20 πρωτότυπες επαναληπτικές ασκήσεις στη γ΄ λυκείου...

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1. http://lisari.blogspot.gr team_ 20 lisariteam 2015 2. 20 - lisari team 2 ..3 lisari team..4 ...... 5 ..6 1 .....26 2 ....30 3 .....34 4 .....38 5 .... 43 6 ..45 7 ..... 49 8 ..... 54 9 ..56 10 ....59 11 ....64 12 ....67 13 ....72 14 ....75 15 ....78 16 ....84 17 ....88 18 ....92 19 ....96 20 ..100 3. 20 - lisari team 3 20 . . , blog http://lisari.blogspot.gr. , , , . . , , [email protected] lisari - 2015- 4. 20 - lisari team 4 lisari team ( - ) ( - . ) ( - ) ( - ) ( ) ( - ) (3 ) ( ) ( - ) ( - ) ( 2 ) ( ) ( 19+ - ) (3 ) ( - ) ( - ) (3 ) (1 ) ( , ) ( - ) ( - - ) ( - ) ( ) ( ) (6 ) ( - ) ( 1 ) ( - ) ( - ) ( ) ( .. - .. ) ( - ) (1 ) ( ) ( ) ( ) 5. lisari team tem_! 20 6. 20 - lisari team 6 1o z 6 z z z z 4 64 0 (1) f :R R 33 f x x z x z . ) f R z 2. ) 3 x x z z z z 1, z . ) (1), 2 z z 4 2 17. z 2 z z 4 2 17; ) 1 2z ,z 6 3 6 3 1 1 1 2 2 2z z 4 z 64 z z 4 z 1) 1 2z z 2 2, 1 2 z i z 2) 2 2 1 2z z 0 , 1 2z z 2 2 7. 20 - lisari team 7 2o f :R R f x 2 1 2 x dt , ln t t 1 1 ln2 x R. (1) ) f x 1 , x R f R . ) f f A 2,f x . ) 2 0x 2, 0x 0 0 f t dt x f x . ) 2x x 0 x f(t 2) 1 lim dt. t ) f / . 8. 20 - lisari team 8 3o A) x x x xe e x x xe R . B) f :R R : f R , x f '' x f ' x e , x R , f . ) x x 1 f x 1, x R e f . ) R 2 2 x 2 1 x e . 3 e ) x 2015 x x k L lim f t dt , y k f . ) ( ) 1 2 x 0 x 0 x 1 x L lim , L lim . f x f x ) 0 x x 1 1 e e R . 9. 20 - lisari team 9 4o R f 1 x t 0 0 f x 1 xte dt (f t f 1 t dt , x R . ) f ' x 1 f x f 1 x , x R f. ) f x 2 2 f x , x R . ) 0 , dx ln f(x) . ( ) --, 141, iv, . 306) ) k x f t x x 0 L lim e dt , k 1,2,... 10. 20 - lisari team 10 5o f,g x f x x 1 0 e 1 x f x f x x x f x , x 0 g x 2 xln x 1 , x 0 . ) x f x x 1 e , x ) g x 0 ) 1 2g x 2 g x 2 2016 x 1 x 1,0 , * 1 2x ,x . 11. 20 - lisari team 11 6o f x x 1 f x . e x ) f . ) 1 1A x ,f x , 2 2B x ,f x 1 2x 1 x , f . ) ,f f ' . ) [,] , 2 , f . ) f . ) 1 23 x ln xx x f x xx x L lim , L lim . f x e e 12. 20 - lisari team 12 7o f x 9 0 f x t t dt , x 0, . 2 ) f . ) f x 0 , x 0, 2 f x f '' x , x 0, . 2 ) m : 0, R 2 f x m x x . ) f 11 f , =2,3, ) , 0, 2 9 t t dt . 13. 20 - lisari team 13 8o ,, 2 z z , z , z , * z C . ) z. ) . ) ,. ) , A. 14. 20 - lisari team 14 9o z,w : 1 1 z Re z , Re z Im z 0 4 4 13 uu 2z i z i. 2 ) M z f x x,x 0, z fC ( d Re(z) 0 , t 0 dt ). ) z, 22 z z 2 z z 2 i . ) 2z 13 5 . ) w ww wu wu 2 0 , : 1) w , 2) w u , 2, u , 3) 2 u w 2 , 4) 2 2 2 u w 2 w u 2. 15. 20 - lisari team 15 10o f R g :R R 1 0 g x f xt dt. ) x 0 1 g x f t dt , x 0 x g 0. ) 1 2 0 1 f t dt f t dt , 1,2 , g f . ) f 0, 2 2 x 0 f x g x lim f 0 f 0 x ) f R , 1) g R . 2) 1 0 f t dt 2 1 f t dt . 3) x 2016 x 0 0 x f t dt x 2016 f t dt , x 0 . 4) fC y f 0 , gC . 5) * z C z 1 0 0 z f z tf t dt zf t dt , (0,1) , z 1 e 1 ln z . 6) h :R R h x f x g x 2016 2 x L lim f 2 f 1 g 1 g 2 x x x . 16. 20 - lisari team 16 11o f : 0, R f x 0, x 0 , 2h 0 1 3 h h lim f 1 2 f 1 h f 1 f o oA x ,f x o o 1 M x ,0 2x . ) f 1 2 1 x f x e , x 0. ) z Im z 0 2 1 z 1 e z 1 , 1 f ' x x lim z 1 0 ) x 2 1 x 1 f x f t 1 dt 1 x t x , x 1 ) x 2x 1 1 1 lim 1 f t dt 2t 2 17. 20 - lisari team 17 12o F,f : 0, R : F x f x , x (0,) x f x x ) x 3 f t 1 x 1 dt ln ln3 , x 0, t 1 x 2 ) 2 ln3 F F 3 3 2 ) f 0, o x 3 , o 2 f x F F 3 3 ) N 1 x 3 , o 1 3f x f x . ) m: 0, R f x m x x , xx x , x 3 2 , 2Eln3 1 f x e i f x ( i ). 18. 20 - lisari team 18 13o f : 0, R f x x f x x , x 0, . ) f 0 f 0 ) f 0, ) f 0 0 f 0, , : 1) 0, , f 0 2) , 0, f f , . 3) 2f 2f f 6 4 2 4) 2 6 3 3f f x dx f 6 2 2 5) f 0 6) , f x x 0, . 19. 20 - lisari team 19 14o f : 0, R 2 : x f x x f x x f x , x x 0, 2 x 3 3x 3 3 lim . x 2f x ) g : 0, R 2 f x g x xx f ) o x 0, 2 of x 0 o x 4 ) 2 xx 2 , x 0, . 2 ) 2 , , 0, 2 20. 20 - lisari team 20 15o ) x e x 1 , x R . ; ) f R , : x 2 e f x f x f x 2 , x R 2 xx 0 x 3x lim 3 f x e f 0 0 ) x f x e x , x R ) 2 x 1 f x f x I dx e 1 ) g 2 xg x f x 1, x R , 1) g 2) 1 g 0 2 3) g R 4) 1 0 5 2e g 0 g x dx g 1 2 5) g 1 e g x ln x , x 1. 21. 20 - lisari team 21 16o f : 0, R x 1 t x e 1 1 ln x 1 f x lim 1 e dt x e t , x 0 (1) . ) 1 x f x xe , x 0 ) f ) f x 1 , x 0 ) , 0 , 1 x 1 x x x e e , x 0 ) 4 3 4x f x 2x 1 M lim 2x 4x ) f A 1,f 1 . : 1) 2 1 2 f x dx e 2) f , xx x 1,x 2 , x x x xx 7E 2 lim 2 3 3) x 1 kx f t dt L lim x , k 1,2,... 22. 20 - lisari team 22 17o x 1 2z 1 i , z 2 x i , x R 0 1 f :R R 1 2f(x) Im z z . : 1 2Re z z 1, x R e 2 e 2 (, R ) , ) e ) , f 0 2 , 1 1 2 2 ) f x 0, ) 1) lnx x 1 , x 0 . ; 2) g :R R f x 1 g x 2 x . E g , xx x 0,x 2 , 1 2 3) x x 0 lim g t dt ) x 1 1 2 f 0x 0 1 Im z x 1 f x 1 e 1 Re z z lim 0 x Im z 23. 20 - lisari team 23 18o ) 3 x x x x 6 , x 0 (*) ) 20163 x x t F x x dt , x 0 18 t 3 x 3 3 x t 19x x x dt t 18 18 0, ) 1 , : 1) x x 0 x t lim dt t 2) x 2 x 0 x t lim dt t 3) x 3 x 0 x t lim dt t ) : 0, R x lnx , xx x 1, x e , e E 1 3 ) 2 0 8 x dx 1 9 2 0 8 x dx 1. 9 24. 20 - lisari team 24 19o f : 0, R : x f xy f 2f x y , x,y 0 f 1 0 f 1 1 . ) 2 1 1 2 f x f x x x , x 0 ) f x lnx , x 0 ) K , 2 e 1 . , f , x 1 . - ) y 0 f A,B. A,B . 25. 20 - lisari team 25 20o f :R R , : x e f x 1, f : y x ) x f x x e , x R ) f x 1 , x R f f ) e m 0: f f m e ) 1 1 1 e 1 ) 2 2 x 3x 2 x 1 e e 3x 1, x R. ) k k e k e k k R: e e e 1 . 26. 20 - lisari team 26 1 ) 2 zz z . 6 6 2 6 3 z z zz 4 64 0 z z z 4 64 0 z z 4 z 64 0 (2) 6 3 x x 4x 64 0 2 . Horner 2 : 1 0 0 1 0 -4 -64 2 2 4 8 18 36 64 1 2 4 9 18 32 0 , : 6 3 5 4 3 2 x x 4x 64 x 2 x 2x 4x 9x 18x 32 . , (2) : 5 4 3 2 z 2 z 2 z 4 z 9 z 18 z 32 0 z 2 0 z 2. (1) z 0 ( (1) z 0 64 0 , ). f , , 2 f x 3x z 0, x . f . ) g : z 1, z , 3 2x g x x z z g z 1, z , : 3 2z g z z z z 0 z 33 2 zz 1 g z 1 z 1 z z 2 2 3 3 z 3 z 1 z z z 2 2 z 2 z 1 0 z z ( 2 2x 2x 1 0, 2 0 . 2 2 z 2 z 1 0 ). Bolzano, 3 3 2o o o o o o x x x z 1, z :g x 0 x z 0 x zz z z ox , 2 3x g x 1 0, z x z 1, z , g . ) 6 z z zz 4 64 0 , ) z 2 . 2 z z 4 2 17 27. 20 - lisari team 27 z x yi, x,y . 2 2 2 2 z 2 x y 4 y 4 x (3) , 22 2 2 2 2 z z 4 x yi x yi 4 x y 2xyi x yi 4 x y x 4 y 2x 1 i , (3)2 222 2 2 2 2 2 2 2 z z 4 x y x 4 y 2x 1 x 4 x x 4 4 x 4x 1 4x 22 2 2 4 2 3 2 2 4 2 3 2x x 8 4 x 4x 1 4x 4x x 64 4x 32x 16x 16x 4 16x 4x x 4x 2 2 17 68 16x 2 17 4x , x 2 2 17 x 2 17 4x , x 2 17 17 , 2 2 , 2 2 8x 8x x 2 2 17 4x 17 4x , 17 x 0 x ,0 . 2 x 0 , 0 2 17 . 2 z z 4 2 17 x Re z 0, z yi,y . z 2 y 2. z 2i ) ) 1 2z z 2 : 1z : 2z AB 2 2 OB 2 , x 17 2 0 17 2 x + - 28. 20 - lisari team 28 2 2 1 2z z OA OB 2OM 2 OM 2 OB BM 2 4 2 2 2 , . 1) 1 2 z w z . : 1 2z z 2 2 1 2 2 z z 1 2 2 z 2z w 1 2 2 2 w 1 2 2 w 1 2 , 1 2z z 2 2 w 1 2 , w 1 w 1 2 2 w 1 w 1 w 1 w 1 w 1 w 1 w w 0 Re w 0 1 2 z Re 0 z 1 2 z ki z , k , w 1 2 ki 1 2 2 k 1 2 2 k 1 k 1 1 2 z i z 2) 2 2 1 2z z 0 2 1 2 1 2z z 2z z 0 2 1 2 1 2z z 2z z 2 1 2 1 2z z 2z z 2 1 2 1 2z z 2 z z 2 1 2z z 8 1 2z z 2 2 , 1 2z z OB 2 , OM 2 2 , : 2 2 MB OB MO 4 2 2 , 1 2z z 2 MB 2 2 29. 20 - lisari team 29 [ : 1. 1) , 2) 2 2 2 2 1 2 1 2 1 2z z z z 2 z 2 z ( ). 1 2z z 1 2z z . 2. 2z 1z , 2AB 2z 4 . 90 2 2 2 2 2 1 2 1 216 z z z z 1) , 2) ] 30. 20 - lisari team 30 2 ) ( ) nx x 1 x 0 ( x 1 ) 1 g(t) nt t 1 (0,1) (1, ) x 2 1 H(x) dt nt t 1 (1, ) ( 1, x g 2 (1, ) ) f (x) 2 1 2 x dt nt t 1 1 n2 x , f (x) 2 1 H(f(x)) dt nt t 1 {x f(x) 1} , x ,f(x) 1 x . g (1, ) 2 (1, ) , f (x) 2 1 H(f(x)) dt nt t 1 , f (x) 2 1 f (x) H(f(x)) dt nt t 1 nf(x) f(x) 1 (1) f (x) 1 nf(x) f(x) 1 f (x) 0 nf(x) f(x) 1 1 n2 n2 1 ( 2 e n2 ne n2 1 n2 1 0 f(x) 1 nf(x) f(x) 1 nf(x) f(x) 1 0 ) f ) f nf(x) f(x) 1 n2 1 , . nf(x) f(x) 1 f (x) n2 1 (2) f (2) 1 f (x) f (x) 1 f(x) f (x) ( f (x)) 0 n2 1 f(x) n2 1 f(x) x 31. 20 - lisari team 31 ( f (x) 0 , n2 1 0 , f(x) 1 f(x) 0 1 f(x) 0 ) f f (x) 2 1 2 x dt nt t 1 1 n2 x 2 f (2) 2 1 dt 0 nt t 1 1 nt t 1 0 0 nt t 1 f(2) 2 f (2) f (2) 2 2 1 1 dt 0 dt 0 nt t 1 nt t 1 , f(2) 2 2 f (2) f (2) 2 1 1 dt 0 dt 0 nt t 1 nt t 1 , f(2) 2 nf(x) f(x) 1 f (x) n2 1 x 2 nf(2) f(2) 1 n2 2 1 n2 1 f (2) 1 n2 1 n2 1 n2 1 fC A 2,2 : y f(2) f (2)(x 2) : y 2 x 2 : y x ) x h(x) f(t)dt x f(x) , x f , x f(t)dt x,f(x) , h , x h (x) f(t)dt x f(x) f(x) 1 f (x) 0 ( f(x) 1 f(x) 1 0 f (x) 0 ) h h [2,] ( ) h(2) h() 0 2 2 h(2) f(t)dt 2 f(2) f(t)dt 0 , f(t) 0 >2 2 f(t)dt 0 2 f(t)dt 0 h() f(t)dt f() f() 0 , f 32. 20 - lisari team 32 , fC , f(x) x x 2 . 2 f() f() 0 Bolzano 0x (2,) 0 0x x 0 0 0 0 0 h(x ) 0 f(t)dt x f(x ) 0 f(t)dt x f(x ) h , 0x . ) x 0 x 0 0 x t 2x f x 2 t 2 2x 2 f(x 2) f(t 2) f(2x 2) :t 0 f(x 2) 1 f(t 2) 1 f(2x 2) 1 f(x 2) 1 f(t 2) 1 f(2x 2) 1 t t t f(t 2) 1 f(x 2) 1 0 t t x 2xf(2x 2) 1 f(t 2) 1 0 t t 2x x f(t 2) 1 f(x 2) 1 dt 0 t t 2x x f(2x 2) 1 f(t 2) 1 dt 0 t t 2x 2x x x f(t 2) 1 1 dt (f(x 2) 1) dt t t 2x 2x x x 1 f(t 2) 1 (f(2x 2) 1) dt dt t t 2x2x 2x x xx f(t 2) 1 (f(2x 2) 1) dt (f(x 2) 1) t [ nt] [ nt] 2x x f(t 2) 1 (f(2x 2) 1)( n2x nx) dt (f(x 2) 1)( n2x nx) t 2x x 2x f(t 2) 1 2x (f(2x 2) 1) n dt (f(x 2) 1) n x t x 2x x f(t 2) 1 (f(2x 2) 1) n2 dt (f(x 2) 1) n2 t x 0 u 2 imf(x 2) imf(u) f(2) 2 ( f 2) x 0 u 0 imf(2x 2) imf(u 2) 2 ( u 2x x 0 u 0 ) , x 0 x 0 im(f(2x 2) 1) n2 im(f(x 2) 1) n2 n2 , 2x xx 0 f(t 2) 1 im dt n2 t ) x f(x) x 0 1 1 0 f(x) x x x 1 im 0 im 0 x 33. 20 - lisari team 33 x 1 im 0 f(x) x x 1 im f(x) im 1 f(x) ( x 1 im 0 f(x) 1 0 f(x) ) y x ( ) Cf x f(x) im x x im[f(x) x] . x im f(x) x im x x x x x f (x) nf(x) f(x) 1 1 nf(x) 1 im imf (x) im im f(x)[ 1 ] x n2 1 n2 1 f(x) f(x) ( x im f(x) x im nf(x) x x ( nf(x)) 1 im im 0, (f(x)) f(x) De l Hospital x nf(x) im 0 f(x) x nf(x) 1 im[ 1 ] 1 f(x) f(x) 1 0 n2 1 ) , De l Hospital x f(x) im x , fC ( ) 34. 20 - lisari team 34 3 . x x , x . : x x x x x xe e xe e x x x x e x e e x x x x x x x x x x x e xe x e e xe e e xe e xe x . ) x x f 0,0 : f(0) 0 (1) f (0) 0 (2) , x : x f (x) f (x) e x e : x x x e f (x) e f (x) 1 e f (x) x , x . : x e f (x) x c , x . (c:) x 0 : 2 0 e f (0) 0 c f (0) c c 0 x x e f (x) x f (x) x e : x x f (x) x e e , x . 1c : x x 1f(x) x e e c x . x 0 : 1 1 1 1f(0) 1 c 0 1 c c 0 . : x x f(x) x e e 1 x x 1 1 f(x) x 1 e e 35. 20 - lisari team 35 x x 1 f(x) 1 e , x . f : : x f (x) x e , x x e 0 , f x . : f 1 x f ,0 f 0 , lim f x 0, A [: x x xx x x x x 1 x 1 1 lim f x lim 1 lim 1 lim x 1 1 1 ] e e e : f 2 x f 0, f 0 , lim f x 0,1 A [ : x xx x xx x 1x 1 1 lim lim lim 0 e ee : x lim f x 0 1 1 ]. f : 1 2f 0, 0,1 0, ) : 2 2 2 2 x 2 2 2 2 2x x 1 x e 1 x 3 1 x 3 3 e e ee e 2 2 2 x 0,2 0 2 2 2x f 1 x 3 1 1 f x f 2 x 2 x 2 2 x 2 ee ) x lim f x 1 , y 1 fC . x 0 f x - + f 36. 20 - lisari team 36 : 1 . x 2015 x 2015 x 2015 x 2015 t t x k x 1 x 1 x 1 f t dt 1 t 1 e dt dt t 1 e dt x 2015 t x 1 x 2015 x 1 t 1 e dt x 2015 x 2015t t x 1 x 1 2014 t 1 e e dt x 2015x 2015 x 1 t x 1 2014 x 2016 e x 2 e e x 2015 x 1 x 2015 x 1 2014 x 2016 e x 2 e e e x 2015 x 1 2014 x 2017 e x 3 e . : x 2015 x 2015 x 1 x x x 1 L lim f t dt lim 2014 x 2017 e x 3 e 2014 0 0 2014 [: x 2015 x 2015x x x x 2015 x 2017x 2017 lim x 2017 e lim lim e e x 2015x 1 1 lim 0 e x 1 x 1x x x x 1 x 3x 3 lim x 3 e lim lim e e x 1x 1 1 lim 0 e ] ) : x 0 lim x 1 1 1 0 f 0 x 0 limf x f 0 0 : 0 0 x xx 0 x 0 x 0 x 0 x 1x 1 x x 1 lim lim lim lim f x f x xe x e 1 1 1 1 : 1L 1 . : 0 0 x x x 0 x 0 x 0 x 0 xx x 1 lim lim lim lim e x 1 f x f x xe x 0 0 x x x 0 x 0 x 0 x 0 xx x 1 lim lim lim lim e x 1 f x f x xe x . , 2L . 37. 20 - lisari team 37 ) 0 . : x x x x 1 1 x 1 1 x 1 1 1 1 f x f e e e e e e 1f , 1x ,0 : 1f x f . 1x f ,0 . , 2f , 0 , 2x 0, : 2f x f . 1 2x 0 x . 38. 20 - lisari team 38 4 ) 1 x t 0 0 f x 1 x te dt f t f 1 t dt 1 x1t t 0 0 0 f x 1 x te e dt f t f 1 t dt x1t 0 0 f x 1 x e e f t f 1 t dt x 0 f x 1 x e e 1 f t f 1 t dt x 0 f x 1 x e e 1 f t f 1 t dt x 0 f x 1 x f t f 1 t dt, x x 0 f 0 1 f t f 1 t x 0 f t f 1 t dt . f : f x 1 f x f 1 x x (1) x 1 x , (1) f 1 x 1 f 1 x f x , x (2) (1) + (2) f x f 1 x 2 f x f 1 x 2x , x c f x f 1 x 2x c (3) x x 1 x , (3) f 1 x f x 2 2x c (4) 3 4 0 2 2c c 1 (4) 1 2 f 1 x f x 1 2x f x 1 1 2x f x x , x 2 1f x x c , x 1c 1f 0 1 c 1 2 f x x 1 , x ) x 2 2 x , x 0 , lnx:"1 1" x 2 x 2 2 x ln 2 ln x xln 2 2ln x ln x ln 2 , x 0 x 2 g: 0, , ln x g x x x 2 2 x , x 0 g x g 2 , x 0 h : 0, h(x) g x g 2 39. 20 - lisari team 39 2 ln x 1 ln x h (x) g x x x , x 0 2 1 ln x h (x) 0 0 1 ln x 0 ln x 1 x e x 2 x 0 2 1 ln x h (x) 0 0 1 ln x 0 ln x 1 0 x e x 2 x 0 2 1 ln x h (x) 0 0 1 ln x 0 ln x 1 x e x x 0 e h x + - h h 2 0 h . 0,e , h x 0 0,e x 2 ln 4 ln 2 2ln 2 ln 2 ln 2 ln 2 h 4 g 4 g 2 0 4 2 4 2 2 2 h . e, , h x 0 e, , x 4 : x 2 2 x ,x 0 x 2 x 4 (5) , 2 2f x 2 x 1 2 2 f x 2 x 1 2 y x 1 0 5 y 2 2 y ,y 0 y 2 y 4 2 2 y 2 x 1 2 x 1 x 1 2 2 y 4 x 1 4 x 3 x 3 f x 2 2 f x x 1, 3 ) t 1 t g x dx lnf x t , t H 1 ln f x , t 1 dx lnf x , 40. 20 - lisari team 40 t 1 1 dx lnf x lnf t . g , , 2 22 2 2 2 2 t ln t 1t lnf t1 1 g t lnf t t t lnf t t ln t 1 , t g , 2 2 t ln t 1 0 t 2 2 v t t ln t 1 t 0 3 2 2 2t 2t v t 2t 0 t 1 t 1 t 0 v 0, t 0 v t v 0 v t 0 g t 0 t g , g()=0 0 g g g 0 1 1 dx 0 dx lnf x lnf x 1 1 1 1 dx dx lnf x lnf x 1 1 1 1 1 dx 0 dx 0 lnf x lnf x x 2 2 1 1 1 1 dx dx 0 dx 0 lnf x x lnf x x 2 2 22 2 2 x ln x 11 1 dx 0 dx 0 xln x 1 x ln x 1 , ln x x 1 , x 0 x 2 x 1 2 2 2 2 ln x 1 x x ln x 1 0 ( x 0 ) , 2 2 2 2 x ln x 1 0 x ln x 1 , 0 2 2 2 2 x ln x 1 dx 0 x ln x 1 ) k k k x f t x xf t x f tx 0 x0 0 e dt e dt e e dt , k 1,2,... e x f t 0x lim e dt 41. 20 - lisari team 41 : 2 t 1 t e , 2 t t 2te , t . 2t 2 2e 0 t t 0 2te 0 t 0 2t 2 2e 0 t t 0 2te 0 t 0 2t 2 2e 0 t t 0 2te 0 t 0 t 0 t - + 0, t 0 e t , 2 t 1 e e , t x 0 2x x t 1 0 0 e dt edt e x x lim ex 2x t 1 0x lim e dt x 0 2y yf t t 1 0 0 y e dt y e dt 0,x 2 t 1 e 0 0,x 0,x 2 x 1 x e 0,x 2 2 2 2 x t 1 x 1 t 1 10 0 e dt x 0 e e dt xe x x (6) , x e 2 2 2 2 0 x 0 x 1 1 x 1 2 2 2 2x 0 1 x 1 1 x 1 e e e xe xe xe 2 2 1 x 1 1 1 1 xe xe xe x 0 42. 20 - lisari team 42 x 1 lim 0 xe 2 x 1x 1 lim 0 xe , 2 1x 1 lim 0 xe 2 1 x lim xe (6) 2x t 1 0x lim e dt 1 2 2 2 2 xx t 1t 1 x 1 0 x x 10 x xx d.L.H. x x xx e dte dt e L lim lim lim lim e e ee 2 u x x 1 u u u lim e 2 2 2 2 2 2 2 xx t 1t 1 x 1 00 x xx d.L.H. x x x x e dte dt e e L lim lim lim lim 0 2xe 2xe e 3 2 2 2 2 3 3 3 3 xx t 1t 1 x 1 0 x x 10 2x 2 xx d.L.H. x x x x e dte dt e 1 L lim lim lim lim e 3xe 3x e e 2 3 2 3 u x x 1 x x 1 u x u u lim e lim e 0 3x 1 lim 0 3x L 0 4 L 0 , , 1 L 0 , =2,3,... 43. 20 - lisari team 43 5 ) g , 0, x 0x 0 x 0 x 0 g 0 lim g x lim g x f 0 lim f x lim 2 xln x 1 f 0 2 h x f x x,x . h ( ) h x 0 , x . h . h 0 f 0 0 2 0. h x 0 , x (1) 2 x (1) 2 x x x x h x e 1h x 1 0 0 h x e 1 0 h x e 1,x e 1 h x e 1 h x , x x f x x e 1 f x x e 1, x ) g 0 f 0 2, 0 g x 0 . g x 0 ,0 , x x e 1 0,x ,0 x x x e g x x e 1 1 0 2 1 e ,0 g . ,0 , g , ,0 x x 0 g ,0 lim g x , lim g x , 2 0 , 2 1 1x 0:g x 0 ( g ,0 ) g x 0 0, , 2 xln x 1 0 , x 0, x g x 2 xln x 1 ln x 1 0 x 1 0, , g . 0, ( x 1 1 ln x 1 0 x 0 x 1 , g , 0, x x 0 g 0, lim g x , lim g x , 2 0 , 2 2 2x 0:g x 0 ( g 0, ) g x 0 . ) ) g x 2, x 0 . 1 2x ,x 0 44. 20 - lisari team 44 1 2g x 2 , g x 2 : 1,0 1 2 x x g x 2 x 1 g x 2 2016x x 1 1,0 , . , 1 2 ( ) ( ) 1 0 2 g x g x 2 0 Bolzano, 1,0 : 0 :( 1) 1 2 g x 2 1 g x 2 2016 1 1 2g x 2 g x 2 2016 1 45. 20 - lisari team 45 6 ) x e x , x , f . 1 : ln x x 1 , x 0 . x x e : x 1 x x x x x x lne e 1 x e 1 e x 1 e x , x . 2 : : , x x e x . x x e 1 , x x 0 x 0 0 , x 0 , x , x x e x 1 0 e x , x . ) f ( ), x x 1 f x e x x x 2x x x 2x x x x x 2 x 1 e x x 1 e x e x e x x 1 e 1 e x e x x e x e 1 e x x x 2 2x x g x2e xe 1 , x e x e x , x x g x 2e xe 1 , x . x 0 x min 46. 20 - lisari team 46 g: g , x g x e 1 x g x 0 1 x 0 x 1 ( x e 0 ). g 1 x g ,1 lim g x , g 1 1 , e 1 A < ( x x x x lim g x lim 2e xe 1 0 0 1 1 , x x xx x x xx x x 1 1 lim xe lim lim lim 0 e e e ). g 2 x x 1 g 1 , lim g x , lim g x 1 , e 1 A ( x x x x lim g x lim 2e xe 1 x x lim e 2 x 1 1 ). 10 A 1 1x 1 : g x 0 . , 1x 1 , g 1 0 . 1x , g , 1 . 1g x 0 1f x 0 . , 1 1A x , f x , 1 1x 1: f x 0 , fC x x . , 20 A 2 2B x , f x , 2x 1: fC x x . ) Rolle f 1 2x , x 1 2 x , x . f 0 , . ) g x 0 , 1 2x , x , 1 21 x , x . x 1 g x g 47. 20 - lisari team 47 g 1 2x , x , . g 1 e 1 0 g x 0 , 1 2x x , x . , f x 0 1 2x , x f 1 2x , x . 2 1x x 2 . Bolzano g 2 , 1 , 1, 2 . , g 2 , 1 , 1 , 2 . 1 2x , x g 1x 2 , 1 2x 1, 2 . , 1 1 2 1 2 2 2 x 1 1 x 2 , x x 2 1 x 2 1 x 2 . , 2 1x x 2 ( 1 2 x , x ). ) f 0 1 f x 0 x 1 x 1 f x 0 0 , 1 . , : 1 1 1 1 1x x x x x x x x 0 0 0 0 0 x 1 x 1 e e x e e 1 E f x dx dx dx dx dx e x e x e x e x x1 1 11 x x 0 0 0 0 e x 1 dx dx x ln e x 1 ln e 1 e x . . 48. 20 - lisari team 48 ) x lim f x . , x x xx x x lim e x lim e 1 1 0 e , xx x x x x lim lim e e xx 1 1 lim 0 e . , x xx x x xx x 1x 1 1 1 lim f x lim lim lim 0 e x e 1 e x . , x lim f x 0 . , 1 3 2x x f x f x1 L lim lim 1 f xf x f x ( x lim f x 0 2x 1 lim f x x f x lim f x x u f x 0 ). , x u 0 f x u lim lim 1 f x u .), 1L . , 2 x ln x x xx x x x xx x xx x x 1 L lim lim lim x lim f x x e e e x e x . , f x x f x , x 0 . , f x f x x f x , x 0 . 2L 0 . 49. 20 - lisari team 49 7 ) 9 g(t) ( t t) 0, 2 , f 0, 2 9 f '(x) ( x x) . , f x 0 x( x) x 4 ( 0 x 2 ) x 0 4 2 h(x) x x - + h(x) x x 0, 4 , 0, 4 , 0 8 8 [ 2 2 1 1 2 2 2 24 2 8 2 2 2 8 2 , 2 2 1 1 2 2 2 24 2 8 2 2 2 8 2 8 8 ] x x 0 , x 0, 4 , 9 f '(x) (x x) 0 x 0, 4 f 0, 4 h(x) x x , 4 2 , 4 2 , 3 3 0 8 8 (1) [ 50. 20 - lisari team 50 3 2 2 8 2 3 2 2 8 2 (1) ] x x 0 9 , f '(x) (x x) 0 4 2 x , 4 2 f , 4 2 : x 0 4 2 f (x) - + f f : f 2 0, 2 , 9 8 8 f ''(x) (x x) ' 9(x x) (x x)' 9(x x) (x x) 0 0, , 4 4 2 f 4 , f 0, 2 ) f 0 0 f 0 2 [ 92 0 f (t t) dt 2 u t du dt 2 t 0 u 2 t u 0 2 , 0 9 9 92 2 0 o 2 f (( u) ( u)) du (u u) du (u u) du f 2 2 2 2 f f f 0] 2 2 2 , f 0 x f(0) f(x) f(x) 0 4 51. 20 - lisari team 51 f x f(x) f 0 f(x) 0 4 2 2 , 94 0 f (t t) dt 0 4 ( g(t) 0 0, 4 0, 4 g 0, ) 4 f(x) 0, x 0, 2 f(x) f ''(x) x 0, 2 : 0, :f() f ''() 2 9 8 0 0 0 0 0 t t dt 9( ) ( ) , f(0) f 0 2 f ''(0) f '' 9 0 2 f(x) f ''(x) 0, 2 ( : f 0 0, , 4 4 2 f 0 0, 2 , f '' 0 f 4 4 ) ) x 0, 2 . f 0,x (0,x) : f(x) f(0) f(x) f '() x x , f ' f(x) 0 x f '() f '(x) f '(x) f(x) xf '(x) 0, x x 0, 2 , 2 f(x) xf '(x) f(x) m'(x) ( )' 0, x x x 0, 2 52. 20 - lisari team 52 m 0, 2 . m 0, 2 ) 1 . m 1 f 1 1 1 f(1) 1 m m(1) f(1) f 1 fC 1 1 M ,f( ) 1 1 1 (): y f f ' x f fC () , 1 1 1 f(x) f f ' x ( 1 x ) x 1 1 1 1 f(1) f f ' (1 ) (1) xf '(x) f(x), x (0, ) 2 1 1 1 x f ' f , (1) 1 1 1 f(1) f f 1 1 1 1 f(1) f f f 1 f(1) f f(1) 1 f ) 0 0 , 0, 2 : x 9 , , (x)= (t t) dt 53. 20 - lisari team 53 , n , : 9 9 9 9 K() () '(n) (t t) dt (n n) (t t) dt n n (2) , (2) 9 9 0 n 1 0 n 1 n n 1 n n 1 (t t) dt 1 9 (t t) dt 0 0 . t , : 9 9 9 (t t) (t t) (t t) 9 9 9 (t t) dt (t t) dt (t t) dt 9 t t 1 9 9 (t t) dt (t t) dt 9 (t t) dt 54. 20 - lisari team 54 8 ) , z i, , . * z i, , 22 2 iz z i, z i, i z i i iz , , . * z , , z z , , . , * z i, , ) , 2 2 AB z z z z z z z z z z zz zz z A z z z z z A zz z z z () = (), . ) , 2 22 OA z OB z z z zz O z zz z OA OB O = , ,, . , xx. () = (), (, ) (, ), . 55. 20 - lisari team 55 ) , 22 2 22 2 2 22 22 2 2 z AB A z z z z z z z z 2 z z z z z z z 2 z z z 2 2 2 2 z z z z 2 z z z 2 z z 0 2 z z 2 z 22 22 2 2 2 2 2 2 2 z z z 2 Re z Im z 2Re z Re z Im z 2Re z Re z Im z Im z Re z z y x, y x , (0, 0), z 0 . 56. 20 - lisari team 56 9 ) z x yi, x,y , x y 0 , , 2 2 2 2 2 2 1 1 1 1 z Re z x yi Re z 4 4 4 4 1 1 x y x 4 4 1 1 x y x 4 4 y x x 0 y x, xy 0 x 0 y 0 z f x x, x 0 . , z t x t y t i x t x t i, t 0 . , 1 x t x t 2 x t , x t 0x t 1 1 Re z t Im z t x t 1 x t , 42 x t 2 x t , 1 1 y t 4 2 0t Re z , Im z 0 1 1 z t i 4 2 . ) z x x i, x 0 , : 222 2 2 2 z z 2 z z 2 i x x i x x 2 2 xi 2 i x x 2x xi x x 4 xi 2 i 2x xi 2x 8 xi 4 x 2x x 8 x 2x 4 x x 4 z 4 2i 57. 20 - lisari team 57 ) : , 2 2 2 2 2 2z 13 5 2 x x i 13 5 2x 13 2 x i 5 2x 13 4x 25 4x 169 52x 4x 25 0 4x 48x 144 0 x 12x 36 0 x 6 0 : , 2 2 2 2z 13 2 x xi 13 2x 13 4x , 2 2 2z 13 5 2z 13 25 2x 13 4x 25 g : 0, 2 g x 2x 13 4x . g 0, , g x 4 2x 13 4 8x 48 8(x 6) , g x 0 x 6 g x 0 0 x 6 H g 6, g 6 25 , x 0 , 2 g x g 6 2x 13 4x 25 1) w x yi u i , u w i x yi x y x y i u w u w x y x y i x y x y i w w w u w u 2 0 x 0 6 g(x) + g min 58. 20 - lisari team 58 2 2 x y 2x 2y 2 0 1 , , 2 2 2 2 2 2z 1313 13 13 5 u u 2z i z i 2z i z i z 2 2 2 2 2 2 u 2 2 2 0 2 , 2 2 2 2 2 2 2 2 2 4 2 4 4 8 4 2 0 (1) , 2 2 2 2 4 2 2 2 2) , 2 2 22 2 w w w u w u 2 0 w w w u w u 2 w w u u w u 2 u u u u u w u w u 2 u w u 2 u 3 , w u , 2, u u . 3) (3) 2 5 u 2 , 2 2 5 1 u w u 2 2 2 2 2 1 2 u w u w 2 2 4) , 2 2 2 2 2 2 2 2 2 2 u w u w 2 u 2 w u 2 u w 2 u 2 w u w u 2 w 2