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Методические указания по выполнению контрольной работы № 2 по математике для студентов инженерно-технических специальностей заочной формы обучения

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2.2. ȼɵɱɢɫɥɟɧɢɟ ɞɥɢɧ ɞɭɝ ɤɪɢɜɵɯ. ȼɵɱɢɫɥɟɧɢɟ ɨɛɴɟɦɨɜ

ȿɫɥɢ ɩɥɨɫɤɚɹ ɤɪɢɜɚɹ ɡɚɞɚɧɚ ɭɪɚɜɧɟɧɢɟɦ y=f(x), ɝɞɟ f(x) – ɧɟɩɪɟɪɵɜɧɨ ɞɢɮɮɟɪɟɧɰɢɪɭɟɦɚɹ ɮɭɧɤɰɢɹ, adxdb, ɬɨ ɞɥɢɧɚ l ɞɭɝɢ ɷɬɨɣ ɤɪɢɜɨɣ ɜɵɪɚɠɚɟɬɫɹ ɢɧɬɟɝɪɚɥɨɦ

		b
	l ³ 1 (yc)2 dx .
		a
	ȿɫɥɢ ɠɟ ɤɪɢɜɚɹ ɡɚɞɚɧɚ ɩɚɪɚɦɟɬɪɢɱɟɫɤɢɦɢ ɭɪɚɜɧɟɧɢɹɦɢ x=x(t), y=y(t) (DdtdE),
ɬɨ l	E	(x c)2	(yc)2 dt .
ɬɨ l	³	(x c)2	(yc)2 dt .
	³	t	t

Ⱥɧɚɥɨɝɢɱɧɨ ɧɚɯɨɞɢɬɫɹ ɞɥɢɧɚ ɞɭɝɢ ɩɪɨɫɬɪɚɧɫɬɜɟɧɧɨɣ ɤɪɢɜɨɣ, ɨɩɢɫɚɧɧɨɣ ɩɚɪɚɦɟɬɪɢɱɟɫɤɢɦɢ ɭɪɚɜɧɟɧɢɹɦɢ: x = x(t), y = y(t), z = z(t), D d t d E

l	E	(x c)2	(yc)2	(zc)2 dt .
l	³	(x c)2	(yc)2	(zc)2 dt .
	³	t	t	t
	D

ȿɫɥɢ ɡɚɞɚɧɨ ɩɨɥɹɪɧɨɟ ɭɪɚɜɧɟɧɢɟ ɤɪɢɜɨɣ U = U(M), D d M d E, ɬɨ

l ³	U	2	c	2	dM .
			(U )

ȿɫɥɢ ɩɥɨɳɚɞɶ S(x) ɫɟɱɟɧɢɹ ɬɟɥɚ ɩɥɨɫɤɨɫɬɶɸ, ɩɟɪɩɟɧɞɢɤɭɥɹɪɧɨɣ ɨɫɢ Ox, ɹɜɥɹɟɬɫɹ ɧɟɩɪɟɪɵɜɧɨɣ ɮɭɧɤɰɢɟɣ ɧɚ ɨɬɪɟɡɤɟ [a, b], ɬɨ ɨɛɴɟɦ ɬɟɥɚ ɜɵɱɢɫɥɹɟɬɫɹ ɩɨ ɮɨɪɦɭɥɟ

V ³S (x )dx . a

Ɉɛɴɟɦ V ɬɟɥɚ, ɨɛɪɚɡɨɜɚɧɧɨɝɨ ɜɪɚɳɟɧɢɟɦ ɜɨɤɪɭɝ ɨɫɢ Ox ɤɪɢɜɨɥɢɧɟɣɧɨɣ ɬɪɚɩɟɰɢɢ, ɨɝɪɚɧɢɱɟɧɧɨɣ ɤɪɢɜɨɣ y=f(x), (f(x)t0), ɨɫɶɸ ɚɛɫɰɢɫɫ ɢ ɩɪɹɦɵɦɢ x=a ɢ x=b (a<b), ɜɵɪɚɠɚɟɬɫɹ ɢɧɬɟɝɪɚɥɨɦ

V S³ f 2 (x)dx .

		ɉɪɢɦɟɪ 2.8. ȼɵɱɢɫɥɢɬɶ ɞɥɢɧɭ ɞɭɝɢ ɤɪɢɜɨɣ y2 x 3 , ɨɬɫɟɱɟɧɧɨɣ ɩɪɹɦɨɣ
x	4	(ɪɢɫ. 2.5).
x	3	(ɪɢɫ. 2.5).
	3
		y		x	4
					4
			Ⱥ		3
			Ⱥ		3
		y2	x3

0	4	ɯ
	3

																						ȼ
																		Ɋɢɫ. 2.5
			Ɋɟɲɟɧɢɟ. Ⱦɥɢɧɚ ɞɭɝɢ ȺɈȼ ɪɚɜɧɚ ɭɞɜɨɟɧɧɨɣ ɞɥɢɧɟ ɞɭɝɢ ɈȺ.
			3		3					1
		x 2 , yc			3
y		x 2 , yc					2 x 2 .					Ɍɨɝɞɚ
						4									2		4							4					1
1					3				§3			·				3			9				4	3	§		9	·	1		§	9		·
1	l	lOA			3				§3			·				3			9		xdx		4	3	§		9	·	2		§	9	x	·
2	l	lOA			³			1 ¨					x ¸		dx	³ 1			4		xdx		9	³	¨1		4	x¸		d¨1		4	x	¸
2					0				©2			¹				0			4				9	0	©		4	¹			©	4		¹
		§		9	·			3
		§		9	·			2
	4	¨1			x¸				4 / 3				8	43 / 2		1		56					56			112 .
		¨1		4	x¸
		©		4	¹															; l		2
	9			3								27						27		; l		2	27
	9			3					0			27						27					27			27
				2																													(t	2	2)sin t 2t cost;
			ɉɪɢɦɟɪ								2.9. ȼɵɱɢɫɥɢɬɶ							ɞɥɢɧɭ				ɞɭɝɢ				ɤɪɢɜɨɣ								2
			ɉɪɢɦɟɪ								2.9. ȼɵɱɢɫɥɢɬɶ							ɞɥɢɧɭ				ɞɭɝɢ				ɤɪɢɜɨɣ					®°x
																															°		(2 t			2	)cost 2t sin t,
																															¯y		(2 t				)cost 2t sin t,

ɟɫɥɢ t ɢɡɦɟɧɹɟɬɫɹ ɨɬ t1=0 ɞɨ t2=S.

Ɋɟɲɟɧɢɟ. Ⱦɢɮɮɟɪɟɧɰɢɪɭɹ ɩɨ t, ɩɨɥɭɱɚɟɦ

xtc	2t sin t (t 2 2)cost 2cost 2t sin t								t 2 cost,
ytc	2t cost (2 t 2 )sin t 2sin t 2t cost								t 2 sin t,
ɨɬɤɭɞɚ	(x c)2 (yc)2		t4 cos2 t t4 sin 2 t						t4 (cos2 t sin 2 t) t2 .
	t	t
		S		t	3	S	S	3
		S			3	S		3
ɋɥɟɞɨɜɚɬɟɥɶɧɨ, l		³t2dt						.
ɋɥɟɞɨɜɚɬɟɥɶɧɨ, l		³t2dt						.
		0	3			0	3

ɉɪɢɦɟɪ 2.10. ɇɚɣɬɢ ɞɥɢɧɭ ɞɭɝɢ ɤɚɪɞɢɨɢɞɵ U=a(1+cosM ), (a>0, 0dMd2S) (ɪɢɫ. 2.6).

Ɋɟɲɟɧɢɟ. ɁɞɟɫɶUc

asin M,

(Uc )2 U2

2a2 (1 cosM)

cos

. ȼ ɫɢɥɭ ɫɢɦɦɟɬɪɢɢ l

dM 8a.

2 2a³cos

O	2	M 0
Ɉ		M 0
Ɉ

M32S

Ɋɢɫ. 2.6

Ɂɚɦɟɱɚɧɢɟ. ɉɨɫɬɪɨɟɧɢɟ ɥɢɧɢɢ ɜɟɞɟɬɫɹ ɜ ɩɨɥɹɪɧɨɣ ɫɢɫɬɟɦɟ ɤɨɨɪɞɢɧɚɬ ɩɨ ɬɨɱɤɚɦ, ɤɨɬɨɪɵɟ ɜ ɞɨɫɬɚɬɨɱɧɨɦ ɤɨɥɢɱɟɫɬɜɟ ɡɚɩɢɫɵɜɚɸɬɫɹ ɜ ɜɢɞɟ ɬɚɛɥɢɰɵ ɢɯ ɤɨɨɪɞɢɧɚɬ.

ɉɪɢɦɟɪ 2.11. ɇɚɣɬɢ ɨɛɴɟɦ ɬɟɥɚ, ɨɛɪɚɡɨɜɚɧɧɨɝɨ ɜɪɚɳɟɧɢɟɦ ɜɨɤɪɭɝ ɨɫɢ Ox

ɮɢɝɭɪɵ, ɨɝɪɚɧɢɱɟɧɧɨɣ ɥɢɧɢɹɦɢ 2y x2 ɢ 2x 2y 3 0 (ɪɢɫ. 2.7).

Ɋɟɲɟɧɢɟ. ɇɚɣɞɟɦ ɚɛɫɰɢɫɫɵ ɬɨɱɟɤ ɩɟɪɟɫɟɱɟɧɢɹ ɤɪɢɜɵɯ

y	x2	ȹ y	3 2x 3 x :				x2		3 x; x2 2x 3 0; x 3, x				2	1.
y		ȹ y	3 2x 3 x :						3 x; x2 2x 3 0; x 3, x					1.
	2		2	2	2			2			1
	2		2	2	2			2
									y

										2 y	x2


						Ⱥ
												2x 2 y 3 0

										ȼ




					-3				O
					-3				O	1		x

Ɋɢɫ. 2.7

ɂɫɤɨɦɵɣ ɨɛɴɟɦ ɟɫɬɶ ɪɚɡɧɨɫɬɶ ɞɜɭɯ ɨɛɴɟɦɨɜ: ɨɛɴɟɦɚ V1 ɬɟɥɚ, ɩɨɥɭɱɟɧɧɨɝɨ

ɜɪɚɳɟɧɢɟɦ ɤɪɢɜɨɥɢɧɟɣɧɨɣ ɬɪɚɩɟɰɢɢ, ɨɝɪɚɧɢɱɟɧɧɨɣ ɩɪɹɦɨɣ y	3	x ( 3 d x d1),
	2

ɢɨɛɴɟɦɚ V2 ɬɟɥɚ, ɩɨɥɭɱɟɧɧɨɝɨ ɜɪɚɳɟɧɢɟɦ ɤɪɢɜɨɥɢɧɟɣɧɨɣ ɬɪɚɩɟɰɢɢ,

ɨɝɪɚɧɢɱɟɧɧɨɣ ɩɚɪɚɛɨɥɨɣ y

( 3 d x d1) . ɂɫɩɨɥɶɡɭɹ ɮɨɪɦɭɥɭ V

S³ f 2 (x)dx ,

ɢɦɟɟɦ

§ x2

§3

·2

V1 V2

S ³

x¸ dx S ³

S ³

x¸

d¨

x¸

§ 3

·3

x ¸

272

S³

2.3. ɇɟɫɨɛɫɬɜɟɧɧɵɟ ɢɧɬɟɝɪɚɥɵ

2.3.1.ɂɧɬɟɝɪɚɥɵ ɫ ɛɟɫɤɨɧɟɱɧɵɦɢ ɩɪɟɞɟɥɚɦɢ (ɧɟɫɨɛɫɬɜɟɧɧɵɟ ɢɧɬɟɝɪɚɥɵ ɩɟɪɜɨɝɨ ɪɨɞɚ)

ȿɫɥɢ ɮɭɧɤɰɢɹ		f (x) ɧɟɩɪɟɪɵɜɧɚ ɩɪɢ a d x f, ɬɨ	ɧɟɫɨɛɫɬɜɟɧɧɵɦ
ɢɧɬɟɝɪɚɥɨɦ ɩɟɪɜɨɝɨ ɪɨɞɚ ɧɚɡɵɜɚɟɬɫɹ ɫɥɟɞɭɸɳɢɣ ɩɪɟɞɟɥ:
f		b
³	f (x)dx lim	³ f ( x)dx .	(2.1)
a	bo f	a
a		a

ȿɫɥɢ ɫɭɳɟɫɬɜɭɟɬ ɤɨɧɟɱɧɵɣ ɩɪɟɞɟɥ ɜ ɩɪɚɜɨɣ ɱɚɫɬɢ ɮɨɪɦɭɥɵ (2.1), ɬɨ ɧɟɫɨɛɫɬɜɟɧɧɵɣ ɢɧɬɟɝɪɚɥ ɧɚɡɵɜɚɟɬɫɹ ɫɯɨɞɹɳɢɦɫɹ; ɟɫɥɢ ɠɟ ɷɬɨɬ ɩɪɟɞɟɥ ɧɟ ɫɭɳɟɫɬɜɭɟɬ ɢɥɢ ɪɚɜɟɧ f, ɬɨ ɪɚɫɯɨɞɹɳɢɦɫɹ.

Ⱥɧɚɥɨɝɢɱɧɨ ɨɩɪɟɞɟɥɹɸɬɫɹ ɧɟɫɨɛɫɬɜɟɧɧɵɟ ɢɧɬɟɝɪɚɥɵ

lim

³ f (x)dx

³ f (x)dx ,

ao f

³ f ( x)dx

lim

³ f (x)dx lim

³ f ( x)dx .

ao f

bo f

ɉɪɢɦɟɪ 2.12. ȼɵɱɢɫɥɢɬɶ

³e 3x dx .

Ɋɟɲɟɧɢɟ. ɂɦɟɟɦ:

³e

dx lim

³e

lim (1

)

lim ¨

bo f

bo f©

3 bo f

		f			dx
ɉɪɢɦɟɪ 2.13. ȼɵɱɢɫɥɢɬɶ ³					dx		.
ɉɪɢɦɟɪ 2.13. ȼɵɱɢɫɥɢɬɶ ³				2			.
		f x			2x 5
Ɋɟɲɟɧɢɟ.	f (x)	1				1			– ɧɟɩɪɟɪɵɜɧɚɹ ɮɭɧɤɰɢɹ
Ɋɟɲɟɧɢɟ.	f (x)	x2 2x	5 (x 1)2						– ɧɟɩɪɟɪɵɜɧɚɹ ɮɭɧɤɰɢɹ
		x2 2x	5 (x 1)2					4

ɧɚ (f;f) ;

2x 5

f x

a 1·

lim

arctg

2 2x 5

4 (x 1)2

f x

ao fa

ao f©

2 ¹

b 1

1 ·

blimo f ³

blimo f

arctg

(x 1)

2 ¹

Ɍɨɝɞɚ ³

. ɂɧɬɟɝɪɚɥ ɫɯɨɞɢɬɫɹ.

2x 5 2

2.3.2. ɂɧɬɟɝɪɚɥɵ ɨɬ ɧɟɨɝɪɚɧɢɱɟɧɧɵɯ ɮɭɧɤɰɢɣ (ɧɟɫɨɛɫɬɜɟɧɧɵɟ ɢɧɬɟɝɪɚɥɵ ɜɬɨɪɨɝɨ ɪɨɞɚ)

ȿɫɥɢ	f (x) ɧɟɩɪɟɪɵɜɧɚ ɩɪɢ a<x<b ɢ ɜ ɬɨɱɤɟ x=b ɧɟɨɝɪɚɧɢɱɟɧɚ, ɬɨ
ɧɟɫɨɛɫɬɜɟɧɧɵɦ ɢɧɬɟɝɪɚɥɨɦ ɜɬɨɪɨɝɨ ɪɨɞɚ ɧɚɡɵɜɚɟɬɫɹ
b		b H
³ f (x)dx lim		³ f (x)dx .	(2.2)
a	Ho 0	a

ȿɫɥɢ	ɫɭɳɟɫɬɜɭɟɬ ɤɨɧɟɱɧɵɣ ɩɪɟɞɟɥ ɜ ɩɪɚɜɨɣ		ɱɚɫɬɢ ɮɨɪɦɭɥɵ (2.2), ɬɨ

ɧɟɫɨɛɫɬɜɟɧɧɵɣ ɢɧɬɟɝɪɚɥ ɧɚɡɵɜɚɟɬɫɹ ɫɯɨɞɹɳɢɦɫɹ; ɟɫɥɢ ɠɟ ɷɬɨɬ ɩɪɟɞɟɥ ɧɟ ɫɭɳɟɫɬɜɭɟɬ ɢɥɢ ɪɚɜɟɧ f, ɬɨ ɪɚɫɯɨɞɹɳɢɦɫɹ.

Ⱥɧɚɥɨɝɢɱɧɨ ɨɩɪɟɞɟɥɹɟɬɫɹ ɢɧɬɟɝɪɚɥ ɢ ɜ ɫɥɭɱɚɟ f (a) rf.

b		b
³ f ( x)dx	lim	³ f (x)dx .				(2.3)
a	Ho 0	a H
a		a H
ȼ ɫɥɭɱɚɟ, ɤɨɝɞɚ f(c)= rf, c (a,b), ɬɨ
b		c H		b
³ f (x)dx	lim	³	f ( x)dx lim	³ f (x)dx .
a	Ho 0	a	Go 0	c G
a		a		c G
				1	dx
ɉɪɢɦɟɪ 2.14. ȼɵɱɢɫɥɢɬɶ ɢɥɢ ɭɫɬɚɧɨɜɢɬɶ ɪɚɫɯɨɞɢɦɨɫɬɶ ³							.
ɉɪɢɦɟɪ 2.14. ȼɵɱɢɫɥɢɬɶ ɢɥɢ ɭɫɬɚɧɨɜɢɬɶ ɪɚɫɯɨɞɢɦɨɫɬɶ ³						2	.
				0 x

Ɋɟɲɟɧɢɟ.

f (x)

–

ɧɟɩɪɟɪɵɜɧɚ ɧɚ

(0,1], lim

f (x) lim

xo 0

xo 0 x2

ɋɥɟɞɨɜɚɬɟɥɶɧɨ,

–

ɧɟɫɨɛɫɬɜɟɧɧɵɣ

ɢɧɬɟɝɪɚɥ

ɜɬɨɪɨɝɨ

ɪɨɞɚ.

0 x

1 dx

lim

f, ɩɨɷɬɨɦɭ, ɢɧɬɟɝɪɚɥ ɪɚɫɯɨɞɢɬɫɹ.

³H x2

0³ x2

H o 0©

3.ɎɍɇɄɐɂɂ ɇȿɋɄɈɅɖɄɂɏ ɉȿɊȿɆȿɇɇɕɏ

3.1.ɉɨɧɹɬɢɟ ɮɭɧɤɰɢɢ ɧɟɫɤɨɥɶɤɢɯ ɩɟɪɟɦɟɧɧɵɯ

ɉɭɫɬɶ D – ɩɪɨɢɡɜɨɥɶɧɨɟ ɦɧɨɠɟɫɬɜɨ ɬɨɱɟɤ n–ɦɟɪɧɨɝɨ ɚɪɢɮɦɟɬɢɱɟɫɤɨɝɨ ɩɪɨɫɬɪɚɧɫɬɜɚ. ȿɫɥɢ ɤɚɠɞɨɣ ɬɨɱɤɟ P(x1,x2,...,xn) D ɩɨɫɬɚɜɥɟɧɨ ɜ ɫɨɨɬɜɟɬɫɬɜɢɟ ɧɟɤɨɬɨɪɨɟ ɞɟɣɫɬɜɢɬɟɥɶɧɨɟ ɱɢɫɥɨ f(P)=f(x1,x2,...xn), ɬɨ ɝɨɜɨɪɹɬ, ɱɬɨ ɧɚ ɦɧɨɠɟɫɬɜɟ D ɡɚɞɚɧɚ ɱɢɫɥɨɜɚɹ ɮɭɧɤɰɢɹ f ɨɬ n ɩɟɪɟɦɟɧɧɵɯ x1,x2,...xn. Ɇɧɨɠɟɫɬɜɨ D ɧɚɡɵɜɚɟɬɫɹ ɨɛɥɚɫɬɶɸ ɨɩɪɟɞɟɥɟɧɢɹ, ɚ ɦɧɨɠɟɫɬɜɨ E={u R|u=f(P), P D} – ɨɛɥɚɫɬɶɸ ɡɧɚɱɟɧɢɣ ɮɭɧɤɰɢɢ u=f(P).

ȼ ɱɚɫɬɧɨɦ ɫɥɭɱɚɟ, ɤɨɝɞɚ n=2, ɮɭɧɤɰɢɸ ɞɜɭɯ ɩɟɪɟɦɟɧɧɵɯ z=f(x,y) ɦɨɠɧɨ ɢɡɨɛɪɚɡɢɬɶ ɝɪɚɮɢɱɟɫɤɢ. Ⱦɥɹ ɷɬɨɝɨ ɜ ɤɚɠɞɨɣ ɬɨɱɤɟ (x,y) D ɜɵɱɢɫɥɹɟɬɫɹ ɡɧɚɱɟɧɢɟ ɮɭɧɤɰɢɢ z=f(x,y). Ɍɨɝɞɚ ɬɪɨɣɤɚ ɱɢɫɟɥ (x,y,z)=(x,y,f(x,y)) ɨɩɪɟɞɟɥɹɟɬ ɜ ɫɢɫɬɟɦɟ ɤɨɨɪɞɢɧɚɬ Oxyz ɧɟɤɨɬɨɪɭɸ ɬɨɱɤɭ P. ɋɨɜɨɤɭɩɧɨɫɬɶ ɬɨɱɟɤ P(x,y,f(x,y)) ɨɛɪɚɡɭɟɬ ɝɪɚɮɢɤ ɮɭɧɤɰɢɢ z=f(x,y), ɩɪɟɞɫɬɚɜɥɹɸɳɢɣ ɫɨɛɨɣ ɧɟɤɨɬɨɪɭɸ ɩɨɜɟɪɯɧɨɫɬɶ ɜ ɩɪɨɫɬɪɚɧɫɬɜɟ R3.

3.2. ɉɪɟɞɟɥ ɢ ɧɟɩɪɟɪɵɜɧɨɫɬɶ ɮɭɧɤɰɢɢ ɧɟɫɤɨɥɶɤɢɯ ɩɟɪɟɦɟɧɧɵɯ

ɑɢɫɥɨ	Ⱥ	ɧɚɡɵɜɚɟɬɫɹ ɩɪɟɞɟɥɨɦ ɮɭɧɤɰɢɢ u=f(P) ɩɪɢ ɫɬɪɟɦɥɟɧɢɢ ɬɨɱɤɢ
P(x1,x2,...,xn)	ɤ	ɬɨɱɤɟ P0(a1,a2,...,an), ɟɫɥɢ ɞɥɹ ɥɸɛɨɝɨ H>0 ɫɭɳɟɫɬɜɭɟɬ ɬɚɤɨɟ
G>0, ɱɬɨ	ɢɡ	ɭɫɥɨɜɢɹ 0 U(P , P )	( x a )2 ... ( x	n	a	n	)2 G ɫɥɟɞɭɟɬ
		1 0	1 1	n		n

| f ( x1 , x 2 ,..., x n ) A | H . ɉɪɢ ɷɬɨɦ ɩɢɲɭɬ:

	A lim	f ( p) lim f (x1 , x2 ,..., xn ) .
	PoP	x oa
	0	x21 oa21
		...
		xn oan
	Ɏɭɧɤɰɢɹ u=f(P) ɧɚɡɵɜɚɟɬɫɹ ɧɟɩɪɟɪɵɜɧɨɣ ɜ ɬɨɱɤɟ P0 , ɟɫɥɢ:
1)	ɮɭɧɤɰɢɹ f(P) ɨɩɪɟɞɟɥɟɧɚ ɜ ɬɨɱɤɟ P0 ;
2)	ɫɭɳɟɫɬɜɭɟɬ	lim f (P) ;
		PoP0
3)	lim f (P)	f (P0 ) .
	PoP0

Ɏɭɧɤɰɢɹ ɧɚɡɵɜɚɟɬɫɹ ɧɟɩɪɟɪɵɜɧɨɣ ɜ ɨɛɥɚɫɬɢ, ɟɫɥɢ ɨɧɚ ɧɟɩɪɟɪɵɜɧɚ ɜ ɤɚɠɞɨɣ ɬɨɱɤɟ ɷɬɨɣ ɨɛɥɚɫɬɢ. ȿɫɥɢ f(P) ɨɩɪɟɞɟɥɟɧɚ ɜ ɧɟɤɨɬɨɪɨɣ ɨɤɪɟɫɬɧɨɫɬɢ ɬɨɱɤɢ P0 ɢ ɯɨɬɹ ɛɵ ɨɞɧɨ ɢɡ ɭɫɥɨɜɢɣ 1)–3) ɧɚɪɭɲɟɧɨ, ɬɨ ɬɨɱɤɚ P0 ɧɚɡɵɜɚɟɬɫɹ ɬɨɱɤɨɣ ɪɚɡɪɵɜɚ ɮɭɧɤɰɢɢ f(P). Ɍɨɱɤɢ ɪɚɡɪɵɜɚ ɦɨɝɭɬ ɛɵɬɶ ɢɡɨɥɢɪɨɜɚɧɧɵɦɢ, ɨɛɪɚɡɨɜɵɜɚɬɶ ɥɢɧɢɢ ɪɚɡɪɵɜɚ, ɩɨɜɟɪɯɧɨɫɬɢ ɪɚɡɪɵɜɚ ɢ ɬ. ɞ.

3.3. Ⱦɢɮɮɟɪɟɧɰɢɪɨɜɚɧɢɟ ɮɭɧɤɰɢɣ ɧɟɫɤɨɥɶɤɢɯ ɩɟɪɟɦɟɧɧɵɯ

3.3.1. ɑɚɫɬɧɨɟ ɢ ɩɨɥɧɨɟ ɩɪɢɪɚɳɟɧɢɹ ɮɭɧɤɰɢɢ

ɉɭɫɬɶ z=f(x,y) – ɮɭɧɤɰɢɹ ɞɜɭɯ ɧɟɡɚɜɢɫɢɦɵɯ ɩɟɪɟɦɟɧɧɵɯ ɢ D(f) – ɨɛɥɚɫɬɶ ɟɟ

ɨɩɪɟɞɟɥɟɧɢɹ.	ȼɵɛɟɪɟɦ ɩɪɨɢɡɜɨɥɶɧɭɸ	ɬɨɱɤɭ P0 x0 , y0 D( f ) ɢ ɞɚɞɢɦ x0
ɩɪɢɪɚɳɟɧɢɟ	'x , ɨɫɬɚɜɥɹɹ ɡɧɚɱɟɧɢɟ y0	ɧɟɢɡɦɟɧɧɵɦ. ɉɪɢ ɷɬɨɦ ɮɭɧɤɰɢɹ f(x,y)
ɩɨɥɭɱɢɬ ɩɪɢɪɚɳɟɧɢɟ:
	'x z 'x f (x0 , y0 )	f ( x0 'x, y0 ) f ( x0 , y0 ).

ɗɬɚ ɜɟɥɢɱɢɧɚ ɧɚɡɵɜɚɟɬɫɹ ɱɚɫɬɧɵɦ ɩɪɢɪɚɳɟɧɢɟɦ ɮɭɧɤɰɢɢ f(x,y) ɩɨ x.

Ⱥɧɚɥɨɝɢɱɧɨ, ɫɱɢɬɚɹ x0 ɩɨɫɬɨɹɧɧɨɣ ɢ ɞɚɜɚɹ y0 ɩɪɢɪɚɳɟɧɢɟ 'y ,

ɩɨɥɭɱɢɦ ɱɚɫɬɧɨɟ ɩɪɢɪɚɳɟɧɢɟ ɮɭɧɤɰɢɢ z=f(x,y) ɩɨ y:

'y z 'y f (x0 , y0 ) f (x0 , y0 'y) f (x0 , y0 ).

ɉɨɥɧɵɦ ɩɪɢɪɚɳɟɧɢɟɦ ɮɭɧɤɰɢɢ z f (x, y) ɜ ɬɨɱɤɟ P0 (x0 , y0 ) ɧɚɡɵɜɚɟɬɫɹ ɩɪɢɪɚɳɟɧɢɟ 'z , ɜɵɡɵɜɚɟɦɨɟ ɨɞɧɨɜɪɟɦɟɧɧɵɦ ɩɪɢɪɚɳɟɧɢɟɦ ɨɛɟɢɯ ɧɟɡɚɜɢɫɢɦɵɯ ɩɟɪɟɦɟɧɧɵɯ x ɢ y:

'z 'f ( x0 , y0 ) f (x0 'x, y0 'y) f (x0 , y0 ) .

Ƚɟɨɦɟɬɪɢɱɟɫɤɢ ɱɚɫɬɧɵɟ ɩɪɢɪɚɳɟɧɢɹ ɢ ɩɨɥɧɨɟ ɩɪɢɪɚɳɟɧɢɟ ɮɭɧɤɰɢɢ z('x z, 'y z, 'z) ɦɨɠɧɨ ɢɡɨɛɪɚɡɢɬɶ ɫɨɨɬɜɟɬɫɬɜɟɧɧɨ ɨɬɪɟɡɤɚɦɢ A1B1,A2 B2 ɢ A3 B3

(ɪɢɫ. 3.1).

'x z

'y z

	y
Ɋ1(x0 'x, y0 )	Ɋ3 (x0 'x, y0 'y)
Ɋ1(x0 'x, y0 )

	x		Ɋ0 (x0 , y0 )	Ɋ2 (x0 , y0 'y)
			Ɋɢɫ. 3.1
	ɉɪɢɦɟɪ 3.1. ɇɚɣɬɢ ɱɚɫɬɧɵɟ ɢ ɩɨɥɧɨɟ ɩɪɢɪɚɳɟɧɢɹ ɮɭɧɤɰɢɢ z xy2 ɜ ɬɨɱɤɟ
P0 (1;2) , ɟɫɥɢ 'x		0,1; 'y	0,2 .
	Ɋɟɲɟɧɢɟ. ȼɵɱɢɫɥɢɦ ɡɧɚɱɟɧɢɹ
'x z	f (1,1;2,0) f ( 1;2)		(x0 'x) y02 x0 y02	'xy02 0,1 4 0,4;
'y z	f (1,0;2,2) f (1;2)		x0 ( y0 'y)2 x0 y02	2x0 y0'y 'y2
2 1 2 0,22 0,84;
'z	f (1,1;2,2) f (1;2) (x0 'x)( y0 'y)2 x0 y02
1,1 2,22 1 22		1,324.
				39

ȿɫɥɢ	u f (x, y, z) , ɬɨ ɞɥɹ ɧɟɟ ɪɚɫɫɦɚɬɪɢɜɚɸɬɫɹ ɱɚɫɬɧɵɟ ɩɪɢɪɚɳɟɧɢɹ
'xu, 'yu,	'zu ɢ ɩɨɥɧɨɟ ɩɪɢɪɚɳɟɧɢɟ 'u .
	3.3.2. ɑɚɫɬɧɵɟ ɩɪɨɢɡɜɨɞɧɵɟ

Ɉɩɪɟɞɟɥɟɧɢɟ. ɑɚɫɬɧɨɣ ɩɪɨɢɡɜɨɞɧɨɣ ɮɭɧɤɰɢɢ z=f(x,y) ɩɨ ɩɟɪɟɦɟɧɧɨɣ x ɧɚɡɵɜɚɟɬɫɹ ɩɪɟɞɟɥ ɨɬɧɨɲɟɧɢɹ ɱɚɫɬɧɨɝɨ ɩɪɢɪɚɳɟɧɢɹ ɮɭɧɤɰɢɢ 'x z ɤ ɩɪɢɪɚɳɟɧɢɸ

ɚɪɝɭɦɟɧɬɚ 'x , ɤɨɝɞɚ ɩɨɫɥɟɞɧɟɟ ɫɬɪɟɦɢɬɫɹ ɤ ɧɭɥɸ:
	lim				f (x0 'x, y0 ) f (x0 , y0 )					.
	lim									.
	'xo0							'x
ɑɚɫɬɧɭɸ ɩɪɨɢɡɜɨɞɧɭɸ ɮɭɧɤɰɢɢ							z	f (x, y) ɩɨ ɩɟɪɟɦɟɧɧɨɣ x ɨɛɨɡɧɚɱɚɸɬ
ɫɢɦɜɨɥɚɦɢ
		wz		; zcx ;			wf (x, y)		; f xc(x, y).
		wx					wx
Ɍɚɤɢɦ ɨɛɪɚɡɨɦ,
wz	lim	'	x	z		lim		f (x0	'x, y0 ) f (x0, y0 )		.
wx	lim	'x				lim			'x		.
wx	'xo0	'x				'xo0			'x

Ɉɩɪɟɞɟɥɟɧɢɟ. ɑɚɫɬɧɨɣ ɩɪɨɢɡɜɨɞɧɨɣ ɮɭɧɤɰɢɢ z=f(x,y) ɩɨ ɩɟɪɟɦɟɧɧɨɣ y ɧɚɡɵɜɚɟɬɫɹ ɩɪɟɞɟɥ ɨɬɧɨɲɟɧɢɹ ɱɚɫɬɧɨɝɨ ɩɪɢɪɚɳɟɧɢɹ ɮɭɧɤɰɢɢ 'y z ɤ ɩɪɢɪɚɳɟɧɢɸ ɚɪɝɭɦɟɧɬɚ 'y , ɤɨɝɞɚ ɩɨɫɥɟɞɧɟɟ ɫɬɪɟɦɢɬɫɹ ɤ ɧɭɥɸ:

wz	lim	'y z	lim	f ( x0 , y0	'y) f (x0, y0 )	.
wy		'y			'y
	'yo0		'yo0

ɉɪɢɦɟɧɹɸɬɫɹ ɬɚɤɠɟ ɨɛɨɡɧɚɱɟɧɢɹ zcy , wf (x, y) , f yc(x, y) . wy

ɑɚɫɬɧɵɟ ɩɪɢɪɚɳɟɧɢɹ ɢ ɱɚɫɬɧɵɟ ɩɪɨɢɡɜɨɞɧɵɟ ɮɭɧɤɰɢɢ n ɩɟɪɟɦɟɧɧɵɯ ɩɪɢ n>2 ɨɩɪɟɞɟɥɹɸɬɫɹ ɢ ɨɛɨɡɧɚɱɚɸɬɫɹ ɚɧɚɥɨɝɢɱɧɨ. Ɍɚɤ, ɧɚɩɪɢɦɟɪ, ɩɭɫɬɶ ɬɨɱɤɚ(x1, x2 ,..., xk ,..., xn ) – ɩɪɨɢɡɜɨɥɶɧɚɹ ɮɢɤɫɢɪɨɜɚɧɧɚɹ ɬɨɱɤɚ ɢɡ ɨɛɥɚɫɬɢ

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