 7 6. Apì èna jèma tou diagwnismoô Putnam tou 947. 'Estw f (z) =z + az + b me a, b C. Na apodeiqjeð ìti an f (z) =gia k je z me z =tìte isqôei a = b =. Upodeixh: Ja prèpei f () = f ( ) =kai f (i) = f ( i) =. 63. Apì tic exet seic tou 5. DÐnontai oi migadikoð arijmoð z, z, z 3 me z = z = z 3 =3.. DeÐxte ìti z = 9 z.. DeÐxte ìti o arijmìc z z + z z eðnai pragmatikìc. 3. DeÐxte ìti: z + z + z 3 = 3 z z + z z 3 + z 3 z 64. Gia touc pragmatikoôc arijmoôc α, γ isqôei <γ<α. JewroÔme ìlouc touc migadikoôc arijmoôc z gia touc opoðouc isqôei z + γ + z γ =α ( ) Poia eðnai h mikrìterh kai poia h megalôterh tim pou mporeð na p rei to mètro tou z? 65. 'Estw ìti z = +ti ti kai z = t+i t i, t R.. BreÐte ta mètra twn z,z. DeÐxte ìti oi z,z èqoun to Ðdio fantastikì mèroc. 3. DeÐxte ìti an gia ton jetikì akèraio m isqôei z m + zm = zm + zm tìte o m eðnai rtioc. 66. BreÐte migadikoôc a, b, c pou èqoun mètro kai a + b + c = abc =. Upodeixh: DeÐxte ìti ab + bc + ca =kai ìti (x a)(x b)(x c) =x 3 x + x. 67. Gia to migadikì arijmì z eðnai gnwstì ìti isqôei z 3 + z 3 =6.. Na apodeðxete ìti z.. MporeÐ rage na eðnai z =. 68. Apì tic exet seic tou 6. DÐnontai oi migadikoð arijmoð z,z,z 3 me z = z = z 3 =kai z + z + z 3 =.. Na apodeðxete ìti: (a ) z z = z z 3 = z 3 z (b ) z z 4 kai Re (z z ) Na breðte to gewmetrikì tìpo twn eikìnwn twn z,z,z 3 sto migadikì epðpedo kaj c kai to eðdoc tou trig nou pou autèc sqhmatðzoun.

30 8.4. Ask seic se ìlo to kef laio 69. DÐnontai oi migadikoð arijmoð z,z,z 3 me z = z = z 3 = kai z + z + z 3. Na apodeðxete ìti:. z z +z z 3 +z 3 z z +z +z 3 =. (z +z )(z +z 3 )(z 3 +z ) z z z 3 R 7. 'Estw λ R stajerìc me λ. Na lôsete sto sônolo twn migadik n thn exðswsh: w + λ w + λ + i = 7. 'Estw oimigadikoð arijmoð u kai v = 3 i u.. Upojètoume ìti u =. DeÐxte ìti h eikìna tou v an kei se kôklo èstw C.. Upojètoume ìti h eikìna tou u an kei sthn eujeða me exðswsh y = x. DeÐxte ìti h eikìna tou v an kei se eujeða èstw ε h opoða ef ptetai ston kôklo C tou erwt matoc (a'). 7. Na apodeðxete ìti oi eikìnec ìlwn twn migadik n thc morf c z = +mi mi, m R ( ) an koun ston monadiaðo kôklo (kèntro h arq twn axìnwn, aktðna ). Na exet sete an oi parap nw migadikoð arijmoð exantloôn} ton kôklo dhlad an k je shmeðo tou monadiaðou kôklou eðnai eikìna k poiou migadikoô thc morf c ( ) 73. Up rqoun rage stajeroð migadikoð α, β ètsi ste gia k je z C na isqôei z = αz + β? 74. Na apodeiqjeð ìti gia k je migadikì arijmì z isqôei +z +z + z 75. Na apodeiqjeð ìti gia k je migadikì arijmì z isqôei +z + +z + z Upodeixh: Na diakrðnete tic peript seic z kai z <. Sthn deôterh na parathr sete ìti +z + +z + z +z z + +z + z 76. Na apodeðxete ìti z + α z + α + z + β z + β

31 Kef laio. MigadikoÐ ArijmoÐ Gia touc migadikoôc arijmoôc z,z,z 3 eðnai gnwstì ìti oi eikìnec touc an koun ston kôklo C me kèntro thn arq twn axìnwn kai aktðna ρ. Na apodeðxete ìti an o migadikìc arijmìc w = z z + z z 3 + z 3 z z + z + z 3 orðzetai tìte h eikìna tou epðshc an kei ston C 78.. Gia to migadikì arijmì z isqôei Re ( ) ( ) z + i z + i =Im z i z i ( ) Na breðte to gewmetrikì tìpo L thc eikìnac tou z.. Gia to migadikì arijmì w isqôei ( ) ( ) w w Re =Im w + w + ( ) Na breðte to gewmetrikì tìpo M thc eikìnac tou w. 3. Na breðte touc migadikoôc pou ikanopoioôn kai tic dôo sqèseic ( ), ( ). 4. 'Enac migadikìc z ikanopoieð thn sqèsh ( ) kai ènac migadikìc w ikanopoieð thn sqèsh ( ). Poia eðnai h mègisth kai poia h el qisth tim pou mporeð na p rei to mètro tou z w? 79. Na apodeðxete ìti an oi eikìnec twn z,z,z 3 orðzoun trðgwno tìte to barôkentro tou eðnai h eikìna tou z +z +z 3 3. DeÐxte akìmh ìti an, epiplèon, oi eikìnec twn z,z,z 3 an koun ston monadiaðo kôklo tìte to orjìkentro tou trig nou eðnai h eikìna tou z + z + z 'Estw H to sônolo ìlwn twn migadik n pou èqoun mètro kai z,w H. Na apodeðxete ìti ja isqôei z + w zw += an kai mìno an isqôei z + w + zw =. BreÐte touc z,w an eðnai gnwstì ìti (z ) (w ) = 8. Se k je pragmatikì arijmì t antistoiqoôme ton migadikì arijmì f (t) = +ti ti. DeÐxte ìti. f (t) = f(t) =. 'Estw ìti t + t = kai t t =. Na breðte thn apìstash twn eikìnwn twn f (t ),f(t ).

32 3.4. Ask seic se ìlo to kef laio 8. Na apodeðxete ìti an z + z +=tìte: α 3 + β 3 + γ 3 3αβγ =(α + β + γ) ( α + βz + γz )( α + βz + γz ) 83. Na apodeðxete ìti an i m = i n (m, n N) tìte i m +4n = i n +4m. 84. EÐnai z =. BreÐte to z (z i) +iz. 85. Na breðte ton gewmetrikì tìpo twn eikìnwn twn migadik n z gia touc opoðouc isqôei z + =. z Upodeixh: ( x + y ) ( 4 x + y ) + ( x y ) += ( x y + y )( x +y + y ) 86. Na apodeðxete ìti an z =tìte Im ( z (z+) ) =. 87. Na apodeðxete ìti gia k je zeôgoc migadik n arijm n z,z isqôoun ta akìlouja: (. z )( z ) =(+z z )(+ z z ) (z + z )( z + z ). z z (+ z )( + z ) 88. An 6z + = 4z breðte to z Gia touc migadikoôc z,z eðnai gnwstì ìti z = z =kai (z z ) = 4. Na apodeðxete breðte touc z,z. 9. Apì tic exet seic tou 96. (Arqitektonik JessalonÐkhc) DÐdetai h exðswsic x +(λ ) x +8λ 4λ = Na orisjeð o λ ste to ρ + ρ na eðnai mègiston. Epexhghsh: To λ jewreðtai pragmatikìc kai ρ,ρ eðnai oi rðzec thc exðswshc. 9. 'Estw ìti z = r kai z = r me < r < r. Na breðte thn megalôterh kai thn mikrìterh timh pou mporeð na p rei h par stash: A = z z + z + z ( ) ( ) ( ) 9. 'Estw ìti z = z = z 3 kai Re z z =Re z z 3 =Re z3 z = m. Na brejeð o m. 93. 'Estw hexðswsh x + αx + β = ìpou oi α, β eðnai pragmatikoð arijmoð. Na breðte poièc sqèseic prèpei na ikanopoioôn oi α, β ètsi ste h exðswsh na èqei m pragmatikèc rðzec me mètro mikrìtero tou.

33 Kef laio. MigadikoÐ ArijmoÐ Na apodeðxete ìti an isqôei z + z =tìte ja isqôei Re (z z ). 95. Gia ton migadikì arijmì z isqôei z + = z. Na breðte thn mègisth kai thn el qisth tim tou mètrou tou z. Upodeixh: DeÐxte pr ta ìti gia z isqôei (z + z) +(z + z)+ z 4 3 z =.Mex =Re(z) èqoume mða exðswsh wc proc x pou prèpei na èqei pragmatikèc lôseic. 96. 'Estw z stajerìc migadikìc kai α stajerìc pragmatikìc. Na breðte ton gewmetrikì tìpo twn eikìnwn twn migadik n arijm n z gia touc opoðouc isqôei Re (z z)=α. 97. 'Estw ìti z =. Na apodeiqjeð ìti 4 z + z 'Estw ìti gia touc migadikoôc arijmoôc z, w isqôoun oi sqèseic: Re (( + i) z) =, wz = i. Na breðte to gewmetrikì tìpo twn eikìnwn twn z,w. Na breðte poiìc apì touc migadikoôc z èqei to mikrìtero dunatì mètro. MporoÔme na k noume to Ðdio gia touc migadikoôc w? 99. Apì tic exet seic tou 9. JewroÔme touc migadikoôc arijmoôc z =(λ +)+(λ ) i, λ R. (a ) Na breðte thn exðswsh thc eujeðac p nw sthn opoða brðskontai oi eikìnec twn migadik n arijm n z, gia tic di forec timèc tou λ R. (b ) Apì touc parap nw migadikoôc na apodeðxete ìti o migadikìc arijmìc z èqei to mikrìtero dunatì mètro.. Na brejoôn oi migadikoð arijmoð w oi opoðoi ikanopoioôn thn exðswsh w + w = z ìpou z = i = omigadikìc arijmìc pou anafèretai sto prohgoômeno er thma. 3. 'Estw mða sun rthsh ϕ : C C me tic akìloujec idiìthtec: ϕ (z + w) =ϕ (z)+ϕ (w) gia ìla ta z,w C ϕ (zw) =ϕ (z) ϕ (w) gia ìla ta z,w C ϕ (z) =z gia ìla ta z R

34 3.4. Ask seic se ìlo to kef laio Na apodeðxete ìti ja isqôei ϕ (z) =z gia ìla ta z C eðte ja isqôei ϕ (z) = z gia ìla ta z C. 3. 'Estw mða sun rthsh α : C R me tic akìloujec idiìthtec: α (z) =z gia mh arnhtikì pragmatikì z α (z z )=α (z ) α (z ) gia k je zeôgoc migadik n z,z. Up rqei jetikìc pragmatikìc M ste α (z) M gia k je z me z = Na apodeðxete ìti α (z) = z gia k je z. 3. Apì tic exet seic tou. DÐnetai h exðswsh ìpou z C me z. z + z =. Na breðte tic rðzecz kai z thc exðswshc.. Na apodeðxete ìti z + z = 3. An gia touc migadikoôc arijmoôc w isqôei w 4+3i = z z tìte na breðte to gewmetrikì tìpo twn eikìnwn twn w sto migadikì epðpedo. 4. Gia touc migadikoôc arijmoôc w tou erwt matoc 3, na apodeðxete ìti 3 w t7 33. Na apodeðxete ìti an o ν > eðnai fusikìc kai isqôei (x + i) ν + (x i) ν =tìte o x eðnai pragmatikìc. 34. 'Estw a, b C me a b = a = b >. BreÐte thn tim thc par stashc ( a ) ( ) 4 b 4 +. b a 35. Gia touc migadikoôc arijmoôc z, z isqôei: z z z 3 = z z 3 kai z z z 3 = z z 3 Na apodeiqjeð ìti z = z.

35 Kef laio. MigadikoÐ ArijmoÐ Apì tic exet seic tou. 'Estw oi migadikoð arijmoð z kai w me z 3i, oi opoðoi ikanopoioôn tic sqèseic: kai z 3i + z +3i = w = z 3i + z 3i. Na breðte ton gewmetrikì tìpo twn eikìnwn twn migadik n arijm n z.. Na apodeðxete ìti z 3i = z 3i 3. Na apodeðxete ìti o w eðnai pragmatikìc arijmìc kai ìti w. 4. Na apodeðxete ìti: z w = z 37. Apì tic exet seic tou. JewroÔme touc migadikoôc arijmoôc z kai w gia touc opoðouc isqôoun oi epìmenec sqèseic: z + z + =4 () w 5 w = (). Na apodeðxete ìti o gewmetrikìc tìpoc twn eikìnwn twn migadik n arijm n z sto epðpedo eðnai kôkloc me kèntro thn arq twn axìnwn kai aktðna ρ =.. An z, z eðnai dôo apì touc parap nw migadikoôc arijmoôc z me z z = tìte na breðte to z + z. 3. Na apodeðxete ìti o gewmetrikìc tìpoc twn eikìnwn twn migadik n arijm n w sto epðpedo eðnai h èlleiyh me exðswsh x 9 + y 4 =kai sth sunèqeia na breðte th mègisth kai thn el qisth tim tou w. 4. Gia touc migadikoôc arijmoôc z, w pou epalhjeôoun tic sqèseic () kai () na apodeðxete ìti: zw Na sqedi sete ton gewmetrikì tìpo thc eikìnac tou z stic akìloujec peript seic:. Re (z) =. Re ( z ) = 3. Re (z) + Im (z) =

36 34.4. Ask seic se ìlo to kef laio 39. DÔo mh par llhlec eujeðec tèmnoun ton kôklo z =sta shmeða pou eðnai eikìnec twn migadik n a, b kai c, d antistoðqwc. Na apodeiqjeð ìti to koinì shmeðo touc eðnai h eikìna tou migadikoô w = a + b c d ab cd

37 Kef laio 'Oria kai Sunèqeia Sunart sewn. Sunart seic. Pr xeic sunart sewn.. A' OMADA 3. Na exet sete an o arijmìc eðnai tim thc sun rthshc f (x) =x 4x 'Estw h sun rthsh f (x) = { x x< x + x Na upologðsete thn tim thc par stashc A = f () + f ( ) + f (f ( )) 3. 'Estw h sun rthsh g (x) =7x 3 x +. DeÐxte ìti g (x) g (x) = x ( 49x ). 33. 'Estw h sun rthsh f (x) =x 3x +. Na apodeðxete ìti gia k je h isqôei f ( 3 + h) = f ( 3 h). 34. 'Estw h sun rthsh f (x) =x + ( x. Na) epalhjeôsete thn isìthta f (x + h) f (x h) =h + (x+h)+ (x h) 35. Gia thn sun rthsh f (x) =αx +βx+γ isqôei f () = 3, f ( ) = 3, f () =. Na breðte ta α, β, γ. 36. Apì ta jèmata tou diagwnismoô Flanders, 99. 'Estw An y = f(x) tìte to x eðnai Ðso me: f (x) =, (x ) x 35

38 36.. Sunart seic. Pr xeic sunart sewn. f(y). f( f(y) y ) 3. y 4. yf(y) 5. yf(y) 37. Poio eðnai to pedðo orismoô twn sunart sewn. f (x) = x + 3 x. g (x) = ln x x x Na breðte ta pedða orismoô twn parak tw sunart sewn:. f (x) =x 4x +3. ϕ (x) = x 4x+3 3. g (x) = x 4x h (x) = x 4x+3 5. ω (x) =ln ( x 4x +3 ) 6. ψ (x) = ln(x 4x+3) 39. Na breðte ta pedða orismoô twn sunart sewn x. f (x) =. g (x) = x x x 3. Poia apì ta shmeða A (, ), B( 3, 6), Γ ( t, 3 t ) an koun sth grafik par stash thc sun rthshc f (x) = x+ x+? 3. Gia poia tim tou t to shmeðo M (t, ) an kei sth grafik par stash thc f (x) =x 3 3x? 3. H grafik par stash thc sun rthshc f (x) =αx + x dièrqetai apì to shmeðo M (, 4). Poioc eðnai o α? 33. H grafik par stash thc sun rthshc f (x) = αx + x + α dièrqetai apì to shmeðo A(, 7). Na breðte to f(7). 34. Na k nete tic grafikèc parast seic twn sunart sewn. f (x) = { x, x < x +, x. f (x) = { x, x < x +, x > 35. DÐnontai oi sunart seic f (x) =x + kai g (x) = 5 4 x x twn opoðwn oi grafikèc parast seic tèmnontai sta A, B.. Na breðte ta A, B.. Na breðte thn exðswsh thc eujeðac AB. 36. Sta epìmena sq mata na ekfr sete to y sunart sei tou x.

39 Kef laio. 'Oria kai Sunèqeia Sunart sewn 37 ff ff ff y ff ff y ff (I) x ff (II) x ff ff y ff y ff x ff O ff ff x (III) ff (IV) 37. Sta epìmena sq mata breðte thn sun rthsh f. O O (a) (b) 38. Sta epìmena sq mata breðte thn sun rthsh f. O O (a) (b) 39. Sta epìmena sq mata breðte thn sun rthsh f (Sto sq ma (a) h C f apartðzetai apì èna hmikôklio kai dôo hmieujeðec). y O Ο x (a) (b)

40 38.. Sunart seic. Pr xeic sunart sewn 33. Sta epìmena sq mata na ekfr sete to d sunart sei tou x. f (x) = 4 x f (x) = x f (x) M (x; f (x)) d f(x) M (x; f (x)) d (a) O x (b) O x 33. Gia poia tim tou λ h sun rthsh f (x) =(λ )x + λx +me pedðo orismoô to R eðnai Ðsh me thn sun rthsh g (x) =3x +x +poô èqei pedðo orismoô epðshc to R? 33. Na apodeðxete ìti oi sunart seic: f (x) = x 3x + x x + x< +x kai g (x) = x +5x 3 x< x x + x eðnai Ðsec Gia poia tim tou λ to shmeðo M (,λ) an kei sth grafik par stash thc f (x) =x 3 + x +? 334. Sto sq ma apeikonðzetai h grafik par stash thc f (x) = (x +3)(x ) (x 4) 6 Ta shmeða A, B, Γ an koun ston xona x x,ta E,H ston y y kai BΔ ΔZ. E A B (; ) H Z Na breðte tic suntetagmenec twn shmeðwn:

41 Kef laio. 'Oria kai Sunèqeia Sunart sewn 39. E. A, B, Γ 3. Z, H 335. DÐnontai oi sunart seic f (x) = 9 x, g (x) = x 3x +. Na orðsete tic sunart seic f + g, f g, f g To jroisma twn f (x) = x kai g (x) = x den orðzetai. GiatÐ? 337. H sun rthsh f eðnai stajer kai 4f (7) =. Poio eðnai to f ()? 338. 'Estw h sun rthsh f (x) = 4x+ x. Na apodeðxete ìti k je arijmìc di foroc tou eðnai tim thc f. EpÐshc na apodeðxete ìti o den eðnai tim thc f MÐa sun rthsh f eðnai - kai f (x ) = f (9). Poioc eðnai o x? 34. Na apodeðxete ìti oi parak tw sunart seic me pedðo orismoô to R eðnai :. f (x) =3x +. g (x) =3 3x s (x) =x g (x) = x Na apodeðxete ìti oi parak tw sunart seic me pedðo orismoô to R den eðnai :. f (x) = x +. g (x) =x + x + 3. s (x) = x x 4. g (x) =e x 34. Na breðte sun rthsh me pedðo orismoô to R tètoia ste gia ìla ta x na isqôei f (x +)=x Poièc eðnai oi antðstrofec twn parak tw sunart sewn?. f : R R, f (x) =e x. g : R R, g(x) = x 3. h :[, + ) R, h(x) =x 344. Na orðsete thn sônjesh g f stic akìloujec peript seic:. f (x) =+ x, g (x) = + x. f (x) =x + x, g (x) = x 5+ x 3. f (x) =x 3 x x +, g (x) =lnx 345. DÐnontai oi sunart seic f, g me f (x) = x + kai g (x) =x + kai pedðo orismoô to D. EÐnai oi sunart seic Ðsec ìtan:

42 4.. Sunart seic. Pr xeic sunart sewn. D = R?. D =(, )? 3. D =(, 4)? 346. Sto epìmena sq mata up rqoun oi grafikèc parast seic twn sunart sewn f kai g, h. Na exet sete an èqoun mègisto el qisto. C f Cg C h 347. JewroÔme dôo sunart seic f, g orismènec sto R kai thn sun rthsh h (x) =Im ((f (x)+ig (x)) ) Na apodeðxete ìti k je rðza thc h eðnai kai rðza thc f thc g DeÐxte ìti h sun rthsh g(x) = x eðnai antistrèyimh. Met na breðte ta koin shmeða thn grafik n parast sewn twn C f, C f MÐa aun rthsh f èqei sônolo tim n to di sthma [, ]. Poiì eðnai to sônolo tim n thc sun rthshc g (x) =f (x)+3? 35. H eujeða y =tèmnei thn grafik par stash thc sun rthshc f se dôo shmeða. EÐnai h f antistrèyimh?.. B' OMADA 35. Na breðte ta pedða orismoô twn sunart sewn:. ϕ (x) = x. σ (x) =ln ( e x 5e x +6 ) 35. Gia thn sun rthsh f (x) = (x+)(x+k) x + [, ] isqôei f (x) [, ]. Na breðte to k. eðnai gnwstì ìti gia k je x 353. Gia mða sun rthsh f : R R isqôei f (x)+x e x =xe x f (x) gia ìla ta x. Na breðte ton tôpo thc f Apì ta jèmata tou diagwnismoô Flanders, 99. 'Estw h sun rthsh f :[, + ) R me f(x) =x x. Tìte to sônolo tim n thc eðnai:. [, + ). R 3. [ 4, + )

43 Kef laio. 'Oria kai Sunèqeia Sunart sewn 4 4. K poio sônolo pou perièqetai se di sthma [α, β] me α, β R 5. [, + ) 355. Pwc prèpei na epilegeð o α ste to pedðo orismoô thc sun rthshc f (x) = x α x na perièqei to di sthma I =[, ]? 356. Na breðte sun rthsh me pedðo orismoô to R tètoia ste gia ìla ta x na isqôei f (x +)=x Na breðte sun rthsh me pedðo orismoô to R tètoia ste gia ìla ta x na isqôei f (x +)=e x + x Gia thn sun rthsh f : R R eðnai gnwstì ìti isqôei f (x + y) =f (x)+f (y) gia ìla ta x, y R. Na apodeðxete ìti isqôei:. f () =. f ( x) = f (x) gia k je x R 3. f (μx) =μf (x) gia k je x R kai k je μ N 4. f ( ) x ν = ν f (x) gia k je x R kai k je ν N 5. f (qx) =qf (x) gia k je gia k je q Q me q> 6. f (qx) =qf (x) gia k je gia k je q Q. 7. Me c = f () isqôei f (q) =cq gia k je q Q Gia thn sun rthsh f : R R eðnai gnwstì ìti gia k je x, y isqôei: Na apodeðxete ìti:. f() = f (xy) =f (x)+f (y). Gia k je x isqôei f ( x) = f (x). ( ) 3. Gia k je x, y isqôei f x y = f (x) f (y) 4. An gia to x isqôei x ν = α (ν N) tìte f(α) =νf(x). 5. f( ) = 6. Gia k je x isqôei f( x) =f(x)

44 4.. Sunart seic. Pr xeic sunart sewn 36. 'Estw h sun rthsh f (x) = αx+β. me αγ. Na apodeðxete ìti ta ex c eðnai isodônama: γx+δ. (f f)(x) =x gia k je x δ γ.. α = δ 36. Gia mða sun rthsh f : R R isqôei f g = g f gia ìlec tic sunart seic g : R R. Na apodeðxete ìti f (x) =x gia k je x R 36. DeÐxte ìti h sun rthsh f (x) =x 3 + x a eðnai antistrèyimh. Met breðte ta koin shmeða thc grafik c thc par stashc thc f kai thc antðstrofhc thc Apì ta jèmata tou diagwnismoô Flanders, 99. Sto epðpedo hexðswsh ( x + y ) =4x y orðzei mða kampôlh. To sônolo twn tetagmènwn y twn shmeðwn thc kampôlhc eðnai. [, ]. R + 3. [, ] 4. [, ] 5. R 364. Apì ta jèmata tou diagwnismoô Flanders, 99. 'Estw α, β R kai oi sunart seic f,f,..., f ν,... pou orðzontai wc ex c: f (x) =αx + β An ν tìte f ν = f ν f H sqèsh f 99 = f 99 f 99 mporeð na isqôei:. Gia mða tim tou α kai mða tim tou β.. Gia dôo timèc tou α kai mða tim tou β. 3. Gia treðc timèc tou α kai mða tim tou β. 4. Gia mða tim tou α kai opoiad pote tim tou β. 5. Gia dôo timèc tou α kai opoiad pote tim tou β Gia mða sun rthsh f : R R up rqoun arijmoð κ, λ ètsi ste gia ìla ta x na isqôei (f f)(x) =κx + λ. Na apodeðxete ìti gia ìla ta x isqôei f (κx + λ) =κf (x)+λ 366. MÐa sun rthsh f orismènh se èna di sthma Δ onom zetai kurt an gi k je x, y Δ kai gia k je α, β [, ] me α + β =isqôei: f (αx + βy) αf (x)+βf (y) Na apodeðxete mða sun rthsh f :Δ R eðnai kurt an kai mìno an gia k je tri da arijm n x,x,x 3 Δ me x <x <x 3 isqôei f (x ) f (x ) x x f (x 3) f (x ) x 3 x

45 Kef laio. 'Oria kai Sunèqeia Sunart sewn Sto epìmeno sq ma na ekfr sete to d sunart sei tou x. y = x + d O x f (x) = x 3 3x + M (x; f (x)) 368. Na breðte tic antðstrofec twn parak tw sunart sewn:. f :(, ] R me f (x) =3x 6x +. g (x) =e x+ x + 3. f :(, ] R me h (x) =e x + e x 369. 'Estw mða sun rthsh f : R R. Na apodeðxete ìti h sun rthsh g (x) = f(x)+f( x) eðnai rtia en h h (x) = f(x) f( x) eðnai peritt. B sei toôtou deðxte ìti k je sun rthsh me pedðo orismoô to R mporeð na grafeð wc jroisma mðac rtiac kai mðac peritt c sun rthshc. y = g (x) y = f ( x) y = f (x) y = h (x) 37. 'Estw mða sun rthsh f orismènh se èna di sthma Δ tètoia ste na paðrnei timèc sto di sthma (, ). 'Estw h sun rthsh g (x) = f (x) +f (x) Na apodeðxete ìti an h f eðnai gnhsðwc monìtonh tìte kai h g eðnai gnhsðwc monìtonh me eðdoc monotonðac Ðdio me thc f. 37. Se poièc apì tic epìmenec peript seic h sun rthsh f eðnai gnhsðwc aôxousa sto di sthma Δ

46 44.. Sunart seic. Pr xeic sunart sewn. f (x) =3x, Δ=(, + ). f (x) =x (x +), Δ=( 3, ) 3. f (x) =x +, Δ=(, ) 4. f (x) = x, Δ=(, + ) 5. f (x) =ημx, Δ=(, + ) 6. f (x) =x, Δ=[, ] 37. DÐnontai oi sunart seic f (x) = ex + e x kai g (x) = ex e x (uperbolikì sunhmðtono kai uperbolikì hmðtono). Na apodeðxete ìti:. H f eðnai rtia kai h g peritt.. IsqÔei f(x) >g(x) gia ìla ta x 3. H g eðnai gnhsðwc aôxousa. 4. IsqÔei f (x) g (x) = gia ìla ta x. Tèloc: 5. Na breðte thn antðstrofh thc g Na apodeðxete ìti an h g f eðnai - tìte kai h f eðnai 'Estw f (x) =x+3. Na upologðsete thn par stash: A = f ( f () + f () ) 'Estw ìti h f eðnai gnhsðwc aôxousa. Na apodeðxete ìti an gia k poio x isqôei f ( x x ) f ( ) tìte ja eðnai x = Gia thn sun rthsh f : (, + ) R isqôei f (e x ) = x +. Na apodeðxete ìti gia k je zeôgoc jetik n arijm n α, β isqôei f (αβ) =lnα + lnβ Gia mða sun rthsh f me pedðo orismoô to R isqôei f (x +) = f ( x ) gia k je x. EÐnai antistrèyimh? 378. Poi eðnai h antðstrofh thc f (x) =x ν+, ν =,,...? 379. Me f (x) = α x breðte ton α ste na isqôei f = f.

47 Kef laio. 'Oria kai Sunèqeia Sunart sewn An oi f g kai g eðnai gnhsðwc aôxousec kai g (D g )=D f eðnai rage kai h f gnhsðwc aôxousa? 38. Gia mða sun rthsh f : R R kai gia treic arijmoôc α<β<γeðnai gnwstì ìti (f (β) f (α)) (f (γ) f (β)) <. MporeÐ h f na eðnai gnhsðwc monìtonh? 38. Na apodeðxete leptomer c ìti mða gnhsðwc monìtonh sun rthsh èqei to polô mða rðza Up rqoun rage gnhsðwc monìtonec periodikèc sunart seic? 384. Oi sunart seic f,g,h eðnai orismènec se èna di sthma D kai paðrnoun jetikèc timèc. Oi dôo pr tec eðnai aôxousec kai h trðth eðnai fjðnousa. Na kajorðsete to eðdoc monotonðac twn sunart sewn:. f + g h. fg h, 3. fg +3h 385. Oi sunart seic f,g, eðnai orismènec se èna di sthma D kai paðrnoun jetikèc timèc. Na apodeðxete ìti:. An h f parousi zei mègisto (antistoðqwc: el qisto) sto x tìte h f parousi zei el qisto (antistoðqwc: mègisto) sto x.. An oi f,g parousi zoun el qisto (antistoðqwc: mègisto) sto x tìte kai h f + g parousi zei el qisto (antistoðqwc: mègisto) sto x 386. Me f (x) = x+ x :. Na apodeðxete ìti h f eðnai antistrèyimh kai isqôei f = f.. Na orðsete thn sun rthsh f (f) Na apodeðxete ìti k je gnhsðwc monìtonh sun rthsh eðnai antistrèyimh kai ìti h antðstrofh thc eðnai epðshc gnhsðwc monìtonh me to Ðdio eðdoc monotonðac H grafik par stash miac sun rthshc f eðnai h eujeða pou dièrqetai apì to shmeðo A(, 3) kai èqei suntelest dieujônsewc. AfoÔ breðte ton tôpo thc f na breðte thn f 'Estw f mða gnhsðwc aôxousa sun rthsh. Na apodeðxete ìti ta koin shmeða twn C f kai C f -ef' ìson up rqoun- an koun sthn eujeða y = x. 39. Na apodeðxete ìti h sun rthsh f (x) =x ημ x συνx sto di sthma Δ= [, π ] eðnai gnhsðwc aôxousa. 39. Na apodeðxete ìti h sun rthsh f (x) = x x + x + x +3 eðnai gnhsðwc aôxousa.

48 46.. Sunart seic. Pr xeic sunart sewn 39. Na apodeðxete ìti an mða sun rthsh f parousi zei el qisto sto x kai h g eðnai aôxousa tìte h g f parousi zei kai aut el qisto sto x. Ti ja sunèbaine an h g tan fjðnousa? MporeÐte me to m ti} na breðte to el qisto thc w (x) = ημx +3 ημx? 393. 'Estw h sun rthsh f (x) = { x λ(x ), x λ, x = Na breðte gia poièc timèc tou λ h f eðnai -. Gia autèc tic timèc na brejeð h f Gia thn sun rthsh f eðnai gnwstì ìti gia ìla ta x R isqôei f 3 (x)+f (x)+x = AfoÔ apodeðxete ìti h f eðnai - na breðte thn antðstrofh kai to eðdoc monotonðac thc f MÐa sun rthsh eðnai -. MporeÐ na eðnai rtia? 396. MÐa sun rthsh f :(, ) R eðnai gnhsðwc monìtonh. 'Eqei mègisto? El qisto? Na apodeðxete ìti h sun rthsh f (x) = ( 3 5) x + ( 4 5) x eðnai gnhsðwc fjðnousa.. Na apodeðxete ìti h exðswsh 3 x +4 x =5 x èqei monadik lôsh ton arijmì Oi sunart seic f(x) =ημx kai g(x) =εϕx eðnai gnhsðwc monìtonec sto di sthma (, π ) kai epomènwc antistrèyimec. Upojèste ìti èqete mða arijmomhqan (kompiouter ki) me to opoðo mporeðte na brðskete to f (x) gia k je x. Pwc ja breðte to g (α)? 399. Gia thn sun rthsh f : R R eðnai gnwstì ìti isqôei f (f (x)) = x gia k je x. Na apodeðxete ìti eðnai. Katìpin na apodeðxete ìti f = f. MporeÐte na breðte mða sun rthsh me aut thn idiìthta? 'Allh? 4. Gia thn sun rthsh f : R R eðnai gnwstì ìti isqôei gia k je x. Na breðte thn f. Upodeixh: Na jèsete ìpou x to x. xf (x)+f ( x) =x + x Gia mða sun rthsh f eðnai ( gnwstì) ìti ta shmeðo thc grafik c thc par stashc thc eðnai ta shmeða M t+ t+, t+ t+3, t, 3. Na breðte thn f.

49 Kef laio. 'Oria kai Sunèqeia Sunart sewn 'Estw h oikogèneia sunart sewn f λ (x) =x + λx +, λ R (Bl. sq ma). = = = = =. BreÐte sun rthsh f λ ste h C λ na dièrqetai apì to shmeðo M(, ).. Up rqei rage sun rthsh f λ thc parap nw oikogèneiac pou mèroc thc grafik c thc par stashc na brðsketai k tw apìton xona x x? 43. MÐa sun rthsh ϕ eðnai gnhsðwc aôxousa. Na lôsete thn exðswsh ϕ ( x +4 ) = ϕ (4x) 44. Apì tic exet seic tou 998, Dèsmh I. H sun rthsh f : R R ikanopoieð th sqèsh: f (f (x)) + f 3 (x) =x +3, x R. Na apodeðxete ìti h f eðnai èna proc èna}.. Na lôsete thn exðswsh: f ( x 3 + x ) = f (4 x), x R 45. 'Estw f (x) = x + x. Na apodeðxete ìti h f eðnai gnhsðwc aôxousa.. Na apodeðxete ìti to sônolo tim n thc f eðnai to di sthma (, ). 3. Na breðte thn f.

50 48.. Sunart seic. Pr xeic sunart sewn 46. Gia thn sun rthsh f : R R eðnai gnwstì ìti gia ìla ta x isqôei: x 3 + x +3x + f (x) x 4 x 3 +x +3x + Na breðte ta f() kai f(). 47. 'Estw h sun rthsh f (x) =x + 4x +. Na apodeðxete ìti gia k je x x isqôei ( ) f (x ) f (x ) x + x = + x x 4x ++ 4x +. Na apodeðxete ìti gia k je x x isqôei f(x ) f(x ) x x > 3. Na apodeðxete ìti h f eðnai antistrèyimh. 4. Na breðte thn antðstrofh thc f. 48. 'Estw f :Δ R mða gnhsðwc monìtonh sun rthsh ìpou to Δ eðnai èna anoiktì di sthma. Na apodeðxete ìti h f den èqei oôte mègisto oôte el qisto. 49. H sun rthsh f (x) =e x den èqei el qisto. GiatÐ? 4. Na apodeðxete ìti an h sun rthsh f èqei mègisth tim to M tìte gia k je x,x D f ja isqôei f(x )+f(x ) M. 4. 'Estw ìti mða sun rthsh eðnai orismènh se anoiktì di sthma kai èqei mègisth tim. Na apodeðxete ìti de eðnai monìtonh. 4. Gia poia tim tou t oi sunart seic f (x) = ( t)x +t x+3 t kai g (x) = (t )x + x+t eðnai Ðsec? 43. JewroÔme dôo sunart seic f,g orismènec sto R. 'Estw I h tautotik sun rthsh tou R (dhlad I(x) =x gia ìla ta x).. Na apodeðxete ìti an g f = I tìte h f eðnai.. Na apodeðxete ìti an f g = I tìte to sônolo tim n thc f eðnai to R. 3. Na apodeðxete ìti an g f = I, f g = I tìte h f eðnai antistrèyimh kai hantðstrofh thc eðnai h g. 44. Na apodeðxete ìti an oi sunart seic f,g eðnai antistrèyimec kai orðzetai h sônjesh g f tìte kai h g f eðnai antistrèyimh kai isqôei (g f) = f g. 45. 'Estw f (x) =x x + kai h (x) =x (x ). Na breðte sun rthsh g (x) =αx + β tètoia ste g f = h.

51 Kef laio. 'Oria kai Sunèqeia Sunart sewn Apì tic exet seic tou 997, Dèsmh I. 'Estw C h gramm tou epipèdou me exðswsh y = αx 3 + βx + γx + δ, ìpouα, β, γ, δeðnai pragmatikoð arijmoð kai α. 'Estw A (x,y ), B (x,y ), Γ(x 3,y 3 ), Δ(x 4,y 4 ) shmeða thc C. Upojètoue ìti to mèso tou eujugr mmou tm matoc AB sumpðptei me to mèso tou eujugr mmou tm matoc ΓΔ kai epðshc upojètoume ìti to mèso autì den an kei sthn eujeða pou èqei exðswsh β +3αx =.. Na apodeiqjeð ìti x x = x 3 x 4. Na apodeiqjeð ìti to shmeðo A sumpðptei me to shmeðo Γ meto shmeðo Δ. 47. EÐnai rage to jroisma dôo antistrèyimwn sunart sewn antistrèyimh sun rthsh? 48. Na apodeðxete ìti gia k je sun rthsh f up rqei α R ètsi ste h sun rthsh g (x) =f (x) α na èqei mða toul qiston rðza...3 G' OMADA 49. EÐnai gnwstì kai mporeðte na to qrhsimopoieðte eleôjera ìti an orðzontai oi sunart seic g f kai h (g f) tìte orðzontai kai oi h g kai (h g) f kai m lista isqôei (h g) f = h (g f). ProspajeÐste d sete mða apìdeixh. 4. 'Estw mða sun rthsh f kai A, B dôo mh ken uposônola tou pedðou orismoô thc. Na apodeðxeue ìti. f (A B) =f (A) f (B). An A B tìte f (A B) f (A) f (B) Na d sete èna par deigma me to opoðo nafaðnetai ìti hisìthta sto (b') den isqôei. 4. Gia mða sun rthsh orismènh se èna di sthma Δ isqôei f (x) f (y) (x y) gia ìla ta x, y Na apodeðxete ìti eðnai stajer. 4. Apì ta jèmata thc Olumpi dac APMO, 989. Na prosdiorðsete ìlec tic sunart seic f : R R gia tic opoðec isqôei: H f eðnai gnhsðwc aôxousa. f (x)+g (x) =x, ìpou g = f

52 5.. 'Oria Sunart sewn. 'Oria Sunart sewn.. A' OMADA 43. Sta epìmena sq mata apeikonðzetai h grafik par stash mðac sun rthshc f me pedðo orismoô to R. Se k je perðptwsh na sumplhr sete tic isìthtec: f () =... lim f (x) =... f (3) =... lim f (x) =... x x 3 O O (a') (b') O O (g') (d') O O (e') (ς ) 44. Na sumplhr sete tic isìthtec:. lim x x = 3. lim x α β = 5. lim x β + β = 7. lim u u =. lim x 3 x = 4. lim x β α 6. lim ( β) = x β 8. lim u v u = 45. Sto sq ma pou akoloujeð up rqei h grafik par stash mðac sun rthshc f.

53 Kef laio. 'Oria kai Sunèqeia Sunart sewn 5 O Na sumplhr sete tic isìthtec:. lim x + f (x) =. lim x f (x) = 3. lim x f (x) = 4. lim x + f (x) = 5. lim x f (x) = 6. lim x + f (x) = 7. lim x f (x) = 8. lim x + f (x) = 46. Sta sq mata pou akoloujoôn up rqei h grafik par stash mðac sun rthshc f. Na breðte k je èna apì ta ìria lim f (x), x lim f (x), lim f (x), lim f (x) + x x + x me dedomèno ìti se k je perðptwsh eðnai +. O O (a') (b') (g') O (d') O 47. Na upologðsete ta parak tw ìria:

54 5.. 'Oria Sunart sewn. lim x ( x ). lim x ( x 3x + ) 3. lim x + x 4. lim ( x +) x 5. lim x α ( x +) 6. lim x x x+ 48. Na upologðsete ta parak tw ìria: x. lim x x x. lim 3x+ x x x 3. lim 3x+ x x 4x+3 x 4. lim 3 4x +x+6 x x x 5. lim 3 +x x x x 4 x 6. lim 3 +3x 4 x x Na upologðsete ta parak tw ìria: (x+). lim 4 x x x x 4+3 x. lim x x 3. lim x 3 9 x x x 7 x x 4. lim +4x+4 x x+ 4x x+ 5. lim x x x+ x lim x x+3 3x 43. Na upologðsete ta parak tw ìria:. lim x x x. lim x+3 3 x x 6 x 3x x x 4 3. lim 4. lim x 3 x 8 x 5. lim x 6 4 x x 4 6. lim 5+x x 4 x +4x 43. H sun rthsh f eðnai orismènh sto R kai isqôei lim lim f (x) =. Na breðte ta parak tw ìria: x. lim x f (x +). lim x f (x ) 43. Na upologðsete to ìrio lim 3. lim f ( ημx) x x 4. lim x f ((x ) e x ) { x, x<. f (x) = x, x ìtan x = { x +, x. f (x) = x x+, x > ìtan x = { x, x 3. f (x) = x +, x ìtan x = 4 { x, x 4. f (x) = x +, x > ìtan x = e 433. Na breðte ta ìria: x f (x) = kai 5. lim x f (x x ) 6. lim x ln (f (x)) x x f (x) stic parak tw peript seic:

55 Kef laio. 'Oria kai Sunèqeia Sunart sewn 53 x. lim +5 x x 3. lim x 3 x 3 x 4 +x + (x ) x x x 3. lim 4. lim +x x x (x+h) 4 x 4 h h 5. lim x 6. lim 3 x x+ x x 3 +x 434. 'Estw h sun rthsh ϕ (x) = x4 5x +6 x 4. Na upologðsete ta ìria: 9. lim x ϕ (x) 3. lim ϕ ( x) 3. lim x 3 x ϕ ( x) Na sumplhr sete tic isìthtec:. lim x 5 x 8 =. lim x 6 x 8 = 3. lim x 7 x 8 = 4. lim x 8 x 8 = 5. lim x 9 x 8 = 6. lim x 7 + x 8 = 7. lim x 8 + x 8 = 8. lim x 9 + x 8 = 436. Na breðte ta parak tw ìria:. lim x +. lim x + 3. lim x + 4. lim x + x x 4 x 4 x lim x + 6. lim x + 7. lim x + 8 x 3 x 3x+ x x+ 8. lim x + x 437. Na breðte ta parak tw ìria:. lim x 3 x+3. lim x 3 + x+3 3. lim x 3 + x 9 4. lim x 3 x lim x 3 + x +x 6 6. lim x 3 + 3x x 438. Na breðte ta ìria:. lim x x 3 (x ) 3. lim x 3+ 7+x 3 x 3 +x 5x+3. lim x x+7 x 3 +x 5x+3 x +x 4. lim x 3 x+3

56 54.. 'Oria Sunart sewn x 5. lim 3x +x x x+ 6. lim x+3+ x+4 x + x Na sumplhr sete tic isìthtec. lim (x +3)= x +. lim x + 3. lim x + x+3 = ( x 9 ) = 4. lim x += x + 5. lim (3 x) = x + 6. lim x + 3 x = 44. Na sumplhr sete tic isìthtec. lim x + 4x+ =. lim x += x + 3. lim x + ( x x ) = 44. Na breðte ta ìria:. lim x + ln ( x + ). lim x ln ( x 4 + ) 3. lim ( x + 3) x 44. Na breðte ta ìria: (. lim x + ( ) x ) x +. lim x +3 x x + 4 x +5 x 3. lim x +3 x x 4 x +5 x 4. lim x (x 3) = x + 5. lim x + ( x) x +4= 6. lim x + ( 3 5 ) x 4. lim ( x + x ) x + 5. lim x (x + x) 6. lim (x + x + ) x + x +x 4. lim x + x+3 x 5. lim 3x +x x + x+ 6. lim x++ x x + x 'Estw h sun rthsh f (x) = x +3x+ x 4 +4x+3. Na breðte to pedðo orismoô thc.. Na breðte ta ìria: (a ) lim f (x) (b ) lim x + f (x) (g ) lim f (x) (d ) lim f (x) x x 5 x 5

57 Kef laio. 'Oria kai Sunèqeia Sunart sewn 55 (e ) lim f (x) (± ) lim x f (x) (z ) lim x + f (x) (h ) lim f (x) x x (j ) lim f (x) (i ) lim x + f (x) (ia ) lim f (x) x x e (ib ) lim f (x) x 444. DÐnetai h sun rthsh f (x) =x x + Na breðte ta ìria:. lim x f (x). lim y 3 f (y) 3. lim x f ( x) 4. lim f (f (x)) x ( ) 5. f lim f (x) x 6. lim x f (x + y) ( ) 7. lim lim f (x + y) x y 3 8. lim y 3 ( 9. lim x ( ) lim f (x + y) x ) lim f (x + y) x 3 ( ). lim lim f (x) t x t 445. DÐnetai h sun rthsh f (x) = { x + px + q x x +3 p+ x x> Na breðte ta p, q ètsi ste h f na èqei ìrio sto DÐnetai h sun rthsh f (x) = { x +4 αν x < 3x αν x (A) Na breðte to ìriothcf ìtan to x teðnei:. sto 5. sto 6 apì mikrìterec timèc 3. sto + 4. sto 5. sto apì megalôterec timèc 6. sto apì mikrìterec timèc 7. sto 8. sto m + (B) Na breðte gia poia pragmatik tim tou α to ìriothcf sto α eðnai:

58 56.. 'Oria Sunart sewn. 4. α 447. Apì tic exet seic tou 978, Poluteqnikìc kai Fusikomajhmatikìc kôkloc. Na breðte, qrhsimopoi ntac gnwstèc idiìthtec, to ìrio thc sun rthshc ϕ (x) = x 3x+ 3x 4 ìtan x Apì tic exet seic tou 984, Dèsmh IV. 'Estw h pragmatik sun rthsh ψ thc pragmatik c metablht c x me ψ (x) = x 5. Na breðte to 5x lim ψ (x). x Na breðte ta ìria (. lim 3x 7x +9 ) x + x. lim x+ x + x + ( 3. lim 3x 8 +3x +6x ) x x 4. lim x+ x x + ( x)(x 6. lim 3) x + x x 5x 7. lim 4 x x 3 9x ( 8. lim (x ) + x + x + 9. lim 5x x + x + (x ) ) 5. lim x + x x 9x 45. Na breðte ta ìria. lim x 3 3x + x +. lim x 3x + x ( x ) 3. lim 3x + x x + ( 4. lim x x 3 + x + ) x x+ x 5. lim x + x. lim 4x 4 x 3 + x 6. lim x + x+ x + x+3 x + x x 7. lim + x + x+ x + 8. lim x + 9. lim x + ( x + x x) ( x ++ x + x ). lim x + x+ x x.. B' OMADA 45. Na apodeðxete ìti an lim x σ ϕ (x) =tìte lim x σ ϕ (x) =. 45. Gia tic sunart seic f,g eðnai gnwstì ìti lim f (x) =α, lim x σ Na apodeðxete ìti lim x σ g (x) =α. f (x) g (x) = x σ

59 Kef laio. 'Oria kai Sunèqeia Sunart sewn JewroÔme tic sunart seic me f,g me pedðo orismoô to R tètoiec - ste na isqôei f (x) g (x) gia ìla ta x. Na apodeðxete ìti an isqôei lim f (x) g (x) =tìte isqôei lim f (x) = lim g (x) =. x σ x σ x σ κ 454. An κ>λ> na breðte to ìrio lim x λ x x + κ x +λ x Gia mða rtia sun rthsh f me pedðo orismoô to R isqôei lim f (x) =a. x + Poiì eðnai to lim (f (x)+f ( x))? x Na apodeðxete ìti an lim x + f(x). α = lim x + x f(x) αx. β = lim x + x 3. γ = lim x + ( f (x) ( αx + βx )) ( f (x) ( αx + βx + γ )) =tìte: 457. Gia tic sunart seic f kai g me pedðo orismoô to R isqôei lim (f (x)+g(x)) = p R, lim x σ (f (x) g (x)) = s R x σ Na apodeðxete ìti: ( lim f (x)+g (x) ) = p s x σ 458. Na apodeðxete ìti an gia thn sun rthsh f (x) gia k je x isqôei ( f (x) ημx) ( ) x f (x) ημx x tìte lim f (x) =. x (λ 3μ)x 459. Gia poièc timèc twn λ, μ isqôei lim +4x+ x + (λ+μ)x+5 =? 46. Na lôsete thn exðswsh lim ημα α α+x x =. 46. Gia thn sun rthsh ϕ : R R eðnai gnwstì ìti isqôei ημx + αx xϕ (x) x 4 +(α +)x gia ìla ta x. Na apodeðxete ìti lim x ϕ (x) =α 'Estw f : R R gia thn opoða isqôei lim lim (x +)f ( ημx) x x Na breðte to ìrio lim x + ημ x x x + x 3 x f (x) =. Na breðte to ìrio

60 58.. 'Oria Sunart sewn 464. Apì tic exet seic tou 987, Dèsmh I. Stic exet seic ekeðnhc thc qroni c tèjhke to ex c z thma: Na breðte to ìrio thc akoloujðac (α ν ) me ( ) α ν = 7ν 4 +6ν +5 7ν 4 +3ν +3 63ν 5ν + Na apant sete to antðstoiqo er thma gia sunart seic dhlad : Na breðte to ìrio giax + thc sun rthshc ( ) f (x) = 7x 4 +6x +5 7x 4 +3x +3 63x 5x Na breðte to ìrio x+ x +3 x lim x + x 3 x 466. Gia mða poluwnumik sun rthsh f(x) eðnai gnwstì ìti:. lim x f (x) = 4 f(x). lim x + x+α = 3. lim x f(x) x+α = β Na brèite thn f kai touc arijmoôc α, β BreÐte to ìrio lim x + x +3 x x +3 x Gia tic di forec timèc tou μ R na breðte ta ìria: (μ )x. lim 3 +4x+ x + μx + (μ )x. lim 3 +4x+ x μx + 3. lim x + ( x +x 3+μx) 469. DÐnetai h sun rthsh f (x) = { x +4, x < 3x, x. Na breðte ta ìriathcf ìtan to x teðnei:

61 Kef laio. 'Oria kai Sunèqeia Sunart sewn 59 (a ) sto 5 (b ) sto 6 apì mikrìterec timèc (g ) sto + (d ) sto (e ) sto apì megalôterec timèc (± ) sto apì mikrìterec timèc (z ) sto. Na breðte gia poia pragmatik tim tou α to ìriothcf sto α eðnai: (a ) (b ) 5 α 47. BreÐte to ìrio ( ) lim x ++μx x 47. Na breðte ta ìria ( x. lim x x 3 ) x +. lim x + x+ x +x+ x+ ( x)(x+) 47. An κ>λ>μ> poio eðnai to ìrio lim x + (κx λ x μ x ) 473. 'Estw h sun rthsh f (x) = (x α)(x β) x ìpou α<β. Na breðte to pedðo orismoô thc.. Na breðte ta ìria: (a ) (b ) (g ) (d ) lim f (x) x + lim f (x) x lim f (x) x α lim f (x) x β + (e ) (± ) (z ) (h ) lim f (x) x α+β lim f (x) x β lim (f (x)+f ( x)) x + lim f (x) x 'Estw h sun rthsh f (x) = x ημ x ημx. Na breðte to ìriothcf gia x.. Na apodeðxete ìti h f den eðnai monìtonh se kanèna apì ta diast mata (,h) me h>.

62 6.. 'Oria Sunart sewn 475. JewroÔme tic sunart seic f (x) =+x kai h (x) = Na breðte an up rqoun timèc tou λ ste na isqôei { x 3, x <, x Na breðte to ìrio lim x + f(x) λ lim = x h(x) ( x 3 + ημ 3 x + x + ημx ) 477. Gia thn sun rthsh f :(, + ) R isqôei ( +x ) f (x)+x x gia ìla ta x. Na breðte to lim f (x). x 'Estw mða sun rthsh me pedðo orismoô to di sthma (σ,σ ) kai P (α, β) èna shmeðo tou epipèdou. Se k je shmeðo M (x, f (x)) thc C f antistoiqoôme thn apìstash tou d(x) apì to P.. Na apodeðxete ìti an σ =+ tìte lim x σ d (x) =+. An isqôei lim x σ d (x) =+ eðnai rage σ =+? 479. 'Estw h sun rthsh ( x + ) y 3 + x y + ( x 3 + x ) y + x 3 + x f (x) = lim y + xy 3 + x 4 x + Na apodeðxete ìti eðnai Ðsh me thn sun rthsh g (x) = x + x. 48. An kai poioc mporeð na eðnai o t? lim f (x) x =t t lim f (ημx) =t x 48. Na breðte gia poièc timèc twn α, β isqôei ( x x αx β) =..3 G' OMADA lim x Na breðte ta ìria:. lim (xν )(xν+ )(xν+ )...(xν+k ) x (x )(x )(x 3 )...(x k ) μ. lim x+α μ α x ν x x+α ν α x

63 Kef laio. 'Oria kai Sunèqeia Sunart sewn 6 (x 3. lim μ ) (x μ )(x ν )+(x ν ) x (x μ ) +(x μ )(x ν )+(x ν ) 483. 'Estw P (x) =x ν + α ν x ν + α ν x ν α x + α Q (x) =x ν + β ν x ν + β ν x ν β x + β kai èstw akìmh H (x) = ν P (x) ν Q (x). Na apodeðxete ìti to pedðo orismoô thc sun rthshc H perièqei k poio di sthma thc morf c (r, + ).. Na apodeðxete ìti lim H (x) =α ν β ν x + ν 484. Sthn skhsh aut ja apodeðxete ìti oi sunart seic συν, ημ kai εϕ den èqoun ìrio gia x + kai gia x. 'Estw f(x) =συνx.. Na apodeðxete ìti gia k je tim tou x isqôei: f (x) =f (x) = f ( x π ). Na apodeðxete ìti an h f èqei ìrio L gia x + tìte to L ja eðnai pragmatikìc arijmìc kai L =L = L 3. Na apodeðxete ìti h f den èqei ìrio gia x Na apodeðxete ìti h f den èqei ìrio gia x. 5. Na apodeðxete ìti h ημ den èqei ìrio gia x + giax. 6. Me thn bo jeia thc, gnwst c, sqèshc συνx = εϕ x +εϕ x ìti h εϕ den èqei ìrio gia x + giax. na apodeðxete.3 Sunèqeia Sunart sewn.3. A' OMADA 485. Se poièc apì tic parak tw peript seic h sun rthsh f eðnai suneq c sto x?

64 6.3. Sunèqeia Sunart sewn f (x ) f (x ) x x (a') (b') f (x ) f (x ) x x (g') (d') 486. Gia thn sun rthsh f eðnai gnwstì ìti: EÐnai suneq c sto f() = 3 Na sumplhr sete tic isìthtec:. lim x (f (x)+x) =. lim x (xf (x)) = 3. lim x ( f (x) 9 ) = 4. lim x f (x) 9 f(x) 3 = 5. lim x f (x) = 6. lim x f (x +)= 7. lim x 4 (f (x ) + x) = ( f(x)+x x f(x ) x 8. lim ) = 487. 'Estw f : R R suneq c sto x me f(x ) = 6. Na brejoôn ta parak tw ìria: f(x)+3. lim x x 3f(x)+. lim x x 5 f(x) + f(x) + (f(x)) 3. lim 3 6 x x (f(x)) Na exet sete se poièc peript seic h sun rthsh f eðnai suneq c sto x =.

65 Kef laio. 'Oria kai Sunèqeia Sunart sewn 63. f (x) =x +. f (x) = { x +, x < x +, x { x 3. f (x) = +, x < x +3, x { x 4. f (x) = +, x < x 3 +, x { x 5. f (x) = +, x x 3 +, x x +, x 6. f (x) = (x +) <x< x 3 +, x 489. Stic epìmenec peript seic na breðte an up rqei tim tou α h sun rthsh f eðnai suneq c sto x =.. f (x) =. f (x) = { x x, x α, x = { x x, x α, x = { αx + x 3. f (x) = 7, x = x +, x < 4. f (x) = α, x = x +, x > 49. Poièc apì tic parak tw sunart seic eðnai suneqeðc? { x, x<. f (x) = x, x { x, x<. f (x) = x, x > { x, x< 3. f (x) = x, x 49. Na apodeðxete ìtihsun rthsh x 5x+6 x 3, x < 3 f (x) =, x =3 x 5, x > 3 eðnai suneq c. 49. Apì tic exet seic tou 983, Dèsmh IV. Na exet sete wc proc th sunèqeia tic sunart seic me tôpouc:. f (x) = { 4x 9x+ x αν x R\{} 7 αν x = sth jèsh x = { x αν x. g (x) = sth jèsh x x x< = 493. Gia thn sun rthsh f eðnai gnwstì ìti eðnai suneq c sto x kai ìti f (x )=8. Na upologðsete ta ìria:

66 64.3. Sunèqeia Sunart sewn. lim x x (f(x) + 3) (. lim f x x+3 x + x+8 3. lim f(x h) h f( x h ) ) 494. Oi sunart seic f kai g èqoun pedðo orismoô to R kai oi sunart seic f +3g, 3f + g eðnai suneqeðc. Na apodeðxete ìti kai h 5f +g eðnai suneq c. Upodeixh: An onom soume f +3g = h, 3f + g = s tìte f = 3 8 s 8 h, g = 8 s h k.t.l Na apodeðxete ìti an oi f, f + g eðnai suneqeðc sto x tìte kai h g eðnai suneq c sto x Apì tic exet seic tou 986, Dèsmh I. Na prosdiorðsete ta α, β ste h sun rthsh 3αe x+ + x αν x f (x) = x αx +3β αν <x< βημx + ασυνx + αν x na eðnai suneq c sto R Apì tic exet seic tou 989, Dèsmh IV. DÐnetai h sun rthsh f me { 3α +, <x x f (x) = 3 x, x > x 4 Na prosdioristeð to α R ste h sun rthsh na eðnai suneq c sto x = Apì tic exet seic tou 98.. Dinetai h sun rthsh f pou orðzei o tôpoc f (x) = x (x ) x gai k je x R kai eðnai f () =. Na exetasteðaneðnai suneq c kai na gðnei to gr fhma thc.. Na brejeð to ìrio thcsunart sewc y = x + x +x +3 ìtan x B' OMADA 499. Oi sunart seic f,g eðnai orismènec sto R kai gia k poio x R isqôei f (x) lim x x g (x) =+ lim g (x) =s x x Na exet sete an mporeð h f na eðnai suneq c sto x. ennoeð grafik par stash

67 Kef laio. 'Oria kai Sunèqeia Sunart sewn Na apodeðxete ìti an mða sun rthsh f eðnai suneq c sto α tìte lim (f (α + h) f (α h)) = h 5. 'Estw mða sun rthsh f :Δ R kai èna kleistì di sthma [α, β]. Na apodeðxete ìti an gia dôo opoiad pote x,x Δ olìgoc metabol c thc f : λ = f (x ) f (x ) x x sta x,x an kei sto [α, β] tìte h f eðnai suneq c. 5. Apì tic exet seic tou 979, Poluteqnikìc kai Fusikomajhmatikìc kôkloc. Na exet sete n h sun rthsh ϕ pou dðnetai apì ton tôpo ϕ (x) =x ημ x, an x kai ϕ () =, eðnai suneq c sto sônolo twn pragmatik n arijm n. 53. Ja lème ìti mða sun rthsh f ikanopoieð mða sunj kh tou: Lipschitz an up rqei jetikìc arijmìc c ètsi ste na isqôei: Rudolf Otto Sigismund Lipschitz f (x ) f (x ) c x x gia ìla ta x,x. Hölder an up rqoun jetikoð arijmoð c kai α ètsi ste na isqôei: f (x ) f (x ) c x x α gia ìla ta x,x Na apodeðxete ìti an h f ikanopoieð mða apì tic parap nw sunj kec eðnai suneq c. An k nate dôo apodeðxeic mei ste tec se mða koit ntac poia apì tic dôo sunj kec eðnai pio genik. Otto Ludwig Hölder Gia thn sun rthsh ϕ eðnai gnwstì ìti lim ϕ(x) α x σ ϕ(x)+α =. Na apodeðxete ìti lim ϕ (x) =α. x σ 55. Na d sete èna par deigma asuneq n sunart sewn me jroisma suneq sun rthsh. 56. 'Estw Φ:R R mða sun rthsh pou eðnai suneq c sto x. 'Estw akìmh mða sun rthsh f : R R tètoia steanisqôei f (x) f (y) Φ(x) Φ(y) gia k je x, y Na apodeðxete ìti h f eðnai suneq c (deðte kai thn skhsh 53). An sth sunj kh tou Hölder to α gðnei megalôtero tou h f eðnai stajer. Autì ja apìdeiqjeð eôkola sto kef laio twn Parag gwn kai, ousiastik, eðnai h skhsh B selðda 57 tou sqolikoô biblðou. To Ðdio apotèlesma mporeð, pio dôskola, na apodeiqjeð kai qwrðc th qr sh twn parag gwn. Prìkeitai gia thn skhsh 4 thc G' Om dac

68 66.3. Sunèqeia Sunart sewn 57.. JewroÔme dôo sunart seic f,g orismènec se èna di sthma Δ kai x Δ ste: oi f,g eðnai suneqeðc f (x) g (x) f (x )=g(x ) Na apodeðxete ìti k je sun rthsh h gia thn opoða isqôei f(x) h(x) g(x) gia k je x Δ eðnai suneq c sto x. Na apodeðxete ìti h h eðnai suneq c sto x stic akìloujec peript seic (a ) x + h (x) (x ) +ìtan x = (b ) x h (x) e x ìtan x = 58. 'Estw f suneq c sun rthsh orismènh sto di sthma [α, β]. Na apodeðxete ìti an f (x) gia k je x [α, β) tìte isqôei f (β). 59. 'Estw f :Δ R. Na apodeðxete ìti:. An Δ, f (x + y) =f (x)+f (y) gia ìla ta x, y kai h f eðnai suneq c sto tìte einai suneq c.. An Δ, f (x + y) =f (x) f (y) gia ìla ta x, y kai h f eðnai suneq c sto tìte einai suneq c. 3. An Δ, f (xy) =f (x) f (y) gia ìla ta x, y kai h f eðnai suneq c sto tìte einai suneq c. 4. An Δ, f (xy) =f (x)+f (y) gia ìla ta x, y kai h f eðnai suneq c sto tìte einai suneq c. 5. Na apodeðxete ìti an oi sunart seic f, g eðnai orismènec se èna di sthma kai suneqeðc tìte oð sunart seic m (x) =min(f (x),g(x)) kai M (x) = max (f (x),g(x)) eðnai suneqeðc. 5. 'Estw hexðswsh x + λx + λ =me gnwsto to x kai h sun rthsh g (λ) = h mikrìterh rðza thc x + λx + λ = DeÐxte ìti h g eðnai suneq c. 5. An h sônjesh g f eðnai suneq c kai h g eðnai suneq c eðnai h f suneq c?

69 Kef laio. 'Oria kai Sunèqeia Sunart sewn Apì tic exet seic tou, Teqnologik KateÔjunsh. DÐnetai h sun rthsh f me { x f (x) = ( 8x +6, <x<5 α + β ) ln (x 5+e)+(α +)e 5 x, x 5. Na brejoôn ta lim f (x), lim f (x) x 5 x 5 +. Na brejoôn ta α, β R ste h sun rthsh f na eðnai suneq c sto x =5 3. Gia tic timèc twn α, β tou erwt matoc (b') na breðte to lim x + f (x). 54. 'Estw mða gnhsðwc aôxousa suneq c sun rthsh f :(α, β) R. 'Estw x (α, β). MporeÐ rage na isqôei f (x) =+? lim x x 55. 'Estw f :Δ R mða sun rthsh, x Δ kai d(x) apìstash tou tuqìntoc shmeðou M(x, f(x)) thc C f apì to shmeðo A(x,f(x )). Na apodeðxete ìti h f eðnai suneq c sto x an kai mìno an lim x x d (x) =..3.3 G' OMADA 56. H sun rthsh tou akeraðou mèrouc. Se k je arijmì x antistoiqoôme 3 ton megalôtero apì touc akeraðouc pou den uperbaðnoun ton x. O arijmìc autìc sumbolðzetai me [x] kai onom zetai akèraio mèroc tou x. Dhlad : [ x ]=o megalôteroc apì touc akeraðouc m me m x. BebaiwjeÐte ìti apì ton orismì prokôptei: [3] = 3, [3, 4] = 3, [, 8] =, [, 8] = 4. Na apodeðxete ìti x < [x] x<[x]+ 3. 'Estw f (x) =[x] (a ) Na apodeðxete ìti h sun rthsh f eðnai stajer se k je èna apì ta diast mata..., [ 3, ), [, ), [, ), [, ), [, ),... (b ) Na apodeðxete ìti h f eðnai aôxousa. (g ) Na apodeðxete ìti h f eðnai suneq c se ìlouc touc arijmoôc ektìc apì touc akeraðouc. (d ) Na k nete thn grafik par stash thc f. 4. Na k nete th grafik par stash thc sun rthshc g (x) =x [x], x [, 3] kai na breðte to sônolo tim n thc. 3 ApodeiknÔetai ìti ènac tètoioc arijmìc up rqei

70 68.4. Sunart seic suneqeðc se di sthma 57. 'Estw Δ=[α, β] kai f :Δ R suneq c. Na apodeðxete ìti up rqei toul qiston mða suneq c sun rthsh g : R R tètoia ste g (x) =f (x) gia ìla ta x, y Δ C g C f ff fi Sth sunèqeia na apodeðxete me èna par deigma ìti to sumpèrasma den isqôei an Δ=(α, β]..4 Sunart seic suneqeðc se di sthma.4. A' OMADA 58. Na apodeðxete stic parak tw peript seic ìti h sun rthsh f èqei mða toul qiston rðzasto di sthma Δ. f (x) =x συνx, Δ= (, π ). f (x) =x + ημx + e x, Δ=( π, π) 3. f (x) =x +lnx x, Δ= (, ) 4. f (x) =x 5 +lnx +, Δ= (, ) e 59. Na apodeðxete ìti se k je perðptwsh h sun rthsh f èqei mða toul qiston rðza sto di sthma Δ.. f (x) =x 3 + x 3 sto Δ=(, ). f (x) =x 3 + ημx 3 sto Δ=(,π) 3. f (x) = x + x sto Δ=(, ) 4. f (x) =x ( x + x) sto Δ=(, ) 5. f (x) = ( x 3 ) ln ( x + ) sto Δ=(, ) 6. f (x) = 3 x + e x sto Δ=(, )

71 Kef laio. 'Oria kai Sunèqeia Sunart sewn Na apodeðxete ìti h exðswsh 7x 9 = x 5 èqei mða toul qiston lôsh metaxô-kai. 5. 'Estw h sun rthsh f (x) =x e x +. EÐnai, profan c f () = kai f (3) = e 3. Me b sh aut ta dedomèna na apodeðxete ìti o arijmìc - eðnai tim thc f. 5. Gia mða suneq sun rthsh f orismènh sto R isqôei f () + f () =. Dèixte ìti ja èqei mða toul qiston rðza sto di sthma [, ]. 53. Oi rðzec thc sun rthshc f (x) =x 4 7x 3 +8x +8x 48 eðnai oi arijmoð,, 3, 4. Ti prìshmo èqei sto di sthma (, )? 54. Na apodeðxete ìti oi parak tw exis seic èqoun mða toul qiston lôsh sto di sthma [, ].. e x = x+. ln x = x 3. x 3 + ημx = x 55. Na apodeðxete ìti h exðswsh ln(x +)=4x èqei mða toul qiston lôsh sto di sthma (,e ). 56. Na apodeðxete ìti oi grafikèc parast seic twn sunart sewn f (x) = συνx kai g (x) =x èqoun èna toul qiston koinì shmeðo. Upodeixh: ErgasjeÐte sto di sthma [, π ]. 57. Na apodèðxete ìti h exðswsh x 4 + ( t ) x +(t ) x t =èqei mða toul qiston rðza sto (, ). Upodeixh: Thn perðptwsh t =na thn exet sete qwrist. 58. MporeÐ rage mða, mh stajer, suneq c sun rthsh f :[, ] R na paðrnei mìno tic timèc, 3? Mìno tic,, 3? Mìno akèraiec timèc? 59. Na sumplhr sete ton pðnaka: f(x) = Δ= f (Δ) = x + [3, 7] x [, 5] x [ 4, [ ] ημx π 6, π ] 3 ln x [,e ) x (, 3] x (, ]

72 7.4. Sunart seic suneqeðc se di sthma.4. B' OMADA 53. Na apodeðxete stic parak tw peript seic ìti h sun rthsh f èqei mða toul qiston rðza sto di sthma Δ. f (x) =x 5 + ημx +, Δ=(, + ). f (x) = x3 +κx+λ x, Δ=(, + ) + 3. f (x) =εϕx +ln ( x + ), Δ= ( π, π 4. f (x) = x 3 α 3 + x 3 β 3, Δ=(α, β) 53. 'Estw oi arijmoð p, q, r me <p<r<q. Na apodeðxete ìti h exðswsh pημx + qσυνx = r èqei mða toul qiston lôsh sto di sthma [, π ]. 53. Na apodeðxete ìti k je polu numo perittoô bajmoô èqei mða toul qiston pragmatik rðza Gia thn suneq ( sun rthsh ) f : R R eðnai gnwstì ìti èqei monadik rðza to α kai ìti f a + >. DeÐxte ìti f (a +)> 'Estw f :[α, β] R mða suneq c sun rthsh kai γ (α, β) tètoio ste (f (α) f (γ)) (f (β) f (γ)) >.. Na k nete merik sqèdia gia to pwc perðpou ja eðnai h grafik par stash thc f.. Na apodeðxete ìti up rqoun dôo arijmoð ènac mikrìteroc tou γ kai ènac megalôteroc tou γ stouc opoðouc h f paðrnei Ðsec timèc MÐa suneq c sun rthsh ϕ eðnai orismènh sto [α, β] kai oi timèc thc sta α, β eðnai rðzec thc exðswshc t 97t 953 =. Na apodeðxete ìti h ϕ èqei mða toul qiston rðza sto (α, β) 536. 'Estw α<βdôo pragmatikoð arijmoð kai mða suneq c sun rthsh f : R R pou den èqei rðzec. Na apodeðxete ìti h sun rthsh g (x) =(x α) f (x α)+(x β) f (x β) èqei mða toul qiston rðza sto di sthma [α, β] 'Estw èna to di sthma [α, β] kai mða suneq c sun rthsh f :[α, β] R f(α)+f(β). Na apodeðxete ìti oi arijmoð, kai kf (α) +( k) f (β) me k [, ] brðskontai metaxô twn f (α),f(β). Na apodeðxete ìti oi oi exis seic (a ) f (x) = f(α)+f(β) )

73 Kef laio. 'Oria kai Sunèqeia Sunart sewn 7 (b ) f (x) =kf (α) +( k) f (β), k [, ] èqoun mða toul qiston rðza sto [α, β] 538. 'Estw P èna polu numo bajmoô megalôterou Ðsou tou. Na apodeðxete ìti an gia mða suneq sun rthsh f me pedðo orismoô to R isqôei P f = tìte h f eðnai stajer DÐnontai oi jetikoð arijmoð c,c,..., c ν kai t <t <... < t ν. Na apodeðxete ìti h exðswsh c + c c ν = x t x t x t ν èqei akrib c ν lôseic, mða se k je èna apì ta diast mata (t,t ),..., (t ν,t ν ). 54. 'Estw mða suneq c sun rthsh f orismènh sto di sthma Δ. kai oi arijmoð α,α,..., α ν pou an koun sto Δ. Na apodeðxete ìti up rqei arijmìc α Δ tètoioc ste na isqôei f (α) = f (α )+f (α ) f (α ν ) ν. Upojètoume epiplèon ìti h f paðrnei mh arnhtikèc timèc. Na apodeðxete ìti up rqei arijmìc α Δ tètoioc ste na isqôei f (α) = ν f (α ) f (α )... f (α ν ) 54. Na exet sete an eðnai suneqeðc oi akìloujec sunart seic:. H sun rthsh pou antistoiqeð se k je forologhtèo eisìdhma ton fìro pou analogeð.. H sun rthsh pou antistoiqeð sto b roc enìc taqudromikoô dèmatoc to analogoôn tèloc. 54. Gia thn sun rthsh f : R R eðnai gnwstì ìti isqôei f (x)+f (x) =x + x + Na brejeð h f stic akìloujec peript seic:. H f eðnai suneq c sto R. H f eðnai suneq c sto (, ) (, + ) 543. 'Estw f mða suneq c sun rthsh orismènh sto di sthma [α, β] kai k>, k. Na apodeðxete ìti an k f(α) = ( ) f(β) k tìte h f èqei mða toul qiston rðza sto [α, β].

74 7.4. Sunart seic suneqeðc se di sthma 544. 'Estw f :(σ,σ ) R suneq c. Na apodeðxete ìti an lim x σ f (x) = kai lim x σ f (x) =+ tìte h f èqei sônolo tim n to R JewroÔme dôo suneqeðc sunart seic orismènec sto di sthma Δ. Na apodeðxete ìti an oi grafikèc parast seic touc den èqoun koinì shmeðo hc f ja eðnai p nw apì thn C g eðte h C g ja eðnai p nw apì thn C f `Estw f :[α, β] R suneq c me f (α) α kai f (β) β. Na apodèixete ìti up rqei x [α, β] ste f (x) =x. Luitzen Egbertus Jan Brouwer 'Estw f :[α, β] R suneq c ste f (α) =f (β).'estw γ = α+β δ = β α. Na apodeðxete ìti up rqei x [,δ] ste f (x + α) =f (x + γ) kai 548. To Je rhma tou Brouwer. Na apodeðxete ìti k je suneq c sun rthsh f apì èna kleistì di sthma Δ ston eautì tou èqei èna toul qiston stajerì shmeðo dhlad up rqei èna toul qiston stoiqeðo x tou Δ tètoio ste na isqôei f (x) =x 'Estw f :(, + ) R me f (x) = x. IsqÔei profan c f (x) > gia k je x. DeÐxte ìti den up rqei jetikìc c tètoioc ste na isqôei f (x) >cgi ìla ta x.. 'Estw f :[α, β] R suneq c tètoia ste f (x) > gia k je x. Na apodeðxete ìti up rqei up rqei jetikìc c tètoioc ste na isqôei f (x) > c gi ìla ta x. 55. 'Estw h sun rthsh f (x) = { (λx) 3 + λx + x< (λx) 5 +3 x Na apodeðxete ìti up rqei toul qiston mða tim tou λ gia thn opoða h f eðnai suneq c. 55. 'Estw f :[α, β] R suneq c. Na apodeðxete ìti up rqei ξ (α, β) ètsi ste f (ξ) = α + ξ α + β + ξ β 55. Gia th suneq sun rthsh f :(, ) R eðnai gnwstì ìti isqôei: ln (x ) + ln ( x) =ln+f (x) Na apodeðxete ìti h f diathreð stajerì prìshmo.

75 Kef laio. 'Oria kai Sunèqeia Sunart sewn 'Estw P (x) èna polu numo bajmoô toul qiston kai f, g dôo sunart seic me pedðo orismoô to R ek twn opoðwn h g eðnai suneq c tètoiec ste Na apodeðxete. H f eðnai suneq c f = P g. An h exðswsh f (x) =P () den èqei lôsh tìte h g diathreð stajerì prìshmo 554. JewroÔme dôo suneqeðc sunart seic f : R R, g : R R tètoiec ste gia k je x R oi arijmoð f(x), g(x) na eðnai eterìshmoi.. Na apodeðxete ìti oi C f, C g den èqoun koin shmeða.. Na apodeðxete ìti an gia k poio x eðnai f(x ) >g(x ) tìte ja eðnai kai f(x) >g(x) gia ìla ta x (deðte kai thn skhsh 545). 3. 'Estw ìti up rqoun α, β ètsi ste f (α) =α kai g (β) =β. AfoÔ deðxete ìti ta α, β eðnai di fora na deðxete ìti up rqei γ metaxô touc ètsi ste na isqôei: f (γ)+g (γ) =γ.4.3 G' OMADA 555. 'Estw Δ èna di sthma kai f mða suneq c sun rthsh orismènh sto Δ.. Na apodeðxete ìti an gia k poia α, β, γ Δ me α < β < γ isqôei f (α) <f(β) kai f (γ) <f(β) tìte h f den eðnai -.. Na katal xete Ðdio sumpèrasma me tou (a') kai gia thn perðptwsh pou isqôei f (α) >f(β) kai f (γ) >f(β). 3. Na apodeðxete ìti an h f eðnai - tìte eðnai gnhsðwc aôxousa Sthn skhsh aut mporeðte na qrhsimopoi sete wc dedomènh twn akìloujh prìtash: MetaxÔ dôo di forwn pragmatik n up rqei ènac toul qiston rhtìc arijmìc. Estw mða suneq c sun rthsh f orismènh se èna di sthma Δ tètoia ste gi na isqôei f (x) =gia k je rhtì arijmì tou diast matoc Δ.. Na apodeðxete ìti an gia k poio x Δ isqôei f (x ) tìte x / Q.. Na apodeðxete ìti an gia k poio x Δ isqôei f (x ) tìte up rqei di sthma I pou perièqei to x kai isqôei f (x) gia k je x I Δ Peter Gustav Lejeuene Dirichlet Na apodeðxete ìti isqôei f(x) =gia ìla ta x Δ.

76 74.4. Sunart seic suneqeðc se di sthma QrhsimopoieÐste to prohgoômeno gia na apodeðxete ìti :. An dôo suneqeðc sunart seic orismènec se èna di sthma paðrnoun Ðsec timèc stouc rhtoôc autoô tou diast matoc tìte eðnai Ðsec.. H sun rthsh tou Dirichlet den eðnai suneq c. f (x) = { x Q x/ Q 3. To prìblhma tou Cauchy. Oi mìnec suneqeðc sunart seic f : R R me thn idiìthta f (x + y) =f (x)+f (y) gia ìla ta x, y eðnai (deðte kai thn skhsh 358) oi sunart seic thc morf c f (x) =αx 4. Na breðte ìlec tic suneqeðc sunart seic f : R R gia tic opoðec isqôei f (x + y) =f (x) f (y) gia ìla ta x, y 557. 'Estw mða suneq c sun rthsh orismènh se èna di sthma Δ kai tètoia ste ta ìria thc sta kra tou Δ na eðnai +. Na apodeðxete ìti h f èqei èl qisto. Augustin Louis Cauchy Apì ta Jèmata thc hc Olumpi dac ESSD, 977. JewroÔme to polu numo x + x 9 + x x + x + DÔo toma paðzoun to parak tw paiqnðdi. O pr toc antikajist opoiond pote apì touc asterðskouc me k poion arijmì, katìpin o deôteroc antikajist me k poion arijmì opoiond pote apì touc upìloipouc asterðskouc, Ôstera o pr toc antikajist ènan apì touc asterðskouc me arijmì k.o.k (Sunolik 9 k- in seic). An to polu numo pou prokôptei den èqei pragmatikèc rðzec kerdðzei o pr toc paðkthc. An èqei toul qiston mða pragmatik rðza kerdðzei o deôteroc. MporeÐ na kerdðsei o deôteroc paðkthc anex rthta apì ton trìpo paiqnidioô tou pr tou? 559. To je rhma stajeroô shmeðou tou Banach. 'Estw sun rthsh f : R R pou ikanopoieð mða sunj kh Lipschitz (deðte skhsh 53) me stajer mikrìterh tou dhlad up rqei jetikìc arijmìc c< ètsi ste na isqôei: f (x ) f (x ) c x x gia ìla ta x,x. Na apodeðxete ìti h f èqei akrib c èna stajerì shmeðo dhlad h exðswsh f(x) =x èqei akrib c mða lôsh. Stefan Banach

77 Kef laio. 'Oria kai Sunèqeia Sunart sewn 75.5 Ask seic se ìlo to kef laio 56. An h f eðnai orismènh sto di sthma [α, β], eðnai gnhsðwc aôxousa sto di sthma [α, x ) kai gnhsðwc fjðnousa sto di sthma (x, β]. EÐnai rage to f(x ) mègisto thc f? 56. Gia thn sun rthsh f isqôei ( f (x) x 8 x 6)( f (x) x 4 x ) gia k je x R.. Na apodeðxete ìti h f eðnai suneq c sto.. Na breðte to lim x + f (x) 3. Na breðte to lim x + ( f (x)+ f (x) ) 56. BreÐte ta ìria lim x, lim log x + x + x JewroÔme touc arijmoôc α kai α <α <... < α ν kai th sun rthsh ϕ (x) =(x α )(x α )... (x α ν )+α. DeÐxte ìti den eðnai Na apodeðxete ìti h sun rthsh f : R R me f (x) = (+x )x x, x kai f () = eðnai asuneq c kai antistrèyimh. DeÐxte ìti h antðstrof thc eðnai suneq c Gia mða sun rthsh f : R R eðnai gnwstì ìti gia k je α<βisqôei f ([α, β]) [α, β]. Na apodeðxete ìti f (x) =x gia ìla ta x 'Estw h sun rthsh f (x) = x+α x α f f f. me α. Na orðsete thn sun rthsh 567. Gia mða sun rthsh f me pedðo orisoô to R isqôei f 3 (x)+f (x) x = gia k je x. AfoÔ apodeðxete ìti gia k je x x isqôei f (x ) f (x ) = deðxte katìpin ìti eðnai suneq c. x x f (x )+f (x ) f (x )+f (x ) DeÐxte ìti an h sun rthsh g f èqei sônolo tim n to R tìte h g èqei sônolo tim n to R. IsqÔei rage to prohgoômeno an antð gia sônolo tim n} eðqame pedðo orismoô}? 569. BreÐte to ìrio ( lim x ++μx) x

78 76.5. Ask seic se ìlo to kef laio 57. 'Estw y = f (x) = αx+β γx α. DeÐxte ìti x = f (y). ( x 57. Gia poièc timèc tou a isqôei lim a+ x + a ) =? 57. H sun rthsh f : R R eðnai suneq c kai isqôei Na breðte to ìrio: f (x) f (x ) lim = x x f (x) f (x ) lim f ( x + h + h ) h 573. Na apodeðxete ìti den up rqoun antistrèyimec periodikèc sunart seic Gia poi tim tou λ h sun rthsh { 3 f (x) = λ ημx x, x < 5 λ+x + λ+x, x eðnai suneq c? 575. 'Estw h rht sun rthsh. Na breðte to pedðo orismoô thc. f (x) = x4 9x 3 +5x 4x +4 x 3 5x +8x 4. Na breðte gia poia σ R {, + } to ìrio thc f gia x σ eðnai pragmatikìc arijmìc 'Estw mða suneq c sun rthsh f :(α, β) R. Upojètoume ìti lim x α f (x) = A R kai lim x β f (x) =B R kai A B. Na apodeðxete ìti to anoiktì di thma me kra ta A, B perièqetai sto sônolo tim n thc f Na apodeðxete ìti den up rqei gnhsðwc aôxousa poluwnumik sun rthsh rtiou bajmoô 'Estw h sun rthsh f (x) =+x + x 3.. Na apodeðxete ìti eðnai antistrèyimh kai ìti to sônolo tim n thc eðnai to R.. Na apodeðxete ìti gia k je α isqôei f ( +ημα + ημ 3 α ) = ημα Apì tic exet seic tou 98. x. Na brejeð to lim x+ x x.

79 Kef laio. 'Oria kai Sunèqeia Sunart sewn 77. DÐnetai h sun rthsh me tôpo y = x3 x +x x. Na brejeð to pedðoorismoô +x thc kai h oriak tim thc, ìtan to x teðnei sto BreÐte to l joc: x lim + x x + x = lim x + ( ) ( ) x x + x) =( x + x = 58. MÐa suneq c sun rthsh èqei timèc se k je di sthma. DeÐxte ìti to sônolo tim n thc eðnai to R. 58. Na sqoli sete to parak tw: DÐnetai h suneq c sun rthsh f : R R kai oi arijmoð α, β, γ, δ R me α<β. An (f(α) γ)(f(β) δ) <, na apodeðxete ìti up rqei θ R tètoio ste h exðswsh f(x) =θ na èqei mða toul qiston rðza sto (α, β). JewroÔme th sun rthsh LÔsh. g (x) =f (αx + β ( x)) γx δ ( x), x [, ] H g eðnai suneq c sto [, ] kai epiplèon g() = f(β) δ, g() = f(α) γ. Epeid g () g () = (f (β) δ)(f (α) γ) < apì to je rhma Bolzano, up rqei ξ (, ) ste g(ξ) = f (αξ + β ( ξ)) = γξ + δ ( ξ) 'Ara, an θ = γξ + δ( ξ), oarijmìc ρ = αξ + β( ξ) ja eðnai rðza thc exðswshc f(x) =θ kai m lista ρ (α, β), afoô ξ (, ) H sun rthsh f eðnai ( gnhsðwc aôxousa. Poiì eðnai to eðdoc monotonðac thc sun rthshc g (x) =f e f( e x ) )? f(x) f(x) 584. Upojètoume ìti lim x x =. BreÐte to ìrio lim x 3x Upojètoume ìti α<γ<βkað ìti dôo sunart seic f,g eðnai orismènec kai suneqeðc sta (, β], [α, + ) antistoðqwc gia k je x [α, β] isqôei f(x) =g(x). Na apodeðxete ìti h sun rthsh eðnai suneq c. h (x) = { f (x), x < γ g (x), x γ

80 78.5. Ask seic se ìlo to kef laio 586. MÐa sun rthsh f orismènh sto R eðnai gnhsðwc aôxousa kai suneq c. Gia dôo arijmoôc a<bisqôei f (a)+f (b)+3=4f (a)+6f (b). BreÐte to f ([a, b]) 'Estw h sun rthsh f (x) = { ( ) (x +) x + x x x = Na apodeðxete ìti h sun rthsh aut, an kai asuneq c (na to elègxete) paðrnei ìlec tic timèc metaxô twn f (),f( ) MÐa om da oreibat n anebaðnei sthn koruf enìc bounoô akolouj ntac mða orismènh diadrom kai sthn sunèqeia katebaðnei akolouj ntac akrib c thn Ðdia diadrom. H an bash kai h kat bash di rkesan perissìtero apì 4 rec. Na apodeðxete ìti up rqei èna shmeðo thc diadrom c sto opoðo h om da brèjhke dôo forèc thn Ðdia ra dôo diaforetik n hmer n MÐa mh mhdenik sun rthsh f èqei pedðo orismoô to R kai isqôei f (x + y)+f (x y) =f (x) f (y) gia ìla ta x,y Na apodeðxete ìti:. f () =. H f eðnai rtia 3. f (x) =f (x) gia ìla ta x. 59. Gia thn sun rthsh f eðnai gnwstì ìti eðnai orismènh kai suneq c sto [, ] kai ìti f() =,f() =. PoioÐ apì touc parak tw isqurismoôc eðnai bèbaion ìti prokôptoun apì tic upojèseic?. Up rqei ξ [, ] ètsi ste gia k je x [, ] na isqôei f (ξ) f (x).. Up rqei ξ [, ] ètsi ste f (ξ) =. 3. Up rqei ξ [, ] ètsi ste f (ξ) = H f eðnai fjðnousa. 5. To sônolo tim n thc f eðnai to [, ]. 6. Up rqei ξ [, ] ètsi ste ( ) lim f (x) = f ( ξ ) x ξ

81 Kef laio. 'Oria kai Sunèqeia Sunart sewn H grafik par stash thc sun rthshc f (x) = αx + x + α dièrqetai apì to shmeðo A(, 7). Na breðte to f(7). 59. 'Estw f mða gnhsðwc fjðnousa kai suneq c sun rthsh orismènh sto sônolo twn pragmatik n arijm n. Na apodeðxete ìti:. HexÐswsh f (x) =x èqei akrib c mða lôsh.. To sôsthma èqei akrib c mða lôsh. x = f (y) y = f (z) z = f (x) 593. 'Estw mða sun rthsh f :Δ R kai x Δ. 'Estw d :Δ R h sun rthsh pou antistoiqðzei k je shmeðo M (x, f (x)) thc C f sthn apìstash tou apì to shmeðo P (x,f(x )). Na apodeðxete ìti:. An Δ=R tìte lim d (x) =+ x +. H f eðnai suneq c sto x an kai mìno an isqôei lim x x d (x) = 'Estw f (x) =x 3x +kai mða sun rthsh g : R R tètoia ste g(x) gia ìla ta x na isqôei (f g)(x). Na apodeðxete ìti lim x + f(x) = 595. Apì tic exet seic tou 995, Dèsmh I. 'Estw mða sun rthsh f : R R suneq c sto [α, β] kai oi migadikoð arijmoð z = α + if (α), w = f (β) +iβ me αβ. An w + z = w z na apodeðxete ìti h exðswsh f (x) =èqei mða toul qiston rðza sto di sthma [α, β] Xèroume ìti an mða suneq c sun rthsh f orðzetai se di sthma Δ kai den eðnai stajer tìte to sônolo tim n thc eðnai di sthma. DeÐxte ìti an to Δ eðnai kleistì di sthma kai to f(δ) eðnai epðshc kleistì di sthma Na breðte ìlec tic suneqeðc sunart seic f : R R gia tic opoðec isqôei (f (x)) 3 = x ( x +7xf (x)+6 (f (x) )) gia ìla ta x 'Estw f :[α, β] R mða sun rthsh. Upojètoume ìti h f paðrnei k je tim akrib c dôo forèc. Dhlad ìti gia k je x [α, β] up rqei akrib c èna x [α, β] ètsi ste f (x )=f (x ). Na apodeðxete ìti h f de mporeð na eðnai suneq c.

82 8.5. Ask seic se ìlo to kef laio 599. Na apodeðxete ìti k je kurt sun rthsh (deðte skhsh 833) eðnai suneq c. 6.. 'Estw f :[, ] R mða suneq c sun rthsh. Upojètoume ìti ( ) f () + 4f + f () = Na apodeðxete ìti h f èqei mða toul qiston rðzasto (, ).. QrhsimopoieÐste to prohgoômeno er thma gia na apodèixete 4 ìti h exðswsh 4αx 3 +3βx +γx = α + β + γ èqei mða toul qiston lôsh sto (, ). 6. 'Estw f : R R mða aôxousa sun rthsh tètoia ste gia ìla ta x na isqôei f (f (x)) = kx 3 ìpou k eðnai stajerìc pragmatikìc arijmìc. Na apodeðxete ìti an eðnai k h f ja eðnai gnhsðwc aôxousa.. Na breðte mða toul qiston sun rthsh me thn parap nw idiìthta stic peript seic ìpou k =, k>. 3. Na apodeðxete ìti den up rqei sun rthsh me thn parap nw idiìthta ìtan eðnai k<. 6. JewroÔme dôo suneqeðc sunart seic f,g orismènec sto di sthma Δ tètoiec ste na isqôei f (x) g (x) gia k je x Δ Rolle. Na apodeðxete ìti jaisqôei f (x) >g(x) gia ìla ta x eðte ja isqôei f (x) <g(x) gia ìla ta x. 'Estw h :Δ R suneq c tètoia ste na isqôei gia ìla ta x. (h (x) f (x)) (h (x) g (x)) = (a ) Na apodeðxete ìti an h C h tèmnei tic C f, C g se shmeða me tetmhmènec x,x tìte i. x x 4 Argìtera ja mporeðte na d sete mia pio eôkolh apìdeixh me b sh to je rhma tou

83 Kef laio. 'Oria kai Sunèqeia Sunart sewn 8 ii. H exðswsh h (x) = f(x)+g(x) èqei mða toul qiston rðza sto anoiktì di sthma me kra ta x,x. (b ) Na apodeðxete ìti jaisqôei h (x) =f (x) gia ìla ta x eðte ja isqôei h (x) =g (x) gia ìla ta x 63. H exðswsh tou Kepler. 'Estw κ, λ pragmatikoð arijmoð me κ, λ <.. Na apodeðxete ìti gia k je x exðswsh y = κx + λημy Johannes Kepler èqei akrib c mða lôsh y.. Na apodeðxete ìti h sqèsh orðzei mða sun rthsh f : R R f (x) =κx + λημf (x) 3. Gia th sun rthsh f tou prohgoumènou erwt matoc (): (a ) Na deðxete ìti eðnai -. (b ) Na breðte ton tôpo thc antðstrofhc thc. (g ) Na apodeðxete ìti gia k je x,x isqôei f (x) f (x ) k λ x x (d ) Na apodeðxete ìti h f eðnai suneq c. (e ) Na apodeðxete f() =. (± ) Na apodeðxete ìti lim f (x) =±, lim x + peript seic an loga me to prìshm tou lim x f (x). (z ) Na apodeðxete ìti to sônolo tim n thc f eðnai to R f (x) = (dôo x 64. 'Estw f mða sun rthsh me pedðo orismoô to R gia thn opoða isqôei gia ìla ta x,x. f (x ) f (x ) = x x. Na apodeðxete ìti gia dojèn x jaeðnai f (x) =f () + x eðte ja eðnai f (x) =f () x.. Na apodeðxete ìti an gia ta x,x isqôei f (x )=f ()+x kai f (x )= f () x tìte k poio apì ta x,x eðnai.

84 8.5. Ask seic se ìlo to kef laio 3. Na apodeðxete ìti: f (x) =f () + x gia ìla ta x eðte f (x) =f () x gia ìla ta x 65. Apì thn Kinèzikh Majhmatik Olumpi da, 983. 'Estw f :[, ] R gia thn opoða isqôei: f () = f () f (x) f (y) < x y gia ìla ta x y Na apodeðxete ìti gia gia ìla ta x, y isqôei f (x) f (y) < 66. Apì tic exet seic tou panepisthmðou tou Berkeley, 99. Na apodeiqjeð ìti an f : R R suneq c kai gia k je x y isqôei: f (x)) f (y) > x y ( ) tìte to sônolo tim n thc f eðnai to R. Upodeixh: DeÐxete ìti an h f ikanopoieð thn sunj kh ( ) to autì isqôei kai me thn h (x) = f (x) f () h opoða dièrqetai apì thn arq twn axìnwn. ArkeÐ na deðxete ìti h h èqei sônolo tim n to R. Efarmìste th sunj kh ( ) gia ta x, gia na deðte to mèroc tou epipèdou ìpou ja brðsketei h C h. Na k nete èna sq ma. DeÐxete ìti h h eðnai -. DeÐxte tèloc ìti dojèntoc m up rqei mða tim thc h pou xepern ei to m mða pou eðnai mikrìter tou. 67. JewroÔme f,g dôo suneqeðc sunart seic me pedðo orismoô to di sthma [α, β].. Na apodeðxete ìti an gia ìla ta x isqôei f (x) <g(x) tìte ja up rqei jetikìc arijmìc c ètsi ste gia ìla ta x na isqôei f (x)+c<g(x). Na apodeðxete ìti an oi f,g èqoun thn Ðdia mègisth tim tìte ja up rqei èna toul qiston x [α, β] ètsi ste f (x )=g (x ). 68. Apì tic exet seic tou panepisthmðou tou Berkeley, 995. 'Estw f,g :[α, β] R dôo suneqeðc sunart seic pou èqoun thn Ðdia mègisth tim. Na apodeðxete ìti up rqei t [α, β] ste: f (t)+3f (t) =g (t)+3g (t)

85 Kef laio. 'Oria kai Sunèqeia Sunart sewn Apì ton diagwnismì Harvard-MIT,. DÔo kôkloi èqoun diamètrouc Ðsec me d kai di kentro d. Me A(d) sumbolðzoume to embadìn tou kôklou me thn mikrìterh aktðna pou mporeð na perièqei touc dôo autoôc A(d) kôklouc. BreÐte to ìrio lim d + d. 6. 'Estw f : R R mða suneq c sun rthsh me lim f (x) = lim f (x) = x x + +. JewroÔme akìmh tic sunart seic : ϕ (x) = x x, x (, ) σ (x) = x + x, x R. Na apodeðxete ìti oi σ, ϕ eðnai antistrèyimec kai ìti h mða eðnai antðstrofh thc llhc. 5. Na apodeðxete ìti h σ eðnai gnhsðwc aôxousa. 3. 'Estw h sun rthsh ω :[, ] R me {, x = ± ω (x) = σ (f (ϕ (x))), <x< (a ) Na apodeðxete ìti h ω eðnai suneq c. (b ) Na apodeðxete ìti h ω èqei mègisth tim kai ìti ω (x) = x = ± (g ) Na apodeðxete ìti h ω èqei el qisth tim èstw m kað ìti m = σ (f (ϕ (x ))) gia k poio x (, ). (d ) Na apodeðxete gia to x tou prohgoumènou erwt matoc isqôei f (ϕ (x)) f (ϕ (x )) gia k je x (, ). 4. Na apodeðxete ìti h f èqei el qisth tim 'Estw f : R R mða suneq c sun rthsh tètoia ste f(x)f(f(x)) = gia k je x R. DÐnetai akìmh ìti f() =.. BreÐte to f(f()).. DeÐxte ìti up rqei a ste f(a) =. 3. BreÐte to f() 5 An èqete pant sei thn skhsh 45èqete apant sei se autì to er thma. 6 To er thma autì mporeð na apanthjeð kai autotel c dhlad qwrðc na qrhsimopoihjoôn oi σ, ϕ all qrhsimopoi ntac ton orismì tou orðou kai jewr mata suneq n sunart sewn.

86 84.5. Ask seic se ìlo to kef laio 6. 'Estw f (x) =x. JewroÔme thn sun rthsh g pou antistoiqðzei k je pragmatikì arijmì t sthn mègisth tim thc sun rthshc sto di sthma [t, t +]. BreÐte thn g kai exet ste an eðnai suneq c. 63. Gia thn sun rthsh f eðnai gnwstì ìti lim [f(x + h) f(x h)] = h ProkÔptei apì aut thn plhroforða ìti h f eðnai suneq c sto x? 64. Gia thn sun rthsh f eðnai gnwstì ìti gia k je t, x, y isqôei DeÐxte ìti eðnai stajer. f (( t) x + ty) =( t) f (x)+tf (y) 65. 'Estw f :Δ R suneq c kai gnhsðwc aôxousa, g :Δ R suneq c kai gnhsðwc fjðnousa me f (Δ) g (Δ). Tìte h exðswsh f (x) =g (x) èqei akrib c mða lôsh.

87 Kef laio 3 Diaforikìc Logismìc 3. Par gwgoi sunart sewn 3.. A' OMADA 66. Na upologðsete to f (x ) stic parak tw peript seic:. f(x) = x x+, x =. f(x) =x + ημx, x = π 4 3. f(x) =x ln x, x = e 4. f(x) = x + x, x = 5. f(x) =x x, x = 6. f (x) =xe x, x = 67. Na brejoôn oi par gwgoi twn sunart sewn: x. f (x) = x +. f (x) = x ex +x ln x 68. Na brejoôn oi par gwgoi twn sunart sewn: 3. f (x) = ex e x e x +e x 4. f (x) =ln x+ x. f (x) =αx 3 + βx + γx + δ. f (x) = β α α x 3. f (x) = β α x α 4. f (x) =(x α)(x β)(x γ) 69. Na brejoôn oi par gwgoi twn sunart sewn. f (x) = x ex +x ln x. g (x) =x + x x 6. Apì tic exet seic tou 98 (Panell niec G' LukeÐou). Oi sunart seic f(x),g(x) èqoun koinì pedðo orismoô to Δ R kai paragwgðzontai pantoô s utì. Epiplèon eðnai g(x) gia k je x Δ. JewroÔme t ra thn sun rthsh φ (x) = f(x) g(x). Na apodeiqjeð ìti an φ (p) = tìte eðnai φ (p) = f (p) g (p). 85

88 Par gwgoi sunart sewn 6. 'Ena par deigma tou Euler. Na apodeðxete ìti an f (x) =e eex tìte f (x) =e eex e ex e x Leonhard Euler Upojètoume ìti oi f,g eðnai dôo paragwgðsimec sunart seic orismènec se èna di sthma Δ. Na epalhjeôsete tic isìthtec:. ( f (x) g (x) ) =f (x) f (x) g (x)+f (x) g (x). ( f (x)+ g(x) g (x) (f (x)+) ) = f(x)f (x)g(x) f (x)g (x) g (x), g(x). 3. ( f (x +) g (x +) ) =f (x +)f (x +) g (x +)g (x +) 4. ( f (ln x) g ( x 3)) = 3f(ln x)g (x 3 )x 3 +f (ln x)g(x 3 ) x 5. ( f (x) e f(x) +lng (x) ) = f (x) e f(x) f (x)+e f(x) f (x)+ g (x) g(x), g (x) > ( 6. (f (x)) g(x)) =(g (x) f (x)lnf (x)+f (x) g (x)) (f (x)) g(x), f (x) > 63. JewroÔme tic sunart seic f,g pou eðnai dôo forèc paragwgðsimec. DeÐxte ìti: (f (g (x))) = f (g (x)) (g ) (x)+f (g (x)) g (x) 64. Sto sq ma apeikonðzetai h grafik par stash mðac sun rthshc f kai h efaptomènh thc C f sto shmeðo thca(, 4). Na breðte thn f (). A O C f 65. Oi sunart seic f, g eðnai orismènec sto R, paragwgðsimec kai isqôei f (x)+g (x) =x +x +4 gia ìla ta x. Na apodeðxete ìti isqôei f (x) f (x)+g (x) g (x) =x +

89 Kef laio 3. Diaforikìc Logismìc Na breðte thn exðswsh thc efaptomènhc thc C f stoshmeðothcm (x,f(x )) stic akìloujec peript seic:. f (x) =x + e x, x =. f (x) =x +lnx, x = 3. f (x) = x+ x, x = 4. f (x) =ημx + συνx, x = π 67. Gia tic paragwgðsimec sunart seic f : R R, g : R R isqôei f (x) g (x) =x. MporeÐ h grafik par stash kai thc f kai thc g na dièrqetai apì thn arq twn axìnwn? 68. Se poiì shmeðo thc grafik c par stashc thc f (x) = x3 h efaptomènh sqhmatðzei me ton xona xx gwnða 45 ; 3 5x +7x Na breðte thn exðswsh thc efaptomènhc thc C f pou dièrqetai apì to shmeðo M stic akìloujec peript seic:. f (x) =x, M (, ). f (x) = x, M (, ) 3. f (x) =x + x, M (, ) 4. f (x) =x + x, M (, ) 63. Na epalhjeôsete ìti:. H f (x) = x +ημx +3συνx ikanopoieð thn f (x)+f (x)+x =.. H f (x) = ln x +lnx +ikanopoieð thn f (x) x + f (x) x =. 3. H f(x) =ημx+συνx+e x ikanopoieð thn f (x)+f (x)+f (x)+f(x) =. 4. H f (x) =e x ikanopoieð thn f(x) f (x) + f (x) f(x) =. 63. An oi sunart seic f,g,h,s eðnai paragwgðsimec se èna di sthma deðxte ìti f (x) g (x) h (x) s (x) = f (x) g (x) h (x) s (x) + f (x) g (x) h (x) s (x) 63. Gia poièc timèc twn κ, λ h sun rthsh { x 3, x < f (x) = κx + λ, x eðnai paragwgðsimh sto α? 633. Gia poièc timèc tou α hklðsh thc grafik c par stashc thc f (x) = αx+α x+α sto shmeðo A (, ) eðnai Ðsh me.

90 Par gwgoi sunart sewn 634. 'Ena par deigma tou Newton. 'Estw ìti oi α, β eðnai stajerèc kai ìti oi paragwgðsimec sunart seic x (t),y(t),z(t) sundèontai me thn sqèsh: Na apodeðxete ìti x 3 (t) x (t) y (t)+α z (t) β 3 = 3x (t) x (t) x (t) y (t) x (t) y (t) y (t)+α z (t) = 635. Na apodeðxete ìti h y = px + q eðnai efaptomènh thc grafik c par stashc thc f (x) =αx + βx + γ se shmeðo me tetmhmènh s an kai mìno an isqôei p =αs + β, q = αs + γ. Sir Isaac Newton Na breðte thn koin efaptomènh twn grafik n parast sewn twn sunart sewn f (x) = 3 4 x kai g (x) = ( 9 x 3 6x +9x 9 ) (Leibniz, 7) 'Estw y (x) =u (x) v (x) ìpou oi u, v eðnai 3 forèc paragwgðsimec. Na apodeðxete ìti y = u v +u v + uv y = u v +3u v +3u v + uv Gottfried Wilhelm von Leibniz Na breðte ta α, β ètsi ste h eujeða y = x+3 na ef ptetai sthn grafik par stash thc sun rthshc f (x) =x +αx +3β sto shmeðo me tetmhmènh x = Na epalhjeôsete ìti oi grafikèc parast seic twn sunart sewn f (x) = x kai g (x) =x x + èqoun èna mìno koinì shmeðo kaiìti oi efaptomènec touc se autì to shmeðo eðnai k jetec. 64. Na apodeðxete ìti an tìte f (x) = συνx ( ) ημx = f (x) f (x) 64. H sun rthsh f eðnai orismènh sto R, dôo forèc paragwgðsimh kai isqôei f () = f () =. 'Estw g (x) =f (e x ). Na breðte to g (). 64. 'Estw h sun rthsh f (x) =αe x + βx. An f () =, f () = 3 na breðte to f () 'Estw h sun rthsh f (x) =x x. 'Estw A, B ta shmeða epaf c twn efaptomènwn thc C f pou dièrqontai apì to shmeðo M (, 3). Na breðte thn exðswsh thc eujeðac AB.

91 Kef laio 3. Diaforikìc Logismìc Na breðte thn exðswsh thc efaptomènhc thc grafik c par stashc thc { 3x f (x) = +, x < 4x x +, x sto shmeðo thc me tetmhmènh 'Estw f (x) =ημ (ln x). Na apodeðxete ìti x f (x)+xf (x)+f (x) = 646. 'Estw f (x) = px+ +e x. Na breðte gia poia tim tou p isqôei f ()+f () =. 3.. B' OMADA 647. 'Estw f (x) = x 3x +. Na exet sete an up rqei h par gwgoc thc f :. Sto x =. Sto x = 648. Gia thn sun rthsh f : R R eðnai gnwstì ìti Na breðte thn par gwgo thc sto. x + ημx f (x) x + ημx + x e x 649. Apì to majhmatikì diagwnismì Harvard-MIT, 4. 'Estw f paragwgðsimh. Upojètoume ìti h sun rthsh f(x) f(x) èqei par gwgo 5 sto x =kai par gwgo 7 sto x =. BreÐte thn par gwgo thc f(x) f(4x) sto x =. 65. Sto sq ma pou akoloujeð emfanðzetai sto Ðdio sôsthma axìnwn h grafik par stash twn sunart sewn f (x) =ημx, g (x) =ημx gia timèc tou x metaxô kai,5. H pr th sun rthsh antistoiqeð ston arijmì x (jewreðtai ìti ekfr zei rad) to hmðtono tou. H deôterh sun rthsh antistoiqeð ston arijmì x to hmðtono twn x moir n. Oi dôo autèc sunart seic eðnai diaforetikèc. L.q. f () = ημ =, 84 en f () = ημ =, 745. y = μx y = μx ffi

92 9 3.. Par gwgoi sunart sewn. Na deðxete ìti g (x) =ημ ( ) πx 8. Na breðte thn par gwgo thc g (x) 65. Na breðte ta α, β an me f (x) =αx + β x eðnai f() =,f () =. 65. Se k je x antistoiqoôme to shmeðo M(x, y) thc eujeðac x +3y =me tetmhmènh x. 'Estw d(x) h apìstash tou M apì thn arq twn axìnwn. Na breðte thn par gwgo thc d(x) Gia mða sun rthsh f : R R isqôei f (x + y) =f (x)+f (y) gia ìla ta x, y. Na apodeðxete ìti an h f paragwgðzetai se èna x paragwgðzetai se k je x Apì tic exet seic tou 999, Dèsmh IV. DÐnetai h sun rthsh f (x) =ημ (αx), x R. Na breðte thn tim tou α ste na isqôei f (x)+4α f (x) = gia k je x R 655. DÐnontai oi f (x) =e x kai g (x) =e x x. Na apodeðxete ìti (f (x) g (x)) = f (x) g (x) 656. 'Estw f (x) =6x 5 x 3. Na apodeðxete ìti ta shmeðaìpouhc f èqei orizìntia efaptomènh eðnai suneujeiak 'Estw topolu numo f (x) =α ν x ν + + α x + α. Na apodeðxete ìti h efaptomènh thc C f sto shmeðo thc me tetmhmènh eðnai h y = α x + α An f,g eðnai dôo sunart seic tìte lème ìti oi grafikèc touc parast seic C f, C g ef ptontai an èqoun koinì shmeðo sto opoðo èqoun koin efaptomènh. Na apodeðxete ìti me oi C f, C g ef ptontai. f (x) =lnx, g (x) = x e 659. Na breðte thn par gwgo thc sun rthshc { x f (x) = ημ x + x x x = 66. Na d sete èna par deigma mðac sun rthshc me pedðo orismoô to R pou:. Na paragwgðzetai se ìla ta x ektìc apì èna.. Na paragwgðzetai se ìla ta x ektìc apì dôo. 3. Na paragwgðzetai se ìla ta x ektìc apì peira.

93 Kef laio 3. Diaforikìc Logismìc Apì tic exet seic tou 997, Dèsmh I. Stic exet seic tou ekeðnou tou ètouc dìjhke to akìloujo jèma: Υποθέτουμε ότι υπάρχει πραγματική συνάρτηση g παραγωγίσιμη στο R τέτοια ώστε, υπάρχει πραγματικός αριθμός α, ώστενα ισχύει g (x + y) =e y g (x)+e x g (y)+xy + α για κάθε x, y R Νααποδείξετεότι i) g () = α ii) g (x) =g (x)+g () e x + x για κάθε x R Ac onom soume me (U) thn sqèsh thc upìjeshc.. Na jèsete x = y =sthn (U) gia na deðxete ìti ja eðnai g() = α. Na paragwgðsete thn (U) mða for wc proc x kai mða for wc proc y gia na deðxete ìti ja isqôei gia ìla ta x, y: g (x + y) =e y g (x)+e x g (y)+y (3.) g (x + y) =e x g (y)+e y g (x)+x (3.) e y ( g (x) g (x) ) e x ( g (y) g (y) ) +(y x) = (3.3) QrhsimopoieÐste ta prohgoômena gia na deðxete ìti: kai ìti gia ìla ta x, y isqôei: g() = (3.4) g (x) =g (x)+g () e x + x (3.5) 'Hdh metaxô llwn ja èqete apodeðxei ìti zhtoôse to jèma. 3. Me thn bo jeia twn (3.3), (3.5) deðxete ìti e y x e x y + y x = (3.6) 4. Na apodeðxete ìti sun rthsh me tic idiìthtec pou anafèrontai sto jèma den up rqei! 66. Na breðte ìlec tic parabolèc y = αx + βx + γ pou ef ptontai sthn eujeða y =3x +sto shmeðo A(, 5) 'Estw f (x) =e x, A shmeðo thc C f kai B h tom thc efaptomènhc thc C f sto A me ton xx. Na apodeiqjeð ìti h probol tou AB ston xx eðnai èna stajerì di nusma Na d sete èna par deigma dôo sunart sewn pou den paragwgðzontai sto x all to jroisma touc na paragwgðzetai sto x.

94 9 3.. Par gwgoi sunart sewn 665. Na breðte, effloson up rqei, koin efaptomènh twn grafik n parast sewn twn f (x) =x 3 8x 9, g (x) =x 3 + x 'Estw f :Δ R. Na apodeðxete ìti:. An Δ, f (x + y) = f (x) +f (y) gia ìla ta x, y kai h f eðnai paragwgðsimh sto tìte einai paragwgðsimh.. An Δ, f (x + y) =f (x) f (y) gia ìla ta x, y kai h f eðnai paragwgðsimh sto tìte einai paragwgðsimh. 3. An Δ, f (xy) =f (x) f (y) gia ìla ta x, y kai h f eðnai paragwgðsimh sto tìte einai paragwgðsimh. 4. An Δ, f (xy) =f (x) +f (y) gia ìla ta x, y kai h f eðnai paragwgðsimh sto tìte einai paragwgðsimh O arijmìc ρ lègetai dipl rðza tou poluwnômou P (x) an up rqei polu numo S(x) ètsi ste na isqôei P (x) =(x ρ) S (x), S(ρ) Na apodeðxete ìti èna polu numo P (x) me pragmatikoôc suntelestèc ja èqei ton ρ R dipl rðza an kai mìno an isqôei P (ρ) =P (ρ) =, P (ρ) 668. Apì tic exetaseic tou 994, Dèsmh I.. 'Estw ρ pragmatikìc arijmìc A(x), B(x) polu numa me pragmatikoôc suntelestèc ste B(ρ) kai to A(x) èqei bajmì megalôtero Ðso tou. Na apodeðxete ìti up rqei polu numo f(x) tètoio ste A (x) B (x) =(x ρ) f (x), an kai mìno an A (ρ) =A (ρ) =.. 'Estw ν akèraioc megalôteroc Ðsoc tou. Na breðte tic timèc twn κ, λ gia tic opoðec to polu numo Q (x) =x ν ( νx 3 + κx + λx +8 ) èqei par gonta to (x ) Na breðte ta shmeða thc grafik c par stashc thc f (x) =ημx ημ x, x [, π] sta opoða h efaptomènh eðnai par llhlh ston xona x x. 67. Gia thn sun rthsh f : R R eðnai gnwstì ìti eðnai suneq c kai ìti BreÐte thn f (). f (x) lim x ημx =5

95 Kef laio 3. Diaforikìc Logismìc Na apodeðxete ìti an gia thn suneq sun rthsh f isqôei lim x α f (x) (x α) = t R tìte h g (x) = f (x) paragwgðzetai sto x = α. 67. Na breðte thn exðswsh thc efaptomènhc thc grafik c par stashc thc { 3x f (x) = +, x < 4x x +, x sto shmeðo thc me tetmhmènh 'Estw h sun rthsh f (x) =αημx + βημx + γσυνx Na breðte ta α, β, γ ètsi ste na isqôei ( f (π) =f (π) =f π ) = 674. Gia tic sunart seic f, g eðnai gnwstì ìti eðnai orismènec sto R, paragwgðsimec sto -kaiìti isqôei gia ìla ta x.. Na breðte ta f ( ),g( ) f (x)+g (x) =x 6 +x 3 +. Na apodeðxete ìti f f(x) ( ) = lim x x+, g g(x) ( ) = lim x x+ 3. Na apodeðxete ìti (f ( )) +(g ( )) = Na apodeðxete ìti h sun rthsh f : Δ R eðnai paragwgðsimh sto x Δ an kai mìno an up rqei α R ètsi ste h sun rthsh { f(x) f(x ) g (x) = x x, x Δ, x x α, x = x na eðnai suneq c sto x 676. H parat rhsh tou Karajeodwr. Na apodeðxete ìti an mða sun rthsh f eðnai paragwgðsimh sto x tìte up rqei sun rthsh f tètoia ste KwnstantÐnoc Karajeodwr H f eðnai suneq c sto x

96 Par gwgoi sunart sewn Gia ìla ta x isqôei f (x) =f (x )+ f (x)(x x ) f(x )=f (x ) Na diatup sete kai na apodeðxete to antðstrofo Na apodeðxete ìti ta shmeða thc grafik c par stashc thc sun rthshc f (x) =6x 5 x 3 sta opoða h deôterh par gwgoc mhdenðzetai eðnai suneujeiak 'Estw h sun rthsh f (x) =εϕx. Na breðte ton suntelest dieujônsewc thc efaptomènhc thc C f se èna shmeðo thctou opoðou h tetagmènh eðnai Pwc prèpei na epilèxoume to α ste h efaptomènh thc grafik c par stashc thc f (x) =x 3 +x x +sto shmeðo A(α, f(α)) na dièrqetai apì to shmeðo M(, 9)? 68. Na apodeðxete ìti oi efaptomènec thc grafik c par stashc thc y = α x sta shmeða tom c thc me thn eujeða y = βx eðnai par llhlec. 68. EÐnai h eujeða y =x +3 efaptomènh thc grafik c par stashc thc y = x Apì to majhmatikì diagwnismì Harvard-MIT, 5. 'Estw f(x) =x 3 + ax + b, me a b, gia thn opoða upojètoume ìti h efaptìmenec thc grafik c par stashc thc sta shmeða me tetmhmènec x = a kai x = b eðnai par llhlec. BreÐte to f() MÐa eujeða me suntelest dieujônsewc λ tèmnei thn grafik par stash thc f (x) =x + αx + β se dôo shmeða me tetmhmènec x,x. Na apodeðxete ìti λ = f (x )+f (x ) 'Estw mða sun rthsh f orismènh se èna di sthma, h opoða paðrnei jetikèc timèc kai eðnai paragwgðsimh sto a. Na apodèixete ìti ln f (x) ln f (a) lim = f (a) x α x a f (a) 685. 'Estw f mða sun rthsh orismènh se èna di sthma kai x èna shmeðo f(x) f(x tou diast matoc autoô. An to ìrio lim ) x x x x up rqei kai eðnai pragmatikìc arijmìc tìte h f lègetai arister paragwgðsimh sto x kai h tim tou orðou arister par gwgoc thc f sto x. An loga orðzetai ènnoia thc dexi paragwgðsimhc sun rthshc.. Na apodeðxete ìti an h f eðnai arister kai dexi paragwgðsimh sto x tìte ja eðnai paragwgðsimh sto x an kai mìno n h arister kai h dexi par gwgoc th f eðnai Ðsec.

97 Kef laio 3. Diaforikìc Logismìc 95. Na apodeðxete ìti an h f eðnai arister kai dexi paragwgðsimh sto x tìte eðnai suneq c sto x 'Estw h sun rthsh f (x) = { x ημ x, x, x =. Na apodeðxete ìti h f eðnai paragwgðsimh me { f xημ (x) = x συν x, x, x =. Na melet sete wc proc thn sunèqeia thn f Apì tic exet seic tou, Jetik KateÔjunsh. Na qarakthrðsete tic prot seic pou akoloujoôn gr fontac sto tetr diì sac thn èndeixh Swstì L joc dðpla sto gr mma pou antistoiqeð se k je prìtash.. An h f eðnai paragwgðsimh sto x,tìte h f eðnai p ntote suneq c sto x.. An h f eðnai suneq c sto x,tìte h f eðnai paragwgðsimh sto x. 3. An h f èqei deôterh par gwgo sto x,tìte h f eðnai suneq c sto x 'Estw èna di sthma Δ kai x Δ. 'Estw f :Δ R mða sun rthsh pou eðnai suneq c sto x kai α>. Na apodeðxete ìti h sun rthsh { x x α f (x), x Δ, x x g (x) =, x = x eðnai paragwgðsimh sto x. DeÐxte akìmh ìti an epiplèon h f eðnai paragwgðsimh se k je shmeðo tou Δ ektìc Ðswc tou x kai α> tìte h g eðnai dôo forèc paragwgðsimh sto x 'Estw f : R R mða rtia sun rthsh h opoða eðnai paragwgðsimh sto x. Na apodeðxete ìti h f eðnai paragwgðsimh sto x kai isqôei f ( x )= f (x ). Se tð aplousteôetai h apìdeixh an gnwrðzoume ìti h f eðnai paragwgðsimh se ìlo to R? Na diatup sete kai na apodeðxete an logh prìtash gia peritt sun rthsh. 69. 'Estw mða sun rthsh f orismènh se èna di sthma Δ kai x Δ. 'Estw ìti to ìrio f (x + h) f (x h) ˆf (x ) = lim (3.7) h h up rqei kai eðnai pragmatikìc arijmìc. O arijmìc autìc onom zetai par gwgoc Riemann thc f sto x.

98 Par gwgoi sunart sewn. Na apodeðxete ìti an h f eðnai paragwgðsimh sto x tìte up rqei h par gwgoc Riemann thc f sto x kai isqôei ˆf (x )=f (x ). 'Estw h sun rthsh f (x) = x. Na apodeðxete ìti an kai h f den eðnai paragwgðsimh sto x isqôei ˆf () =. 3. Upojètoume ìti oi par gwgoi Riemann twn f,g up rqoun sto x. Na apodeðxete ìti to autì isqôei gia thn sun rthsh h = f + g kai ìti ĥ (x )= ˆf (x )+ĝ (x ) Georg Friedrich Bernhard Riemann 'Estw ìti h sun rthsh f eðnai orismènh se èna di sthma Δ eðnai treic forèc paragwgðsimh. Gia ta x pou de mhdenðzetai h f orðzetai h sun rthsh: f (x) = f (x) f (x) 3 h opoða onom zetai par gwgoc Schwarz thc f.. Na breðte thn f ìtan f (x) =e x ( f ) (x) f (x). Na breðte thn f ìtan f (x) =x +kai Δ=(, + ) 3. Na apodeðxete ìti an gia thn sun rthsh g isqôei f (x) g (x) =gia k je x Δ tìte ja isqôei f (x) =g (x). Hermann Amandus Schwarz 'Estw h kampôlh C me exðswsh x+ y = α kai h eujeða ε : x p + y q = Gia thn ε eðnai gnwstì ìti eðnai efaptomènh thc C. Na upologðsete to jroisma p + q O GalilaÐoc sto èrgo tou Di logoi perð dôo nèwn episthm n} (638) upèdeixe thn akìloujh mèjodo gia thn kat prosèggisin q raxh mðac parabol c: Krem me mða lept alussðda apì dôo karfi A,B kai apotup noume to sq ma thc (alussoeid c kampôlh). All zontac to m koc kai tic jèseic st rixhc ja èqoume diaforetikèc parabolèc.

99 Kef laio 3. Diaforikìc Logismìc 97 5 qrìnia met ènac apì touc adelfoôc Bernoulli, o Jacob Bernoulli br ke thn akrib majhmatik èkfrash thc alussoeidoôc kampôlhc: y = cx + c x, c >, c Na apodeðxete ìti h parap nw exðswsh de mporeð na eðnai exðswsh parabol c. Dhlad na apodeðxete ìti den up rqei parabol y = αx + βx + γ ètsi ste αx + βx + γ = cx + c x gia k je x R Gia mða sun rthsh f : R R isqôei f (x + y) f (x) =αx y + βxy + γy 3 Galileo Galilei gia k je x, y (α, β, γ stajerèc). Na apodeðxete ìti eðnai paragwgðsimh JewroÔme dôo sunart seic f,g pou eðnai paragwgðsimec sto α me g (α) f(x) f(α). Na apodeðxete ìti lim x α g(x) g(α) = f (α) g (α) 696. Na apodèixete ìti an h sun rthsh xf(x) èqei par gwgo sto x kai h f eðnai suneq c sto x tìte h f eðnai paragwgðsimh sto x. Pou qrhsimopoi sate ìti h f eðnai suneq c sto x? EÐnai aut h upìjesh aparaðthth? Jacob (Jacques) Bernoulli Apì tic exet seic tou 99, Dèsmh IV. DÐnetai h sun rthsh g h opoða eðnai orismènh sto R, dôo forèc paragwgðsimh s utì kai isqôei ìti g ( ) = 7. An h f eðnai mða sun rthsh me f (x) =3(x ) g (x 5) na apodeðxete ìti h f eðnai dôo forèc paragwgðsimh sto R kai na upologðsete thn f () Apo to shmeðo A (x,y ) gontai oi efaptomènec AB, AΓ proc thn grafik par stash thc sun rthshc f (x) =x. Poièc sunj kec prèpei na ikanopoioioôn oi x,y ste na isqôei AB = AΓ? 699. An f (x) = x x poia eðnai h exðswsh efaptomènhc thc C f sto shmeðo thc me tetmhmènh -? 7. H par gwgoc thc f eðnai gnwst. Poia eðnai h par gwgoc thc f ( ημ x ) + f ( συν x )?

100 O rujmìc metabol c 7. EÐnai gnwstì ìti gia x isqôei +x + x x ν = xν+ x B sei toôtou na upologðsete to jroisma: +x +3x νx ν 7. Na apodeðxete ìti h efaptomènh thc èlleiyhc x α thc M (x,y ), y > eðnai h eujeða x x α + y y β =. + y β =sto shmeðo 73. BreÐte efaptomènh thc grafik c par stashc thc y = e kx pou dièrqetai apì thn arq twn axìnwn. 74. Gia poièc timèc twn κ, λ, μ oi grafikèc twn sunart sewn f (x) =x + κx + λ g (x) =x 3 + μx èqoun koin efaptomènh sto shmeðo A(, 3)? 75. Na apodeðxete ìti oi efaptomènec twn grafik n parast sewn twn sunart sewn f (x) = x, g (x) =log x sta shmeða touc A (, ), B (, ) tèmnontai k jeta G' OMADA 76. 'Estw f mða sun rthsh pou eðnai treðc forèc paragwgðsimh se èna di sthma Δ kai thc opoðac oi par gwgoi paðrnoun jetikèc timèc. 'Estw u (x) = (f (x)) kai v (x) =f (x)(f (x)). Na apodeðxete ìti: 77. 'Estw f (x) = αx+β f isqôei u (x) u (x) v (x) v (x) = γx+δ. Na apodeðxete ìti gia thn ni-ost par gwgo thc f (ν) (x) = 3. O rujmìc metabol c 3.. A' OMADA ( βγ αδ γ ( ) ν ν! x + δ ) ν γ 78. Na breðte to rujmì metabol c tou y wc proc x ìtan:

101 Kef laio 3. Diaforikìc Logismìc 99. x +3y =4. xy =3 3. x + y =4,y > 4. e x + e y = 5. ln x +lny = 6. ( x + )( y + ) =,y > 79. Ta shmeða A, B kinoôntai stouc jetikoôc hmi xonec Ox, Oy kai h jèsh touc kat thn qronik stigm t eðnai A (x (t), ),B(,y(t)). y (t) B d(t) E(t) A O x (t). 'Estw ìti x (t) =3t, y (t) =4t. Na breðte to rujmì metabol c tou m kouc d (t) =AB kai tou embadoô E (t) =(OAB).. Na k nete ton Ðdio upologismì genik dhlad ìtan den xèrete tic sunart seic x (t), y (t). (Sto telikìapotèlesma ja up rqoun oi x (t), y (t) kai oi par gwgoi touc x (t), y (t)). 7. H jèsh enìc kinhtoô sto epðpedo eðnai to shmeðo M (x (t), y(t)) ìpou oi x (t), y(t) eðnai paragwgðsimec sunart seic. Poia sqèsh sundèei ta x (t),y(t),x (t),y (t) ìtan:. To kinhtì kineðtai ston kôklo x + y =. To kinhtì kineðtai sthn eujeða x + y =. 3. To kinhtì kineðtai sthn eujeða sthn kampôlh me exðswsh e x + e y =

102 3.3. Ta jewr mata Fermat, Rolle, Lagrange 3.. B' OMADA 7. Na breðte poia sqèsh sundèei thn taqôthta tou autokin tou kai tou rumoulkoumènou kat thn qronik stigm pou to deôtero apèqei apìstash s apì ton toðqo. s h 7. 'Ena ulikì shmeðo kineðtai ston xona xx ètsi ste gia k je qronik stigm t h jèsh tou eðnai s (t) = k t + c ìpou k, c eðnai jetikèc stajerèc. Na apodeðxete ìti h taqôthta tou υ(t):. EÐnai mikrìterh tou k.. TeÐnei sto k ìtan t Apì tic exet seic tou 993, Dèsmh I. DÐnetai orj gwnða xoy kai to eujôgrammo tm ma AB m kouc m tou opoðou ta kra A, B olisjaðnoun p nw stic pleurèc Oy kai Ox antistoðqwc. To shmeðo B kineðtai me stajer taqôthta v =m/ sec kai h jèsh tou p nw ston xona Ox dðnetai apì thn sun rthsh s (t) =vt, t [, 5] ìpou t o qrìnoc (se deuterìlepta). Na brejeð to embadìn E (t) tou trig nou AOB wc sun rthsh tou qrìnou.. Poiìc eðnai o rujmìc metabol c tou embadoô E(t) th stigm kat thn opoða to m koc tou tm matoc OA eðnai 6 m? 74. Apì tic exet seic tou 998, Dèsmh I. 'Enac gewrgìc prosjètei x mon dec lip smatoc se mia agrotik kallièrgeia kai sullègei g(x) mon dec tou paragìmenou pro ìntoc. An g (x) =M + M ( e μx ) ìpou M, M kai μ eðnai jetikèc stajerèc na ekfr sete to rujmì metabol c tou paragìmenou pro ìntoc wc sun rthsh thc g. Poia eðnai h shmasða thc stajer c M? 3.3 Ta jewr mata Fermat, Rolle, Lagrange 3.3. A' OMADA 75. Sta epìmena sq mata faðnetai h grafik par stash twn sunart sewn f, g. BreÐte se poia shmeða mhdenðzetai h par gwgoc touc kai poi apì aut eðnai topik akrìtata.

103 Kef laio 3. Diaforikìc Logismìc O C f C g O 76. Gia tic parak tw sunart seic na gr yete èna kat logo pijan n jèsewn akrot twn sômfwna me to je rhma tou Fermat.. f (x) = x x. f (x) = 3 x3 x 3. f (x) = 4 x4 3 x3 x +x 4. f (x) = x 4x +6ln(x +) 5. f (x) = ex 3e x +x 6. f (x) =lnx x +ln(x +) 77. AfoÔ elègxete stic epìmenec peript seic, ìti h f ikanopoieð tic upojèseic tou jewr matoc tou Rolle na breðte ta ξ pou mhdenðzoun thn f.. f :[, ] R, f (x) =x +. f :[ 3, ] R, f (x) =x 3 +3x + 3. f : [, ] R, f (x) = x +x+ x + 4. f :[, ln ] R, f (x) =e x 3e x +, 5. f :[, ] R, f (x) = x x { x = 6. f :[, ] R, f (x) = x ln x <x 78. AfoÔ elègxete stic epìmenec peript seic, ìti h f ikanopoieð tic upojèseic tou jewr matoc mèshc tim c tou Lagrange na breðte ta ξ gia ta opoða f (ξ) = f(β) f(α) β α.. f :[, 4] R, f (x) =x +. f :[, 4] R, f (x) =x 3 + x + 3. f :[, 4] R, f (x) = x+3 x+ 4. f :[, 4] R, f (x) = x x 5. f :[, 4] R, f (x) =x +

104 3.3. Ta jewr mata Fermat, Rolle, Lagrange 6. f :[, 4] R, f (x) = x +5x Apì tic exet seic Mathematical Tripos, Cambridge, 935. Na breðte to ξ tou jewr matoc mèshc tim c gia thn sun rthshsh f (x) = x (x ) (x ) sto di sthma [, ] B' OMADA 7. DÐnetai h sun rthsh f (x) = x + x. Na apodeðxete ìti eðnai paragwgðsimh kai na breðte thn par gwgo thc.. Na efarmìsete to je rhma mèshc tim c sto di sthma [, ] 7. Na apodeðxete ìti an dôo sunart seic ikanopoioôn tic upojèseic tou jewr matoc tou Rolle sto di sthma [α, β] kai den èqoun par llhlec efaptomènec tìte ja èqoun toul qiston mða koin efaptomènh. 7. Sto sq ma up rqei h grafik par stash twn shmeðwn M(x, y) pou ikanopoioôn thn sqèsh: (y +) 3 = x () O. H sqèsh () orðzei mða sun rthsh y = f(x). Na breðte ton tôpothc.. Na breðte se poia x paragwgðzetai h f. 3. Sta kra k je diast matoc thc morf c [ α, α] h f paðrnei Ðsec timèc. EntoÔtoic den èqei efaptomènh par llhlh ston xx. (Na elègxete autì ton isqurismì exet zontac an h f mhdenðzetai). 'Erqetai to parap nw se antðjesh me to je rhma tou Rolle? 73. 'Estw f, g dôo paragwgðsimec sunart seic me pedðo orismoô to [, ] tètoiec ste: f() = g() = f(x), g(x) gia k je x (, )

105 Kef laio 3. Diaforikìc Logismìc 3 Na apodeðxete ìti up rqei ξ (, ) ètsi ste: f (ξ) f (ξ) + g (ξ) g (ξ) = 74. Apì tic exet seic tou PanepisthmÐou tou DelqÐ, 984. Na apodeðxete ìti den up rqei k tètoioc ste h exðswsh x 3 x + k =na èqei dôo di forec rðzec sto (, ). 75. Apì tic exet seic Mathematical Tripos, Cambridge, 99. Na apodeðxete ìti an tìtehexðswsh èqei mða toul qiston rðza. α n + + α n α n + α n =, α x n + α x n α n x + α n = 76. 'Estw f (x) =x ln x kai arijmoð α, β me <α<β. Na apodeðxete ìti to ξ tou jewr matoc mèshc tim c gia thn f sto di sthma [α, β] eðnai o arijmìc ξ = e β β β α α α β α 77. 'Estw f (x) =x (x ) (x ) (x 3). Na apodeðxete ìti h par gwgoc thc f èqei akrib c treic rðzec pou an koun sto di sthma (, 3). 78. Na apodeðxete ìti h exðswsh e x =(e ) x èqei akribwc treic rðzec tic dôo profaneðc kai akìmh mða sto di sthma (, 3). 79. Na apodeðxete ìti an h par gwgoc mðac sun rthshc orismènhc se èna di sthma de èqei rðzec tìte h sun rthsh eðnai Apì tic exet seic tou 983, Dèsmh I. H sun rthsh f orismènh kai suneq c sto kleistì di sthma [α, β] èqei par gwgo sto anoiktì di sthma (α, β) kai f (α) =f (β) =. Na apodeiqjeð:. 'Oti gia th sun rthsh F (x) = f(x) x c,ìpouc/ [α, β] up rqei c (α, β) tètoio ste F (c )=.. An c / [α, β], ìti up rqei c (α, β) ste h efaptomènh sto shmeðo (c,f(c )) thc gramm c me exðswsh y = f (x) dièrqetai apì to shmeðo (c, ). 73. SthriqjeÐte sto je rhma tou Rolle ìpwc diatup netai sto biblðo sac gia na apodeðxete thn akìloujh genðkeush tou:

106 Ta jewr mata Fermat, Rolle, Lagrange Estw mða sun rthsh orismènh kai paragwgðsimh sto di sthma (α, β) (α, β R) tètoia ste lim x α x β f (x) = lim f (x) =L + ìpou L R L = ±. Tìte up rqei ξ (α, β) ètsi ste f (ξ) =. Sth sunèqeia:. SkefjeÐte pwc rage ja mporoôse na apodeiqjeð to je rhma tou Rolle ìpwc diatup netai sto biblðo xekin ntac apì thn parap nw genðkeush.. SkefjeÐte pwc ja mporoôsame sthn genðkeush na sumperil boume kai diast mata ìpou k poio apì ta kra eðnai ±. 73. 'Estw h sun rthsh f (x) =(x α) μ (x β) ν me μ, ν jetikoôc akeraðouc kai x [α, β]. Na apodeðxete ìti gia k je x (α,β) isqôei: ( μ f (x) =f (x) x α + ν ) x β. AfoÔ epalhjeôsete ìti h f ikanopoieð tic upojèseic tou jewr matoc tou Rolle na apodeðxete ìti to ξ pou mhdenðzei thn f eðnai to ξ = ν μ + ν α + μ μ + ν β 733. 'Estw ìti h par gwgðsimh sun rthsh f èqei sto di sthma Δ toul qiston ν rðzec. Na deðxete ìti h sun rthsh ( f (x) ) èqei toul qiston ν rðzec Apì tic exet seic tou 99, Dèsmh I. JewroÔme th sun rthsh f me ( ) f (x) = αx3 β δ x +(γ δ) x + δ ìpou oi α, β, γ, δ eðnai pragmatikoð arijmoð kai isqôei 3 + β + γ =. Na apodeðxete ìti up rqei ξ (, ) tètoio ste h efaptomènh thc grafik c par stashc thc f sto shmeðo (ξ,f (ξ)) na eðnai par llhlh proc ton xona x x Apì tic exet seic tou 99, Dèsmh I.. Na apodeiqjeð ìti mða sun rthsh f orismènh sto R èqei thn idiìthta f = f an kai mìno an f (x) =ce x,ìpouc pragmatik stajer. α

107 Kef laio 3. Diaforikìc Logismìc 5. Na brejeð sun rthsh g orismènh sto di sthma ( π, π ), h opoða ikanopoieð tic sqèseic g (x) συνx + g (x) ημx = g (x) συνx kai g () = Apì tic exet seic tou 996, Dèsmh I. DÐnontai o pragmatikèc sunart seic f,g pou èqoun pedðo orismoô to sônolo R. An oi f kai g èqoun suneqeðc pr tec parag gouc kai sundèontai metaxô touc me tic sqèseic f = g, g = f tìte na apodeðxete oti up rqoun oi sunart seic f kai g kai eðnai suneqeðc. ApodeÐxte akìma ìti isqôoun oi sqèseic f + f = g + g = kai ìti h sun rthsh h = f + g eðnai stajer To je rhma mèshc tim c tou Cauchy. 'Estw f, g dôo sunart seic orismènec sto di sthma [α, β] pou ikanopoioôn tic upojèseic tou jewr matoc mèshc tim c. Epiplèon upojètoume ìti h par gwgoc thc g de mhdenðzetai kai ìti g (α) g (β). Na apodeðxete ìti up rqei èna toul qiston ξ (α, β) tètoio ste f (β) f (α) g (β) g (α) = f (ξ) g (ξ) 738. Gia thn paragwgðsimh sun rthsh f orismènh sto [α, β] isqôei f(α) = f(β) =, Na apodeðxete ìi up rqei ξ (α, β) ste f (ξ) =f(ξ). Augustin Louis Cauchy 'Estw f, g orismènec kai paragwgðsimec sto Δ gia tic opoðec isqôei f (x) g (x) f (x) g (x) gia ìla ta x. Na apodeðxete ìti metaxô dôo opoiwnd pote riz n thc f up rqei rðza thc g. 74. 'Estw f sun rthsh orismènh sto [α, β] h opoða ikanopioeð tic upojèseic tou jewr matoc mèshc tim c. Na apodeðxete ìti up rqei, kat perðptwsh, ξ [α, β] ètsi ste:. f (β) e β ξ f (α) e α ξ =(β α)(f (ξ)+f (ξ)). f (ξ) f(ξ) = ξ α + ξ β ìpou f (x) sto (α, β) Upodeixh:. Efarmìste to je rhma mèshc tim c sthn f (x) e x. Efarmìste to je rhma Rolle sthn f (x)(x α)(x β) 74. 'Estw f : R R mða paragwgðsimh sun rthsh. Na apodeðxete ìti an oi f kai f eðnai rtiec sunart seic tìte h f eðnai stajer.

108 Ta jewr mata Fermat, Rolle, Lagrange 74. Apì tic exet seic tou PanepisthmÐou tou DelqÐ, 989. DeÐxte ìti an h sun rthsh f eðnai paragwgðsimh sto [α, β] kai h par gwgoc thc eðnai suneq c sto [α, β] kai paragwgðsimh sto (α, β) tìte up rqei ξ (α, β) ètsi ste f (β) =f (α)+(β α) f (α)+ (β α) f (ξ) G' OMADA 743. Upojètoume ìti oi c,c,..., c ν eðnai ana di foroi tou kai oi arijmoð t,f,..., t ν eðnai ana dôo di foroi. Na apodeðxete ìtihexðswsh c x t + c x t c ν x tν = èqei to polô ν rðzec sto di sthma (, + ). Brook Taylor To je rhma tou Taylor. 'Estw mða sun rthsh f orismènh sto di sthma Δ=[x h, x + h] h opoða eðnai n forèc paragwgðsimh.. JewroÔme thn sun rthsh n F (t) =f (x + h) i! f (i) (t)(x + h t) i, t Δ Na apodeiqjeð ìti: i= F (t) = f (n) (t) (n )! (x + h t) n. JewroÔme thn sun rthsh ( ) x + h t n G (t) =F (t) F (x ) h Na apodeiqjeð ìti: G (x + h) =G (x )= 3. Na apodeiqjeð ìti up rqei ξ (x h, x + h) tètoio ste f (x + h) =f (x )+! f () (x ) h+...+ (n )! f (n ) (x ) h n + n! f (n) (ξ) h n (TÔpoc tou Taylor) 4. Na efarmosjeð to prohgoômeno gia na apodeiqjeð ìti up rqei ξ metaxô tou kai tou x ètsi ste e x =+! x +! x + 3! x (n )! xn + n! xn e ξ

109 Kef laio 3. Diaforikìc Logismìc O rìloc thc pr thc parag gou 3.4. A' OMADA 745. Na apodeðxete ìti oi parak tw sunart seic eðnai gnhsðwc aôxousec:. f (x) =x 3 +3x +6x +. f (x) = ln x + x 3. f (x) =x 3 +6e x + e 4. f (x) = 3e x e x x xe x 746. Na breðte ta diast mata sta opoða oi parak tw sunart seic eðnai gnhsðwc aôxousec.. f (x) =3x 4 x +. f (x) =e x e x +8x + 3. f (x) =x 3 +3x +6x +ln x 4. f (x) = x8 4x Na breðte th mègisth kai thn el qisth tim thc sun rthshc f (x) = x 8 + x me pedðo orismoô to di sthma [, 6] 748. Na breðte th mègisth kai thn el qisth tim thc sun rthshc f (x) = x 5 x 3 + x +sto di sthma [, ] Na breðte th mègisth kai thn el qisth tim thc sun rthshc f (x) = x 5x +6 sto di sthma [, ]. 75. 'Estw h sun rthsh f :[, 4] R me f (x) =x +3x +.. Na apodeðxete ìti h f eðnai gnhsðwc aôxousa.. Na breðte to sônolo tim n thc f. 75. AfoÔ melet sete wc proc th monotonða th sun rthsh na breðte to sônolo tim n thc. f (x) =x 3 9x +x +, x [, 3] 75. Na apodeðxete ìti gia thn sun rthsh f (x) = x +x+ isqôei f (x) 3. x Na apodeðxete ìti gia k je x [ 3 4, ] isqôei 3 x x

110 O rìloc thc pr thc parag gou 754. Poiì eðnai to sônolo tim n thc sunarthshc f (x) = x x 3x+3? 755. Na apodèixete ìti gia k je x [, π ] isqôei συνx ημx Upodeixh: Me f (x) =συνx ημx, x [, π sun rthsh f. ] eðnai f (x) = 3συ νx. BreÐte th monotonða thc ημx 756. Na apodeðxete ìti gia ìlec tic pragmatikèc timèc tou x isqôei x +x 757. Na breðte th mègisth tim thc sun rthshc f (x) =+4x ln 9 3 x 3 3 x 758. Apì tic exet seic tou 98.. Na brejoôn oi pragmatikoð arijmoð α, β ste h sun rthsh y = αx 3 + βx 6x +na dèqetai topik akrìtata sta shmeða x =kai x =. Na melethjeð h monotonða thc parap nw sun rthshc, afoô antikatastajoôn ta α, β me tic timèc touc Na breðte thn el qisth tim thc sun rthshc f (x) = x + x B' OMADA 76. Gia mða sun rthsh f : R R eðnai gnwstì ìti eðnai paragwgðsimh kai ìti isqôei f (x) > gia ìla ta x. Na apodeðxete ìti h f den èqei mègisth tim. IsqÔei to prohgoômeno sumpèrasma an antð tou R èqoume èna anoiktì di sthma Δ. 'Ena kleistì? 76. Na breðte shmeðo sthn grafik par stash thc y = x pou apèqei apì to shmeðo A(, ) el qisth apìstash. 76. DÔo jetikoð arijmoð èqoun jroisma tetrag nwn Ðso me 5. Pìte to jroisma touc gðnetai mègisto? 763. 'Estw mða paragwgðsimh sun rthsh f :Δ R ìpou to Δ eðnai èna anoiktì di sthma. 'Estw A èna shmeðo tou epipèdou. Na apodeðxete ìti an èna shmeðo B thc C f apèqei apì to A el qisth apìstash tìte h efaptomènh thc C f sto B eðnai k jeth sthn AB DÐnontai oi sunart seic f (x) =x g (x) =px

111 Kef laio 3. Diaforikìc Logismìc 9. Na apodeðxete ìti gia k je tim tou p oi grafikèc touc parast seic tèmnontai se dôo shmeða A, B.. Na ekfr sete thn apìstash AB wc sun rthsh tou p. 3. Na breðte gia poia tim tou p h apìstash AB gðnetai el qisth MÐa etaireða stamat k je diaf mish pro ìntwn thc. Diapist jhke ìti oi pwl seic twn pro ìntwn thc rqisan na pèftoun kai pio sugkekrimèna ìti: O rujmìc pt shc twn pwl sewn kai oi pwl seic kat opoiad pote qronik stigm t èqoun stajerì lìgo. Thn stigm pou stam thse h diaf mish h epiqeðrhsh pwloôse 5 mon dec pro ìntoc thn hmèra. Met apì hmèrec oi pwl seic èpesan stic 3 mon dec. Pìsec ja eðnai oi pwl seic met apì 3 hmèrec; 766. O plhjusmìc Π twn katoðkwn mðac pìlhc aux netai wc proc ton qrìno me rujmì ( ln 5 )( ) t 5 Π 4 4 ìpou Π eðnai o plhjusmìc thc pìlhc kat thn qronik stigm t =. Se pìsa qrìnia o plhjusmìc ja eðnai eπ? 767. H k je selðda enìc biblðou prèpei na èqei KeÐmeno-eikìnec embadoô S Arister -Dexi perij rio α 'Anw-K tw perij rio β Pwc prèpei na epilegoôn oi diast seic thc selðdac ètsi ste na èqei to el qisto dunatì embadìn? 768. JewroÔme jetikoôc arijmoôc x, y me jroisma k kai m, n jetikoð akèraioi. Na apodeðxete ìti h mègisth tim thc par stashc x m y n eðnai Ðsh me mm n n k m+n (m+n) m+n Na melet sete wc proc th monotonða thn sun rthsh f (x) =(+x) n + ( x n ), ìpou n jetikìc akèraioc. 77. Apì tic exet seic tou 996, Dèsmh I. Na apodeðxete tic anisìthtec:. ημx < x, x>. ημx > x x3 3, x> 77. Na apodeðxete ìti gia k je x> isqôei: ln ( + x) <x x + x3 3

112 3.4. O rìloc thc pr thc parag gou 77. Stic epìmenec peript seic ta x, y eðnai jetikoð arijmoð kai metab llontai ètsi ste na ikanopoieðtai mða sqèsh. ZhteÐtai na exet sete an h paratijèmenh par stash twn x, y parousi zei mègisto el qisto. Sunj kh Par stash x + y = α xy xy = α x + y x + y = α x + y x + y = α x + y x + y = α x + y x + y = α x + y 773. MÐa paragwgðsimh sun rthsh f èqei pedðo orismoô to R, èqei suneq par gwgo kai den èqei akrìtata. Na apodeðxete ìti eðnai monìtonh Sto sq ma apeikonðzetai mða teqnht lðmnh sq matoc orjogwnðou. x P B xx xx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx ff xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx A fi 'Enac ajlht c o opoðoc mporeð na kolump sei me taqôthta υ kai na trèxei me taqôthta υ prìkeitai na metakinhjeð apì to A sto B. Pwc prèpei na epilegeð h diadrom AP B ste na apaithjeð o el qistoc qrìnoc? 775. 'Estw h sun rthsh g (x) =(+x) x. Na kajorðsete to pedðo orismoô thc D g. Na melet sete thn g wc proc th monotonða. 3. Na apodèixete ìti gia k je x D g isqôei ( + x) x 776. Apì tic exet seic tou PanepisthmÐou tou Berkeley, 996. Na apodeðxete ìti gia th jetik stajer t isqôei h isodunamða: e x >x t gia ìla ta x> t<e 777. Poio shmeðo thc kampôlhc y = x +7x+ eðnai plhsièstera sthn eujeða y =3x 3?

113 Kef laio 3. Diaforikìc Logismìc y = x +7x + P ( ; 8) y = 3x Na breðte tic timèc twn α, β R gia tic opoðec h sun rthsh f (x) =α (ln x + x)+βx +3x + parousi zei topik akrìtata sta x =,x =. Na kajorðsete to eðdoc twn akrot twn Na apodeðxete ìti gia k je α (, + ), n [, + ) isqôei h anisìthta tou Bernoulli: ( + α) n +nα 78. Apì tic exet seic tou, Jetik KateÔjunsh. H sun rthsh f eðnai paragwgðsimh sto kleistì di sthma [, ], kai isqôei f (x) > gia k je x (, ). An f() = kai f() = 4, na deðxete ìti:. h eujeða y =3tèmnei th grafik par stash thc f sflena akrib c shmeðo me tetmhmènh x (, ). up rqei x (, ), tètoio ste f (x )= f ( / 5) + f ( / 5) + f (3 / 5) + f (4 / 5) 4 3. up rqei x (, ), ste h efaptomènh thc grafik c par stashc thc f sto shmeðo M(x,f(x )) na eðnai par llhlh sthn eujeða y =x Gia th sun rthsh f : R R èinai gnwstì ìti eðnai treic forèc paragwgðsimh kai ìti isqôei f (x) =x ( +f (x) ) gia ìla ta x Na apodeðxete ìti h f eðnai stajer. (Prosèxte idiaitèrwc thn tim x =). An epiplèon f() = breðte thn f. 78. Apì tic exet seic tou 999, Dèsmh IV. 'Estw h sun rthsh f paragwgðsimh sto [, + ), gia thn opoða isqôei [f (x)] 5 +[f (x)] 3 +3f (x) =(x +)ln(x +) 4 5 x x + x gia k je x [, + ). x [, + ). Na apodeðxete ìti h f eðnai gnhsðwc aôxousa sto

114 3.4. O rìloc thc pr thc parag gou 783. Apì tic exet seic tou, Jetik KateÔjunsh. Th qronik stigm t =qorhgeðtai sflenan asjen èna f rmako. H sugkèntrwsh tou farm kou sto aðma tou asjenoôc dðnetai apì th sun rthsh αt f (t) = ), t +( t β ìpou α kai β eðnai stajeroð jetikoð pragmatikoð arijmoð kai o qrìnoc t metr tai se rec. H mègisth tim thc sugkèntrwshc eðnai Ðsh me 5 mon dec kai epitugq netai 6 rec met th qor ghsh tou farm kou.. Na breðte tic timèc twn stajer n α kai β.. Me dedomèno ìti h dr sh tou farm kou eðnai apotelesmatik, ìtan h tim thc sugkèntrwshc eðnai toul qiston Ðsh me mon dec, na breðte to qronikì di sthma pou to f rmako dra apotelesmatik Na lôsete thn exðswsh 9 x +3 x =5 x +7 x 785. Apì tic exet seic tou 994, Dèsmh I. An h sun rthsh f eðnai paragwgðsimh sto di sthma [,e] me <f(x) < kai f (x) gia k je x [,e], na apodeðxete ìti up rqei ènac mìno arijmìc x (,e) tètoioc ste f (x )+x ln x = x 786. Na lôsete to prìblhma 774 sthn perðptwsh pou h lðmnh èqei sq ma kôklou aktðnac ρ kai ta A, B eðnai antidiametrik. xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx P B xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx ρ xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx O xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx ρ xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx A 787. Pìsec lôseic èqei h exðswsh x 3 7x +4x 7=? 788. Pìsec rðzec èqei to polu numo f (x) =x 4 +4x +? 789. Na apodeðxete ìti h exðswsh x 3 3x += sto di sthma (, ) èqei akrib c mða rðza.

115 Kef laio 3. Diaforikìc Logismìc 'Estw f (x) =x e x Na apodeðxete ìti:. f (x) =f (x)+xe x +. lim f (x) =+ x + 3. lim f (x) = x 4. H f èqei dôo mìno rðzec. 5. H f èqei sto R mða mìno rðza. 79. 'Estw h sun rthsh f (x) =x λ e λ x me λ>.. Na apodeðxete ìti h mègisth tim thc f eðnai (λe) λ. Pwc prèpei na eklegeð o λ ètsi ste h mègisth tim thc f na eðnai h el qisth dunat? 79. 'Estw f (x) =e x 3 x. Na melethjeð h f wc proc th monotonða kai na brejeð to sônolo tim n thc.. Na brejeð to pl joc twn riz n thc exðswshc e x 3 x λ = 793. Na breðte to pl joc twn riz n thc exðswshc: x 3 x = λ 794. Na apodeðxete ìti paragwgðsimec sunart seic f : R R pou èqoun thn idiìthta f + f = eðnai akrib c oi sunart seic thc morf c ìpou c R. f (x) =ce x

116 O rìloc thc pr thc parag gou 795. Apì tic exet seic tou 984, Dèsmh IV. 'Estw h pragmatik sun rthsh y thc pragmatik c metablht c x me y = x + 4 x.. Na breðte to pedðo orismoô kai to sônolo tim n thc y.. Na exet sete thn y wc proc th monotonða se k je èna apì ta diast mata (, ] kai [, + ) 'Estw f :[α, β] R paragwgðsimh tètoia ste h f èqei dôo mìno rðzec x,x me α<x <x <β.. Na apodeðxete ìti se k je èna apì ta diast mata h f èqei to polô mða rðza. [α, x ], [x, x ], [x, β]. Na apodeðxete ìti an to x eðnai rðza thc f tìte h f den èqei rðza sto [α, x ) (x,x ]. ('Omoia an to x eðnai rðza thc f tìte f den èqei rðza sto [x,x ) (x,β]) 3. Na apodeðxete ìti an to α eðnai rðza thc f tìte h f den èqei rðza sto (α, x ] ('Omoia an to β eðnai rðza thc f tìte h f den èqei rðza sto [x,β) ) 4. Na apodeðxete ìti h f èqei rðza sto (α, x ) an kai mìno an f(α)f(x ) < 797. Na breðte pìsec rðzec thc sun rthshc f (x) =x 3 6x +9x an koun sto di sthma [, 3] Na apodeðxete ìti gia k je zeôgoc migadik n arijm n α, β kai gia k je jetikì pragmatikì r> isqôei: α + β r r ( α r + β r ) Upodeixh: Na exet sete qwrist thn tetrimènh perðptwsh ìpou β =. Me β, lìgw thc ( trigwnik c anisìthtac arkeð na apodeðxoume ìti r α β r ) ( α r + β +). ErgasjeÐte me thn sun rthsh f (x) = r (x r +) (x +) r, x EÐnai dunatìn na apodeiqjeð me mejìdouc thc 'Algebrac (mhn epiqeir sete na to apodeðxete) ìti exðswsh x 5 5x 4 +5= (E) den epilôetai (dhlad den eðnai dunatì na broôme tôpouc pou na parèqoun tic rðzec thc me qr sh twn tess rwn pr xewn kai exagwg riz n). Na apodeðxete ìti h (E) èqei treic pragmatikèc rðzec.

117 Kef laio 3. Diaforikìc Logismìc 5 8. Na breðte to pl joc twn riz n thc exðswshc συνx = x. 8. Na breðte to pl joc twn riz n thc exðswshc ln x = x. 8. Na apodeðxete ìti. x x>xln x>x. x > x ln x>(x ), x > 83. Apì tic exet seic tou 999, Dèsmh IV. Na apodeðxete ìti gia k je x [, + ) 84. Na apodeðxete ìti h ln (x +)>x x 5 f (x) =x 4 +x 6x + èqei to polô dôo pragmatikèc rðzec. 85. DÐnontai oi sunart seic f (x) =e x +ln x+x +5, g (x) = x +x+3. Na breðte thn mègisth tim thc sun rthshc f g. 86. Na lôsete thn exðswsh lim y x y ex = 87. Na apodeðxete ìti gia α> h sun rthsh eðnai gnhsðwc aôxousa. ϕ (x) = eαx x + α 88. Apì to diagwnismì Putnam, 948. Poia eðnai h mègisth tim tou z 3 z + ìtan z eðnai migadikìc arijmìc me z =? 89. 'Ena komm ti qartìni sq matoc orjogwnðou parallhlogr mmou èqei diast seic α < β. Apì autì apokìptoume tèssera tetr gwna pleur c x (blèpe sq ma) kai kataskeu zoume èna koutð anoiktì ep nw. Pwc prèpei na epilegeð to x ste to koutð na èqei thn mègisth dunat qwrhtikìthta?

118 O rìloc thc pr thc parag gou α β x x β x α x x α x β x 8. H arq tou Fermat. 'Ena kinhtì mporei na kinhjeð sto mèso M me taqôthta υ kai sto mèso M me taqôthta υ. Ta dôo mèsa diaqwrðzontai apì thn eujeða ε. A M x " S y M B Na apodeðxete ìti an kinhjeð apì to A sto B kat th diadrom ASB o qrìnoc kðnhshc ja eðnai el qistoc efìson to shmeðo S epilegeð ètsi ste na isqôei: Pierre Fermat 'Estw λ R kai h sun rthsh ημx ημy = υ υ f (x) =λx 3 + x +( 3λ) x +λ. Na apodeðxete ìti anex rthta apì to poioc eðnai o λ h grafik par stash thc f (x) dièrqetai apì dôo stajer shmeða ta opoða kai na prosdiorðsete.. Na melet sete wc proc th monotonða kai ta akrìtata thn f (x) gia tic di forec timèc tou λ R.

119 Kef laio 3. Diaforikìc Logismìc 7 8. Apì tic exet seic tou PanepisthmÐou tou Berkeley, 977 kai 98. 'Estw f : R R paragwgðsimh. Upojètoume ìti f (x) >f(x) kai ìti f (x )=. DeÐxte ìti f (x) >f(x ) gia ìla ta x>x. Upodeixh: ErgasjeÐte me thn sun rthsh g (x) = f(x) e x. 83. MÐa sun rthsh, orismènh se kleistì di sthma, mhdenðzetai sta kra tou kai eðnai dôo forèc paragwgðsimh. Na apodeðxete ìti an h sun rthsh paðrnei mða jetik tim tìte ja up rqei èswterikì shmeðo tou diast matoc ìpou h deôterh par gwgoc thc ja eðnai arnhtik. 84. 'Ena prìblhma tou Johann Bernoulli. To hmikôklio tou sq matoc èqei di metro kai to M eðnai metablhtì shmeðo tou. Na brejeð to x ste to embadìn tou orjogwnðou OMNT na eðnai mègisto. Johann Bernoulli Apì tic exet seic I.I.T, IndÐa, 994. O kôkloc x + y = tèmnei ton x- xona sta P, Q. 'Enac lloc kôkloc me kèntro to Q kai metablht aktðna, tèmnei ton pr to kôklo, p nw apì ton x- xona sto R kai to eujôgrammo tm ma PQ sto S. Na breðte to mègisto tou embadoô tou trig nou QSR. 86. Apì tic exet seic tou PanepisthmÐou tou Berkeley, 987. Na apodeðxete ìti h exðswsh αe x =+x + x

120 O rìloc thc pr thc parag gou ìpou α jetikìc èqei akrib c mða pragmatik rðza. Upodeixh: Na onom sete f (x) =αe x (+x + x ). H perðptwsh ìpou α eðnai eôkolh diìti h f eðnai gnhsðwc aôxousa. H perðptwsh ìpou <α< jèlei megalôterh prosoq. Sthn perðptwsh aut h f èqei dôo rðzec. DeÐte ti timèc paðrnei h f e autèc tic rðzec. 87. Apì to majhmatikì diagwnismì Harvard-MIT,. 'Estw f paragwgðsimh sun rthsh tètoia ste f(x)+f (x) gia ìla ta x R me f() =. Poia eðnai h megalôterh dunat tim tou f()? (Upìdeixh : Na jewr sete th sun rthsh e x f(x).) G' OMADA Jean Gaston Darboux To Je rhma tou Darboux. 'Estw f :Δ R mða paragwgðsimh sun rthsh. Na apodeðxete ìti h par gwgoc f thc f èqei thn diìthta twn endiamèswn tim n, dhlad k je arijmìc y metaxô dôo tim n y = f (x ), y = f (x ) eðnai tim thc f. Upodeixh: ErgasjeÐte me thn perðptwsh pou x <x, y <y<y (Oi upìloipec peript seic antimetwpðzontai an loga). 'Estw g (x) =f (x) y (x x ) Na apodeðxete ìti h g eðnai suneq c sto [x,x ] kai epomènwc èqei el qisth tim se k poio x [x,x ]. DeÐxte ìti autì to x ja prèpei na eðnai eswterikì tou diast matoc [x,x ]. Sthn sunèqeia qrhsimopoieðste to je rhma tou Fermat gia na sumper nete ìti ja eðnai f (x )=y. 89. H anisìthta tou Cauchy. Na apodeðxete ìti an α,α,..., α ν eðnai ν mh arnhtikoð arijmoð tìte isqôei α + α α ν ν ν α α... α ν Upodeixh: QrhsimopoieÐste epagwg sto ν. Upojèste ìti to apodeiktèo isqôei gia ν kai gia na apodeðxete ìti isqôei gia ν + na ergasjeðte thn sun rthsh f (x) = α +α +...+α ν+x ν+ ν+ α α... α ν x Edmund Landau H anisìthta tou Landau. 'Estw f :[, ] R mða dôo forèc paragwgðsimh sun rthsh gia thn opoða gia ìla ta x isqôoun: f (x) A f (x) B Na apodeiqjeð ìti gia ìla ta x isqôei f (x) A + B 8. 'Estw α> me α. Gia thn sun rthsh f : R R isqôei f (x) f (y) α x y gia ìla ta x, y. Na apodeiqjeð ìti hexðswsh f (x) =x èqei mða toul qiston lôsh. H upìdeixh dìjhke stouc diagwnizìmenouc

121 Kef laio 3. Diaforikìc Logismìc O rìloc thc deôterhc parag gou 3.5. A' OMADA 8. Na apodeðxete ìti stic parak tw peript seic h sun rthsh f strèfei ta kurt proc ta k tw se ìlo to pedðo orismoô thc:. f (x) =x. f (x) =e x 3. f (x) =log x 4. f (x) =x +x + 5. f (x) =x x 6. f (x) = x 83. Se k je mða apì tic parak tw peript seic na breðte ta diast mata ìpou h f eðnai kurt koðlh:. f (x) = 6 x3 x +. f (x) =x 4 6x 3 +x 3. f (x) = x x ln x + x 4. f (x) =x + x 5. f (x) =ημx 6. f (x) =x ln x x x4 84. Na breðte ta shmeða kamp c twn parak tw sunart sewn:. f (x) =x 3 3x +8. f (x) =x ln x 5x + 3. f (x) =x 4 6x f (x) =x e x 84xe x + 68e x x 4 +6x 3 x + 5. f (x) =3x 5 x 4 +x f (x) =e x x 3.5. B' OMADA 85. Gia poièc timèc tou α sun rthsh f (x) =x 4 4αx 3 +8x + strèfei ta koðla proc ta nw se ìlo to R? 86. To shmeðo A(, 3) eðnai shmeðo kamp c thc f (x) =αx 3 + βx BreÐte ta α, β.

122 3.5. O rìloc thc deôterhc parag gou 87. H m gissa} thc Agnesi 'Estw o kôkloc me di metro OA ìpouoeðnai h arq twn axìnwn kai A(,α),α >. Se k je shmeðo T(x,α) thc eujeðac y = α antistoiqoôme to shmeðo tom c M thc eujeðac OT me thn kôklo. 'Estw y htetagmènh tou M. OrÐzetai tìte mða sun rthsh y = f (x). Α Τ Μ Ο. Na brejeð o tôpoc thc f. Na brejoun ta shmeða kamp c thc f Maria Gaetana Agnesi Apì tic exet seic tou 985, Dèsmh I. DÐnetai h sun rthsh f me f (x) =x (x 3)+4, gia x R. Onom zoume x,x ta shmeða sta opoða h f parousi zei topik akrìtata kai x 3 to shmeðo stoopoðo parousi zei kamp. Na apodeiqjeð ìti ta shmeða tou epipèdou (x,f(x )), (x,f(x )), (x 3,f(x 3 )) eðnai suneujeiak. 89. Apì tic exet seic tou 99, Dèsmh I.. DÐnetai h sun rthsh f orismènh kai dôo forèc paragwgðsimh sto di sthma Δ me timèc sto (, + ). Na deiqjeð ìti h sun rthsh g me g (x) = ln f (x) strèfei ta koðla nw an kai mìno an isqôei f (x) f (x) (f (x)).. Na brejeð to mègisto di sthma, sto opoðo h sun rthsh g me g (x) = ln ( x + ) strèfei ta koðla nw. 83. Na apodeðxete ìti h sun rthsh èqei trða suneujeiak shmeða kamp c. f (x) = x + x + versiera apì to latinikì vertere pou shmaðnei strèfw}. Wstìso versiera tan kai suntìmeush thc lèxhc avversiera pou shmaine kai m gissa}. Apì èna metafrastikì l joc sta Agglik apodìjhke wc witch kai ètsi èmeine sthn bibliografða.

123 Kef laio 3. Diaforikìc Logismìc 83. 'Estw ìti h f eðnai orismènh kai dôo forèc par gwgðsimh sto [α, β], ìti f (α) =f (β) =m kai f (x) < gia ìla ta x. Na apodeðxete ìti isqôei f (x) m gia ìla ta x. 83. Apì tic exet seic tou 996, Dèsmh IV. DÐnetai h sun rthsh f, dôo forèc paragwgðsimh sto R gia thn opoða isqôei f (x) gia k je x R kai h sun rthsh g tètoia ste g (x) f (x) =f (x) gia k je x R. Na apodeðxete ìti, an h grafik par stash thc f èqei shmeðo kamp c to A (x,f(x )), tìte h efaptomènh thc grafik c par stashc thc g sto shmeðo B (x,g(x )) eðnai par llhlh sthn eujeða y x +5= 'Estw f :[α, β] R me f (x) > gia k je x [α, β] (a ) Na apodeðxete ìti ( ) α + β f < f (α)+f (β) (b ) Genikìtera na apodeðxete ìti an p >, q > me p + q = tìte isqôei f (pα + qβ) <pf(α)+qf (β) f(β) pf (α)+qf (β) f (pα + qβ) f(α) α pα + qβ β (g ) Pwc ja diatup nontan oi anisìthtec twn prohgoumènwn erwthm twn an f (x) < gia k je x [α, β]?. Na apodeðxete tic parak tw anisìthtec: (a ) e α+β < eα +e β (b ) ( α + β, (α <β) ) ν α ν + β ν (g ) ημ ( 3 3 x y) > 3 8 3ημx + 3ημy <x<y<π 834. Apì tic exet seic tou 986, Dèsmh I. 'Estw h sun rthsh f me f (x) = ( α ) ( x 3 α + ) x x +7 3 x R Na breðte to α R ste h f na parousi zei kamp sto x = 3. Met gia thn tim aut tou α, nasqhmatðsete ton pðnaka metabol n thc f.

124 3.5. O rìloc thc deôterhc parag gou 835. Apì tic exet seic tou 3. Stic exet seic tou ètouc 3 dìjhke to akìloujo jèma: Εστω μία συνάρτηση f συνεχής σ ενα διάστημα [α, β] που έ- χει συνεχή δεύτερη παράγωγο στο (α, β). Αν ισχύει f (α) = f (β) =και υπάρχουν αριθμοί γ (α, β), δ (α, β) έτσι ώστε f (γ) f (δ) < να αποδείξετε ότι:. Υπάρχει μία τουλάχιστον ρίζα της εξίσωσης f (x) =στο διάστημα (α, β).. Υπάρχουν σημεία ξ,ξ (α, β) τέτοια ώστε f (ξ ) < και f (ξ ) >. 3. Υπάρχει ένα τουλάχιστον σημείο καμπής της γραφικής παράστασης της f. AfoÔ diab sete ta sqetik me to shmeðo kamp c apì to biblðo sac:. Na apant sete ta dôo pr ta erwt mata.. Na jewr sete thn sun rthsh f :[, ] R me x 3 6x 5x, x f (x) = x, <x< x 3 +6x 5x +, x Ο kai sth sunèqeia: (a ) Na apodeðxete ìti 6x x 5, x f (x) =, <x< 6x +x 5, x (b ) x, x f (x) =, <x< x +, x

125 Kef laio 3. Diaforikìc Logismìc 3 (g ) Na apodeðxete ìti h f den èqei shmeða kamp c. 3. Apì to b') sun getai ìti to er thma g') tou jèmatoc eðnai l joc. GiatÐ? 836. Apì tic exet seic tou 999, Dèsmh I DÐnetai sun rthsh f : R R duo forèc paragwgðsimh h opoða se shmeðo x R parousi zei topikì akrìtato to kai ikanopoieð th sqèsh gia k je x R. f (x) > 4 ( f (x) f (x) ). Na apodeðxete ìti h sun rthsh g (x) =f (x) e x eðnai kurt R.. Na apodeðxete ìti eðnai f (x) gia k je x R Apì tic exet seic tou 99, Dèsmh I.. Na melethjeð wc proc th monotonða kai ta koðla h sun rthsh f me f (x) =α x x, x R kai <α<.. Na brejoôn oi pragmatikèc timèc tou λ gia tic opoðec isqôei h isìthta α λ 4 α λ = ( λ 4 ) (λ ) 838. 'Estw h sun rthsh f (x) =αx 3 + βx + γx + δ me α kai β 3αγ >. Na apodeðxete ìti. H f parousi zei topikì el qisto se k poio x,topikì mègisto sek poio x kai kamp se k poio x 3.. DeÐxte ìti Γ(x 3,f(x 3 )) eðnai mèso tou tm matoc me kra A (x,f(x )) kai B (x,f(x )) Upodeixh: (gia to ( b') ) DeÐxte pr ta ìti f (x )+f (x ) ( x + x f ) = 8 (x x ) (3α (x + x )+β) 839. Apì tic exet seic tou 997, Dèsmh I. DÐnetai h pragmatik sun rthsh g, dôo forèc paragwgðsimh sto R, tètoia ste g (x) > kai g (x) g (x) [g (x)] > gia k je x R. Na apodeðxete ìti. H sun rthsh g g eðnai gnhsðwc aôxousa kai. g ( x + x ) g (x ) g (x ) gia k je x,x R.

126 Oi kanìnec Bernoulli-De l Hospital Upodeixh: (gia to ( b') ) ArkeÐ na exasfalðsoume ìti ln g ( x + x ) ln g (x ) g (x ) isodônama ìti ln g ( x + x ) ln g (x )+ ln g (x ). MporeÐte na ergasjeðte ìpwc sthn skhsh 833 me th sun rthsh f (x) =lng (x) 'Estw f :[α, + ) R paragwgðsimh me par gwgo gnhsðwc aôxousa kai jetik sto α. DeÐxte ìti f(x) =+. lim x +. 'Estw f : R R mða sun rthsh pou eðnai dôo forèc paragwgðsimh kai gia thn opoða isqôei G' OMADA lim f(x) = lim f(x) R x x + Na apodeðxete ìti h f de mporeð na eðnai kurt. 84. Na apodeðxete ìti an h sun rthsh f eðnai dôo forèc paragwgðsimh sto di sthma Δ kai isqôei f (x) > gia ìla ta x tìte gia k je x,x,..., x ν Δ kai λ,λ,..., λ ν [, ] me λ + λ λ ν =isqôei f (λ x + λ x λ ν x ν ) λ f (x )+λ f (x ) λ ν f (x ν ) 3.6 Oi kanìnec Bernoulli-De l Hospital 3.6. A' OMADA 84. Na upologðsete ta parak tw ìria: ln(+x). lim x e x. lim x x e x +x x ημx 3. lim x x 4. lim x + ln x x x x 5. lim α x x 6. lim x ln x x α 7. lim x ln(x ) x +x ημ(αx) 8. lim x ημ(βx) 843. Na upologðsete ta parak tw ìria: e. lim x+ x + x. lim x + x ln(x +) ln(x+) 3. lim x + ln(x+) 4. lim x + x++ x+ ln(x+)+ln(x+) 5. lim x + ln(x+3)+ln(x+4) x+ x 6. lim x + x+ x 7. lim x + 8. lim x 4 x+ 3 x+ ln( x) x ln( x) x

127 Kef laio 3. Diaforikìc Logismìc Na upologðsete ta parak tw ìria: ( ). lim x ln(+x ) x. lim x + x ln x 3. lim x + xx 4. lim x + x x 5. lim x (συνx) x 6. lim x + x ln x 7. lim x ( π ) ( π x) εϕx ( ) x 8. lim x+ x + x 845. BreÐte to ìrio lim θ π ημ (θσυνθ) συν (θημθ) 846. Apì to majhmatikì diagwnismì Harvard-MIT, 6. BreÐte to ìrio 3.6. B' OMADA e xσυνx x lim x ημx 847. 'Estw h sun rthsh { x f (x) = + x + λ x x λ x x> ìpou λ>. Na apodeðxete ìti h f eðnai suneq c.. Up rqoun rage timèc tou λ gia tic opoðec h f eðnai paragwgðsimh? 848. 'Estw ϕ èna opoiod pote polu numo me bajmì megalôtero Ðso tou.. Na apodeðxete ìti isqôei: ϕ (x) ln x lim x + e x = lim x + ϕ (x) =. Poièc apì tic parak tw isìthtec eðnai alhjeðc? ( ) (a ) lim ϕ(x) x + e ln x x ϕ(x) = ϕ (b ) lim (x) e x ln x x + e x ln x = (g ) lim x + ( ϕ (x) e x ln x ) =

128 Oi kanìnec Bernoulli-De l Hospital 849. Na apodeðxete ìti ( lim + α ) x = e α x + x 85. Gia poi tim tou λ h sun rthsh { λ, x = f (x) = x x, x (, + ) eðnai suneq c? 85. Apì tic exet seic tou 997, Dèsmh I. DÐnontai oi pragmatikèc sunart seic f,g me pedðo orismoô to R, pou èqoun pr th kai deôterh par gwgo kai g (x) gia k je x R. 'Estw α pragmatikìc arijmìc. Jètoume A = f(α) g(α) kai B = f (α) Ag (α) g(α). An φ eðnai mða pragmatik sun rthsh orismènh sto R\{α}, tètoia ste f (x) (x α) g (x) = A (x α) + B x α + φ (x) g (x) gia k je x R\{α} na apodeiqjeð ìti up rqei to lim x α φ (x). Upodeixh: LÔnontac thn dojeðsasqèshwcprocφ (x) ja breðte ìti φ (x) = f(x) Ag(x) Bg(x)x+Bg(x)α x αx+α. Katìpin breðte to ìrio lim φ (x) me thn bo jeia tou kanìna tou De l Hospital. H tim pou x α f prèpei na breðte gia to ìrioeðnai (α)g (α) f(α)g (α)g(α) g (α)f (α)g(α)+f(α)(g ) (α) g alli c (f (α) Ag (α) Bg (α) (α)). 85. Apì tic exet seic Mathematical Tripos, Cambridge, 95. Na apodeðxete ìti an h f èqei deôterh par gwgo sto a tìte f (a) = lim h f (a + h) f (a)+f (a h) h 853. Gia thn sun rthsh f eðnai gnwstì ìti lim x σ f (x) =kai ìti den mh- e denðzetai kont sto.breðte to ìrio lim f(x) +ημf(x) x σ f(x) x 854. Na apodeðxete ìti gia k je α> isqôei lim α x + α x = G' OMADA 855. An oi α, β, γ eðnai jetikoð brèite to lim x ( α x3 + β x3 + γ x3 α x + β x + γ x ) x 856. Me <α<β<γbreðte to ìrio lim x + (αx + β x + γ x ) x.

129 Kef laio 3. Diaforikìc Logismìc AsÔmptwtoi 3.7. A' OMADA 857. Na breðte tic katakìrufec asumpt touc twn sunart sewn:. f (x) = x. ϕ (x) = x+ x 3x+ 3. σ (x) = x x 3x+ 4. g (x) = x x 3 x x + 5. h (x) = x x 6. θ (x) = x +x Na breðte tic asumpt touc gia x + thc sun rthshc f stic peript seic:. f (x) = x + x+ 4. f (x) = 4 x x+ + x. f (x) = x + 5. f (x) = x+ x 3 x+ 3. f (x) = x + x+ + x 6. f (x) = (x+)x x B' OMADA 859. Na breðte tic asumpt touc thc C f ìtan x 5x f (x) = x 'Estw. Na breðte tic asumpt touc thc f (x) =x 3 6x +x 6 g (x) = f (x). Na apodeðxete ìti h efaptomènh thc C f sto A(, 6) èqei llo èna koinì shmeðo methnc f tou opoðou na upologðsete tic suntetagmènec. 86. Na breðte tic asumpt touc thc sun rthshc f (x) =ln x x. 86. 'Estw f (x) =e x

130 AsÔmptwtoi. Na brejoôn oi asômptwtoi thc C f.. Up rqoun rage koin shmeða thc C f kai twn asumpt twn thc? 863. Apì tic exet seic tou 994, Dèsmh IV. 'Estw ìti h eujeða y =x +5eÐnai asômptwth thc grafik c par stashc thc sun rthshc f sto +. f(x). Na breðte ta ìria: lim x + x kai lim (f (x) x). x +. Na breðte ton pragmatikì arijmì μ an lim x Gia tic sunart seic f,g eðnai gnwstì ìti lim (f (x) g (x)) = x + μf(x)+4x xf(x) x +3x = Na apodeðxete ìti an h y = αx + β eðnai asômptwtoc thc f gia x + tìte ja eðnai kai asômptwtoc thc g gia x MÐa sun rthsh f : R R eðnai paragwgðsimh. 'Eqei katakìrufec asumpt touc? 866. Gia thn sun rthsh f : R R eðnai gnwstì ìti up rqei θ ètsi ste gia k je x R na isqôei: f (x) θ 'Eqei rage h f katakìrufec asumpt touc? G' OMADA 867. JewroÔme touc jetikoôc arijmoôc α,α,..., α ν kai touc opoiousd pote arijmoôc β,β,..., β ν, γ,γ,..., γ ν. Na brejoôn oi arijmoð A, B ètsi ste ( αx + β x + γ + α x + β x + γ α νx + β νx + γ ν Ax B) = lim x Gia k je sun rthsh ϕ pou èqei asômptwto sto + thn eujeða y = κx + λ me ˆϕ sumbolðzoume thn sun rthsh ˆϕ (x) =κx + λ. Na apodeðxete ìti an H sun rthsh f èqei sto + asômptwto me jetikì suntelest dieujônsewc H sun rthsh g èqei sto + asômptwto tìte kai h sun rthsh g f èqei sto + asômptwto isqôei ĝ f =ĝ ˆf 869. Ac upojèsoume ìti mac endiafèrei h asumptwtik prosèggish sunart sewn gia x ± ìqi apì eujeðec y = αx + β all apì sunart seic thc morf c y = αe x + β. Gr yte mða jewrða} gia autì to eðdoc asumpt twn. BreÐte kai merik paradeðgmata.

131 Kef laio 3. Diaforikìc Logismìc Melèth kai grafik par stash A' OMADA 87. Na gðnei h melèth kai h grafik par stash thc f (x) =x +3x Na gðnei h melèth kai h grafik par stash thc f (x) =x 4 3x Apì tic exet seic tou 983, Dèsmh IV. DÐnetai h sun rthsh me tôpo f (x) =x x. Na gðnei h melèth kai prìqeirh grafik par stash thc sun rthshc aut c Na gðnei h melèth kai h grafik par stash thc sun rthshc f (x) = ( x 3 4x +7x 6 ) 874. Na gðnei h melèth kai h grafik par stash thc sun rthshc f (x) = x(x +) x B' OMADA 875. Apì tic exet seic tou 997, Dèsmh I. Upojètoume ìti up rqei pragmatik sun rthsh f orismènh sto R, dôo forèc paragwgðsimh tètoia ste up rqoun pragmatikoð arijmoð α kai β ste (x ) f (x)+ ( αημx βx ) f (x) =e x gia k je x R. 'Estw ìti up rqei pragmatikìc arijmìc ρ ste f (ρ) =. Na exet sete an to f (ρ) eðnai topikì el qisto thc sun rthshc f 'Estw ìti gia th sun rthsh f isqôei f (x )=f () (x )=f () (x )=... = f (ν) (x )=kai f (ν+) (x ) kai h f (ν+) eðnai suneq c. Na epalhjeôsete ta sumper smata tou akìloujou pðnaka: f (ν+) (x ) / ν an o ν eðnai rtioc an o ν eðnai perittìc an f (ν+) (x ) > tìte h f parousi zei kamp sto x tìte h f parousi zei el qisto sto x an f (ν+) (x ) < tìte h f parousi zei kamp sto x tìte h f parousi zei mègisto sto x G' OMADA 877. Gia mða sun rthsh orismènh kai paragwgðsimh sto R eðnai gnwstì ìti. f (x) =e x (x ) ( x) gia ìla ta x. lim f (x) = lim f (x) = x x + Na melet sete thn f ìso kallðtera mporeðte. Mhn epiqeir sete na thn breðte!!

132 Ask seic se ìlo to kef laio 3.9 Ask seic se ìlo to kef laio 878. 'Estw h sun rthsh f (x) =x 4 3x +. Na breðte ta diast mata monotonðac thc f.. Na breðte ta topik akrìtata thcf. EÐnai olik? 3. Na upologðsete to ìrio f (x) lim x + e x 4. Na breðte thn par gwgo thc sun rthshc h (x) = f (x). 5. Gia poièc timèc tou λ isqôei gia ìla ta x? f (x) λ + 6. Na apodeðxete ìti gia k je eujeða (ε) me exðswsh y = αx + β up rqei efaptomènh thc C f pou èqei ton Ðdio suntelest dieujônsewc me thc (ε) Apì tic exet seic tou 995, Dèsmh IV. DÐnetai h sun rthsh f (x) =x 4 x + α α R. An A (x,f(x )), B (x,f(x )), Γ(x 3,f(x 3 )) eðnai topik akrìtata thc grafik c par stashc thc f kai x <x <x 3, na apodeðxete ìti h eujeða AB eðnai k jeth sthn eujeða BΓ.. An <α<, na apodeðxete ìti h exðswsh f (x) =èqei akrib c mða lôsh sto di sthma (, ). 88. 'Estw h sun rthsh f (x) =x 3 x +. Na brejoôn ta diast mata monotonðac kai ta akrìtata thc f.. Pìsec rðzec èqei h f? 3. Up rqoun rage pragmatikoð arijmoð t tètoioi ste f(t) =t? 4. Na upologðsete thn par gwgo thc sun rthshc g (x) =ημf (ημx)

133 Kef laio 3. Diaforikìc Logismìc 'Estw h sun rthsh f (x) =x ln x. Na breðte ta topik akrìtata thc f.. Na apodeðxete ìti up rqei efaptomènh thc C f h opoða dièrqetai apì thn arq twn axìnwn. 3. Na breðte to oriothcf(x) ìtan x Apì tic exet seic tou 989, Dèsmh I. 'Estw f,g sunart seic me pedðo orismoô èna di sthma Δ gia tic opoðec upojètoume ìti: i) EÐnai dôo forèc paragwgðsimec sto Δ ii) f = g ii) Δ kai f() = g(). Na deiqjeð ìti:. Gia k je x Δ,f(x) g(x) =cx ìpou c R.. An h f(x) =èqei dôo rðzec eterìshmec ρ <ρ tìte h g(x) =èqei toul qiston mða rðza sto kleistì di sthma [ρ, ρ ] Na apodeðxete ìti gia k je x> kai k je jetikì akèraio ν isqôei 3 e x > Na upologðsete to ìrio: lim x ν k= x k k! ( x ) ημx. JewroÔme thn sun rthsh f :[ π, π ] R me { f (x) = x ημx x x = Na apodeðxete ìti h f : (a ) EÐnai suneq c (b ) EÐnai paragwgðsimh (g ) 'Eqei akrib c mða rðza DÐnontai oi sunart seic f (x) =x 4x + g (x) = x3 3 αx βx An k eðnai jetikìc akèraioc to k! eðnai k! =... k. EpÐshc orðzetai! =

134 Ask seic se ìlo to kef laio. Na brejoôn oi timèc twn α, β ste oi dôo sunart seic na èqoun jèsh topikoô akrot tou sto x R me f (x )=g (x )=. Gia thn tim twn α, β pou br kate sto prohgoômeno er thma: (a ) Na breðte to pl joc twn riz n thc sun rthshc h(x) =f(x)g(x) (b ) Na breðte to ìrio: lim x + e f(x) g (e x ) (g ) Na apodeðxete ìti h sunarthsh r(x) =f(g(x)) èqei el qisto MÐa sun rthsh f èqei pedðo orismoô to R, eðnai kurt kai gia k poio zeôgoc arijm n α < βisqôei f (α) < f(β). Na apodeðxete ìti lim f (x) = x Apì tic exet seic tou 995, Dèsmh I. DÐnontai oi pragmatikoð arijmoð κ, λ me κ<λkai h sun rthsh f (x) =(x κ) 5 (x λ) 3 me x R. Na apodeðxete ìti:. f (x) f(x) = 5 x κ + 3 x λ gia k je x κ kai x λ.. H sun rthsh g (x) = ln f (x) strèfei ta koðla proc ta k tw sto di sthma (κ, λ) 'Estw h sun rthsh f (x) =x + ημx. Na apodeðxete ìti h f eðnai gnhsðwc aôxousa.. Na apodeðxete ìti h C f tèmnei thn eujeða y = x se peira shmeða pou apèqoun apì thn arq twn axìnwn apost seic νπ, ν =,,,... kai eðnai shmeða kamp c thc f. 3. EÐnai h y = x asômptwtoc thc C f? 4. Poia eðnai h mègisth apìstash enìc shmeðou thc C f apì thn y = x? 889. DÐnetai h sun rthsh f (x) = { x e x ln x, x >, x =. Na apodeðxete ìti h f eðnai paragwgðsimh sto shmeðo x =. Na breðte to ìrio lim x + f (x)

135 Kef laio 3. Diaforikìc Logismìc Gia tic sunart seic f, g isqôei gia k je x>. f (x)+ln x + e x =e x f (x) g (x)+g (x)lnx. Na breðte tic f, g f(x). Na breðte to ìrio lim x + g(x). 89. Na apodeðxete ìti an h ϕ (x) =αx 3 +βx +γx+δ parousi zei akrìtata se shmeða pou brðskontai se eujeða h opoða deièrqetai apì thn arq twn axìnwn tìte ja isqôei βγ =9αδ. 89. Apì tic exet seic tou 996, Dèsmh IV. JewroÔme tic paragwgðsimec sunart seic f, g pou èqoun pedðo orismoô to di sthma [, + ) gia tic opoðec isqôei h sqèsh: f (x) =g (x)+ημ x + e x gia k je x [, + ). Na apodeðxete ìti f ()+g (x) <g()+f (x) gia k je x (, + ) Apì tic exet seic tou 4. DÐnetai h sun rthsh f me tôpo f (x) =x ln x.. Na breðte to pedðoorismoô thc sun rthshc f, na melet sete thn monotonða thc kai na breðte ta akrìtata.. Na melet sete thn f wc proc thn kurtìthta kai na breðte ta shmeða kamp c. 3. Na breðte to sônolo tim n thc f Na apodeðxete ìti h grafik par stash thc sun rthshc f (x) =x + x x èqei asômptwto thn pr th diqotìmo twn axìnwn DÐnontai oi sunart seic f (x) =ημx + συνx, g (x) =εϕx + σϕx me x (, π ).. Na deðxete ìti oi f,g parousi zoun mègisto kaiel qisto antistoðqwc ìtan x = π 4.. Na deðxete ìti gia k je x (, π ) isqôei εϕx + σϕx > ημx + συνx Gia poi tim tou α h grafik par stash thc f (x) =x 3 αx + ef ptetai sthn y = BreÐte th mègisth tim thc f (x) =x 3 x.

136 Ask seic se ìlo to kef laio 898. 'Estw hoikogèneia twn sunart sewn { λx + x ln x, x > f (x) =, x =. Na deðxete ìti oi sunart seic f eðnai suneqeðc.. Na deðxete ìti gia k je λ h f qei mða mìno orizìntia egaptomènh kai ìti to shmeðo epaf c an kei sthn eujeða y = x Oi sunart seic f,g eðnai orismènec sto di sthma [α, β] kai dôo forèc paragwgðsimec. EpÐshc isqôoun: f (α) =f (α) = f (x) g (x) =f (x) g (x). Na apodeiqjeð ìti f (x) g (x) =f (x) g (x). Na apodeiqjeð ìti an kanènac arijmìc tou [α, β] den eðnai rðza thc g tìte k je arijmìc tou [α, β] eðnai rðza thc f, 9. Oi arijmoð α, β, x eðnai jetikoð, oi α, β stajeroð kai o x metablhtìc, Poi eðnai h el qisth tim pou mporeð na p rei h par stash αx + β x? 9. 'Ena liont ri kineðtai sthn kuklik pðsta enìc tsðrkou me taqôthta km/h me shmeðo to ekkðnhshc A. Sto kèntro thc pðstac brðsketai ènac probolèac pou fwtðzei to liont ri. 'Ena pètasma ef ptetai tou kôklou sto shmeðo A kai se autì faðnetai h ski tou liontarioô. Me ti taqôthta kineðtai h ski th qronik stigm pou to liont ri èqei dianôsei to 8 tou kôklou? A 9. Apì tic exet seic tou 6. DÐnetai h sun rthsh f (x) = x + x ln x. Na breðte to pedðo orismoô kai to sônolo tim n thc sun rthshc f.. Na apodeðxete ìti h exðswsh f(x) =èqei akrib c rðzec sto pedðo orismoô touc.

137 Kef laio 3. Diaforikìc Logismìc An h efaptomènh thc grafik c par stashc thc sun rthshc g (x) =lnx sto shmeðo A(α, ln α) me α> kai h efaptomènh thc grafik c par stashc thc sun rthshc h(x) =e x sto shmeðo B(β,e β ) me β R tautðzontai, tìte na deðxete ìti o arijmìc α eðnai rðza thc exðswshc f(x) = 4. Na aitiolog sete ìti oi grafikèc parast seic twn sunar sewn g kai h èqoun akrib c dôo koinèc efaptomènec. 93. Gia th sun rthsh h (x) =ax 3 + bx + cx + d eðnai gnwstì ìti isqôei x h (x) x + gia ìla ta x kai ìti to <<=>> sthn pr th anisìthta isqôei gia x = kai sth deôterh isqôei gia x =. Na brejeð h h(x). 94. Apì to majhmatikì diagwnismì Harvard-MIT, 5. 'Estw f : R R mða leða 4 sun rthsh tètoia ste (f (x)) = f(x)f (x) gia ìla ta x. Upojètoume ìti f() = kai f (4) () = 9. BreÐte ìlec tic dunatèc timèc tou f (). 95. Gia th sun rthsh f : R R eðnai gnwstì ìti eðnai dôo forèc paragwgðsimh kai ìti gia k poia x,x,x 3 me <x <x <x 3 isqôei f (x i )=x i, i =,, 3.. Na apodeðxete ìti up rqoun t,t me <x <t <x <t <x 3 ètsi ste f (t i )= f(t i) t i, i =,.. Na apodeðxete ìti h exðswsh f (x) =èqei dôo toul qiston lôseic Na apodèixete ìti gia k je x R up rqei akrib c èna y ètsi ste y 3 + y = e x + x. H sun rthsh f orðzetai apì th sqèsh f 3 (x)+f (x) =e x + x x R Na apodeðxete ìti: (a ) Gia k je x isqôei f (x) f (x )= f (x)+f (x) f (x )+f (x )+ ((ex + x) (e x + x )) (b ) Gia k je x isqôei f (x) f (x ) e x + x e x x (g ) H f eðnai suneq c. 4 Dhlad èqei parag gouc ìlwn twn t xewn.

138 Ask seic se ìlo to kef laio (d ) H f eðnai paragwgðsimh kai isqôei f (x) = ex + 3f (x)+. (e ) H f eðnai gnhsðwc aôxousa kai to sônolo tim n thc eðnai to R. 97. 'Estw f(x) =lnx kai <α<β.. Na breðte to (monadikì) γ [α, β] pou epalhjeôei to je rhma mèshc tim c gia thn f sto α<β.. Na apodeðxete ìti αβ < γ < α+β 'Estw f :Δ R mða paragwgðsimh sun rthsh me suneq par gwgo. 'Estw α, β, γ Δ tètoia ste α<β<γ. 'Estw ìti (f (β) f (α)) (f (γ) f (β)) < Na apodeðxete ìti up rqei ξ (α, γ) ètsi ste f (ξ) =. Na apodeðxete to Ðdio apotèlesma qwrðc ìmwc thn upìjesh suneqeðac thc f. 99. 'Estw h sun rthsh { x x f (x) = x x >. Na breðte se poia x h f eðnai suneq c.. Na breðte se poia x h f eðnai paragwgðsimh. 3. Na k nete, prìqeira, thn grafik par stash thc sun rthshc f. 4. Up rqei rage sun rthsh F : R R tètoia ste F = f? 9. Gia thn sun rthsh f : R R eðnai gnwstì ìti f() =, eðnai paragwgðsimh sto,f () = kai ìti gia ìla ta x, y isqôei: f (x + y) = f (x)+f (y) +f (x) f (y). Na apodeðxete ìti f( x) = f(x) gia ìla ta x.. Na apodeðxete ìti h f(x) kai f(x) gia ìla ta x. 3. Na apodeðxete ìti h f eðnai paragwgðsimh kað ìti gia ìla ta x isqôei: f (x) = ( f (x) ) 4. Na apodeðxete ìti h f eðnai gnhsðwc monìtonh,

139 Kef laio 3. Diaforikìc Logismìc Na apodeðxete ìti gia ìla ta x isqôei: (f (x)+) f (x)+ (f (x) ) f (x) = 6. Na breðte thn f. 9. Apì tic exet seic tou. Gia mða pragmatik sun rthsh f, pou eðnai paragwgðsimh sto sônolo twn pragmatik n arijm n R, isqôei ìti: f 3 (x)+βf (x)+γf (x) =x 3 x +6x gia k je x R, ìpou β,γ pragmatikoð arijmoð me β < 3γ.. Na deðxete ìti h sun rthsh f den èqei akrìtata.. Na deðxete ìti h sun rthsh f eðnai gnhsðwc aôxousa. 3. Na deðxete ìti up rqei monadik rðza thc exðswshc f(x) =sto (, ). 9. Apì tic exet seic tou. 'Estw oi sunart seic f, g me pedðo orismoô to R. DÐnetai ìti h sun rthsh thc sônjeshc f g eðnai -.. Na deðxete ìti h g eðnai -.. Na deðxete ìti h exðswsh: g ( f (x)+x 3 x ) = g (f (x)+x ) èqei akrib c dôo jetikèc kai mða arnhtik rðza. 93. Apì tic exet seic tou 995, Dèsmh IV. DÐnetai h sun rthsh f dôo forèc paragwgðsimh sto R gia thn opoða isqôei f (x) gia k je x R kai h sun rthsh g tètoia ste g (x) f (x) =f (x) gia k je x R. Na apodeðxete ìti an h grafik par stash thc f èqei shmeðo kamp c to A(x,f(x )) tìte h efaptomènh thc grafik c par stashc thc g sto shmeðo B(x,g(x )) eðnai par llhlh sthn eujeða y x +5= 'Estw h sun rthsh f (x) = + x + + x 5 'Otan telei sete th lôsh apant ste sta erwt mata: Pìte dôo eujeðec eðnai p r llhlec? Pìte oi eujeðec y = αx + β, y = α x + β eðnai par llhlec? Dokim ste f (x) = e x + 5 e, x =, g (x) = f(x) f (x). er thma tou jèmatoc thc skhshc 78? TÐ èqete na peðte gia to teleu io

140 Ask seic se ìlo to kef laio. Na apodeðxete ìti gia ìla ta x isqôei ( ) ( ) f + x = f x. Na breðte th mègisth tim thc f. 95. Apì to diagwnismì Putnam, 973. Pìsec rðzec èqei h sun rthsh f (x) = x x? 96. 'Estw h sun rthsh (sun rthsh tou Cauchy): { f (x) = e x, x, x =. Na brejoôn oi f, f.. Na brejoôn ta shmeða kamp c thc f. 97. 'Estw f :[α, β] R mða paragwgðsimh sun rthsh. Na apodeðxete ìti an h f parousi zei el qisto sto α tìte isqôei f (α). 98. Apì tic exet seic tou 8. DÐnetai h sun rthsh: f (x) = { x ln x, x >, x =. Na apodeðxete ìti h sun rthsh f eðnai suneq c sto.. Na melet sete wc proc th monotonða thn sun rthsh f kai na breðte to sônolo tim n thc. 3. Na breðte to pl joc twn diaforetik n jetik n riz n thc exðswshc x = e α x gia ìlec tic pragmatikèc timèc tou α. 4. Na apodeðxete ìti isqôei f (x +)>f(x +) f (x) 99. Apì tic exet seic tou 9. DÐnetai h sun rthsh f (x) =α x ln (x +), x > ìpou α> kai α.. An isqôei f (x) gia k je x>, na apodeðxete ìti α = e.. Gia α = e

141 Kef laio 3. Diaforikìc Logismìc 39 (a ) na apodeðxete ìti h sun rthsh f eðnai kurt. (b ) na apodeðxete ìti h sun rthsh f eðnai gnhsðwc fjðnousa sto di sthma (, ] kai gnhsðwc aôxousa sto di sthma [, + ) (g ) an β,γ (, ) (, + ), na apodeðxete ìtihexðswsh f (β) x + f (γ) x = èqei toul qiston mia rðza sto (, ). 9. Apì tic exet seic ASEP tou 9. 'Estw h sun rthsh ( f (x) = x + ) α x <x< kai α>.. ApodeÐxte ìti h sun rthsh f(x) eðnai kurt sto di sthma (, ).. ApodeÐxte ìti gia ìla ta x, y (, ) kai α> isqôei ( ) x + y f (x)+f (y) f 3. ApodeÐxte ìti, an x>,y >,α> kai x + y =,tìte isqôei: ( x + x) α ( + y + ) α 5α y α 9. Apì tic exet seic tou. 'Estw h sun rthsh f (x) = ( x + x) α, <x< kai α>.. ApodeÐxte ìti h sun rthsh f(x) eðnai kurt sto di sthma (, ).. ApodeÐxte ìti gia ìla ta x, y (, ) kai α > isqôei f ( x+y ) f(x)+f(y). 3. ApodeÐxte ìti, an x>,y >,α> kai x + y =,tìte isqôei: ( x + x) α ( + y + ) α 5α y α 9. Gia thn sun rthsh f : R R eðnai gnwstì ìti EÐnai dôo forèc paragwgðsimh. H deôterh par gwgoc thc eðnai suneq c. IsqÔei f (x) f (x) =gia ìla ta x.

142 Ask seic se ìlo to kef laio BreÐte poia mporeð na eðnai h f. 93. Na breðte paragwgðsimh sun rthsh f : R R gia thn opoða isqôei f (x) =cf (x) ìpou c eðnai stajer. 94. JewroÔme thn sun rthsh BreÐte thn f () f (x) = { x (x x ), x >, x = 95. Gia mða suneq sun rthsh f orismènh se di sthma eðnai gnwstì ìti den mhdenðzetai kai ìti h sun rthsh (f (x)) eðnai paragwgðsimh. Na apodeiqjeð ìti kai h f eðnai paragwgðsimh. 96. DÐnetai h sun rthsh h (x) = x + x + x. Na melethjeð wc proc tic asumpt touc. 97. Apì tic exet seic tou. DÐnetai h sun rthsh f : R R, dôo forèc paragwgðsimh sto R, me f () = f() =, h opoða ikanopoieð th sqèsh: e x ( f (x)+f (x) ) = f (x)+xf (x) gia k je x.. Na apodeðxete ìti: f(x) =ln(e x x) x R.. Na melet sete th sun rthsh f wc proc th monotonða kai ta akrìtata. 3. Na apodeðxete ìti h grafik par stash thc f èqei akrib c dôo shmeða kamp c. 4. Na apodeðxete ìti h exðswsh ln (e x x) =συνx èqei akrib c mða lôsh sto di sthma (, π ). 98. 'Estw paragwgðsimh kai kurt sun rthsh f : R R me lim x ± f(x) x =. Na apodeiqjeð ìti eðnai stajer.

143 Kef laio 4 Oloklhr tikìc Logismìc 4. Aìristo Olokl rwma: Ta Basik 4.. A' OMADA 99. BreÐte apì mða par gousa twn parak tw sunart sewn:. e x+. e x 3. e x 4. e x+ 93. BreÐte apì mða par gousa twn parak tw sunart sewn:. x x + 3. x 3 + x. x + ex 4. ημx + συνx 93. Na sumplhr sete ton pðnaka: Sun rthsh f g f + g fg+ gf fg Fg G 3fF +4gG 3 MÐa par gousa F G 93. BreÐte apì mða par gousa F thc f (x) =x +tètoia ste F () = H sun rthsh f èqei par gousa thn F (x) = x e x +(x +). Na breðte mða par gousa G hc f tètoia ste G (ln ) = 934. MporoÔn rage oi sunart seic F (x) =e x+ + x, G (x) =e x+ x na eðnai par gousec thc Ðdiac sun rthshc; 4

144 4 4.. Aìristo Olokl rwma: Ta Basik 935. Sto sq ma pou akoloujeð emfanðzontai merikèc par gousec thc Ðdiac sun rthshc f dhlad h grafik par stash merik n apì tic sunart seic pou apartðzoun to f (x) dx. BreÐte poi par gousa. 'Eqei rðza to 4.. Dièrqetai apì to shmeðo A(, ) Na exhg sete giatð up rqei to polô mða par gousa thc f pou èqei rðza ton arijmì B' OMADA 937. EÐnai gnwstì ìti ìtan èna s ma kineðtai ston aèra dèqetai antðstash apì ton aèra thc opoðac o rujmìc aôxhshc eðnai an logoc thc taqôthtac tou. Gia èna sugkekrimèno s ma h antðstash eðnai 4N ìtan kineðtai me taqôtha 3m/sec. Poia ja eðnai h antðstash tou s matoc ìtan kineðtai me taqôthta 6m/sec? 938. 'Ena mègejoc Ξ(t) metab lletai sunart sei tou qrìnou t kai o rujmìc metabol c tou k je qronik stigm eðnai dipl sioc thc tim c tou megèjouc. BreÐte thn tim tou megèjouc gia t =3an gnwrðzete ìti h tim tou gia t = eðnai Na breðte paragwgðsimh sun rthsh me pedðo orismoô to R h opoða èqei thn akìloujh idiìthta: O suntelest c dieujônsewc thc efaptomènhc thc C f sto shmeðo M (x,f(x )) eðnai x. 94. Na apodeðxete ìti: x + kdx= [ x ( x + k + k ln x + )] x + k + c

145 Kef laio 4. Oloklhr tikìc Logismìc Na apodeðxete ìtihsun rthshf : R R { xημ f(x) = x συν x x x = an kai asuneq c sto èqei par gousa thn F : R R me { x F (x) = ημ x x x = en h g : R R me g(x) = { x x < x + x h opoða eðnai epðshc asuneq c sto den èqei par gousa. 94. Na brejeð paragwgðsimh sun rthsh me pedðo orismoô to R gia thn opoða isqôei f() = kai f (x) =e x f(x) gia ìla ta x 'Estw f suneq c me pedðo orismoô to to Δ. Na apodeðxete ìti an èna stoiqeðo F tou f (x) dx dhlad gia mða par gousa èqei sônolo tim n to R tìte kai k je llh èqei sônolo tim n to R 'Estw f suneq c me pedðo orismoô to Δ. Na apodeðxete ìti mða par gousa F thc f eðnai gnhsðwc aôxousa tìte eðnai kai oi upìloipec Na brejoôn ìlec oi sunart seic f me pedðo orismoô to R pou h efaptomènh se k je shmeðo thcc f dièrqetai apì thn arq twn axìnwn. 4. Aìristo Olokl rwma: Teqnikèc 946. Na upologðsete ta oloklhr mata:. (x +)dx 3.. ( x + ) x dx 947. Na upologðsete ta oloklhr mata: x+ x dx 4. x 3 +dx x 6. x +dx. ημ (x) dx 3. x e x dx 4. e x dx 948. Na upologðsete ta oloklhr mata:. 3 x 8 dx. dx + x 3. (x +)(x +)dx 4. x+ x+ dx

146 Aìristo Olokl rwma: Teqnikèc 949. Na upologðsete ta oloklhr mata:. ( x + y ) dx. ( x + y ) dy 3. ( x+ + ημx ) dx 4. 3x +dx 95. Na upologðsete ta oloklhr mata:. x xdx. x+α x+β dx 3. x+y x y dx 4. x+y x y dy 95. Na upologðsete ta oloklhr mata:. (x +)(x +)(x +3)dx. 3x +x+ x 3 +x +x+ dx 3. (x m + x n ) dx 4. ημ ( πx + π 3 ) dx 95. Na upologðsete ta oloklhr mata:. xe x dx. ( x + x ) dx 3. xημxdx 4. ( x + y ) dx 953. Na upologðsete ta oloklhr mata:. e x+ (x ) dx. xe x+ dx 3. x e x+ dx 4. x dx 954. Na upologðsete ta oloklhr mata:. e x συνxdx. ( e x + x ) dx 3. ημtσυνtdt 4. e x e+ dx 955. Na upologðsete ta oloklhr mata:. x3 x dx. ( p p+q ) t dt Na upologðsete ta oloklhr mata: ( p p+q ) t dq 4. x 3x+ dx

147 Kef laio 4. Oloklhr tikìc Logismìc 45. x+ x 3x+ dx. 6x 3 +x x 3x+ dx 3. x x dx 4. (e x + x e ) dx 957. Na upologðsete ta oloklhr mata:. x ( x +) ( x + ) dx. συν 3x dx 3. 5x +3x+ x dx 4. 5x +3x+ (x )(x ) dx 958. Na upologðsete ta oloklhr mata:. xe 3x dx. ( x 3 + e 3 + ημx ) dx 3. ημ (x) ημ (3x) dx 4. συν (3x + π) dx 959. Na upologðsete ta oloklhr mata:. συν (3x + π) ημ (x π) dx. e x + e x dx 3. ( x +3 x ) dx 4. ( + 4x) 3 5 dx 96. Na upologðsete ta oloklhr mata:. x 3 e x4 dx. x 3 e x dx 96. Na upologðsete ta oloklhr mata:. ( ) + dx (x ) (x 3). (x ) (x 3) dx 3. (ημx +) 4. ( (x ) +(x 3) ) dx 3. e αx+βx dx 4. x 3 x 5 3x dx 96. Na upologðsete ta oloklhr mata:. u 5 3u 6 +8 du. x ln x 3. (αx + β) γ dx 4. ( 4 7) t dt 963. Na upologðsete ta oloklhr mata:. { x x < f (x) dx ìtan f (x) = x x. x + x dx 964. Na upologðsete ta oloklhr mata:

148 Aìristo Olokl rwma: Teqnikèc. (x α)(x β) dx. e x e x dx 3. x 3dx 4. x dx 965. Na upologðsete ta oloklhr mata:. (ln(x)) 5 x dx. ( x 3 ) 3 x dx x x dx 4. xe 4x +5 dx 4.. B' OMADA 966. 'Estw x,x,x 3 oi rðzec tou poluwnômou x 3 7x +6x. AfoÔ prosdiorðsete arijmoôc α, β, γ ètsi ste gia k je x di foro twn x,x,x 3 na isqôei x 3 7x +6x = α x x + β x x + katìpin na upologðsete to olokl rwma: x 3 7x +6x dx 967. 'Estw h sun rthsh f (x) = (x ) (x ) 3. γ x x 3. Na breðte arijmoôc A, B, Γ, Δ,E ètsi ste f (x) = A Δ + E (x ) (x ) 3.. Na upologðsete to olokl rwma f (x) dx 968. Na upologðsete to olokl rwma εϕ xdx 969. Na upologðsete to olokl rwma αx+β γx+δ dx. 97. Na upologðsete to olokl rwma (x )(x )(x 3) dx x + B + Γ (x ) x Na brejeð sun rthsh f me pedðo orismoô to (, + ) tètoia ste:. f (x) x =. H grafik par stash thc f na èqei efaptomènh sto shmeðo (,f()) thn eujeða 3y 3x +5= 97. Na upologðsete to olokl rwma ln x x dx 973. Na upologðsete to olokl rwma x 4 +x 5 dx 974. QrhsimopoieÐste ton kajolikì metasqhmatismì tou Euler, t = εϕ x, gia upologðsete to olokl rwma συνx dx

149 Kef laio 4. Oloklhr tikìc Logismìc Na upologðsete to olokl rwma x x x+3 dx 976. Me thn bo jeia tou metasqhmatismoô u = x + x olokl rwma: ( ( ) ) e x+ x dx x na upologðsete to 977. 'Estw ìti Na apodeðxete ìti: I ν = J ν = ημ ν xdx συν ν xdx.. I ν = ν συνx ημν x + ν I ν ν J ν = ν ημxσυνν x + ν J ν ν 978. Na upologðsete to olokl rwma x megalôteroc tou. (αx 3 ν dx ìpou ν eðnai fusikìc +β) 979. 'Estw f (x) =α x + β. Na upologisjoôn ta α, β ètsi ste na isqôei f () = kai 3 f (x) dx = BreÐte to ln ( +x ) dx. 98. Na upologðsete to olokl rwma x α+ x β dx. 98. Na upologðsete ta oloklhr mata:. x 3 +x 4 dx. 3x+ 3x +x+ dx 3. (x 3) x 3x +dx 983. Na upologðsete ta oloklhr mata:. (συνx + ημx) συνx ημxdx. ημx (+συνx) dx 3. xσυνx (xημx+συνx) dx 984. Na upologðsete to olokl rwma x ημ ( x) συν ( x) dx Na upologðsete to olokl rwma συν( x) xημ ( x) dx.

150 Aìristo Olokl rwma: Teqnikèc 4.. G' OMADA 986. 'Estw I n = e ax συν n βxdx. Na ekfr sete to I n sunart sei tou I n 'Eqei apodeiqjeð ìti to olokl rwma ln x dx pou onom zetai logarijmikì olokl rwma denupologðzetai dhlad molonìti up rqei de mporeð na ekfrasjeð sunart sei twn gnwst n stoiqeiwd n sunart sewn. Na sthriqjeðte sto parap nw gia na apodeðxete ìti kai to olokl rwma e x ln xdx den upologðzetai To pio aplì montèlo gia thn perigraf thc an ptuxhc enìc plhjusmoô P (t) prokôptei an upojèsoume ìti o rujmìc metabol c tou eðnai an logoc tou dh up rqontoc plhjusmoô dhlad ìti P (t) =kp(t) (Ekjetikì montèlo). Suqn ìmwc oi plhjusmoð de mporoôn na uperboôn èna mègisto L pou kajorðzetai apì tic ek stote sunj kec. Se mia tètoia perðptwsh jewroôme ìti o rujmìc metabol c tou plhjusmoô eðnai an logoc ìqi mìno tou up rqontoc plhjusmoô all kai twn uparqìntwn perijwrðwn aôxhshc dhlad tou L P (t). AntÐ dhlad thc sqèshc P (t) =kp(t) tou ekjetikoô montèlou èqoume thn sqèsh: P (t) =kp (t)(l P (t)). Na apodeðxete ìti P (t) P (t)(l P (t)) dt =. Qrhsimopoi ntac ton metasqhmatismì u = P (t) na apodeðxete ìti P (t) P (t)(l P (t)) dt = du u (L u) 3. UpologÐzontac to olokl rwma du u(l u) kdt na apodeðxete ìti ìpou c stajer. 4. Na apodeðxete ìti P (t) = LeL(kt+c) +e L(kt+c) ìpou P = P () P (t) = LP P +(L P ) e Lkt

151 Kef laio 4. Oloklhr tikìc Logismìc Orismèno Olokl rwma: 'Ennoia, Idiìthtec 4.3. A' OMADA 989. An f (x) dx = 4, 4 f (x) dx =64kai 3 f (x) dx =na breðte to 4 3 f (x) dx. 99. An α (f (x)+3g(x)) = 5 kai na breðte ta kai 4 α f (x) dx +5 α α α f (x) dx g (x) dx g (x) dx =7 99. 'Estw ìti h f eðnai orismènh sto R kai suneq c. Na apodeðxete ìti: 4 3 f (x) dx 3 f (x) dx = 99. An gia thn suneq f isqôei β α f (x) dx =4na upologðsete thn par stash: 5 β α βf (x) dx β α αf (x) dx β α f (x) dx f (x) dx α β 3f (x) dx 993. 'Estw f (x) =e x. Na lujoôn oi parak tw exis seic. t f (x) dx =3. +t f (x) dx =3 3. f (t + x) dx =3 4. tf (x) dx = Gia poioôc arijmoôc b> isqôei b (b 4x) dx 6 5b 995. Apì tic exet seic tou PanpisthmÐou tou Toronto, Na upologðsete to F (x) = x π (t +)συν (t) dt.

152 Orismèno Olokl rwma: 'Ennoia, Idiìthtec. Na EpalhjeÔsete thn ap nthsh sac paragwgðzontac Na apodeðxete ìti 3 ( x + βx + γ ) ( dx = x + βx + γ ) ( dx + x + βx + γ ) dx 4.3. B' OMADA 997. Na breðte tic timèc tou α gia tic opoðec isqôei ) (t log α) dt =log ( α 998. Gia mia suneq sun rthsh f :[α, β] R eðnai gnwstì ìti β α f (x) dx = Na apodeðxete ìti h f èqei mða toul qiston rðza sto [α, β] Na apodeðxete ìti an α<βkai gia thn suneq sun rthsh f isqôei β tìte f(x) =gia ìla ta x [α, β].. Na apodeðxete ìti. Na apodeðxete ìti α f (x) dx = < e x dx < e ln 4 4 < 4 3 ln t t dt < ln 3 3. 'Estw f kai g dôo suneqeðc sunart seic orismènec sto di sthma Δ. Upojètoume ìti gia dôo upodiast mata [α, β] kai [κ, λ] tou Δ isqôei: β α f (x) dx > β α g (x) dx kai λ κ f (x) dx < λ κ g (x) dx. Na apodeðxete ìti oi (C) f, (C) g è- qoun èna toul qiston koinì shmeðo. 3. 'Estw f : R R mða sun rthsh gia thn opoða isqôei f (x) > gia ìla ta x. Na lôsete thn exðswsh e x x f (t) dt =.

153 Kef laio 4. Oloklhr tikìc Logismìc G' OMADA 4. Apì tic exet seic tou PanepisthmÐou tou Berkeley, 98. 'Estw f :[, ] R suneq c. BreÐte to ìrio lim t H sun rthsh F (x) = x f (t) dt α 4.4. A' OMADA 5. Na paragwgðsete tic sunart seic:. f (x) = x et dt. g (x) = x tdt 3. h (x) = x +x+ t dt 4. s (x) = 4x x+ e t dt 5. w (x) = x ( axe t + b ) dt x t f (x) dx 6. 'Estw ϕ (x) = x ημtdt. BreÐte ta ϕ ( π 4 7. 'Estw h sun rthsh F (x) = x ), ϕ ( π ), ϕ ( ) π 4,ϕ ( ) π. (t 5) dt. Na breðte tic exis seic twn efaptomènwn thc C f sta shmeða me tetmhmènec kai Na epalhjeôsete thn isìthta: αx dt x α t = dt t 9. Na breðte sun rthsh f me pedðo orismoô to R tètoia ste na isqôei f (x) =xημx kai f() =.. 'Estw F (x) = x et dt, G (x) = x ( 3 t + ) dt. Na epalhjeôsete tic isìthtec:. (F (x)+g(x)) = e x x. (F (x) G (x)) = e x x ( 3 t + ) dt x x et dt x 3. (F G) (x) = ( x + ) x e 3 (t +) dt. BreÐte to x x Upodeixh: ParagwgÐste. e t +t e t + dt. x. BreÐte to lim ημ(t )dt x x( συνx) et dt

154 Hsun rthsh F (x) = x α f (t) dt 3. Gia thn paragwgðsimh sun rthsh f me pedðo orismoô to R isqôei x = f(x) +4t dt Na deðxete ìti h f eðnai dôo forèc paragwgðsimh kai f (x) =4f (x) B' OMADA 4. 'Estw suneq c sun rthsh f :[, + ) (, + ). Na apodeðxete ìti gi ìla ta x> isqôei x x f(t)dt > x tf(t)dt 5. H sun rthsh I :[, ] R me I (x) = x t+ dt t eðnai suneq c +5t+6 (giati?) kai epomènwc parousi zei (olikì) mègisto kai(olikì) el qisto. BreÐte ta. 6. Gia thn suneq sun rthsh f : [, + ) R eðnai gnwstì ìti de mhdenðzetai kai ìti gia ìla ta x kai isqôei Na breðte thn f. x f (t)(t +)dt = f (x) 7. Na apodeðxete ìti ( β(x) α(x) f (t)dt) = β (x) α (x) f (α (x)) f (β (x)) 8. Na melet sete wc proc th monotonða thn sun rthsh e x ln t f (x) = t dt 9. Apì tic exet seic tou 995, Dèsmh IV. An G (x) = x f (t) dt ìpou f (t) = 3t e u u du kai x>, t> na breðte:. G (). to lim xg (x) 3 x + x+. Na melet sete wc proc th monotonða thn sun rthsh g (x) = ln tdt x

155 Kef laio 4. Oloklhr tikìc Logismìc 53. Na brejeð h par gwgoc thc f stic akìloujec peript seic:. f (x) = x. f (x) = x ημt t dt ημtx tx dt. Apì tic exet seic tou 995, Dèsmh I. JewroÔme touc pragmatikoôc arijmoôc α, β me <α<βth suneq sun rthsh f :(, + ) R gia thn opoða β α f (t) dt =kai th sun rthsh x g (x) =+ x α f (t) dt, x (, + ) Na apodeðxete ìti up rqei èna toul qiston x (α, β) tètoio stenaisqôoun:. H efaptomènh thc grafik c par stashc thc sun rthshc g sto shmeðo (x,g(x )) na eðnai par llhlh ston xona x x.. g (x )=+f (x ) 3. 'Estw h sun rthsh f (x) = x x+ ln tdt.. Na breðte to pedðo orismoô thc.. Na breðte ta diast mata sta opoða eðnai kurt -koðlh. 4. 'Estw f (x) = x t e t dt. Na breðte ta diast mata monotonðac thc f. Na upologðsete thn el qisth tim thc f 5. 'Estw f : R R mða suneq c sun rthsh. Me thn bo jeia thc g (x) =( x) x f (t) dt na apodeðxete ìti h exðswsh x f (t) dt =( x) f (x) èqi mða toul qiston rðza. 6. Na brejeð suneq c sun rthsh f : R R gia thn opoða gia ìla ta x isqôei: x e t f (x t) dt = ημx 7. Na apodeðxete ìti h sun rthsh F (x) = x gnhsðwc aôxousa. ( t 4 t 3 t + ) dt eðnai

156 Hsun rthsh F (x) = x α f (t) dt 8. Na apodeðxete ìti gia k je α (, + ) isqôei: α dx +x = α dx +x 9. Na paragwgðsete thn sun rthsh x 3 a3 dt. 3. Gia poia tim tou x to olokl rwma x+3 Poia eðnai h mègisth tim? 3. Na apodeðxete ìti h sun rthsh f (x) = x gnhsðwc aôxousa. Upodeixh: SthriqjeÐte sthn anisìthta e x x + x x t (5 t) dt gðnetai mègisto? ( e ημt ημt ) dt eðnai 3. Na breðte sun rthsh me pedðo orismoô to R gia thn opoða, gia ìla ta x, isqôei x f (t) dt = xe x + d 33. 'Estw I (x) = x ( + t)ν dt. Na apodeðxete ìti ( + x) I (x) νi (x) = 34. Na beðte se poia x parousi zei akrìtato kamp h sun rthsh: ϕ (x) = x (t ) (t ) 3 dt 35. Na breðte thn paragwgðsimh sun rthsh f an eðnai gnwstì ìti x f (t) dt = f (x) e x 36. 'Estw f : R R mða suneq c sun rthsh tètoia ste na isqôei f (x + y) =f (x)+f (y) gia ìla ta x, y. Na apodeðxete ìti gia k je α> isqôei α α f (x) dx =. 37. Na breðte suneq sun rthsh f orismènh sto di sthma [, π ] gia thn opoða isqôei x α f (t) dt =ημx ìpou α [, π ]. Sth sunèqeia na breðte thn tim tou α. 38. 'Estw mða suneq c sun rthsh f orismènh se èna di sthma Δ kai F (x) = x+α x α f (t) dt. Na apodeðxete ìti F (x) =f (x + α) f (x α).

157 Kef laio 4. Oloklhr tikìc Logismìc H sun rthsh f : R R eðnai suneq c kai gia ìla ta x isqôei x Na breðte ton tôpothcg (x) =f ( x ). f (t) dt = xημ (πx) 4. Apì to majhmatikì diagwnismì Harvard-MIT, 6. Upojètoume ìti oi f kai g eðnai paragwgðsimec sunart seic tètoiec ste xg (f (x)) f (g (x)) g (x) =f (g (x)) g (f (x)) f (x) gia ìla ta x. Epiplèon, h f eðnai mh arnhtikh kai h g jetik. Akìmh gia k je α α f (g (x)) dx = e α Dojèntoc ìti g(f()) = na upologðsete to g(f(4)). 4. 'Estw f :[, ) [, ) mða suneq c sun rthsh h opoða gðnetai mhdèn mìno sto mhdèn. 'Estw. Na breðte to lim x + g (x). g (x) = x x f (t) dt tf (t) dt. Na apodeðxete ìti h g eðnai gnhsðwc fjðnousa. 4. 'Estw α R stajerìc. Gia thn suneq sun rthsh f : R R eðnai gnwstì ìti αx α f ( t x α) dt = f (t) dt gia ìla ta x. Poia mporeð na eðnai h sun rthsh f? Upodeixh: Na metasqhmatðsete to olokl rwma αx f (t) dt. α 43. 'Estw g : R R mða suneq c sun rthsh gia thn opoða isqôei g (x) > gia ìla ta x. 'Estw ìti gia thn sun rthsh f : R R isqôei f (x) = x g (f (t)) dt. Na apodeðxete ìti h f eðnai paragwgðsimh kai na breðte thn par gwgo thc.. Na apodeðxete ìti h f eðnai antistrèyimh. 44. Na breðte th suneq sun rthsh f me pedðo orismoô to R gia thn opoða isqôei: x f (x) = f (t) dt)+

158 Orismèno Olokl rwma: Teqnikèc 45. Apì to majhmatikì diagwnismì Harvard-MIT, 3. Gia poia tim tou α> to a x ln x 3 dx gðnetai el qisto? a 46. Apo tic exet seic tou 993, Dèsmh I. Na brejeð suneq c sun rthsh f : R R gia thn opoða isqôei: me x, α R. x α e t f (t) dt = e x e α e x f (x) 47. Na breðte to α ste h sun rthsh { x ημ xdx f (x) =, x x 3 α, x = na eðnai suneq c. Charles Hermite Apì tic exet seic tou 999, Dèsmh I. 'Estw h :[, + ) R suneq c sun rthsh pou ikanopoieð th sqèsh h (x) = 999 (x ) + gia k je x. Na apodeðxete ìti x h (t) dt t. h(x) = 999x ln x, x. Jacques Salomon Hadamard H h eðnai gnhsðwc aôxousa sto [, + ). 49. Na apodeðxete ìti an h suneq c sun rthsh f : R R eðnai rtia tìte h sun rthsh F (x) = x f (t) dt eðnai peritt G' OMADA 5. To je rhma Hermite- Hadamard. Na apodeðxete ìti an h f eðnai kurt sto [α, β] tìte isqôei: ( ) α + β f < β f (α)+f (β) f (t) dt < β α α 4.5 Orismèno Olokl rwma: Teqnikèc 4.5. A' OMADA 5. Na upologðsete ta oloklhr mata:

159 Kef laio 4. Oloklhr tikìc Logismìc xdx. x+ dx 3. x dx 4. π π ημxdx 5. Na upologðsete ta oloklhr mata:. x +7dx. 4 (ex e x ) dx 3. 3 x + x dx 4. 3π συνxdx 53. Na upologðsete ta oloklhr mata:. β α xdx. π 4 π ημxdx 6 3. λ λ ln xdx 4. p p x dx 54. Na upologðsete ta oloklhr mata:. 6 x dx. 3 x dx e e x+ x+ dx x dx 55. Na upologðsete ta oloklhr mata:. t ( s u + ) du. ( x ) dx Na upologðsete ta oloklhr mata: 4. x dx e x + dx. ( x (3t +)dt) dx. ( ) (3t +)dt dx 3. x x 4. π 4 π 4 (t ) dt συν x dx 57. Na upologðsete ta:. ( ) 4 3 (x +y) dx dy. ( 4 ) 3 (x +y) dx dy 3. ( ) 4 3 (x +y) dy dx 4. ( 4 ) 3 (x +y) dy dx 58. Na upologðsete ta oloklhr mata:. 4 ( 3 ) x 4 dx. s 3 s x ln xdx 3. 3 ( x 4x +3 ) dx 4. 3 ( x 4x + κ ) dx

160 Orismèno Olokl rwma: Teqnikèc 59. Na qrhsimopoi sete thn teqnik thc olokl rwshc kat par gontec gia na upologðsete ta oloklhr mata:. xe x dx. π x συνxdx 3. π 3 π 4 x ημ x dx 4. e x3 ln xdx 6. Na qrhsimopoi sete thn teqnik thc allag c metablht c gia na upologðsete ta oloklhr mata:. e. 3 ln x x dx x x dx 3. e e 4. e y ln y dy ημ ln y y dy 6. Na upologðsete ta oloklhr mata:. π π ημ xdx. ρ ρ f (x) dx ìpou f (x) =x +x 4 kai ρ <ρ eðnai oi rðzec thc f. 3. n m ( mx + nx ) dx 4. x x tdt 6. DÐnetai h sun rthsh f (x) = { x αν x< x αν x. Na apodeðxete ìti h f eðnai suneq c.. Na upologðsete to olokl rwma 63. Na upologðsete to olokl rwma: 5 f (x) dx ( x 3 + x 4 ) dx 64. Apì tic exet seic tou 988, Dèsmh IV. Na upologðsete to olokl rwma: x 3 5x + dx x

161 Kef laio 4. Oloklhr tikìc Logismìc Na apodeðxete ìti: α β xσυνxdx + xσυνxdx + α β 66. Na breðte ton λ (, π ( x + xσυνx ) dx = β3 3 ) π ètsi ste 6 (ημx συνx) dx = λ π 6 ημxdx. 67. 'Estw f (x) =αx + βx + γ. Na breðte gia poièc timèc twn α, β, γ isqôoun f () = 8, f () + f () = 33 kai f (x) dx = Na lôsete thn exðswsh x+ ( t 4t + 5 ) dt =3 x Apì tic exet seic tou 993, Dèsmh IV. An h sun rthsh g(x) èqei suneq par gwgo sto [, ] kai ikanopoieð thn sqèsh na breðte to g(). xg (x) dx = 993 g (x) dx 7. Apì to diagwnismì Putnam, 973. 'Estw f : [, ] R paragwgðsimh me suneq par gwgo ste f () = kai f (x) gi aìla ta x. Na apodeðxete ìti ( f (x) dx) f 3 (x) dx Upodeixh: ErgasjeÐte me th sun rthsh h (x) = ( x f (t) dt) x f 3 (t) dt. 7. An t ( 3x +x + ) dx =breðte to t ( 3x x + ) dx. 7. Gia th suneq sun rthsh f eðnai gnwstì ìti f( x)+f(x) =x gia f (x) dx. ìla ta x R. BreÐte to 73. Apì to majhmatikì diagwnismì Harvard-MIT,. MÐa suneq c pragmatik sun rthsh f ikanopoieð thn tautìthta f (x) =3f (x) gia ìla ta x. An f (x) dx =breðte to f (x) dx.

162 Orismèno Olokl rwma: Teqnikèc 4.5. B' OMADA 74. Na upologðsete to olokl rwma 75. Na upologðsete to olokl rwma: 6 x + x dx dx x +9 x 76. Na apodeðxete, qwrðc na upologðsete ta oloklhr mata, ìti: π 8 π Na upologðsete to olokl rwma x ημ 9 x = x 7 3x 5 +7x 3 x συν dx = x e συνx dx = 78. 'Estw I ν = α e x x ν dx. Na upologðsete to I. ( ν ) k x dx k=. Na apodeðxete ìti I ν = νi ν α ν e α. 3. Na upologðsete to I 'Estw I ν = π συνν xdx.. Na apodeðxete ìti I ν+ = ν+ ν+ I ν.. Na apodeðxete ìti (a ) I k = (k ) (k) π (b ) I k+ = e συνx dx 8. Na upologðsete to olokl rwma π 4 π (εϕx + σφx +lnx) dx Na upologðsete to olokl rwma ln x x dx

163 Kef laio 4. Oloklhr tikìc Logismìc 6 8. Gia th suneq sun rthsh f isqôei αf (x)+βf ( x) =((β α) x + α) gia ìla ta x. Na apodeðxete ìti an α β tìte f (x) dx =. 83. Na upologðsete to olokl rwma π π 4 ημ x x dx 84. 'Estw f :[, ] R suneq c sun rthsh tètoia ste f ( x ) dx =. Na upologisjeð to olokl rwma I = f(x ) dx, <α. Upodeixh: Na gr yete I = f(x ) α x + dx + x = u. 85. BreÐte to olokl rwma x xdt. α x + f(x ) α x dx kai na jèsete sto pr to olokl rwma Na apodeðxete ìti an f (x) = b a ημ (tx) dt tìte f (x) = b a (συνtx) tdt. 87. Na apodeðxete ìti dx x n 3 ( + x ) n = (n ) ( + x n + ) (n ) kai katìpin na upologðsete to dx (+x ) Na apodeiqjeð ìti an μ, ν eðnai jetikoð akèraioi tìte ) (x μ ν ν + x μ dx = dx ( + x ) n 89. 'Estw λ jetikìc arijmìc kai f mða sun rthsh suneq c sto di sthma [,λ].. Na apodeðxete ìti λ f(x) λ f(x)+f(λ x) dx = f(λ x) f(x)+f(λ x) dx. Na apodeðxete ìti to olokl rwma eðnai anex rthto thc f. λ f(λ x) f(x)+f(λ x) dx

164 Orismèno Olokl rwma: Teqnikèc 3. Me thn bo jeia tou prohgoumènou erwt matoc me llo trìpo na upologðsete to olokl rwma: x 4 x 4 +( x) 4 dx 9. Na upologðsete ta oloklhr mata I = I = ( x + x +)dx ( x ln x + 3x ) dx 9.. 'Estw ìti x α = u x. Na apodeðxete ìti x = α +u u.. QrhsimopoieÐste ton metasqhmatismì x α = u x gia na upologðsete to olokl rwma I = x α dx 3. An F (x) = x t dt na upologðsete to el qisto thcf. 9. Me thn bo jeia tou metasqhmatismoô x = αημt na upologðsete to olokl rwma α α α x dx 93. Na lujeð h exðswsh α ( συνx + α ) dx = συνα + α 'Estw f mða suneq c sun rthsh me pedðo orismoô to R. Na apodeðxete ìti β β α f (x) dx = f (α + x) dx α 95.. 'Estw α> kai f :[,α] R suneq c. Na apodeðxete ìti α f (α x) dx = α f (x) dx. 'Estw f :[,α] R kai g :[,α] R me tic akìloujec idiìthtec: f,g suneqeðc gia ìla ta x [,α] isqôei f(α x) =f(x) up rqei stajer c ètsi ste gia ìla ta x [,α] na isqôei g(x)+ g(α x) =c.

165 Kef laio 4. Oloklhr tikìc Logismìc 63 Na apodeðxete ìti α f (x) g (x) dx = c α f (x) dx 3. Na upologðsete to olokl rwma π xημx +συν x dx 96. Na upologðsete to olokl rwma ( ) 4 3 dy (x+y) dx. 97. Upojètoume ìti h f eðnai duo forèc paragwgðsimh sto [, ] kai ìti h f eðnai suneq c. 'Estw ìti f (e) =f(e) =f() = kai e f(x) dx = x. BreÐte to e f (x)lnxdx. 98. Na upologðsete to olokl rwma π 3 π 6 ημ xσυν x dx 99. Na apodeðxete ìti an f : R R eðnai mða suneq c sun rthsh tìte α α f (x) dx = α (f (x)+f ( x)) dx. Apì to majhmatikì diagwnismì Harvard-MIT, 4. 'Estw f : R R mða paragwgðsimh sun rthsh me f (x) f (x) dx = kai f (x) f (x) dx =8. Poiì eðnai to f 4 (x) f (x) dx?.. Na breðte arijmoôc α, β, γ ètsi ste gia k je arijmì x na isqôei x (x +) = α x + βx + γ x +. Na upologðsete to olokl rwma x (x +) dx. Apì tic exet seic tou 999, Dèsmh I. DÐnetai h sun rthsh f (t) = t+3 t+, t [, 4].. Na upologðsete to olokl rwma I = 4 f (t) dt.. 'Estw h sun rthsh g (x) = 4 x+ f (t) x+ e t x dt, x>. (a ) Na apodeðxete ìti e x (b ) Na upologðsete to e t x e 4 x lim g (x) x + gia k je t [, 4] kai x>.

166 Orismèno Olokl rwma: Teqnikèc 3. An I ν = ln ν xdx na apodeðxete ìti I ν = x ln ν x νi ν 4. Na apodeðxete ìti gia k je zeôgoc jetik n akeraðwn μ, ν isqôei x μ ( x) ν dx = 5. Na upologðsete to olokl rwma e 6. Na epalhjeôsete thn isìthta β+ α+ xe x dx β α e x ν ( x) μ dx +ln x 3+x ln x dx xe x dx = βe β+ αe α+ + βe β+ αe α+ 7. Na epalhjeôsete thn isìthta α β dα + α β dβ = β+ β + + α α ln α 8. 'Estw I ν = π 4 εϕν xdx. Na apodeðxete ìti gia k je ν 3 isqôei: I ν = ν I ν 9. Na apodèixete ìti gia α> kai n 3 isqôei α α x n +x n dx =.. Apì tic exet seic tou panepisthmðou thc Oxfìrdhc, 889. BreÐte èna anagwgikì tôpo gia to olokl rwma I ν = e αx συν ν xdx ìpou o ν eðnai ènac jeikìc akèraioc kai sth sunèqeia upologðste to e αx συν 4 xdx.. Apì tic exet seic tou 99, Dèsmh I. An I ν = π 4 εϕν xdx, ν N,tìte:. Na apodeðxete ìti gia k je ν> isqôei I ν = ν I ν,. Na upologðsete to I 5.. 'Estw Na apodeðxete ìti I ν,μ = I ν,μ = x ν ( x) μ dx μ ν + I ν+,μ

167 Kef laio 4. Oloklhr tikìc Logismìc Na upologðsete to x ln xdx. Na apodeðxete ìti h sun rthsh { x f (x) = ln x <x x = eðnai suneq c 3. Na upologðsete to f (x) dx 4. Me thn bo jeia tou metasqhmatismoô x = εϕt na upologðsete to olokl rwma dx +x 5. 'Estw f : R R mða rtia suneq c sun rthsh. Na apodeðxete ìti:. ( α α f (x)ln x + ) +x dx =. α α f (x) e kx + dx = α 6. An f (x) dx (f (x)+x) dx =4na breðte to f (x) dx. 7.. DeÐxte ìti β α f (x) dx = β α f (a + β x) dx. DeÐxte ìti π ημν xdx = π συνν xdx, ν N 3. BreÐte ta π ημ xdx, π συν xdx 8. Na breðte tic timèc tou jetikoô arijmoô α gia tic opoðec isqôei α ( 4x +3x ) dx α 9. 'Estw f : R R tètoia ste gia ìla ta x na isqôei f (α x)+f (α + x) =β ìpou α, β eðnai stajeroð arijmoð. Na apodeðxete ìti α. JewroÔme th sun rthsh g (x) =. DeÐxte ìti h g eðnai suneq c. f (x) dx = αβ { ημx x, x, x =

168 Orismèno Olokl rwma: Teqnikèc. Na melet sete wc proc th monotonða th sun rthsh f :[ π, π] R me f (x) = π π g (x + t) dt. 'Estw f mða suneq c sun rthsh orismènh sto di sthma [ α, α]. Na apodeðxete ìti:. An h f eðnai rtia tìte α α f (x) dx = α f (x) dx. An h f eðnai peritt tìte α α f (x) dx =.. UpenjumÐzetai ìti mða sun rthsh f onom zetai periodik an up rqei arijmìc T pou onom zetai perðodoc thc f ètsi ste: gia k je x D f eðnai kai x + T D f gia k je x D f isqôei f(x + T )=f(x). Na apodeðxete ìti an h f : R R èinai mða suneq c periodik sun rthsh me perðodo T tìte gia k je x R isqôei Upodeixh: x = T u x+t x f (t) dt = T f (t) dt 3. Na apodeðxete ìti an h suneq c sun rthsh f : R R eðnai gnhsðwc aôxousa tìte x r x r f (t) dt + f (t) dt f (t) dt 4. Na apodeðqjeð ìti an h suneq c rtia sun rthsh f : R R eðnai periodik me perðodo T tìte gia k je suneq sun rthsh g : R R isqôei T xg (f (x)) dx = T T g (f (x)) dx 5. H f :[, ] R eðnai suneq c. BreÐte to π f (ημx) συνxdx. 6. H sun rthsh f eðnai orismènh sto di sthma [α, β] kai èqei suneq deôterh par gwgo. Na apodeðxete ìti an f (α) =f (α) kai f (β) =f (β) tìte isqôei β e x ( f (x) f (x)+f (x) ) dx = α

169 Kef laio 4. Oloklhr tikìc Logismìc Apì to diagwnismì Putnam, 979. 'Estw ìti <a<b. BreÐte to ( ) lim (bx + a ( x)) t t dx t 8. Gia th suneq sun rthsh f eðnai gnwstì ìti gia ìla ta x isqôei: f (x) =P (x)+ x e t f (x t) dt ( ) ìpou P (x) eðnai èna polu numo me rðza to.. Na apodeðxete ìti f (x) =P (x)+p (x). Na exhg sete giatð h prohgoômenh sqèsh mac odhgeð sthn eôresh thc f kai giatð h sun rthsh f pou ikanopoieð thn ( ) eðnai monadik. 9. 'Estw ìti <α<βkai ìti gia th suneq f :[α, β] R isqôei β α f (xt) dt β α f (t) dt gia k je jetikì arijmì x. Na apodeðxete ìti: β α f (t) dt = βf (β) αf (α) 3. Apì tic exet seic tou 996, Dèsmh IV. JewroÔme th sun rthsh f (x) = +x + λx, λ R. f(x). Na upologðsete thn tim tou λ an eðnai gnwstì ìti lim x + x =.. Gia thn tim tou λ pou br kate parap nw na upologðsete to olokl rwma I = x f (x) dx. 3. JewroÔme dôo sunart seic f, g suneqeðc sto [α, β].. Na apodeðxete ìti gia k je λ R isqôei β α (f (x) λg (x)) dx. Na apodeðxete ìti gia k je λ R isqôei ìpou A= β α Aλ +Bλ +Γ β β g (x) dx, B= f (x) g (x) dx, Γ= f (x) dx α α

170 Orismèno Olokl rwma: Teqnikèc 3. Na apodeðxete ìti isqôei : ( β ( β f (x) g (x) dx) α α )( β ) f (x) dx g (x) dx α (Anisìthta Cauchy-Schwartz-Bunyakovsky) 3. JewroÔme f,g suneqeðc sunart seic me pedðo orismoô to R tètoiec ste gia k je x na isqôoun: f ( x) =f (x)+κ g ( x) = g (x)+λ ìpou oi κ, λ eðnai stajeroð pragmatikoð arijmoð. Na apodeðxete ìti: α α f (x) g (x) dx = λ α f (x) dx κ α g (x) dx + κλα Augustin Louis Cauchy 'Estw ν jetikìc akèraioc. Na apodeðxete ìti gia k je x R up rqei akrib c ènac arijmìc y ètsi ste y ν+ + y = x kai epomènwc orðzetai mða sun rthsh x f (x) =y me pedðo orismoô to R kai tètoia ste gia k je x. Katìpin: f ν+ (x)+f (x) =x () Hermann Amandus Schwarz Na apodeðxete ìti f() =.. Na apodeðxete ìti h f eðnai suneq c. 3. Na apodeðxete ìti h f eðnai paragwgðsimh. 4. Na apodeðxete ìti h f eðnai gnhsðwc aôxousa. 5. Na apodeðxete ìti to sônolo tim n thc f eðnai to R. 6. Na ekfr sete wc par stash thc f to x f (t) dt. Viktor Yakovlevich Bunyakovsky

171 Kef laio 4. Oloklhr tikìc Logismìc G' OMADA 34. Apì to majhmatikì diagwnismì Harvard-MIT, 4. 'Estw P (x) =x 3 3 x + x + 4. 'Estw P [] (x) =P (x) kai gia ν èstw P [ν+] (x) =P [ν] (P (x)). Na upologðsete to P [4] (x) dx. 35. (H anisìthta tou Jensen) 'Estw f : R R mða kurt sun rthsh kai g : R R suneq c. Na apodeiqjeð ìti: ( ) f g (x) dx f (g (x)) dx 36. (Oi anisìthtec tou Young) 'Estw f :[,γ] R mða mða paragwgðsimh sun rthsh me f (x) > gia ìla ta x kai f() =. JewroÔme α [,γ], β [,f(γ)]. Na jewr sete wc dedomèno ìti h f eðnai suneq c gia na deðxete ìti: α β f (x) dx + f (x) dx αβ Na apodeðxete ìti to <Ñson>> isqôei mìno ìtan β = f (α). Sth sunèqeia na apodeðxete ìti gia touc jetikoôc arijmoôc α, β, p, q me p + q =isqôei: α p p + βq q αβ 4.6 Embad 4.6. A' OMADA Johan Ludwig William Valdemar Jensen Na upologðsete to embadìn S sto parak tw sq ma: William Henry Young y = x S y = x Na upologðsete to embadìn S sto parak tw sq ma:

172 Embad x + y = y = x S 39. Na upologðsete to embadìn S sto parak tw sq ma: y = x S y = x 4. Na upologðsete to embadìntou qwrðou pou perikleðetai apì tic grafikèc parast seic twn sunart sewn ln x kai ln x. 4. Apì tic exet seic tou 996, Dèsmh IV. Na upologðsete to embadìn tou qwrðou pou perikleðetai apì tic grafikèc parast seic twn sunart sewn g (x) = x, f (x) =x kai thn eujeða x =. 4. Apì tic exet seic I.I.T, IndÐa, 989. Na breðte to megisto kai to el qisto thc sun rthshc f (x) =x (x ), x Sth sunèqeia na breðte to embadìn tou qwrðou pou perikleðetai apì thn kampôlh y = x (x ),ton y- xona kai thn eujeða y =. 43. Poiì eðnai to embadìn pou perikleðoun oi grafikèc parast seic twn y = x, y = x? 44. To grammoskiasmèno embadìn tou sq matoc eðnai. Poiì eðnai to α?

173 Kef laio 4. Oloklhr tikìc Logismìc Apì to majhmatikì diagwnismì Harvard-MIT,. ProsdiorÐste th jetik tim α pou eðnai tètoia ste h parabol y = x + na diqotomeð to embadìn tou orjogwnðou me korufèc (, ), (α, ), (,α +), kai (α, α +). 46. 'Estw f (x) = x +4x 3. Na upologðsete to embadìn tou qwrðou pou perikleðetai apì: thn C f thn efaptomènh thc C f sto shmeðo thca(, 3) kai thn efaptomènh thc C f sto shmeðo thcb(3, ) B A y = x + 4x 3

174 Embad 47. Na upologðsete to embadìn tou qwrðou pou orðzetai apì thn grafik par stash thc f (x) =(x ) (x ) (x 3) (x 4) kai ton x xona. 48. To qwrðo tou sq matoc perikleðetai apì tic grafikèc parast seic twn sunart sewn f (x) = x +6x 5 g (x) = x +4x 3 h (x) =3x 5 Na upologisjeð to embadìn tou. 49. Apì tic exet seic tou 99, Dèsmh I. DÐnetai h sun rthsh f me f (x) =3x + x.. Na breðte tic asômptwtec thc grafik c par stashc thc sun rthshc f.. Na upologðsete to embadìn E (α) tou qwrðou pou perikleðetai metaxô thc grafik c par stashc thc sun rthshc f, thc eujeðac me exðswsh y =3x, kai twn eujei n me exis seic x =kai x = α me α>. 3. Na upologðsete to ìrio tou embadoô E (α) tou anwtèrw qwrðou ìtan to α teðnei sto peiro B' OMADA 5. Na upologðsete to embadìn pou perikleðetai apì tic grafikèc parast - seic twn sunart sewn f (x) =x ν, g (x) = ν x. 5. Apì tic exet seic tou 99, Dèsmh I. DÐnetai h sun rthsh f me f (x) =(x +4)e x, x R. Na upologisteð to embadìn tou qwrðou pou orðzetai apì ta shmeða (x, y) me x, y f (x). 5.. Na apodeðxete ìti h exðswsh x +=x èqei akrib c dôo (pragmatikèc) rðzec.. Na apodeðxete ìti to embadìntou grammoskiasmènou qwrðou sto parak tw sq ma

175 Kef laio 4. Oloklhr tikìc Logismìc 73 eðnai Ðso me 3 (q q ) ( q +q + q q +q + q ) ìpou q <q eðnai oi rðzec thc exðswshc tou erwt matoc (a'). 53. DÐnetai h sun rthsh f(x) = x + x SumbolÐzoume me E(a) to embadìn pou perikleðetai apì thn grafik par stash thc f,touc xonec kai thn x = α. Na upologisjeð to ìrio lim α α E(α) 54. Sto sq ma up rqei h grafik par stash enìc tritobajmðou poluwnômou me rðzec, 3, 4, Na apodeðxete ìti E = E. ProspajeÐste na genikeôsete. 55. Na upologðsete to embadìn S tou qwrðou pou perikleðetai apì tic x =, x =, y = x, y =x x. 56. 'Estw f(x) =e x. Na breðte gia poia tim tou p to embadìn S, pou perikleðetai apì tic C f, y =, x =,x = p eðnai Ðso me to embadìn S pou perikleðetai apì tic C f, y =, x =,x = p.

176 Embad 57. Gia poia tim tou λ to embadìn tou qwrðou pou perikleðetai apì tic grafikèc parast seic twn f (x) =x λx, g(x) = x λ 58. Gia to parak tw sq ma na apodeðxete ìti E = E. eðnai mègisto? 59. JewroÔme tic sunart seic f(x) = ex +e x, g(x) = ex e x. 'Estw E(t) to embadìn tou qwrðou pou perikleðetai apì tic grafikèc parast seic twn f,g, ton y- xona kai thn eujeða x = t. Na breðte to ìrio lim E(t). t + 6. Apì tic exet seic tou 6. JewroÔme th sun rthsh f (x) = +(x ) me x.. Na apodeðxete ìti h f eðnai -.. Na apodeðxete ìti up rqei h antðstrofh sun rthsh f thc f kai na breðte ton tôpo thc. 3. (a ) Na breðte ta koin shmeða twn grafik n parast sewn twn sunart sewn f kai f me thn eujeða y = x. (b ) Na upologðsete to embadì tou qwrðou pou perikleðetai apì tic grafikèc parast seic twn sunart sewn f kai f. 6. Apì tic exet seic tou 989, Dèsmh I. DÐnetai h sun rthsh f me f (x) =ημ ( x + π ) kai pedðo orismoô to di sthma [ π 4, π 4 ].. Na brejeð h exðswsh thc efaptomènhc thc grafik c par stashc thc f sto shmeðo x = π 8.. Na upologisjeð to embadìntou qwrðou pou perikleðetai apì thn parap nw efaptomènh, thn grafik par stash thc f kai touc jetikoôc hmi xonec Ox, Oy. 6. Gia poia tim tou λ h eujeða pou:

177 Kef laio 4. Oloklhr tikìc Logismìc 75 dièrqetai apì to shmeðo A(, ) kai èqei suntelest dieujônsewc λ sqhmatðzei me thn kampôlh y = x +qwrðo embadoô 6? 63. Apì tic exet seic tou 997, Dèsmh IV. An oi sunart seic f, g eðnai duo forèc paragwgðsimec sto R kai ikanopoioôn tic sqèseic : f (x) g (x) =4gia k je x R f () = g () kai f() = g(). Na breðte th sun rthsh t(x) =f(x) g(x), x R. Na breðte to embadìn tou qwrðou pou perikleðetai apì tic grafikèc parast seic twn sunart sewn f kai g. 64. JewroÔme tic sunart seic m (x) =συνx +, n (x) =ημx Na breðte thn suneq sun rthsh f pou èqei thn idiìthta: Gia k je α>to embadìn S pou perikleðetai apì tic x =, x = α, C m, C f eðnai Ðso me to embadìn S pou perikleðetai apì tic x =, x = α, C n, C f. y = ffflνx + S y = f (x) S y = μx x = ff 65. Na upologðsete to embadìn tou grammoskiasmènou qwrðou: 66. 'Estw C h grafik par stash thc sun rthshc f :[, + ) R me f (x) =x α, α>. 'Estw M tuqìn shmeðo thc C kai M,M oi probolèc tou M stouc xonec x x, y y antistoðqwc. Onom zoume E,E taembad tou

178 Embad miktogr mmou OM M kai tou parallhlogr mmou OM MM. Na apodeðxete ìti: E = α + E 67. JewroÔme tic sunart seic kai thn eujeða y = x. f (x) = +x, g(x) = x, x >. DeÐxte ìti h C f eðnai k tw apì thn C g.. Se k je α (, ) antistoiqoôme to shmeðo A(α, g (α)). Apì to A fèrnoume k jeth sthn y = x. DeÐxte ìti aut ja tèmnei p ntote thn C g se èna kai monadikì shmeðo B(β,g (β)) me β (, + ). 3. DeÐxte akìmh ìti to embadìn pou perikleðetai apì tic x = α, x =, y =kai C f eðnai Ðso me to embadìn pou perikleðetai apì tic x = β, x =, y =kai C f. 68. Up rqoun peirec, to pl joc, suneqeðc sunart seic f pou orðzontai sto [, + ) kai oi grafikèc parast seic touc perièqontai sto pr to tetarthmìrio kai dièrqontai apì thn arq twn axìnwn. Gia par deigma merikèc eðnai oi f (x) =e x, f (x) =ln(x +), f (x) =x, f (x) = x ìlec gia x>. ZhteÐtai na breðte poièc apì autèc èqoun thn akìloujh idiìthta: Gia k je shmeðo A C f,meprobol ston x x to shmeðo B, to embadìn tou trig nou OAB eðnai Ðso me to dipl sio tou miktogr mmou qwrðou OAB.

179 Kef laio 4. Oloklhr tikìc Logismìc Apì tic exet seic tou 99, Dèsmh IV. DÐnetai h sun rthsh f me tôpo f (x) = { e x e, x < ln x x, x Na apodeðxete ìti h f eðnai suneq c kai na upologðsete to embadìn tou qwrðou, to opoðo perikleðetai apì th grafik par stash thc f,ton xona x x kai tic eujeðec me exis seic x =kai x = e. 7. 'Estw C h parabol me exðswsh y = αx (α>). 'Estw A, B dôo di fora shmeða thc C. 'Estw Γ to shmeðo tom c twn efaptomènwn thc C sta A, B. 'Estw tèloc Δ ekeðno to shmeðo thc parabol c pou èqei thn Ðdia tetmhmènh me to Γ. SumbolÐzoume: Me E to embadìn tou miktogr mmou qwrðou pou perikleðetai apì tic AΓ, BΓ kai thn C. Me E to embadìn tou trig nou ABΓ.

180 Embad Me E 3 to embadìn tou trig nou ABΔ. Me E 4 to embadìn tou miktogr mmou qwrðou pou perikleðetai apì thn AB kai thn C. Na apodeðxete ìti:. x 3 = x +x. Hefaptomènh thc C sto Δ eðnai par llhlh sthn AB. 3. Apì ìla ta shmeða tou parabolikoô tìxou AB to Δ eðnai ekeðno pou apèqei apì thn AB th megalôterh apìstash. 4. (Arqim dhc) E 4 = 4 3 E 3 5. (Cavalieri) E = 3 E Arqim dhc 87- p.q G' OMADA 7. Sthn skhsh aut ja upologðsete to embadìn thc èlleiyhc kai tou uperbolikoô qwrðou. Bonaventura Francesco Cavalieri Embadìn thc èlleiyhc: 'Estw h èlleiyh x α + y β =. Gia na broôme to embadìn thc arkeð na broôme to embadìn thc pou perikleðetai sto pr to tetarthmìrio kainapollaplasi soume epð 4. To qwrðo autì perikleðetai apì thn grafik par stash thc y = α β α x, x [,α] kai ton x x. To embadìn tou eðnai Ðso me to I = α β α α x dx. Gia na upologðsete autì to olokl rwma na jèsete x = αημt. Ja breðte I = παβ 4. Telik to embadìn thc èlleiyhc eðnai E = παβ.. Embadìn uperbolikoô qwrðou: 'Estw h uperbol x y α β =. Gia na upologðsoume to embdìn tou qwrðou pou perikleðetai apì thn uperbol kai thn eujeða x = t > αarkeð na upologðsoume to dipl sio tou oloklhr matoc I = t β α α x α dq. Gia ton upologismì mporeðte na qrhsimopoi sete O Cavalieri up rxe majht c tou GalilaÐou.

181 Kef laio 4. Oloklhr tikìc Logismìc 79 (a ) Thn antikat stash u = α ( x + x) (b ) thn antikat stash u = α (ex + e x ) (g ) thn antikat stash u = Prèpei na breðte α συνx I = β ( ( t t α α α ln t + ) ) (t α)(t + α) + α ln α 7. M koc kampôlhc. Sthn skhsh aut mporeðte na deðte, p c, me mikr tropopoðhsh twn ide n pou qrhsimopooôme gia ton upologismì twn embad n mporoôme na upologðzoume m kh kampul n. Ja ergasjoôme sthn eidik perðptwsh mðac kampôlhc y = f (x) ìpou h f :Δ R eðnai paragwgðsimh kai h f eðnai suneq c. 'Estw [α, β] Δ. Ja broôme to m koc L thc kampôlhc apì to A =(α, f (α)) sto B =(β,f (β)). Gia to skopì autì qwrðzoume to di sthma [α, β] se ν upodiast mata pl touc β α. 'Eqoume ètsi ν touc arijmoôc x = α, x = α + β α,..., x ν = α ν kai me th bo jeia touc ta shmeða X = A =(x,f(x )), X =(x,f(x )),... X ν = B =(x ν,f(x ν)) 'Otan to ν aux nei aperiìrista dhlad ìtan ν + to m koc thc tejlasmènhc X X...X ν X ν teðnei sto L dhlad lim ν + X X...X ν X ν = L All X X...X ν X ν =(X X )+(X X )+... +(X ν X ν)= (x x ) +(f (x ) f (x )) + + (x x ) +(f (x ) f (x )) (x ν x ν) +(f (x ν ) f (x ν)) An efarmìsoume to je rhma mèshc tim c gia thn f se k je èna apì ta diast mata [x,x ], [x,x ],... [x ν,x ν] brðskoume ìti f (x ) f (x )=f (ξ )(x x ) f (x ) f (x )=f (ξ )(x x )... f (x ν ) f (x ν)=f (ξ ν)(x ν x ν)

182 Embad 'Ara X X...X ν X ν = x x = β α ν +(f ν (ξ k )) k= Epomènwc L = lim ν + β α ν +(f (ξ )) x ν x ν +(f (ξ ν)) = ν k= +(f (ξ k )) = β α +(f (x)) dx 'Ara L = β α +(f (x)) dx Efarmìste ton parap nw tôpo gia na breðte to m koc L stic akìloujec peript seic: Upodeixh: Gia to olokl rwma +x dx na jèsete x = εϕt.

183 Kef laio 4. Oloklhr tikìc Logismìc Ask seic se ìlo to kef laio 73. Na sumplhr sete tic isìthtec:. f (x) dx = 4. ( β α f (x) dx ) =. ( f (x) dx ) = 3. ( x a f (t) dt) = 5. β α f (x) dx = 6. x α f (t) dt = 74. Gia mða suneq sun rthsh f :[, ] R isqôei Na apodeðxete ìti. (f (x) x) dx. f (x) =x gia k je x xf (x) dx 3 f (x) dx 75. Me thn bo jeia tou metasqhmatismoô u =+lnx na upologðsete to olokl rwma: ( + ln x) dx x 76. 'Estw suneq c sun rthsh f orismènh sto di sthma Δ kai F (x) = x+α x α f (t) dt Na apodeðxete ìti F (x) =f (x + α) f (x α). 77. Apì tic exet seic tou 996, Dèsmh I. H sun rthsh f eðnai suneq c sto di sthma [α, β] kai isqôei ìti α f (x)+f (α + β x) =c ìpou c stajerìc pragmatikìc arijmìc. Na apodeðxete ìti: β ( ) α + β f (x) dx =(β α) f = β α (f (α)+f (β)) 78. Na breðte ton α an eðnai gnwstì ìti 6α α α 3 6 e x dx = π ημ (3x) dx

184 Ask seic se ìlo to kef laio 79. Na upologðsete to ìrio kai met to ìrio 8. Na upologðsete to ìrio x+ lim x + x x+ lim x + x ln tdt e t ln tdt x+ ημt lim x + x t dt 8. Na brejeð suneq c sun rthsh f :(, + ) R tètoia ste: f (x) > gia ìla ta x 3 f ( 3 xu) du = f (x) f () = e 8. Apì tic exet seic tou. 'Estw mða pragmatik sun rthsh f, suneq c sto sônolo twn pragmatik n arijm n R gia thn opoða isqôeoun oi sqèseic: i) f (x),giak jex R. ii) f (x) = x tf (xt) dt, gia k je x R. 'Estw akìmh g h sun rthsh pou orðzetai apì ton tôpo gia k je x R. g (x) = f (x) x. Na deðxete ìti isqôei f (x) = xf (x). Na deðxete ìti h sun rthsh g eðnai stajer. 3. Na deðxete ìti o tôpoc thc sun rthshc f eðnai f (x) = +x 4. Na breðte to ìrio lim (xf (x) ημx) x 'Estw g : R R mða gnhsðwc aôxousa sun rthsh me sônolo tim n to R.

185 Kef laio 4. Oloklhr tikìc Logismìc 83. Na apodeðxete ìti an gia th sun rthsh f : R R isqôei gia ìla ta x tìte f = g. (f g)(x) x (g f)(x). Gia th sun rthsh f : R R eðnai gnwstì ìti isqôei gia ìla ta x, ìpou eðnai to uperbolikì hmðtono. ϕ (sinh (x)) x sinh (ϕ (x)) sinh (x) = ex e x (a ) Na breðte thn ϕ kai na apodeðxete ìti eðnai suneq c. (b ) Na upologðsete to olokl rwma ϕ (x) dx. 84. 'Estw h sun rthsh kai C f h grafik thc par stash. f (x) =x 3 3x + x. Na breðte ta diast mata ìpouhf strèfei ta koðla proc ta nw proc ta k tw kai na prosdiorðsete, an up rqoun, ta shmeða kamp c thc.. Na upologðsete to embadìn tou qwrðou pou perikleðetai apì thn C f kai thn efaptomènh thc C f pou dièrqetai apì to shmeðo P (, ). 85. Apì tic exet seic tou 997, Dèsmh IV. 'Estw f mða pragmatik sun rthsh suneq c sto R tètoia ste f (x) gia k je x R. JewroÔme th sun rthsh g (x) =x 5x + x 5x f (t) dt,. Na apodeðxete ìti g ( 3) g () < x R. Na apodeðxete ìti h exðswsh g (x) = èqei mða mìno rðza sto di sthma ( 3, ). 86. Na epalhjeôsete thn isìthta: x+ (x +)x+ ln tdt = ln x x x

186 Ask seic se ìlo to kef laio 87. Apì tic exet seic tou.. JewroÔme dôo sunart seic h, g suneqeðc sto [α, β]. Na apodeðxete ìti an h (x) >g(x) gia k je x [α, β], tìte kai β α h (x) dx > β α g (x) dx.. DÐnetai h paragwgðsimh sto R sun rthsh f, pou ikanopoieð tic sqèseic: f (x) e f(x) = x, x R kai f () = (a ) Na ekfrasteð h f wc sun rthsh thc f. (b ) Na deðxete ìti x <f(x) <xf (x) gia k je x>. (g ) An E eðnai to embadìn tou qwrðou Ω pou orðzetai apì th grafik par stash thc f, tic eujeðec x =, x =kai ton xona x x na deðxete ìti 4 <E< f () 88. Apo to diagwnismì tou ASEP, (MajhmatikoÐ). An f (x) =e x, g (x) = ln x kai I = f (x) dx + e g (x) dx tìte:. I =. I = e 3. I = e 4. I = e 89. Na breðte to ìrio: lim x x t t 4 + dt x 3 9. Na apodeðxete ìti an h suneq c sun rthsh f : R R eðnai gnhsðwc aôxousa tìte κ+λ f (t) dt κ λ f (t) dt + 9. H grafik par stash C f thc sun rthshc f (x) =e x ημx, x tèmnei ton xona xx sta peira to pl joc shmeða M k (kπ, ), k =,,,... f (t) dt Poio eðnai rage to embadìn tou qwrðou pou perikleðetai apì thn C f kai ton xx? 9. Na breðte tic asumpt touc thc g (x) = x t t dt (t )

187 Kef laio 4. Oloklhr tikìc Logismìc DÐnetai h sun rthsh f orismènh kai paragwgðsimh sto [, + ) kai tètoia ste x + gia k je x.. Na brejeð h f(x). x f(t)dt =(x +)f(x). Na brejeð h F (x) = x f(t)dt 3. Na brejoôn ta shmeða tom c twn grafik n parast sewn twn f,f. 94. Apì tic Exet seic tou 994, Dèsmh IV. 'Estw mða sun rthsh f orismènh sto R, h opoða èqei suneq deôterh par gwgo sto R, parousi zei topikì akrìtato sto shmeðo x =kai h grafik thc par stash dièrqetai apì to shmeðo A(, ). An isqôei na upologðsete to f(). [ xf (x)+3f (x) ] dx = Na lôsete thn exðswsh x (3t ) dt = 96. Na breðte to ìrio lim α + α ημ xdx 97. JewroÔme α < β kai dôo suneqeðc sunart seic f kai g me pedðo orismoô to R gia tètoiec ste gia ìla ta x isqôei Na apodeðxete ìti α 3 f(α x)+f(β + x) =g(α x)+g(β + x) β α β f(x)dx = α g(x)dx Upodeixh: Ja isqôei α β (f(α x)+f(β + x))dx = α β (g(α x)+g(β + x))dx 98. 'Estw h sun rthsh f : R R me { e x αν x f (x) = +lnx αν x >. Na apodeðxete ìti h f eðnai suneq c.

188 Ask seic se ìlo to kef laio. Na upologðsete to olokl rwma: f (x) dx 3. Na apodeðxete ìti gia k je α> h exðswsh f(x) =α èqei lôsh. 99. 'Estw sun rthsh f : R R h opoða eðnai dôo forèc paragwgðsimh kai h f eðnai suneq c. 'Estw π ( I = f (x)+f (x) ) ημxdx Na apodeðxete ìti I = f () + f (π). 'Estw { x f(x) =, x x, x >. Na melet sete wc proc thn sunèqeia thn f kai met na qar xete thn grafik thc par stash.. (a ) Na exet sete an h sun rthsh g : R R me g(x) =xf(x) eðnai fragmènh dhlad an up rqoun arijmoð α, β ètsi ste na isqôei α g (x) β gia ìla ta x. (b ) Na lôsete thn exðswsh x+ x f(t)dt = 6. Poièc apì tic parak tw prot seic alhjeôoun?. An f (x) g (x) gia k je x [α, β] kai oi f,g eðnai paragwgðsimec tìte isqôei f (x) g (x) gia k je x [α, β].. An f (x) g (x) gia k je x [α, β] kai oi f,g eðnai suneqeðc tìte isqôei β α f (x) dx β α g (x) dx. 3. An f (x) g (x) gia k je x [α, β] kai oi f,g eðnai suneqeðc tìte isqôei f (x) dx g (x) dx (Dhlad gia k je arqik F thc f kai gia k je arqik G thc g isqôei F (x) G (x) gia k je x [α, β]). 4. An f (x) g (x) gia k je x [α, β] kai oi f,g eðnai suneqeðc tìte up rqoun arqik F thc f kai arqik G thc g ste F (x) G (x) gia k je x [α, β]. 5. An f (x) g (x) gia k je x [α, β] up rqoun ta ìria twn f,g sto x tìte isqôei lim f (x) lim g (x). x x x x

189 Kef laio 4. Oloklhr tikìc Logismìc 87. Na apodeðxete ìti gia k je zeôgoc pragmatik n arijm n α, β isqôei: β ( ) e x e β + e x x e x α dx = α 3. Apì tic exet seic tou, Dèsmh IV. JewroÔme suneq sun rthsh f : R R pou ikanopoieð thn isìthta: x ( +t ) f (t) dt = x + 6x ( t + t ) dt. Na apodeðxete ìti f (x) = x+5 x + x R. Na breðte thn exðswsh thc efaptomènhc thc grafik c par stashc thc f sto shmeðo thca (,f()). 4. Gia thn suneq sun rthsh f :[α, β] R isqôei β α f (t) dt =. Na apodeðxete ìti up rqei x [α, β] tètoio ste x α f (t) dt =. 5. Apì tic exet seic tou 996, Dèsmh I. Na brejeð suneq c sun rthsh f gia thn opoða isqôei h sqèsh: gia k je x R. e x f(x)dx = f (x)+e x 6. 'Estw mða suneq c pragmatik sun rthsh f gia thn opoða isqôei f (x) > gia ìla ta x. Na apodeðxete ìti gia k je β (, ) isqôei: β β f (x) dx β f (x) dx 7. Gia mða sun rthsh f orismènh kai paragwgðsimh sto [, + ) isqôei. Na brejeð h f(x). x + x. Na brejeð h F (x) = x f (t) dt. f (t) dt =(x +)f (x) 8. Apo ton diagwnismì tou ASEP, (MajhmatikoÐ). Na apodeðxete ìti gia k je oloklhr simh sun rthsh f : R R, ìpou R to sônolo twn pragmatik n arijm n kai gia k je zeug ri pragmatik n arijm n α, β isqôei ìti: β [f (x) f (α + β x)] dx = gia sac: suneq α

190 Ask seic se ìlo to kef laio 9. Apo tic exet seic tou 993, Dèsmh I. DÐnetai h sun rthsh f (x) = +4me +x x> 4. Na exet sete th monotonða thc sun rthshc f.. Na upologðsete to x+ lim x + x f (t) dt. 'Estw f mða suneq c sun rthsh orismènh sto [α, β] kai g (x) = β α (f (t) x) dt. Na apodeðxete ìti h g gðnetai el qisth ìtan x = β α f (t) dt β α. Na apodeðxete ìti g (x) > gia k je x.. Gia tic suneqeðc sunart seic f,g,h isqôei f (x) dx = α, g (x) dx = α, h (x) dx = α 3 ìpou α + α + α 3 =. Na apodeðxete ìti up rqei α (, ) ste f (α)+g(α)+h(α) =α Upodeixh: Na jewr sete thn sun rthsh ϕ (x) = x f (t) dt + x f (t) dt + x f (t) dt x. Apo tic exet seic tou 995, Dèsmh I. H sun rthsh f : R R eðnai paragwgðsimh kai isqôei f (x) > gia k je x R. Na apodeðxete ìti h sun rthsh F (x) = β α f (x t) dt, x R eðnai paragwgðsimh kai ìti an up rqei x R me F (x )=tìte F (x) = gia k je x R. 3. Apo tic exet seic tou 3. 'Estw h sun rthsh f(x) =x 5 + x 3 + x.. Na melet sete thn f wc proc thn monotonða kai ta koðlakainaapodeðxete ìti h f èqei antðstrofh sun rthsh.. Na apodeðxete ìti f(e x ) f( + x) gia k je x R. 3. Na apodeðxete ìti h efaptomènh thc grafik c par stashc thc f sto shmeðo (, ) eðnai o xonac summetrðac twn grafik n parast sewn thc f kai thc f.

191 Kef laio 4. Oloklhr tikìc Logismìc Na upologðsete to embadìn tou qwrðou pou perikleðetai apì th grafik par stash thc f,ton xona twn x kai thn eujeða me exðswsh x =3. 4. Na apodeðxete ìti an gia k je x eðnai f (x) > kai F (x) = x f (t) dt tìte gia k je x (, + ) isqôei x F (x) <F (x). 5. Gia thn sun rthsh f : R R eðnai gnwstì ìti f (x) +3f ( x) = e x e x e x +e x.. Na breðte ton tôpo thc f.. Na upologðsete to olokl rwma f (x) dx. 6. Na apodeðxete ìti an h sun rthsh f me pedðo orismoô to R eðnai paragwgðsimh tìte kai h sun rthsh g (x) = f (tx) dt eðnai paragwgðsimh. 7. Na apodeðxete ìti an h sun rthsh f eðnai suneq c sto [, ] tìte isqôei f ( x) dx = f (x) dx. 8. Na breðte to ìrio lim x x e t dt x Mhn epiqeir sete na upologðsete to olokl rwma tou arijmht. 9. Na apodeðxete ìti xα ( x) β dx = xβ ( x) α dx.. Apì tic exet seic tou 998, Dèsmh I. DÐnetai h paragwgðsimh sun rthsh f :(, + ) R gia thn opoða isqôoun f (x) >, x > f (x)+xf (x) =, x > kai h grafik thc par stash dièrqetai apì to shmeðo A (, ).. Na deðxete ìti h par gwgoc thc f eðnai suneq c sto anoiktì di sthma (, + ) kai na breðte th sun rthsh f.. Na deðxete ìti 3. Na breðte th sun rthsh x x x f (x) < f (t) t dt < x, x > F (x) = x ( + ) t f (t) dt, x >

192 Ask seic se ìlo to kef laio 4. Na apodeðxete ìti e x gia k je x megalôtero tou èna. e t dt <. 'Estw f :[, + ) R paragwgðsimh kai tètoia ste f (x) >xgia k je x> f () = Na apodeiqjeð ìti gia k je α> isqôei α xf 3 (x) dx ( α xf (x) dx). Na apodeðxete ìti den up rqei sun rthsh f me suneq par gwgo sto di sthma [, 4], gia thn opoða na isqôoun f (x) 3, f () = kai f (4) = 'Estw h suneq c sun rthsh f :[,α] R gia thn opoða isqôei f (α x) =f (α) f (x) gia ìla ta x. Na apodeðxete ìti α f (x) dx = αf(α).. Na upologðsete to olokl rwma π 4 ln ( + εϕx) dx. Pafnuty Lvovich Chebyshev H anisìthta tou Chebyshev. Na apodeiqjeð ìti an oi sunart seic f,g eðnai suneqeðc kai monìtonec me to Ðdio eðdoc monotonðac kai α<β tìte isqôei: ( β )( β ) β f (x) dx g (x) dx (β α) f (x) g (x) dx α α α 5. Apì tic exet seic tou 4. DÐnetai h sun rthsh g (x) = e x f (x) ìpou h sun rthsh f eðnai paragwgðsimh sto R kai f () = f ( 3 ) =.. Na apodeðxete ìti up rqei èna toul qiston ξ (, 3 ) tètoio ste f (ξ) = f (ξ).. E n f (x) =x 3x na upologðsete to olokl rwma I (α) = α g (x) dx, α R 3. Na breðte to ìrio lim α I (α). 6. 'Estw g mða suneq c sun rthsh orismènh sto (, + ). Na apodeiqjeð ìti gia k je ξ> isqôei ξ f (x)+x ξ +x dx + f ( x) x +x dx =lnξ Upodeixh: Sto deôtero olokl rwma tou a' mèlouc na jèsete u = x.

193 Kef laio 4. Oloklhr tikìc Logismìc 9 7. 'Estw f :[α, β] R mða suneq c kai m stajer sun rthsh. GnwrÐzoume ìti to sônolo tim n thc f eðnai k poio kleistì di sthma èstw [f (x ε ),f(x μ )].. Na apodeðxete ìti 3 f (x ε )(β α) < β. Apì thn prohgoômenh sqèsh sun goume ìti α f (x) dx < f (x μ )(β α) f (x ε ) < β α f (x) dx β α <f(x μ ) B sei aut c deðxte ìti 4 up rqei ξ (α, β) ste β α f (x) dx = f (ξ)(β α) ApodeÐxte to Ðdio apotèlesma efarmìzontac to je rhma mèshc tim c gia thn paragwgðsimh F (x) = x α f (t) dt. 3. An èqete dôo suneqeðc sunart seic skefjeðte ti mporeðte na epitôqete efarmìzontac to je rhma mèshc tim c tou Cauchy Apì tic exet seic tou 4. 'Estw h suneqhc sun rthsh f : R R tètoia ste f () =. An gia k je x R, isqôei g (x) = x 3 ìpou z = α + βi, meα, β R,tìte: z f (t) dt 3 z + z (x ),. Na apodeðxete ìtihsun rthshg eðnai paragwgðsimh sto R kai na breðte th g.. Na apodèixete ìti z = z +. z 3. Me dedomènh th sqèsh tou erwt matoc b na apodeðxete ìti Re ( z ) =. 4. An epiplèon f () = α>, f (3) = β kai α>β na apodeðxete ìti up rqei x (, 3) tètoio ste f (x )=. 3 DeÐte prosektik thn skhsh Gii. sel. 353 kai entopðste omoiìthtec - diaforèc pèra fusik apì to sumbolismì f (x ε) antð m kai f (x μ) antð M. 4 Prìkeitai gia eidik perðptwsh tou jewr matoc mèshc tim c tou oloklhrwtikoô logismoô 5 DeÐte thn sullog Par gwgoi} enìthta 3.

194 Ask seic se ìlo to kef laio 9. Apì tic exet seic tou 5. DÐnetai h sun rthsh f me tôpo f (x) =e λx, λ >. DeÐxte ìti h f eðnai gnhsðwc aôxousa.. DeÐxte ìti h exðswsh thc efaptomènhc thc grafik c par stashc thc f, h opoða dièrqetai apì thn arq twn axìnwn eðnai h y = λex BreÐte tic suntetagmènec tou shmeðou epaf c M. 3. DeÐxte ìti to embadìn E (λ) tou qwrðou, to opoðo perikeleðetai metaxô thc grafik c par stashc thc f, thc efaptomènhc thc sto shmeðo M kai tou xona y y,eðnai E (λ) = e λ 4. UpologÐste to λ E (λ) lim λ + +ημλ 3. Apì tic epanalhptikèc exet seic tou 5. DÐnetai h suneq c sun rthsh f : R R gia thn opoða isqôei. Na deðxete ìti (a ) f() = (b ) f () =. Na breðte to λ R ètsi ste f (x) x lim x x = 5 x + λ (f (x)) lim x x +(f (x)) =3 3. An epiplèon h f eðnai paragwgðsimh me suneq par gwgo sto R kai f (x) >f(x) gia k je x R, na deðxete ìti: (a ) xf (x) > gia k je x (b ) f (x) dx < f () 3.. Na lôsete thn exðswsh x (x + t) dx =. Na apodeðxete ìti an gia th suneq sun rthsh f : R R isqôei f (x) g (x) dx = gia k je suneq sun rthsh g : R R tìte f (x) =gia k je x.

195 Kef laio 4. Oloklhr tikìc Logismìc 'Estw ìti to polu numo f (x) =αx + βx + γ èqei thn idiìthta f (x) g (x) dx = ìtan g eðnai opoiad pote apì tic sunart seic, x,x. DeÐxte ìti f (x) = gia k je x. 3. 'Estw Na apodeðxete ìti: [ x f (x) = g (x) = ] e t dt e x (+t ) +t dt. f (x)+g (x) =, gia k je x. f (x)+g (x) = π 4 x 3. lim x + e t dt = π 33. Apì tic exet seic tou 5. 'Estw mða sun rthsh f paragwgðsimh sto R tètoia, ste na isqôei h sqèsh gia k je x R kai f () =. f (x) =e x f(x). Na deiqjeð ìti:. Na brejeð to: ( ) +e x f (x) =ln lim x x f (x t) dt ημx 3. DÐnontai oi sunart seic h (x) = x x t5 f (t) dt kai g (x) = x7 DeÐxte ìti h (x) =g (x) gia k je x R 4. DeÐxte ìti h exðswsh x x èqei akrib c mða lôsh sto (, ). t 5 f (t) dt = 8 7.

196 Ask seic se ìlo to kef laio 34. Apì tic exet seic tou 998, Dèsmh IV. DÐnetai h sun rthsh h (x) = ( e 4x e αx), x ìpou α pragmatikìc arijmìc megalôteroc tou 4.. Na deðxete ìti lim h (x) =h () =. x +. Na melet sete wc proc ta akrìtata th sun rthsh h(x). 3. An x eðnai rðza thc pr thc parag gou kai x eðnai rðza thc deutèrac parag gou thc h(x) na breðte th sqèsh pou sundèei ta x, x. 4. Na upologðsete to olokl rwma M = ln h (x) dx ìtan α = 'Estw f :[α, β] R dôo forèc paragwgðsimh, me f suneq, tètoia ste f(x) kai to <<=>> isqôei mìno gia x = α kai x = β. 'Estw M mègisth tim thc f.. Na apodeðxete ìti β α f (x) dx.. 'Estw x èna opoiod pote shmeðo tou (α, β) pou mhdenðzetai h f (tètoia shmeða up rqoun apì to je rhma tou Rolle). Na apodeðxete ìti up rqoun ξ (α, x ), ξ (x,β) ètsi ste f (ξ ) f (ξ ) 4f (x ) β α 3. Na apodeðxete ìti up rqoun p, q me α<p<q<β tètoia ste 4. Na apodeðxete ìti β α f (p) f (q) 4M β α f (x) M + f (x) dx β α 36. Apì tic exet seic tou GallikoÔ Baccalauréat, 986. JewroÔme ta oloklhr mata: I = π 4 π dx συν x, J = 4 dx συν 4 x. (a ) Poia eðnai h par gwgoc thc sun rthshc efaptomènh? (b ) Na upologðsete to I.

197 Kef laio 4. Oloklhr tikìc Logismìc 95. (a ) 'Estw h sun rthsh f : [, π ] 4 R me f (x) = ημx συν 3. DeÐxte ìti h x f eðnai paragwgðsimh sto [, π ] 4 kað ìti gia ìla ta x apì autì to di sthma isqôei f (x) = 3 συν 4 x συν x (b ) Me th bo jeia twn prohgoumènwn upologism n na breðte mða sqèsh pou sundèei ta I, J kai sth sunèqeia na upologðsete to J. 37. Apì tic exet seic tou GallikoÔ Baccalauréat, 6.. 'Estw h sun rthsh f (x) =x e x,orismènh sto R. 'Estw C h grafik thc par stash se èna orjokanonikì sôsthma axìnwn ìpou h mon da mètrhshc eðnai cm. (a ) Na breðte ta ìria thc f sto kai sto +. P c ermhneôontai grafik ta sumper smata sac? (b ) Na aitiolog sete giatð h f eðnai paragwgðsimh. BreÐte thn par gwgo thc f. (g ) Na k nete èna pðnaka metabol n thc f kaj c kai th grafik thc par stash.. 'Estw ν ènac jetikìc akèraioc. 'Estw I ν = xν e x dx. (a ) BreÐte th sqèsh pou sundèei ta I ν+,i ν. (b ) Na upologðsete to I kai met to I. (g ) Na ermhneôsete grafik ton arijmì I se sqèsh me th grafik par stash tou erwt matoc ) g 3. (a ) Na apodeðxete ìti gia k je x [, ] kai k je jetikì akèraio ν isqôei x ν x ν e x ex ν. (b ) Me th bo jeia tou krithrðou thc parembol c na breðte to ìrio tou I ν ìtan to ν teðnei sto Apì to diagwnismì Putnam, 5. Na upologðsete to Upodeixh: x = u +u ln (x +) x + dx 39. Estw f : R R suneq c me f () = 7 gia thn opoða isqôei gia ìla ta x, y f (x + y) =f (x)+f (y) 6 H er thsh aut zht ei ìrio akoloujðac pou de ja èqete duskolða na to apant sete

198 Ask seic se ìlo to kef laio. Na apodeðxete ìti f () =.. Na apodeðxete ìti gia k je m R isqôei: x+m x f (t) dt = mf (x)+ 3. Na apodeðxete ìti h f eðnai paragwgðsimh. 4. Na apodeðxete ìti isqôei f (x) =f (). 5. Na breðte thn f. m f (t) dt 4. Apì tic exet seic tou 997, Dèsmh IV. 'Estw f pragmatik sun rthsh orismènh sto R pou eðnai duo forèc paragwgðsimh kai isqôei f (x) > gia k je x R. 'Estw α, β R me α<β. Na apodeiqjeð ìti :. f (x) f (α) f (x)(x α) gia k je x [α, β].. β α f (x) dx f (β)(β α) +f (α)(β α) 4. An h f eðnai ìpwc sthn skhsh 4 deðxte ìti β α f (x) dx < f (α)+f (β) (β α) Na d sete mða gewmetrik ermhneða ìtan h f paðrnei mh arnhtikèc timèc. 4. Na brejeð to pedðo orismoô thc sun rthshc (x )(x 4) ln (xt) dt (x )(x 3) 43. Apì tic exet seic tou PanepisthmÐou tou Berkeley, 977. Na apodeiqjeð ìti h sun rthsh F (k) = dx, k< kσυν x eðnai gnhsðwc aôxousa. 44. 'Estw f (y) =e x y, y>.. (a ) DeÐxte ìti f (y). (b ) DeÐxte ìti gia k je y,y isqôei f (y) f (y ) y y.. JewroÔme α<β. Na apodeðxete ìti h sun rthsh eðnai suneq c. F (y) = β α e x y dx

199 Kef laio 4. Oloklhr tikìc Logismìc JewroÔme thn oikogèneia twn sunart sewn f k (x) = k 49k x + x, k [, ] k. Na breðte th mègisth tim thc f k (x).. Poia apì tic f k (x) sqhmatðzei me ton xona x x to megalôtero embadìn? 46. Apì tic exet seic tou PanepisthmÐou tou Berkeley, 98. OrÐzoume BreÐte thn F (). F (x) = συνx ημx e t +xt dt 47. Apì tic exet seic tou 7. 'Estw f mia suneq c kai gnhsðwc aôxousa sun rthsh sto di sthma [, ] gia thn opoða isqôei f() >. DÐnetai epðshc sun rthsh g suneq c sto di sthma [, ] gia thn opoða isqôei g(x) > gia k je x [, ]. OrÐzoume tic sunart seic: F (x) = x G (x) = x f (t) g (t) dt, x [, ] g (t) dt, x [, ]. Na deiqjeð ìti F (x) > gia k je x sto di sthma (, ].. Na apodeiqjeð ìti: gia k je x sto di sthma (, ]. f(x) G(x) >F(x)

200 Ask seic se ìlo to kef laio 3. Na apodeiqjeð ìti isqôei: gia k je x sto di sthma (, ]. 4. Na brejeð to ìrio: lim x + F (x) G (x) F () G () ( ( x ) f (t) g (t) dt) x ημ tdt ( x g (t) dt) x Apì tic exet seic tou 8. 'Estw f mia sun rthsh suneq c sto R gia thn opoða isqôei f (x) = ( x 3 +3x ) f (t) dt 45. Na apodeðxete ìti f (x) =x 3 +6x 45. DÐnetai epðshc mia sun rthsh g dôo forèc paragwgðsimh sto R. Na apodeðxete ìti g g (x) g (x h) (x) = lim h h 3. An gia th sun rthsh f tou erwt matoc () kai th sun rthsh g tou erwt matoc () isqôei ìti g (x + h) g (x)+g(x h) lim h h = f (x)+45 kai g() = g () =, tìte (a ) na apodeðxete ìti g(x) =x 5 + x 3 + x + (b ) na apodeðxete ìti h sun rthsh g eðnai Apì ton Et sio Diagwnismì ApeirostikoÔ LogismoÔ, x HPA, 999. BreÐte to ìrio lim x + 4+t dt x Apì ton Et sio Diagwnismì ApeirostikoÔ LogismoÔ, HPA,. Upojètoume ìti oi f,f kai f orðzontai sto [, ln ], ìti h f eðnai suneq c, ìti f () =, f () = 3, f (ln ) = 6, f (ln ) = 4 kai ln e x f (x) dx =3. Na upologðsete to ln e x f (x) dx.

201 Kef laio 4. Oloklhr tikìc Logismìc Gia thn suneq sun rthsh f : R R gia k je x isqôei: x+α x f (t) dt = b ìpou α, β eðnai pragmatikoð arijmoð di foroi tou mhdenìc. Na apodeiqjeð ìti h f eðnai periodik. 5. Apì tic exet seic tou DÐnetai h sun rthsh f(x) =x + ln(x +), x R.. Na melet sete wc proc th monotonða th sun rthsh f.. Na lôsete thn exðswsh: ( x 3x + ) =ln [ ] (3x ) + x Na apodeðxete ìti h f èqei dôo shmeða kamp c kai ìti oi efaptìmenec thc grafik c par stashc thc f sta shmeða kamp c thc tèmnontai se shmeðo tou xona y y. 4. Na upologðsete to olokl rwma xf(x)dx 53. Apì tic exet seic tou. DÐnetai h suneq c sun rthsh f : R R h opoða gia k je x R ikanopoieð tic sqèseic: f(x) x f(x) x =3+ x t f(t) t dt. Na apodeðxete ìti h f eðnai paragwgðsimh sto R me par gwgo f (x) =. Na apodeðxete ìti h sun rthsh, x R, eðnai stajer. 3. Na apodeðxete ìti f(x) f(x) x,x R g(x) =(f(x)) xf(x) f(x) =x + x +9,x R

202 4.7. Ask seic se ìlo to kef laio 4. Na apodeðxete ìti x+ x f(t)dt < x+ x+ f(t)dt, x R 54. Upojètoume ìti h sun rthsh f eðnai orismènh sto R kai dôo forèc paragwgðsimh. Na brejeð h deôterh par gwgoc thc sun rthshc x+ ( y ) F (x) = f(z)dz dy x 55. Apo tic exet seic tou 9. 'Estw f mða suneq c sun rthsh sto di sthma [, ] gia thn opoða isqôei: OrÐzoume tic sunart seic: G(x) = H(x) = { H(x) x x (t ) f(t)dt = x tf(t)dt, x [, ] x f(t)dt +3, x (, ] 6 lim t, x = t o t. Na apodeðxete ìti h sun rthsh G eðnai suneq c sto di sthma [, ].. Na apodeðxete ìti h sun rthsh G eðnai paragwgðsimh sto di sthma (, ) kai ìti isqôei: G (x) = H(x) x, <x< 3. Na apodeðxete ìti up rqei toul qiston ènac arijmìc α (, ) tètoioc ste na isqôei H(α) =. 4. Na apodeðxete ìti up rqei toul qiston ènac arijmìc ξ (,α) tètoioc ste na isqôei ξ α α tf(t)dt = ξ f(t)dt 56. 'Estw f : R R mða paragwgðsimh sun rthsh me: f() = f (x) > gia k je x Na apodeðxete ìti f (x) > x f(t)dt gia k je x>.

203 Kef laio 4. Oloklhr tikìc Logismìc 57. 'Estw f : R R suneq c sto R kai I : R R me ( I(x) = f (t) xt f(t)+x t 4) dt Na deðxete ìti h I parousi zei el qisto sto shmeðo x =5 t f(t) dt. 58. Apì tic exet seic tou. DÐnontai oi suneqeðc sunart seic f,g : R R, oi opoðec gia k je x R ikanopoioôn tic sqèseic: i) f(x) > kai g(x) >. ii) f(x) x e = t e x g(x+t) dt iii) g(x) x e = t e x f(x+t) dt. Na apodeðxete ìti oi sunart seic f kai g eðnai paragwgðsimec sto R kai ìti f(x) =g(x) gia k je x R.. Na apodeðxete ìti: f(x) =e x, x R. 3. Na upologðsete to ìrio: ln f(x) lim x f ( ) x 4. Na upologðsete to embadìn tou qwrðou pou perikleðetai apì th grafik par stash thc sun rthshc F (x) = x f ( t ) dt me touc xonec x x kai y y kai thn eujeða me exðswsh x =. 59. Apì tic exet seic tou. DÐnetai h sun rthsh f(x) = (x ) ln x, x>.. Na apodeðxete ìti h sun rthsh f eðnai gnhsðwc fjðnousa sto di sthma Δ =(, ] kai gnhsðwc aôxousa sto di sthma Δ =[, + ). Sth sunèqeia na breðte to sônolo tim n thc f.. Na apodeðxete ìti h exðswsh x x = e 3, x> èqei akrib c dôo jetikèc rðzec. 3. An x, x me x <x eðnai oi rðzec thc exðswshc tou erwt matoc, na apodeðxete ìti up rqei x (x,x ) tètoio, ste f (x )+f(x ) =

204 4.7. Ask seic se ìlo to kef laio 4. Na breðte to embadìn tou qwrðou pou perikleðetai apì th grafik par stash thc sun rthshc g(x) =f(x)+ me x>, ton xona x x kai thn eujeða x = e. 6. 'Estw f :[a, b] R suneq c tètoia ste up rqei c : [a, b] ste c a f(t)dt b c f(t)dt <. Na apodeiqjeð ìti up rqei di sthma [p, q] [a, b] me q p f(t)dt =. 6. Apì tic exet seic tou. 'Estw h suneq c sun rthsh f : (, + ) R, h opoða gia k je x> ikanopoieð tic sqèseic: f (x) x x+ f(t)dt x x e ln x x = ( x ln t t f(t) dt + e ) f(t). Na apodeðxete ìti h f eðnai paragwgðsimh kai na breðte ton tôpo thc. An eðnai f (x) =e x (ln x x), x>, tìte:. Na upologðsete to ìrio: [ lim (f(x)) ημ ] x + f(x) f(x) 3. Me th bo jeia thc anisìthtac lnx x, pou isqôei gia k je x>, na apodeðxete ìti h sun rthsh F (x) = x α f(t)dt x > ìpou α >, eðnai kurt. Sth sunèqeia na apodeðxete ìti: F (x) + F (3x) > F (x), giak jex>. 4. DÐnetai o stajerìc pragmatikìc arijmìc β >. Na apodeðxete ìti up rqei monadikì ξ (β,β) tètoio ste: F (β)+f (3β) =F (ξ)

205 Apant seic a)b)g). 8+4i. + 3i 3. 3 i i i 6. +3i i 8. i i i i 4. + i i i 4. 5 i. 8 i i i i i 9 x =, y = x =, y = 3

206 7 kai y o +i 4 4 i 3 68 i λ = z =5+3i, z = i 8 (a') (x + iy)(x iy) (b') (x + iy)(x iy) 9 Mìno an oi z,z eðnai pragmatikoð. z =+( t)i, w =5+ti, t R x = 8, y =8 κ = 4 5,λ = 5 3 z = i 4. ± 3i. 3. ±3i ±i συνθ ± iημθ 5 a =, b =3 a = 3, b = 3 9 Im(z)+Re(z) 3 z = + i 3, z 3 =, z 6 =. 3 x = ±3 3 K je arijmìc pou den uperbaðnei to 5 35 x = 3+i, x = 3 i 36 Nai-'Oqi 4

207 38 w = z + z 39 A = αν ν =4k αν ν =4k + +i αν ν =4k + i αν ν =4k +3 4 z = p q p +q 4 Up rqoun peirec timèc tou z: z =( t)+ti, t R 43 κ =kai λ =3 κ = kai λ = κ = 3 7,λ = 7 47 x 3y = 48 Prosèxte. H skhsh den lèei ìti ta α, β eðnai pragmatikoð. 49 x =64+64i 9 5 i 5 7, 9 5 w = ± i, w = ±i 5 Prèpei o z na eðnai pragmatikìc Re(z) =Re(u). 53 Re ( z ) =3, Im ( z ) = 4 54 x, = 4 ± 4 i 3 55 z = ± ( + 3i) 57 ( ν ν ) + ν i 'Oqi. MporeÐ na eðnai dôo pragmatikoð arijmoð. 6 HexÐswsh: a) EÐnai adônath an α ( 4β γ αδ ) <. b) 'Eqei mða lôsh thn z = δ β i an α ( 4β γ αδ ) α(4β =kai g) 'Eqei dôo lôseic z, = ± γ αδ ) + δ α β β i an α ( 4β γ αδ ) >. 63 A ν =, ν =6κ, ν =6κ +, ν =6κ +, ν =6κ +3, ν =6κ +4, ν =6κ +5 5

208 64 Ektìc apì ton +i lôseic eðnai kai oi i ( 3 ) 3+ i, + ( 3+ ) 3 i a) x>,y =b) x< kai y =3x 75 'Otan n =4m a),b)5,g) 79, 4, a) ±,b), z = z = z 3 = 85 kai i Mìno gia touc pragmatikoôc. 9 z = ±, z = 3 i, z = 3 i 93 z =6+7i z =6+8i 94 z =, z =, z = ± i κ + λ z = 7 +7i 3 6

209 4 4 z = ± i 3 6 rz = r z = apìluth tim tou z z = r z 9 3, (γ α ) +(γ+β) α (γ β) 4 kai- 3 z = +i 5 Gia kanèna. 7 Gia ton z =+i 3-45 (b') ii. 48 α = 3 5 Katal goume sta qwrða:.. 7

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

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 UpenjumÐseic gia thn Jetik kai Teqnologik KateÔjunsh Kajhght c: N.S. Maurogi nnhc 1 Tautìthtec - Anisìthtec 1. (α ± ) = α ± α +. (α ± ) 3 = α 3 ± 3α +3α ± 3 3. α 3 ± 3 =(α ± ) ( α α + ) 4. (α + + γ) =

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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

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