25 askiseis algebra_a_lykeiou

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  • 1. - www.mathematica.gr : - 2011
  • 2. 1. www.mathematica.gr | : | - | 3 : : 1. 2. 3. 4. 5. 6. 7. parmenides51 mathematica.gr 2. 3. 4. 5. 6. parmenides51
  • 3. 3: 24 2011 - 22:37 f ^ x h = m x 2 - 2 ^m + 2 h x + 8 m ! R. 1) m yly A. 2) m, xlx . 3) m f^xh = 0 . 4) m, f y = 0. 5) m ! R -"0,2, B, C f xlx : ) ABC 8 2 E - . m m = ) m E = 24. 6) m = 2 f . : 4: : f (x) x ) m d R x x x - 1 6 0 0 < ao ao 2 m m $ = + - 24 2011 - 23:33 25 2011 - 13:59 1) m f (0) = f^-8h. 2) m = 3 : ) 33 f^18h . ) ^-3, f^-3hh ^0, f ^0hh. ) f xlx yly. ) f^xh # 9. ) f . ) f . 1: 23 2011 - 16:33 f x 2 x 2 x 2 2 ^ h =^m - h - m + m + m ! R -"2,. 1) f^xh = 0 m ! R -"2,. 2) m f (x) m ! R -"2, x0 2 = . 3) m = 4: ) 8 ^ h . f x 2 x #- ) f xlx yly. ) f^xh = 2x - 2. ) f y = 6. ) f^xh f^xh yly. : 2: ^ h . f x x - 2 = x 1 x 1 - 23 2011 - 17:22 23 2011 - 21:56 1) . 2) f . 3) f . 4) ( ) Cf x = 0, x =- 1, x = 1 y = 0. 5) f^xh $ 1. 6) ^-2, f^-2hh, O^0,0h ^2, f^2hh . 7) f . : 25 2011 - 14:33 : 23 2011 - 21:56 4 | - | : | www.mathematica.gr www.mathematica.gr | : | - | 5
  • 4. 7: f x = 1 ^ h . - x 4 x 26 2011 - 15:24 1) f . 2) f . 3) f (2) 1 f ( 2) 1 2 2 ^ - h + ^ - - h . 4) ( ) Cf . 5) O ^0,0h A^-1, f (- 1)h B^1, f (1)h. ^ h = 1 . 6) f x x 7) Cf 2 4 . : 8: 2 2 g ^ h = 1 - x + x + , a 1 2 a a ! - + a a 4 8 4 1) x g(a) = 0. 2) x g^2h = g^-1h. 3) g^ah > 0 6x d R. 4) 1 - + 26 2011 - 15:51 26 2011 - 21:56 + , a ! 1. a 1 > 0 1 1 a - 5) . 5: 25 2011 - 21:04 f x 2x 1 x 2 2 2 ^ h = -^m + h +^l - lh x d R l,m ! R. x = 1 f -3. 1) l m. 2) l = 1 m = 3 : ) f x 2x 2x f x 2 2 ^ h- = - ^ h. ) g g x x f x 4 2 ^ h = - ^ h+ . ) f . ) f^1 - xh = f^1 + xh. ) g . ) f^xh g Ox . : , 25 2011 - 21:55 / 22:08 6: : , 26 2011 - 01:35 / 15:16 f ^ x h = 2x 2 - 2 ^m - 5 h x -^m - 5 h, m d R. 1) D = 4^m - 5h^m - 3h. 2) m . 3) x1,x2 , m : x1 2 + x 2 . 2 4) m ! R : f^xh = f^xh 6 x d R. 5) 2x1x2 x1 x {^mh = + 2 + 2 . 2 1 6) f^xh 2 x = : ) m. ) f^xh ^1,0h ^2, f (2)h . : 26 2011 - 22:17 : 26 2011 - 02:01 / 16:43 6 | - | : | www.mathematica.gr www.mathematica.gr | : | - | 7
  • 5. 11: (irakleios) 27 2011 - 13:16 3x 4 x 12 2 2 -^l - h = (1) 1) 8 160 4 2 D = l - l + . 2) (1) l. 3) l , (1) . ; 4) x1, x2 . l x1 1 x2 x2 0 ^ + h+ 1 ; : 12: 27 2011 - 16:55 27 2011 - 19:32 f ^ x h = m x 2 +^m - 1 h x + m - 1 m ! R. 1) m f^xh = 0 . 2) m ! 0 m, f^xh > 0 x ! R. 3) f^xh = 0 x1, x2 : ) m x1 + x2 x1 $x2 . ) x1 2 $ x + x $ x 2 . 2 1 2 # 0 ) x x x x 1 2 $ + $ # m. 4 2 1 2 1 2 2 {^mh = - $ - $ . 4) m < 0, x1 2 x x x 2 1 2 5) m f^xh xlx. ^ = 1 6) m h x ^ f x h h R. 9: f f x + - - x x 1 1 ^ h . + + - x x 1 1 = 1) . 2) f x ao # ao x x 1 x x 1 > 1 ^ h = * 3) f . 4) f^xh $ 1. 1 ^ h+ ` j $ x ! 0. 5) f x f 2 x 27 2011 - 01:25 6) f . 7) y = f^xh. 8) y = f (x) . : 10: 2 1 2 ^ h = + + ! . f f x 1 , 0 x x x 1) x f x 7 x 1 8 2 ^ h- + = . 2) ) f^xh = 0, ) f^xh = 1 ) f^xh =- 1. ^ + 1 = 3) f x 0 ^ f x h h . 4) f 1 2 ` j = ^ h x ! 0. x x f x 5) ) f^ x h $ 0 ) f^ x h $ 4. 27 2011 - 02:34 27 2011 - 12:11 : 27 2011 - 20:55 : 27 2011 - 13:18 8 | - | : | www.mathematica.gr www.mathematica.gr | : | - | 9
  • 6. 15: (irakleios) 28 2011 - 16:59 : x x 0 (1) 2 a + b + c = a + b + c = 0 a ! 0. 1) (1) . 2) (1) ; 3) E 1 x 1 x 2 2 2 ^l + h =^l - l + h + l x , l, . 4) x1, x2 o (1). ) : x1 $ x2 -^x1 + x2h. ) x1 x2 + (1), : A = x1 ^ x2 2011 - 1 h+ x ^ x 2011 - 1 h. 2 1 : , 16: 28 2011 - 18:28 / 21:28 29 2011 - 04:16 f f x 1 x x 1, R 2 ^ h =^a - h + a + a + a d a ! 1. 1) a ) f^xh = 0 , ) f^xh = 0 . 2) S, P f^xh = 0 D > 0, a S = P. 2 3 3) g, g^xh = f^xh , 3 a > . 4) ) f 1 f 1 0 2011 6 ^ h ^- [email protected] = , ) ^ f^1h- 1hf^1h = 0, ) f 1 81 0 2 2^ h- a = . 5) x ! 0 ) 1 ` j, ) 1 f x x 2 ` j = a . x f x ) x : 1 f x 2 ` j = ^ h. x f x 13: 28 2011 - 02:47 f x x x 2 ^ h = a + b + c g x f 0 x f 1 x 2 ^ h = ^ h +^ ^ h- bh + a, a, b, c ! R a ! c ! 0. 1) g^xh = 0 (1) . a 2) (1) x1 1 =- x2 c =- . 3) f^xh = 0 : ) ) x2 (1). 4) f^xh = 0 t1,t2 , b t1 t2 + = t1 t2 a D - = . a : (irakleios) 14: 28 2011 - 12:14 28 2011 - 14:52 f^xh = a 2 - x, a ! R, ^-1, 3 h. 1) a = 1. 2) f . 3) : f^3 4h > f^ 3 h. 4) f . 5) f^xh = x f^xh = x. 6) g^xh = f^xh+ f^-xh yly. 2 2 + - ^ ^ hh ^ - ^ - hh 7) 1 f 1 f 4 1 1 4 3 + = . : 29 2011 - 15:43 : 28 2011 - 15:20 10 | - | : | www.mathematica.gr www.mathematica.gr | : | - | 11
  • 7. 17: f x x 3x 1 2 ^ h = - + f^xh = 0 ^1h. 1) ^1h x1, x2 . 2 2 2) : A , , = + = + C = 1 - 2 . x x 2 2 B x x x x 1 2 1 2 30 2011 - 12:43 3) Ax + By + C = 0 xlx yly. 4) ^A, Bh, ^B, Ah ^C,Ch . 5) 2x 5 x 6 0 2 - b + c = ^1h, b,c. 6) : + ^ h 0 2 ^ h - + f f 5 0 2 1 f ^ - ^ h h - + 1 3 4 f + = . : 18: 30 2011 - 01:49 30 2011 - 04:25 g x x 1 x 1, R 2 ^ h = m -^m - h - me m !- 1, m ! 0 f x x1 x x2 ^ h = - , x1, x2 g^xh = 0. 1) x1, x2 . 2) x1 < x2 me^-1,0h: ) m f^0h = f^1h. ) m f^0hf^1h > 0. ) f^xh+ f^-xh = g(1) + 4. ) g f . ) (f):y = x1 x + x2 xlx yly. 19: (irakleios) 30 2011 - 20:42 : x a 4 x 4a 0 2-^ - h - = (1), a . 1) . 2) . a ; 3) t1 = 2^1 - 3kh, t2 = 3^1 + 2kh (1): ) a. ) ^1 + 2kh^1 - 3kh. ) k. ) (1) 4) : ) 1 x 5 x 41 2 ^ + h - = . ) 2 x x5 26 2 2 2 ^ + h + = . : 20: 30 2011 - 21:59 30 2011 - 21:27 f (x) = (l - 1)x + m, k, m ! R. 1) A(0,7) B^1, 5h, k, m ! R. 2) k =- 1 m = 7 : ) N : f (x) = 5. ) : f (x) < 3. f x ( ) ) : 2 3 # 0 . 2 + - x x : 28 2011 - 15:20 : , 30 2011 - 22:16 / 22:18 12 | - | : | www.mathematica.gr www.mathematica.gr | : | - | 13
  • 8. 21: 30 2011 - 22:40 {^ x h = x 2 - m x + m - 1, m ! R . 1) {^xh = 0 m ! R. 2) A x1,x2 {^xh = 0 : ) m x1 2 + x 2 . 2 $ 1 2 + - 14 $ 1 2 2 - - - . x x x x 4 3 2 ) m : 5 3 4 + = 2 m= + - + . ) f ( ) 3 x1 2 x 2 5 2012 ) f . 1 3) { 2 x = : ) m = 1. ^ - h { x ) 1 3 h . = + 2 2 x { ^ : 22: (irakleios) 30 2011 - 23:15 01 2011 - 19:07 m 1 x mx 1 0 2 ^ - h - + = (1). 1) (1) m. 2) m (1) x1, x2 x1 x2 0 - ! ; 3) m !! 1 S, P (1) . m $ S $ P + 1. 4) m (1) 1; 5) m 20112012 = , (1) , . 23: 2 ^f h = m + ^f2h:y =^5m - 6hx + 8, m d R. 1 :y x 6 01 2011 - 21:58 1) m . 2) ^f1h ^f2h : ) J K ^f1h ^f2h yly xlx . ) m ^JKh ^JKh = 10. 3) m = 1: ) M ^f1h xlx. ) JKL . ) JKL. ) ^f3h J K. ) ^f1h ^f3h yly. : 24: 01 2011 - 23:24 01 2011 - 23:14 (;... ) 5 . Am Bm , m ! "1,2,3,4,5,. ^ m - 1 h 2 x + m - 1 Am,Bm f ^ x h = 20 h , -10 1 x 1 10, : x 2m m - + + 7 1 20 ^ g x = ^ h ) m Am,Bm ; ) m ; ) m ; ) ^1, 6h, ; ) A2 B3, ; : ^xh ^-h . : 01 2011 - 19:54 : parmenides51 02 2011 - 20:27 14 | - | : | www.mathematica.gr www.mathematica.gr | : | - | 15
  • 9. 25: parmenides51 02 2011 - 02:01 f ^ x h = m x 2 + 2 ^m - 1 h x + m + 1 m ! R. 1) f^xh m. 2) m f^xh = 0 t1, t2 , t1 + t2 =-t1 - t2 . 3) g^xh h^xh f^xh m = 0 m =- 1 , h ^ x 1 ^ g x ^ $ f - h h h. 4) m = 2 f x 8x 12x 29 2 3 ^ h # + + . 5) ^f1h: f^-1hx + f^0hy = f^1h ^f2h: f^0hx - f^1hy = 3f^-1h, f^1h = f^-1h. : 02 2011 - 02:52 16 | - | : | www.mathematica.gr www.mathematica.gr | : | - | 17
  • 10. 1: 1) : 23 2011 - 17:22 23 2011 - 21:56 2 4 2 2 4 4 4 4 4 16 16 > 0 2 2 2 2 2 D =^- m h - ^m - h $ ^m + h = m - ^m - h = m - m + = m ! R -"2,, . 2) f : m > > > 2 0 * 2 * b m - + + m - b m - = = =- =- x = x =- =- x x 0 min 0 min a ^ h ^ h 2 2 2 2 2 0 2 a 2 2 2 2 m m - - = > 2 > 0 > > > > m * m ) ) 3) ) m = 4 : f ^ x h = 2x 2 - 8x + 6 , : m m m + + + = = - = - m 2 2 2 0 2 4 2 0 4 m m f ^ x h ^ h * * * * x x 2 2 2 - + x x 2 8 6 - - + - + G- G G - G - + + ++ + x ! ! ! ! x x x 2 4 3 x x x x 4 3 x x 2 8 0 2 8 0 2 8 0 8 0 2 2 2 - + x x 4 3 - ++ + + G G G 8 0 + * * * + + + + ! ! ! 0 x 4 x 3 8 x 0 0 x x 4 3 0 0 x x x x x x ^ 2 ^ ^ h h h + + + + G G $ $ $ x x x x 4 3 0 ) ) @ 6 h ! 0 ! 3) ) x = 0 & f^0h = 6 & Cf + yly = K^0, 6h . f ^ x h = 0 + 2x 2 - 8x + 6 = 0 + 2 ^ x 2 - 4x + 3 h = 0 + x 2 - 4x + 3 = 0 + + Cf + x l x =" A1 ^ 1, 0 h , A2 ^ 3,0 h, . 3) ) : f ^ x h = 2x - 2 + 2x 2 - 8x + 6 = 2x - 2 + 2 $ x 2 - 4x + 3 = 2 $ ^ x - 1 h + x 2 - 4x + 3 = x - 1 + x x x x 1 3 0 ! ^ - - - f 3 , + + + 0 , 3 1,0 x 2 - + = - x x x x 4 3 1 ^ Z ]]] ]] Z ]]] ]] ) ) @ 6 @ 6 @ 6 - + - + = ^ - + = = ^ h h ! 3 , 1 , 3 , 3 3 3 3 3 2 Z ) 0 - + - = - x x x x 4 3 1 1 , 3 5 4 0 , , , 2 x x x 0 f ! [ + + 2 - + = x x x 3 2 0 1 , 3 x , x x 1 4 - + , , , 1 3 0 ! = = ar t x o x x 1 ( .), 2 1 , 3 1 3 ! ! ! ^ ^ ^ h h h h ]]] ]] [ ] ] [ ] ) ) ) 3) ) : f ^ x h > 6 + 2x 2 - 8x + 6 > 6 + 2x 2 - 8x > 0 + 2x ^ x - 4 h > 0 + x ! ^-3 ,0 h,^ 4, +3h 3) ) b b M c- , f c- mm " ^ 2, f ^ 2 hh " M ^ 2, - 2 h 2 a 2 a yly ( ) K^0,6h. ( ) () : ^fh:y = ax + b:^1h M^2, - 2h ! ^fh + 2a + b =- 2:^3h K^0,6h ! ^fh + b = 6:^3h. (2) (3) () b = 6 a =- 4. y =- 4x + 6 . 2: 1) : ) ) ) = -"- , , , x x x x x x A R ! 0 1 0 0 1 0 1 ! ! ! 2 2 f 1 0 1 + + & & ! - ! ! & Af =^-3, - 1h,^-1,0h,^0,1h,^1, + 3h - ^ h ^ h ^ h ^ h ^ h = 2 ! -3 - , - , , + 3 & = x , 1 1,0 0,1 1, fx 1 x 1 x x - - - !! ^ h = ! , x x x x 2 2 1 & = = 2 f $ ^ x x 1 1 x f x x x A 1 1 1 2 - - h Af f . 2) f , x ! Af &- x ! Af 1 1 f ^ - x h = =- =- f ^ x h, f . - x x 3) x1,x2 ! ^-3, - 1h 1 0 1 1 x x 1 2 - & & ^ h ^ h & 1 < 2 < < > f x > f x f x x 1 2 ^-3, - 1h. x1,x2 ! ^-1,0h 0 1 1 x x 1 2 & & ^ h ^ h & 1 < 2 < > f x > f x f x x 1 2 ^-1,0h. * : 1 0 0 1 1 2 - & - & ^ h ^ h & , f 1 < < 2 < > > 1 > f x > f x f 1 x x x x 1 2 ^-3, - 1h,^-1,0h . x1,x2 ! ^0,1h 0 1 1 1 2 & & ^ h ^ h & ^0,1h. < x 1 < x 2 > f x > f x f x x 1 2 x1,x2 ! ^1, + 3h 0 1 1 1 1 2 & & ^ h ^ h & < < x 1 < x 2 > f x > f x f x x 1 2 ^1, + 3h. ** : 0 1 1 1 2 & & ^ h ^ h & , f < 1 < < 2 > 1 > f x > f x f 1 x x x x 1 2 ^0,1h,^1, + 3h . *** !!! f ( ! 1 ^-3, - 1h,^-1,0h ^0,1h,^1, + 3h ), 1 1 x 1 < 0 < x & < 0 < & f ^ x h 1 < f ^ x h. 2 x x 2 1 2 18 | - | : | www.mathematica.gr www.mathematica.gr | : | - | 19
  • 11. 2: 23 2011 - 21:56 24 2011 - 23:33 4) -1,0,1 ! Af Cf x =- 1,x = 0,x = 1, f ^ x h = 1 ! 0, 6x ! Af , Cf x y = 0 ( xlx). 5) 1 ^ h H * * ) f x 1 x 1 H H G x x x x 1 1 + + + + ! , ! ! ! , ! , ! x 0 1 0 1 1 0 1 - 1 G x G 1 - ) ) ^- h ^ h . + + + ! ! ! 0 1 , , , , x x 1 1 x x < < 0 1 0 0 1 ! 1 ^ f x x = h 6) 2, 2 2, 1 hh j 2 ^ ^ ` - - & - - A f A 1 ^ f x x = h , 2, 2 2, 1 hh j 2 ^ ^ ` B f & B O^0,0h . () O^0,0h 2, 1 B` j. 2 () O^0,0h yly ( O B ) ^fh:y = mx:^1h 1 1 1 1 ^ 1 h B 2, 2 :y x : ` ! ^f & = m & m = & ^f = ^ 2 j h h h 2 4 4 2 1 2 1 - = ^- h +- =- A 4 2 1 2 1 2 ^fh ) OB, O, A, B . 6) 2 - : f A^-2, f^-2hh, B^2, f^2hh O^0,0h, f^-2h =- f^2h. A, O, B . 7) : 3: 1) f^0h = 8 & Cf + yly = A^0, 8h = ct . 2) 2 2 32 4 2 32 4 2 8 4 2 0, R 2 2 2 2 D = 6- ^m + [email protected] - m = ^m + h - m = 6^m + h - [email protected] & fD = ^m - h H 6m ! f^xh = 0 x1 , x2 ^ + h ^ - h ^ + h ^ - h ! ! m m m 2 2 4 2 2 m m 2 2 2 2 m 2 2 & = = 4 h h * $ . ^ ^ ^ ` h h B , h , C , j Z [ x 1 x x x = = & & + l = C xx ^ ^ + - - m m 2 2 2 2 m 2 + + - m m 2 2 2 2 m 2 4 m 2 f 2 0 0 2 1 2 m = = ]] ]] f xlx 4. c 3) f^xh = 0 : 0 8 m . > " > 0 & > 0 a m 4) f xlx : D = 0 + 4 ^m - 2 h 2 = 0 + m = 2 . 5) ) 1 8 x x 4 4 - ^ h ^ h ^ h ^ h ABC = OA BC = - = - = ABC 2 1 m = 2 1 $ $ $ $ $ & f $ 2 2 4 2 4 8 2 m m - m m ) : m - 2 2 m =- = ) * - = - =- m m m m m m - = = - = + $ + + + + 1 24 8 24 3 m m 2 3 2 3 2 3 1 2 m m E = 6 b 6) m = 2 : f x 2x 6x 8 2 ^ h = - + 2 4 3 - = = , m > 0 : 2 a `-3 B f , 3 2 3 8 , + 3j f 2 3 2 3 x0 = 2 7 f` j = f . 2 20 | - | : | www.mathematica.gr www.mathematica.gr | : | - | 21
  • 12. 4: 25 2011 - 14:33 25 2011 - 21:55 1) f 0 f 8 6 9 6 9 0 3 0 3 0 3 2 2 2 ^ h = ^- h + m - m = + m - m + = +^m - h = + m - = + m = . x x - + 2) m = 3 : 1 , 0 ^ h ) , : x x < H 9, 0 f x = ) f^18h = 18 + 9 = 27 & 3 f^18h = 3 27 = 3 & 3 $ 3 f^18h = 9 = 3. ) A^-3, f^-3hh " A^-3, 4h B^0, f^0hh " B^0, 9h AB 3 0 9 4 AB 34 2 2^ h = ^- - h +^ - h & f^ h = ) : f^0h = 9 & Cf + yly = K^0, 9h ^ h 0 ) ) f xlx. - = + = x x 1 0, 0 + + x x ar t = =- < > < H H 9 0, 0 x o x x o x 1 0( .), 0 9 0 ( .), 0 f x < < ar t = ) : 9 x G - 1 9 H Z ]]] ]] ) < < 0 x x x x 6 h 6 @ Z ]]] ]] [ [ ) * ^ h - x ! - - G + 0 + 0 + 0 + ! f x 8,0 G 9 9 0 8 0 0 0 8,0 0 x x x x x + H G H = ] ] ) ) ) x < 0 f -1, f ^-3,0h. x H 0 f 1 f 60, + 3h. ) : 5: b m = m + + 1) : 2 4 1 1 xmin =- = m = = " & & f x x 1 3 4 1 min min a m= =- 3 ^ h ^ h 2 . 3 2 minf = = = fl - l - & l - l - =- min f f x min f 1 2 2 2 2 3 2) ) 1, 3 f x 2x 4x 1 2 l = m = & ^ h = - - , : f ^ x h= 2x 2 - 4x - 1 f x 2x 2x f x f x 2x f x 2x 0 2 2 2 2 ^ h- = - ^ h =-^ ^ h- h + ^ h- G 4 1 0 1 4 + f- x - G + f x H- ) : g x x 2x 4x 1 4 g x x 4x 5 2 2 2 ^ h = -^ - - h+ & f ^ h = - + + g , : x 4x 5 0 x 1,x 5 2 - + + = + =- = 6 @ x 4x 5 0 x 1,5 2 - + + H + ! - c- - E ^- @ ) f , b 3 " 3,1 2 a b ;- , + 3 m " 6 1, + 3 h . 2 a ) : f ^ x h = 2x 2 - 4x - 1 = 2x 2 - 4x + 2 - 3 = 2 ^ x - 1 h 2 - 3 & 2 2 2 ^ ^ ^ ^ ^ h h h h h - = - - - = - - = - + = + - - = - f x x x x f x x x 1 21 1 3 2 3 2 3 1 21 1 3 2 3 & ) & f ^ 1 - x h = f ^ 1 + x h = 2 x 2 - 3 . 2 2 ) g^0h =f 5 & Cg + yly = K^0, 5h 2 2 - x + x + = - - x + x + = - ^ h ) ) =- = x x = g x 0 1, 5 4 5 0 + + + & G G G G x 1 5 4 5 0 x 1 5 & Cg + xlx = "K^-1,0h,M^5,0h, . ) N^1, - 3h N M. ^fh:y = ax + b:^1h ( N M). : b ^ - h ^ h + =- ^ - ^ h ^ h ! ! f f + =- = =- h N 1, 3 a b 3 ^ h = - 15 M 5,0 a b ) ) ) * + + + f & y x + = 5 0 3 a b a 4 3 15 4 f 3 : 4 3 4 4 a = 22 | - | : | www.mathematica.gr www.mathematica.gr | : | - | 23
  • 13. 5: 26 2011 - 02:01 / 16:43 1) f x = 1 -3 f^1h =- 3. b f x x x 2 ^ h = a + b + c 2a 25 2011 - 22:08 D - 4a - . + 4 1 = 1 & m = 3 - 2 + l 2 - 2 l =- 3 & ^l - 1 h 2 = 0 & l = 1. m 2) ) l = 1 m =- 3 f x 2x 4x 1 2 ^ h = - - . f x 2x 2x f x 4x 1 4x 1 2 2 ^ h- = - ^ h & + = + , 4 - 1 . x $ ) g x x 4x 5 2 ^ h = - + + . x 4x 5 0 x 1,5 2 - + + $ & d6- @. ) f (- 3,1] , [1, + 3) ^1, - 3h. ) f x 1 f 1 x 2x 3 2 ^ + h = ^ - h = - . ) x = 0 g^0h = 5 . g yly ^0, 5h. y = 0 x 4x 5 0 x 4x 5 0 x 1 x 5 2 2 - + + = &- + + = & =- 0 = . g xlx ^-1,0h,^5,0h. ) y = ax + b . A^1,- 3h B^5, 0h , . 3 + =- + = a b a b 3 , ) 3 4 5 0 - 15 . 4 a = b = 6: 1) 2 5 4 2 5 4 5 3 2 D = 6- ^m - [email protected] - $ $ 6-^m - [email protected] = f = ^m - h^m - h. 2) : ^ ^ h h m - m - = +m = m = = 4 5 3 0 3, 5 a ^ ^ h h D + m - m - + > 0 4 5 3 > 0 4> 0 fm ! ^-3,3h,^5, +3h . 3) : x1 2 + x 2 = 0,75 +^ x + x h 2 - 2 1 2 2x 1 x = 2 0,75 b + =- + = - m x x " x x 5 1 2 1 2 a c m = =- $ " $ x x x x - 2 5 1 2 1 2 a 2 3 ^m - h + m - = + + 5 5 4 3 m- =n ^ h 2 2 + n + n - = + n + n - = + 4 0 4 4 3 0 5 :1 3 * m- =n 2 * * . n m =- - =- f + + f 1 2 5 3 2 - = 5 1 2 7 2 11 2 5 n m m m = = = 4) : 2 0 = + m = m = = D a > 4> 0 a = f ^ x h = f ^ x h + f ^ x h H 0, 6 x ! R + D G 0 + 4 ^ m - 3 h^ m - 5 h G 0 + fm ! 63,[email protected] 5) 2x1x2 x1 x x x x x 0 3, 5 z^mh = + 2 + 2 & z^mh = ^ + h 2 & z^mh = + &^ h = - & = . 2 1 2 1 2 z m m 5 A R z 6) ) f b m - = - b 2 a 2 m 5 - =- m = = & &f x 6 a 2 2 5 1 2 . b ) 6 f x 2x 2x 1 2 m = & ^ h = - - 2 1 b ` j - = =- a 2 2 1 2 3 2 f " b & c c ` mm j = - - = - K , f K 1 , a 2 2 3 2 a K^1,0h,M^2, f^2hh " M^2,3h . () K M. () y = ax + b:^1h ( K M ). ^ h ^ h ^ - - = $ - ^ h ^ h K + = : 1,0 ) ) ) ) 2,3 0 a b a b ^ ^ h = - h f f + = = =- = h + + + & & + = 2 3 0 a b a 3 3 3 5 2 3 1 2 3 y x K : 3 3 M b a ! ! ! f f , , . 24 | - | : | www.mathematica.gr www.mathematica.gr | : | - | 25
  • 14. 7: 26 2011 - 22:17 - - - ! ! ! ! ! H # ## 26 2011 - 15:51 $ ! ! , x x 1) : 4 0 ) ) ' =^- h ^ h . x 4 0 x x x 0& 4 4 x x 0& 4 4 4 4,0 0,4 + + & Af x =^- 4,0 h,^ 0,4 h A f 2) x ! A - f & x ! A f 1 - = ^ - h = & ^ h ^ h, - -- x 4 x 1 - x 4 x f x - =- f x f x x x =- f . 3) K f 2 1 f 2 1 f 2 1 f 2 1 f 2 1 f 2 1 2 2 = ^ ^ h- h + ^ ^- h- h = ^ h- + ^- h- = ^ h- + - ^ h- = 2 1 K f 2 1 f 2 1 2 2 2 = - + - - = - + + & = ^ ^ h- h + ^ ^- h- h = . 4 1 2 4 1 1 2 4 2 4 4) 0 g Af & Cf yly. ^ f^xh ! 0,6x ! Af & Cfh xlx. 5) f A^-1, f^-1hh,B^1, f^1hh O^0,0h A, O, B O AB. 6) : 1 1 1 = - = - = = ^ h * ) ) ) = x x x x x x x x ! = - f x 3 + + + + + x 4 ! ! ! ! ! ! ! 0, 4 4 1 0, 4 4 1 0, 4 3 0, 4 x x x ! 7) f^xh =- x ^-4,0h,^0,4h. 1 ^ h * * ^ h . 4 =- - f x x x x ! =- - - ! =- + + & ! ! 0, ! 4 ! x 0 < f x x 2 x x x 4 0 1 0, 4 x x f . 8: 1) 0 1 2 ^ h = g x x x x 0 2 2 a + a $ + $ + 2 - + a a 4 8 4 = - + + 0 1 2 1 2 a a = - + + - 1 1 1 1 0 1 2 2 2 + a $ + $ $ = - + - + = ^ h - + + a a 1 x x 0 x x : 1 a - a ] 1 : 1 4 1 3 1 < 0 2 2 2 D = - a - - a =- - a , R. 2) : g ^ 2 h = g ^- 1 h + x 2 + x + 1 = 2 x 2 + x + + x 2 + x + = x 2 + x + + f x 2 = + x =! 1 2 2 2 2 4 2 1 2 1 2 2 3) : 2 a! 1 1 x x x x x x 1 0 2 2 a + f + > > 1 1 > 2 - + - a a 2 - + a a 4 8 4 1 2 1 2 2 - + - a a - - + - + f x 1 x 1 x 1 2 2 ^ h = - a + - a + a ] 1 1 a! D = 1 - a 2 - 4 1 - a 2 =- 3 1 - a 2 < 0 , f^xh > 0,6x ! R 1 - a 2 > 0 . 4) 1 a- a- = -a -a =~ - + a a ~ ~ + - + - + + + a + + 1 1 1 0, 1 1 > > 2 1 0 2 1 > 0 1 1 1 > 0 1 > 0 a - : D=- a= 3 < 0 , 1 > 0 ^ ~ h = ~ 2 + ~ + & ^ ~ h 6~ ! ^ +3 h 3 . f 1 f > 0, 0, R 5) 1 - + + R e a ! 1 a 1 > 0 1 1 a - . M 4 5 ( ). . M x > 0 1 + x $ 2 . a - 1 = x > 0 : x 1 - + + . a 1 $ 3 > 0 1 1 a - 26 | - | : | www.mathematica.gr www.mathematica.gr | : | - | 27
  • 15. 9: 27 2011 - 02:34 1) x + 1 + x - 1 ! 0 + x ! R : + H - H x 1 0, x 1 0 + + - = + + = - = + f =- x 1 x 1 0 x 1 x 1 0 1 1 6x ! R & x + 1 + x - 1 > 0 & x + 1 + x - 1 ! 0, Af R = 2) : x x + = + - = - x x x x ^ + - - 27 2011 - 02:34 x x H G - + - h ) ) ^ h = ^ h = ^ h G + G G & & & & f f x x x 1 1 x 1 : 1 1 0 1 0 1 1 1 1 f x 1 1 1 1 + + - x x 1 - > > < < 1 1 1 1 - + + - x x x x 1 0, 1 0 1 0, 1 0 - = - + = + + =- - - = - x 1 x 1, x 1 x 1 x 1 x 1, x 1 1 x x x x > > > < < ) ) ) + + & & - ^ + - - x x h 1 1 1 + + - x x 1 1 x h h h ^ ^ ^ - - - - x x = 1 1 1 - - + - x 1 1 x 1 , 1: 2 ^ f x ^ f x x = & f x x > x = = = h h Z [ ]] ]] ^1h, ^2h : ao G ao x x , 1 1 , 1 ^ h = * . f x x x > 3) f R =^-3, +3h x ! R &- x ! R - + - - - x x 1 1 - + = - - - = + ^ h = ^ h ^ h f . (2 ): : x ! 6-1,[email protected] &- x ! 6-1,[email protected] f^-xh =- x =- f^xh 1 1 x ! ^-3, - 1h,^1, +3h &- x ! ^-3, - 1h,^1, +3h - - + =- + - - - = & - + + - - x x 1 1 x x 1 1 - + + x x 1 1 x x 1 1 + + - x x 1 1 f x - =- f x f x x 1 x 1 , x 1 x 1 ^ - h = =- =- f ^ x h, f x x x - x ! R &- x ! R f^-xh =- f^xh . 9: 4) x x x Z " = = x H G Z ]]] ]]] Z ]] ]] ]] ]] ) " ^ h = H x ! ! f x 1 1 1 1 + + + + 1 H 1 1 1 [ G G 1 1 1 1 1 adyoax o ( ) x x x x x x > ) > > [ [ * ) 5) 1 1 1 H & ^ h = & ^ h+ ` j = + x fx x f x f x x x < x < & f ^ x h = & ^ h+ ` j = + = + , 0 1 1 1 1 1 x f x f x x x x x x R* ! 1 1 ^ h+ ` j = + . f x f x x x : > 2 = 2 1 1 1 2 2 x 0, x x x + + x 2 2 + H + H + H + + H + +- H + x x x x x 2 2 2 1 2 1 2 0 x x x x 1 0: 3 2 + ^ - h H ^ h . (3) . 6) x G 1 & f^xh = x ( 1 3 ), f 6-1,[email protected] 1 1 a > & ^ h = ( g x , 1 x fx x ^ h = a = ) x ^-3, - 1h ^1, + 3h (!!! ). 7) 8) 1 2 . 28 | - | : | www.mathematica.gr www.mathematica.gr | : | - | 29
  • 16. 10: f : ( ) 27 2011 - 16:55 2 2 + + x x ` x + j . 2 1 1 = = $ ! x 2 f x 0, 0 x x x + 2 2 1 2 1) x f (x) 7 | x 1 | 8 x ` j - 7 | + 1 | = 8 - + = + + x x 27 2011 - 13:18 + (x + 1) 2 - 7 | x + 1 | - 8 = 0 + | x + 1 | 2 - 7 | x + 1 | - 8 = 0 y = x + 1 : y 2- 7y - 8 = 0 , y = 8 y =- 1. x + 1 = 8 + x + 1 =! 8 + x = 7 x = 9 x + 1 =- 1 . 2)) f (x) = 0 x ! 0 f (x) $ 0,6x ! 0 + + f (x) 0 0 0 1 ` j = =- . = + + + x x 1 2 1 x x x = ) ( ) 1 1 x x 2 + + ` j = + = + = = + = + + + f x 1 1 | 1 | | | 1 1 x x x x x x x + 1 =- x + x =- . ) f (x) =- 1 . 1 2 3) f x + 1 x x j j . x ! 0 x !- 1. h ` ` ^ + = + = 0 f x x 2 2 x 1 1 0 + + ^ h : 1 x + = 0 1 x + x = 0 x =- 1 x = 0, . x 4) 1 1 N 2 2 2 + + J KKK ^ OOO 1 1 h f c x m = ^ h x 2 f x 1 x x = + = x x x x 1 2 2 = L ^ P h 5) ) x ! 0 : ^ x h c m , x ! 0. ) x ! 0 : 2 + $ + $ 0 0 x x f 1 1 2 + 2 2 2 $ + $ + $ + # f x x x x ^ h c m ^ + h - - x 4 4 1 4 3 2 1 0 x x -1 , : | x | 1 3 1 - # # 1, x # 1 +- 1 # x # 1. 3 x ! 0, : x ! [- 1,0) , (0,1] 11: 1) 3x k 4 x 12 1 2 2 -^ - h = ^ h 3x k 4 x 12 3x k 4 x 12 0: 2 2 2 2 2 -^ - h = + -^ - h - = ^ h : k 4 4 3 12 k 4 144 k 8k 16 144 k 8k 160 2 2 2 2 4 4 D = 6-^ - [email protected] - $ $ ^- h =^ - h + = - + + & D = - + 2) k 2 4 h 2 0, k R - H 6 ! ^ k 4 8k 160 k 4 8k 16 144 ^ k 2 4 h 2 144 144 > 0 D = - + = - + + = - + & D H , (2) R. 3) : minD = 144, k 4 0 k 4 0 k 4 k 2 2 2 2 2 ^ - h = + - = + = + =! k =! 2 3x 12 0 x 4 x 2 2 2 - = + f = + =! . 4) : k 4 ^ k2 & =- - = 3 2 b b a h c ^ m h + =- + = - x x x x : 3 3 4 1 2 1 2 a =- = 12 3 c c a x1 x2 x x 4: 4 ` j ^ h. = =- 1 2 $ & $ a : ^ ^ h h $ ^ + h + + + + +^ + h + + x1 1 x2 x2 < 0 x1 x1x2 x2 < 0 x1 x2 x1x2 < 0 3 , 4 2 2 - k k - - - . + + + + + + 3 4 2 2 2 k > 0 < < k < k < k < k < 4 0 3 4 4 4 12 16 16 4 30 | - | : | www.mathematica.gr www.mathematica.gr | : | - | 31
  • 17. 12: 27 2011 - 20:55 1) f^xh = 0 : = ^ h f x 0 m = 0 f ^ x h =- x - 1 + - x - 1 = 0 + x =- 1 27 2011 - 20:55 , m . m ! 0 ( ) 0 1 4 1 0 1 1 4 0 2 D = +^m - h - m^m - h = +^m - h $ ^m - - mh = + m m ^ - h ^ - - h = * . = =- + m 1 $ 3 m 1 0 + f 1 1 3 2) m ! 0 m m > > > m Z [ ]] ]] * * . 6 ! f m ^ h f x > 0, x R 1 0 & + + + + > ^ ^ h h D 0 - - - - 0 & $ m m 1 3 1 0 0 & 1 1 3 < < m m > < 3) f^xh = 0 : ^ ^ h h - - - * 8 j @ D m 0 $ H m m 1 3 1 0 ! - f , ^ + & + ,0 0,1 0 1 3 ! H m b - a m - - ) 1 1 m + =- + = c - a m m - ^ h 1 1 x1 x2 x1 x2 : 1 & m m = = ^ h x1 x2 x1 x2 : 2 $ & $ m m h h m 8 j @ ^ 1 , ^ 2 c ^ h m - m - m - 2 m ) 2 2 1 1 1 1 - H 0, 6 ! , 1 ,0 0,1 3 2 2 2 ^ c + = + =- =- + - m m m $ $ $ & $ $ G x1 x 2 x 1 x 1 x 1 x 2 x 1 x 2 x 1 x 2 x 1 x 1 0 m m m 1 1 - m m - c Z ]] ]] ) : & Z 2 2 1 4 G G G G - m m - m m + c ^ m m h [ ]] ]] [ Z ]] ]] [ $ $ + + + + 8 , 0 j , ^ 01 , @ ! 8 - j , ^ @ ! 8 - j , @ 1 1 4 & , , 1 1 2 2 & , , 2 x x x x 4 1 3 1 3 0 01 2 1 3 ^ 0 01 1 2 2 1 1 ! m m m - Z [ Z [ 2 2 2 2 - + m m m 2 4 2 G & , , G - + m m 4 2 0 & , , G - + m m 4 4 2 & , , Z [ ]] ]] ]] ]] ]] ]] + + + + 1 8 j ^ @ 8 j ^ @ 8- j ^ @ 3 ! ! - , , , 0 01 1 3 0 01 1 3 0 01 ! m m m - Z [ 2 2 G h 2 2 & G G m m m - - - ^ ^ h Z [ ]] ]] ]] ]] Z [ ]] ]] + + + + 1 ,0 0,1 8 @ 8 j @ 8 j @ 3 2 2 & 1 ,0 0,1 ! ^ ^ ! - - ^ , , , 3 2 2 & 1 ,0 0,1 3 ! m m m - j Z [ - - G m G 2 2 2 & - + G G G G m m - ]] ]] Z [ ]] ]] + + + 2 2 1 1 ,0 0,1 8 j ^ @ 8 j ^ @ 3 2 2 2 2 & 1 ,0 0,1 ! - , , 3 ! m m - . 12: 4) ^ ^ h 2 2 2 h + =- $ $ z m = - x1 $ x - x $ x = - x $ x + x $ x & 2 1 1 1 2 2 1 1 1 2 x1 x x x 2 2 1 1 2 - m m c m m m - - 8 j j @ @ ` m! - , ^ m &m! ^ ^ h = c m ^ h = ^ h = - m ^ & z m & & ! 1 , 01 , 1 2 , , , , 3 0 0 1 > 0 0 1 1 1 1 m z m m z m m @ z^mh ( 1 ^ h = ) . z1 m m 5) f xlx 0 1, 1 D = + fm = m =- . 3 6) ^ = 1 h x ^ f x h h ft~x . 2 ^ h R : f ^ x > 0, x R ^ 1, h h 6 ! + m ! +3 . 32 | - | : | www.mathematica.gr www.mathematica.gr | : | - | 33
  • 18. 13: (irakleios) 28 2011 - 12:14 28 2011 - 15:20 1) f (0) = c f (1) - b = a + b + c - b = a + c , g(x) x (a )x a 2 = c + + c + . g^xh = 0 (a ) 4a (a ) > 0 2 2 D = + c - c = - c a ! c, g^xh = 0 . - + - (*) ! ! D c c b a a ( ) ( ) - 2) x 2 2 a 1,2 c = = - + + - c c a a ( ) ( ) 2 c = = - - + - - c c a a a a ( ) ( ) =- 2 x 1 1 c c 2 - 2 x = 2 c =- c c = (*) !| x |=! x. , . 3) (1) , Vieta b = 0 ax2 =-c c c x = - , . 0 a ! - , x2 . a > a $ c, . , D > 0 -a $ c > 0. b 4) Vieta a b t1 + t2 =- , a b t1 + t2 = - = . a D - = : 2 a t1 t2 2 + 2 - = . D 1 2 2 t t t t 1 2 a b + 2 - = - Vieta S 2 - 4P = S 2 - 4P , . 2 t t t t ( ) 4 4 c 1 2 1 2 2 a a 14: =a - ^ h f x 2 x 1) ^ - 1, 3 h ! C + f ^ - 1 h = f 3 + a 3 = 3 + a = 1 . 2) : 2 - x H 0 + x G 2 + Af =^-3,[email protected] . x1, x2 ! ^-3,[email protected] x1 < x2 2 x1 > x2 2 2 x1 > 2 x2 0 2 x1 > 2 x2 f x1 > f x2 G &- - H- & - - H & - - & ^ h ^ h . f ( ). f . 3) 16 < 27 & 4 2 < 3 3 & 6 4 2 < 6 3 3 & 3 4 < 3 & f 3 4 > f 3 ^ h ^ h . 4) a = 1 & f^xh = 2 - x f^0h = 2 & Cf + yly = M^0, 2h f^xh = 0 + 2 - x = 0 + x = 2 & Cf + xlx = N^2,0h . , ( ) : MN 2 2 MN 6 2 2 ^ h = ^ h + & f^ h = . = ^ h f x x - = x x 2 Z 2 - = ]] ]] * * 5) & & & = ^ h f x x - = ) + + + + - = x x x x 2 - = x x x x x x 2 2 0 2 0 2 2 H H [ ) 2 + - = = =- x x , + ) + ) + = x 6) : f^xh = 2 - x, x G 2,& f^-xh = 2 + x, x H- 2 & Ag = 6-2,[email protected] ( ). x ! 6-2,[email protected] +- x ! 6-2,[email protected] g^-xh = f^-xh+ f^xh = f^xh+ f^-xh & g^-xh = g^xh , g yly. x x 1 2 H H x x 2 0 0 0 1 1 1 2 2 + - ^ ^ hh ^ - ^ - hh + ^ - h - ^ - h 7) : 1 f 1 f f f 4 1 1 4 1 1 2 2 + = + = + 1 3 - + - = . = f 3 2 1 3 2 1 3 2 3 1 2 + = + 34 | - | : | www.mathematica.gr www.mathematica.gr | : | - | 35
  • 19. 15: 28 2011 - 18:28 28 2011 - 21:28 1) : D = b2 - 4ac = b2 - 4a^-a - bh = b2 + 4a2 + 4ab =^2a + bh2 $ 0, . 2) D = 0 +^ 2 a + bh 2 $ 0 + b =- 2 a. b - = 1 . 2 a 3) ( 1)x ( 1)x ( 1)x ( 1)x 0 2 2 2 2 2 2 l + = l - l + + l + l + - l - l + - l = ( 1) ( 1) 0 2 2 l + - l - l + - l = , 1 0 2 l + ! . 1o . l 2 - l + 1 =- 2 ^l 2 + 1 h + l 2 - l + 1 =- 2 l 2 - 4 l - 2 + 3 l 2 + 3 l + 3 = 0 . ( l 2 + 1)x 2 = ( l 2 - l + 1)x + l l. 4) ) 1 (1), x1 . x1x2 - (x1 + x2) = x1x2 - x1 - x2 = x2 - 1 - x2 =- 1 . ) x1,x2 (1) x1 + x2 (1), x1 = x1 + x2 + x2 = 0 x2 = x1 + x2 + x1 = 0 . . c = 0 b =-a a x 2 - a x = 0 + a x ^ x - 1 h = 0 , . x1 = 0 x2 = 1 x1 = 1 x2 = 0 . A = x1 (x2 2011 - 1) + x (x 2011 - 1) 2 1 A =- 1. 0 a+b+c= 15: 1) f x x x , 0 f 1 f 1 0 2 ^ h = a + b + c a ! & ^ h = a + b + c & ^ h = , 1 (1) (1) . 2) (1) ( 1 ) 1 : x x x 1 , x R x x x 2 x 0, x R 2 2 2 2 a + b + c = a^ - h 6 ! + a + b + c = a - a + a = 6 ! + f^b + 2ah $ x = a - c:^2h,6x ! R (2) x, : + = - = + + = b a a c a b c 2 0 * 0 ) ) 0 + = = b a a c =- = b a c a 2 0 2 + + 3) 1 x 1 x 1 x 1 x x 1 x 1 x x 1 0 2 2 2 2 2 2 2 2 2 ^l + h =^l - l + h + l +^l + h =^l + h - l + l +^l + h -^l + h - l^ - h = ^ 2 h x ^ x h ^ x h ^ x h 6^ 2 h x @ * = = l + ! 6l! + - - - = - + - = l + l 1 1 l 1 0 + 1 $ l 1 l 0 + 1 1 R 2 t t 1 0, 1 2 2 l + > R 2 2 h l- 1 H 0, l + 1 0, 6l! ^ 2 ^ 2 1 h 2 2 2 1 2 1 ^ 1 h 2 2 1 2 ^ 2 1 h 2 > 0 l + - l = fl - l + + l + = l - + l + & l + - l > 2 1 0 l l + > > > 2 + t t t t + fl 1 l & 1 & & ! 2 1 2 1 2 1 l + . 1 x 1 x 2 2 2 ^l + h =^l - l + h + l l. 4) x1 1 = : 1 x1 = ^ h , x2 1 = . ) x1x2 - x1 + x2 = x - 2 1 - x =- 2 1 ) x1 + x2 & 1 + x2 (1) 1 + x2 = 1 + x2 = 0 , 1 + x2 = x2 + 1 = 0 ... (). ** . x1 = 1, x2 = 0 , ( ) ( x1, x2 ) , : x1 = 1, x2 = 0 A = - x =- 1 1 . 36 | - | : | www.mathematica.gr www.mathematica.gr | : | - | 37
  • 20. 16: 29 2011 - 15:43 29 2011 - 15:43 1) D = a2 - 4^a - 1h $ ^a + 1h = a2 - 4^a2 - 1h & fD = 4 - 3a2 . ) f^xh = 0 : D < 0 + 4 - 3 a 2 < 0 + fa 2 > + a 2 > + a > + a < - a > + 4 3 4 3 2 3 3 2 3 3 2 3 3 + a ! c-3 , - m , c , +3m . ) 2 3 3 2 3 3 0 2 2 2 - - ) * * * * a H G G G G G 4 3 0 a a a a a a a + + f + + + + ! 1 ! ! ! ! 4 3 1 4 3 1 2 3 3 1 2 3 3 2 3 3 1 H a a D 2 3 + a ! ;- m , c E . 3 ,1 1, 2 3 3 a D > + fa ! c- m , c m 1 2) 0 2 3 3 ,1 1, 2 3 3 = + =- S x1 x2 S & a - + 1 . 1 = = p x1 x2 p $ & a a - 1 a a = + - 1 1 c H + h h - h H a a a a a a - a ^ ^ ^ a a a a - = h h ^ ^ c c + - - a a - - - = + - + m Z [ ]] ]] Z ]]] ]]] [ Z ]]] ]] [ + + + + c c m m , 1 1 , , 1 2 1 1 1 1 0 2 2 3 1 2 1 1 2 2 1 1 0 2 2 3 ! m m ] , a , , , , , = S p 2 3 3 2 3 3 a - 2 3 c c 3 m m 1 1 3 3 1 1 2 3 3 ! ! a a - - 2 2 1 ^ = + - a a a = + a a = + + =- a a a a h Z [ ]] ]] Z [ ]] ]] Z ]]] ]] [ ) 2 2 + + + + H 1 0 H c c c c c c m m ! m m m m H a a - , , , 1 1 2 3 1 1 2 3 2 3 1 1 1 2 3 a , , , , , , 2 3 3 3 3 1 1 3 3 1 1 2 3 3 ! ! a a - - ] adyo = =- 0 1( .) 1 2 a Z ]]] ]]] [ * + + ^ ^ c c h h m m - - + - 3 3 , , , 1 1, 2 3 3 ,1 1, 2 3 3 1 ( .) =- a ar t 2 2 3 c ^ m h - - + 3 o , , 1 1, 3 ! ! ! a a a * * a S = p. 16: 2 3 3) 3 a > + a ! c-3 , - m , c , +3m 1o D < 0, 2 3 3 2 3 3 x d R f x 1 x x 1 2 ^ h =^a - h + a + a + a - 1. g : 2 3 > < 0 2 3 3 , 3 + + + a +D 3 c c m m - - + H a ! 3 a ! a f x > , * * m . h c 0 3 > 1 0 , 2 3 3 2 3 3 , ^ 1, 2 3 3 3 ! a a - + + ^ h 4) ) f^1h = a - 1 + a + a + 1 = 3a f^-1h = a - 1 - a + a + 1 = a, f 1 f 1 0 3 0 3 0 0 2011 2 2011 2 6 ^ h $ ^- [email protected] = +^ a h = + a = + a = . = = ^ ^ h h ^ h ^ h * . ) 1 1 1 0 3 1 3 0 0 a a - = - = f $ f + a $ a + 1 3 1 2 2 a = a a! ^ h ^ h ) . ) 1 81 0 9 9 0 9 9 0 9 1 0 = =- 0 1 f2 2 2 2 - = - = - = - = a + a a + a a + a a + a a 5) ) 0 - a a 1 1 ` j = + + a + . x x x 2 ! & x f 1 - a a ) 1 1 ` j c + + + m = - + +^ + h = x x x x 2 2 $ = a + a a + a a a 2 a + x f 1 1 1 x x x 2 x x 6 Z [ ]] ]] * . h h + = - a a x 2 1 1 =- = f aoa adyo i x 2 ) 1 & 0 2( .) 2 + ! ao a - = ii ) 1 x - = iia x iib 1 a ) + + 1 1 1 6 ! a - , 1,1 ! a a ! - + ^ ^ h h - - + 6 3 , 3 adyo a ) ( .), , 1 1, a + ^ - a a ) 1 1 ` j ^ h c + + + m =^ - h + + + x x 2 2 $ = + a a 2 a a + x f 1 1 1 x f x x 2 x x 1 x 1 x 1 x x 1 1 x 1 x 2 2 2 2 2 + a - + a +^a + h =^a - h + a + a + +^a + h -^a - h = + 1 1 x 2 1 1 x 2 2x 2 x 1 x 1 2 2 2 2 + 6^a + h-^a - [email protected] = + 6a + - a + @ = + f = + = + =! . 38 | - | : | www.mathematica.gr www.mathematica.gr | : | - | 39
  • 21. 30 2011 - 01:49 30 2011 - 18:04 c a 17: 1) D = 9 - 4 = 5 > 0 & f^xh = 0 x1 $ x2 = 1 > 0 , x1, x2 . 2) 2 2 2 + = 3 1 x x x x 1 2 1 2 ^ x + x h 2 $ 3 , B x1 x x x 2x x B 7 1 2& & $ = + = f = = A 6 A A 1 x x x x 1 2 1 2 2 2 $ = ^ h = + = + - $ & = 2 1 2 2 1 2 2 ^ h 2 . 2 2 = + = B x x 7 x x = = x - x > $ 1 C ! x1 x2 0 1 2 1 2 1 2 x x x x x x 2x x 5 5 2 C = - = - = + - $ & C = & C = 1 2 1 2 2 1 2 2 1 2 2 = = C= A 3, B 7, 5 3) :Ax By 0 :3x 7y 5 0: 1 ^ f h + + C = + ^ f h + + = ^ h ^ 1 h 0 7 5 0 5 = & + = & =- & ^f + l = c - h m x y y yy K 7 0, 5 7 ^ 1 h 0 3 5 0 5) = & + = & =- & ^f + l = Kc - h m y x x xx 3 0, 5 3 4) A1 A,B , A2 B,A , A3 , A1 6,7 , A2 7,6 , A3 5, 5 " ^ h ^ h ^C Ch, " " ^ h ^ h ^ h, : A1A3 6 5 7 5 7 5 6 5 A A ^ h = ^ - h 2 +^ - h 2 = ^ - h 2 +^ - h 2 =^ h , 2 3 A3 . 5) t1, t2 2x 5 x 6 0 2 - b + c = , : 5 = + = + + = t t x 1, x 1 x x b 1 1 2 2 b b 1 2 b = + = ++ + = + + = = = t t Z S x x x x 2 1 1 5 f & & & & 1 2 1 2 2 2 5 2 5 2 b 5 5 6 1 2 = + = + t t x x 1, 1 1 1 2 2 + = $ = ^ ^h h = = + + = + + += = f c $ & & $ & & t t c c c p x x xx x x 2 1 1 3 1 3 5 3 3 1 2 1 2 1 2 1 2 3 1 x x x x 1 2 1 2 ]]] ]] [ ] 6) : + ^ h 0 2 ^ h - + f f 5 0 2 ^ - ^ ^ h h h = - = f 1 f 0 1, f 1 5 ^ h - + 1 3 3 - 5 3 5 + 5 3 f + = + = 4 $ + ^ 3 5 3 ^ ^ h h h - + $ 5 3 5 3 $ - ^ 5 5 3 ^ ^ h h h - + $ 5 3 5 3 + 15 3 2 - 5 15 2 + + - 15 3 5 15 2 = + = + = = + ^ h 0 2 ^ h - + f f 5 0 2 1 f ^ - ^ h h - + 1 3 4 f & + = . 18: (irakleios) 1) x = 1 . - . x2 . = 1 x2 m 2) ) E m ! (- 1,0) 1 1 < - , 1, m 1 = =- . x1 x2 m ( ) 1 1 1 = - - = + , -m . f x x x m m = = + . f (0) f 1 & (1) 1 1 m m H f^0h = f^1h 1 = 1 + , 1 = 0 (). 1 m m ) I f (0) f (1) $ 0 f (0) f (1) > 0. f (0) f (1) ! 0, 2) ) m. m ! (- 1,0) . ) g^1h = 0 : 1 1 + - + = = , x x 4 + 4 2 m m m 1 -1 . m =! , 2 2 ) g^xh- f^xh = 0 . : 1 1 + - - = ( 2) > 0 2 D = m + x x 0 2 m m m m ! (- 1,0) , . ) 0, 1 1 - ` j `- j, E . . A , B ,0 1 m m = = x n 2 2 m 40 | - | : | www.mathematica.gr www.mathematica.gr | : | - | 41
  • 22. 19: 30 2011 - 01:49 30 2011 - 22:16 1) : 4 4 1 4 4 16 8 16 16 4 2 2 2 2 D = 6-^a - [email protected] - $ $ ^- ah =^a - h + a = a - a + + a & fD =^a - h . 2) R 4 0 2 a ! & D =^a + h H (1) a ! R. (1) 0 4 0 4 0 4 2 D = +^a + h = + a + = + a =- . 3) ) p1 2 1 3k , p2 3 1 2k p1 p2 2 1 3k 3 1 2k = ^ - h = ^ + h & + = ^ - h+ ^ + h & b - + = a-a p1 p2 4 & + = - + + & + = & a - = & fa = p1 p2 2 6k 3 6k p1 p2 5 4 5 9 . ) 1 3 1 2 36 $ p p p p 9 c a a= ` j $ =- a & $ =- p1 p2 4 p 1 p 2 36 - k k 1 2 1 2 ^ ^ ^ ^ h h h h - $ + = $ = = & f $ - + =- k k 1 3 1 2 6 2 3 6 6 . ) 1 3 1 2 6 1 2 3 6 6 6 7 0 1 k k ^ - h ^+ h =- + + - - 2 =- + f 2 + - = + f * - . = = k k k k k k k 7 6 ) 1 ^ ^ h h = - =- = + = $ $ 2 1 3 1 4 3 1 2 1 9 k p p 7 ) 1 6 2 & = ` ` ` ` jj jj = - - = 2 1 3 7 6 9 * . = + - =- 3 1 2 7 6 4 k p p 1 2 & $ $ =- 4) ) 1 x 5 x 41 1 x 5 x 41 0 1 x 5 x 5 36 0 2 2 2 ^ + h - = +^ + h - - = +^ + h - - - = + 2 ^ ^ h h + - + - = + = + =- = ) ) = . ^ h cka a= & " + + x x + f + + f ! 1 9 1 4 8 x x 1 9 1 1 5 1 36 0 x =- G adyo ( .) x x x x 0 5 8 ) 0 2 2 x2 =~H 2 x 5x 26 2 5 26 4 4 5 26 0 2 2 2 ^ h ^ h + + = + + ~ + ~ = + ~ + ~ + + ~ - = + 2 G =- ~ ar t =~ ) . 2 2 + - = ~ = = = + ~ 9 ~ 22 0 + f + & + ! o ( .) x x x 2 2 0 11 2 2 ~ = M 4) ) - ( ): (2 x ) 5x 26 ( 2 x ) 5( 2 x ) 36 0 2 2 2 22 2 + + = + - - - - - - = (1) -4, x 2 4 x 2 2 + = + =! . 20: : 1) f^0h = 7, m = 7 f^1h = 5 l =- 1. 2) ) f^xh = 5 + -2x + 7 = 5. -2x + 7 = 5 + x = 1 -2x + 7 = 5 + x = 6. ) f^xh < 5 + -2x + 7 < 5. -3 < - 2x + 7 < 3 +- 10 < - 2x < - 4 + 2 < x < 5. ) : - + x 2 7 ^- + h^ + - h ^- + h^ - h^ + h 6- @ + . 2 # + # + # + e , 3 x x x x x x x 2 + - x x 2 3 0 2 7 2 3 0 2 7 1 3 0 3,1 [ 7 , ) 2 42 | - | : | www.mathematica.gr www.mathematica.gr | : | - | 43
  • 23. 20: ^ ^ h h ! A C f ^ ^ ^ h h h = - + = f l m m f x x f 1 & 0 30 2011 - 23:15 = = m l m 1) 0,7 0 7 7 30 2011 - 22:18 * ) ) ^ h ^ h ^ h ^ h ^ ^ ^ h h h ! B C f + + = - + = - + f l m l m f x x f 1 1 1 f & = -+ = 1,5 1 5 1 5 7 m = + +f h - + = 1 7 5 = =- 7 1 f + + l m l ^ 2) l =- 1,m = 7 & f^xh =- 2x + 7: ) - + = x x = 2 7 5 1 * * . 6 =- + ^ h f x 2x 7 ^ + = - + = + 0 + 0 h f x x - + =- x x 5 2 7 5 2 7 5 = =- + - ^ h f x 2x 7 7 ) f x < 3 2x 7 < 3 3 < 2x 7 < 3 3 7 < 2x < 3 7 + - + +- - + +- - - - + ^ h - -^ h x : 2 < 0 - - - - - + + f < < > > < x < x 2 5 10 2 4 10 2 2 2 4 2 - - - . ^ h - + + - = + = =- ^- + h ^ + - h f x 2 2 3 0 1 3 2 $ ! ! x x x x G 2 7 x x x , x 2 7 2 3 0 ) G 0 G 0 ) + + + 2 2 + - + - x x x x & 2 3 2 3 x x 1 3 ^ ^ h h - + $ + - = +f = =- = ^ 8 . h j + ff x ! - 3,1 , , +3 7 2 x x x x x x 2 2 7 2 3 0 1, 3, 7 2 21: 1) 4 1 4 4 1 0, R 2 2 2 D =^-mh - ^m - h = m - m + =^m - h & D H 6m ! . b - m 1 - =- a 2) x1, x2 z^xh = 0 : x1 + x2 = m : 1 - 1 1 c m = a ^ h x1 $ x2 = m - 1 : 2 ^ h. ^ ^ h h 1 , 2 ) x1 2 + x 2 =^ x + x h 2 - 2x x & x 2 + x 2 = m 2 - 2 ^- h & 2 + 2 = 2 - + 2 1 2 1 2 1 2 m 1 f x 1 x 2 m 2 m 2 . ) - - - = + + + + ^ h ^ h - -- +- x1 x2 2 x1 x2 2 1 , 2 $ + - x1 x2 x1 x2 x x 14 x x 2 4 3 2 3 + = 5 14 2 4 4 3 2 1 2 1 2 + = + + 3 5 4 m - + - m m m ~ ~ 4 1 3 2 + = + + - + = + - + = m+ =~H + + + + 3 5 14 2 4 4 2 2 3 5 14 2 4 2 0 4 2 14 3 5 4 m+ =~ 2 + ~ - + = ~ + ~ - ~ = + ~ = + ~ = H + m + = + 16 8 60 42 42 16 52 26 52 2 0 2 2 - = - =- 2 2 2 2 4 0 m m ) ) . + + m m = = 2 + =m - m+ x x 2 2 1 2 2 2 ) : f ^ h 3 x 2 x 5 2012 f ^ h 3 2 2 5 2012 m = + - + & m = m - m + - + & 1 2 2 2 f 3 2 3 2012 2 & ^mh = m - m - + . f : m - m- = +m=- m= 2 3 0 1, 3 2 ^ @ 6 h . m 2 - 2 m - 30 H + fm G- 1 0 m H 3 & A = -3 , - 1 , 3, + 3 f ) f ( x -x ) f , . 3) ) { : b m - = ap o vynn = b m m 1 a 2 2 o . . x 1 2 =- = = m = x & x & & 1 a 2 2 2 2 . ^ ^ h h = - - = 2 { & { 3) ) m = 1 : x x x 1 2 ) : = ^ h 2 2 { x = k 6 x ! = > 0 , 0 + + + + + 2 x 3 1 = + = + = + - - = k 2 2 k k k k k 2 3 2 0 1 > * 2 2 3 2 3 2 0 x 2 2 k 2 0 k =- ( ar o t .) 2 < = = = x x x x 2 = k > 6 x ! 0, 0 x ! = * ) ) = . k 2 x 1 2 2 0 1 0 1 0 + + + + + x x ! ! ! ! 44 | - | : | www.mathematica.gr www.mathematica.gr | : | - | 45
  • 24. 22: 01 2011 - 19:54 01 2011 - 23:24 1) m - 1 ! 0 + m ! 1 + m !! 1 m 4 m 1 m 2 0 2 2 D = - ^ - h =^ - h $ . (1) . m = 1 (1) x = 1, m =- 1 (1) x =- 1. m (1) . 2) (1) m ! 2 + m !! 2, D 2 0 x x + - - + m !- 2. m m m m 1 2 2 4 2 ! + ! + ! + ! m m 2 2 2 2 m 3) m !! 1 1 , - . 1 1 S m P m = - = 1 2 2 1 m m $ + + $ + + - - $ + mS P 1 0 - - m m 1 1 1 1 m - 1 1 m - 1 2 ^ - h , m. m m - + - + $ + $ + $ m 0 0 0 m m m m 1 1 1 1 - - 4) x = 1 (1) m - 1 - m + 1 = 0 + m = m, m $ 0. ! ^ - h . m m 5 ) m 20112012 = D 2011 2 > 0 2012 = - , (1) 2 2 x 1,2 = ^ h . 2012 2012 - - 2011 2011 2 m 20112012 = 2 2 = = = x 1 2 23: 1) (: 2x2 ) : 2 2 = + m m y x 6 ) ) ^ h + =^ - h + = - + y x = + m y x 6 2 + + m m x x 5 6 8 6 5 6 8 6 2 = + m y x + ) + ) m x + 6 - ^ 5 m - 6 h x - 8 = 0 ^ 2 - + h = ^ h ^f1h / ^f2h (1) , ( ax = b b = 2 ! 0 ). 2) ^f1h,^f2h , : 6 2 = + m y x x m m 5 6 2: 1 2 0 ! * & . 2 ! m m - 5 6 0 ! ! m m 0& 6 5 + ) : 2 = x 0 h f =m + & = 1 :y x 6 y 6 ^ ^ l ^ h h = f + & K 1 y y K 0,6 8 6 = m! 0 5 y ^ ^ ^ h h h = - + - + = =- f m & m + 2 : y 5 6 x 8 5 6 x 8 0 x 8 m - 5 6 ^ l c 5 6 h m K = K - f + & 2 x x ,0 m - . ) : O K ti c ^ OK o o O ( .), 0,0 2 2 2 2 h K 10 K 10 OK O 100 y x 100 2 2 ^ h ^ h ^ h ^ h K K = + K = + + K = + K + = + * 6 * * * = m- 8 m m - 5 6 2 2 2 100 6 m m - m m = - = - - = ! ! ! ! 5 8 5 6 100 36 + + + + 6 5 8 5 6 64 6 5 8 5 6 8 6 5 2 8 5 6 > 0 m m + - 8 - m m 5 6 8 6 - m m = Z [ ]] ]] ! ! 5 1 5 6 = 1 - = m m - = - =- m m m m m m = = = = Z [ ]] ]] Z ]] ]] ) * [ * + + + + + f , 1 6 5 5 6 1 6 5 5 6 1 5 6 1 6 ! ! 5 7 5 1 6 5 7 m m 5 ! Z [ ]] ]] : . 46 | - | : | www.mathematica.gr www.mathematica.gr | : | - | 47
  • 25. 23: 02 2011 - 20:27 3) m = 1 : ^f1h: y = x + 6 ^f2h: y =- x + 8 . 0 y = ) : 1 : y x 6 0 x 6 x 6 x ^ f h = + " = + + =- & ^ f h + . 1 8 ) m = 1 : K^0,6h, M^-6,0h 5 6 m= 1 " 01 2011 - 23:24 c m ^ h , ^KKh = 10 K - K ,0 8,0 m - ( m = 1 ^KKh = 10). ^ KM h = x 2 + y 2 & f^ KM h = 36 + 36 = 36 $ 2 & f^ KM h = 6 2 M K KM = x - xM = 8 - -6 & f KM = 14 K ^ h ^ h ^ h . ^KKh ! ^KMh ! ^MKh ! ^KKh, . 1 ) 2 1 ^KKMh = $ ^KMh $ ^OKh = $ $ yK = $ 14 $ 6 & ^KKMh = 42 x.n , 2 14 1 2 . ) ^f3h : ^f3h: y = ax + b. : = + = + $ $ ^ ^ ^ ^ h h h h ! ! f a b f a b 0,6 6 0 8,0 0 8 = 6 + = + + b a b a = =- ) ) * h . 3 + + & y x 3 8 6 0 6 f ^ 3 : 4 3 =- + 4 6 K 3 K h f = + 1 y x : 6 ^ ) x ,y y x 6: 3 0 0 0 1 M ! f & = + ^ ^ ^ h h h 0 0 . x ,y M0^ 0 0h yly ^ f 3 h : y =- x + h 3 h 3 4 6 Ml ^- 0 0h, 4 x ,y 0 ^ f & = + ^ 3 y 0 x 0 6: 4 . (3) (4) ( ). : = + = + 6 y 0 x 0 y x * 3 * ) ) ) . 0 0 4 6 6 = + = - y x = + = - = + = = ^ ^ ^ h = h h ^ - h 0 1 3 + + + + & f + f 0 3 0 0 x x 0 0 4 6 y x 0 0 x x 0 4 3 6 0 6 0 0,6 y x x y x M 0 0 0 0 0 0 0 = 24: parmenides51 1) f^xh,g^xh , , , : 2 ^ m - 1 h > 0 + ^ m - 1 h 2 > 0 + m - 1 ! 0 + m ! 1 20 m ! "1,2,3,4,5,, m ! "2,3,4,5,. 2 m - 20 7 > 0 7 > 0 > 7 > 7 > 4 2 - = 2 2 + + + m m m m ! "1,2,3,4,5,, m ! "3,4,5,. 2) , , : - h m 1 m 20 2 2 - 20 7 h , . 2 2 2 2 + + + f + - = - - + = - = m m m m m m 1 7 2 1 7 4 = ^ ^ A4 B4 , . 3) , : - - m m ( 1) 20 2 2 2 2 2 2 + + + + - - - + - - - - m m m m m m m > ( 1) > 7 2 1 > 7 2 > 1 8 < 4 20 7 m ! "1,2,3,4,5, m ! "1,2,3,,. A1, A2, A3 B1, B2, B3 . ` 6 j 4) 1, 20 . : 6 ( m - 1 ) 2 + m - 1 , (1) 6 ` j = = ! + + + 1 C f 6 20 20 f - + + - m m m 2 1 1 = - - = =- = m ! "1,2,3,4,5, m = 3. 2 + + + 20 6 20 m m m m 6 0 2 3 2 : 6 1, ` j = 20 - + - m m = ! g + + + (1) 6 20 7 1 20 6 20 C g 2 12 0 3 4 2 + m + m - = + m = m =- m ! "1,2,3,4,5, m = 3. 48 | - | : | www.mathematica.gr www.mathematica.gr | : | - | 49
  • 26. 24: parmenides51 5) A2 : x x 2 h . - + - ^ 2 1 2 1 20 + 20 1 ^ f x = = h B3 : h . 2 - x + + x ^ 3 7 3 1 20 + 2 4 20 ^ g x = = h 02 2011 - 20:27 , f^xh = g^xh + x x = + + = + =- , . + + x x + x 20 1 2 4 20 1 2 4 3 ^ h =- =- . f- 3 = - + 3 1 20 2 20 1 10 3, 3 3, 1 ^- f^- hh = `- - j. 10 .1 1: : 4/5 3/5 . .2 3: : 3/5 . .3 4: , . .4 1 2: . : ) 25: 02 2011 - 02:52 1) m ! 0 : x = 0 f^0h = m + 1, yly m ! R* . f^xh = y = 0 m x 2 + 2 ^m - 1 h x + m + 1 = 0 . D = 4 ^m - 1 h 2 - 4 m^m + 1 h + D =- 12 m + 4 . . ) 1 0 D > + m < , m ! 0, xlx ,0 0, 3 1 m ! ^-3 h,` j. 3 ) 0 1 D = + m = , xlx 3 ) 0 1 D < + m = , xlx. 3 m = 0 f^xh =- 2x + 1, xlx yly. 1 2) f^xh = 0 D > 0, 3 m < m ! 0 t1 + t2 =-t1 - t2 t1 + t2 # 0 ( a =- a + a # 0). Vieta - 2 ^ m - 1 h 2 1 t1 + t2 = . m h h 6 h. ^ ^ - - - + # 0 + 2 1 # 0 + ! 3,0 , 1, 3 ^ - - m m m m m h ^ ` ^ h j h h - - + 3 , 3 , 3 ! ! m m ,0 0, 1 * 3 4 . 6 ,0 1, 3 ! m ^ - & ,0 h 3) m = 0 g^xh =- 2x + 1 m =- 1 h x x 4x 2 ^ h =- - , f^-1h = 3. 1 2 ^ h x 4 2 2 - - x x - - x x 4 h x ! : 3 3 0 ^ g x ^ h $ - + $ + $ + f - + x - + x 1 2 1 2 1 - h 2 - - + - x x x 4 6 3 ^- + - h^- + h ` + j. 2 + $ + $ + ! 3 - + x 2 1 x x x x 0 2 3 2 1 0 1 , 2 4) m = 2 f ^ x h = 2x 2 + 2x + 3 . : f 2 ^ x h # 8x 3 + 12x + 29 + 4x 4 + 4x 2 + 9 + 8x 3 + 12x 2 + 12x # 8x 3 + 12x + 29 + x 4 + 4x 2 - 5 # 0 . x 2 = ~ $ 0 ~= 1 () ~=- 5. 1 2 ^ x 2 - 1 h^ x 2 + 5 h # 0 + x ! 6- 1,1 @. 5) f^-1h = 3, f^0h = m + 1, f^1h = 4m - 1 f^1h = f^-1h + m = 1, f^-1h = 3, f^0h = 2, f^1h = 3. (e1), (e2) : 3 2 3 + = 2 3 9 x y x y 6 4 6 6 9 27 =- + = y x y 13 21 3 2 3 21 =- $ + ^ h 2 13 + + + + ) 3 ) 3 ) 3 * 4 * 4. 81 - + =- - = $ x 3 39 21 - 13 27 13 + = - = x y x y y x y = = 50 | - | : | www.mathematica.gr www.mathematica.gr | : | - | 51