Добавил:

Upload Опубликованный материал нарушает ваши авторские права? Сообщите нам.

Вуз:

Национальный минерально-сырьевой университет «Горный»

Предмет:

[НЕСОРТИРОВАННОЕ]

Файл:

attachments_22-12-2011_21-07-09 / razdel 4

.pdf

Скачиваний:

Добавлен:

02.04.2015

Размер:

738.49 Кб

Скачать

☆

<<< < Предыдущая 1 2 3 45 / 75 6 7 > Следующая >>>

M & M0 ,

!(M0, M ) 0 . # ! " M0M & &c

M 0T

M0M M 0T ,

M 0T

( l

M0 .

H !

M0 (

M 0T ,

& " ( Ox (

0 ,

0 = lim

tg 0 = lim tg .

(4.2)

8 .4.13 ,

6x 0

f ( x0 + 6x) f ( x0 )

tg =

M0 P

, !

( (

M 0T

f ( x0 + 6x) f ( x0 )

k0 = tg 0 = lim

= f 8( x0 ).

(4.3)

6x 0

> ( ! k0 =

f 8(x0 )

( M 0T M 0 (x0 , y0 ),

, $ ! ( (

y y0 = f 8(x0 )(x x0 ).

(4.4)

y = f (x) x0

A, f (x)

& ( ( ! ( ! (

& ( x0 .

C !. Q " ( f 8(x) x ! " ( (

y = f (x) ! ( ( & ( x ). ! (

( , Oy ,

& ( ( – , Oy .

y = f (x) ( X

x0 . > ! ( " 6x 0 ,

& x0 + 6x X .

. @ y = f (x)

! x0 , " ! ( , "

" 6x ,

157

	6y = A6x + +(6x) 6x ,			(4.5)
A - , +(6x)	- 6x , &			6x 0 ,
+(6x) 0 6x 0 .
>, A ( x0
6x .		x0	A ,	&" ,
.
Q
" .
?. C , & y = f (x) &
x0 , & , & ! & ! (
( (.		A ,
% ,	[3],	A ,	"	(4.5),
	f 8(x0 ) (	f 8(x)	x0 .	!

6y = f 8( x	) 6x + +(6x) 6x,		+(6x) 0		6x	0	.	(4.6)
0							.	(4.6)

C !. H &

" (

( ( .

H . ?. # y = f (x) x0 ,

! ( .

C !. :& ,

( , &" ,

! ( . I, ,

, . 4.14. :& !

x0 , & ! ( . L ( M0 (x0 , y0 ) Oy ,

y y

O	x0	x	O	x
)			x0
		I 4.14.	&)

, & x0 & ( (. 5

158

( x0 &" ( (, ! x0 ( ( .

. 7, & x0 . 4.14 , !

, & x0 . 4.14 & – + !.

. @ y = f (x)

(a,b) ( &), !

( ! .

: !, " X

(

). H ,

a b (a < b)

lim f (a + 6x) f (a)

6x 0 6x

6x>0

X = [a,b] - ( ,

", ,

lim		f (b + 6x) f (b)	.
lim			.
6x	0	6x
6x	<0

5 , ( ,

$ & . 7 &

y = f (x) x0 . 7 x0

& 6x 0 $ (4.6):

6y = f 8( x	)6x + + (6x)6x	,	+ (6x) 0		6x	0	,
0		,					,

& ( . !

f 8( x	)6x	( ( ( "				6x	,
0
+(6x)6x - ( ( ( 6x .
	.			f 8(x0 )6x , " & ( (
			6x		"	x0 ,
					y = f (x) ! ( &
dy df (x0 ).
	A,			dy = f 8( x0 )6x.
							(4.7)
>,			"		6x x , (

, & & dx

. E

, f (x) = x ([3]).	7 &,
(4.7) $ "
dy = f 8(x0 )dx,	(4.8)

159

(

( ! ( (")

( (.

A (4.8) ,	f 8(x0 ) = dy	. 7 &,
	dx

( (. M dydx

& ( y ( x .

H y = f (x) , .4.13. L , y = f (x) x0

& ( " ( ! (

M0 (x0 , y0 ) [x0, x0 + 6x][3].

: (

! ( . :&" &

( &b ( . H (4.8)

(

6x 9 dx.

% 2. 8 ( " y = 3x2 + x

x = 1, 6x = 0,1 .

8 ( B " :

6y = 3( x + 6x)2 + ( x + 6x) 3x2 x = 6x6x + 3(6x)2 + 6x = (6x + 1)6x + 3(6x)2 .

7 dy = (6x + 1)6x . H 6y dy x = 1, 6x = 0,1

6y = 7 , 0,1 + 3 , 0,01 = 0,73; dy = 7 , 0,1 = 0,7.

% , ! +

8 ( $

(, & . H

[3].

?. # u = u(x)							v = v(x) x ,
! (				(u ± v)8 = u8 ± v8 ,
				(u ± v)8 = u8 ± v8 ,					(4.9)
					8		8	8	(4.10)
				(uv)		= u v + v u ,			(4.10)
8			8		8
" u #		=	u v v u				v = v(x) 0 .		(4.11)
%	&
%	&			v	2
' v (				v

Q ! dx , ,

d (u ± v) = du ± dv ,

(4.12)

160

d (uv) = vdu + udv ,								(4.13)
	" u		#	vdu udv
d %			& =				.	(4.14)
				v	2
	' v		(					y = C ( -
(
x X ) .
7 &, x X
C	8	= 0;				8		(4.15)
				dC = C dx = 0 .
C (, & 7
, 6y 9 0						&		x 6x ,
x, x + 6x X. :,						(		,

(4.15).

@ (4.9) & &" ( &

u = C , C const , (4.10) (4.13), (4.15),

8	8
: (Cv)	= Cv , d (Cv) = Cdv. 7 , (

( .

% ,

A ( [3]).

(sin x)8 = cos x,

(cos x)8

= sin x,

(tgx)8

(ctgx)8

cos2

sin

(loga

x)8

loga e =

(ln x)8

x ln a

& ' !.

C ! 1. H

f (x) =

+ x + 2

x = 3.

f (x) ,

8 (

" 1

= %

+ x + 2 &

= %

+ %

f (x)

+ x

+ 2 .

' 4

(

' 4

(

H ( (:

" 1

x4 &

+ %

+ x8 + 28

(x4 )

(x2 ) + x8 + 28,

' 4

(

' 2

(

( ( ,

161

14 (x4 )8 12 (x2 )8 + x8 + 28 = 14 , 4x3 12 , 2 , x + 1 + 0 = x3 x + 1.

A, f 8(x) = x3 x + 1.

8 ( ( x = 3,

x 3:

f 8(3) = 33 3 + 1 = 25.

C ! 2. 8 ( f (x) = (2x2 + 1), cos x .

! ( (,

f 8(x) = -1/(2x2 + 1)cos x.208 = (2x2 + 1)8 cos x + (2x2 + 1)(cos x)8 .

( , 1,

, (cos x)8 = sin x :

(2x2 +1)8 cos x + (2x2 +1)(cos x)8 = (2,2x + 0)cos x + (2x2 +1),( sin x) = 4xcos x (2x2 +1)sin x.

7 &,

+ 1)sin x .

f (x) = 4x cos x (2x

C ! 3. 8 ( df (x)

f (x) = tgx x =

8 ( , df (x) = f

(x)dx .

7 (tgx)8 =

cos2

df (x) =

, dx

cos2 x

, , cos =

= 2dx

df %

4 (

(

C ! 4. M ( f ( x) = sin x

x0 = .

Q ( ( x0 , f (x0 ))

y f (x0 ) = f 8(x0 )(x x0 ) .

8 ( " :

f (x0 ) = sin = 0

f 8(x) = cos x, f 8(x0 ) = cos = 1.

162

7 y = ( 1)(x ) , y = x - (

8 4.5

1.? ( ?

2.H ( (?

3.L ( ( ?

4.L (: ) ; &)

5.? ?

6.L ( ?

7.? ) ; &) ; )

8.C ( (.

4.6.% , !

. % '

( H "

•B .

•% .

•B . %

+ ! .

•? .

•% ' .

•B , !.

$ . # H &

, & [3], 3, . 67-73 –

M -( ( & $

( & ^ 2 ^

71-80, 81-90, 91-100.

B . %

y x y = f (u),

u = (x) .

? ( ). #

163

y = f (u), u = (x) " u

x , f [ (x)]	x ,
y8 = f 8(u)u8		y8x = y8u , u8x.	(4.16)
Q (4.16)		dx ,

( dy = f 8(u)du.

C !. C y = f (u) & (

, & u & (, (

(. H !

( ) $ . M

, u - , du = 6u

", u - ( (

), du - ! ( , ,

" " 6u.

M " (

( ( [3]:

1) (x

+ 1

; 2) (a

x 8

ln a;

3) (e

) = + x

) = a

)

= e

x < 0 ,

! x+

C !.

( + !

(

y = u(x)v(x) .

%. 8 ( y = xsin x .

7 y = ex y = ln x &,

& A

A = eln A .

y = xsin x = e	ln(xsin x )	= esin x,ln x .
y = xsin x = e		= esin x,ln x .
7
y8 = (esin x,ln x )8 = esin x,ln x , (sin x , ln x)8 =
		"	sin x #
= xsin x ((sin x)8 ln x + sin x , (ln x)8 )= xsin x % cos x , ln x +			&.
		'	x (

> & (

164

B . %

+ !

& y = f (x)												x = ( y) .
?	(										).				#
y = f (x),	x = ( y)						(&),							x f (x)
, f 8(x) 0,									" (								y
( y) ( y ),
					8				1	.							(4.17)
					8				8
						( y) =			8
									f (x)
M " ! (
& (,[3]:
	1)	(arcsin x)8	=		1			,		2)	(arccos x)8 =			1			,

					1 x2									1 x2
					1 x2									1 x2
	3)	(arctgx)8 =			1	,				4)	(arcctgx)8 =			1		.
	3)	(arctgx)8 =		1 + x2		,				4)	(arcctgx)8 =		1 + x2			.

C & (

1.	(C)8 = 0,		C - ,						2.	(x+ )8 = + x+ 1,
3. (ex )8 = ex ,									4. (ax )8 = ax ln a, a > 0, a 1,
5.	(ln x)8 =		1	,					6.	(loga x)8 =				1	, a > 0, a 1,
5.	(ln x)8 =			,					6.	(loga x)8 =			xln a		, a > 0, a 1,
			x										xln a
7.	(sin x)8 = cos x ,								8.	(cos x)8		= sin x ,
9.	(tgx)8 =		1			,			10.		(ctgx)8	=		1		,


		cos2 x												sin2 x
11. (arcsin x)8 =					1			,	12.		(arccos x)8			=	1			,

						1 x2									1 x2
						1 x2									1 x2
13. (arctgx)8 =					1		,		14.		(arcctgx)8		=		1		.

					+ x2										+ x2
			1		+ x2									1	+ x2

165

% '

	y = f (x) .
7	f 8(x) ! ( , " +

, & ( ( x , ( ! , ,

, . E

+ y = f (x) &

y88, yxx88, f 88(x), y(2) , f (2) (x).

( 2- +

y = f (x) &

y888, yxxx888, f 888(x), y(3) f (3) (x) .

1 4- , 5- $ .

. % n -+ y = f (x)

( ( n 1)-

( ! . n -

& : y(n) , f (n) (x).

y = 5x3 + 6x 1, y8 = 15x2 + 6, y88 = 30x, y888 = 30, y(4) = L = y(n) = 0.

:, & n -(

, (n + 1) - , .

% 3.

y = eax , y8 = aeax, y" = a2eax, L , y(k ) = akeax.
.	@	f (x) k
( (	),					! (				(	!
) f (x), f			8		88		(3)	(x),L,	f	(k 1)	(x).
) f (x), f			(x), f		(x), f			(x),L,	f		(x).
C		y = f (x) ,	x	- ,
( " 1-+ , (
		8
		dy = f (x)dx,
dx = 6x -				" x .

> dx; dy & ( x . C ! (

2-+ y = f (x) &

d 2 y d 2 f (x). dx ( ), d 2 y = d(dy) = d[ f (x)dx] = [ f 8(x)dx]8dx = f 88(x)dx2 .

C d 2 y 3-+


y = f (x)			& d3 y d3 f (x).
d	3	y = d(d		2	88	2	] = [ f	88	2	8	888	3	.
d		y = d(d			y) = d[ f (x)dx		] = [ f	(x)dx		] dx =	f (x)dx		.
.					B			n -+				y = f (x)
,					&

166

<<< < Предыдущая 1 2 3 45 / 75 6 7 > Следующая >>>

Соседние файлы в папке attachments_22-12-2011_21-07-09

#
02.04.2015494.5 Кб10kontrol 1.pdf
#
02.04.2015465.65 Кб10razdel 1.pdf
#
02.04.2015254.75 Кб10razdel 2.pdf
#
02.04.2015467.57 Кб10razdel 3.pdf
#
02.04.2015738.49 Кб11razdel 4.pdf
#
02.04.20151.03 Mб11semestr1_1.pdf
#
02.04.20151.18 Mб9semestr1_2.pdf
#
02.04.2015228.88 Кб16test11.pdf
#
02.04.201586.53 Кб26titul list.pdf