Γενικά Θέματα Στην Κατεύθυνση - Το Κλασικό Αρχείο!
45
Γενικά Θέματα στην Κατεύθυνση της Γ΄ Αγαπητοί συνάδελφοι - Φίλοι μαθητές ! Προσπάθησα να συγκεντρώσω ηλεκτρονικά μερικά γενικά επαναληπτικά θέματα που έφτιαξα ο ίδιος ή συνάντησα , σε ξένα κυρίως βιβλία ή περιοδικά . Η πληκτρολόγηση διαρκεί δυστυχώς πολύ και για το λόγο αυτό ο αριθμός των θεμάτων είναι σχετικά περιορισμένος. Τα θέματα λύθηκαν όλα κατά την προηγούμενη σχολική χρονιά μαζί με το υς μαθητές μου , αλλά και από άλλους συναδέλφους .Έχω και τις λύσεις όλων των ασκήσεων , αλλά αυτές δυστυχώς δεν είναι πληκτρολογημένες. Ωστόσο μερικές από αυτές μπορείτε να τις βρείτε στα βιβλία μου , αν και οι περισσότερες ασκήσεις είναι νέες . Μπάμπης Στεργίου - Μαθηματικός Α. ΜΙΓΑΔΙΚΟΙ ΑΡΙΘΜΟΙ Θέμα 1 ο ∆ίνονται οι διαφορετικοί μεταξύ τους μιγαδικοί αριθμοί α ,β, γ που έχουν ίσα μέτρα και α + β + γ = 0 Να αποδείξετε ότι α) αβ + βγ + γα = 0 β) α 3 = β 3 = γ 3 γ) Οι εικόνες των αριθμών α , β, γ είναι κορυφές ισοπλεύρου τριγώνου δ) Οι εικόνες των αριθμών αβ , βγ, γα είναι επίσης κορυφές ισοπλεύρου τριγώνου. Θέμα 2 ο ∆ίνονται οι μιγαδικοί αριθμοί α, β, γ οι οποίοι έχουν εικόνες τα διαφορετικά σημεία Α, Β, Γ αντίστοιχα. Αν α β +β γ +γ α = 1 και w = α γ α β − − , να αποδείξετε ότι: α) w w = . β) Ο w είναι πραγματικός αριθμός. γ) Τα σημεία Α, Β, Γ είναι συνευθειακά. Θέμα 3 ο ∆ίνονται οι μιγαδικοί αριθμοί z, w με z,w ≠ 0, w 2 = z και z 2 = – w . A. Να αποδείξετε ότι: α)|z| = |w| = 1. β) z 1 z = και w 1 w = . γ)z = – w. δ) w 3 = – 1 και z 3 = 1.
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Γενικά Θέματα Στην Κατεύθυνση - Το Κλασικό Αρχείο!
Transcript of Γενικά Θέματα Στην Κατεύθυνση - Το Κλασικό Αρχείο!
-
- !
, .
.
, .
, .
, .
-
. 1 ,, + + = 0
) + + = 0 ) 3 = 3 = 3
) , ,
) , , .
2
, , , , .
+ + = 1 w =
, :
) w w= . ) w . ) , , .
3
z, w z,w 0, w 2 = z z 2 = w . A. :
)|z| = |w| = 1. ) z1z =
w1w = .
)z = w. ) w 3 = 1 z 3 = 1.
-
. ) z, w
) z2004.
4
I. , , || = || = || = 1
:
) V =
+++++
U =
++ .
) w ++
++= 1 , , , w .
) | + 1 | + | + 1| + | + 1| 2 , | | 2 Re( ) 1 ) | + 1 | + |2 + 1| + |3 + 1| 2 ) | + - | + | + - | + | + | 3 II. , , + + = 0.
. :
) + + = 0 , 2 + 2 + 2 = 0 |2 + 2 + 2 | = | 2 2+ 2 2+ 2 2|
) 2 = , 2 = , 2 = 3 = 3 = 3
) Re( ) = Re( ) = Re( ) = 21
B. :
) , , .
) | | = | | = | | = 3
. :
i) , , , , .
ii) , , = 2 , = 2 , = 2 ,
.
iii) .
5
, , , , , ,
.
. + = + , , , ,
)
-
)
. + = + . + = +
. + = + , + = +
6
f: RR ( ff D )(x) = 2 x, xR. :
) f(1)=1. ) f R.
) f . ) f
7
f: (0,+)R : xylnxy yf(x) + xf(y) f(xy), x,y > 0.
:
) f(1) = 0. ) f
x1
= 21x
f(x), x > 0. ) f(x) = xlnx, x>0.
8 f: RR f(xy) = f(x)f(y) f(x+y) = f(x)+f(y)+2xy, x,yR. :
) f(0) = 0, f(1) = 1 f(1) = 1.
) f .
) f(x) = x2, xR.
9
w = zzzz
+
2
2
44
, z C
) 24z + z =0
-
) w = zzzz
+
2
2
44
, zC , C
z ) z1 , z2 , w1, w2
, x(0,1) |xz1-z2|+|xw1-w2|= x
.
10
f: IRIR f(xy) + f(x) + f(y) + 3 = x + y + xy, x,y R. ) : f(1) = 0.
) f. ) 1)x(fx2xlim 2x ++ .
11
f: RR : f 3 (x) + f 2 (x) + f(x) = x5 + 5x 3 20x + 4,
xR, ,R 2 < 3. ) f.
) f .
) f(x) = 0 (0,1).
12
f(x) = x3 x 2 lnx , x > 0.
) f . ) : x3 x2 + lnx, x > 0. ) x3 x2 + lnx, x > 0, : = 1.
13
f x0=0, f (0) = 1 f(0) = 0, :
-
) = 2x
0x xxexlim .
) = x
)x(flim0x .
) 1xex)x(xflim x0x = .
14
f : IR IR , , f(x + y) = f(x)f(y) , x , y IR. ) f(0) = 1
) Cf xx
) f(x) > 0 , x IR ) f 0 , IR
) f , IR
) f 0 f(0) = 3 , :
i) f IR f(x) = 3 f(x) ii) f
iii) g(x) = f(x) e-3x IR f(x) = e3x , x IR. iv) Cf () : y = 3x + 1
15
f(x) = 2ln 2ln - x
+ax
, > 0
) f ) f
) > , ln
< 2 2 - 2
16
f(x) = (ex+1) x1
, x > 0 g(x) = xex (1+ex)ln(1+ex).
-
:
) g . ) g(x) < 0, x>0.
) f (0,+).
17
f:(0,+)R f(xy) + f
yx
= 2f(x), x , y > 0.
f(1) = 0 f (1) = 1, :
) f(x) +
x1f = 0, x > 0.
) f (x) + 2x1
f
x1
= x2
, x > 0. ) f(x) = lnx, x > 0.
18
f: (0,+)R f(1) = 0,
:
f(f (x))+f(x) = 0, x > 0.
:
) f (1)=1. ) ( ff D )(x) = x, x>0.
) xf (x)+f (x) = 0, x>0. ) f(x) = lnx, x > 0.
19
f : IR R 2f(x 3) f 2 (x) 1 x IR
) f (x) = 0 ( - 1 , 1)
) f (- 1) = f (0) = f (1)
) f 4
( - 1 , 1).
-
20
f(x) = (x - 2)e - x + 2x2 3x + 2
) f , f, f
) Na f(x) > 0 , x > 0 .
) f
) f
) f
) f(x) = 0 f
) Cf
) f f
21
f : IR IR f(x) - f(x)e = x 1 , x IR. ) f (x) f (x) f(x)
)
i ) x + 1 = xe , ii ) f(0) = 0 ) f(x) .
) 2x
)()( ' xfxxf , x IR. 22
f : IR IR f 3 (x) + 2f(x) = x , x IR ) f
) f IR
) f ) f
) f .
23
f: IRR , f(0) = 1 , : f() f (-) = 1 , R . :
-
) f () f(-) = f() f (-) , R. ) g(x) = f() f(-) IR. g();
) f () = f(), R . f() ;
24
f(x) = x + x2 x - 1.
) f (x) f (x)
) Cf (0 , f(0))
) f (x)
) f(x) ) f(x)
) x + x2 = x +1.
) x - 1 x (1- x ) , xIR
25
f f(x) = 12xx43x
) f
) Cf
) Cf
) Cf ) f
) f
) x 3 x 2 4x + = 0 , R
.
26
f:IR R f(1) = 3. F f
f(x) F
x1
= x3 , x > 0, :
) h(x) = F(x)F
x1
, (0,+).
) F(x)F
x1
= 1, x > 0.
-
) (x) = 3F(x)x
(0,+).
) f f(x) = 3x2 , x IR. 27
f(x) = x3 x2 + x + 2
) f .
) f ,
.
) f -1 (x) = x y = x
Cf-1
) f-1(x) = 1
) f(x) = 0 ( -1 , 0)
) f(x) = 0
) Cf , xx
x = 0 x = 1.
)
(2 - 3)3 = (2 3)2 - 2 + 4
) = 3 1
2f (x)dx
28
f , , :
f(1) = 1 f (x) = x(4 - f (x) ) x IR . :
) xf (x) = 2x2 , x IR ) f(x) = x2 , x IR . :
) Cf ( 2 , - 5).
) Cf .
-
29
f : IR IR
f 3 (x) + f(x) = x , x IR ) . ) f ) f(x) = 0 f(x). )
f(x) = 3x f (x) 2 f(x) f (x) , x IR )
=x0
2
4)x(f)x(xf3dt)t(f
) f IR f -1
30
f(x) = ex + x - 1
) f .
) f ,
.
) f -1 (x) = x y = x
Cf-1
) f - 1(x) = 1 ) f(x) = 0
) Cf , xx
x = 0 x = 1.
) = e 1
0f (x)dx
) () Cf , (1 , e) ) Cf , () yy.
-
31
f(x) =
1x
2
1 e ,x 0x
0 , x 0
> =
) f . ) f.
) I(x) = 1
xf (t)dt , x > 0
) Cf , xx x = 1.
) Cf.
32
f , IR ,
+x0 2 dt(t))f1(t = 2 0 dt(t)tfx - 4 10 dt(x)tfx , f(0) = 0 f (0) = 2
) f(x) = 1
22 +xx
) () C f , = 0 , = , .
) 3
10 cm / sec ,
ln10
. 33
, , ++ 0. 2 + 2 + 2=0 , , , :
) | + + |2 = 21
1
1 ++ . )
1
1
,1
=== . ) w = + + 2 , |++|=2.
-
34
f(x) = ln2
x + - +x
x ln - +
x
ln , > 0
) f
) 0 < < , ln2+
< +
ln + +
ln
35
f(x) = , 0 < < 1.
) f(x)
) 2x - 2 +2 ( +2)
) 2x 4 - -2 = (2 - 4) ( -2)
36
f(x) = xxln
, x > 0
) f(x) ) e > e
) e x x e , > 0 ) +1 > (+1) , > e ) 2 = x2 , x > 0
37
f: IR IR 2f(x) + f(1 x) = 3x2 2x + 4 x .
. :
) 2f(1-x) + f(x) = 3x2 4x +5
) f f(x) = 2 + 1
B. Cf ( 3 ,9). :
) Cf )
-
38
= xx x dxe 1
+ . :
) = dx1exxe
x
x + . ) = .
39
f: RR* : xe2
)x(f1
)x(f1 =+ , xR. f(0) = 1:
) += dxe))x(f)x(f( x . ) f(x) = ex, xR. 40
f: RR f(x)= 1 x0
x f(xt)dt e+ , xR. : ) f .
) g(x) = e xf(x) x 1 R.
) f(x) = (x + 1)ex, xR.
41
f : (0,+)R f(x) =
x
1f ( t )dt
e , x > 0.
A. N : ) f . ) f (x) = f 2 (x), x > 0.
) f(x) = x1
, x > 0.
. Cf . : ) = . ) , .
-
42
f, g f g g(x) + g( x) = 1, xR. ) : = 00 dx)x(g)x(fdx)x(g)x(f . ) : = 0 dx)x(fdx)x(g)x(f . ) : = dx
e1x2
2 x2 + .
43
f(x) = 2x4 tt dte , xR. ) f (x). ) f .
) = 2
03f(x)dx .
44
:
f(x)=42
2
x1)x1(x1++
g(x)= x
1
xdt)t(f , x0.
) : g(x)=0, xR*. ) =
1
dx)x(f , R*.
45
f IR < .
f(x t)dt+ f(x)dx , xR , :
) f(t)dt++ f(x)dx , R.
) g(x) = f(x t)dt + - f(t)dt
) g
) f() = f()
) f , R , f() = 0
-
46
f : (0, + ) IR f(x y ) = f(x) + f(y) + xy - x - y , x , y > 0 . ) f(1)
) f 1, (0, + ) ) f > 0, (0, + ) . f xo = 1 , f (1) = 2 ,
) f (0, + ) ) f ) = f(x)dx
47
F f : RR , F 2(x) F(x) F( -x) R , 0 . : ) F(0) = F() ) f(x) = 0 IR
48
f [0 , ] , 0 dx)x(f = 2 F f ) F(0) F()
) (0 , ) , f() = 49
f: IR IR g(x) = dt)tx(f , x R < .
) g ) g() =g()
) ( , ) , f(-) = f( ) 50
= ( )1
20
ln(x 1) dxln 2 x x
++ .
)
= 1
0dxln(2 x)2ln(2 x x )
+
)
-
51
, 0 < 1 x
x >0 , .
)
g(x) = x -
, > 0
) g(x) ) = e 52
f (x) = 2 + ln(x 1) g(x) = 2 - ln(x 1) , > 1,
Cf Cg .
A. :
) Cf Cg ) f g
. Cf Cg .
53
800 .
. 4 25 .
2 . o
20 .
40 ,
.
) ()
() = 40 ( + 16
+ 80 ) ,
.
) , ;
) .
-
54
f , g : IR IR , f(x) g(x) = x 4 , IR . y = 3 7 Cf + . ) :
i)x
)x(glim
x + ii)
12x3)x(xfx2x3)x(g
limx +
+++
) y = 2 3 C g + 55
f : (1 ,+ ) IR , f(x) = 2000 + )1xln( c 2000. y = c Cf
(1 , y1) (2, y2), 1 2
) N Cf .
56
f(x) = x 2 (e- x 2) g(x) = x 2 (x 2 e x)
) f(x) g(x) , x R ) f(x) = g(x) ) Cf , Cg x = 1
57
f(x) = 2x1x32x3x ++
) f ) f
) Cf ) Cf
) f ) f
)
x 3 + (1 - )x 2 + 3x 1 = 0 , IR.
-
) Cf , Cf
x = 1
58
f(x) = ++x
x t
t
dte1te
, xIR
) f (x) ) f (x) = x + x , xIR ) f -1 : [ , ] [ , ] ) Cf , Cf 1 x = 0 , x =
= 4 ..
59
f(x) = (x -1) lnx , x > 0 .
) f
) f ,
) f
) Cf
) = dx)x(f ) Cf , = e
60
f(x) = x1 2x
t ln tdt
t 1++ , x 0
) f(x) = ln x , x 0 ) Cf , y =1
. 61
f f(1) = f(1) = 0 )
1
0f(x)dx = 1 201 x f(x)dx2
) f(x) = 1
63 + , =
1
0f(x)dx
. 62
f(x) = xx2 + , x 0. ) f(x) ) f
-
) f 1
) = 2 1
0xf (x)dx
63
g(x) = 1e
xxx + f(x) =
x2
0t)dtg(x
. ) f g
) f ) f
. ) f = [- , ]
) f .
) Cf , xx
x = - , x =
) = -1
-f (x)dx
64
f x)f(e(x)'f 2x = x IR f(0) = 1 ) f (x) = 2 f (x) - f (x) x IR ) f ) () Cf ) Cf , () yy
65
f: RR : x1
x
1e)1x(dt)t(f , xR.
) f(1). ) f (1)=2, Cf (1,f(1)).
) =1xx)x(flim
3
1x
.
-
66
f: RR, f(x)f(y) = 2f(x + y), x,yR. ) f(0) = 0, f .
) f , f(0).
) R, f() = 0, f(x) = 0, xR ) f(0) = 2, f(x) > 0 , xR ) f x = , f() 0.
:
i) f x0 = 0. ii) f R
67
f: RR : f(x) + f( x) = 2003, xR. :
) dxx)x(fdx
x)x(f 4
4 2
4
4 2 = . ) 2003dxx)x(f4
4 2 =
68
f(x) = x 3 3x 2 + 3x .
i) f .
ii) Cf : y = x.
iii) Cf Cf 1.
69
f(x) = 2(x) + 2(x).
) : = 2
0
0dx)x(f2dx)x(f 2
0 0f(x)dx 2 2 f(x)dx
=
) = 2
0dx)x(f . ) :
2dx)x(xf
2
0= .
-
[email protected] 21 45
70
f: (1,+)R x>1 :
=
+ x0 t02 dtdu)u(f2x ++ x0 22 dt)t(f)t2t()x(f)x2x( . ) :
)x(f)1x(dt)t(fxx
0+=+ , x > 1.
) f.
) Cf , x = 0 ,
x = 1
71
f(x) = 23
x4x
.
) f .
) f .
) f. ) Cf.
) Cf f.
) R
x3 x2 4 = 0.
) f.
72
f: R R* : f(x) + f(x) = f(x)f(x), xR.
:
) f(x) 1, xR. ) 2dx
)x(fx
2= .
-
73
f(x) = x
0(xt)dt .
) f
) f
) f IR f (x) = 2x2 - f(x) ,x 0x
) f ) f.
74
f(x) = (1 x)lnx + 1 2xx+
) f
) f
) f ) f
) f(x) = 0 f
) Cf ) f
) (x x 2)lnx - x = 2x x 2 1 , IR ) Cf , xx x = 2
75
= x1 u1 t dt)du6e(f(x) 3 ) f(x)
) f
) f(x)
) f
) f(x) f(x) = 0
) = 10 f(x)dx
-
76
f: IR IR xf(x) +
x
1f(t)dt = f (x)
x IR . Na : ) f
) ( 0 , 1) , f () = f(1) + 10f(t)dt
) f(0) = f(1)
) ( 0 , 1) , f() = 2f(0)1 +
77
. f = (0 , + ) f() = + x1 2 dtxtf(t)x1 > 0
) f = (0 , + ) ) f B. f(x) =
x1xln +
, > 0
) f
) Cf ) f
) f Cf
) Cf ,
= 1 , = e
78
f : (0,+)R f(x) =
x
1f ( t )dt
e , x > 0.
A. N : ) f . ) f (x) = f 2 (x), x > 0.
) f(x) = x1
, x > 0.
. Cf . : ) = . ) , .
-
79
f : IR IR g(x) = x 1x
f(t)dt+ .
Cf f + y = 2x + 3 , :
) x
f '(x)A limx+
= = xlim (f '(x) 2x)+
) g g f
) (x) (x , x + 1) g(x) = f((x)) )
xlim f(x)+ =+ ) xlim g(x)+ =+
) 2xg(x)limx+
= 80
f : IR IR g(x) = x 1x
f(t)dt+ .
Cf f + y = 2x + 3 , :
) x
f '(x)A limx+
= = xlim (f '(x) 2x)+
) g g f
) (x) (x , x + 1) g(x) = f((x)) )
xlim f(x)+ =+ ) xlim g(x)+ =+
) 2xg(x)limx+
= 81
f(x) = 2lnx + x 2 1
) f .
) f .
) f .
) f .
-
) :
f(x) + f(x 2) = f(x 5) + f(x 10)
) , > 0 2 e( )( )( ) + = , = .
)
) 0 1( , ] 1[ , )+ . (1,0) ) 0 1( , ) 1( , )+ ) IR ) = 1 . x > 1 , x < x 2 , x 5 < x10 f(x)+ f(x 2) < f(x 5) + f(x 10) . 0 < x >1, x > x2 x5 > x10 ,
f(x)+ f(x 2) > f(x 5) + f(x 10)
) f() = f()
82
, , 3 = 3 = 3. :
) .
) 2 + + 2 = 0 + + = 0
) , , .
) 2 , 2 , 2 .
) 2 + + 2 = 0 ( - ) 2 = - 3 , :
| ( - ) 2| = |- 3| | | = 32 , = ||
| | = 32 | | = 32
| | = | | = | | =32
83
w = zzzz
+
2
2
44
, z C
) 24z + z =0
) w = zzzz
+
2
2
44
, zC , C
z
-
) z1 , z2 , w1, w2
, x(0,1) |xz1-z2|+|xw1-w2|= x 84
f: IR IR xf(x) +
x
1f(t)dt = f (x)
x IR . Na : ) f
) ( 0 , 1) , f () = f(1) + 10f(t)dt
) f(0) = f(1)
) ( 0 , 1) , f() = 2f(0)1 +
85
f g 1 IR, :( , ) * + f(0) = g(0) = 1 2 22f x f x g x 2g x g x f x 0'( ) ( ) ( ) '( ) ( ) ( )+ = + = , x 1> .
) f , g f = g
) f , g
) f
.
) ()
f , xx x = , x = +1 >0 , :
A Elim ( )+=
) f (-1,+ ) f(x) 0 ( 1, )x + , f (-1,+ ). f(0) =1 >0 , f . g .
( ) 0, ( ) 0> >f x g x ( 1, )x + .
2 2 22 ( ) ( ) ( ) 0 2 ( ) ( ) ( ) ( ) 0f x f x g x f x g x f x g x + = + = (1)
-
2 2 22 ( ) ( ) ( ) 0 2 ( ) ( ) ( ) ( ) 0g x g x f x g x f x g x f x + = + = (2) (1) , (2)
2
( ) ( ) ( ) ( ) ( )2 ( ) ( ) 2 ( ) ( ) 0 0 0( ) ( )
f x g x g x f x f xf x g x g x f xg x g x
= = =
c\ , ( )( )f x cg x
= , 1>x
0x = 1c = , ( ) 1 ( ) ( )( )f x f x g xg x
= = 1>x .
f gD D= , f = g ( 1, )x + .
) 22 ( ) ( ) ( ) 0f x f x g x + = 2 3
33 2
2 ( ) ( ) ( ) 0 2 ( ) ( ) 0
2 ( ) 12 ( ) ( ) 1 ( )( ) ( )
f x f x f x f x f x
f xf x f x xf x f x
+ = + = = = =
c\ , 21( ) x cf x = + , 1> x .
x = 0 , c = 1. 21 1 0,( )
= + >xf x
21( )
1f x
x= + , 1> x .
f , 1( )
1f x
x= + , 1> x ,
1( )
1g x
x= + , 1> x .
) f (-1,+ ) 2
( 1) 1( ) 0( 1) 2( 1) 1
+ = =
-
x -1. x = - 1 .
1 1lim ( ) lim lim 01 11
x x xf x
x xx
+ + += = =+ + .
y = 0 f + . )
1 1 1
1
1 1( ) ( )1 1
1(2 1) 2 1 2 2 2 1 . .
+ + +
+
= = = =+ +
+ = + = + = + +
a a a
a a a
a
a
E a f x dx dx dxx x
ax dx x a a
a
2 2( 2) ( 1)lim ( ) lim (2 2 2 1) 2 lim2 1
12 lim 02 1( 1 1 )
+ + +
+
+ += + + = =+ + +=
+ + +
a a a
a
a aE a a aa a
aa a
86
f(x) = lnx - xln 0 < < 1
) f , .
) x = x > , >
) Cf
) E() Cf xx x = ,
> 0 .
)
0
limE( ), lim E( ) + 87
IRIRf : xyxyffyxff 2))(())(( =+ IRyx , . .) f(0) f(1) = 1 f(1) = - 1
-
) f 1-1 , )(
1)1(xfx
f = , 0x
. f(1) = 1 ,
) xxffxf 2))(()( =+ IRx . ) f 1 1 ) f(x) = x , IRx . . f(1) = - 1 , :
) f(-1) = 1 ) xxffxf 2))(()( =+ IRx ) f 1 1 ) f(x) = - x , IRx .
.) x = y = 0. y = x x = 1 , x = f(1)
) f(x(f(x)) = x 2 x x1
)1(1x
fx
y =
.) y = 1 .
) .
) x x1
) .
. ) f(xf(x)) = x 2 x -1 x = - f(-1).
f(-1) = -1 f(-1) = 1 . f(-1) = -1 , f(- f(-1)) = 1 f(1) = 1 , .
) y = -1
) .
) x x1
).
88
f F IR :
f(x)F(y) f(y)F(x) x , y IR. f(0)=2 , F(0) = 1 , :
) f(x) = 2F(x) IR.
) f(x) = 2e2x , x IR.
) f
,
y'y. ''
-
89
2f x 1 x 1 x( ) = + + . ) f.
) f .
) 21
1
xI dxf x( )
=
x = - u
2006 20062005 2005
2005 2 2005 2
u xI du dx1 u 1 u 1 x 1 x
= =+ + + + + +
2006 20062005 2005
2005 2 2005 2
2006 2006 20072005 2005 2006
2005 2 2 2005
x x2I I I dx dx1 x 1 x 1 x 1 x
x x 2005dx x dx 220071 x 1 x 1 x 1 x
( ) ....
= + = + =+ + + + += + = = =+ + + + +
20072005I2007
=
:
, :
.
2006
2
xf x1 x 1 x
( ) = + + x
xI x f t dt( ) ( )= , :
x 2006
x 2 2
22006 2006
2 2 2
1 1I x f t dt f x f x x1 x 1 x 1 x 1 x
2 1 x 1x x I 0 01 x 1 x
'( ) ( ( ) )' ( ) ( ) ( )
( ) , ( )( )
= = + = + =+ + + + ++ + = =+ +
, , :
-
2007 2007x 2005I x I I 20052007 2007
( ) ( )= = =
90
z1 z2 2 1z = , f(x)= xzxz 21 + f(x)1 x. : 1. f2(x)= ( )2xzzRexxz 212221 ++ 2. O 1
2
zwz
= . 3. O z1 x2+2=1 .
4. f(x)=x
2 ,0 .
91
f + = 3x+4. :
) ( )xxflim
x + ) ( )[ ]x3xflimx +
) ( )x
lnxxflimx
++ )
( )( ) xx2f
x4xflimx +
++
)
+ x
lnx-x1xflim
0x )
+ x2007xflim
0x
92
f(x)=x -x
lnx.
1. f.
2. 1-xx1
ex x>0. 3. Cf (0, 1).
92
IRIRf : f F )(xfe )(xFe .
)()( xfxF = IRx (1)
-
( ) )()( xfxF ee = IRx )()()( xfxF eexF = , (1) )()()( xfxF eexf = IRx (3) 0)( >xf IRx . (3) ( ) )()( ln)(ln xfxF eexf = )()()(ln xfxFxf =+ IRx . ( ) ( )=+ )()()(ln xfxFxf )()(
)()( xfxf
xfxf =+
)()()()( 2 xfxfxfxf =+ )()()()( 2 xfxfxfxf = 1)()(
)()(
2 =xfxf
xfxf
( )=
+ x
xfxf
)(1)(ln IRx .
IRc cxxf
xf +=+)(
1)(ln IRx .(4)
x
xxg 1ln)( += 0>x . 22 111)( xx
xxxg == , 0>x . g
1 1)1( =g , 11ln)( +=x
xxg 0>x . (4) 1+ cx IRx .
93
f [ 0 , 1 ] 0 < f ( x ) 4 x [ 0 , 1 ] (1) , f ( 0 ) + f ( 1 ) = 0 (2) f ( x ) 2 x x [ 0 , 1 ] (3) .
[0 , 1] 0 (0 , 1) :
f(0)= 01
)0()1( ff
= )0()1( ff = )0(2 f = )1(2 f (1) : 4)(0 0 < xf , )0(20 f< 4)1(2 f
02)0(
=x
xffxf .
xfxf 4)0()( 0)1()(
-
f ( x ) 2 x x [ 0 , 1 ] . =1. 94
:f IR IR : (0) 0f = 2 1( ) 3 ( ) 1f x f x = + x IR . ). 3( ) ( ) ( )h x f x f x x= + , x IR . ).
3( )limx
f xx
). . : 3( ) ( ) ( ) 0h x f x f x x= + = , x IR .(1) ). (0) 1f = , f IR 0x > ( ) 0f x > . ( )22
6 ( ) ( )( ) 03 ( ) 1
f x f xf xf x
= , f [0, )+ .
f (0,0) y x= , [0, )+ , f ,
( )f x x< 0x > 3 2( ) 10 f xx x< < 0x > . 3( )lim 0
x
f xx
=
(1) 3
3 2
( ) 1 ( )f x f xx x x
+ = 0x > 3( )lim 0
x
f xx
= ( )lim 0
x
f xx
= .
(1)3( ) ( )1f x f xx x
= 0x > 3 ( )lim 1
x
f xx
=
f ,
lim ( ) lim ( )x xf x l IR f x
+ += = +
f 3 (x) + f(x) = x . ( )
lim ( )xf x
+= + (1)
-
3 2
22
2
3 33 113 1 3
( ) ( )lim lim ( ( ) '( ) lim lim( )
( )+ + + += = = =+ +x x x x
f x f xf x f xx f x
f x
2o
f 3 (x) + f(x) = x 2 1( )( ( ) ) ( )x f x f x f x= + x , f(x) >0 > 0. 0 ( )f x x< < , kser . 3o
f - 1 + + . f + + . (1) . .
95
f 0f (x)f (x)e x , x D [ , )= = + 0f (D) [ , )= +
0b . 0a , f (a) b= . xh(x) xe= 0x .( f ) . h , 1 1.
ba be= f (a) b= . : f (a) b bf (a) b h(f (a)) h(b) f (a)e be a be= = = = ,
. 0f (D) [ , )= + .
0f (x) 0x . 0b ba be= . f (a) b> , f (a) be e> 0f (a) b> , ( )
f (a) b bf (a)e be a be> > , ! f (a) b< , f (a) be e< 0 f (a) b < , ( )
-
f (a) b bf (a)e be a be< < , !
f (a) b= . 96
f [ )0,= + f(2) = 0 3f (x)f '(x) x
x= x > 0
) 0f (x)g(x) , xx
= > )
26 3f (x) x x ,x = ) Cf
xx
)
.
) g, 6 3g(x) x= , g(2) = 0 ) ).
0
0x
f ( ) limf (x)= = = 0 , f 0.
) () =2 2
06 3 4( x x )dx =
)
6
230
6 3 2( x x x)dx
= , (6 )2
3 36 108 6 3 4( ) ... = =
-
97
. 2
4 14)(xxexf
x = , 0x . . 0)( >xf 0x . )(lim xf
x , )(lim0 xfx .
. . x : 14234 +++ xxxe x , : 0 8
. 148332 234 +++ xxxe x , x .
. . 1)( = xexg x , IRx . 1)( = xexg , IRx 01)( >= xexg 0> x 01)( > xggxg 0> xggxg , 0x 0)( >xg 01 > xe x (1). xx 4: (1) 0144 > xe x 0x (2) 0
14)( 24
>=xxexf
x
0x .
4xg x e 4x 1( ) =
. 00000141lim)(lim 2
42 ==
= xxexxf
x
xx
de LHospital :
( )( )
( )( )
82
16lim
2
44lim2
44lim14lim14lim)(lim
4
0
4
0
004
02
4
0
00
2
4
00
==
==
==
x
x
x
x
x
x
x
x
x
xx
e
x
ex
e
x
xexxexf
. . x : 14234 +++ xxxe x ,
234 14 xxxe x + IRx .
-
2x 0x + xxf )( 0x . 00 0x ,
( (2) ). h 0 h IR . 0)0( =h . 0>x 0)()0()( >> xhhxh 0
-
) f . ) g ((f g)(x) (g f )(x)=D D x IR , g(x) x= . ) Cf Ch ,
2 2h(x) x x= ) Cf , Ch yy.
100
f = 0[ , )+ f (x)f (x)e x= 0x . ) f.
) f f -1 , f()= 0[ , )+ )
1
0I f (x)dx= , f(1) = . :
2x x
0
2xI e 1 dx( ) =
x = u = - .
: = 0
101
f IR. :
) x 2
0t f (t)dt 0> , x 0>
) x x
0 0t f (t)dt x f (t)dt
) x 2
x 0
0
1h(x) t f (t)dt , x 0f (t)dt
= : i) (0, )+ .
-
ii) ( ,0) . )
x
x 0
0
1h(x) f (t) t dtf (t) t dt
=
(0, )2 .
) f , :
i) x 2
02 t f (t)dt x f (x) (0, )+ .
-
102
f :(0, ) (0, )+ + f(1) = 2 F f : F(f(x)) + F(x) = 0 x > 0 .
) f
) f
) F
) :
x
A lim (f (x) 3x)+
=
103
f [,] 0 < < , f ( ) f ( ) 0 = = f ''(x) xf '(x) f (x)+ = x [ , ] , :
i. f (,).
ii. f (x) 0= x [ , ]. i. A f (,), . Yt
f(x) > 0, :
f ''(x)+x f '(x) > 0 2x2
e f ''(x) + x 2x2
e f '(x) > 0 [ 2x2
e f(x)] > 0,
x(, ). g(x) = 2x2
e f(x) . [, ]. . Rolle f [, ] (, ) f() =0. g() = 0 :
< < :
g(x) < g() g(x) < 0 f() < 0. f . [,], f(x) > 0 = f().
ii.
f ''(x) xf '(x) f (x) f ''(x) f (x) xf '(x)+ = = ,
-
f . : 2 2x x
2 2f '''(x) xf ''(x) 0 (f ''(x)e ) ' 0 f ''(x) ke+ = = = (1).
:
2 2
x x
2 2 2
f ''(x) xf '(x) f (x)f ''(x) xf '(x) f (x) f ''(x) f (x) xf '(x)x x
f ''(x) f (x) f (x) 1 1( ) ' f ''(t)dt f (x) x f ''(t)dtx x x t t
+ = = =
= = =
f() = 0 . 2tx2
2
1f (x) k e dtt
x
= , x[, ]. (2) f() = 0 , k = 0 (
, ) (2)
: f(x) = 0, x[, ].
104
E f (x)2f '(x) x R , f '(0) 1
1 e= =+
1.) Na f. 2.) Na f(x)=0 3.) f f 4.) y=x f 0 5.) f (x) x , x R 6.) f (x)f (x) e 2x 1 , x R+ = + 7.) f (x) 2x 1 , x R< + 8.)
xlim f (x)
9.) f (x) ln(x 1) , x 1 + > 10.)
xlim f (x)+
11.) f 12.) f 13.) f 14.) f + 15.)
ef ( ) 12
= 16.) :
e / 2
0f (t)dt f , f 1
-
1. ( ) ( ) ( )f x2f x 0 I x1 e = > + \ ( )f 0 1 = f & R 2. x = 0 () :
( ) ( ) ( ) ( ) ( )f 0 f 0f 02f 0 1 e 2 e 1 f 0 01 e = + = = =+ f & 1-1 0 . 3. ) 1 2x , x \ 1 2x x< ( ) ( )1 2f x f x< ( f & R) ( ) ( )1 2f x f xe e< ( xy e= & R). ( ) ( )1 2f x f x0 1 e 1 e< + < +
( ) ( )1 2f x f x1 1
1 e 1 e>+ + ( ) ( )1 2f x f x
2 21 e 1 e
>+ + f ' R f ) 3 ( )( ), f ( )( ), f ( )( ), f ( ) ( ) ( ) ( )f f f f = =
f [,] [,] :
( ) ( ) ( ) ( )1 1 f f, : f = ( ) ( )( ) ( )
2 2
f f, : f
= ( ) ( )1 2f f = 1 2 f ' 1-1 4. f 0 : ( ) ( ) ( )( ): y f 0 f 0 x 0 = ( )f 0 0 = 2: ( ) ( )f 0 0 : y x= = 5. f y = x f 0 ( )f x x 6. ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( )( )f x f x f xf x2f x f x f x e 2 f x e 2 f x e 2x f x1 e = + = = = +
( ) ( )f xe 2x f x c= + x = 0 f(0) =0 1 = c
( ) ( ) ( ) ( )f x f xe 2x f x 1 f x e 2x 1 x = + + = + \ 7. ( ) ( ) ( ) ( ) ( )f x f xe 0 f x e f x 2x 1 f x x> + > + > \ , 8.
xlim (2x 1) + = ( )f x 2x 1 x< + \
( )xlim f x =
9. ... ( ) ( )f x ln x 1 x 1 + > (5) : ( ) ( ) ( ) ( ) ( ) ( ) ( )6 f x f x f xf x x 0 x 1 e 0 x 1 e ln x 1 ln e ln x 1 f (x) + + + +
10. ( )xlim ln x 1+ + = + (9) ( ) ( )f x ln x 1 + ( )xlim f x+ = +
11. f /\&
-
( ) ( )( ) ( )( ) ( )8x x 10f (A) lim f x , lim f x , += = + 12. ( ) ( )
00x x
lim f x f x = f ( )0x xlim f x 13.
( ) ( )( )
""
f xx 4 x x
f x f x 2 2lim lim lim 2x 1 1 01 e
+ = = = = =++
( ) ( ) ( )6 f xx xlim f x 2x lim 1 e 1 0 1 = = = = y 2x 1= + .
14. ( ) ( )
( )"
"
f xx 1 x x
f x f x 2lim lim lim 0x x 1 e
+ + + += = =+
( )xlim (f x 0x)+ = + +
15. (6) ef2e e ex f e 2 1
2 2 2
= + = + ef2ef e e 1
2
+ = + ef2
= e e 1 e e 1 0 + = + + =
( )g e e 1 = + ( )g e 1 0 = + > g & 1-1 ( )g 1 0= 1 ef 12
=
16. ( )e2
0
f t dt ( )1e ef 1 f 12 2 = = : ( ) ( ) ( )f x6 f x e 2x 1 x + = + \ x f -1(x)
( )( ) ( )( ) ( )1f f x1 1f f x e 2f x 1 x + = + \ ( )x 1x e 2f x 1+ = +
( ) x1 x e 1f x x2
+ = \
Cf -1 y=x e/2 1
0 1
Cf
-
f , f -1 :
( ) ( )e
1 1 t21 1 0
0 0 0
e e t e 1 e 1 1f t dt 1 f t dt dt e 1 0 e 02 2 2 2 2 2
e e 3 32 2 8 8
+ = = = + + = = + =
105
f [1,2], f(2) > 4 (1)
f(2) f(1) = 37
(2). :
i) (1,2) f() = 2 ii) (1,2), f() = 2.
i) h(x) = f(x) 3x3
, x[1, 2]. H h [1, 2]
(1, 2) . : h(2) h(1) = f(2) 38
f(1) + 31
= f(2)
f(1) 37
= 0 (2).
: h(2) = h(1), Rolle (1, 2) : h() = 0 f () 2 = 0 f () = 2. ii) g(x) = f(x) 2x, x[, 2]. f [, 2], f [1, 2], g [, 2] . :
g() = f() 2 i= 2 2 = ( 2) < 0, 1 < < 2.
g(2) = f(2) 4 > 0, (1). : g()g(2) < 0, . Bolzano (, 2) g() = 0 f() = 2.
-
106
:
ln x , x (0,1) (1, )f (x) x 1
1 , x 1
+= =
) f . ) f 1. ) f . ) :
2x
xx 1
1 ln tA lim dtx 1 t 1
=
) x 1 x 1
ln xlim f (x) lim ... 1 f (1)x 1
= = = =
) . 1f '(1)2
=
) g(x) x 1 x ln x= 2g(x)f '(x) x(x 1)= . f . ) F f (0, )+ ( , f (0, )+ ) , :
2 2x
xx 1 x 1
1 ln t F(x ) F(x)A lim dt lim ... 1x 1 t 1 x 1
= = = = F (0, )+ , : 2
x 1lim(F(x ) F(x) F(1) F(1) 0 = =
f: IRR , f(0) = 1 , : f f(x) = 800 . . 4 25 . 2 . o 20 . 40 , . ) () () = 40 ( + + 80 ) , f , g : IR IR , f(x) g(x) = x 4 , IR . f(x) = x 3 3x 2 + 3x .
, 2007
