Elementary Calculus i
58
22/03/2015 Μαθηματικά Κατεύθυνσης Γ΄ Λυκείου Ασκήσεις για τον επιμελή μαθητή Μπάμπης Στεργίου Μάρτιος 2015 Λυμένες Ασκήσεις 1 Δίνονται οι μιγαδικοί αριθμοί α, β, γ που είναι διαφορετικοί μεταξύ τους ανά δύο με α = β = γ . Αν οι αριθμοί α + βγ , β + γα , γ + είναι πραγ- αβ ματικοί, να αποδειχθεί ότι: α) , όπου 2 αβγ(βγ + α)= ρ (βγ + αρ ) 1. 2 ρ = α . β) . 2 2 αβγ(γ 1) = ρ (γ ρ ) γ) . αβγ =1 Λύση α) Επειδή και α β α β γ , είναι αβγ 0 . Έτσι α β γ ρ 0 , οπότε: 2 ρ α α , 2 ρ β β , 2 ρ γ γ . 2 2 2 ρ ρ ρ α βγ α βγ α βγ α βγ α βγ α βγ α βγ 2 2 2 2 ρ (βγ αρ ) αβγ(α βγ) αβγ(βγ α) ρ (βγ αρ ) . β) Έχουμε κυκλικά τις σχέσεις: , 2 2 αβγ(βγ α) ρ (βγ αρ ) 2 2 αβγ( γα β) ρ ( γα ρβ) Σελίδα 1 από 33 Μπάμπης Στεργίου – Μάρτιος 2015
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Transcript of Elementary Calculus i
-
22/03/2015
2015
1 , , = = . + , + , +
-
, : ) , 2( +) = ( + )
1.
2 = . ) . 2 2( 1) = ( ) ) . = 1
) , 0 . 0 , :
2 , 2 ,
2 .
2 2 2
2 2 2 2 ( ) ( ) ( ) ( ) . ) : , 2 2( ) ( ) 2 2( ) ( )
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, , : 2 2 2( ) ( ) 2 2 ( ) ( ) ( ) ( ) 2 2( )( 1) ( )( )
2
2
2
(1)
2( 1) ( ) ) () ( ) :
(2) 2( 1) ( ) (1), (2), :
2 2 2( 1 1) ( )
(3)
2( ) ( ) , , :
2 2 3 2
0
( 0 1) 1 , (3) 1 .
21. a, b, c, , x > 0 ,
1z , 2z , 3z 1 :
3 3 3f(x) = x + a + b 3abx
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2 3 3 11 22 2 2
1 2 2 3 3 1
z z z zz z+ +
(z z ) (z z ) (z z )= 1
: ) 1 2 2 2
1 2 1 2
z z 1=(z z ) z z
2 2 2 2 2 21 2 2 3 3 1 1 2 3 1 2 3z z + z z + z z = 3 z + z + z z + z + z 2 . ) 2 2 21 2 2 3 3 1z z + z z + z z 9 . ) , .
f
) 3 3 3a + b + c 3abc 3x + y + z 3 xyz x, y, z > , .
0a = b = c x = y = z
) , , .
1z 2z 3z
) 1 2 2
1 2 1 2
z z 1(z z ) z z
2 1z z w , , :
1 2
1 2 1 2 1 2 1 2 1 12 2 21 1 1 11 11
z z z z z z1 1 1 z z w ww w w ww ww
1 2 1 2 1 2 1 2 1 1 2 2 2 1z z (z z ) z z z z z z z z z z 1 2z z 1 . .
2 2 21 2 1 2 1 2 1 2
1 2 1 2 1 1 2 21 2 1 2 2 1
(z z ) z z 2z z z z2 z z z z z z z z
z z z z z z
21 2 1 2 1 2(z z )(z z ) z z :
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1 2 2 21 2 1 2
z z 1(z z ) z z
) : 2 21 2 2 3 3 1z z z z z z 2
2 2 2 2 2 2 21 2 3 1 2 3 1 2 33 z z z z z z 3 z z z 9 ) , . f : 3 3 3f (x) x a b 3abx x 0 2f (x) 3x 3ab : f (x) 0 x ab , f (x) 0 x ab , f (x) 0 x ab f x a b :
3 23 3f ab ab a b 3ab ab a a b b 0 ) () , :
3 3 3f (c) 0 a b c 3abc
a a b b c a b , a b c 3a x , 3b y , 3c z , x, y, : z 0 3x y z 3 xyz x y z . ) () : 2 2
1 2 2 3 3 1
1 1 1 1z z z z z z 2
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1 2a z z , 2 3b z z 3 1c z z , a, b, c 2 2 21 1 1 1a b c . () :
32 2 2 2 2 2a b c 3 a b c 32 2 2 2 2 21 1 1 13a b c a b c
: 32 2 2 2 2 2 2 2 232 2 2 2 2 21 1 1 1(a b c ) 3 a b c 3 a b c 9a b c a b c
() 2 2 2a b c 9 , 2 2 2a b c 9 . , a b c , C .
3 z : (z 3 4i)(z 3 + 4i) + 3 z 3 4i = 4 1.
) z. ) 4 z 6 . ) , 1z 2z 1 2z z = 2 , 1 2 = z + z 1 2E = z + z 16i .
) z x . :
yi
2(z 3 4i)(z 3 4i) (z 3 4i)(z 3 4i) z 3 4i : 2z 3 4i 3 z 3 4i 4
2z 3 4i 3 z 3 4i 4 0 (1)
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z 3 4i 0 , :
2 3 4 0 ( 1 4) z 3 4i 0 , 1 , z 3 4i 1 . z (3,4) R 1 . ) 2 2OK 3 4 5 , :
OK R z OK R, R 1 4 z 6 . : :
1 z 3 4i z (3 4i) z 3 4i z 5. :
1 z 5 z 6 .
1 z 3 4i z (3 4i) 3 4i z 5 z . 1 5 z z 5 1 z 4 .
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) 1 2z z 2 2R , , , C(K,R) (3,4)
1z 2zR 1 .
. :
1z 2z
O 2
1 2 z z O 2
2 2 5 10 :
1 21 2
z z z z 16i 2 8i2
2 (3 4i) 8i 2 3 4i 2 5 10
1 2z z2
, .
4 2f(x) = x + 1 + x xg(x) = e x 1 . ) g . ) . (f o g)(x) = 1
1.
) g A . :
. x xg (x) (e x 1) e 1 . xg (x) 0 e 1 x 0
g, g
. ( , 0
][0, )
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, f g(x) : 0x x,
0f (x ) . -
f g(x) f (x) , , f 1-1, g(x) x .
)
, f x g(x). f g
1 f (0) , : f g(x) 1 f g(x) f (0)
f 1-1,
. g(x) 0:
222 2
2x x x 1f (x) x 1 x 12 x 1 x 1
2 2x x 1 x x x x x ( x) 0 , 2x x 1 0 x .
, f . f 1-1, :
f (x) 0
f g(x) f (0) g(x) 0 () :
x 0 : g . (x) g(0) g(x) 0 x 0 : g . (x) g(0) g(x) 0
g(x) 0 x g( . .
x) 0x 0x 0
2
2
x+ x +1f (x)=x +1
2f (x)= 0 x+ x +1 = 0 . :
x 0
2 2x+ x +1 = 0 x +1 = x 1= 0 ,
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f , . f (0)= 1> 0 , f (x)> 0 x .
5 2f(x) = x + 1 + x . ) f . ) . 2xx f(t)dt 0) . 22x+4 x +2x+2 xf(t)dt > f(t)dt) . 2 22 2 +3 +7 +4f(t)dt < f(t)dt
1.
. ) f fD , :
222 2 2
2x x x x 1f (x) x 1 x 1 12 x 1 x 1 x 1
.
x 02 2 2 2f (x) 0 x x 1 0 x 1 x x 1 x 1 0 , . f , .
f (0) 1 0 , f (x) 0 x f .
f x x , : 2 2x x 1 x x x x x ( x) 0 2x x 1 0 , f (x) 0 x . f :
f . f .
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2
2 22
2 2x x x x
x 1 x 1lim f (x) lim x 1 x lim lim 0,x 1 x x 1 x
2xlim x 1 x .
2x xlim f (x) lim x 1 x .
f ff (D ) (0, ) . ) . : 2 2f (x) x 1 x x x x x x x 0 f (x) 0 x . ,
( ), : f (x) dx 0 2
x 2x
f (t)dt 0 x x x(x 1) 0 x [0, 1 ] ) . 2. , xx 2
xg(x) f (t)dt
. : . x 2 x 2 x
x 0 0g(x) f (t)dt f (t)dt f (t)dt
. x 2 x0 0g (x) f (t)dt f (t)dt f (x 2)(x 2) f (x) f (x 2) f (x)
x 2 x f , . g . : g (x) 0
2
2
x 4 x 2 2x 2 x
f (t)dt f (t)dt g(x 2) g(x )
g
2 2 2x 2 x x x 2 x x 2 0 x ( 1, 2)
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) g. : 2 2 2 2g( ) g( 4) 4 . ... : F f.
F 2 2 , 3 , 2 2 4, 7 . F .
... 2 21 , 3 2 22 4, 7 , : 22
2 2 2 2 31 2 2
F( 3) F( ) F( 3) F( ) 1F ( ) f (t dt )3 3 3
22
2 2 2 2 72 2 2 4
F( 7) F( 4) F( 7) F( 4) 1F ( ) f (t)dt . 3 3( 7) ( 4)
2 , f . F (x) f (x) 1 ( 2 2 4) 3 1F ( ) F ( )2 ,
.
6 2x 4 . f(x) = 1 + e ) , , 1f . )
3
1
1 dxf(x)
.
1.
) f fD . 2x 4 2x 4f (x) 1 e 2e 0 ,
, f . f 1-1, . x
1f 1f f. : f .
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f . 2x 4
x xlim f (x) lim 1 e 2x 4x xlim f (x) lim 1 e 1 .
f (1, ) , 1f
. 1f
D (1, )
) . :
3 3
2x 411
1 1I dxf (x) 1 e
dx
, : x 1 3 u 4 u
31 3 2u 4
2(4 u) 4 4 2u 2u 43 1 1
1 1I ( du) du e d1 e 1 e e
u1
3 2x 4 3 32x 4 2x 4 111
1 e2I I I du 1dx x 3 1 2e 1 e
.
, 2I 2 I 1 . :
333 2x 4 2x 4
2x 4 2x 4 2x 41 1 1
1 e 1 e 1I dx dx21 e e 1 e 1
dx
32x 4 2 211 1 1ln e 1 ln(e 1) ln(e 1)2 2 2
2 2
2 22 2
2
1 1 e 1 1 e 1 1 1ln ln(e 1) ln ln e 2 12 2 2 2 2e 1 e
e
71. : f :
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2f (x) + 2xf(x) = x (3 f (x)) x 1f(1) =
2. :
) 32xf(x) = x +1 , x . ) 10 1f(t)dt = (1 ln2)2 . ) (0, 1) , :
3 2 30 f(t)dt = (3 1)f( )
) : 2 2f (x) 2xf (x) x (3 f (x)) f (x) 2xf (x) 3x x f (x) 2
2 2 2 2 3f (x) x f (x) x f (x) 3x f (x) x f (x) x
(1) 2 3f (x) x f (x) x c
1f (1)2
, (1) x 1 :
1f (1) f (1) 1 c 2 1 c c 02
, (1) :
2 3 2 3f (x) x f (x) x (1 x )f (x) x
3
2xf (x)
1 x ,
x ) :
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3 3 22 2 2 2xx 1 . x (x x) x x(x 1) x x
x 1 x 1 x 1
111 2 21 2
2 20 00 0
x 1 (x 1) x 1f (x)dx x dx x dx ln(x 1)2 2 2x 1 x 1
1 ln 22
.
) 32xf ( x) f (x)x 1 ,
3 3f ( ) f ( ) . :
3 2 3
0f (t)dt (3 1)f ( )
(1) 3 2 3
0f (t)dt (3 1)f ( ) 0
x , (1) :
3x x 2 3
0f (t)dt x(3x 1)f (x x) 0
3x x 3 3
0x f (t)dt xf (x x)(x x) 0
3 3 3x x x x x x
0 0 0x f (t)dt x f (t)dt 0 x f (t)dt 0
, 3x x
0h(x) x f (t)dt
x . h [0,1] ,
3 2x x0
(x) f (t)dt 3f (x x)(3x 1) , .
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h 3x x 2 30
h (x) f (t)dt x(3x 1)f (x x) (2).
. h(0) 0 h(1) Rolle (0, 1) , :
3(2) 2 3
0h () 0 f (t)dt (3 1)f ( ) 0 (1).
8 2f , x > 0 . (x) = ln2x ln(x +1)) f . ) .
x1F(x) = f(t)dt) , : > 1
5 2F() + (1 )f() x ( 1)(x +1)+ =x 1 x 3
1.
0
(1,3).
) f A (0, ) : 2 21 1f (x) ln 2x ln(x 1) 2 2x2x x 1
2 2
2 2 21 2x x 1 2x 1 xx x 1 x(x 1) x(x 1)
2
, f (0,1] [1, ) .
, f (f (1) 0 x) 0 . ,
x (0, ) ) 0f (x x (0, 1) (1, )
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f (1) 0 . ) F A (0, ) fD (0, ) . : f1 D x1F (x) f (t)dt f (x) , x 0 F ( x) f (x) 0 x (0, 1) (1, ) F 1. F .
, F . F(1) 0
) 1, 3 [1,3], :
5 2g(x) (x 3) F() (1 )f () x (x 1)( 1)(x 1) , x 0 g [1,3] . , 1 . 23) 2( 1) 3 18( 1) 0 g( g(1) 2 F() (1 )f () . F() 0 , 1 0 f () 0
. ... F [1,] (1,). ... (1, ) :
F() F(1) F()
F ( ) 1 1 (1)
F () f () f [1, ) . , 1 :
(1) 1F()f () f () f () F() ( 1)f () 0 1
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F() (1 )f () 0
g(1) 0 , Bolzano , , (1,3).
g(x) 0
9 f : (0, + )
x
f(x) f(t)
1
1 12f(x) + x + e = e f (t) t + dt + 2x t
, x > 0 . : ) 22xf(x) = ln x +1 , . x > 0) f . )
2x x
1x
tf dt tf(t)dtx
, .
1.
x > 0
) , :
f (x) f (x) f (x)1 12f (x) x e x e f (x) e f (x) xx x
1x
f (x) f (x)1 12f (x) x e 0 2f (x)e x 0x x
2f (x) f (x)1 1 x 1f (x)e x e2 x 2x
2
f (x) x 1e c2x
(1)
: x 1
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(2) f (1) f (1)2f (1) 2e 2 f (1) e 1
, xg(x) x e 1 xg (x) 1 e 0 , g . g 1-1. (2) :
f (1) f (1)f (1) e 1 f (1) e 1 0
g f (1) 0 g f (1) g(0) f (1) 0 (1) x 1 :
f (1)e 1 c 1 1 c c 0 (1) :
2f (x) f (x)
2x 1 2xe e
2x x 1
22xf (x) ln
x 1 ,
x 0
) 22xf (x) ln x 1 A (0, ) :
2
2 22x x 1 2xf (x) ln
2xx 1 x 1
2 2 2
2 2 2 2x 1 2(x 1) 2x 2x 2 2x 1 x
2x (x 1) 2x(x 1) x(x 1)
2
f ( . f (x)
x) 0 x 1 21 x
. f (0,1],
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[1, ) f (1) ln1 0 f. ) :
2x
xx
1t tf (t)dt xtx 0f d , (3)
t , t xux u dt . : xdu
2x x
1x
tf dt xfx
(u)du
t)dt
(3) : x x
1 1x f u du tf (
x f
x x
1 1(u)du tf (t)dt 0 (4)
, . h :
x x
1 1h(x u tf (t)dt x 0) x f (u)d
x x x x1 1 1 1h (x) x f (u)du f (u)du xf (x) xf (x) f (u)du tf (t)dt
h (1) 0
.
, x 0 . h (x) f (x)
f ( . h .
:
x) f (1) 0 x 1
x 1 h (x)
h (1) 0
, h (x) 0 .
x 1 h (x) , h (x) 0 .
h 0x 1 , :
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x x
1 1h(x) h(1) x f (u)du tf (t)dt 0
10 f , f(0) = 1 f(x) 0 x . fC , x = t , x = t tf(t) e t . ) f( , xex) = x . )
1 2
1
3xI = dxf(x) +1
.
) g(x) = f x , xx, g , . x = 0 x = 1
1.
) f f (x) 0 x , f (0) 1 0 , f (x) 0 x . . :
. , ( ) :
xg(x) f (x)e1 2x x1 2x , x 1f (x ) f (x ) 2 2
2
1x xe e
1 2x x1 2 1f (x )e f (x )e g(x ) g(x ) : x 0 x xg(x) g(0) f (x)e 1 f (x) e : x 0 x xg(x) g(0) f (x)e 1 f (x) e
t 0 , tt
x tf (t) e ,
E f (x)d t ttf (x)dx f (t) e t 0 .
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t 0 , t dx tf (t) e ,
tE f (x)
t
tf (x)dx f (t) e
t t tt
f (x)dx e f (t) t 0
t ttf (x)dx f (t) e t . ,
:
t t tt tt 0 0
f (t) f (x)dx e f (t) f (x)dx f (x)dx e
t 0 2 , f . :
tf (t) f (t) f ( t) e (1) . : t tf ( t) f ( t) f (t) e . : t t t t tf (t) f ( t) e e f (t) f ( t) e e , c , : t tf (t) f ( t) e e c
. t f (0) 1f (0) f ( 0) 2 c c 0 , : tf (t) f ( t) e e t t . (1) : t t tf (t) e e e e t
1c , t 1f (t) e c f (0) 0 , .
tf (t) et xf (x) e x .
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) 1 2
x1
3xIe 1
dx
x
. , u dx du . x 1 u 1 x 1 .
: u 1
1
1 1 12 2 2
x u u1 1 1
u1
3x 3( u) 3u 3u eI dx du du2 u
1e 1 e 1 e1e
du1
1 2 x
x1
3x e dx Je 1
.
1 1 12 2 x 2 x 11 2 3 1x x x 11 1 1
3x 3x e 3x (e 1)I J dx dx dx 3x dx x 2.e 1 e 1 e 1
, 2I 2 I J 1 . ) xg(x) e 0 , 1 x1 0E e d x . 2u x x u ,
, , dx 2udu x 0 u 0 x 1 u 1 . : .. 1 11 1u u u u0 01 0 0E 2ue du 2ue 2e du 2e 2e 2 ) tg(t)= f(t) e ( ), , g(0)= 0 . ) t tt f(x)dx = f(t) e () t
: t t tt tt tf(x)dx = f( t) e f(x)dx = f( t) e
t tt f(x)dx = f( t) e () (),() :
t t t tf(t) e = f( t) e f(t)+ f( t)= e +e (2)
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(1) : tf(t)+ f( t)= f (t)+e (2) : t t t tf (t) e = e +e f (t)= e , ..
11 f : f (1) = 0 . g(x) = f(x)f (x) fC , xx x = 1 , x = t 1 g(t)
6 t .
) , 2f(x) = (x 1) x . )
2
x2
f xA = dx
2 +1.
1.
) g(1) 0 g(x) f (x)f (x) , g(g(x) 0 x 1 x) 0 x 1 . :
t1
1 1f (x)dx g(t) f (x)f (x)6 6
, t 1 (1)
1t
1 1f (x)dx g(t) f (t)f (t)6 6
, t 1 (2). :
t
1
1f (x)dx f (t)f (t)6
t
1
1f (x)dx f (t)f (t)6
, t 1 (3) (1) (3) :
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t
1
1f (x)dx f (t)f (t)6
(4) t f , f(t). f (t) 1
g(t) f (t)f (t) ( 1) 12 . , t
1f (x)dx 1 , (4) 1 2 1 2 .
, 2f (x) x x 0 f (x) 2x , (4) :
t3 22
1
x x 1x (t t )(2t )3 2 6
3 2
2 3 2 2 2t t 1 (2 t t 2t t 2t )3 2 3 2 6
3 2 2 3 2 22t 3t 6t (2 3 6) 2 t 3t ( 2)t
2
02
2
2 2 13 3 46 2
3 6 2 2 3 6
24
, :
2 2 3 6 2 04 4
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3 2 3 6 12 8 0 ( 2) 0 2
, 1 2 2 14
, 2 2f (x) x 2x 1 (x 1) . ) 2f (x) (x 1) 2f x x 1 . x u , :
2 22 u xu u xx 2 dx . 2 2 2
f u f u 2 fA ( du) du
2 1 1 2 1 2
2 22 x xx x x2 2 2
f x 2 f x (1 2 )f x2A A A dx dx dx
1 2 1 2 1 2
2 0 2 0 22 2 0 2
f x dx f x dx f x dx (x 2x 1)dx 0 23 32 2 2 2
0 2 0
x x 8(x 2x 1)dx x x x x 0 4 23 3 3
8 8 84 2 0 2 23 3 3
163
.
162A3
, 8A3
.
12 g f .
f, g :
) . x0 g(t)dt > 0 x > 0)
xx 00
1h(x) = f(t)g(t)dtg(t)dt
, ,
(0
1.
x 0
, + ) ( , 0) . ) , h() h(). < 0 <
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) , x0
G(x) g(t)dt x . G , g . : , G (x) g(x) 0 x G . , :
x 0
x
0G(x) G(0) g(t)dt 0
) f, g ,
. h(x)
x
0f (t)g(t)dt
x
0g(t)dt
:
x x
0 02x
0
f (x)g(x) g(t)dt g(x) f (t)g(t)dth (x)
g(t)dt
x
2 0x
0
g(x) f (x)g(t) f (t)g(t) dtg(t)dt
x
2 0x
0
g(x) g(t) f (x) f (t) dtg(t)dt
(1)
i) . f , 0 tx 0 x : f (t) f (x) f (x) f (t) 0 g(t) 0 t
: ) f (x) f (t) g(t (t) [0,x]. (t) 0 t [0, x) .
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(t) 0 t [0, x] . [0,x] [0,x], . (1)
, h (0 .
(t) 0x
0(t)dtx
0
00
x 0h (x) 0 , )ii) . x x t 0 , : f (x) f (t) 0 g(t) f (x) f (t) 0 . , : (t) 0
0 x
x 0(t)dt 0 (t)dt 0
x
0(t)dt 0
( ,h (x) 0 0) . h . ( , 0 )
) h ( , 0) (0, ) . :
x 0 x 0
00xx 0 x 0 x 00
f (t)g(t)dt f (x)g(x)h(x) lim h(x) lim limg(x)g(t)dt
x 0lim f (x) f (0)
, , , 0
0 f
0, x 0lim f (x) f (0) .
x 0 . x 0 x 0
h(x) lim h(x) lim f (x) f (0)
0 h() f (0) 0 f (0) h() . . h() h ()
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13 f : . :
1 1f(x) 21 0f(x)ln 1 + e dx = f (x)dx 1.
f , f ( x) f (x) , x (1) :
1 f (x)1
I f (x) ln 1 e dxu
, x dx du . : 1 1f (x) f ( u)
1 1I f (x) ln 1 e dx f ( u) ln 1 e ( du)
1 f (x)1 f (u)f (x)1
1
1 ef (u)ln 1 e du f (x) ln dxe
1 1f (x) f (x) 21 1
f (x) ln e 1 dx f (x) ln e dx I f (x)dx 11
:
1 1 12 21 1 1
1I I f (x)dx 2I f (x)dx I f (x)dx2
2
u
(2)
:
(3) 1 0 12 2 21 1 0
A f (x)dx f (x)dx f (x)dx x , :
0 02 21 1
f (x)dx f ( u)d( u)
28 33 2015
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22/03/2015
1 12 2
0 0f (u) du f (x)dx
, (3) :
0 12 21 0
A f (x)dx f (x)dx
1 1 12 2 20 0 0
f (x)dx f (x)dx 2 f (x)dx , , (2) :
1 1 12 21 0 0
1 1I f (x)dx 2 f (x)dx f (x)2 2
2 dx :
f , . 0f (x)dx 2 f (x)dx f , f (x)dx 0 .
: 1 f (x)1 f (x) f (x)1
1
1 e2I f (x) ln 1 e dx f (x) ln dxe
1 f (x)
f (x)f (x)
1
1 ef (x) ln 1 e ln dxe
1 f (x) f (x)
f (x)1
e 1 ef (x)ln dx1 e
1 1f (x) 21 1
f (x) ln e dx f (x)dx
29 33 2015
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22/03/2015
14 f : [0, + ) f(0) = 1 , f (x) > 0 :
2x 2f(x) + lnf (x) = e + x + ln2x , x > 0) , . 2xf(x) = e x 0) f , . ) :
i) f() 1 f() 1 0 <
ii) x
1
1 dt 1 .
) +x 01A = lim xfx
.
) . 02 f(t)dt 1 + f()
2
2
0
) 2 2x 2 f (x) x 2f (x) ln f (x) e x ln 2x ln e ln f (x) e x ln 2x 22 xf (x) x 2 f (x) e x ln 2xln e f (x) e x ln 2x e e 2 2x 2 x 2 xf (x) e x f (x) e x ee e e 2x e e e e
c
2xf (x) ee e
f 0, : 2xf (x) e f (0)
x 0 x 0lim e lim e c e e c c 0
, , f (2xf (x) e x 0 0) 1 , .
30 33 2015
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22/03/2015
) . f .
2xf (x) 2xe f (0) 1
2 2 2x 2 x x 2f (x) 2e 4x e 2e (1 2x ) 0 . f . f (A) [1, ) , A [0, ) .
) i) f (x) 1g(x)x . :
2 2xf (x) f (x) f (0)xf (x) f (x) 1g (x) x x
2xf (x) f ()(x 0) f (x) f () 0
xx
f (0, x) . ( h(x) xf (x) f (x) 1.) : g() g() ii) , 2t 2t 1 2t2t 1e e . : 2 xx xt 1 2t 1 2t 11 1 1e dt e dt e2 1 2x 1 2x1 1 1 1 1 1e e
2 e 2 e 2 e
12e
x 1 2 2 2x xt t x
1 1
1 1 1e dt te dt e2 e 2e
.
) 22 221
1 1x1 x x3x
x 0 x 0 x 0 x 0 x 02
2e1 e xA lim xf lim xe lim lim 2 lim1 1x xx x
e
21x
x 0
12 lim e ( ) ( )x
.
31 33 2015
-
22/03/2015
) : (): f () f (0)y f (0) (x 0) 0
f () 1y 1 x
.
A f () 1f (x) y f (x) 1 x , x [0, ] . :
20 00
f () 1 f () 1 tf (t)dt 1 t dt 2
2 f () 1f () 1 2 2
2 f () f () 1 f ()2 2 2
15 f : (0, + ) : x1 f(t)dtf(x) = e , x > 0
: ) f . ) , x > . 2f (x) = f (x) 0) 1f(x) =
x, . x > 0
. :
fC
) MA . = MB) OAB , .
1.
) , , f(t) , f(x) .
32 33 2015
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22/03/2015
33 33 2015
(x)
) : (1)
x
1f (t)dt 2f (x) e f (x) f (x) f (x)
.
x
1f (t)dt
e f
) f (x) 0 , (1) :
x 0
2f (x) 1 11 x
f (x) f (x)f (x)
x c (2)
f (1) 1 . x 1 (2) c 0, : 1f (x)
x , x 0
) 21f (x) x . C f 0 01M x ,x
: 0x 0
02 20 00 0
1 1 1y (x x ) y xx xx x
2
y 0 0x 2x . . 0A(2x , 0) x 0 0
2yx
,
0
2B 0,x
. 20 2
0
1(MA) xx
20 20
1(MB) xx
MA MB, .
( , A B
M 0
x xx x
2 , A BM
0
y y 1y2 x .)
) 00
1 1(OAB) (OA) (OB) 2x 22 2
2x
.. 0(x 0).
-
27/03/2015
2
2015
16 : f : 3f (x) + f(x) = x ,
1. x
) f(0). ) f . fC) f( f(x). x) = 0) f(x) = 3xf (x) 2f(x)f (x) x . )
x2
0
1f(t)dt = 3xf(x) f (x)4
.
) f . 1f
) : . x 0 3 2f (0) f (0) 0 f (0) f (0) 1 0 f (0) 0 ) : 2 2
13f (x)f (x) f (x) 1 f (x) 03f (x) 1
1 25 2015
-
27/03/2015
2 25 2015
0) 1
. f . x , (): f ( y f (0) f (0)(x 0) y x . ) f ( f , 0) 0 x 0 . f f (0) 0 :
. x 0 f (x) f (0) f (x) 0
. x 0 f (x) f (0) f (x) 0
: ) f (x) 0 . x (0, , 0) f (x) 0x ( .
) , f(x) : 23f (x)f (x) f (x) 1
(x)
33f (x)f (x) f (x)f (x) f (x) , : 3f (x) x f 3x 3f (x) f (x) f (x)f (x) f (x)
x)
(1) f (x) 3xf (x) 2f (x)f (
) (1):
x x x
0 0 0I f (t)dt 3tf (t)dt 2 f (t)f (t)dt
x2xx 2
0 0 0
f (t)3tf (t) 3f (t)dt 2 3xf (x) 3I f (x)2
f ( 0) 0 g(t) f (t)f (t) 2f (t)G(t . : )
2
2 21I 3I 3xf (x) f (x) I 3xf (x) f (x)4
-
27/03/2015
3f (x)+ f(x)= x f (x) 0 : 3f (x)f (x)+ f(x)f (x)= xf (x)
4 2f (x) f (x)+ + f(x)= xf(x)4 2
:
x x4 2 x x
0 000
f (t) f (t)dt + dt + f(t)dt = tf(t) dt4 2
:
x x4 2 4 2 x x
00 00
f (t) f (t) f (x) f (x)f(t)dt = + tf(t) = + xf(x)=4 2 4 2
3 2 2x f(x) f(x)f (x)f(x) f (x) f (x)= + xf(x)= + xf(x)=4 2 4 2
2 2 24xf(x) xf(x)+ f (x) 2f (x) 3xf(x) f (x)= =
4 4
) . f () . . : 3 3f () f () 3 . : 3 3f () f ()
2 0 2 2f () f () f () 1 0 f () . 2f () f () 1 3f (x) f (x) x x ,
: 1f ( x)
x 31 1 1 1 3f f (x) f f (x) f (x) f (x) x x
3 25 2015
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27/03/2015
i) 3f (x)+ f(x)=
1
x , , f . , , x , 2 x 1 2x < x 1 2f(x ) f(x ) . 3 31 2f (x ) f (x ) , : 3 31 1 2 2 1 2f (x )+ f(x ) f (x )+ f(x ) x x , . f 3f (x)+ f(x)= x 3 0 0 0f (x )+ f(x )= x . ii) :
f 3g(x)= x + x . g , 1-1. g f(x) = x , x . 1f(x)= g (x) . 1g
g, . f . ( :
1-11 1g(f(x))= x = g(g (x)) f(x) = g (x) .)
17 :
1. f, g : ( 1, + )
x0 22 + f(x t)dt = g(x) , x
0
22 + g(x t)dt =f(x)
. x > 1
g
t
) f, g . ) . f =) f. ) , x 21h(x) = f(t )d x , xx, yy. fC
4 25 2015
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27/03/2015
5 25 2015
u) , x t dt du . :
. x 0 x
0 x 0f (x t)dt f (u)( du) f (t)dt
. x 0 x0 x 0
g(x t)dt g(u)( du) g(t)dt :
x
x0
0
2 22 f (t)dt g(x)g(x) 2 f (t)dt
f , g .
x
0f (t)dt
f .
) x0
22 f (t)dtg(x
) , : 2
2gf (x)g ( (x)x)
22f (x)g(x) f (x)
, . : 22f (x) f (x)g( x) 22g (x) g (x)f (x) 2f (x) f (x)g(x)
f (x) , 2g (x) f (x)g(x)
g(x)
: f (
f (x) g (x)x) g(x) ,
(1)
x 1
x 0 f (0) g(0) 1 , f ( g( , . (1) :
x) 0 x)x
01
, ln(f (x) ln(g(x) ln f (x) ln g(x) c x 1
-
27/03/2015
6 25 2015
x 0
x)
x 1 f g
g f
l . : n f (0) ln g(0) c ln1 ln1 c c 0
ln f (x) ln g(x) f (x) g( , . ) : 22f (x) f (x)g(x)
3 2 22f (x) 1 11 ( x) xf (x) f (x) f (x)
c
x 0 21 0 c c 1f (0) . 21 x 1
f (x) f
, : 1 x 1 f (x)
x 1 ,
x0
22 f (t)dtg(x)
, . :
x xx00 0
12 f (t)dt 2 dt 2 2 t 1t 1
2 22 2 t 1 2 2 t 1 1 g(x)t 1
) 2h (x) f (t ) 0 x , h . , x 01 h(x) h(1) h(x) 0 . : x [0, 1]
1 1
0 0E h(x)dx x h(x)dx
1 11 20 0 0xh(x) xh (x)dx 0 xf (x )dx
-
27/03/2015
1 1
202
0
1x dx x 1 2x 1
1
18 f : 3 , 2 = . : f()f()f() = 0 < < <
) f( x . x
1.
) > 0 ) [, ] , . f() =
x
) f( [,]. x) =
) f (x) 0 x) 0
f , . f (x , ,
f (f (
f () ) 0
0 ) 0 , f ()f ()f () 0 , . ) , [, ] f () . f (x)
. g(x [, ] x) f (x) : g [,] . g(x) 0 x [, ]
x))
g(
g [,]. g( , f ( . :
0x x [, ]
, 3f ()f ()f () x) 0 , x [, ] . ) . : f (x) x
]h(x) f (x) x x [, , . h [,] . h(x) 0 x [, ] .
7 25 2015
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27/03/2015
h [,]. h(x) 0 , ,
f (x), ]
xx [
f () , f () , f () . 2 3 , . f ()f ()f ()
, h(x) 0 x [, ] .
1
9 ,
1. f, 0) =2xe
g : f(x R
g(0) = 1f (x)g(x) 2xf (x)g (x) e . ) . f = g
2xe 0
) f(x). ) . f(3) + f(7) > 2f(5)
) : f ( (1) 2xf (x)g(x) e 2xx)g (x) e , g(x) 0 x . (1) : f (x)g(x) f (x)g (x) f (x)g(x) f (x)g (x) 0
2f (x)g(x) f (x)g (x) f (x) f (x)0 0 c
g(x) g(x)x
x 0
f (0) 1c 1g(0) 1
. :
f (x) 1 f (x) g(x)g(x)
, x ) f , (1) : g 2x 2xf (x)g(x) e f (x)f (x) e
8 25 2015
-
27/03/2015
2x 2 2x2f (x)f (x) 2e f (x) e 2 2xf (x) e c , cx 0 c2f (0) 1 0 . : 2 2x xf (x) e f (x) e , x f f (x) 0 x ( (1)). f f (0) 1 , . : f (x) 0
f (x) 0x xf (x) e f (x) e
, x
) 3 72 5
1 (3(5, 7)
, f ... [3,5] [5,7]. H f [3,5] [5,7] (3,5) (5,7). ... , :
, 5)2
1 f (5) f (3) f (5) f (3)f ( ) 5 3 2 .
2 f (7) f (5) f (7) f (5)f ( ) 7 5 2
.
xe 2f (x) 1 . : 1 2 e e 1 2
f (5) f (3) f (7) f (5)f ( ) f ( )2 2
f (5) f (3) f (7) f (5) f (3) f (7) 2f (5) 3 7 5f (3) f (7) 2 f (3)f (7) 2 e 2e 2f (5) .
9 25 2015
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27/03/2015
20 f : (0, + ) f( : 1. 1) = 0 f(x)x +1xf (x) = e +1 , x (0, + )
: ) 1-1. xg(x) = e + x) , . f(x) = lnx x > 0) 11 f(x)
x x 1 , . x > 0
) x 1
f(x)limx 1 = 1 .
) , 1x 2 gx D 2 1x x . 1 2x xe e 1 2x x1 2e x e x . , g . g 1-1.
1g(x ) g(x 2 )
g (x
0
g , .
x) e 1 0 x
) : x f (x)f (x)
x 1xf (x) xe f (x) xf (x) x 1e 1
f (x) f (x)1e f (x) f (x) 1 e f (x) x ln xx
f (x)e f (x) x ln x cx 1 0
n x
, : f (1) f (1)e f (1) 1 ln1 c 1 1 c c 0 , () : f (x)e f (x) x l f (x) ln xe f (x) e ln x g f (x) g(ln x) ,
g:1 1f (x) l
n x x 0
10 25 2015
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27/03/2015
) x 1 .
11 25 2015
x 1 . f (t) ln t [1,x] ..., (1, x) :
f (x) f (1) 1 ln xf ()
x 1 x 1 (1)
1 11 xx 1, (1) :
x 11 ln x 11 1 ln x x 1
x x 1 x
1
, , ... 0 x (x, 1) , : f (1) f (x) ln xf ()
1 x 1 x
1f ( ) 0 x 1 , :
1 1 ln x 11 1 x 1 x x
x 1ln x 1 11 x ln x
x 1 x x
1
11 f (x) x 1x
. , .
) 11 ln x x 1 , : x
x 1 : x 1 1 ln xln x x 1 1
x x x 1
x 1lim
ln x 1x 1
.
-
27/03/2015
12 25 2015
1 , 0 x x 1 1 ln xln x x 1 1x x x 1 ,
x 1
ln xlim 1x 1
.
x 1
f (x)li . m 1x 1
f(x)= lnx . , 1(lnx) =
x.
"", .
2
1 , F f
f : [0, 1] F(0) = F(1) = 0 2(x ) = F(x)f (x)
1. xf
[0, 1]
f(
. x
) , , 1 20 f (x)dx = 0 x) = 0 x [0, 1] . ) . 1 20 xf(x )dx = 0) f.
) f , 2f (x) 0 0x [0, 1] , , .
0x )
0
f (
x) 0
01 2f (x)dx 0 2f (x) 0 f ( x [0,, 1] .
) 2x u . 2xdx du , : 1 12
0 0
1 1xf (x )dx f (u)du F(1) F(0) 02 2
) , : 2xf (x ) F(x)f (x)
-
27/03/2015
(1)
1 120 0
xf (x )dx F(x)f (x)dx :
1 1100 0F(x)f (x)dx F(x)f (x) F (x)f (x)dx F(1)f (1) F(0)f (0) 1 12 20 0
f (x)dx f (x)d x .
13 25 2015
)0 . (1 2
0xf (x)dx
(1) :
()1 20
f (x)dx 0 f (x) 0 f f (x) 0 , x [0, 1] .
2
2 f : F f : 1. F(x + ) Ff(x) =
(x)
x . ) f . ) , f(x +1) = f(x) x . ) G(x) = F(x +1) F(x) . ) , f. f(0) = 2014
) : F(x ) F(x)f (x)
(1)
(1), , x . F f, F . f,
(1), :
-
27/03/2015
F(x ) F(x) f (x ) f (x)f (x) , x
) 2 : F(x 2) F(x)f (x) 2f (x) F(x 2) F(x)
2 (2)
1 : f (x) F(x 1) F(x) (3) (2) : 2f (x) F(x 2) F(x 1) F(x 1) F(x)
(3)
f (x 1) f (x) , 2f (x) f (x 1) f (x ) f (x) f (x 1) . ) G : , ()G (x) F(x 1) F(x) f (x 1) f (x) 0 x G ) , : G(x) c
2014
(3)
F(x 1) F(x) c f (x) c , f (0) f (x) 2014 , x .
f (x) 0 , f . : f Bolzano [,]. F f Rolle [,].
14 25 2015
-
27/03/2015
15 25 2015
11 2[ , ]
F f [,] Bolzano , F [,]. Rolle F f (
2
x) 0 (,).
23 .
3 24x 15x 18x = 1 ( 1, 1
1. )
1
.
: 3 2 3 24x 15x 18x 1 4x 15x 18x 1 0 : 3 2f (x) 4x 15x 18x Bolzano [ 1, 1]
f [ 1, 1] . ff ( 1) 4 15 18 1 2 0 (1) 4 15 18 1 30 0 .
f ( 1)f (1) 0 , Bolzano , . f, F F (x) f (x) , x . , :
4 3 2
4 3 24x 15x 18xF(x) x x 5x 9x x4 3 2
F Rolle [ 1, 1] . : F( 1) 1 5 9 1 2 F(1) 1 5 9 1 14 F( 1)
-
27/03/2015
Rolle F . f. 4 3 2G(x) x 5x 9x x 1 [ 1, 0] [0, 1] :
0 . G( 1) 1 5 9 1 1 1 1 0 . G(0) . G(1) 1 5 9 1 1 13 0
G [ 1, 0] , [0 , G(
, 1]G( 1)G(0) 0 0)G(1) 0 .
Bolzano, 1 ( 1, 0) , : 2 (0, 1)
1G( ) 0 2G( ) 0 :
G 1 2[ , ] . 1 2( , ) . . 1 2G( ) G( ) 0
Rolle, 1 2 ( , ) , : G () 0 f ( ) 0 f (x) 0 , .
2
4 x
21
2f(x) = dt1 + t
.
) f f .
1.
16 25 2015
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27/03/2015
17 25 2015
x) = 0) f ,
f. f() . 10I = f(x)dx
) 2 D2(t) 1 t ,
x
1
2f (x)1 t
2 dt .
, f , f :
x
2 21
2f (x) dt1 t 1 x
2 x,
) 22f (x) 1 x 0 x , f
. 1
21
2f (1) dt 01 t
x 1
1
,
. f ,
. f (x) 0
x f , :
. x 1 f (x) f (1) f (x) 0
. x 1 f (x) f (1) f (x) 0 f .
) x
21
2f (x)1 t
dt , :
1 x1
20 10
2I f (x)dx dt dx1 t
-
27/03/2015
11 x x 1
2 2 201 1 00
2 2 2x dt dx x dt x dx1 t 1 t 1 x
11 2
2 20 0
2x (1 x )(0 0) dx dx1 x 1 x
120ln(1 x ) ln 2 ln1 ln 2
: 1 1 1100 0 0I f (x)dx x f (x)dx xf (x) xf (x)dx
1 1
2 20 0
2 20 x dx dx1 x 1 x
x ..
. .
2
5 : f : [0, 1] 1 1 2 30 03 3f(x)dx = + f (x )dx10 2
) . 1 40 x dx) , x3 2x [0, 1]g(x) = f(x ) , .
1.
10 2g (x)dx = 0) f(x).
)
151 40 0
x 1x dx5 5
. ) ,
1 2 30
f (x )dx .
18 25 2015
-
27/03/2015
10
f (x)dx 3x t ,
0 0f (x)dx f (t )3t dt
3t )dt 3
dx :
23t dt 1 1 3 2
1 12 20 0
3 t f ( x f ( 3x )dt :
1 12 3 2 3f (x )dx 0 0
3 33 x f (x )dx10 2
19 25 2015
1 12 3 1 1x f (x )dx f 2 3(x )dx 0 010 2
1 1 20 0
2 x f2 3f (x )dx 3(x )dx 1 05
(1) ,
(1)
1 40
x dx (:
15
). (1)
1 2 3 2 3 4x f (x ) x dx 0
0f (x ) 2
1
0 23 2 dxf (x ) x 0
1 20
g (x)dx 0
,
2 , x [0, 1]
(2)
) ) 0 ) (2) 2g (x(x) 0 g(x
2g (x2 3f (x )
g ) 0 x .
1 20
g dx 0 ,
.
x 0 0g(x ) x [0, 1] 0 , 3 x ( [0, 1] x 3 x [0, 1] ), : 2 3 2 , x [0, 13f x f(x) (x) x ]
-
27/03/2015
20 25 2015
() .
) = 1
26 F f : f(1F(x)
1. f(x) = 2xe x .
) = 0 .
: ) f(0) f . ) 2xf (x) (x) = f(x) xf x . ) f.
) C xx x =
)
) ( f , 0 . , >
f , f ( x) f (x) x, . x 0 :
(1) F (
f (0) f (0) 2f (0) 0 f (0 ) 0 ) F(x)f (x) 2xe x) f (x) . (1)
F(x)
,
x
f (0
) 0 . F(e
x) ex ( xe
. f,
F(x)), F(x)2xe (1), . ) (1), , :
(1)
F(x) F(x) f (x)f (x) 2e 2xe f (x) f (x) f (x)f (x )x
(2)
0 , (2) 2xf (x) f (x) f (x)
f (0) x 0 (2)
) . (2) : x .
-
27/03/2015
2f (x) xf (x) xf (x)3x f (x) xf (x) xf (x) , x 0 (3)
(1) f (x) 0 x 0 . x 0 (3) :
2
2x f (x) xf (x) x xx
21 25 2015
f (x)f (x) 2
, x 0
2
1
2
2
x c , x 02x
f (x) x c , x 02
1 11 1 1c c
f (1) 2 2 , x 1 :
2
2x x 1 2xf (x)
f (x) 2 x 1 , x 0
x 1 2 21 1 1c cf ( 1) 2 2 , f ( 1) f (1) 1 . :
22xf (x)
x 1 , x 0
) 0 , 22xf (x) x 1 , x . f (0 ) x 0 0 , :
:
f (x)
f (x)dx ln(x ( 2 200E() 1) ln 1)
)
2 ln( 1)lim e E() lim e
-
27/03/2015
2 2
22 1lim lim lim e 0 0 0
e 1
2 2 2 2 2lim lim lim 0 1 .
27 ) f . ) f 1-1. )
1. xf(x) = x + ln(e +1) .
2x+2
2x
e +1x x 2 = lne +1
.
xe 1 0 x
) , f
f , . 2
. A 1x , 2x 1x x . :
1xe 1 e 1 , 1 2x x1) ln(e 1)2xe , 1 2x xe ln(e . 1 2x x1 2x ln(e 1) x ln(e 1 2
) f 1-1.
1) f (x ) f (x ) . f . f ,
2
x 2
x
e 1 0e 1
) , x . . :
2
x 22
22 25 2015
xe 1
e 1x x 2 ln
22 x 2 xx x 2 ln e 1 ln e 1 2x 2 x 2ln e 1 x ln e 1 (x 2)
-
27/03/2015
f: 1 12 2 2f (x ) f (x 2) x x 2 x x 2 0
23 25 2015
x 2)
, , .
28 f [,],
(x 1 1.
+ 2 f(x)dx f(x)dx2 . 1
. xg(x) f (t)dt , x [, ] . g [,] :
xg (x) f (t)dt f (x ) , x [, ] . g (x) f (x) 0 , x [, ] .
g . g ,
2 ,
2 ,
' . g ,
2 ,
, 2 .
... 1 , 2
2 , , 2 :
1 2 g g()g g()
22g ( ) 2
-
27/03/2015
2 2 g(g() g
2
) g22g ( )
g 1 2 , :
1 2
2 g g() 2 g( g2 2g ( ) g( )
)
1g g() g() g g g() g()2 2 2 2
2 f (t)dt f (t)dt2
1
dt 0 .
29
g() f (t)
f :
e , f(1) f(0) = 1
1
x0
f(x) f (x) dxf(x) + e
. I =
. :
xx x xx x x x
e f (x)f (x) f (x) f (x) e e f (x) e f (x)1 1f (x e f (x) (x) e e f (x)
. ) e f
1
xe f (x)) 1 1x0x x
0 0
f (x) f (xI dx 1 dx x ln(e f (x)f (x) e e f (x)
1 f (0) 1 ln(e f (1)) 0 ln(1 f (0) 1 ln(e f (1)) ln(1 f (0)) 1 lne f (1) .
1.
24 25 2015
-
27/03/2015
25 25 2015
f (0) f (1) 1 e , 1 f (0) e f (1) . I 1 ln1 1 .
