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Одесский национальный политехнический университет

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<<< < Предыдущая 12 / 62 3 4 5 6 > Следующая >>>

2.' ,’, . / ( % / .

2.1.$,. 0 1 – 0 + .

$*5% / 5 8 ( 6 6 /( $ m 5 8 ' & ! % 7 # n ! / / .

a11x1 + a12 x2 + ... + a1n xn = b1

+ a

+ ... + a

= b

a , i = 1, m, j = 1, n , #$ !#@ 76% " + 2 N #/

21 1

2 , 65#

...

+ a

+ ... + a

= b

xi , i =

- ! / 6 6 / , bi , i =

- ! 57 5 .

6 6 / ,

1, n

1, m

$*5% / ! /# & 2 .

...

a21

...

a2n

A =

...

, /# & 2% ( ! & # $ " [+ 2 ! .& ! / ', #$ !#N 76%

...

am1

am2

...

amn

6 ! @.

...

C = a21

a21

...

a2n

b2 , /# & 2% 1 ( ! & # "5# # %/ /# & 2 2 6 !.2%

...

! 57 ' 5 !. # & 2% #$ !#N 76% & $L & @.

$!’%$" / 6 6 / #$ !# 6% n $ # 7 ! / ' xi = ci , i = 1, n , .& . 6 # !2 %" ' ! 6 6 /( !6 & ! % % 6 6 / . & ! &@@ 76% ! = & ! 6 .

6 /# #$ !#N 76% ! $ # @ " 5 ! # /#N N 8 & $!’%$ " ! $ # @ , " 5 ! # /#N - 57L * & $!’%$"(.

- .

5% * , > - 6 6 /# 5 8 ' #5* -, ' & ! % 7 -(5# 6(/ 6 @, -' 6 #-7 , > - * ,, & $L & , /# & 2 & ! @!#! *( 6 ! , /# & 2 .

O"> * 6(/ 6 , 6 6 / & ! @N 65( ! / ', 6 6 /# /#N N & L %.

O"> * 6(/ 6 , 6 6 / / L = 65 ! / ', 6 6 /# /#N 6" ( " 57" 6 7 & $!’%$" !

2.2. 0.

$*5% / 6 6 /( $

a11x1 + a12 x2 + ... +
a	21	x	+ a	22	x	2	+ ... +
	21	1		22		2
					...		...
...					...		...
a	n1	x	+ a	n2	x	2	+ ... +
	n1	1		n2		2

n & ! % 7 # n - ! / '.

a1n xn = b1

a2n xn = b2 .

...

ann xn = bn

a		a
	11	12
a21		a21
6 ! # /# & 2% 6 6 / : A =
... ...
		an2
an1		an2
a	a	...
11	12
a21	a21 ...
$L & # /# & 2%: C =	... ...
...	... ...
	am2 ...
am1	am2 ...

...		a
...		1n
		a2n
...		...	.

...
		ann
a		b
	1n	1
a2n		b2
	...		.

a	mn	b
		n

'#8 det A 0 , ( A) = (C) = n .

<# & / @ & " - #. 55 6 6 /# /#N N 8 & $!’%$ ". < #8 / 2 8 & $!’%$ " $#

+ 0.

$ # / det A = ;

	a11	a12	...	a1n
=	a21	a21	... a2n		- ! $ # " 6 6 / .
	... ... ... ...
	an1	an2	...	ann

"5# / ! 57 ' 5

1 = b...2 bn

n ! $ # " ! !.

a12	...	a1n
a21	...	a2n	,
... ... ...			,
... ... ...
an2	...	ann

i , i = 1, n $#/ @@ . &L 8, &(* 8, n -8 6 !. ", 6 !.2 /

		a11	b1	...	a1n		a11	a12	...	b1
2	=	a21	b2	...	a2n	,..., n =	a21	a21	...	b2
		... ... ... ...					... ... ... ...
		an1	bn	...	ann		an1	an2	...	bn

& $!’%$ " 6 6 / $ #' / $# + &/(5#/ :
x		=			1		, x	2	=		2		,..., x		n		=	n	.
x		=					, x		=				,..., x				=		.
1
<#. 6# + &/(5 #$ !#@ 76% + &/(5#/ / .
																				x + 2x			2		3x		3	= 0,
																					1		2				3
							1. $!’%$# 6 6 /( & ! % 7													$# + &/(5#/ / 2x1 x2 + 4x3 = 5,
																					3x	+ x		2	x	3		= 2.
																					1			2		3
$!’%$# %: "5# / ! $ # " $ " N+ 2 ! .& ! / '.
=				1			2			3		= 10 0
				1			2			3
				2			1				4
				3			1			1
6 /# /#N N 8 & $!’%$ " {x1 x2																				x3}, $ #8 / 8 * $# + &/(5#/ / :
x =	1		, x			2	=	2		, x =				3		.
x =			, x				=			, x =						.
1										3

$ # " 1 , 2 , 3 - & /# $#/ @ ( ! $ # "( . &L * , &(* * M & 7 * 6 !. " !, 6 !.2 / ! 57 ' " + 2 N !.

< #8 / 2 ! $ # " :

1 =		0				2 3							= 5, 2 =		1	0	3	= 20, 3 =	1	2	0	= 15.
		0				2 3									1	0	3		1	2	0
		5				1					4				2	5	4		2	1	5
		2				1 1									3	2	1		3	1 2
x	=		1			=		5		=		1	,
x	=					=				=			,
1							10					2
							10					2
									20
	=				2 =				20		= 2,. & L % -( N /.
x2
x2									10
									10			3
	=			3		=	15			=		3
x3				3										.
x3							10					2		.
							10					2

2.3. ' ,’, , 3 4.

/ $ #8- 57L ( ! &6#57 ' # + " ! ' / ! & $!’%$# % 6 6 / 5 8 ' #5* - -, ' & ! % 7 N / K#(6#. <# / / K#(6# -' . 65 ! ! "5@ # ! / .

$*5% / / K#(6# (! "5@ % ! / ') 5% 5% & $!’%$# % 6 6 / $ &7 ' & !-% 7. 6 . , %" ! " (@ 7 # 6 6 / @ -( / $#. 6(!# ( /# & /( ! *5% .

														2x + y + 3z = 2
		1. $!’%$# 6 6 /( $# / / K#(6#												3x 2 y z = 1
														x y + z = 1
		$!’%$# % :						/# & /( ! *5% 6 6 /# $#. 6(N 76% #" / /:
	2		1		3	x					2
	3								=		1
	3		2 1 y						=		1
	1		1 1
	1		1 1			z				1
		$ 5 / . &L( 6 & "( /# & 2 .! , 5 ! , #6 # " + 2 N 2 .& x & /#N/ :
		1	1		3						1
		1	2		2	x					1
		3	2		2						1
		3	2 1 y =								1

		1	1 1
		1	1 1			z					1


		65% ! /# % $ &(* * # & 7 * &% "# . &L * , . / = * # 3 # # 1 & /#N-
/ :				1			3
			1	1			3

				2			2					1
				2			2		x			1			7
					7		11								7
			0						y =			2	& $ 5 / &(*( 6 & "( #
			0						y =			2	& $ 5 / &(*( 6 & "( #
					2			2				2			2
					3			1		z		2
			0		3			1
			0		2
					2			2

							1								3
1																		x													1
1							2								2			x													1																																			3
															11																4																																			3
	0						1											y								=							. / = / &(*( 6 & "( #																																		! /#N/ & $(57 # $
	0						1											y								=							. / = / &(*( 6 & "( #																																		! /#N/ & $(57 # $
								3							7																7																																			2
	0							3									1	z										2

								2									2
								2									2
& 7 ,																											1
				1								3															1
				1								3
1																x
1				2								2				x
																											4																																			13
											11																4																																			13
	0 1															y						=										& $ 5 / & @ 6 & "( #																															, & /# N/ :
	0 1											7				y						=					7					& $ 5 / & @ 6 & "( #																															, & /# N/ :
												7															7																																			7
										13						z												8
	0 0									13																		8
	0 0											7																	7
												7																	7

				1								3
1																x												1
1				2 2												x												1
										11																		4
	0			1												y						=
	0			1								7				y						=						7
												7																7
	0 0 1															z													8


																												13
! / 6 6 /(:
x +			1			y			+				3		z = 1																x =					1	y						3			z + 1 =										1		20		+		3 8		+ 1 x =					15

			2																																	2																				2 13						2 13
			2									2																								2						2														2 13						2 13			13
					y +						11				z =				4											y =				4					11		z =						4			+			11	8			=		20							y =		20
					y +					7					z =			7												y =											z =									+							=		13							y =
										7								7				8										7						7						7									7 13						13						13
														z =								8
														z =							13
																					13
#" / / : x =																								15			,		y =				20			, z =									8				.
#" / / : x =																							13				,		y =			13				, z =													.
																							13									13												13
<& - / . & ! &"(, . 6 #! / x, y, z ! & ! % % 6 6 / :
		15							20														8															30						20									24										26
2							+								+ 3													=			2,											+													= 2,											= 2, 2 = 2
2		13					+		13						+ 3								13					=			2,							13				+		13									13		= 2,								13			= 2, 2 = 2
		13							13						20								13															13						13									13										13
		3		15					2						20										8			=			1,							45						40							+		8		= 1,								13		= 1, 1 = 1

				13											13																							13						13									13										13
				13											13							13													15			13						13									13								13		13
15								20												8															15					20								8													13
												+												= 1,																												= 1,											= 1, 1 = 1
												+												= 1,																												= 1,											= 1, 1 = 1
								13										13																	13					13								13													13
13								13										13																	13					13								13													13

3.$% 5 (.

3.1.1 / 1 1 / 64 1..

& - 65@!# * 2 ! " & 6 !(@ 7 & / $ #& +/ ' . 8 # * -

2%/ , ! $ # * 2 , # #" = !5#6 ! 6 6" /#5 ' lim f (x) = 0 ) # 6"

x x0

! 5 " ' ( lim f (x) = ).

x x0

1. O"> f (x)	6" /#5# +( "2 % ! 2 x0	, (x) =		1	- -
1. O"> f (x)	6" /#5# +( "2 % ! 2 x0	, (x) =	f (x)		- -
	x0 .		f (x)
6" ! 5 "# +( "2 % ! 2	x0 .			1
2. O"> f (x)	6" ! 5 "# +( "2 % ! 2 x0 , (x) =			1		- -
2. O"> f (x)	6" ! 5 "# +( "2 % ! 2 x0 , (x) =			f (x)		- -
	x0 .			f (x)
6" /#5# +( "2 % ! 2	x0 .

3. O"> (x) 6" /#5# +( "2 % ! 2 x0 , # a(x) - -/ = # (%"-

> 6 (N lim a(x) = 0 , a(x) - -/ = #), f (x) = (x)a(x) - 6" /#5# +( "-

x x0

2 % ! 2 x0 .

4. O"> (x) (x) 6" ! 5 " +( "2 , ! 2 x0 ,

f (x) = (x) (x) , 6" ! 5 "# +( "2 % ! 2 x0 .

5. O"> (x) 6" ! 5 "# +( "2 % ! 2 x0 , # (x) #"#, > ! -

%" 8 " 5 2 x0 (x) > k > 0 , f (x) = (x) (x) - 6" ! 5 "# +( "2 % ! 2

x0	(O"> lim (x) = a 0 , 2 (/ ! ! " # ).
	x x0

O"> (x) - 6" ! 5 "# +( "2 % ! 2 x0	, lim (x) = 0 , /#N/ ! $ # -
6 7 0 .	x x0
6 7 0 .

" & 6 !(@ 7 #" = . & ! % % "! !#5 ' 6" /#5 ' +( "2 8. O"> (x)

(x) - 6" /#5 +( "2 , ! 2 x lim (x)			= 1, (x) (x) #$ !#@ 7
0	x x0	(x)

6" /#5 "! !#5 +( "2 , ! 2 x0 . $ # #N 76%: (x) ~ (x) . & - 65@- !# * 27 6" /#5( +( "2 @ $#/ @N/ # "! !#5 @.

3.2. 7 6 + (.
#8 6 (@ 7 * 2 lim		f (x) = a , lim (x) = b . ! " (@ 76% #" & / .
	x x0	x x0
1. 6 (N	lim ( f (x) ± (x)) = lim		f (x) ± lim (x) = a ± b .
	x x0	x x0	x x0
2. 6 (	lim f (x) (x) = lim f (x) lim (x) = ab .
	x x0	x x0	x x0

	3. O"> b 0 , 6 ( lim	f (x)
	3. O"> b 0 , 6 ( lim	(x)
	x x0	(x)

	lim f (x)	a
	x x0	a
=		= b .
=	lim (x)	= b .
	x x0

! . O"> b = 0 , . . (x) - 6" /#5# +( "2 % ! 2

x0 , & /( 3

$#6 6 !(!# / = #. O"> > a 0 ,

lim

f (x)

= lim

f (x)

;

- 6" -

(x)

x x0 (x)

x x0

! 5 "# +( "2 % ! 2 x

( 5#6 ! 6 7 1)

lim

f (x) = a 0

lim

f (x)

= a =

x x0

(x)

( 5#6 ! 6 7 5).

O"> a = 0 , /#N/ ! $ # 6 7

" .

x0 , & /# 3 #" = $#6 -

O">

f (x) (x) - 6" ! 5 " +( "2 , ! 2

6 !(N 76% /#N/ ! $ # 6 7 %

" .

lim (x) = b 0 ,

#8 f (x) - 6" ! 5 "# +( "2 %, #

f (x)

x x0

lim

= lim f (x)

, lim

0 ( 5#6 ! 6 7 5)

(x)

x x0

lim

f (x)

= .

(x)

x x0

f (x) - 6" ! 5 "# +( "2 %, # (x) - 6" /#5#,

lim

f (x)

= lim f (x)

- 6" ! 5 "# (!5#6 ! 6 7 2)

(x)

x x0

lim f (x)

= (!5#6 ! 6 7 4)

(x)

x x0

#8 lim f (x) = a , %" 8 / = & ! @!# (5@,

(x) - 6" ! 5 "# +( "2 % !

x ,

lim

= 0 ( 5#6 ! 6 7 2). & /# 2

6 (N

lim

f (x)

= a 0 = 0 .

x x0 (x)

x x0

(x)

a xn

+ a xn 1 + ... + a

$*5% / & L % .& "5# ! .(: lim

+ b xm 1 + ... + b

x b xm

.& "5# #' 1, 2, 3, .& x 6 57 "

$ #/ " 6" ! 5 " +( "2 ,, -

. $ #" / * 2 /#N/ ! $ # 6 7

' $

. 6 / ( 6 57 "( # $ #/ "( x (

& #

#8- 57L /( 6 . @

6" & (N/ & - . " & 6 !(@ !5#6 ! 6 7 2, O"> x - 6" -

! 5 "# +( "2 %,

- 6" /#5# +( "2 %,

lim

= 0 ,

#" = & / 6(/

# #6 " * 2 .

x xn

+ 7

3x3 + 6x + 7x4

= '

lim

$ = lim

x 5x

+ 3x + x

x3 + 5x + 7

' $

1 +

lim

= lim

= 0 , (!5- 7 3).

x 7x

x x

+ 3x + 1 & #

x2 + 4x3 + x4

+ 1

lim

= lim

= lim x

= , (!5- 7 5).

x 3x

+ 8x + 4 &

< & $*5% ( ' .& "5# ! / = # $& - #6 (. 8 ! 6 ! ".

m = n;

, 65

+ a x

n 1

+ ... + a

65 m > n;

lim

+ ... + b

x b xm + b xm 1

m < n.

- 65 lim

x3 + 3x2 + 2x

x2 x

x 2

$!’%$# %: & x 2 , 6 57 " # $ #/ " 0 , /#N/ ! $ # 6 7 %

" . $-

"5# / 6 57 " # $ #/ " # / = " # 6" & / & - # 6. 57 8 / = ".

5 / 6 57 " # (x + 2) #5 & $"5# / 8 * # / = " .

x3 + 3x2 + 2x

x + 2

x2 + x

x2 + 2x2

___________

x3 + 3x2 + 2x = x(x + 2)(x +1) .

x2 + 2x

___________

<# & / @ # " & & ! % % x2 x 6 = 0 ,

= 2, x

= 3 #" / / 6 57-

" $#. L / ( ! *5% : x2 x 6 = (x + 2)(x 3) .

lim

x3 + 3x2 + 2x

= lim

x(x + 2)(x

+1)

2 x 6

(x + 2)(x 3)

x 2

5. - 65 lim

x 1

x 5

$!’%$# %: & x 5 & /#N/ ! $ # 6 7 %

. $-#! / 6% ! 6 57 "( !

& #57 6 , ! " & 6 !(@ + &/(5(: a2 b2 = (a b)(a + b) , #" / / . / = -

/ 6 57 " $ #/ " # x 1 + 2 .

(

2)(

+ 2)

x 1

lim

= %

= lim

x 5

(x 5)(

x 1 + 2)

(x 5)( x 1 + 2)

x 5

= lim

x 5

= lim

(x 5)(

x 5

x 1 + 2)

x 5 (

x 1 + 2)

6. - 65 lim	x2		x	.
6. - 65 lim				.
x 1		x 1

$!’%$# %: & x 1 & /#N/ ! $ # 6 7

((

)3 1)

= lim

lim

= %

= lim

(

x 1)

x 1

(x +

+ 1)= 3

= lim

x 1

! " & 6 !(@ & /( 1,2.

'0$ % " . &0#

	(		1)(x +				+ 1)=
x	(	x	1)(x +			x	+ 1)=
			(		1)
			(	x	1)

3.3. 8 , (.

& - 65@!# * 27 ! $ !, %" /#@ 7 & * / & +( "2 , ! " & 6 !(@ 7

* 2@ lim	sin(x)	= 1 , %" 8 #$ !#N 76%	. &L @ ! $ # @ * 2 @.
	x
x 0
< . &L , ! $ # , * 2 /#N/ #"			* 2 :

lim

tg(x)

= 1;

2. lim

arcsin(x)

= 1;

lim

arctg(x)

x 0

lim

sin(ax)

= a ;

5. lim

tg(ax)

= a ;

lim sin f (x)

x 0

f (x)

= 1

= 1, lim f (x) = 0 .

x 0

#", 5% * 2 lim

sin(x)

= 1 '#" & ! L % 6 (6# 6" /#5 * "( #

x 0

6#/ /( "( #. /( (6%"# * 2% ! ( lim sin f (x)

= 1, O"> lim f (x) = 0 .

x a

f (x)

x a

#.& "5# , " = # $ * 27 ! ( lim

sin(7x)

lim

sin(x ) )

, lim

sin(x2 3x + 2)

x )

3x + 2

x 0

x )

x 1

. &L# ! $ # # * 2% /( & ! @N 1. < #' = % * 27

lim

sin(7x)

, lim

sin 2

(x)

sin(x)

x 0

lim

! /#*#N $#6 6(!# % #5* -, ' # & * / & '.

< . &L , ! $ # , * 2

/#N/ #"

"! !#5 6" /#5 .

1. sin(x) ~ x (x 0);

2. tg(x) ~ x(x 0);

3. arcsin(x) ~ x (x 0);

4. arctg(x) ~ x (x 0);

1. - 65 lim

1 cos(x)

x 0

2sin

cos(x)

$!’%$# % : lim

= %

= lim

x 0

		x	2
1	sin				1

	lim	2		=		.
		x			2
2 x 0

		2

" & 6 !(@ "! !#5 6" /#5 +( "2 , - 65 / #6 (. * 2 .

2. - 65 lim

arctg 2 (2x)

x 0 cos(7x) cos(3x)

arctg 2 (2x)

$!’%$# %: lim

= %

= lim

x 0 cos(7x) cos(3x)

x 0 2sin(5x)sin(2x)

(2x) ~ 4x

= %arctg

= lim

20x2

&sin(5x) ~ 5x, sin(2x) ~ 2x#

x 0 2 5x

x 0

3. - 65 lim

tg(x)

x 0 sin(x)

cos(x)

$!’%$# %: lim

[ ]= lim

sin(x)

x 0 sin(x)

tg(x)

x 0 sin(x)

2sin 2

'sin

1 cos(x)

= lim

2 2

= lim

x 0

sin(x)

x 0

2sin(x)

x 0

%sin(x) ~ x

= lim

0 = 0

x 0

3.4. 4( , (.

lim(1 + f (x))

lim 1 +

= e ,

lim(1 + x)x

= e ,

f ( x)

= e O"> lim f (x) = 0

x 0

x a

1, .& x , # ! $ 1 +

1 , .& x . <# . / * @ &(* , ! -

$ # , * 2

/ =5 ! & $"& !# ! $ # 6 7 [1 ].

lim(1 + 3x)

= e .

#.& "5# ,

x 0

; < =.

< &(* , ! $ # , * 2 :

1. lim

ln(1 + kx)

= k

2. lim

ln(1 + x)

= 1,

3. lim

ax 1

= ln a, 4. lim

ex 1

= 1.

x 0

2 x

1. - 65

lim

+ 4

$!’%$# %:

x 3x

3x+4

2 x

3x 1 +

4 4

2 x

3x+4

lim

= [1

]= lim

= lim % 1

+ 4

3x +

x 3x

x %

3x + 4

= lim e

10 x

= e

lim

10 x

= e

3x+4

x 3x+4

; < =.

& - 65 * 27 $# . / * @ &(* , ! $ # , * 2

$#6 6 !(@ 7 &(* 8 / -

& L %.

lim( (x)), ( x)

= [1 ]. - $ # / (x) = 1 + C(x)

'#8 lim, (x) = ,

C(x) 0

x a

.& x a ,

$C( x), ( x)

, ( x)

lim( (x))

[1 ]= lim(1 + C(x))

= lim%(1 + C(x))C( x) "

x a

x a&

lim C( x), ( x)

lim ( ( x) 1), ( x)

= ex a

lim ( ( x) 1), ( x)

& /(N/ + &/(5( + &/(5(: lim( (x)), ( x) = ex a

x a

4x 2 2 x 1

2. - 65 : lim

$!’%$# % :

x 4x +

4x 2

2 x 1

4 x 2

4x 2

(x) =

1 (2 x 1)

= [1

]= %

lim

4x + 3

= ex

4 x+3

x 4x + 3

&, (x) = 2x 1

	4 x 2 4 x 3	(2 x 1)
lim			=

= ex	4 x+3

	5(2 x 1)		10 x 5				5
lim		lim
				4 x+3	= e
= ex	4 x+3	= ex				2

3.5. 4 1 / 1 / 64 1..

5% & $"& % ! $ # 6		'0	$	$#6 6 !(@ 7 .& 2 . $#/ 6" /#5 ' "! -
		%	"

		&0	#
!#5 ' 6" /#5 / , # #" = !5#6 ! 6 6" /#5 ' +( "2 8.
O"> lim (x) = 0 , lim (x) =			0	lim	(x) = 1, (x) (x) - "! !#5 6" -
x a	x a			x a	(x)

/#5 : (x) ~ (x) .

& ! / #8!#=5 ! L 6" /#5 , %" $#6 6 !(@ 76% .& - 65 * 27.

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