II. .dimitrmp@sch.gr,2010: . . : - - , -- - 2007. . , G. Polya, (1998, .161) : O . . ( ).. , , , ( ) . .

. . . - . . . - : - - 2

, , 1

( 1.1 - 1.8)

. ( 1.1-1.3)

1. . , , o, , , . . . . 2. . ) , y = f(x): x y, .. x(t), (), g(y), Q(P) . , . ) y = f(x), x = f-1(y), x ( y) . .. , , . ( - , . ), . 3. : , ( ) . , f, g ( . ) ( ) f() g(). 4. , . : , .. (x) (x), , D , D , , ()(x) = ((x))

. . . - . . . - : - - 3

: , x (), (. ) (x) ( !), D = {xD (x) D} , : ()(x) = ((x))= , . . ()(x) D . 5. () . ) ( ). ex, lnx . ) ( ). 6. . . , , . . . (. ) ( , )

A o , () < () < , () < () > () = () = (1-1).

( , ) . . ( , , , , ). , . 7. () . , ( ) (, ], (, ), . (. ) (, ] (, ), ..

(x) =x1 x > 0 (x) = -x x 0

:

(. ) (, ], (, ) , . (. ) (, ] (, ) = (, ) ( . 11, V.1) .

. . . - . . . - : - - 4

8. () () : ) () :

2 + 2 2 ( + ) 2 4 ( = )

> 0 21 + ( = 1).

.. 2x9x12)x(f

+= , 9 + x2 6|x| |f(x)| 2

x = 3, -3 . ) .

.. f(t) = 1t2 = [1, 5], . , 1 t 5 f(5) f(t) f(1), f , . (1, 5) f(5) < f(t) < f(1) .

: , . ( )

) , .. () = [2, +), 2 ( x (x) = 2 ) . 9. - 152 1-1 . 153 ( , ) . 10. 1-1 , .. (x) = ex + 5x. 11. . :

() . .

. . . - . . . - : - - 5

: A - . () 1-1 . (x) = 2+ 1x . = [1, +). yR x y = (x), y = (x) x ( y). y = (x) y = 2+ 1x y - 2 = 1x x 1 = (y - 2)2 y 2 x = 1 + (y - 2)2 , y 2. y 2 x = 1 + (y - 2)2 1 (x) y = (x). ([1, +)) = [2, +) ( [2, +) ([1, +)) ([1, +)) [2, +)). , y 2 x = 1 + (y - 2)2 y = (x), 1-1, -1(y) =1 + (y - 2)2 , y 2. . () , - - (=..) : x, y, . . , (.. S = 2t + 3, t , S ( t, S, 2, 3). , , , . .. . . (.. x y = f(x), .. f(x) = xex) , , , . 12. . . , (x), xA, .. . , (), < . , A = (), = (). < () < () > -1() > -1 ().

. . . - . . . - : - - 6

13. , -1 y = x. . . , -1, : ) (x) = x (x) = -1 (x) (xA()). , . . (x) y = x

, -1 , y = x. ) , (x) = -1 (x) (x) = x (xA()). ( , ) (. ) y = x. A .. (x) = xex-1, x 0, ) , . 1

f(x) =x1 , x > 0, ()

f-1 (x) = x1 , x > 0 .

2 f(x) = x1 , x 1, f-1 (x) = 1 - x2, x 0, ( [0, 1] )

(0,1), (1,0),

2

15,2

15 .

( f(x) = f-1 (x), x [0, 1] (y = f(x), x = f(y)). 1 f, f-1 y = x. , , , , , , . site 2 f (.. f3(x) + f(x) = 3x) f 1-1. y = f(x) x = g(y), f-1 (y) = g(y). : x = g(y) y = f(x) f.

. . . - . . . - : - - 7

: =R 3(x) + (x) = x xR. ) N 1-1, ) . ) . ) , yR xR= y = (x) , xR y = (x). y = (x) y3 + y = x yR xR= y = (x) x ( -1(y)= y3 + y, yR) ( x, , , x = x =..). : x, yR x = y3 + y y = (x) x, yR y3 + y = x, y3 + y = 3(x) + (x). y = (x), g(t) = t3 + t 1-1 R , , y = (x), y = (x) x = y3 + y x, yR yR () x = y3 + yR y = (x). R, ( 1-1) -1(y) = y3 + y, yR , g(t) = t3 + t 1-1 (. 17,18). : , g R g((x)) = x xR. R ) g, 1 -1, ) g ( ) ( )

. . . - . . . - : - - 8

:

1. N x(t) = t1t1ln

+

. (. .) x(t) . 2. . . g(x) = x3 x + 1, h(x) = 2x + 3.

3. (y) = ylny1

ylny + y = 1 , .

4. () =

74

73

+

- 1.

) ( ), ) 3x + 2 2x = 7x x = 1. ) (x 3 + x) = (3 - x). 5. ) 2000 () 2007 R, : . . . . . ) M [0, 1) 1-1. : . . . . ) g 1-1, (gg-1) () = A. . . g(A) ) f 1-1, ff-1, f-1of . . . . 6. . : . (, ) . . (2 - ) = 2 - () , (2 - ) . = . . (2 - ) = () , (2 - ). . . . ; 7. 2 +2 2, +1/ 2 , (>0),

x(t) = 2t1t4

+.

f(x) = 22

x11x1+

++ . f(x); (.2, 2, -2, )

. . . - . . . - : - - 9

8 |||||| ( ,

) h(x) = |x|+ 2x8 . (. 4) 9. x(t) = t2 - 4t +6, t 2 . . . x(t) .

10. 1e1e)y(x y

y

+= ,

.

11. (y) = ln(-y + 2y1+ ). ) R , ) 1-1 , ) . 12. f, g 1-1 . , ) fog, gof 1-1, ) (fog)-1 = g -1o f -1, (gof)-1 = f -1o g -1 .

13*. x(t) = ee

et

t

+, tR.

) x (0, 1) tR x = ee

et

t

+.

) f() = x() + x(1- ) , R ,

) =

++

+

109x...

102x

101x . (.9/2)

14. y = 1/x . 15*. f, g R g(R) = R (fog)(x) = x x R. , ) f, g 1-1, ) f =g -1, g = f -1. 16**. f, g R | f(t) - g(t)| < tR . . f2(t) + g2 (t) + 2f(t)g(t). 17*. f R f(x) > 0 xR,

x)x(fln)x(f1 = xR,

(t) = tlnt1 . ) f, ,

) f , ) f(1). (. 1) 18**. ) (t) = et/t, t 1 . ( ). ) f(x), x e, ef(x) = x f(x) f(x) 1 x e . f 1-1 .

. . . - . . . - : - - 10

. ( 1. 4 - 1. 7 )

1. . :

f R{+, - }: f , . |x|xlim 2

0x

, lim

+.

f R{+, - }: ( f )

( ), .. |x|

xlim0x

. ,

. , , ( ) .

. . . . . , . , , ( ). . R{+, -}, ( ). ( ). ( ), ( ) . ..

602lim

1 +

+.

( R, ). 6

x 0lim( 1) 1 0

+ = > , 2010 1 + (>) 0 0 .

2. 1 (.165). : .. x2 > 0 0 (.. (-1, 0)(0, 1)), ( ) 0xlim 2

0x=

,

. . . - . . . - : - - 11

3. 2 (.166). f(x) < g(x) , )x(glim)x(flim

xx < ,

( ) )x(glim)x(flim

xx ( f(x) g(x)).

.. x2 < 2x2 , x 0, 0x2limxlim 20x

20x

==

.

4. To ( ) 166, (.169) f x0 . . , .. )x(glim)x(flim))x(g)x(f(lim

xxx = ,

1)x

|x|(x

|x|(limx

=

, x

|x|lim0x

,

x|x|lim

0x.

, .. f + g f ,

g f - g . ( ) . 5. ( 5, .166). |||)t(f|lim

t=

)t(flim

t=

( ).

.. |1||x|lim 21x

=

1xlim 21x

, |1|x

|x|lim0x

=

x

|x|lim0x

.

, 0|)t(f|limt

=

0)t(flimt

=

(-|f(t)| f(t) |f(t)| ) .

6. , xR 1x

xlim0x

=

.

R 180

lim

0=

(

x

180

= )

7. fog x0 (.173) . g(x)uo x0, (. 128 . . 2007-2008). , : , , 8. ,xlim,xlim

xx ( -

. - ). , .

. . . - . . . - : - - 12

*: xlimx

=+

R{+, -}. -1x1

+ -, R. ulim)x(lim

ux==+

++ R.

=++

)x(limx

)xx(limx

=++

R

= - xlimx

=+

, = 0.

= /2 0xlimx

==+

x, x . xlimx +

2x = x. -.

9. f(x) g(x) R{+, -},

+=

)x(flimx

+=

)x(glimx

.

=

)x(glimx

=

)x(flimx

.

. 10. : , . .

1x

1xlim

2

1x

,

|u|

ulim

0u

( 2) , . 11. , ( R{+, -}) :

(+) + (+) = +, (-) + (-) = -, =+=+ 0

1,01 , 01,01 =

=

+ .

12. x, > 1 0 < < 1, lnx, !.

. . . - . . . - : - - 13

13. : , : : )x(flim

x=

R, g(x)limf(x)g(x)lim

xx = (), ..

1 = 0x

1x0limx

1xxlim0x0x

=+

=+

, .

: 0 g(x) , . () 14. . , John Wallis 1655. 1.000. O Wallis Arithmetica infitorum 15. :

. A Rx)x(flim

x=

0)x(flim

x=

.

: )x(x)x(f)x(f

= .

x)x(f)x(g

= ,

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

x=

+ 0 < < 1.

. . . - . . . - : - - 14

. ( 1. 8)

1. . (. ) (, ], (, ) , . (. ) (, ] (, ) = (, ) ( .3, .7, ). * . (, ], (, ). . (, ), , < < () > () ( ). < < . < < < () > () > () ,

)()(lim

+

() () > (). () > () (

). 2*. 1-1. 1-1 , ( ) ( ). 3. Bolzano. Bernard Bolzano (1781 1848) , . , ; 1824, Niels Abel (1802-1829) , ( 4 R - ) 5 .

, , :

; x- ;

; , . , , Bolzano. , ( ), , . 4. A () . lzano ( Rolle) 2 (supremum) . ( ).

. . . - . . . - : - - 15

. lzano : (, ), f()f() < 0, .. f() = 2 - 1, [-2, 2].

: f()f() < 0 ( ) . - , .

lzan: ( . lzan) ( ) .

. lzan. f (, ) , R{-. +}, ( ). (, ) f () = 0. (. A . lzan) 5. (...) , -, . lzan ( ). 6. To ... . , Darboux (, 1842-1917) , ..., . .., () . , . . . : [, ]. ( ) () () = (), [, ] (x) = () x(, ], (, ) () (). [, ] , ... [, ]. 7. : olzano, [, ], f(), f(): + -. .. f(t) = t2007 + t2000 + 3t + > 0, +=

+)t(flim

t , >

0 f() > 0. =

)t(flimx

, < 0 f() < 0.

[, ] f()f() < 0 8. ( ) ( .194) . .195, (. . . .) ... 9. ... () .

. . . - . . . - : - - 16

. (x) = x + 1 0 < x 1 (x) = x2 -1 x 0 [-1, 1] 2 0. (x) = x (0, 2) . 10. .... , : f . [, ] f([, ]) = [f (), f()], f . f([, ]) = [f()], f ()] ( . ). () . 11. (.196) (. ) , f (, ), , R{-. +}, ( R{-, +}) = )x(flim

x +, = )x(flim

x .

() , . . 12. : f . [, ) f([, )) = [f(), ), f . f([, )) = ( , f()] (, ]. 13. (, )(, ), ((, ) (, )) = ((, )) ((, )). ( ) 14. (.201-203) , . 15. . Bolzano: , ( ). ... . . 7(ii) .200 : f 2 (x) = x2 xR. ( ) f (x) = x, xR ( f).

. . . - . . . - : - - 17

x > 0. f(x) 0 x (0, +) ( . Bolzano) f(x) = x x > 0 f(x) = - x x > 0.

x < 0. f(x) 0 x(-, 0) ( . Bolzano) f(x) = x x < 0 f(x) = - x x < 0. x = 0 f(x) = 0.

4 : f(x) = x, xR f(x) = -x, xR f(x) = |x| , xR f(x) = -|x|, xR . R 2(x) = ex(x) xR. xR, , (x)((x) - ex) = 0 (x) = 0 (x) = ex (1) (x) = 0 xR,

. R () 0, (1) () = e.

(x) = ex xR. () e , () = 0. < .

() = 0 < e < e = (), .. = e (, ) e = (). (1) () = e , = . > . (x) = ex xR. , = 0 (x) = ex , xR. ( : ((x) - ex/2)2 = (ex/2)2 , (x)=((x) - ex/2) 0,) : : , R ( ) (x)((x) - (x) = 0 xR. (x) 0 xR, = 0 = . ( : (x)((x) - (x) = 0 ((x)- (x)/2)2= ((x)/2)2 , (x)- (x)/2 , )

1. , 3 . 2. , (. 28). 3. (. .7). 4. 1-1 . 1 . . , .

. . . - . . . - : - - 18

: 1. . 2005)t(ftlim 2

0t=

, =

)t2(flim

0t

. 0 . 2005 . + .- .

. 0)x(f

1lim 20x=

=

|)x(f|ln

0x 3elim

. 0 . + . - . . )t(glim)t(flim

tt < f(x) < g(x) x : - .

2. f [, ] m, , i) [, ] f( ([, ] A.E. . lzano .. . . ii ) f(([, ]) [, ] A.E. .lzano .. . . 3. H () =52008 -61453 + 1, 0 1, 1/2008 . . . 4. f (, ) < = )x(flim

x +, = )x(flim

x f

. . . . . . 5. (, ) [, ], : . . . . . . 6. (, ) (, ) (, ) . . . . . 7. (x) [0, 1] (0) (x) (1) x[0, 1] . ([0, 1]) [(0), (1)] . ([0, 1]) = [(0), (1)] .[(0), (1)] ([0, 1])

8. 15223e7lim

2007xx2xxx6lim x

xx

xn

25

x +

=

+

+++

, n N,

. n < 5 B. n > 5 . n = 5 . n = 3 9.

=22t

2)-(t+

t = 2,00001010. (.4)

. . . - . . . - : - - 19

10. (x) = 2x1x ++ . ) 1-1, ) , ) (x) + . 11. (y)= 12y8y2 + , y 6. ) ( ) ) 1-1 -1(x) = x, ) f(y) = (y)/-1(y) 0 +. (.() .., 1) 12. (t) = 2 ( )t5tt + . )

+110 ,

) , ) +.

13. A zC t t(x) = x

|zx| x > 0

t(x) = )1x|z|x)(i2z( 23 + x 0, , z

.

14. . R 2x x 3

lim 2x 3 x 3

=

. (. )

. x22

6x3)x(f= 2

[0, +); (. -6)

15. A. 0)2)x(f)x(f2)x(f(lim 231x

=+++

)x(flim1x

. (A.-2)

B. f R t3 + 2 + tf(t) = 2f(t) + 5t tR. 16. ) (x) = x - e-x , ) R xex =1+ex , ) . . -1(x) x- . 17. z(x) = 1+ (1 - x)i, x 1 (x) = |z(x)| 2, x 1. ) (x). ) 1-1 . ) z(x), x 1, . ) , -1. 18. ex + x1897 + 1 = 0, , .

. . . - . . . - : - - 20

19. ( , !...). 8 5 . ( ) . , 8 5 . , . 20. : . x3 = 2008 , *. x = , > 0, *, ( - ). 21. () y = f(x) 4x2 - y2 = 4 x . ; 22. . f(t) = 8ln(t + 1) - 2t + c t , t 0, cR . ) f(t). ) t = 8 , t = 10 . ( ln11 2,4). ( . 2000) 23*.) . ) h(t) = t2007 - 1707t + 1913 R 24. g

1xxxlim)x(g 2

212

v +

+=

+

+, x 0.

25.) = 222)(

+

+ , , R* 2.

; ) f, g R +=+

))x(g)x(f((lim

x, R{+, -

}. ) +=+

))x(g)x(f((lim 22x

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

)x(g)x(flim 22x

=+

+

26. (x) R (0) + (1)+ (2) + (3) = 2010. ) (0), (1), (2), (3) 502,5 . ) 502,5 . ) !. 27. f [, ] , [, ]. (7f() + 3f())/10 f. () f ;

. . . - . . . - : - - 21

28*. R. |( + 1)| 2 R, (1) 1.

1x)x(lim

1x .

29*. [0, 1] (0) = (1).

) (x) = (x +31 ) [0,

32

],

) (x) = (x +41 ) [0,

43

] .

30*. R f f 2(t) = 2tf(t) tR. (:.13, 15.A/ . 0, 2t, t+|t|, t-|t|) 31. >1 R () R (1) = 2. ) ()/ , > 0, (, ), ) +. 32*.) ()0 (x)0 . ) h R h(x)x = 0 xR. h . 33*. (x) R (x)(-x) < 0 x 0. (0) = 0. (: ) 34*. R (() - e)(() -1- e ) = 0 R. () = e R () = 1+ e R. (. . 14, ) 35*. [, ] 1-1 () < (). < x < () < (x) < (). ; (: , ...) 36*. [, ] () = (). , [, ] ( - )/2 () = (). (. [, ] )

37*. R 0x

)x(limx

)x(lim 1913x1913x==

+.

(x) + x1913 = 0 . 38*. f R y = x + > 0. ) =

+)x)x(f(lim

x,

) Cf . -

* * * *