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Национальный минерально-сырьевой университет «Горный»

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d ( n) y = d (d n 1 y). A,

d n y = f (n) (x)d n x.

(4.18)

7 &, n -

( n - ! ( n -

( (.

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

dy = f 8(u)du

( (, f 8(u) du = 8(x)dx

x . ! $

du (, (4.5)

. ! $

( $ . :,

$ ( $

( &.

A (4.18) : f (n) (x) = d n y , , n - dxn

$ n -

n -( ( (.

d n y

! dxn & ( n -

y ( x .

B , !

y = f (x)

x = (t), y = ;(t), t T ,	(4.19)
T - (	t , (t)
(&) T .

yx8 y ( x ,

(4.19),

y = f (x) ( .
?. # T x = (t)								(&)
t < (t) ;(t)	,							8(t) 0 ,
" ( x		y		& ( (
x . !
		;	8	(t)		y 8
yx8	=	t			=	t	.	(4.20)
yx8	=	t		(t)	=		.	(4.20)
		8		(t)		x 8
		t				t

167

																					yxx88																																								x		yx8 .
(4.20) y , yx8 ,
																														yxx88 =					(yx8 ) 8
																														yxx88 =									x			8			t .																		(4.21)
																																									t
! (, ( &
y ( x .
% . H yx8 yxx88 ,
				x = a				(	t sin t									)	,				y = a 1 cost												), a - .
								(										)								(									), a - .
A ( (4.20)																																						t									t
														y8									asin t								2sin							t			cos						t								t
																															2sin										cos
																																					2										2
							y8x				=					t		=													=						2										2		= ctg								.
							y8x				=				8			=		a(1 cost )											=										2 t								= ctg						2		.
														x t																			2sin
														x t																												2
																																										2
H$ ( (4.21),
					1								1
	(yx8 ) 8											,
							sin	2		t		,	2													1																												1									1
							sin	2					2													1																												1									1
yxx88 =		t	=						2							=																								=																					=					.
yxx88 =		t	=		a(1 cost )											=									2			t												=												2		t					2 t		=			4 t		.
	xt	8			a(1 cost )																2asin				2			t	(1 cost )																	2asin						2		t		,		2sin	2 t			4asin		4 t
	xt																				2asin						2		(1 cost )																	2asin							2			,		2sin		2		4asin			2
																											2																										2							2					2
																							& ' !
												f 8(x) ,												f (x) = 3																				.
C ! 1. 8 (												f 8(x) ,												f (x) = 3									6x2 5											.														f (x)
C			&																																																							f (x)
			1
f (x) = (6x2 5)						(
f (x) = (6x2 5)					3	(
( :
																							1		8																	1
													"						2					#					1			2											1							2						8
													"						2				3	#					1			2										3	1							2						8
					8						= %(6x										5) &					=					(6x						5)									, (6x 5)												=
					8
							f (x)																						3
													'											(					3
																1	(6x					2	5)			2																4x
											=					1						2				3 ,12x =																4x									.
																3																		3
																3																		3		(6x2 5)2
																								f (x) = arctg																								.
C ! 2. 8 (												f 8(x) ,												f (x) = arctg																x2 + 1								.

( &

, :

168


					8							8			1													8
					8				2			8			1											2		8
				f (x) = (arctg x						+ 1) =														, (	x		+ 1) =
				f (x) = (arctg x						+ 1) =													2	, (	x		+ 1) =
												1 + (					x2 + 1)
	1			1		1	1					1			1								1						x
	1			1			1					1			1								1						x
=			,		, (x2 + 1)2			, (x2	+ 1)8			=				,			,						, 2x =						.
		1 + x2 + 1		2										x2 + 2				2											(x2 + 2)
		1 + x2 + 1		2										x2 + 2				2				x2 + 1							(x2 + 2)	x2 + 1
		C ! 3. 8 ( y8x y88xx y ( x , (
:					x = acost,			y = bsin t, a b - , t - .
		(4.20) :
								y8x =		yt8	=		bcost		=					b	ctgt.
								y8x =			=		asin t		=						ctgt.
										xt8			asin t							a

A (4.21), (

y88 = ( yx8 )8t

xx xt8

	"		b	#8
	%			ctgt &
			a
=	'			( t
		a sin t

	b	"	1		#
		, %				&		b
	a		sin	2
=		'			t (		=		.
		a sin t						a2 sin3 t

8 4.6

1.L ( ( ?

2.M ( ( & ( .

3.& .

4.? ( ( ?

5.? ( n - ?

6.H ( k ?

7.? & ?

8.H (

9.H , (

C !

A ( ( & (

( " ,

3.3. EFG:( G % # :HG

, & & [1], [2], [3].

169

3.4. "I ##J&HK

J + – ! , ( .

J + ! G ,

& ! a,b G , ,

& ( ! c G .

J + ! Aik , aik n -+

	,		Dik ! !
( 1)i+k ,		Aik = ( 1)i+k Dik.	( e1 , e2 , e3
:

, ( .

e1, e2 , ( .

8 AB	( ,
A , - B .			a ( b
8
( c , ( :
) c a b
	r	r	r
, . .\| c \|=\| a \|,\| b \|,sin( a=b ) ,
&) c a b ,
	r r	.
) ( a,b, c
8	.	# D( A) = 0 ,		A

( & (.

" ,

( :	x2	y2	= 1,	a b & .
	a2	b2

" ! ,

(	x2	y2	= z , a b -
	a2	b2

" + ( &

! a11, a22 ,..., ann , , " (

( (, !,

( , ! +

" , " (

& .

" - Oxy

(x, f ( x)), x X .

170

" ( (

& ( (, & " ( !

$ (, & ! " & (

B + ,

(			x2	+	y	2	z2	= 1, a,b,c
(			a2	+	b2		c2	= 1, a,b,c
.			a2		b2		c2
.
B + a
(	Oxyz
r	r					r
(x, y, z) , x = ira , y = rj a ,		z = kra .

(

B + ! M

( Oxy : x - /

Ox y- / Oy. :

- OM .

B + , !

(, ( , :

	a11	0	0	, , ,	0
	a11	0	0	, , ,	0
	0	a22	0	, , ,	0
	0	0	a33	, , ,	0
	... ... ... ... ...
	0	0	0	, , ,	ann
B 3-+					y = f ( x) , & d3 y

d 3 f ( x) , d 2 y : d3 y = d (d 2 y) = f ''( x)dx3 .

B n -+ y = f ( x) ,

d (n) y = d (d(n 1) y).

B , & dx ,

" 6x x , dx 9 6x .

B y = f ( x) x0 , &

dy df (x0 ), f '(x0 )6x , " & (

( 6x " , dy = f '( x0 )6x .

( ! ( .

B ! x0 y = f ( x) ,

" ! ( , " " 6x

171

,		: 6y = f '( x0 ) 6x + +(6x) 6x ,
+(6x) 0	6x	0 .
B (a,b)				(
&)			y = f ( x) ,	!

( ! .

B AB AB . G ( ,

G ! En ( E )

n - , ! ( :

	1	0	0	, , ,	0
	1	0	0	, , ,	0
	0	1	0	, , ,	0
E =	0	0	1	, , ,	0	.
	... ... ... ... ...
	0	0	0	, , ,	1
< ( M0			, (

" M0M M M0 .

< ! ( (

x1, x2 ,..., xn.

< . % (,

< , ( (

& & ,

& ( (, &

< , ( (

< ! ! f 8( x0 )

, & (.

< + ,

(	x2	+	y2	z2	= 0 , a,b,c -
	a2		b2	c2

< + ,

( Oxy

( : Ax2 + 2Bxy + Cy2 + Dx + Ey + F = 0, &

A, B, C .

172

I V ,

" & ( (,		>(a + b) = >a + >b,		(> + 3) a = >a + 3a,
>(3a) = (>3) a,	1a = a >,3 a,b V.
I		A ,	& " (
V	U , , " (
x V			( !	A(x) U ,

A(>x + 3y) = >A(x) + 3A( y) x, y V & >,3 .

I + !

L m × n &

aik (i = 1, 2,..., m; k = 1, 2,..., n), "

& :

a11

a12

...

a1n

a21

a22

...

a2n

aik

(i = 1, 2,..., m;

k = 1, 2,..., n).

...

am1

am2

...

amn

L n ( ( n

n , ! ( !

a11	a12	...	a1n
a21	a22	...	a2n	.
...	... ... ...			.
an1	an2	...	ann

L Dik , aik n -+

(n 1) - , " ( (

i -( k - o &,

! !.

( * a

+, 3 @ , ! & Ox, Oy Oz

( + - e .

( ( ) A ,

( D( A) .

( ( ( (

, " & $ .

( ( ( ( ,

" $ .

173

(

n = ( A,B )

( ! ( (.

2 1

(

& ( A n & (

A11

A21

, , ,

An1

A11

A21

, , ,

An1

D( A)

A12

A22

, , ,

An2

A 1 =

A12

A22

, , ,

An2

D( A)

, , ,

A1n

A2n

, , ,

Ann

A1n

A2n

, , ,

Ann

D( A)

Ax + By + Cz + D = 0 , (x, y, z)- " (

, A, B C & " .

* ( Oxy

( x y: Ax + By + C = 0 , A B & " .

+ ,

(	x2	+	y2	z2	= 1, a,b,c -
	a2		b2	c2

. N (

, & ( ! .

" [a,b] , "

	lim		f (a + 6x) f (a)		lim		f (b + 6x) f (b)		.
			6x
	6x	0			6x	0		6x
	6x	>0			6x	<0
!:		-		(				x0

x , !(x, x0 ) x0 $ . : ( x0 & ,

" - x0 .

( ( (

, " $ .

+ (

) , ( ! (

& ( (, &

a11 a12 a21 a22 .

174

7 &, ,	a11	a12	= a a	22	a	a .
	a21	a22	11	22	12	21

n -+ ,

( ! ( &

	a11	a12	...	a1n
	a21	a22	...	a2n	= a A	+ a A + , , , + a A		.
	...	...	...	...	= a A	+ a A + , , , + a A		.
	...	...	...	...	11 11	12 12	1n 1n
	an1	an2	...	ann		D( A) = 0 ,
.		#				D( A) = 0 ,	A

( & (.

% ,

( : y2 = 2 px , p - & ,	0.
% ! +. M !	a1n , a2 n 1,..., an1 ,

( , & (

% (,

( ( ! & (.

% ! M : r

M ( l OM ,

( ( .

% ,

% - ,

% ! ': ,

, + , & &$(

% ! – ,

" .

% *: {xn}

" (, & n $ xn+1 > xn ,

", " ( &$ ".

	% – " & ".				{xn}
	%	+ !:			{xn}
+ !, & M > 0 (
n0	,		xn	> M .
n0	,		xn	> M .
			0

% + !: {xn}

+ !, " M > 0, , n

xn M .

175

% *: {xn}

*, & n $ xn+1 < xn ,

", " ( $ ".

% !: xn

&, & M (

( N , n > N | xn |> M .

% !: a

{xn} , & (

( N , " ( , n > N

xn a < .

% : &

) = )( ), " , x R) (a)
( x a , a	- ) f (x) R ( A) , A
f (x)	! a ( x , " a ).

% : &

) = )( ) , " ,

x R) (a) f (x) R ( A) ,	A
f (x) ! a .	&
% :	&

) = )( ) , "

x R

(a)

f (x) R ( A) , A

! a .

f (x)

y = f ( x) , & (

% +

( x), y

(2)

, f

(2)

( x) ,

(

y , yxx , f

! ( .

y = f ( x)

% n -+

(

n 1

! (

) -

& y(n)

f (n) ( x) .

% + y = f ( x)

f '( x) ! ( .

% + y = f ( x)

( ! ( .

% y = f (x) ( x x0 , & (

y8 f 8( x0 ) , ( & (

$ " x0 "

, :

176

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