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) = α + + γ +α +γ +γα 5. α ν ν =(α ) ( α ν 1 + α ν + + α ν + ν 1) 6. α 3 + 3 + γ 3 3αγ = =(α + + γ) ( α + + γ α γ γα ) = = 1 (α + + γ) ( (α ) +( γ) +(γ α) ) α = = γ 7. α 3 + 3 + γ 3 =3αγ η α + + γ =0 x = α 8. (x α) +(y ) +(z γ) =0 y = z = γ 9. α + ±α 10. α ± α + 0 11. (1 + α) ν > 1+να, 1 <α 0, ν Dun meic - RÐzec - Log rijmoi 1. x = ν α 0 α x 0 x ν = α. ν α ν = α, ν α ν = α 3. An α, 0 tìte ν α = ν α ν 4. An α 0, > 0 tìte ν α = ν α ν, 5. An α 0, > 0 tìte ν α μ = ν α μ ν, μ α = νμ α, 6. An α>0 tìte α μ ν = ν α μ = ν α μ νλ α μλ = ν α μ 7. An α>0, α 1tìte α x 1 α x = α x 1+x, αx 1 α x = αx 1 x, (α x 1 ) x 8. An α, > 0 tìte (α) x = α x x, = α x 1x ( ) x α = α x x 9. An α>0, α 1tìte α x 1 = α x x 1 = x 10. An α>1 tìte α x 1 <α x x 1 <x 11. An α<1 tìte α x 1 <α x x 1 >x α>0, α 1 1. log α x = y x>0 α y = x 13. log x =log 10 x, ln x =log e x 14. log α x = ln x ln α 15. An α>0, α 1tìte x 1 = x log α x 1 =log α x 16. An α>1 tìte x 1 <x log α x 1 < log α x 17. An 0 <α<1 tìte x 1 <x log α x 1 > log α x 18. x 1 = x ln x 1 =lnx 19. x 1 <x ln x 1 < ln x 0. 0 <x<1 ln x<0 x>1 ln x>0 x =1 ln x =0 1. An x 1,x > 0, α>0,α 1tìte log α (x 1 x )=log α x 1 +log α x log α ( x1 x ) =log α x 1 log α x log α x k 1 = k log α x 1 1
ln (x 1 x )=lnx 1 +lnx ln x k 1 = k ln x 1 3 Apìlutec Timèc 1. α = { α α < 0 α α 0. α = α α 0 3. α = α α 0 4. x = α x = ±α 5. α α α 6. α = α 7. α = α 8. α ν = α ν 9. α ν = α ν 10. α α ± α + 11. α + = α + α 0 4 OrÐzousec kai Grammik Sust mata 1. α γ δ = α γ δ = αδ γ. 'Estw to sôsthma } α 1 x + 1 y = γ 1 (Σ) α x + y = γ ìpou k poioc apì touc α 1, 1,α, eðnai di foroc tou 0. 'Estw D = α 1 1 α, D x = γ 1 1 γ, D y = α 1 γ 1 α γ. Tìte: (a ) An D 0to (S) èqei mða mìno lôsh (x, y) me x = D x D, y = D y D (b ) An D =0 kai k poioc apì touc D x, D y eðnai di foroctou mhdenìcto (S) eðnai adônato. (g ) An D = D x = D y =0tìte to (S) èqei peirec lôseic (x, y). 5 Deuterob jmio Tri numo 'Estw h sun rthsh f (x) = αx + x + γ, α 0, Δ= 4αγ. 1. Prìshmo-RÐzec (a ) An Δ > 0 tìte h f èqei dôo nisec rðzec ρ 1, = ± Δ α. 'Otan to x eðnai ektìc twn riz n h f(x) eðnai omìshmh tou α en ìtan eðnai metaxô twn riz n eðnai eterìshmh tou α. (b ) An Δ=0tìte h f èqei mða dipl rðza ρ = α. 'Otan to x eðnai di foro thcdipl crðzach f(x) eðnai omìshmh tou α (g ) An Δ < 0 h f en èqei rðzeckai eðnai omìshmh tou α gia ìlecticpragmatikèctimèctou x. Mègista-El qista (a ) An α>0 h f èqei el qisth tim thn Δ 4α gia x = α. (b ) An α<0hf èqei mègisth tim thn Δ 4α gia x = α. 3. Sqèseic tou Vieta (a ) An eðnai Δ 0 tìte to jroisma kai to ginìmeno twn riz n thc f eðnai S = ρ 1 + ρ = α, P = ρ 1 ρ = γ α (b ) An dôo arijmoð èqoun jroisma S kai ginìmeno P tìte eðnai rðzecthcexðswshc x Sx+P =0.
> 0 =0 < 0 ff>0 4ff ff ff ff 4ff 4ff ff ff ff<0 ff 4ff 6 TrigwnometrÐa 1. π π π π 0 6 4 3 1 ημ 0 3 1 συν 1 3 1 0 εϕ 0 3 3 1 3 * σϕ * 3 1 3 3 0. ημx + συνx =1 3. εφx σϕx =1 εφx = ημx συνx 4. συν x = 1 1+εφ x ημx = εφ x 1+εφ x σφx = συνx ημx 5. συν ( x) =συνx ημ ( x) = ημx εϕ ( x) = εϕx σϕ ( x) = σϕx 6. συν (π x) = συνx ημ (π x) =ημx εϕ (π x) = εϕx σϕ (π x) = σϕx 7. συν (π + x) = συνx ημ (π + x) = ημx εϕ (π + x) =εϕx σϕ (π + x) =σϕx 8. συν ( π x) = ημx ημ ( π x) = συνx εϕ ( π x) = σϕx σϕ ( π x) = εϕx 9. συν (α ± ) =συνασυν ημαημ ημ (α ± ) =ημασυν ± ημσυνα εϕ (α ± ) = εϕα±εϕ 1 εϕα εϕ 10. ημα =ημασυνα συνα = συν α ημ α =συν α 1=1 ημ α 11. εϕα = εϕα 1 εϕ α ημα = εϕα 1+εϕ α 1. συν α = 1+συνα ημ α = 1 συνα 13. Se k je trðgwno ABG isqôoun α = + γ γσυνa α ημa =R (Nìmoc twn hmitìnwn, R h aktðna tou perigegrammènou kôklou) συνα = 1 εϕ α 1+εϕ α (Nìmoc twn sunhmitìnwn) 7 Merikec exis seic 1. x ν = α. x = α ν rtioc ν perittìc α 0 x = ± ν α x = ν α α<0 adônath x = ν α 3. ημx = α α 0 α<0 x = ±α adônath α 1 α > 1 x = α +kπ, k Z, x = π α +kπ, k Z adônath 3
α 1 α > 1 x = α +kπ, k Z, x = α +kπ, k Z adônath α>0 α 0 x = ln ln α adônath 5. εϕx = α 7. ln x = α x = α + kπ, k Z x = e α 8 Embad 1. To embadìn E trig nou ABG eðnai E = 1 αυ α = 1 γημa = = τ (τ α)(τ )(τ γ) = 1 D ( ) ìpou D =det AB, AΓ kai τ = α++γ. An to trðgwno eðnai isìpleuro pleur c α tìte E = α 3 4.. To embadìn parallhlogr mmou eðnai b sh Ôyoc tou tetrag nou pleur c α eðnai α kai tou rìmbou me diagwnðouc δ 1,δ eðnai 1 δ δ. To embadìn trapezðou me b seic B, kai Ôyoc υ eðnai B+ υ. 3. To embadìn kôklou aktðnac ρ eðnai πρ (to m koctou eðnai πρ). Gia to embadìn tomèa kai to m koctìxou gwnðac ϕ èqoume: gwnða f se aktðnia gwnða f se moðrec m koctìxou ρϕ πρϕ 180 embadìn tomèa ρ ϕ πρ ϕ 360 9 Suntetagmènec Estw ta shmeða A(x 1, y 1 ), B(x, y ), Γ(x 3,y 3 ). 1. Hapìstash twna,beðnai d = (x 1 x ) +(y 1 y ). To mèso tou tm matocab eðnai to M ( x 1 +x, y 1+y ) 3. O suntelest c dieujônsewc tou AB kaj c kai thc eujeðacab (efoson x 1 x )eðnai λ = y y 1 x x 1 4. Estw D = x x 1 y y 1 x 3 x 1 y 3 y 1 (a ) Ta A, B, G eðnai suneujeiak an kai mìno an D =0. (b ) An D 0tìte to embadìn tou trig nou ABG eðnai 1 D. 10 DianÔsmata An α =(x 1,y 1 ), =(x,y ) tìte: 1. To jroisma-diafor touceðnai α ± =(x1 ± x,y 1 ± y ) 4
κ α +λ =(κx 1 + λx,κy 1 + λy ) α α + 3. To eswterikì ginìmeno touceðnai α ) α = x1 x + y 1 y = α ( συν α, 4. To mètro tou α eðnai α = α α = x 1 + y 1 5. IsqÔei α α ± α + 6. α + = α + α = α α 7. α // x 1 y 1 x y =0 (efoson orðzontai oi suntelestècdieujônsewc) λ α = λ 8. α α =0 x 1 x + y 1 y =0 (efoson orðzontai oi suntelestècdieujônsewc) λ α λ = 1 11 EujeÐa-KÔkloc 1. H genik exðswsh eujeðac eðnai h Ax +By +Γ =0 me6. 'Estw ènac kôkloc me kèntro K kai aktðna ρ kai d h A + B 0. An B 0 h eujeða èqei suntelest apìstash tou K apì mða eujeða ε. dieujônsewc λ = A B = εϕω ìpou ω eðnai h gwnða pou sqhmatðzei o xonac x x me thn eujeða. ω x x ω x x. H apìstash tou shmeðou M (x 0,y 0 ) apì thn eujeða Ax +By +Γ=0eÐnai d = Ax 0+By 0 +Γ. A +B 3. Mia eujeða me suntelest dieujônsewc α èqei exðswsh thcmorf cy = αx +. Oi y = α 1 x + 1, y = α x + tèmnontai an kai mìno an α 1 α kai eðnai k jetec an α 1 α = 1. An α 1 = α = λ oi eujeðecèqoun thn Ðdia dieôjunsh kai apìstash 1 1+λ kai an epiplèon 1 = tìte sumpðptoun. 4. O kôklocme kèntro to K (x 0,y 0 ) kai aktðna ρ èqei exðswsh (x x 0 ) +(y y 0 ) = ρ. An to K sumpðptei me thn arq twn axìnwn tìte h kôkloc gr fetai x + y = ρ kai h efaptomènh tou se tuqìn shmeðo tou P (x 1,y 1 ) eðnai x 1 x + y 1 y = ρ. 5. HexÐswsh x +y +Ax+Ay+Γ = 0 eðnai exðswsh kôklou an kai mìno an A +B 4Γ > 0. To kèntro tou eðnai to K ( A, B ) 7. 'Estw dôo kôkloi me kèntra K 1, K kai aktðnec ρ 1 >ρ kaihaktðna tou eðnai ρ = A +B 4Γ. kai d hapìstash twn kèntrwn touc(di kentroc). 5
apì dôo stajer shmeða A, B eðnai stajerìckai Ðsocme λ 1eÐnai kôkloc(kôkloctou ApollwnÐou) me di metro pou èqei kra ta shmeða ta opoða diairoôn to tm ma AB eswterik kai exwterik se lìgo λ. 1 Kwnikèc Tomèc 6
1. y = αx + 1 <α<e 1 e e e <α<1. y = αx α = e e 3. y = α x 6. y = ημx, y = συνx 4. y = αx 3 5. 1 y = α x, y =log α x 7. y = εϕx α>e 1 e α = e 1 e 1 Gia leptomèreiec: Mp mphc Toum shc: <<Pìso kal èqoume katano sei thn ekjetik kai logarijmik sun rthsh?>> EUKLEIDHS B', 1994 t.3, 5-55 7