MV3047

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Одесский национальный политехнический университет

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[НЕСОРТИРОВАННОЕ]

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2. 5% +( "2 , f (x) =

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<<< < Предыдущая 1 23 / 63 4 5 6 > Следующая >>>

						21
1. sin(x) ~ x (x 0);					6. ex	1 ~ x (x 0);
2.	tg(x) ~ x(x 0);				7. a x	1 ~ x ln a (x 0);
3.	arcsin(x) ~ x (x 0);				8. ln(1 + x) ~ x (x 0);
4.	arctg(x) ~ x (x 0);				9. loga (1 + x) ~ xloga e (x 0);
5. 1 cos(x) ~		x2	(x 0);		10. (1 + x)k 1 ~ kx, k > 0 (x 0)
		2

1. -65 lim				2x sin(x)			.

			x 0	1 cos(x)

$!’%$# %: " & 6 !(@ #-5 2@ "! !#5 ' 6" /#5 ' ,

2x sin(x)

2x x

4x2

sin(x) ~ x, 1 cos(x) ~

& /#N/: lim

= %

= lim

1 cos(x)

x 0

1 cos(10x) .

2. -65

lim

x 0

$!’%$# %:

(5x) ~ 2

cos(10x) = 2sin

(5x)

= 50x

lim

1 cos(10x)

= lim

50x

= %

1 ~ x2

x 0

#%ex

x 0

=4 .

=50

3. -65

lim ln(1+ sin x) .

$!’%$# % :

x 0

sin(4x)

$'ln(1+ sin x) ~ sin x$

lim ln(1+ sin x) =

= lim sin x = '

0$[sin x ~ x]= lim

= 1 .

x 0

sin(4x)

x 0

#&sin(4x) ~ 4x

4. -65 lim

32 x 7x

x 0 arcsin(3x)

32 x 1 (7 x 1)

$!’%$# %: #/ # ! / / ! 6 57 "( 2@. lim

= %

arcsin(3x) 5x

x 0

' 2 x

1 ~ 2xln 3, arcsin(3x) ~ 3x

2x ln 3 xln 7

x(ln 9 ln 7)

= %

" = lim

= lim

1 ~ xln 7,

3x 5x

%7x

x 0

ln(1 + 2x)

5. -65

lim

x 0

e3x 1

ln(1 + 2x)

'0$

'ln(1 + 2x) ~ 2x,$

$!%$# %: lim

= %

" = lim

1 ~ 3x,

x 0

&0#

x 0

4. 9 + 64 1 .

4.1. $, +.
'#8 +( "2 % f (x) 6 (N ! 2	x0 %" 8 ,, " 5 . f( "2 % f (x) #$ !#N 76% . & -
&! @ ! 2 x0 , %"> 6 (N * 2% 2 N, +( "2 , ! 2			x0 ! # & ! @N $ # @
+( "2 , ! 2 8 2 .	lim f (x) = f (x0 )
	lim f (x) = f (x0 )
	x x0
.%. lim x = x0 , . . & @ & ! 6 7 / = # $#. 6# ( ! *5% :
x x0
lim f (x) = f ( lim x) = f (x0 )
x x0		x x0
; < =. O"> +( "2 % . & &! # ! 2 x0 , ! " (N 76% & ! 6 7:
lim f (x) =	lim	f (x) = lim	f (x) = f (x0 )
x x0	x x0 0	x x0 +0

4.2. 1 , 4 / 1 6 1.

" , ! %" ' ! " (N 76% . & &! 6 7 +( "2 ,, #$ !#@ 76% "#/ & $& !( +( "2 ,.$*5% / " & $& !( +( "2 , + 8 ( 4( ( & (. "# & $& !( x0 #$ !#N 7-

6% " @ & $& !( . &L * & ( +( "2 ,					f (x) , O"> ! 2 8 2 6 (N " 2 ! * 2 +( "2 ,
$5 !# # 6.!#, -			lim	f (x) = A	lim	f (x) = A . & 27 /( :
		x x0 +0		1 x x0 +0		2
)	O"> A1 = A2 , "# x0 #$ !#N 76% " @ 4 4 ( , 4;
Z)	O"> A1 A2 , "# x0 #$ !#N 76% " @ 1 ( , 4;
5 (		A1 A2	#$ !#@ 7 %1 64 1 ! 2 8 2 & $& !( . &L * & (.

"# x0 #$ !#N 76% " @ & $& !( 4( ( 4 +( "2 , f (x) , O"> # $ 6 & -

' * 27 ($5 !# 6.!#) 6 (N & ! @N 6" 6 .

1. $ # . & $& !( +( "2 , y = e x 2 ! 2 x = 2 .

$!’%$# %: $*5% /

f (2 0) =		1			= 0,
	lim	e	x 2
x 2 0					,
		1
f (2 + 0) =						=
	lim	e		x 2
	x 2+	0

#" %" * 2% +( "2 , 6.!# & ! @N 6" 6x = 2 - "# & $& !( &(* * & (.

& $& !( ! %! ,' ..

$!’%$# %: f( "2 % ! $ # # # !6 8 65 ! 8 6

"& / " x = 1.

1, .& x > 1 f (x) =

1, .& x < 1

	23
#" / /, lim	f (x) = 1, lim f (x) = 1 /( ! 2 x = 1 +( "2 % /#N & $& !
x 1 0	x 1+0

. &L * & (. & - " +( "2 , ! 2 8 2 & ! @N 1 ( 1) = 2 .

5% +( "2 ,

$!’%$# %:

sin(x)
		, .& x 0	$ #8 " & $& !( ! %! ,' ..
	x
f (x) =	x
	2,	.& x = 0

sin(x)
		, .& x
	x
5% +( "2 , f (x) =
	2, .& x

lim f (x) = lim f (x) = 1 f (0) .

x 0 0 x 0+0

5 = / f (x) = 1 ($#/ 6 7 f (x) = 2

& &! @ ! 2	x = 0 .
& /#N/ !( +( "2 @ . & &! ( !
x = 0 :
sin(x)
		, .& x 0
	x
F(x) =
	1, .& x = 0

0 x0 = 0

) .& x = 0

"# (6(! * & $& !( #" %"

& $& ! (6(!#N 76%, +( "2 % 6 #N . -

4. 65 +( "2 @ # . & &! 6 7

	x 1, .& x > 1	.
$ #8 " & $& !(: f (x) =		.
x 1, .& x 1

	$!’%$# %: f( "2 % f1(x) = x 1 f2 (x) = x 1
. & &! # 5% (6 ' x . 65 / . & &! 6 7 +( "2 ,
! 2 x = 1.
	f ( 1) = 1 1 = 0,
	f ( 1 + 0) = lim				(x 1) = 2,
		x 1+0
	f ( 1 0) =		lim		( x 1) = 0.
		x 1 0
	#" %" f ( 1+ 0) f ( 1 0) " 2 ! , +( "2 %
/#N & $& ! . &L * & (. & - " +( "2 , & ! @N
	f ( 1+ 0) f (1 0)			= 2
	f ( 1+ 0) f (1 0)			= 2
	5 65 +( "2 @ # . & &! 6 7					$ #8 " & $& !(:
		2x			, .& x 0
		1	x , .& 0 < x 3.
	f (x) = 1+	2	x , .& 0 < x 3.
		2			, .& x > 3
	1				, .& x > 3

$!’%$# %: 65 / . & &! 6 7 +( "2 , ! "#' x1 = 0 x2 = 3.

a) x = 0,	f (0) = 20 = 1,
f (0 0) =	lim	2x = 1,
	x 0 0
f (0 + 0) =	lim		1		= 1
f (0 + 0) =	lim	1+	2	x	= 1
	x 0+0		2

f (0) = f (0 0) = f (0 + 0)

f (x) . & &! #

b) x = 3,	f (3) = 1+		1 x				=	5	,

			2		x=3			2

f (3 0) =		+	1			=	5	,	f (3 + 0) = 1
	lim 1		2	x			2
	x 3 0

f (3 0) f (3 + 0) f (x) /#N & $& ! I - * & ( ! 2 x = 3

& - " +( "2 , & ! @N

f (0 + 0) f (0 0)

6. 65 +( "2 @ # . & &! 6 7

$ #8 " & $& !(:

, .& x

< 0

, .& 0

x < 3 .

f (x) =

x 3

, .&

x - 3

$!’%$# %: 65 / . & &! 6 7 +( "2 , ! "#' x1 = 0 x2 = 3. a) x = 0, f (0) = 2x x=0 = 0,

f (0 0) = lim 2 = ,

x 0 0 0

f (0 + 0) = lim (2x) = 0

x 0+0

f (0 0) = , +( "2 %

/#N & $& ! &(* * & ( ! 2 x = 0 . b) x = 3, f (3) = x 3 x=3 = 0,

f (3 0) = lim (2x) = 6

x 3 0

f (3 + 0) = lim (x 3) = 0

x 3+0

f (3 0) f (3 + 0)

f (x) /#N & $& ! . &L * & ( ! 2 x = 3

& - " +( "2 , & ! @N f (0 + 0) f (0 0) = 6 0 = 6 .

5. / 64 1.

5.1 $, + /.

'#8 +( "2 % y = f (x) ! $ # # ( &!#5 (a,b) , "# x0 .(a,b) .

' @ +( "2 , y = f (x) ! 2							x0 #$ !#N 76% * 2%
</ = lim		f (x0 + x) f (x0 )				#-	f /(x0 ) = lim	f (x) f (x0 )	.

	x 0				x		x x0	x x0
L & . $ # % . ' , +( "2 , f (x) :
/	/	/	,	dy	/

f (x), fx		, y		dx	, yx .

5.2. 6 5.

'#8 +( "2 % u = u(x) # v = v( 6 (@ 7 #" .! 5# + & 2 @!# %:

1.(u ± v)/ = u/ ± v/;

2.(u v)/ = u/v + uv/;

3.(cu)/ = cu/, c = const;

4.(u v w)/ = u/v w + u v/ w+ u v w/;

/		/	/
u	=	u v uv		, v 0;
5.	=	v2		, v 0;
v		v2

x) - !

6.	u /


	c
7.	c /


	v

8. (uv )/

+ & 2 !# +( "2 , ! &!#5 (a,b) .

=1c u/;

=cv/

v2 .

=vuv 1u/ + uv ln u v/

5.3. / 1 % 64 1.

'#8 y = f (x) # u = (x) , y = f ( (x)) - 6"5# # +( "2 %.
/	! 2 x , # +( "2 %	y = f (u) /#N . ' (	/	!
O"> +( "2 % u = (x) /#N . ' ( ux			yu

2 u = (x) , 6"5# # +( "2 % y = f ( (x)) /#N . ' ( y/x ! 2 x , %"( $ #' / $# + &/(5 @.

/	/ /		/	/	/
yx = yuux		#- yx		= f (u)u (x)
O"> +( "2 % y = f (x) / # # &!#5				(a,b)	/
					/#N f (x) 0 , - & # +( "-
/
2 % x = ( y) #" = /#N . ' ( ( y) .& /(:
/( y) =		1	5 x/y =		1	.
		f /(x)
					y/x

5.4. % + / 1 64 1.

					/	= 0, C const																			10.		(arcctg u)/x							=							1			u/x ,
					/
		1. (1)																																						1	+ u	2
		2.		(u )/x = u 1u/x ,																																				1	+ u	2
		2.		(u )/x = u 1u/x ,																					11.		(au )/x = au ln a u/x ,
		3. (sin u)/x							= cosu u/x ,																12.		(eu )/x = eu u/x ,
		4. (cosu)/x							= sin u u/x ,																12.		(eu )/x = eu u/x ,
		4. (cosu)/x							= sin u u/x ,																13.		(ln u)/x			=		1	u/x ,
		5. (tg u)/x =									1							u/x ,														1
											1
																																u
																																u
										cos2 u															14.		(loga u)/x =							1			ln a u/x ,
									=							1																		1
		6.		(ctg u)/x												1				u/x ,
																																		u
														sin2 u																				u
														sin2 u											15.		(shu)/x			= chu u/x ,
		7.		(arcsin u)/x							=					1						u/x ,			15.		(shu)/x			= chu u/x ,
																1									16.		(chu)/x			= shu u/x ,


																1 u2									17.		(th u)/x			=	1								u/x ,
		8. (arccosu)/x =																		1				u/x ,							1
																				1
																																ch 2u

																				u			2
																1				u			2		18.		(cthu)/x =							1							u/x ,
			(arctg u)/x								=					1					u/x ,													1
		9.														1																				sh2u
		9.												1+ u2															)/x =		1					sh2u
														1+ u2											19.		(											u/x ,
																												u

																															2				u
																															2				u
		< #8 . ' #6 (. ' +( "2 8.
		1.		y = cos3 (3x2 1)
		y/ = 3cos2 (3x2 1)sin(3x2 1)6x .
											3		)(						+ 2x)
		2.	y = (5x2 x									2	)(				x		+ 2x)
		2.
		$!’%$# %: $*5% / . ' ( -( "( ! ' ! $ !.
				2				3																2	3					/											3		3		1
						2																			2																		2		1
	/					2			/																2																		2
y	/								/							+ 2x)+ (5x x										)( x + 2x) = 10x 2 x																				( x + 2x) +
y		= (5x x ) ( x														+ 2x)+ (5x x										)( x + 2x) = 10x 2 x																				( x + 2x) +

			2			3			1
			2						1
+ (5x				x		2 )									+ 2

											x
							2				x
		3.	y =			x +			5x2

							cos(5x)
							cos(5x)

$!’%$# %: $*5% / . ' ( & -(.

(x +

5x 2 )/

(x + 5x2 )(

y/ =

cos(5x)

(

cos(5x)

(1 + 5

2x) cos(5x) (x + 5

ln 5

( sin(5x)) 5

)

cos(5x)

4. y = arcsin sin(3x)

		/	=				1									1

	y					1 sin(3x) 2										sin(3x) cos(3x)3.
	y					1 sin(3x) 2										sin(3x) cos(3x)3.
5.			y = ln							4
								x +
												x2 1								/										/
												x2 1

							1										4					x + x2				x2
							1										4					x + x2	1 4(x +			x2		1)
	y	/	=																		=										=


							4																						2
							4							x		+ x2			1			4		(x +	x2		1)
														x		+ x2			1			4		(x +	x2		1)
					x +				x2	1
					x +				x2	1
																1
											4 1			+		1				2x
											4 1			+						2x
	x +			x		2	1								2 x		2	1
=	x +			x			1								2 x			1			.
			4																	2
																				2
													(x + x2 1)


6.			y = (x2 +1)sin(2 x)

y/ = (x2 +1)sin(2 x) 1(2x) + (x2 +1)sin(2 x) ln(x2 +1) cos(2x) 2

5.5 ' .

< * / & , " $&( . ' # +( "2 , ! 2 x0 , N "( ! 8 " + 2 N (# * 6 "( # #' 5() , .& ! , *+ "# +( "2 , y = f (x) ! 2 $ #-62 6 @ x0 ,

k= f /(x0 ) 0 .

! % % , ! 2 M (x0 , y0 ) :

y y0 = f /(x0 )(x x0 ) .

#" %" &/#57 . &. "(5%& # , ! 2 "(, "( ! 8 " + 2 N &/#-

5 & ! @N	1	.
	f /(x0 )

! % % &/#5 ! 2 M (x0 , y0 ) :

y y0 =	1	(x x0 ) .
	f /(x0 )

F(x, y) = 0 , $ %" *

y ( %! /(

28
1. "5#6 & ! % % , # &/#5 "& ! , y = x	2 + 5x 1 ! 2 x = 1.
$!’%$# %: < #8 / y0 = 1+ 5 1 = 5	0
$!’%$# %: < #8 / y0 = 1+ 5 1 = 5
< # % . ' , .& x = x0 . y/ = 2x + 5, f /(x0 ) = 2 1+ 5 = 7 .

! % % ,: y 5 = 7(x 1) 7x y 2 = 0

! % % &/#5 : y 5 = 17 (x 1) 7 y 7 + x 5 = 0 x + 7 y 12 = 0

5.6 / 64 1 , .

f( "2 % y = f (x) $# # # %! , " 5 6 (N & ! % % ! *5% $ 27 * & ! % % ! 5 / =5 ! .

O"> +( "2 %

& -# .& + & 2 @!#-

y = f (x) /#N . ' ( , 5% $ #' = % y (x)

F(x, y) = 0 . x , & $*5% #@ y %" +( "2 @ ! x

& $!’%$# & /# & ! % %

F (x, y) = 0 ! 6 y/x .

< #8 . ' ( +( "2 , $# # , %! .

y + y4 xy + 2e x = 0 .

$!’%$# %: f( "2 % y $# # # %! . < #' / . ' ( ! 6 x . 5#*#@ , > <

+( "2 % ! F.

y/ + 4 y3 y/ y xy/ 2e x = 0 ! 6 / y/

$# (=" .

y/(1+ 4 y

x) = y + 2e

y/ =

y + 2e x

1+ 4 y3 x

2. arctg( y) y + x = 0 .

$!%$# %:

+1 = 0

1+ y

2 1 = 1 y

1+ y

= 1 y

5.7. 6 5 64 1 . , / +.

'#8 $#5 = 6 7 / = #&*(/ / x

+( "2 N@ y $# # # .#/ & ( ! *5% ! ' & -

! 6 8

x = x(t),

t - .#/ &.

y = y(t),

' # +( "2 , $# # , .#/ & $ #' 76% $# + &/(5 @:						y/x =	yt/	, xt/ 0 .
' # +( "2 , $# # , .#/ & $ #' 76% $# + &/(5 @:						y/x =	xt/	, xt/ 0 .
< #8 . ' +( "2 8					$# # ' .#/ & .		xt/
< #8 . ' +( "2 8					$# # ' .#/ & .
			2	+1,	. < #8 / . ' : xt/ = 4t, yt/ = 3 + 3t2 .
1. x = 2t				+1,	. < #8 / . ' : xt/ = 4t, yt/ = 3 + 3t2 .
	y = 3t + t3.
y/x =	yt/	= 3 + 3t3
y/x =	xt/	= 3 + 3t3
	xt/	4t

x = 2(t sin t),

.< #8 / . ' : xt/ = 2(1 cos t), yt/ = 2sin t .

y = 2(1 cos t).

yt/

2sin t

2sin

cos

y/x =

= ctg

2(1 cos t)

2sin

6. ; + .

$*5% / 6. 6 - & $"& % ! $ # 6 ! (

0 , 6 ! @ %" * N $#6 6(!# % . ' -

O"> +( "2 , f (x)

(x) + & 2 8 !# ! " 5 " x0 , $# ! % " / ' -#, > 6#/ ,

f (x0 ) = 0,

lim (x0 ) = 0

f (x)

" x0 , .& /( (x)

0 %">

lim

* 2% lim

/(x)

x x0

f (x)

6 (N. ! 5 . #5% & $"& % ! $ # 6 ! *5% ( 0

lim

f (x)

x x0

(x)

x x0

/(x)

; < = 1. .! 5 6 (N #" = " 5

f (x)

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

x0 .

.! 5 $#6 6 !(N 76% 5% & $"& % ! $ # 6

; < = 2. .! 5 $#6 6 !(N 76% %">

lim

f (x)

x x0

/(x)

& ! 5 . #5%, $#6 6 !(N 76% 5% & $"& % ! $ # 6 8 ! (

, %" #$ -

!#@ 7 6 ! / .

$"& % ! $ # 6 8 ! ( 0 , , 1 ,

0 , 00 $! / ! ' 6 ! ' $#

. / * @ = ' . & ! & 7.

'#8 f (x) 0, (x) .& x 0 lim ( f (x) (x)) = [0 ]. $& - / #"

(x)

. & ! & %. lim ( f (x) (x))

= [0 ]= lim

= '0

x x0

(x)

$ .

lim ( f (x) (x)) = [0 ]= lim

x x0

f (x)

'#8 f (x) , (x) .& x 0 lim ( f (x) (x)) = [ ]. .

x x0

(x)

f (x)

$ .

lim ( f (x) (x)) = [ ]= lim

= lim

x x0

x x0 1

x x0

f (x)

f (x) (x)

(x)

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