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-1-

&

()

3:

555 1

3.1

1.

. -

.

2. H

.

, 1f(x)x

(x) F ln x ,

A ( ,0) (0, ) F(x) = f (x) x A , :

( ,0) , f 1 1

F(x) c , c .

(0, ) , f 2 2F(x) c , c .

A ( ,0) (0, ) , f F(x) c , c

3. 1 2

F ,F f, ,

c. :

1 2F (x) F (x) c, c R .

4. , .

1, 1 x 0

f x1, 0 x 1

, = (1,1),

F = (1,1) F(x) = f (x) .

5. : f , :

) .

) .

6. f , . ()

7. f ,

1 1

2x , x 0f(x) x x

0, x 0

,

:2

1x , x 0F(x) x

0, x 0

.

31

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8. f ,

, 0

x .

9. -

.

x

.

1

/

1 f(x) 0 G(x) c, c R

2 f(x) 1 G(x) x c, c R

3 1f(x)x

G(x) ln x c, c R

4 f(x) x 1x

G(x) c, c R 1

51

f(x)2 x

G(x) x c, c

6 f(x) x G(x) x c, c R

7 f(x) x G(x) x c, c R

8 21

f(x) x

G(x) x c, c R

9 21

f(x) x

G(x) x c, c R

10 xf(x) e xG(x) e c, c R

11 xf(x) , 0 1 xG(x) c, c R

ln

10.

:

F G f g * ,:

i) F G f g

ii) F f .

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: 3 2012-2013

-3-

2

/

1 k(x) f (x) g (x) K(x) f(x) g(x) c, c

2 *k(x) f (x), *K(x) f(x) c, ,c

3 k(x) f (x) g(x) f(x) g (x) K(x) f(x) g(x) c, c

4 2f (x) g(x) f(x) g (x)

k(x)g (x)

f(x)k(x) c, c

g(x)

5 k(x) g f(x) f (x) K(x) g f(x) c, c

11. -

-

f .

:

x .

3

/

1 g(x) f (x) G(x) f(x) c, c

2f (x)

g(x)f(x)

G(x) ln f(x) c, c

3 g(x) f (x)f (x), 1 1f (x)

G(x) c, c , 1 1

4f (x)

g(x)f(x)

G(x) 2 f(x) c, c

5 g(x) f (x) f(x) G(x) f(x) c, c

6 g(x) f (x) f(x) G(x) f(x) c, c

7 2f (x)

g(x) f(x)

G(x) f(x) c, c

8 2f (x)

g(x) f(x)

G(x) f(x) c, c

9

f(x)g(x) f (x) e f(x)

G(x) e c, c

10 f(x)g(x) f (x) f(x)

G(x) c, cln

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: 3 2012-2013

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[2-7, 1-4 308-9]

1. f(x) 2x ,

yy 2e .

2. , -

A(0,1) .

) xf

f(x) e 2x x, D

) 3 f1

f(x) x x , D ( 1, )2 x 1

) x ff(x) 3 4 ln 4 2, D

) x x ff(x) e x e x, D

3. ( ) -

6

5 10N (t) t , (0 t 6 )9 3

.

6

3 7,5 .

4. t sec

21(t) 1 m / sect 1

0 t 5 . t 1 sec ,

.

5. :) 3 2

1f(x) 8x 3x 1 , x 0

x

)2

x 2 g(x) x ,x ,

7 2 2 x

) x(x) (x 1)e

)2

xx xf(x)

x

) xx x

g(x) e

) (x) (3x 5)

) 5x 3f(x) 4e

) 42xg(x) ( )1

)1 1

(x) , x ,2x 1 2

)3

1f(x)

(2x 1)

) g(x) 2x 1

)1

(x)2x 1

) x 3f(x)x 2

6. F 25 xf(x) 4x 3xe , F( 1) F(2) .

7. f , f(1) 3 , f -

0

x 1 , 4f (x) 30x 2, x .

8. f : xf x f x( ) ( ) 1 e , x ,

0

x 0 2.

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: 3 2012-2013

-5-

9. ( ) -

15 4x , x -

2x 5 . :

) ,

) .

10. -:

) 2 3 2 33x 2xf(x) x x 1( )( )

)3ln x

g(x)x

)2

x 2(x)

x 4x 7

)2

2xf(x)

1+ x

) x,g(x) x 0,

)x

1(x)

1 e

) 3

f(x) x x, x ,2 2

)x

1 xg(x)

e x

11. f : f(0) 0 f(1) 1 .

0

x (0,1) 0x

0f (x ) e 2 e .

12. 1

2x 2 02 x

(0,1) .

13. f(x) x 1 2 .

[ f F], F f .

14. f : F .

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

15. *f : ,

F 2F(x)F(1 x) F(x ) x .

16. f : f(0) 1 F -

: ( )fx 1(F x) , x .

) F(0) .

) (x)g(x) F F x( ) .

) f .

17. f : F .

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

) f 2 x F 1( ) (x) x .

) F 2 x F 1( ) (x)

x

.) f .

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: 3 2012-2013

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&

()

3:

555 1

3.4

3.5 -

1. . -

f(x)dx 0

.

2. .

.11 2

0 02xdx x 1 0 1

11 2

0 02xdx x 1 2 1 1 .

3. f [,] .

4.

.

f (x)dx f() f()

f (x)dx f () f ()

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

f(x)dx g(x)dx .

: ( )

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

f(x) g(x) dx 0 f(x)dx g(x)dx 0 f(x)dx g(x)dx .

6.

f(x)dx

f

f.

.

f(x)dx f(t)dt f(y)dy .

7.

1

f (x)dx

f(x)dx .

f 1f

: x f u (1), dx f (u)du (2).

1

u 2

u (3), 1

u 2

u -

1

f(u ) 2f(u ) f

1-1.

32

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-2-

1f f

: 1(v)x f (1), 1dx f (v)dv (2).

1v

2v (3),

1v

2v -

1 1f (v ) 1 2f (v )

-

1f 1-1.

: 2

1

0f (x)dx 5 3f(x) x x .

f(u) x (1) dx f du(u) (2)

f(u) 0 f(u) 2 .

5 3 3 2 3f(u) 0 u u 0 u (u 1) 0 u 0 u 0 5 3f(u) 2 u u 2

u =1 f 1-1.

1

u 0 2

u 1 (3)

(1),(2),(3) :

2 1 1 1

1 1 4 2

0 0 0 0

19f (x)dx f f(u) f (u)du u f(u)du u (5u 3u )du

12 .

. [ ]

f(x)dx F (x)dx F(x) F() F()

. :

f (x)g(x)dx f(x)g(x) f(x)g (x)dx ,

f , g .

:

i)

x

P(x)e dx , P(x) x ,

* .

ii)

P(x)(x )dx , P(x) x ,

* .

iii)

P(x)(x )dx , P(x) x ,

* .

iv)

P(x)ln(x )dx , P(x) x ,

* , x 0 .

v)

x

e (x )dx ,

*, , .

vi)

x

e (x )dx

, *

, , .

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: 3 2012-2013

-3-

-

, :

[1] x e

[2] (x ) (x )

[3] P(x) ()

1x

x

[4] ln(f(x)) , -

.

, :

x

P(x)e dx P(x) -

0

-

, ,

-

.

P(x)(x )dx

P(x)(x )dx

P(x) -

0

-

, ,

-

P(x)ln(x )dx P(x) -

0

-

,

x

I e (x )dx

x

I e (x )dx

0 -

, 2 ,

-

-

,

I.

, ,

I.

2

P(x)dx

(x )

2

P(x)dx

(x )

P(x) -

1 0

-

, 1 ,

-

2

1

x .

21

x .

, :) ,

) .

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: 3 2012-2013

-4-

.

f g(x) g (x)dx -

:

2

1

u

uf g(x) g (x)dx f(u)du ,

f,g , u g(x), du g (x)dx 1 2u g(), u g() .

, [

3, 31] u f(x) .

*******

.

P(x)dxQ(x) , -

:

[1] P(x) Q (x) , ,

u Q(x)

Q(x)dx ln Q(x)

Q(x)

.

[2] , , P(x) Q (x) ,

) < :

Q(x) x *(x ) , N 2x x 2 4 0 .

P(x)

Q(x)

:

x A

x .

(x ) :

1 2

2

AA A

...x (x ) (x )

.

2x x

2

Ax B

x x

.

, -

.

:

331 2

3 2 2 3 2

x x 2 x

x 1 x 2(x 1)(x 2) (x x 2) (x 2) (x 2) x x 2

, [1].

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) :

P(x):Q(x) : P(x) Q(x) (x) (x) ,

(x),(x) , , (x)