ii. ΣΥΝΑΡΤΗΣΕΙΣ ‐ ΟΡΙΑ ‐...
TRANSCRIPT
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II. [email protected],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 . -
* * * *