Матанализ (2 сем) - Фаддеев (печатный)

, 2 .

I. .

§1.	.
				. . 552 2 .)
.	f(x) [a,b]				b>a.
			b
f (x)dx b f (x)dx :
a			a
			b
		f (x)dx blim f (x)dx
		a	a
, f (x)dx					,
			a

, .

1 dx;

1 x p

p 0

1

b

1

, p 1

1

p 1

1

1

1

1

b

p 1

dx lim

dx lim

lim

1

1 x p

b 1 x p

b 1

p x p 1

1 b 1

p b p 1

, p 1

p 1

1

dx lim b

1

dx lim ln(b)

1 x p

b 1 x p

b

1. f [a,b] b a c a f (x)dx f (x)dx

.

a

c

b

c

b

c

b

f (x)dx blim f (x)dx blim f (x)dx f (x)dx f (x)dx blim f (x)dx

a

a

a

c

a

c

2.

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

f (x) g(x) dx, ,

.

a

a

a

b

b

b

f (x) g(x) dx blim f (x) g(x) dx blim

f (x)dx g(x)dx

f (x)dx g(x)dx

a

a

a

a

a

a

. f (x)dx

, g(x)dx - , f (x) g(x) dx - .

a

a

a

3. (

).

		b2
f (x)	, 0 c 0 b1 , b2 c	f (x)dx
a		b1

1

, 2 .

.

b

F(b) f (x)dx

z

f (x)dx

F (b) A F (b)

: 0 c 0 :

b

z

b1 , b2

F (b2 ) F (b1 )

.

b2

b1

F (b2 ) f (x)dx; F (b1 ) f (x)dx

a

a

4.

« ».

, « »

.

f (x) , > a

f (x) 1.) , clim f (x) 0 .

a

c

:

c

c

1.

f (x)

f (x)dx

f

0,

f (x)dx

f (x)dx,

(x)dx

a

a

a

a

4.

, .

5.

.

.

f(x)

[a,b]

b>a

f(x) 0 x>a.

b

f (x)dx , M 0 b a f (x)dx M .

a

a

.

F(b) [a, ), F(b1) F(b2), b1 > b2,

F(b) b +			, F , .
M > 0, F(b) M b > a.
	b
F (b) f (x)dx
	a
	b2
b1 b2 , F (b1 ) F (b2 ) f (x)dx 0
	b1
6. .
	f(x) g(x)	[a,+ ) [a,b] b>a.
x>a f(x) g(x).
1.	g(x)dx ,	f (x)dx .
	a	a
2.	f (x)dx ,		g(x)dx .
	a		a

.

5.

2

, 2 .

b b b

1) g(x)dx , M 0 b a g(x)dx M , f (x)dx g(x)dx M . . f(x)

a

a

a

a

0, 5.

f (x)dx .

a

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

f (x)dx ,

a

a

a

.

.

.

f(x) g(x)

[a,+ ) [a,b] b>a.

lim

f (x)

C 0 .

x g(x)

1.

g(x)dx ,

f (x)dx .

a

a

2.

f (x)dx ,

g(x)dx .

a

a

.

1) . lim

f (x)

C , . 0 x0

x x0

f (x)

C

f (x)

C .

x g(x)

g(x)

g(x)

.2.

.

a x c

f (x) 2Cg(x) , .

= ). .

g(x)dx

c

2 g(x)dx ,

f (x)dx

( . 6.). . 1. ,

c

g(x)dx g(x)dx

,

f(x).

a

2)

.

.

?

f(x) g(x)

[a,+ ) [a,b] b>a.

>a C 0, x c f(x) C g(x). g(x)dx , f (x)dx .

.

a

a

.

.

f(x)

-

[a,+ ),

[a,b] b>a.

C

1.

p>1, c>a, C 0,

f (x)

x c.

f (x)dx .

p

x

a

C

2.

p 1, c>a, C>0,

f (x)

x c.

f (x)dx .

p

.

x

a

dx

a>0

p>1,

p 1.

p

a

x

		C
	1) g(x)		,	p 1, g(x)dx , f (x) g(x).	6 .: f (x)dx .
		p
		x		a	a

3

, 2 .

C

2) g(x)

,

p 1, 0,

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

f (x)dx .

p

x

a

a

.

.

f (x)dx

,

a

f (x)

dx .

a

.

f (x)dx

,

a

,

f (x)

dx .

a

7.

, .

.

f (x)dx , .

f (x)

dx .

a

a

.3

b2

b2

f (x)

dx

0 C

b1 , b2 C

f (x)dx ,

f (x)

0

f (x)

dx 0,

a

b1

b1

b2

b2

b1>b2. f (x)dx

f (x)

dx

: f (x)dx

,

b1

b1

a

, .

8. – I.

f(x) g(x) -

[a,+ ).

1.

f(x) [a,+ )

f (x) 0 ;

x

2.

g(x) , . ,

g(x)dx

M ;

a

f (x) g(x)dx .

a

.

: f(x) g(x) –

, f(x) – ;

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

b2

x

f (x) g(x)dx ,

, G(x) g(x)dx , :

b1

a

b2

b2

f (x) g(x)dx f (b1 ) g(x)dx f (b2 ) g(x)dx, f (b1 ) G( ) G(b1 ) f (b2 ) G(b2 ) G( )

b1

b1

2M

2M

b 2

2M

f (x) g(x)dx

f (b1 )

f (b2

b 1

0 C b C

f (b)

4M

4

, 2 .

b2

f (x) g(x)dx

, b1, b2 > C,

2M

, , 8.

4M

b1

4M

f (x) g(x)dx

.

a

.

sin( x)

1)

:

>0, >0

x

p

0

.

x

g(x) sin( x),

G(x) cos(x) sin(t)dt,

M 1,

cos(x)

1 ;

1

2

f (x)

,

f

, f (x) 0 ;

x p

x

sin( x)

sin( x)

1

dx

p>1, .

,

.

x

p

x

p

x

p

x

p

0

sin( x)

0

1.

( .

).

x

p

0

9.

– II.

f (x) g(x)dx .

f(x) g(x) -

[a,+ ). f(x) –

a

,

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

.

a

a

b2

b2

c

: b1 , b2

c (b1 , b2 ) f (x) g(x)dx

f (b1 ) g(x)dx

f (b2 ) g(x)dx .

b1

b1

c

1) >0 b [0,+ ) f(b) C;

2)

>0 b b1, b2 >b

g(x)dx

- .

a

b2

f (x) g(x)dx

2C .

b1

.

5

. 2 .
	§2 .
.				f(x)				.
	a
f (x)dx f (x)dx f (x)dx .
			a
						a	b
					f (x)dx clim f (x)dx blim f (x)dx
						c	a
.				f (x)dx			,

blim f (x)dx V . p. f (x)dx .

b

.

x

:

dx .

1 x 2

0

x

:

x

dx

dx .

1 x 2

1 x 2

0

x

dx

1

d (1 x 2 )

1

ln

1 x 2

2

2

b

1

x

2

1 x

2

x

1

b

1

dx

ln

1 x 2

ln

1 b 2

2

1

x

2

0

2

b

0

.

0

x

dx .

1 x 2

x

dx .

1 x 2

b

f (x)dx 0, f ( x) f (x), V . p.

x

dx 0

1 x 2

b

1

. 2 .

§3. .

.

f(x) [a+ , b] >0.

b

b

f (x)dx lim f (x)dx .

a

a

b

c

1.

f [a,b] b a c a f (x)dx f (x)dx

a

a

b

c

b

f (x)dx f (x)dx f (x)dx, c b.

a

a

c

b

b

b

2.

f (x)dx g(x)dx ,

f (x) g(x) dx ;

a

a

a

3. .

: F(x) a >0

>0 x1, x2: |x1-a|< , |x2-a| < |F(x1)-F(x2)| < .

b

f (x)dx

,

a

x2

0 0 x1 , x2

a x1 x2 a

f (x)dx

.

x1

b

c

4.

f (x)dx ,

lim

f (x)dx 0

c a 0

a

a

.

b

def

b

f (x)dx lim f (x)dx;

a

a

x2

0 0 x1 , x2

a x1 x2 a

f (x)dx

x1

x2

def

x2

f (x)dx;

f (x)dx

lim

x a 0

a

1

x1

x2

f (x)dx

.

a

5.

f(x) (a,b] b>a f(x) 0

x (a,b]. f (x)dx

a

b

, 0 a b f (x)dx .

6. .

f(x) g(x)

(a,b])

(a,b] b>a.

x 0 f(x) g(x).

b

b

1.

g(x)dx ,

f (x)dx .

a

a

1

. 2 .
				b					b
	2.	f (x)dx , g(x)dx .
				a					a
7.									, .
b		def c			b			x c
	f (x)dx			f (x)dx		f (x)dx,
								f (x) 0
a			a		c
	b			c			b		x a
V				0	f	(x)dx		f (x)dx ,
	. p.	f (x)dx lim							f (x) 0
	a			a			c
			b
f (x)dx f (x)dx f (x)dx
a			a		b

2

, 2 .

				II.							.
		§1.				,								.
												. . 27 2 .)
.													Sn ,
				1
n			def
Sn	, .	ak	lim Sn .								,
1		k 1		n
				,										.
1
.
.
1.
					, .
	.

b .

1 1

N k N ak bk

n

N

n

ak ak ak Sn (a) , n N

k 1

k 1

k N 1

n

N

n

bk bk bk Sn (b)

k 1

k 1

k N 1

N

ak

bk

Sn (a) Sn (b)

n

k 1

Sn (a) Sn (b) .

def

,

ak

lim Sn .

1

k m

n

n m, Sn am am 1 an

.

.

m 1

ak ak ak ,

k 1

k 1

k m

lim S

,

S

a a

a

a

n

n

2

n

k

n

1

k 1

~

~

lim

,

a a

a

a

S

n

S

n

k

n

m

m

1

n

km

~

Sn ak Sn

k 1

1

, 2 .

bk ,

2.

b

. ,

1

1

1

bk ak bk .

1

k 1

k 1

.

Sn (a) a1 a2 an

Sn (b) b1 b2 bn

S

n

(c) a

k

b

k

Sn (c) c1 c2 cn a1 a2 an b1 b2 bn Sn (a) Sn (b)

3.

.

m

.

, 0 N

0 m, n N

ak

.

1

k n

.

def

Sn a1 a2

an

1

Sn

,

.

0 N 0 m , n N

S m S n

.

m

Sm Sn a1 a2 am a1 a2 an an 1 an 2 am ak

k n 1

m

Sm Sn

ak .

kn 1

4..

.				,
		. .			, ak			0.
				1				k
	.
												n
							0			N 0 n N		ak	,
1												k n
												an
, an			0.
.			n
					1
						, , .
				1	k

.

ln(1 x) x, x 0 f (x) ln(1 x)

	1
x
ln(1 x)	0	5	10
		x	2