Μαθηματικά θετικού προσανατολισμού ΣΥΝΕΧΕΙΑ

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ΣΥΝΕΧΕΙΑ Μαθηµατικά Γ’ Λυκείου mathhmagic.blogspot.gr ΕΠΙΜΕΛΕΙΑ : Ο ΑΛΙ ΜΠΑΜΠΑ ΚΑΙ ΟΙ ΣΑΡΑΝΤΑ ΚΛΕΦΤΕΣ………………. 0 ΜΑΘΗΜΑΤΙΚΑ Γ ΛΥΚΕΙΟΥ ΣΥΝΕΧΕΙΑ ΟΜΑΔΑ ΠΡΟΣΑΝΑΤΟΛΙΣΜΟΥ ΘΕΤΙΚΩΝ ΣΠΟΥΔΩΝ, ΟΙΚΟΝΟΜΙΑΣ ΚΑΙ ΠΛΗΡΟΦΟΡΙΚΗΣ ΕΠΙΜΕΛΕΙΑ: Μήταλας Γ , Δρούγας Α. Χάδος Χ. Γερμανός Ξ. Πάτσης Σ. Ο ΤΣΕΛΕΜΕΝΤΕΣ ΤΟΥ ΥΠΟΨΗΦΙΟΥ ΣΤΑ ΜΑΘΗΜΑΤΙΚΑ ΘΕΤΙΚΟΥ ΠΡΟΣΑΝΑΤΟΛΙΣΜΟΥ
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  1. 1. mathhmagic.blogspot.gr : .0 , : , . . . .
  2. 2. mathhmagic.blogspot.gr : . 1 . , , , . , ., viral, on line . , . . . . , !! .... 1 1 1 4 1/2 400 , 4 , 3/4 , , 1 , 1 , , 1 , , . , , , . , , . , ( ) , , , , . . . 3 . . ( ). ( ) . (). 10 . , . . 180 30 15 .
  3. 3. mathhmagic.blogspot.gr : .2 1) , , ,. ... 2) , ., 3) , . , 4) , , 5) , ., 6) , ., 7) , ., 8) , . , 9) , , 10) ,. . , 11)- . , 12)- .. , 13)- .., 14) 1,2 , ., 15) 1000+1 , ., 16) , & ., 17) , .... 18) 19), . 20) . ,., 21) ,-, 22) .. . 23),., 24),. 25) , Spivak M. , . 26) . , 24)Problems in Calculus ,..Maron Mir Publisher 25) , . , 26) 27) , , 27) , ., 28) , .., ., ., . 29)Problem book: Algebra and Elementary functions, Kutepov A.,Rubanov, MIR Publishers 30) , ( qr-code) .
  4. 4. mathhmagic.blogspot.gr : . 3 46. : 1. 0x . 0x x= fC ( ) ( )0f x ( ) . 0x ( ) ( ) ( ) 0 0 0 x x x x lim f x lim f x f x + = = . 1: (xo =1 2 3) f. (1). (2). ( ) ( ) ( ) + = = = x 1 x 1 lim f x lim f x f 1 2 f =0 x 1 ( ) ( ) + = = x 2 x 2 lim f x 2.6 5 lim f x f =0 x 2 ( ) ( ) ( ) + = = = x 3 x 3 lim f x lim f x 2.6 7 f 1 f =0 x 3 ( ) ( ) = = x 1 lim f x 2 2.6 f 1 2 f =0 x 1 ( ) ( ) ( ) + = = = x 2 x 2 lim f x 2 5 lim f x f 2 f =0 x 2 ( ) ( ) ( ) + = = x 3 x 3 lim f x lim f x f 1 f =0 x 3 Cf 7 5 2.6 1 2 3 2 Cf5 2.6 1 2 3 2 . . , , . :http://www.wolframalpha.com/input/?i=f%28x%29%3Dxsin%281%2Fx%29
  5. 5. mathhmagic.blogspot.gr : .4 2: f 0 fx D . f 0x ( ) ( ) 0 0 x x lim f x f x = . ). , ( ) ( ) 0 0 x x lim f x f x . ). 0 fx D ( ( )0f x f . ). ( ). ). . 3. (, ) (, ). f [, ], ( ), ( ) ( ) ( ) ( )x x lim f x f lim f x f + = = . : 0x . f : 2,3 2,4 . f . . ( ) = x 1 lim f x 0 ( )+ = x 1 lim f x 2 ( ) ( )+ + x 1 x 1 lim f x lim f x . f = 0 x 1 . ( ) = x 1 lim f x 3 ( )+ = x 1 lim f x 1 ( ) ( )+ + x 1 x 1 lim f x lim f x . f =0 x 1 46-1. . . O 1 1 3 2 32 x y O 1 1 1 2 3,532 x y
  6. 6. mathhmagic.blogspot.gr : . 5 47 . 1. . 0x ( ) 0x x lim f x = , 0x x . ( )0 f x= f 0x , ( )0 f x f 0x ( ). 1: ( ) = 3 x 1 f x x 1 , ( ) =f 1 2 . 2: f 4 ( )f 4 , x 0 ( ) ( )4 x 8 x 4 f x x 4 . 47-1. f 3 ( ) ( ) 2 x 3 f x x 9 = x R . ( )f 3 . f 3 ( ) ( )x 3 limf x f 3 = . ( )x 3 limf x . x 3 ( ) ( )( )2 x 3 x 3x 9 f x x 3 x 3 x 3 + = = = + . ( ) ( )x 3 x 3 lim f x lim x 3 6 = + = . ( )f 3 6= . 47-2. f : R R : () R () ( ) x 0 xf x x lim 2 x + = . ( )M 0,1 . ( )f 0 1= . ( ) ( ) ( )x 0 x 0 x 0 x 0 lim xf x x lim x lim f x lim x 0 f 0 0 0 + = + = + = x 0 lim x 0 = . ( ) ( ) ( ) ( )x 0 x 0 x 0 x 0 xf x x xf x x x lim lim limf x lim f 0 1 x x x x + = + = + = + (1). ( ) ( )f 0 1 2 f 0 1+ = = . 47-3. f 0x 0= ( ) 2x 0 x xf x 3lim R x = , ( )f 0 . ( ) ( ) 2 x xf x 3g x ,x 0 x = , ( )x 0 limg x R = . : ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 2 x 0 2 2 x x x g x x x 3 3xf x x g x xf x x g x f x f x xg x 3 3 x x + = = + = = + . ( ) ( )x 0 x 0 x 3limf x lim xg x 0 x 3 3 = + = + = . f 0, ( ) ( )x 0 f 0 limf x 3 = = .
  7. 7. mathhmagic.blogspot.gr : .6 47-4. f ( ) ( )4 x-2 f x +3x-1=0 x 2 . f 2. 47-5. 0 = . f A R= . ( )1,+ f 2 ln x 2x 3 . ( ).1 f x 2x 3 . 0x 1= ( )f 1 1= ( ) ( ) ( ) 2 x 1 x 1 x 1 x 1 lim f x lim ln x 2x 3 1 x lim f x lim 1 2x 3 + + = + = = = f 0x 1= . f .
  8. 8. mathhmagic.blogspot.gr : . 7 47-11. f ( ) ( ) 2 23 xx, x 0 f x x 1, 0 x x 1, x = < < . : x 0< f . 0 x < < f . x< f . 0 0x 0 x = = . 0x 0= : ( ) ( ) ( ) ( ) ( ) 2 2 x 0 x 0 x 0 x 0 lim f x lim x 1 0 1 1 1 0 lim f x lim x x 0 0 0 f 0 0 0 0 + + = = = = = = = = = f 0. 0x = : ( ) ( ) ( ) ( ) ( ) ( ) 2 23 3 x x 22 2 x x lim f x lim x 1 1 1 0 1 1 lim f x lim x 1 1 1 1 0 f 1 + + = = = = = = = = = = f . 47-12. ( ) x , x 0 f x x 1, x 0 = = . f 0 ( ) ( ) x 0 x 0 x 0 x f 0 1 limf x lim 1 x = = = . ( ) ( )x 0 f 0 limf x = , f 0. ( ) ( ),0 0, + f x x . f R. 47-13. ( ) { }f x max 7 3x,5x 11= + . { }max , . { } , max , , = : { } 1 max 7 3x,5x 11 7 3x 7 3x 5x 11 8x 4 x 2 + = + . { } 1 max 7 3x,5x 11 5x 11 5x 11 7 3x 8x 4 x 2 + = + + . ( ) 1 7 3x, x 2 f x 1 5x 11, x 2 = + . ( ) ( )1 1 x x 2 2 1 17 lim f x lim f x f 2 2+ = = = , f 1 2 . 1 1 , , 2 2 + f .
  9. 9. mathhmagic.blogspot.gr : .8 47-14. ( ) 2 x 5 x f x 2x 3 x = + . ( )0U x 0x ( ) 0x x lim f x x Q . ( ) 0x x lim f x x R / Q 0x . f 0x ( ) ( ) ( ) 0 0 0 x x x x x Q x R / Q lim f x lim f x f x = = . x ( ) ( ) 0 0 0 x x x x lim f x lim 2x 3 2x 3 = + = + . x R / Q ( ) ( )0 0 2 2 0 x x x x lim f x lim x 5 x 5 = = . 2 2 0 0 0 0 0 0x 5 2x 3 x 2x 8 0, x 2, x 4 = + = = = . f 0 0x 2, x 4= = 47-15. f ( ) 2 f x = -x +3x , 0 0x =0 x =3 . 47-16. 0x : i) 2 3 x +4, x1 x-1 ii) x , x3 x+1-2 f x = 8, x=3 8x-24 , x0 x f x = , x=0 2x+1 , x . , f . f R = . , R f ( ) ( ) ( ),1 , 1,2 2, + , . f 1 2. ( ) ( ) ( ) ( ) ( ) ( ) x 1 x 1 x 2 x 2 lim f x lim f x f 1 (1) lim f x lim f x f 2 (2) + + = = = = . : ( )f 1 2= ( ) ( )x 1 2 x 1 x 1 lim f x lim e x 1 = + = + ( ) ( )( )x 1 x 1 lim f x lim x 2 0 2 2+ + = + = + = ( )f 2 2= ( ) ( )( )x 2 x 2 lim f x lim x 2 0 2 2 = + = + = ( ) ( ) ( )( )x 2 x 2 lim f x lim ln x 1 2 x 0 2 2+ + = + = + = + (1) (2) 1 2 4 2 2 3 + = = + = = . 4 3= = . 47-24. , ( ) 2 x 4 , x 0 f x 2x , x 0 + = = R. f x 0 , . R f , 0x 0= . ( ) ( )x 0 limf x f 0 = . : ( )f 0 =
  10. 11. mathhmagic.blogspot.gr : .10 ( )x 0 x 0 2 x 4 lim f x lim 2x + = ( )x 0 x 0 lim 2 x 4 2 2 0 lim 2x 0 + = = = . 0 0 , ( )f x ( ) ( ) 4 x 4 2 x 4 x2 x 4f x 2x 2x 2x 2 x 4 + + += = = + + . ( ) ( ) ( ) ( ) x 0 x 0 x 0 x 0 x 1 1 1 1 limf x lim lim lim 2x 2x 2 1 4 82x 2 x 4 2 x 4 2 2 x 4 x 2x = = = = = + + + + + + ( )x 0 x 0 2x lim 1 lim 2 x 4 4 2x = + + = . 1 8 = . 47-25. R ( ) 3 2 2 x 3x 1, x 1 f x x 2x 6, 1 x 3 x x 3, 3 x + < = + + + < . { }R 1,3 . x 1= ( ) ( ) ( )3 2 x 1 x 1 f 1 1 2 6 lim x 3x 1 3 1, lim x 2x 6 1 2 6 + = + + = + + = + . 3 1 1 2 6, 3 0 + = + = (1). x 3= ( ) ( ) ( )2 2 x 3 x 3 lim x 2x 6 9 6 6 lim x x 3 9 3 3 f 3 9 6 6 + + = + + + = + + = + . 9 6 6 9 3 3, 5 2+ = + + = (2). (1) (2) 3 1= = f R. 47-26. ( ) 2 x 7-4 , x 3 f x = x-3 , x=3 + , ( )x 3 limf x , f . 47-27. ( ) 2 2 3x -5x-2 , x 2 f x = x-2 -+5, x=2 . 2. 47-28. ( ) 3-x- 3 , x 0 f x = x 3 3, x=0 . f 0 . 47-29. ( ) 2 2 2 2 4 2 2 2 x x , x xf x , x = + = . ( )2x limf x . f 2.
  11. 12. mathhmagic.blogspot.gr : . 11 ( ) 2 2 2x x , x 0 f x x x 3, 0 x + < = + . 47-30. (x-)(x+) , x 2 f(x)= x+5 , x>2 , , f 0x =2 . 47-31. 2 2 x +x-12 , x1 , , f 0x =1. 47-32. , : ( ) 1+x , x< i. f x = x- x+2, x ( ) ( ) ( ) 2 2 2 2 5 2 x , x . . iii). f ( ) ( ) 2 2 g x x h x x 1 = = + . 47-38. f : R R R ( )( )f f x x 0+ = x R . f : ( )( )f f x x 0+ = x R . ( ) ( )1 2f x f x= , ( )( ) ( )( )1 2f f x f f x= . ( )( ) ( )( )1 1 2 2 1 2f f x x f f x x x x+ = + = . f 1-1. f R 1-1, . ( ) f . 1 2x x< , ( ) ( )1 2f x f x< ( )( ) ( )( )1 2f f x f f x< . fof . f . 1 2x x< , ( ) ( )1 2f x f x> ( )( ) ( )( )1 2f f x f f x< . fof . ( )( )f f x x fof= . , g : R R ( )g x x= R. f R , . 47-39. : i) f(x)=(x) ii) 2 f(x)=ln(x +x+1) iii) 2 1 f(x)= x +1 iv) x f(x)=e v) f(x)=ln(lnx) . 47-40. : . f+g f-g , f g . . ( )x f x + x 0, f 0.
  12. 14. mathhmagic.blogspot.gr : . 13 48. 0x 1. ( ). . 1: f : 0x 0 0x . 2: f g x R ( ) ( ) ( ) ( )2 2 2 f x g x x 2x f x x g x x+ + = + R. 48-1. f 2 ( ) h 0 f 2 h lim 3 h + = , ( ) ( ) x 2 f x f 2 L lim x 2 = . ( ) ( )f 2 h g h h + = , ( ) ( )f 2 h hg h+ = ( )h 0 limg h 3 = . ( ) ( )h 0 h 0 limf 2 h lim hg h 0 3 0 + = = = . f 2 ( ) ( )h 0 f 2 limf 2 h 0 = + = . ( )f 2 0= L ( ) x 2 f x L lim x 2 = . ( ) ( )f x x x 2 = , ( ) ( ) ( ) ( ) ( ) ( ) x 2 x 2 h 0 h 0 f 2 h f 2 hf x 2 h L lim lim x lim 2 h lim 3 h x 2 h + + + = = = = + = = . L 3= . 48-2. f : R R ( ) ( ) ( )f x y f x f y+ = x,y R , , , R. ( ) ( )x limf x f = . f R, 0x R . ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 0 0 0 0 0 0 x x h h lim f x lim f x h lim f x f h f x f f x f x = + = = = + = . f 0x R . 48-3. f 2 ( ) x 2 f x x 2 3 lim x 2 4 + = , ( ) ( ) x 2 f x f 2 lim x 2 . ( ) ( ) f x x 2 h x x 2 + = ( ) ( )( )f x h x x 2 x 2= + + , ( ) ( )( )x 2 x 2 limf x lim h x x 2 x 2 0 2 2 = + + = + = . x 2= ( ) ( )x 2 limf x f 2 2 = = . : ( ) ( ) ( ) ( ) x 2 x 2 x 2 f x x 2 f x 2 x 2 2 f x f 23 3 x 2 2 3 lim lim lim x 2 4 x 2 4 x 2 x 2 4 + + + + = = = . ( ) ( ) ( ) f x f 2 x 2 2 g x x 2 x 2 + = , ( ) ( ) ( ) f x f 2 x+2-2 g x x 2 x-2 = + , ( ) ( ) x 2 f x f 2 lim 1 x 2 = .
  13. 15. mathhmagic.blogspot.gr : .14 48-4. ( )f : 0, R+ ( ) ( )3 f x f x ln x+ = (1). f 0x 1= . x 0> ( ) ( ) ( ) ( )( ) ( ) ( ) 3 2 2 lnx f x f x ln x f x f x 1 lnx f x 1 f x + = + = = + . ( ) ( ) ( )2 lnx f x lnx ' ln x f x lnx 1 f x = + (2). ( )x 1 x 1 lim ln x lim ln x 0 = = . (2) ( )x 1 lim f x 0 = . (1) x 1= ( ) ( ) ( ) ( )( ) ( )3 2 f 1 f 1 ln1 f 1 f 1 1 0 f 1 0+ = + = = . ( ) ( )x 1 limf x f 1 = , f 0x 1= . 48-5. f : R R ( ) ( ) 2 xf x 2 f x 3x 1 = + x R . f , . x R ( ) ( ) ( ) ( )2 2 xf x 2 f x 3x 1 x 1 f x 2 3x 1 = + = + . x 1 ( ) 2 2 3x 1 f x x 1 + = . f 1 ( ) ( ) ( )2 2x 1 x 1 x 1 3 x 12 3x 1 3 f 1 lim f x lim lim x 1 22 3x 1 + + = = = = + + . ( ) 2 2 3x 1 , x 1 x 1f x 3 , x 1 2 + = = . : f , , 0x , . 48-6. f ( ) ( ) ( )f xy f x f y= + (1) x,y R : i). f 1, f R. ii). f 1 , f R. i). f 0x R . f 1 ( ) ( )x 1 limf x f 1 = . x y 1= = (1) ( ) ( ) ( ) ( )f 1 f 1 f 1 f 1 0= + = . ( )x 1 limf x 0 = . ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 0 0 x x t 0 0 0 0 0 x x t 1 t 1 t 1 lim f x limf x t lim f x f t f x limf t f x 0 f x = = = + = + = + = . f 0x . ii). f 1. ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) x t 1 t 1 t 1 t 1 t 1 limf x f limf t f lim f f t f f limf t f limf t 0 limf t f 1 . = = + = + = = = f 1. 48-7. ( ) h 0 f 3+h lim =5 h f 3, ( ) ( ) x 3 f x -f 3 lim x-3 A = . 1). ( ) ( ) ( ) ( ) x 3 x 3 f x x x 3 f 3 2 lim 4 x 1 2 = = + f x 3=
  14. 16. mathhmagic.blogspot.gr : . 15 2). f 3 ( ) x 0 f x 3 lim 5 x + = , ( ) ( ) x 3 f x f 3 lim x 3 . 48-8. f 2 ( ) 2x 2 f x -5x lim 3 x -4 = , f 2. 48-9. f 0 ( ) 2x 0 xf x -3x lim =2 x +x , ( )f 0 . 1). f ( ) ( ) ( )f x y f x f y+ = + (1) x,y R . f 0 R. 2). f x,y 0> ( ) ( ) ( )f xy f x f y= + . x 1= * R+ . 48-10. f : R R ( ) ( ) ( )f x y f x f y+ = + x,y R . : i). ( )f 0 0= ii). f 0, . 2. x R , x 0x ( )0f x . ( ) ( ) 0x x lim f x . ( ) ( ) 0 0 x x lim f x f x = . 3: f 0x , x R : (1). ( ) 2 0f x x 9 x 3, x 0 + = , (2). ( ) 0f 3x 5 3x 7 , x 2 = . 48-11. f : R R ( ) ( )2f x x f x= + x R . : i). ( )f x x x R . ii). f 0x 0= . i). x R ( ) ( ) ( ) ( )2f x x f x x f x x f x= + + + , x x , x R . ( ) ( ) ( )2 f x x f x f x x + . ii). x R ( ) ( )f x x x f x x (1). x 0= (1) ( )0 f 0 0 . ( )f 0 0= . ( )x 0 x 0 lim x lim x 0 = = . (1) ( )x 0 lim f x 0 = . ( ) ( )x 0 lim f x f 0 = f 0.
  15. 17. mathhmagic.blogspot.gr : .16 48-12. f : R R ( ) ( )f f 2015 , , R . f . ( ) ( ) 0 0 0 x x lim f x f x x R = . 0x,x R ( ) ( ) ( ) ( )0 0 0 0 0 f x f x 2015 x x 2015 x x f x f x 2015 x x . ( ) ( )0 0 0 0x x x x lim 2015 x x 0 lim 2015 x x = = . ( ) ( )( ) ( ) ( ) 0 0 0 0 0 x x x x x x lim f x f x 0 lim f x lim f x = = . : f : R R 0x R 0 > ( ) ( )0 0f x f x x x ( )0 0x x ,x + , f 0x . 48-13. f : R R , ( )2 2 1 x f x 1 x + x R . : i). f 0x 0= . ii). ( ) x 0 f x x 1 1 lim x 2 + = i). ( )2 2 1 x f x 1 x , x R + 1 ( )x 0 : 1 f 0 1= , ( )f 0 1= . 2 ( )x 0 limf x 1 = , . 1 2 ( ) ( )x 0 limf x f 0 = , f 0x 0= . ii). : ( ) ( ) ( ) ( ) ( ): x ,x 0 2 2 2 2 2 f x 1 f x 1 1 x f x 1 x x f x 1 x f x 1 x x x x x x + , ( ) x 0 f x 1 lim 0 x = (1), . ( ) ( ) ... x 0 x 0 f x x 1 f x 1 x 1 1 1 lim lim x x x 2 + + = = . 48-14. f : R R , ( )x f x 4x x R . ( )f 0 . : x 0> , ( ) ( ) 4x x f x 4x f x x , ( ) ( ) x 0 x 0 4x lim f x lim f 0 4 x+ + (1). x 0< , ( )f 0 4 (2). (1) (2) ( )f 0 4= . 48-15. ( ) 2 5 f x x -6x+14 x , f 3. 48-16. f 1 ( ) ( )2 2 3x-x -2 x-1 f x x -x x ( )f 1 . 48-17. f: 0x =0 x 0 : ( ) 2 2 4 2 4 2 2x - x x +4-2 f x 1+ x +x x . ( )f 0 .
  16. 18. mathhmagic.blogspot.gr : . 17 48-18. f,g: ( ) ( )f x -x g x , x . g 0x =0 , f 0x =0 . 48-19. f ( )xf x x-x , x f 0 ( )f 0 . 48-20. f ( )( ) 2 2 f x x- x +x-2 , x . ( )f =3 . 48-21. i) f 0x =0 . f(0) , * x xf(x)=x-1. ii), g(0) g 0x =0 x 2 |xg(x)-x| x . - . , ; : . ( ) ( ) ( )= +2 1 f x / ,0 0, x . f ( ) ( ) +,0 0, , , . (((( )))) (((( )))) (((( ))))2 1 f x / ,0 0, x = += += += +
  17. 19. mathhmagic.blogspot.gr : .18 49. BOLZANO : f, , . : f , ( ) ( )f f 0< ( )0x , , ( )0f x 0= . ( )f x 0= ( ), . : 1). .Bolzano 2). f , ( ) ( )f f 0< , . 3). f , ( ) ( )f f 0< f ( ), . 4). . .Bolzano, : f , ( ). ( ) ( )f ,f . .Bolzano ( )0x , 1: ( )f x ln x 3= , . Bolzano ( )4 1,e 2: ( ) 2 2 3x x 1, 0 x 1 f x 5x 4x, 1 x 2 = < . .Bolzano [ ]0,2 ( )0x 0,2 , ( )0f x 0= . f() f() Cf f() f() Cf xo
  18. 20. mathhmagic.blogspot.gr : . 19 3: ( ) ( ) 1 x 2, 2 x 1 f x 2 , x 1 x 1 , 1 x 2 x 1 + + < = = + < + . .Bolzano [ ]2,2 . 94-1. t 0= , 257m (. ). 3 , 555m 2 . 3 , , 257m . ). ( )y t [ ]t 0,8 . ( )y t . ). 3 . ). ( )y t . . ). 1 2t t< 3 2 1t t 3= + . 20 t 8 10 t 5 , [ ]1t 0,5 . 1 2t ,t ( ) ( )1 2y t y t= . ( ) ( ) ( ) ( ) ( ) ( )1 2 1 2 1 1y t y t y t y t 0 y t y t 3 0= = + = . ( ) ( ) ( )f t y t y t 3= + 1t ( )0,5 . f [ ]0,5 . ( ) ( ) ( ) ( )( ) ( ) ( )( ) ( )( )f 0 f 5 y 0 y 3 y 5 y 8 257 555 555 257 88804 0= = = < . : f [ ]0,5 . ( ) ( )f 0 f 5 0< . Bolzano, [ ]1t 0,5 ( ) ( ) ( ) ( ) ( )1 1 1 1 1f t 0 y t y t 3 0 y t y t 3= + = = + , 2 3 ( )1 1 t ,t 3+ . 94-2. ( ) [ ) [ ) 3 2 2 x 3x 7x 5, x 2,1 f x x x 1, x 1,4 + + + = + + . f ( )2,4 . ( ) ( )3 2 x 1 x 1 lim f x lim x 3x 7x 5 1 3 7 5 16 = + + + = + + + = ( ) ( )2 x 1 x 1 lim f x lim x x 1 1 1 1 3+ + = + + = + + = . f 1. ( ) ( ) ( ) ( )3 2 f 2 2 3 2 7 2 5 8 12 14 5 5 0 = + + + = + + = < .
  19. 21. mathhmagic.blogspot.gr : .20 ( ) x 1 lim f x 16 0 = > . 1, ( ) 1 ,1 0 > , ( )f 0> . [ ]2, ( ) 3 2 f x x 3x 7x 5= + + + . f Bolzano [ ]2, , ( ) ( ) 2, 2,4 ( )f 0= 94-3. Bolzano ( ) 3 2 f x x 3x x 3= + [ ]0,2 . 94-4. Bolzano ( ) 3 3 3x 5x 6 1 x 3 f x 2x x 3 3 x 4 = + + < [ ]1,4 . 94-5. Bolzano ( ) 1 x 3x 1 3 x 1 f x 2 x 1 x 1 1 x 5 + = + + < [ ]3,5 . 94-6. , Bolzano ( ) 2 2 2 2 3 x 2x 4 0 x 1 f x 2 x x 6 1 x 2 = + + < [ ]0,2 . 94-7. Bolzano ( ) 3 2x 4 0 x 3 f x x 1 3 x 4 = < [ ]0,4 . BOLZANO M Bolzano Scientific American . (Problem solving) K. Duncker. : . , . . . . .
  20. 22. mathhmagic.blogspot.gr : . 21 . , - - . . . . S(t) t ( t 0,T , ) (t) t ( t 0,T' , ). , ( ) ( )f 0 f 1 0 < . Bolzano ( ) x f x 0 xe x x 0= = ( )0,1 . 95-2. [ ]f : 0,4 R ( ) ( )f 0 f 4= ( ) ( ) ( )h x f x f x 2= + . i). h. ii). [ ] 0,2 ( ) ( )f f 2= + . i). 0 x 4 0 x 2 0 x 2 4 + . h [ ]0,2 = . ii). ( ) ( ) ( ) ( ) ( )f x f x 2 f x f x 2 0 h x 0= + + = = [ ]0,2 . h [ ]0,2 . ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( ) h 0 f 0 f 2 h 2 f 2 f 4 f 2 f 0 f 0 f 2 = = = = . ( ) ( ) ( ) ( )( ) 2 h 0 h 2 f 0 f 2 = . ( ) ( )f 0 f 2= , ( ) ( )h 0 0 ' h 2 0= = , ( )h x 0= 0 2. ( ) ( )f 0 f 2 , ( ) ( )h 0 h 2 0 < , Bolzano ( )h x 0= ( )0,2 ( )h x 0= [ ]0,2 .
  21. 24. mathhmagic.blogspot.gr : . 23 95-3. , 0> x 2 2x + = + . ( ) x x 2 2x= + , 0, + . ( ) ( ) ( ) ( ) ( )( ) 0 0 2 2 0 2 0 2 2 2 0 = + = + > + = + + + = + < . Bolzano 0, + ( ) ( ) 0 0 + < . ( ) 0, + ( ) 0 2 2= + = . 95-4. ( ) ( ) ( )2 f x x 3x 2 g x= + x R , g R. 1 2 ( )f x 0= , ( ) ( )g 1 g 2 0 . . ( ) ( )g 1 g 2 0 < . g R, [ ]1,2 . g Bolzano [ ]1,2 , ( ) 1,2 ( )g 0= . ( ) ( ) ( ) ( )2 2 f 3 2 g 3 2 0 0= + = + = ( ) 1,2 ( )f x 0= . 1 2 . , ( ) ( )g 1 g 2 0 . 95-5. x-x+1=0 (0,) . 95-6. 3 x -5x+3=0 ( ) ( )0,2 . 95-7. : . 2 x -1=0 ( )0,2 . . 3 2 2x -5x +7=0 . 95-8. x=3-2x ( ) , 4 2 . 95-9. f, , (,+1) f(x)=0 i) 3 f(x)=x +x-1 ii) 5 f(x)=x +2x+1 iii) 4 f(x)=x +2x-4 iv) 3 f(x)=-x +x+2 . 95-10. : ) 4 6 x +1 x +1 + =0 x-1 x-2 ) x e lnx + =0 x-1 x-2 , , (1,2). 95-11. f [ ]1,2 ( )f x 3x 2= (1). ( ) ( )f 1 1 f 2 4< > , (1) ( )1,2 = . 95-12. f ( )0
  22. 25. mathhmagic.blogspot.gr : .24 95-13. f, , ( ) ( ) ( )3 2 3 2 f x +f x +f x =x -2x +6x-1, x , 2 ,, , , (1) ( )1,2 = . : . 1 2 3 ( ) ( ) ( )f f 0 , ). ( ) ( ) ( ) ( )f f 0 f f 0= = = , 0 0x ' x = = . ). ( ) ( )f f 0< Bolzano ( ) ( )0 0x , : f x 0 = . () , () . , . 3: f R ( ) ( )f 0 f 2= . ( ) ( )f x f 2 x= , 0, 2 . 95-15. f ( ) ( )f x 2 f x 0+ + = x R , [ ] 0,2 ( ) ( )f f 1= + . ( ) ( )f x f x 2= + ( ) ( ) ( ) ( ) x 0 f 0 f 2 (1) x 1 f 1 f 3 (2) = = = = . [ ] 0,2 ( ) ( )f x f x 1= + . ( ) ( ) ( )h x f x f x 1= + R : ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( ) 1 2 2 h 0 f 0 f 1 f 2 f 1 h 0 h 2 f 1 f 2 0 h 2 f 2 f 3 f 2 f 1 = = = + = = + . ( ) ( )f 1 f 2= ( ) ( )h 0 h 2 0 = h 0 2. ( ) ( )f 1 f 2 ( ) ( )h 0 h 2 0 < h ( )0,2 . ( )h x 0= [ ]0,2 . 95-16. f [ ]0,3 ( )0 f x 3< [ ]x 0,3 . [ )0x 0,3 ( ) ( )2 0 0 0f x 3f x x 0 + = . ( ) ( ) ( )2 x f x 3f x x= + [ ]x 0,3 . [ ]0,3 . ( ) ( ) ( ) ( ) ( )( )2 0 f 0 3f 0 f 0 f 0 3= = . ( )0 f x 3< [ ]x 0,3 , ( ) ( )f 0 0 f 0 3 0> . ( ) 0 0 . ( ) ( ) ( )2 3 f 3 3f 3 3= + . ( ) ( )2 f 3 3f 3 3 + ( )f 3 , ( )2 3 4 3 9 12 3 0 = = = < . ( ) ( )2 f 3 3f 3 3 0 + > . ( ) 3 0> . ( ) ( ) 0 3 0 . : ( ) ( ) 0 3 0 < Bolzano [ ]0,3 . ( )0x 0,3 ( ) ( ) ( )2 0 0 0 0 x 0 f x 3f x x 0= + = . ( ) ( ) ( ) 0 3 0 0 0 = = ( ( ) 3 0> ). 0x 0= . [ )0x 0,3 ( ) ( )2 0 0 0f x 3f x x 0 + = . 95-17. f : , R ( ) ( ) ( )2 f f f 0+ = .
  23. 26. mathhmagic.blogspot.gr : . 25 , ( )f 0= . ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( )2 f f f 0 f f f 0 f 0 ' f f + = + = = = . ( )f 0= ( )f x 0= ,= ( ) ( )f f = , ( ) ( ) ( )2 f f f 0 = . : ( ) ( )f f 0 < f Bolzano , . ( ) , ( )f 0= . ( ) ( ) ( )f f 0 f 0 = = . ( )f x 0= ,= . , ( )f 0= . 95-18. f , ( ) ( )f +f =0 >0 , 0x , ( )0f x =0 . 95-19. ( ) 2 f x =x +x+, 0 . 5+3+3=0 , ( )f x =0 [ ]0,2 . : BOLZANO , . 1 2 3 . , R , 1 2x , , x , , ( ) ( )1 2f x f x 0 < , Bolzano [ ]1 2x ,x . 4: 3 2 x 3x x 3 0 + = ( )1,3 . 95-20. 6 x 16x 11 = R. ( ) 6 x x 16x= R, [ ]0,2 , . ( ) ( ) 6 0 11 0, 2 2 16 2 11 64 43 21 0= < = = = > , ( ) ( ) 0 2 0 < . Bolzano [ ]0,2 , [ ] 0,2 ( ) 6 0 16 11= = . 95-21. f, g : ). R. ). ( ) ( )f x 1 2g x< < , x R . ). , R 0 < < ( ) ( )f 2 g = = . R ( ) ( )f g = . ( ) ( ) ( ) x f x g x x= , x , . , . ( ) ( ) ( ) ( ) ( )( ) f g 2 g 2g 1= = = . 0> ( ) ( ) ( )2g x 1 2g 1 2g 1 0> > > . ( ) 0> . ( ) ( ) ( ) ( ) ( )( ) f g f f 1= = = . ( ) ( ) 0 f x 1 f 1> < < . ( ) 0< . ( ) ( ) 0 < . Bolzano , . ( ) , R , ( ) ( ) ( ) 0 f g = = .
  24. 27. mathhmagic.blogspot.gr : .26 95-21. =ln(2x) x (0,1) = f(x) ln(3x) x , >x 0 : f 1 ,1 3 . = = < 1 1 1 1 f ln(3 ) 0 3 3 3 3 , = >f(1) ln 3 1 0 . ( ) < 1 f f 1 0 3 , Bolzano ( ) =f x 0 ( ) 1 ,1 0,1 3 ( )0,1 : = = =x x ln(3x) x e 3x 3x e 0 . = x g(x) 3x e 0,1
  25. 28. mathhmagic.blogspot.gr : . 27 ( ) ( ) ( ) ( ) ( ) ( ) g f 1 2 g 0 f 0 1 1 g f 1 2 = = = = = = ( ) ( ) ( ) ( ) ( ) ( ) g g 0 0 . . g ,0 g g 0 0 . . g 0, < < ( )g x 2 ( ), . ( ) ( )h x f x 2= [ ],0 0, f : ( ) ( ) ( ) ( ) ( ) ( ) h f 2 1 h 0 f 0 2 h f 2 1 = = = = = = ( ) ( ) ( ) ( ) ( ) ( ) h h 0 0 . . h ,0 h h 0 0 . . h 0, < < ( )h x 2 ( ), . 4 ( ), . 95-27. ,,, R 0, 0 0 > + + + < + > . 3 2 x x x 0+ + + = 2 ( )1,1 . ( ) 3 2 f x x x x = + + + R, [ ] [ ]1,0 , 0,1 . ( )f 1 = + + . 0 + + < + + + < + + > , . ( ) f 1 + + < + + + < + + + . 0+ + + < , ( )f 1 0 < . ( )f 0 0= > . f Bolzano [ ]1,0 , ( )1 1,0 ( )1f 0= . [ ]0,1 ( ) ( )f 0 0 f 1 0= > = + + + < f Bolzano [ ]0,1 . ( )2 1,0 ( )2f 0= . 3 2 x x x 0+ + + = 2 ( )1,1 . 95-28. 4 x =11-2x . 95-29. . ( ) ( )( ) ( )( ) ( )( )f x =3 x- x- +4 x- x- +9 x- x- 0 +