1 -

1

.

1

+ + - 2000-2015

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2

+ ....................................................................3

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: www.mathp.gr

: ,

1 -

3

+

1. f ;

R. () f , x A y. y f x f(x).

, : f : A R x f (x)

2. f ;

M(x, y) y f x , , ( )M x f x , x A , f fC .

3. f g ;

f g :

x A ( ) ( )f x g x .

f g f g .

4. f :

0x A () ,

0x A () ,

f :

1 -

4

0x A () , 0f x , :

0f x f x x A

0x A () , 0f x ,

0f x f x x A

5. :f :g , f f ;

f , g , , f g , gof , : ( ) ( )gof x g f x .

gof x f ( )f x g . :

1 / ( )A x A f x B gof A1 , f(A)B .

6. f :

' ;

' ;

f :

1 -

5

' , 1 2,x x 1 2x x 1 2( ) ( )f x f x .

' ,

1 2,x x 1 2x x 1 2( ) ( )f x f x .

7. f :

0x A () ; 0x A () ;

f :

0x A () , 0( )f x , 0( ) ( )f x f x x A . 0x A () , 0( )f x , 0( )f x x A . 8. :f 11;

:f 11, 1 2,x x A

:

A 1 2x x , 1 2( ) ( )f x f x

: 1 2, fx x D .

9. :f ;

f : A R. 1-1, y

, f(A) , f x

f(x) = y. g : f(A) R

1 -

6

y f(A) x A g y x

g :

f(A) f,

f

: f x y g y x

, f x y, g y x

. g f. g

f f 1 . :

1f x y f y x

10. 1, f fC C 1,f f

; .

C C f 1f

y = x xOy xOy.

1 -

7

1-1 f C

C f f 1 (. 37). 1f x y f y x

M(, ) C f, (,)

C f 1 . , ,

xOy xOy.

11. 11 1 0...vP x a x x a x a , :

0

0limx x P x P x

:

12. .

, ,f g h .

( ) ( ) ( )h x f x g x 0x

1 -

8

0 0

lim ( ) lim ( )x x x x

h x g x l

0

lim ( )x x

f x l

13. 0x ;

f x0 x0 . f

x0, :

14.

,a ;

, ;

f (, ),

(, ).

f [, ],

(, ) :

lim ( ) ( ) lim ( ) ( )x a x

f x f a f x f

15. Bolzano.

f , [, ]. :

f [, ] , ,

f() f() < 0 . , , x0 (, ) , f(x0) = 0

, , , f(x) = 0 (, ) .

16. .

f , [, ]. :

f [, ]

f() f()

1 -

9

, f() f() , x0 (, )

, f(x0) = .

f() < f(). f() < < f() (. 67).

g(x) = f(x) , x [, ], :

g [, ]

g() g() < 0 , g() = f() < 0 g() = f() > 0 .

, Bolzano, x0 (, ) ,

g(x0) = f(x0) = 0, f(x0) = .

17. .

f [, ] , f [, ]

m.

18. Cf (x0, f(x0)) ;

f (x0, f(x0)) Cf . 0

0

0

( ) ( )limx x

f x f xx x

, Cf

, .

1 -

10

, (x0, f(x0)) :

0 0( ) ( )y f x x x , 0

0

0

( ) ( )limx x

f x f xx x

19. f ' x0 ; f ' x0 ,

0

0

0

( ) ( )limx x

f x f xx x

f x0 f (x0).

f x0, R :

.

20. : f ' x0,

.

x x0

f x0. ,

00lim ( ) ( )x x f x f x , f

x0.

21. f(x) = c, c R. f

R 0f x , 0c

1 -

11

x0 R, x x0 :

:

0c .

22. f(x) = x. f R

1f x , 1x

:

x0 R, x x0 :

:

1x . 23. f(x) = x , N-{0, 1} . f

R 1vf x x ,

1vx x

x0 R, x x0 :

1 -

12

:

:

( x ) = x 1.

24. f x x . f (0, +)

1

2f x

x ,

12

x x

x0 (0, +) , x x0 :

:

:

.

25. f, g x0, f + g

x0 0 0 0( ) ( )f g x f x g x .

x x0, :

f, g x0, :

1 -

13

:

0 0 0( ) ( )f g x f x g x

26. f(x) = x , N*. f R*

1( )f x x , 1x x

x R* :

27 . f(x) = x. f R1 = R {x |

x = 0} 21( )f x

x , 2

1x x

x R1 :

28. ( ) , af x x a , (0, +) : 1( ) af x x ,

1a ax x

y = x = e lnx u = lnx, y = e u. ,

1 -

14

29. f(x) = x , > 0, R ( ) lnxf x a ,

lnx xa a

y = x = e xln u = xln, y = e u. ,

30. f(x) = ln |x|, x R*, R* 1ln x x

x > 0, 1ln lnx x x

x < 0, ln |x| = ln (x), , y = ln(x) u = x , y = lnu. ,

1ln x x

.

31. y = f(x) x x0

x , y y = f(x) , f

x0, y x

x0 0( )f x .

32. Rolle

1 -

15

:

f :

],[ ),( ( ) ( )f f , , ( , ) , :

( ) 0f

, , , ( , ) , fC ))(,( fM x.

33.

:

f : ],[

),( , , ( , ) , :

( ) ( )( ) f ff

, , , ( , ) , f ))(,( fM .

y

O x

(,f ())

(,f ()) (,f ())

18

(,f ())

a x

y

M(,f ())

A(a,f (a))

20

1 -

16

34. f .

f ( ) 0f x x , f .

1 2,x x 1 2( ) ( )f x f x .

1 2x x , 1 2( ) ( )f x f x .

1 2x x , ],[ 21 xx f . , 1 2( , )x x ,

2 12 1

( ) ( )( ) f x f xfx x

. (1)

, ( ) 0f ,, (1),

1 2( ) ( )f x f x . 2 1x x , 1 2( ) ( )f x f x . , , 1 2( ) ( )f x f x .

35. gf , .

gf , ( ) ( )f x g x x , c , x :

( ) ( )f x g x c

f g x

( ) ( ) ( ) ( ) 0f g x f x g x . , , gf . , C , x ( ) ( )f x g x c , ( ) ( )f x g x c .

y

O x

y=g(x)+c

y=g(x)

22

1 -

17

36. f, .

( ) 0f x x , f .

( ) 0f x x , f .

x1 x2 x1 < x2. f(x1) < f(x2). , [x1,x2] f ... , (x1,x2) ,

, f(x2) f(x1) = f () (x2 x1 ( ) 0f x2 x1 > 0, f(x2) f(x1) > 0, f(x1) < f(x2).

37. f, Ax 0 ;

f, , Ax 0 , 0 ,

0( ) ( )f x f x 0 0( , )x A x x .

0x , )( 0xf f.

38. f, 0x A ;

f, , 0x A , 0 ,

0( ) ( )f x f x , 0 0( , )x A x x .

0x , )( 0xf f.

39. Fermat .

1 -

18

f 0x . f 0x , :

0( ) 0f x

f 0x . 0x f , 0 ,

0 0( , )x x

0( ) ( )f x f x , 0 0( , )x x x . (1) , , f 0x ,

0 0

0 00

0 0

( ) ( ) ( ) ( )( ) lim limx x x x

f x f x f x f xf xx x x x

.

,

0 0( , )x x x , , (1), 00

( ) ( ) 0f x f xx x

,

0

00

0

( ) ( )( ) lim 0x x

f x f xf xx x

(2)

0 0( , )x x x , , (1), 00

( ) ( ) 0f x f xx x

,

0

00

0

( ) ( )( ) lim 0x x

f x f xf xx x

. (3)

, (2) (3) 0( ) 0f x .

.

40. f ),( , 0x , f . :

i) ( ) 0f x ),( 0x ( ) 0f x ),( 0 x , )( 0xf f. ii) ( ) 0f x ),( 0x ( ) 0f x ),( 0 x , )( 0xf f.

iii) A ( )f x 0 0( , ) ( , )x x , )( 0xf f ),( .

y

O

f (x0)

x0 x0+ x0 x

33

1 -

19

i) E ( ) 0f x 0( , )x x f 0x , f ],( 0x .

0( ) ( )f x f x , 0( , ]x x . (1)

( ) 0f x 0( , )x x f 0x , f ),[ 0 x . :

0( ) ( )f x f x , 0[ , )x x . (2)

y

O

f(x0)

f0

a x0 x

y

O

f0

a x0 x

35a

f (x0)

, (1) (2), :

0( ) ( )f x f x , ( , )x ,

)( 0xf f ),( . ii) .

y

O

f0

a x0 x

y

O

f0

a x0 x

35

iii)

( ) 0f x , 0 0( , ) ( , )x x x .

y

O

f>0

f>0

a x0 x

y

O

f >0

f >0

a x0 x

35

f 0x ],( 0x ),[ 0 x . , 1 0 2x x x 1 0 2( ) ( ) ( )f x f x f x . )( 0xf

f. , , f ),( . , ),(, 21 xx 1 2x x .

1 2 0, ( , ]x x x , f ],( 0x , 1 2( ) ( )f x f x .

1 -

20

1 2 0, [ , )x x x , f ),[ 0 x , 1 2( ) ( )f x f x .

, 1 0 2x x x , 1 0 2( ) ( ) ( )f x f x f x .

, 1 2( ) ( )f x f x , f ),( .

, ( ) 0f x 0 0( , ) ( , )x x x .

41. :

. f ;

. f ;

f . :

f , f .

f , f .

42. f ;

f .

( ) 0f x x , f .

( ) 0f x x , f .

43. ))(,( 00 xfxA

f;

f ),( , 0x .

f ),( 0x ),( 0 x , ,

1 -

21

fC ))(,( 00 xfxA , ))(,( 00 xfxA f.

44. ))(,( 00 xfxA f f

0( )f x ;

))(,( 00 xfxA f f

, 0( ) 0f x .

45. f ;

0

lim ( )x x

f x

,

0

lim ( )x x

f x

, 0x x

f.

46. f ; ( );

lim ( )x

f x

( lim ( ) )x

f x

, y f ( ).

47. y x f , ;

y x f , ,

lim [ ( ) ( )] 0x

f x x

,

lim [ ( ) ( )] 0

xf x x

.

48. de l Hospital

1 -

22

1 ( 00 )

0

lim ( ) 0x x

f x

, 0

lim ( ) 0x x

g x

, 0 { , }x 0

( )lim( )x x

f xg x

(

), :

0 0

( ) ( )lim lim( ) ( )x x x x

f x f xg x g x

.

2 ( )

0

lim ( )x x

f x

, 0

lim ( )x x

g x

, 0 { , }x 0

( )lim( )x x

f xg x

( ), :

0 0

( ) ( )lim lim( ) ( )x x x x

f x f xg x g x

.

49. f ;

f . f

F

F'(x) = f(x) , x .

50. f . F f , :

G(x) = F(x) + c , c R , f

G f G(x) = F(x) + c , c R

G(x) = F(x) + c, c R f ,

:

G'(x) = (F(x) + c)' = F'(x) = f(x), x .

G f . x F'(x) = f(x)

G'(x) = f(x), G'(x) = F'(x), x

1 -

23

, c , G(x) = F(x) + c, x

51. -, :

. ...f x dx f x dx

. ...

a f x dx

. 0f x , f x dx

... ... . f [,]. f(x) 0 x

[,] f ,

f x dx

... ...

f x dx f x dx

0a

af x dx

0f x , 0f x dx

f [,]. f(x) 0 x [,]

f , 0f x dx

52. f x dx

0f x ;

:

f(x) 0 x [,], f x dx

()

f xx

x = x = (. 11). , f x dx E

1 -

24

53. cdx

c > 0;

cdx c .

54. . , :

f , , , :

...

...af x dx f x dx f x dx

. f(x) 0 < < , ;

. af x dx f x dx f x dx

. () = (1) + (2), 1 f x dx

, 2 f x dx

a f x dx

( 13)

1 -

25

55. , : f ,

, xF x f x dx x

f .

...x f x dx

x

x f x dx f x

56. :

:

f ' [,]. G f

[, ], :

f x dx G G a

, xF x f x dx

f [,]. G f [,], c

R , :

G(x) = F(x) + c (1)

(1), x = , :

a

aG a F a c f x dx c c

c = G().

,

G(x) = F(x) + G(),

, x = , :

aG F G a f x dx G a

1 -

26

:

a f t dt G G a

57. ;

a f x g x dx f x g x f x g x dx

, f, g [,].

58.

;

2

1( ) ( ) u

a uf g x g x dx f u du

, f , g , u = g(x) , du = g(x)dx u1 = g() , u2 = g()

59. g(x) 0 x [,]

g, x = x =

x = x = ;

g x dx

60.

f , g x = x = ;

:

(i) f(x) g(x) x [,]

(ii) f, g [,].

1 -

27

, :

( )f x g x dx

f(x)g(x) [,], f , g

x = x = :

( )f x g x dx

1 -

28

- 2000-2015

,

, .

1. f x0 g x0 , gof x0.

2. f, g IR fog

gof, .

3. f: IR 11, y f(x)=y

x .

4. f : R 1-1 ,

x1 , x2A : x1 = x2 , f(x1)=f(x2) 5. f f y = x, 1f

6. f:AR 11,

1f f -1( f(x))=x, xA f( f -1(y))=y, yf(A) 7. f:AR 11,

8. f [, ] (, ],

f [, ] .

9. C C f 1f

y = x xOy xOy.

10. C C f -f

xx.

11. 11, .

12. , 1-1 ,

.

13. f (,),

1 -

29

(,),

x xlim f (x) = lim f (x)

14. f

f .

15. f()

f .

16. f [, ] x0(,)

, f(x0)=0, f() f()0.

17. f

, x x , .

18. f ()

0x , 0( ) ( )f x f x x . 19. f ,a 0f ,

( ) 0f , 0f .

20. x x x

21. : x

xlim 1x

22. : x 0

x 1lim 1x

23. 0 0, ,x x . :

0 0x x x xlim f (x) lim (f (x) ) 0

24. 0 0, ,x x .

:

1 -

30

0 0 0x x x x x xlim f (x) ( lim f (x) lim f (x) )

25. f x0 0

lim 0x x

f x

0

lim 0x x

f x

26. 0x x

lim

(f(x)+ g(x)) 0x x

lim f (x)

0x x

lim

g(x)

27. 0

lim 0x x

f x

, f(x) > 0 x0 . 28.

0

lim 0x x

f x

, f(x) < 0 x0

29. > 1, xxlim 0

30. 0

1 -

31

42. f IR. ,

[, ] , f

Rolle.

43. f

x . f

f(x) > 0 x .

44. f

. f ,

.

45. f

. f

,

.

46. f () ,

.

47. f [, ]

x0[, ] f . T

f(x0)=0.

48. f

. f(x)>0

x , f .

49. f R

, f(x)>0 ,

x.

50. f (, )

xo. f (, xo)

1 -

32

(xo, ) , (x0 , f(x0)

f.

51. f ,

f

.

52. f x0

. f x0 f(x0)=0, f

x0 .

53. f ' (, ),

x0 , f . f(x) > 0

(, x0) f(x) < 0 (x0 , ), f (x0)

f .

54. f [ , ]a [ , ]x a 0f x , 0f x dx

.

55. f , , , :

af x dx f x dx f x dx

56. f x dx xx

xx.

57. f [, ]

0f x [ , ]x a ,

f , ,x a x xx a f x dx

58. f

, :

x f t dt f x

1 -

33

59. f, g , :

a f x dx f x g x f x g x dx

60 . :

( ) ( ) ( ) ( )g x

af t dt f g x g x

61. f R , :

( ) ( )aa f x dx xf x xf x dx

62. ox x

lim f x

ox x1lim 0

f x

63. f () ,

.

64. 2 . 65. f

. f ,

.

66. f, g fog gof,

fog=gof.

67. x (x)= x.

68. f [, ]. f(x)0

x[, ] f ,

f (x)dx 0

69. ox x

lim f (x) 0

f(x)>0 xo, ox x

1limf (x)

.

70. f (,x0)(x0,)

:

1 -

34

o 0 0x x x x x x

lim f x lim f x lim f x

71. 0 < < 1 , x

xlim 0

.

72. f

. f , f (x) > 0

.

73. :

g(x)

f(t)dt f g(x) g (x)

.

1 -

35

- . .

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