ΗΛΙΑΣΚΟΣ ΦΡΟΝΤΙΣΤΗΡΙΑ. Θετικής - Τεχνολογικής Κατεύθυνσης Μαθηματικά Γ Λυκείου Συναρτήσεις ΥΠΗΡΕΣΙΕΣ ΠΑΙΔΕΙΑΣ ΥΨΗΛΟΥ ΕΠΙΠΕΔΟΥ ΣΤΕΦΑΝΟΣ ΗΛΙΑΣΚΟΣ

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1 ΗΛΙΑΣΚΟΣ ΦΡΟΝΤΙΣΤΗΡΙΑ ΥΠΗΡΕΣΙΕΣ ΠΑΙΔΕΙΑΣ ΥΨΗΛΟΥ ΕΠΙΠΕΔΟΥ Θετικής - Τεχνολογικής Κατεύθυνσης Μαθηματικά Γ Λυκείου Συναρτήσεις ΣΤΕΦΑΝΟΣ ΗΛΙΑΣΚΟΣ info@iliaskos.gr

2 ..1! A y! B! A y! B fa : " B D fa f! A f] Ag f] Ag = " y y = f] g,! A, f] Ag B y = f] g- - f] g f^ 0 h0! A- + y = 1 - f ' ^h = 1 - ' g^h =- 1- R R {] g- {] g H 0 49

3 {] g- - {] g > 0 {] g- {] g! lπ + π, l! Z {] g- {] g! lπ, l! Z f] g = y f] Ag. - f^- h= f ^ h! A c yy ' f^- h=-f ^ h! A c - f ^ + nth= f ^ h n = 0,! 1,!, f! A 50

4 .! Af] g= g] g - C = A+ B C = A+ B! Y 0 f+ g ^ f+ gh] g= f] g+ g] g f- g ^ f- gh] g= f] g-g] g f$ g ^ f$ gh] g= f] g$ g] g g f g f f] g b l] g = g] g! C g] g = 0 f ' ] g = 0 yy ' = 0- f] 0g ' f] g > 0 ' f] g < 0 - f] g- g] g - 51

5 -.4 f% g ^ f% gh] g = f^g] gh Af% g= "! Ag g] g! Af, g% f^ g% fh] g = g^f] gh Ag% f= "! Af f] g! Ag, - h^h = h ^ hf^h = 1 - g^h = h^h= gf ^ ^ hh f% g- g] g g% f- f] g f^g ^ hhg^h f^h 5

6 y = g^h f^yh- f^h f^g ^ hhf^h g^h- f ^ h g ^ h fg ^ ^ hh g ^ h g ^ h - lf] g + mf^g] gh = n! A- g] g f] gf^g] gh f] glm g^h- gg ^ ^ hh = - a g^h = a -g^h = g^h = - 1 l = m lf ^ g] gh + mf^h] gh = n y = g] g - lf^yh+ m f^{ ^yhh = n {^h 5

7 .5 f^h y = f^h- y = f^h y = f^h + ll > 0 l y = f^h y = f^h y = f^h + ll < 0 l y = f^h y = f^+ hhh < 0 h y = f^h- y = f^+ hhh > 0 h.6 f] g 1,! D 1 < f] 1g< f] g 1,! D 1 < f] 1g> f] g - 1,! D1 < f] 1gG f] g. 1,! D1 < f] 1gH f] g K K 54

8 0! Af] 0 gf] g G f] 0 g! A 0! Af] 0 gf] g H f] 0 g! A f ^ h- f ^ 1h m = m > 0 m < 0 - f+ g- f$ g f$ g f$ g f$ g f$ g f$ g f% g f% g 55

9 .7 f] g 1,! A1! f] 1g! f] g f] g 1,! Af] 1g= f] g1 = y! f] Agf] g = y0! A y = y 0 - g% f g% f fa : " R- 1,! Af] 1g= f] g- 1 = - 1,! A 1! f] 1g= f] g.8 y! f] Ag! A- f - f] g = y + f 1 ^yh = - 1 f ^f] gh =! Af^f ^yhh = yy! A f y = f - 56

10 - y = M^ab, h N^ba, hf - f] ag = b + f 1 ^bh = a f - f] g = y- - 1 f ] g f - 1 ] gf ] g f - f] g= f ] g - 1 f] g = f ] g = f - 1 ] g! f ] A g - 1 f ] g.9 0! A " 1-1" f - f - 57

11 .10 R R fa : " R ' A0 ^, h A1 ^, h f ^ h = 1 y = = f^h! 0! R ' f+ g f^h< f0 ^ h < 0 < 0 Rf^1h = 1f^009h = 009 R f :y = ^, 1h- fa : " R0! A g] g= f ] g,! A 0- R % R f ^ h = c f^r h = " c, R " 1-1" f - f y = 58

12 R f% g R R f: R " R f f f: R " RR f R - f: R " R R +.11 f ^ h = f ^ h = f ^ h = f ^ h = vyo f ^ h = f^h = f ^ h = - + f ^ h = hn - f ^ h = f ^ h = f ^ h = f^h = f ^ h = - f ^ h = + 59

13 f ^ h = - + f ^ h = f ^ h = 1 - f^h = f ^ h = f ^ h = e - 1 e - f ^ h = f ^ h = f ^ h = f^h = f ^ h= ln^4 -h f ^ h = ln^ h f^h = ln f ^ h = ln - f ^ h = ln + ln f^h= f{ ^-πh f ^ h = ln^ h f ^ h = ln ln + 5 f ^ h = ^ln -h^ln h r f ^ h = v{ ` + j f ^ h = f{ - +,! 6, 5h f ^ h = * + 7,! 656, h + 1,! ^100, h 60

14 f ^ h = ln^4- h f ^ h = f ^ h = ^ - 9h f ^ h = f ^ h = f^h = f ^ h = f ^ h = ln f ^ h = f ^ h = f ^ h = ^ -h - 1- f ^ h = ln + m! R f ^ h = ln^m + m+ m hr f ^ h = + 1 f ^ h = e - f ^ h = - f ^ h = ln f ^ h = - + f^h = f ^ h = f ^ h = f ^ h = + e - e 61

15 f ^ h = + vyo f ^ h = 6hn + vyo+ 1 f 1 e e + ^ h = - f ^ h = f ^ h = f ^ h = ln`1 - e 1 j -, G f ^ h = ( ln^- 1h, > - 5, < 1 f ^ h = ( ln -, H -, G f ^ h = ' - e, > l! R M^ll-, 6h f^h = -4 f ^ h = g ^ h = ln h ^ h = ' - ' f ^ h = - 1 g^h =- f ^ h = e g^h = 5e -6 g ^ h= f ^h- f ^ h + +! R g^h fa : " R f ^h-f ^ h- 1 = ^ h! A f ^ h ' 6

16 f ^ h= g ^ h+ ln - 1! R f^hg^h f ^h+ g ^hg ^f$ gh^h! R f^hg^h f ^h- f ^h+ f ^ h = - +! R- f^h ' - vyo f ^ h = 4 - f ^ h = hn^ - 5hn+ h f ^ h = f ^ h = ln^+ + 1h e e f = e ^ h - e ^f+ gh^h ^f$ gh^h f ^ h^g% fh^hr- g^h f% gg% f f% gg% f f: R " R f ^ + yh= f^h+ f^yh y,! R f0 ^ h = 0 f: R " R f ^ + yh+ f^- yh= ffy ^ h ^ h y,! R f0 ^ h = 1 f ^ h = + - 4, H 0 f ^ h = ' , < 0 Z ]-, < f ^ h = [ +, G G ] \ -, > 6

17 f ^ h = ln^e h f 1 - ^ h = f ^ h = f ^ h = + - f ^ h= ln^+ 1h f^h =-e f ^ h = Rf^h= g ^ h f ^ h = g^h= ^h f ^ h = -g^h = ^ -h f ^ h= ^ hg^h = - 4 f ^ h = g^h = + - f ^ h = + 1-1g^h = f^h = g^h = f ^ h = - g^h = - m! R m f + m ^m- 1h + m ^ h = g^h = + 1 -m + m fgr, : " R - ^f + g h^hg ^f+ gh^h -! R f ^ h= g ^ h fgr, : " R ^f+ gh ^h-^f-gh ^h- 4 H ^f+ gh^h6 ^f+ Rf^h= g ^ h f ^ h = - g ^ h = - 9 f f+ gf- gf$ g g +, H f^h = - g ^ h = ( - 1, < f+ gf- g 64

18 f^h f ^ h = f ^ h = ^ + - 1h f ^ h= hn 4 ^+ 5h f ^ h = ln^vyo^ + 1h+ h f ^ h = vyo^ln^1 + hh 1 f ^ h = g ^ h = - f% gg% f% % f^h = + g^h = - f% gg% f% % f ^ h = + g ^ h = -4 - f% gg% f% % f^h = g ^ h = ln - f% gg% f% % f ^ h = - g ^ h f e g ^ h ^ h = + 4! R g^h f^h = 1 - g^h f^g ^ hh = vyo! R g^h f ^ h = + g^h fg ^ ^ hh = - hn+! R g^h g ^ h = + f ^ h f^ln^g^hhh = -! R- f^h g^h = + f ^ h f^g ^ hh = -! R f^h g^h =- + 1 f ^ h fg ^ ^ hh = 5+! R- f^h f ^ h ff ^ ^ hh =! Rf^0h f ^ h ff ^ ^ hh = 5 5 5! Rf^ h = f ^h 65

19 f^ hf^1-h- f ^ h= + 1! Rf^h f^ hf^- h+ f ^ h= 5hn! Rf^h - f^ hf^-h- 4f ^ h= e -e! Rf^h f^ hf^h+ f^- h=! Rf^h f ^ h f ^ h-f^- h= -! R -",,- f^h - 1 f^hf^h - f` j = + 1! R * -",f^h f ^ h f^1- h+ ^1- f h ^ h = 0! R f^h f: R " R f^h+ f^1 - h=! R f: R " Rf^f ^ hh = -! R f^- h= f ^ h- f^h y = 1 f: R " R! R f^h-fy ^ hg -y f ^ hg + f^0h! R f ^ h= + f^0h! R f ^ h = f ^ h = ` j - 1 f ^ h = + f ^ h = , G f ^ h = ' - + 7, > - f- g 66

20 fgr, : " R f% g g^h f^h = -vyo- 8 π 0, B π 0 < a < b < vyoa -vyob > a -b 7 5 f^h = ^- 4h + ^- 4h > 1 f: ^0, + h " Rf ^ h = ln f^1h = 0 ^0, + h C f ' 1 1 ln - > ln - > 0 f: R " R- A4 ^, h B51 ^, h f^+ f^ hh < 1 f: R " Ry = g ^ h= 4 + ^f ^ h- h! R f ^ h -f ^ h f^h R f ^ h R f^h > 1f ^h-f ^ h- = 0! R- f^f ^ hh > R g ^ h= f^-h-f^1-h! R R fa : " R f ^ h-fy ^ h < -y y,! A! yg^h= f ^ h- K 67

21 f ^ h = ln g^h = f% g = g% f g% f g% f a! R ^g% fh^h= a g% f f ^ h = f ^ h = - f ^ h = - > 1 f 1 ^ h = - - f ^ h = + 5 f^h = f ^ h = f ^ h = f ^ h = - - f ^ h = f e e ^ h = f ^ h = ln f ^ h = ln 1 - f e - ^ h = 1 + 1, < f ^ h = ' - 5, H - 1 f ^ h = g ^ h = - - ^f% gh 1 g % f f ^ h = g ^ h = + - ^f % gh g % f f^h = e + + f^h" 1-1" 9 e + = e + 5 f^h = + - 1

22 f ^h= f ^ h π f ^ h= vyo^h-! `0, j f ^h + = 0 f ^ h = + +! R - f ^h = π f ^ h = hn+! 80, B - f ^h - = 0 f^hg^h" 1-1" ^f% gh^h^g% fh^h" 1-1" f: R " RR " 1-1" - f: R " R f ^hg ff ^ h ^1 - h! R f ^ h f: R " R f^ h + f ^ h = -! R f ^ h f: R " R ff ^ ^ hh + f^h = -! Rf^h = f^h" 1-1" f ^h = f: ^-, 0h" ^-, 0h ln^f^hh = f^h < 0 f^h ef ^h =- f ^ h + 1 < 0 f: R " R ff ^ ^ hh + f ^h = +! R f^h" 1-1" f^ h f^4 h! R f: R " R f^rh = R ff ^ ^ hh = f^h +! R 1 f ^h= ^f ^ h-h! R f^h = 4f^4hf ^8h f: R " R ff ^ ^ hh =! R 69

23 f^h f^h = f ^- 1h+ f ^1h= f^0h f^8h = 64f^h f^hr A1 ^, h B0 ^, h f ^f^ + h+ h= 1 f^f ^ + h-h> f^hr A^, h B0 ^, h f^fe ^ -h- h= f ^f ^ln + 1h + h > - 1 f^hr A ^, h B59 ^, h f^+ f ^ + hh = 9 f^f ^ -8h-h< f^hr A4 ^, h B61 ^, h f^+ f ^ - hh = 9 f ^f ^ h- h< fgr, : " R f^rh = R ^f% f+ g% fh^h=! R f^h f ^h= f ^ h+ g ^ h ^f% fh^h= + f ^ h! R " 1-1" f0 ^ h = 0 f ^ h= + f ^h! f^rh f: R " R f ^h+ f ^ h+ = 0! Rf^h- f ^h f: R " R fa ^ h = R f 5 ^h+ f ^h+ + = 0! R f^h f^ + - h = f^ - h f: R * " R f ^ y h= f ^ h+ f ^ y h y,! R * 70

24 f1 ^ h = 0 f ` 1 j =-f ^ h R *! f^h = 0 " 1-1" f ^+ yh= f ^h$ f ^yh y,! R f: R " R f ^ h! 0! R f^+ yh= f^fy h ^ h y,! R f0 ^ h = 1 ff ^ h ^- h= 1! R f ^ h = 1 f: R " R ff ^ ^ hh = 9+ 8! R f^rh = R 1 f ^h= 9 6 f ^ h-8@! R f^9 + 8h= 9f^h+ 8! R f^- 1h = f: R " R ^f ^ hh + = 8! R f^h = a f^rh = R f 1 f: R " R f^f ^ hh + f^h =- 4! R f: R * " R f ^ h- fy ^ h= f` y j y,! R * f ^ h = 0 f^h = 0 f^h f ^ h + f ^ + h= f ^ + 1h+ f ^ + 1h 71

25 y,! R f ^ + yh= f^fy h ^ h f0 ^ h! 0 y = 1- C f ' y,! f^rhf ^yh= f ^h+ f ^yh f: R " R^f% fh^h= -! R f1 ^ h = 1 f ^ h+ f ^h=! R 7