Γιώργος Μιχαηλίδης-Επαναληπτικές ασκήσεις Γ Λυκείου

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44 θέματα - κανόνια από τον συνάδελφο Γιώργο Μιχαηλίδη, μια ευγενής προσφορά του ιδίου και των εκδόσεων «Μαυρίδη». Οι εφ' όλης της ύλης ασκήσεις που θα δείτε, θα σας αφήσουν κάτι παραπάνω από ικανοποιημένους!

Transcript of Γιώργος Μιχαηλίδης-Επαναληπτικές ασκήσεις Γ Λυκείου

  • 1. 9 786188 021433 ISBN: 978-618-80214-3-3

2. 623 1 z z 2= z 2. : [ )f : 0, + ( ) x f x e= [ )F: 0, + ( ) ( ) x 0 F x z 2f t dt x= + ) F . ) z : ( ) x 0 F x lim 3 x = ) x 0> ( ) 0,x : ( ) F x x xe . z 2 = 3 { } z / z 1= > [ )f : 1,+ : ( ) x z f x ln x 2 . x z = + ) z A ( )x lim x f x + . ) z A : ( ) z 1 f x 2 z 1 + x 1 . ) : z 1 z 1 B , z A , z A . z 1 z 1 = = + + 3. 624 5 * 1 2z , z , 1 2z z = = f : ( )f 2 1.= g : : ( ) ( ) ( ) ( ) x 1 22 g x f t Re z z dt 2 x .= + + ) ( ) ( )g 2 , g 2 . ) ( )g x 0 x . i) ( ) ( )1 2Re z Re z .+ = ii) ( ) ( ) 2 2 1 2 m z Im z .= + 6 z , ( )Im z 1.= f : : ( ) ( )x f x x ln e z .= + ) ( )x lim f x . + ) f . ) x , : ( ) ( )x 1 x z z f x 1 f x e z e z+ < + < + + . ) z : ( ) ( ) ( ) 1 2 0 x f x f x 2x f x dx ln 2. + = 7 * z ( )Re z 1.= : ( ) 3 P x x 3x z= + . ) ( )m z 3< ) ( )P 1 0= ( )2 0 P x I dx x 2 = + . ) ( )P z 0,= 100 50 z 2 .= 4. 625 9 { }* z i . f : . ) x e x z + x , z . ) x 0 : ( ) x x 0 z f t dt e 1 0 , ( ) 1 f e z . ) x : ( ) x x 0 z f t dt e 1, = z ( ) x e f x . z i = 11 ( )f : 0, + : ( )f 1 0= ( ) ( )xf x 2f x x = x 0> . ) ( ) ( ) 2 f x h x , x = 1 1 ( )0, .+ ) z, w z w : ( ) ( )w w f z z z f w = . z w Re 0. z w + = ) ( ) ( )h x , f x . ) ( ) ( ) x 1 2x 1 f t dt L lim . ln x = 5. 626 13 z ( ) ( )Re z Im z= [ )f : 1, ,+ ( ) ( )2 f x x z 2ln x .= ) z, f 4 0x e .= ) z , : ( ) ( ) 3f f 4e , 1 .< < ) z f [ )1, .+ 15 z z 1.> : z 1 ln z = f : , : ( ) x 2 f x z x x . : ) f 0x 0= ( )f 0 . ) 1.> ) f , : ( ) 1 0 1 f x dx . 3 17 z , z 3 1. = f : : ( ) ( )2 f x ln x x z= + + . ) : ( )x lim x f x ( )x lim x f x . + 6. 627 ) z 2. ) 2 2 > ( )f x x= . ) z : 1 2 1 dx ln3. x z = + 19 z * f : - : ( ) ( )f x z f x x (1) ) z 1 i= + , ( ) 2 1 f x x 2 = + (1). ) : i) g : ( ) ( )x z g x e f x = . ii) ( ) ( ) f 2 z ln . f 1 > iii) ( ) ( ) x z f x f 0 e x 0. 21 ) : ( ) ( )2 2 1 ln 1 + + 0 . ) [ )f : 1, + : ( ) ( ) 1 f x x ln 1 x ln x = + . i) N f [ )1, + . 7. 628 ii) f. iii) , : x A x B 1 1 1 e 1 x x + + + + x 1 . 23 ( )f : 0, + : ( ) 2 f x 3ln x x 5x m, m .= + + ) m fC ( )( )A 1, f 1 . ) m = 4, ( )F 0, + : ( ) ( ) x 1 F x f t dt= . i) F . ii) ( )F 0 1= > : 2 3 x 5x 4 x e 1 + = [ ]1, . 25 f: ( ) xx 1 f x e , x 1 x 1 = + ) f. ) N ( ) 1, + , ( )f 0.= ) [ ]g, h : 1,1 : ( ) ( ) ( ) ( )2013 2013 g x 2ln x , h x 2ln x= =+ : ( ) 1 : y x 2014. = + 8. 629 i) N ( )( )M , g gC - ( ) . ii) ( )( ) , h hC - ( ) . 27 f, g : , : ( ) ( ) x 2 x 2x e 1 f x , g x . 1 x e 1 = = + + ) f, g. ) , : 2 2 e 1 . 1 e 1 = + + ) x, y x 0, y 0> > : ( ) ( )f ln x lnf y 1.+ = ) : ( ) ( ) 1 x x 2 1 1 f t f t dt dt 0, x 0. t + = > 29 f : ( ) ( )( )( )f x x 1 x 2 x 3= g : A ( ) { } 1 1 1 g x , A 1, 2, 3 . x 1 x 2 x 3 = + + = ) g . ) ( ) ( ) ( ) f x g x . f x = ) ( ) ( ) ( ) 2 f x f x f x > x A. 9. 630 ) : ( ) ( )f x f x 0 = . 31 f : : ( ) x f x e x , = . ) f . ) ( )f x 0 x . ) ( ) ( ) ( ) ( )x f x f x lim f x f x+ . ) i) A 0 1,< + < ( )x e 1 x+ = ( )0,1 . ii) N ( )( )0 0A x , f x ( )0x 0,1 , fC ( )M 0, . 33 ( )f : 0, + , x 0, y 0> > : ( ) ( ) ( )2 2 f xy x f y y f x .= + f 1 ( )f 1 1. = ) x 0> : ( ) ( ) ( ) h 1 f xh f x 2f x lim x. xh x x = + ) f ( )0, + x 0> : ( ) ( ) 2 x f x 2f x x . = + 10. 631 ) : ( ) 2 f x x ln x, x 0.= > ) ( ) x 0 f x lim . x 35 f : : ( )f 0 1= ( ) ( )f x f x 1 x + =+ ) ( )f x . ) f. ) N ( ) x x 1 e 1 0 x . + ) ( )f x 1.= ) , 0 : ( ) ( )f 1 f 1 0 x 2 x 1 + = ( )1, 2 . 37 ) ( ) 0,1 , : ln 1 0.+ + = ) ( )f : 0,+ ( ) xln x f x . x 1 = + 11. 632 i) N f - x 0> ( )f x . ii) ( )h, g : 0, + ( ) ( ) ( ) ( )f x h x xf x , g x x = = . g hC , C ( )( )M , g , ( )( ) , h . 39 f : ( ) 2x 2x e 1 f x . e 1 = + ) f. ) ( ) x f x, x 1,1 . 2 = ) g : : ( ) ( )g x g x f 2 = x g . ) : ( ) ( ) 2 f x 1 f x x . = ) : ( ) 1 2 0 A f x dx.= 12. 633 41 ( ) ( ) [ ) 2 x f x x 2 e , x 0, .= + + ) f . ) : ( )1 f 4 x 1 f 1 0. e = ) : ( ) ( ) ( )f x 1 f t f x 1 + [ ]t x 1, x 1 , x 1 + ( ) 2 x 1 t x x 1 lim t 2 e dt + + + . ) ( ) ( )2 f 2x x f x 4+ = + [ ]0, 2 . 43 : ( ) 3 2 f x x x 3x 1, x= + + . ) f "1 1". ) Rolle ( ) ( )f x 2 F x e x x 3.= + + ) f 0x 1,= . ) 0= i) f . ii) f y 9x 2014.= + 13. 634 45 ( ) ( )4 f x x 1 x, x= + , : ( )f x 1 x . : ) 4.= ) f . ) ( ) ( )5 2 g x x 1 10x 5x= + - [ )0, .+ ) A ( ) ( ) 2 x 1 f x Ax x 1+ + + x 0. 47 f : : ( ) x 3 f x e x x.= + + ) f - 1 f . ) ( )1 f x 0 = . ) : ( ) ( ) x 0 f x f x lim . 5x ) ( ) f e= ( ) f e ,= : 0.= = ) : ( ) 1 3x 3 1 1 f e f x ln x ln x xx + + = + + ( )1, e . 14. 635 49 [ ]f : 0,1 [ ]0,1 . ) ( ) ( )f 0 f 1 0,= = : ( )f x 0 [ ]x 0,1 . ) : ( ) ( ) ( ) ( )( )f x f 0 x f 1 f 0 + [ ]x 0,1 . ) : ( ) ( )g x lnf x= ( ) ( )2 1 h x ln , x 0,1 . x x = ( )c 0,1 g hC , C c . ) : ( )x 0 lim x h x .+ 51 ( ) 1 1 f x x x, x 2 3 = + ( ) ( )g x x f x , x= : ) ( ) 5 f x 6 x . ) ( ) ( ) 5 f x f y x y 6 x, y . ) 0> ( )g 0.= ) : ( ) ( ) 1 f 1. 3f + = 15. 636 53 f : : ( )f 0 1= ( )f x 0, x ( )( ) ( )( )x x f x e x f x e 1 x = ) N ( )f x . ) ( ) 0, 2 , fC - ( )( )A , f ( )P 1, 0 . ) ( )f x 1 x . ) ( ) 2 2x 1 2 0 x f t dt x x 1 0, + = ( )0,1 . 55 ( ) ( )f x x 4 ln x 3x 4, x 0= + > . ) f ( ]1 0,1= [ )2 1, .= + f. ) x 4 2014 3x x e , x 0 = > . ) , < , ( ) , : ( ) ( )f f 2010. = ) g : ( ) x 1 g x 3x 4 lim 0. x 1 + = 16. 637 ( ) gC ( )( ) 1, g 1 ( )f x ( ) . 57 [ )f : 0, + : ( ) ( )x f x ln x 1 = + x 0 . ) ( ) ( )ln x 1 , x 0 f x x 1 x 0 + > = = ) 1 f . ) ( ) ( )g x xf x , x 0= [ )0, .+ , gC ( )( )A 1, g 1 . ) ( ) 1 x xf x ln 2, x 0 2 2 + = + . 59 f : : ( )f x 0> x ( ) ( ) ( )( )2 2 x 1 f x f x x x 1+ = + x ( )f 0 1= 17. 638 ) : ( ) x 2 e f x , x x 1 = + f . ) : ( ) 1 2 0 f x dx ln 2> ) : ( ) ( )2 x 1 4 f 2 e f x 1 < + . ) ( ) 0,1 , : ( ) ( ) ( ) 2 0 f t dt 2 1 2 1 ln 2.+ = 61 ( )0, + f ( ) 3 4 f x dx 1= ( ) 5 4 f x dx 3.= g: ( ) ( ) x 2 x 1 g x f t dt, x 0. + + = > ) g . ) ( ) 2, 3 : ( ) ( )f 2 f 1 4.+ + = ) f 1 2 ( ) ( )f 1 f 1 1= = ( ) ( )f 2 f 2 2= = ( ) ( ) 2 1 2x 0 g x f x dx x lim x . 18. 639 ) : ( ) ( ) 2 2 2 2 x 3 x 3 x 4x 2 f t dt f t dt 2 + + ++ = + ( )0, .+ 63 [ ]f : 0,1 , - [ ]g : 0,1 : ( ) ( ) ( ) ( ) ( ) 2x x 2 0 0 1 g x t 1 f t dt t 1 f t dt 2 = + + . ) ( )g x ( ) 1 f x . x 1 = + ) ( )f x : ( ) 1 0 f x x 1 < + [ ]x 0,1 , : i) g . ii) ( ) 0,1 : ( ) 1 g g . 2014 > ) f g . 65 f : ( ) x 2 0 1 f x dt. t 1 = + ) x : ( ) ( ) 2 1 f x 1 f x f x x 1 + = + + . ) ( ) ( )( )x lim f x 1 f x + + 19. 640 ) ( ) 1, 2 , : 1 3 2 2 0 1 1 dt 1 1 t = + + . ) [ ]x 0,1 , ( )f x x . ) ( ) 1 2 0 1 f x dx . 3 < 67 ( )g : 0, + ( ) 2 1 g x 2x f x = x 0> , f [ )0, + ( )f 0 0.= ) y 2014x = + gC + , - fC ( )( )A 0, f 0 . ) y 1007x= gC , m , : 1 1 mf f . m m = ) : 1 21 1 1 f dx 0, xx = < 0x , ( )0f x 0. = ) ( )x, y 0, + x y< ( ) ( )2 2 y f x x f y> : ( ) ( )g x g y .< 69 g : f : ( )g x 0> ( )f x 0> x . 20. 641 F: : ( ) ( ) ( )x g x 0 t F x f dt g x = . ) x ( ) ( ) ( ) x 0 F x g x f u du.= ) ( ) x g x e= ( ) x f x e = , F. ) ( )F x x x : ( ) ( )g 0 f 0 1. = ) : ( ) ( ) ( ) ( )F 1 g 2 F 2 g 1 . < 71 F: ( ) ( )x 1 f t F x x dt, x 1 t = f [ )1,+ ( )f 1 0= : ( ) ( )f x xf x 0+ x 1. ) ( )F x 0 x 1. ) ( ) ( ) 2 1 f t dt f 2 t = ( ) 1, 2 , : ( ) ( )f f 0.+ = ) ( ) ( ) 2 2 f x 1 f 2x , x 1. 2x x 1 + = + ) F ( )f x ln x, x 1.= 21. 642 73 f : ( ) ( ) ( )f x f x f x 0, x + + = ) g ( ) ( )( ) ( )( ) ( ) ( ) 2 2 g x f x f x 2f x f x = + + , f . ) : ( ) ( )f 1 f 1 1 + = ( ) ( )f 0 f 0 1 + = ( ) 1 0 f x dx. ) : ( ) ( ) ( ) ( ) ( ) ( )ln f f f ln f f f + + + + = , < , ( )c , , : ( ) ( )f c f c . = ) ( )f x 0 > x : ( ) ( ) ( )( ) ( ) ( )2 x f x f f x f 2 + + x . 75 f [ )1, + ( )f x 0> x 1. : ( ) ( ) x 2 1 G x t f t dt, x 1= ( ) ( ) x 1 H x tf t dt, x 1.= ) : ( ) ( ) ( )P x xH x G x , x 1.= : i) ( )P x 0 x 1. ii) ( )P x [ )1, + . 22. 643 iii) x 1> : ( ) ( ) ( )P x P x 2 P x 1 . 2 + + + < ) N : ( ) ( ) ( ) G x F x H x = ( )1, .+ ) : ( ) ( ) ( ) 2 x 1 2x 1 H x ln tdt L lim . G x x 1 + = 77 f : x : ( )x e f x 1 0 ( ) ( ) x t x 0 e t e x dt 1 f x f t = + ) f . ) f . ( ) x x f x 1 , x e = , : ) f . ) ( ) 0,1 fC , 0x = y y, : E 1 .= ) : ( ) ( ) ( ) 2 x 1 1 lim f x 1 . f x f x 23. 644 79 g : : ( ) ( ) x x x e x g t dt 0 = x . f : : ( ) ( )( )( ) ( ) ( ) ( ) g x 2 g x f t f t 1 dt f x f x x + += x ( )f x 0 x . ( )f 0 1.= ) ( )g x . ) ( )f x . ) i) N : ( ) ( ) ( ) ( )2 f x xf x f x f x = f . ii) : ( ) 1 0 2 f x dx 2 > . ) N : ( ) ( )x 1 lim lnf x . f x+ ) : ( ) 2 x x f t dt 0. = 24. 645 81 [ )f : 0,+ : ( ) [ )f x x x 0,< + ( ) ( ) x ux 1 11 x 1 t f t dt dt du 2 t f t = + x 0 ) f ( ) ( ) x f x x f x = x 0. ) f . ) : ( ) ( ) 2 4 4 2 2 x 1 f f x f x x f x < ) : i) ( ) ( )xf x f x x 0 ii) ( ) 2 1 1 1 f x dx 2 f 2 2 < . 83 f : : f ( )f x 0> x ( ) x 2x x 1 e 1 x f 2x t dt 0 x ) : i) ( ) 1 0 f t dt 2= ii) ( ) 0,1 , ( ) 0 f t dt 1= iii) ( ) ( ) ( ) x 0 x x f t dt x 0 f t dt xf x e e + = ( )0, . 25. 646 ) ( ) ( ) ( ) x 0 g x f t f 1 t dt, x= + i) g. ii) g . iii) ( )g x , x x x 0= , x 1= 4 . 85 f : , : ( ) 2 1 x t x 0 f 0 lim x e dt + = ( )f 0 0 = ) ( )f x 1 x ) ( )f 0 0 = ( ) 2x 0 f x 1 lim x . f g : : ( ) ( )( ) ( ) ( )g x 2 3 2 g x f x 2x 2x f x x t 1dt, x , += + + ( ) 2 x 2 f x e x= x . ) ( ) ( ) 2x 2 2x h x f t x dt + = , [ )x 0, + ( ) ( ) 2 2 x 2x 3 4 x 2x 1 6 f t dt f t dt 0. + + + + + <