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Белорусский национальный технический университет

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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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ɨɩɪɟɞɟɥɟɧɢɹ ɮɭɧɤɰɢɢ u f (x1, x2 ,..., xn ) . ɉɪɢɞɚɜɚɹ ɡɧɚɱɟɧɢɸ ɩɟɪɟɦɟɧɧɨɣ xk (k 1,2,..., n) ɩɪɢɪɚɳɟɧɢɟ 'xk , ɪɚɫɫɦɨɬɪɢɦ ɩɪɟɞɟɥ

lim	f (x1 ,..., xk 'xk ,..., xn ) f (x1 ,..., xk ,..., xn )	.

'xk o0	'xk

ɗɬɨɬ ɩɪɟɞɟɥ ɧɚɡɵɜɚɟɬɫɹ ɱɚɫɬɧɨɣ ɩɪɨɢɡɜɨɞɧɨɣ (1–ɝɨ ɩɨɪɹɞɤɚ) ɞɚɧɧɨɣ ɮɭɧɤɰɢɢ ɩɨ ɩɟɪɟɦɟɧɧɨɣ xk ɜ ɬɨɱɤɟ (x1, x2 ,..., xn ) ɢ ɨɛɨɡɧɚɱɚɟɬɫɹ

					wu	ɢɥɢ	f xk c(x1 , x2 ,..., xn ) .
					wxk	ɢɥɢ	f xk c(x1 , x2 ,..., xn ) .
					wxk
ɉɪɢɦɟɪ 3.2. ɇɚɣɬɢ wu , wu , wu						, ɝɞɟ u x2 yz3 x y2 .
			wx wy wz
Ɋɟɲɟɧɢɟ. Ⱦɥɹ			ɧɚɯɨɠɞɟɧɢɹ			wu	ɫɱɢɬɚɟɦ y,	z ɤɨɧɫɬɚɧɬɚɦɢ, ɚ ɮɭɧɤɰɢɸ
						wx
u x2 yz3 x y2		– ɮɭɧɤɰɢɟɣ ɨɞɧɨɣ ɩɟɪɟɦɟɧɧɨɣ x. Ɍɨɝɞɚ
wu	(x2 yz3	x y 2 )cx		(x2 yz3 )cx (x)cx ( y 2 )cx
wx
2xyz3 1		0	2xyz3 1.
Ⱥɧɚɥɨɝɢɱɧɨ ɦɨɠɧɨ ɩɨɥɭɱɢɬɶ:				wu		x2 z3 2 y, wu		3z2 x2 y .
				wy			wz
ɑɚɫɬɧɵɦɢ		ɩɪɨɢɡɜɨɞɧɵɦɢ				2-ɝɨ	ɩɨɪɹɞɤɚ	ɮɭɧɤɰɢɢ u f (x1, x2 ,..., xn )

ɧɚɡɵɜɚɸɬɫɹ ɱɚɫɬɧɵɟ ɩɪɨɢɡɜɨɞɧɵɟ ɨɬ ɟɟ ɱɚɫɬɧɵɯ ɩɪɨɢɡɜɨɞɧɵɯ ɩɟɪɜɨɝɨ ɩɨɪɹɞɤɚ. ɉɪɨɢɡɜɨɞɧɵɟ ɜɬɨɪɨɝɨ ɩɨɪɹɞɤɚ ɨɛɨɡɧɚɱɚɸɬɫɹ ɫɥɟɞɭɸɳɢɦ ɨɛɪɚɡɨɦ:

(x1, x2 ,..., xn );

fxk xk

wxk ©wxk ¹

wxk

(x1, x2 ,..., xn ),

wx wx

¨wx

fxi xk

i ©

ɢ ɬ. ɞ.

Ⱥɧɚɥɨɝɢɱɧɨ ɨɩɪɟɞɟɥɹɸɬɫɹ ɢ ɨɛɨɡɧɚɱɚɸɬɫɹ ɱɚɫɬɧɵɟ ɩɪɨɢɡɜɨɞɧɵɟ ɩɨɪɹɞɤɚ ɜɵɲɟ ɜɬɨɪɨɝɨ.

ɉɪɢɦɟɪ 3.3. ɇɚɣɬɢ ɱɚɫɬɧɵɟ ɩɪɨɢɡɜɨɞɧɵɟ ɜɬɨɪɨɝɨ ɩɨɪɹɞɤɚ ɞɥɹ ɮɭɧɤɰɢɢ

Ɋɟɲɟɧɢɟ.

; wz

;

w2 z

·c

w2 z

·c

;

wywx

¹ x

w2 z

·c

w2 z

2x ·c

;

wxwy

2 ¸

¹ y

3.3.3. ɉɨɥɧɵɣ ɞɢɮɮɟɪɟɧɰɢɚɥ ɮɭɧɤɰɢɢ

ɉɨɥɧɵɦ ɩɪɢɪɚɳɟɧɢɟɦ ɮɭɧɤɰɢɢ f (x1, x2,..., xn ) ɜ ɬɨɱɤɟ P(x1, x2 ,..., xn ) ,

ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɦ ɩɪɢɪɚɳɟɧɢɹɦ ɚɪɝɭɦɟɧɬɨɜ 'x1, 'x2 ,..., 'xn , ɧɚɡɵɜɚɟɬɫɹ ɪɚɡɧɨɫɬɶ 'u f (x1 'x1, x2 'x2 ,..., xn 'xn ) f (x1, x2 ,..., xn ) . Ɏɭɧɤɰɢɹ u=f(P) ɧɚɡɵɜɚɟɬɫɹ ɞɢɮɮɟɪɟɧɰɢɪɭɟɦɨɣ ɜ ɬɨɱɤɟ (x1, x2 ,..., xn ) , ɟɫɥɢ ɜ ɧɟɤɨɬɨɪɨɣ ɨɤɪɟɫɬɧɨɫɬɢ ɷɬɨɣ

ɬɨɱɤɢ		ɩɨɥɧɨɟ	ɩɪɢɪɚɳɟɧɢɟ		ɮɭɧɤɰɢɢ		ɦɨɠɟɬ	ɛɵɬɶ ɩɪɟɞɫɬɚɜɥɟɧɨ ɜ	ɜɢɞɟ
'u	A1 'x1 A2 'x2 ... An 'xn o(U) ,
ɝɞɟ	U	'x2	'x2	... 'x2	;	A , A ,.., A		– ɱɢɫɥɚ, ɧɟ ɡɚɜɢɫɹɳɢɟ	ɨɬ
		1	2	n		1 2	n

'x1, 'x2 ,..., 'xn .

ɉɨɥɧɵɦ ɞɢɮɮɟɪɟɧɰɢɚɥɨɦ du 1-ɝɨ ɩɨɪɹɞɤɚ ɮɭɧɤɰɢɢ u f (x1, x2 ,..., xn ) ɜ

ɬɨɱɤɟ (x1, x2 ,..., xn ) ɧɚɡɵɜɚɟɬɫɹ ɝɥɚɜɧɚɹ ɱɚɫɬɶ ɩɨɥɧɨɝɨ ɩɪɢɪɚɳɟɧɢɹ ɷɬɨɣ ɮɭɧɤɰɢɢ ɜ ɪɚɫɫɦɚɬɪɢɜɚɟɦɨɣ ɬɨɱɤɟ, ɥɢɧɟɣɧɚɹ ɨɬɧɨɫɢɬɟɥɶɧɨ 'x1, 'x2 ,..., 'xn , ɬɨ ɟɫɬɶ

du A1'x1 A2'x2 ... An'xn .

Ⱦɢɮɮɟɪɟɧɰɢɚɥɵ ɧɟɡɚɜɢɫɢɦɵɯ ɩɟɪɟɦɟɧɧɵɯ ɩɨ ɨɩɪɟɞɟɥɟɧɢɸ ɩɪɢɧɢɦɚɸɬɫɹ ɪɚɜɧɵɦɢ ɢɯ ɩɪɢɪɚɳɟɧɢɹɦ:

dx1 'x1, dx2 'x2 ,..., dxn 'xn .

Ⱦɥɹ ɩɨɥɧɨɝɨ ɞɢɮɮɟɪɟɧɰɢɚɥɚ ɮɭɧɤɰɢɢ u f (x1, x2 ,..., xn ) ɫɩɪɚɜɟɞɥɢɜɚ

ɮɨɪɦɭɥɚ

...

ɉɪɢɦɟɪ 3.4. ɇɚɣɬɢ ɩɨɥɧɵɣ ɞɢɮɮɟɪɟɧɰɢɚɥ ɮɭɧɤɰɢɢ z

ln(y x2 y2 ) .

Ɋɟɲɟɧɢɟ.

x2 y 2 y x2 y 2

y x2 y 2

2 x2 y 2

¨1

y x

xdx

x2 y 2 y x2 y 2

x2 y 2

ɉɨɥɧɵɣ ɞɢɮɮɟɪɟɧɰɢɚɥ ɢɫɩɨɥɶɡɭɟɬɫɹ ɞɥɹ ɩɪɢɛɥɢɠɟɧɧɵɯ ɜɵɱɢɫɥɟɧɢɣ

ɡɧɚɱɟɧɢɣ ɮɭɧɤɰɢɢ. Ɍɚɤ, ɧɚɩɪɢɦɟɪ,

ɞɥɹ ɮɭɧɤɰɢɢ ɞɜɭɯ ɩɟɪɟɦɟɧɧɵɯ z f (x, y) ,

ɡɚɦɟɧɹɹ 'z | dz , ɩɨɥɭɱɢɦ

f ( x0 'x, y0 'y) | f ( x0 , y0 ) df ( x0 , y0 ) .

ɉɪɢɦɟɪ 3.5. ȼɵɱɢɫɥɢɬɶ ɩɪɢɛɥɢɠɟɧɧɨ ɫ ɩɨɦɨɳɶɸ ɩɨɥɧɨɝɨ ɞɢɮɮɟɪɟɧɰɢɚɥɚ

arctg§¨1,97 1·¸. ©1,02 ¹

	§	x	·
Ɋɟɲɟɧɢɟ. Ɋɚɫɫɦɨɬɪɢɦ ɮɭɧɤɰɢɸ	f (x, y) arctg¨	x	1¸	. ɉɪɢɦɟɧɹɹ
Ɋɟɲɟɧɢɟ. Ɋɚɫɫɦɨɬɪɢɦ ɮɭɧɤɰɢɸ	f (x, y) arctg¨		1¸	. ɉɪɢɦɟɧɹɹ
	¨		¸
	© y		¹

ɜɵɲɟɭɤɚɡɚɧɧɭɸ ɮɨɪɦɭɥɭ ɤ ɷɬɨɣ ɮɭɧɤɰɢɢ, ɩɨɥɭɱɢɦ

§ x 'x

§ x

·c

arctg¨

1¸

| arctg¨

1¸

arctg¨

1¸

'x ¨arctg

1¸

¹ x

¹ y

ɢɥɢ, ɩɨɫɥɟ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɯ ɩɪɟɨɛɪɚɡɨɜɚɧɢɣ,

§ x 'x

§ x

'y .

arctg¨

1¸ | arctg¨

1¸

y2 (x y)2

y 2 (x y)2

ɉɨɥɨɠɢɦ ɬɟɩɟɪɶ x=2, y=1, 'x=–0,03, 'y=0,02. Ɍɨɝɞɚ

§2 0,03

1( 0,03)

arctg¨

1¸ | arctg¨

1¸

0,02

12 (2 1)2

(2 1)2

arctg1

0,03 0,02

0,015 0,02 | 0,75.

3.3.4. Ⱦɢɮɮɟɪɟɧɰɢɪɨɜɚɧɢɟ ɫɥɨɠɧɵɯ ɢ ɧɟɹɜɧɵɯ ɮɭɧɤɰɢɣ

Ɏɭɧɤɰɢɹ z=f(u,v), ɝɞɟ u=M(x), v=\(x), ɧɚɡɵɜɚɟɬɫɹ ɫɥɨɠɧɨɣ ɮɭɧɤɰɢɟɣ ɩɟɪɟɦɟɧɧɵɯ x ɢ y. Ⱦɥɹ ɧɚɯɨɠɞɟɧɢɹ ɱɚɫɬɧɵɯ ɩɪɨɢɡɜɨɞɧɵɯ ɫɥɨɠɧɵɯ ɮɭɧɤɰɢɣ ɢɫɩɨɥɶɡɭɸɬɫɹ ɫɥɟɞɭɸɳɢɟ ɮɨɪɦɭɥɵ:

wz	wz	wu	wz	wv	;
wx wu			wv
		wx		wx
wz	wz	wu	wz	wv .
wy wu			wv
		wy		wy

ȼ ɫɥɭɱɚɟ, ɤɨɝɞɚ u=M(x),

v=\(x), ɛɭɞɟɬ:

f (M(x),\(x))

– ɮɭɧɤɰɢɹ ɨɞɧɨɣ

ɩɟɪɟɦɟɧɧɨɣ ɢ, ɫɨɨɬɜɟɬɫɬɜɟɧɧɨ,

wz du

wu dx

ɉɪɢɦɟɪ 3.6. ɇɚɣɬɢ ɱɚɫɬɧɵɟ ɩɪɨɢɡɜɨɞɧɵɟ ɮɭɧɤɰɢɢ z

arctg

, ɝɞɟ u=x+y,

v=x–y.

Ɋɟɲɟɧɢɟ. ɉɨ ɮɨɪɦɭɥɟ wz

ɢɦɟɟɦ:

u v

;

u2 v2

Ⱥɧɚɥɨɝɢɱɧɨ

u v

( 1)

u2 v2

ȿɫɥɢ ɭɪɚɜɧɟɧɢɟ F(x,y)=0 ɡɚɞɚɟɬ ɧɟɤɨɬɨɪɭɸ ɮɭɧɤɰɢɸ y(x) (ɜ ɧɟɹɜɧɨɦ) ɜɢɞɟ ɢ

Fyc(x, y) z 0 , ɬɨ

dy	Fxc(x, y)	.
dx
	Fyc(x, y)

ȿɫɥɢ ɭɪɚɜɧɟɧɢɟ F (x, y, z) ɡɚɞɚɟɬ ɮɭɧɤɰɢɸ ɞɜɭɯ ɩɟɪɟɦɟɧɧɵɯ z(x, y) (ɜ

ɧɟɹɜɧɨɦ ɜɢɞɟ) ɢ Fzc(x, y, z) z 0 , ɬɨ ɫɩɪɚɜɟɞɥɢɜɵ ɮɨɪɦɭɥɵ:

wz	Fc(x, y, z)		wz	Fyc(x, y, z)
	x	;			.
wx	Fzc(x, y, z)	;	wy	Fzc(x, y, z)	.
wx	Fzc(x, y, z)		wy	Fzc(x, y, z)

ɉɪɢɦɟɪ 3.7. ɇɚɣɬɢ ɱɚɫɬɧɵɟ ɩɪɨɢɡɜɨɞɧɵɟ ɮɭɧɤɰɢɢ z, ɡɚɞɚɧɧɨɣ ɧɟɹɜɧɨ ɭɪɚɜɧɟɧɢɟɦ xyz x3 y3 z3 5 0 .

Ɋɟɲɟɧɢɟ.
wz	yz 3x2	3x2 yz		;
wx	xy 3z2	3z2	xy

wz	xz 3y2	xz 3y2		.
wy	xy 3z2	3z2	xy

3.4. Ʉɚɫɚɬɟɥɶɧɚɹ ɩɥɨɫɤɨɫɬɶ ɢ ɧɨɪɦɚɥɶ ɤ ɩɨɜɟɪɯɧɨɫɬɢ

ȿɫɥɢ ɩɨɜɟɪɯɧɨɫɬɶ ɡɚɞɚɧɚ ɭɪɚɜɧɟɧɢɟɦ z=f(x,y), ɬɨ ɭɪɚɜɧɟɧɢɟ ɤɚɫɚɬɟɥɶɧɨɣ ɩɥɨɫɤɨɫɬɢ ɜ ɬɨɱɤɟ M 0 (x0 , y0 , z0 ) ɤ ɞɚɧɧɨɣ ɩɨɜɟɪɯɧɨɫɬɢ ɢɦɟɟɬ ɜɢɞ:

z z0 f xc(x0 , y0 )(x x0 ) f yc(x0 , y0 )( y y0 ) ,

ɚ ɭɪɚɜɧɟɧɢɟ ɧɨɪɦɚɥɢ, ɩɪɨɜɟɞɟɧɧɨɣ ɱɟɪɟɡ ɬɨɱɤɭ ɬɚɤɨɜɨ:

x x0		y y0
f xc(x0 , y0 )		f yc(x0 , y0 )

M 0 ( x0 , y0 , z0 ) ɩɨɜɟɪɯɧɨɫɬɢ,

z 1z0 .

ȼ ɫɥɭɱɚɟ, ɤɨɝɞɚ ɧɟɹɜɧɨɟ ɭɪɚɜɧɟɧɢɟ ɩɨɜɟɪɯɧɨɫɬɢ ɬɚɤɨɜɨ: F(x,y,z)=0, ɬɨ

ɭɪɚɜɧɟɧɢɟ ɤɚɫɚɬɟɥɶɧɨɣ ɩɥɨɫɤɨɫɬɢ ɜ ɬɨɱɤɟ M 0 ( x0 , y0 , z0 )

ɢɦɟɟɬ ɜɢɞ

Fxc(x0 , y0 , z0 )( x x0 ) Fyc(x0 , y0 , z0 )( y y0 ) Fzc(x0 , y0 , z0 )(z z0 ) 0 ,

ɚ ɭɪɚɜɧɟɧɢɟ ɧɨɪɦɚɥɢ

x x0

y y0

z z0

Fxc(x0 , y0 , z0 )

Fyc(x0 , y0 , z0 )

Fzc(x0 , y0 , z0 )

ɉɪɢɦɟɪ 3.8. Ɂɚɩɢɫɚɬɶ ɭɪɚɜɧɟɧɢɹ ɤɚɫɚɬɟɥɶɧɨɣ ɩɥɨɫɤɨɫɬɢ ɢ ɧɨɪɦɚɥɢ ɤ

ɨɞɧɨɩɨɥɨɫɬɧɨɦɭ ɝɢɩɟɪɛɨɥɨɢɞɭ x2 2 y 2 z2 5 0 ɜ ɬɨɱɤɟ P0(2;–1;1).

Ɋɟɲɟɧɢɟ.

Fxc(x0 , y0 , z0 ) 2x

Fyc(x0 , y0 , z0 ) 4 y

Fzc(x0 , y0 , z0 ) 2z

ɉɨɷɬɨɦɭ ɭɪɚɜɧɟɧɢɟ ɤɚɫɚɬɟɥɶɧɨɣ ɩɥɨɫɤɨɫɬɢ ɤ ɞɚɧɧɨɣ ɩɨɜɟɪɯɧɨɫɬɢ ɡɚɩɢɲɟɬɫɹ

ɜ ɜɢɞɟ 4(x 2) 4( y 1) 2(z 1)

0 ɢɥɢ 2x 2 y z 5

0, ɚ ɭɪɚɜɧɟɧɢɟ ɧɨɪɦɚɥɢ

– ɜ ɜɢɞɟ

x 2

y 1

z 1

ɢɥɢ

x 2

y 1

z 1

3.5. ɗɤɫɬɪɟɦɭɦ ɮɭɧɤɰɢɢ ɧɟɫɤɨɥɶɤɢɯ ɩɟɪɟɦɟɧɧɵɯ

Ɏɭɧɤɰɢɹ u		f ( p)		ɢɦɟɟɬ ɦɚɤɫɢɦɭɦ (ɦɢɧɢɦɭɦ) ɜ ɬɨɱɤɟ P ( x0			, x0 ,..., x0 ) ,
						0 1	2	n
ɟɫɥɢ ɫɭɳɟɫɬɜɭɟɬ		ɬɚɤɚɹ		ɨɤɪɟɫɬɧɨɫɬɶ	ɬɨɱɤɢ P0, ɞɥɹ ɜɫɟɯ ɬɨɱɟɤ	P( x1 , x2 ,..., xn )
ɤɨɬɨɪɨɣ,	ɨɬɥɢɱɧɵɯ		ɨɬ	ɬɨɱɤɢ P0,	ɜɵɩɨɥɧɹɟɬɫɹ ɧɟɪɚɜɟɧɫɬɜɨ	f (P0 ) ! f (P)
(ɫɨɨɬɜɟɬɫɬɜɟɧɧɨ: f (P0 ) f (P) ).
ɇɟɨɛɯɨɞɢɦɨɟ ɭɫɥɨɜɢɟ ɷɤɫɬɪɟɦɭɦɚ. ȿɫɥɢ ɞɢɮɮɟɪɟɧɰɢɪɭɟɦɚɹ ɮɭɧɤɰɢɹ								f (P)
ɞɨɫɬɢɝɚɟɬ ɷɤɫɬɪɟɦɭɦɚ ɜ ɬɨɱɤɟ P0, ɬɨ ɜ ɷɬɨɣ ɬɨɱɤɟ ɜɫɟ ɱɚɫɬɧɵɟ ɩɪɨɢɡɜɨɞɧɵɟ 1-ɝɨ
ɩɨɪɹɞɤɚ	f xck (P0 )	0,	k	1,2,..., n .
Ɍɨɱɤɢ, ɜ ɤɨɬɨɪɵɯ ɜɫɟ ɱɚɫɬɧɵɟ ɩɪɨɢɡɜɨɞɧɵɟ ɪɚɜɧɵ ɧɭɥɸ, ɧɚɡɵɜɚɸɬɫɹ
ɫɬɚɰɢɨɧɚɪɧɵɦɢ ɬɨɱɤɚɦɢ ɮɭɧɤɰɢɢ u					f (P) .

Ⱦɨɫɬɚɬɨɱɧɵɟ ɭɫɥɨɜɢɹ ɷɤɫɬɪɟɦɭɦɚ. ȼ ɫɥɭɱɚɟ ɮɭɧɤɰɢɢ ɞɜɭɯ ɩɟɪɟɦɟɧɧɵɯ ɞɨɫɬɚɬɨɱɧɵɟ ɭɫɥɨɜɢɹ ɷɤɫɬɪɟɦɭɦɚ ɦɨɠɧɨ ɫɮɨɪɦɭɥɢɪɨɜɚɬɶ ɫɥɟɞɭɸɳɢɦ ɨɛɪɚɡɨɦ.

ɉɭɫɬɶ P0 ( x0 , y0 )	– ɫɬɚɰɢɨɧɚɪɧɚɹ ɬɨɱɤɚ ɮɭɧɤɰɢɢ z		f (x, y) , ɩɪɢɱɟɦ ɷɬɚ ɮɭɧɤɰɢɹ
ɞɜɚɠɞɵ ɞɢɮɮɟɪɟɧɰɢɪɭɟɦɚ ɜ ɧɟɤɨɬɨɪɨɣ ɨɤɪɟɫɬɧɨɫɬɢ ɬɨɱɤɢ P0 ɢ ɜɫɟ ɟɟ ɜɬɨɪɵɟ
ɱɚɫɬɧɵɟ ɩɪɨɢɡɜɨɞɧɵɟ ɧɟɩɪɟɪɵɜɧɵ ɜ ɬɨɱɤɟ P0. Ɉɛɨɡɧɚɱɢɦ
cc	cc	cc		2	.
A fxx (x0 , y0 ), B fxy (x0 , y0 ), C f yy (x0 , y0 ), D AC B
Ɍɨɝɞɚ:
1) ɟɫɥɢ D>0, ɬɨ ɜ ɬɨɱɤɟ P0 ( x0 , y0 )		ɮɭɧɤɰɢɹ z	f (x, y)	ɢɦɟɟɬ ɷɤɫɬɪɟɦɭɦ, ɚ

ɢɦɟɧɧɨ: ɦɚɤɫɢɦɭɦ ɩɪɢ A<0 (C<0) ɢ ɦɢɧɢɦɭɦ ɩɪɢ A>0 (C>0);

2)ɟɫɥɢ D<0, ɬɨ ɷɤɫɬɪɟɦɭɦ ɜ ɬɨɱɤɟ P0 ( x0 , y0 ) ɨɬɫɭɬɫɬɜɭɟɬ;

3)ɟɫɥɢ D=0, ɬɨ ɬɪɟɛɭɟɬɫɹ ɞɨɩɨɥɧɢɬɟɥɶɧɨɟ ɢɫɫɥɟɞɨɜɚɧɢɟ.

ɉɪɢɦɟɪ 3.9. ɂɫɫɥɟɞɨɜɚɬɶ ɧɚ ɷɤɫɬɪɟɦɭɦ ɮɭɧɤɰɢɸ z x3 y3 3xy .

Ɋɟɲɟɧɢɟ. ɇɚɣɞɟɦ ɱɚɫɬɧɵɟ ɩɪɨɢɡɜɨɞɧɵɟ 1-ɝɨ ɩɨɪɹɞɤɚ ɢ ɩɪɢɪɚɜɧɹɟɦ ɢɯ

ɧɭɥɸ.

3(x2 y) 0;

3( y2 x) 0.

ɉɨɥɭɱɚɟɦ ɫɢɫɬɟɦɭ:

x2 y

2 x

¯y

Ɋɟɲɚɹ ɫɢɫɬɟɦɭ, ɧɚɣɞɟɦ ɞɜɟ ɫɬɚɰɢɨɧɚɪɧɵɟ ɬɨɱɤɢ P1 (0,0) ɢ P2 (1,1) . ɇɚɣɞɟɦ

ɱɚɫɬɧɵɟ ɩɪɨɢɡɜɨɞɧɵɟ 2–ɝɨ ɩɨɪɹɞɤɚ:

w2 z

6x;

w2 z

6 y .

wx2

wxwy

wy2

Ɂɚɬɟɦ ɫɨɫɬɚɜɢɦ ɞɢɫɤɪɢɦɢɧɚɧɬ D

AC B2 ɞɥɹ ɤɚɠɞɨɣ ɫɬɚɰɢɨɧɚɪɧɨɣ ɬɨɱɤɢ.

Ⱦɥɹ ɬɨɱɤɢ P :

w2 z

0 ; B

w2 z

3 ;

w2 z

0 ; D

9 0 .

wx2

wxwy

wy 2

ɋɥɟɞɨɜɚɬɟɥɶɧɨ, ɷɤɫɬɪɟɦɭɦɚ ɜ ɬɨɱɤɟ P1 ɧɟɬ.

Ⱦɥɹ

ɬɨɱɤɢ

P :

w2 z

6 ;

w2 z

2 z

6 ;

wx2

wxwy

wy 2

36 9 ! 0;

A ! 0 . ɋɥɟɞɨɜɚɬɟɥɶɧɨ, ɜ ɬɨɱɤɟ P2

ɮɭɧɤɰɢɹ ɢɦɟɟɬ ɦɢɧɢɦɭɦ, ɪɚɜɧɵɣ

zmin

1 1 1 3

3.6. ɇɚɢɛɨɥɶɲɟɟ ɢ ɧɚɢɦɟɧɶɲɟɟ ɡɧɚɱɟɧɢɹ ɮɭɧɤɰɢɢ ɧɟɫɤɨɥɶɤɢɯ ɩɟɪɟɦɟɧɧɵɯ ɜ ɡɚɦɤɧɭɬɨɣ ɨɛɥɚɫɬɢ

Ɏɭɧɤɰɢɹ z f (x, y) , ɨɩɪɟɞɟɥɟɧɧɚɹ ɢ ɧɟɩɪɟɪɵɜɧɚɹ ɜ ɡɚɦɤɧɭɬɨɣ ɨɛɥɚɫɬɢ D ɫ

ɝɪɚɧɢɰɟɣ Ƚ ɢ ɞɢɮɮɟɪɟɧɰɢɪɭɟɦɚɹ ɜ ɨɬɤɪɵɬɨɣ ɨɛɥɚɫɬɢ D, ɞɨɫɬɢɝɚɟɬ ɫɜɨɟɝɨ ɧɚɢɛɨɥɶɲɟɝɨ ɢ ɧɚɢɦɟɧɶɲɟɝɨ ɡɧɚɱɟɧɢɢ (ɝɥɨɛɚɥɶɧɵɯ ɷɤɫɬɪɟɦɭɦɨɜ).

Ɍɨɱɤɢ ɝɥɨɛɚɥɶɧɨɝɨ ɷɤɫɬɪɟɦɭɦɚ ɫɥɟɞɭɟɬ ɢɫɤɚɬɶ ɫɪɟɞɢ ɫɬɚɰɢɨɧɚɪɧɵɯ ɬɨɱɟɤ ɮɭɧɤɰɢɢ f ɜ ɨɬɤɪɵɬɨɣ ɨɛɥɚɫɬɢ D ɢ ɫɪɟɞɢ ɬɨɱɟɤ ɝɪɚɧɢɰɵ Ƚ.

ɉɪɢɦɟɪ	3.10. ɇɚɣɬɢ ɧɚɢɛɨɥɶɲɟɟ	ɢ ɧɚɢɦɟɧɶɲɟɟ ɡɧɚɱɟɧɢɹ ɮɭɧɤɰɢɢ
z e x3 3x2 6 y2	ɜ ɨɛɥɚɫɬɢ x2 y 2 d1.
Ɋɟɲɟɧɢɟ. Ƚɪɚɧɢɰɚ ɨɛɥɚɫɬɢ D x2 y 2		1 – ɨɤɪɭɠɧɨɫɬɶ ɪɚɞɢɭɫɚ 1. ɋɞɟɥɚɟɦ
ɱɟɪɬɟɠ (ɪɢɫ. 3.2).

Ɉɤɪɭɠɧɨɫɬɶ ɪɚɡɛɢɜɚɟɬ ɩɥɨɫɤɨɫɬɶ ɧɚ ɞɜɟ ɱɚɫɬɢ. Ʉɨɨɪɞɢɧɚɬɵ ɬɨɱɟɤ ɤɪɭɝɚ ɭɞɨɜɥɟɬɜɨɪɹɸɬ ɧɟɪɚɜɟɧɫɬɜɭ x2 y 2 d1. ɇɚɣɞɟɦ ɫɬɚɰɢɨɧɚɪɧɵɟ ɬɨɱɤɢ ɮɭɧɤɰɢɢ z ɜ

ɤɪɭɝɟ.

(3x

6x)e

x3 3x2 6 y 2

6x 0;

°zx

12 yex

6 y

° zc

¯y 0.

Ɇ2(-2;0) -1

-1

Ɋɢɫ. 3.2

Ɋɟɲɚɹ ɷɬɭ ɫɢɫɬɟɦɭ, ɧɚɯɨɞɢɦ ɞɥɹ ɮɭɧɤɰɢɢ z ɞɜɟ ɫɬɚɰɢɨɧɚɪɧɵɟ ɬɨɱɤɢ M1(0;0)

ɢ M 2 ( 2;0) . Ʉɪɭɝɭ ɩɪɢɧɚɞɥɟɠɢɬ ɬɨɱɤɚ M1(0;0) ; z(M1)

ɇɚɣɞɟɦ

ɧɚɢɛɨɥɶɲɟɟ ɢ

ɧɚɢɦɟɧɶɲɟɟ ɡɧɚɱɟɧɢɟ ɮɭɧɤɰɢɢ z ɧɚ ɨɤɪɭɠɧɨɫɬɢ

x2 y 2

ɇɚ

ɧɟɣ

1 x2 ; x [ 1;1];

z(x) ex3 3x 2 6 .

ɂɦɟɟɦ

z( 1) e

; z(1)

Ⱦɚɥɟɟ,

ɪɟɲɚɹ ɭɪɚɜɧɟɧɢɟ

(3x

6x)e

3x2

0 ,

z (x)

ɧɚɯɨɞɢɦ ɫɬɚɰɢɨɧɚɪɧɭɸ ɬɨɱɤɭ: x

0 ( 1;1); z(x )

z(0)

e6 .

ɂɬɚɤ,		ɩɨɥɭɱɢɦ		ɫɥɟɞɭɸɳɢɟ		ɡɧɚɱɟɧɢɹ	ɮɭɧɤɰɢɢ	z:
z(M1 )	1; z( 1;0)		e2 ; z(1;0) e4 ;		z(0;1) e6 .	Ɉɬɫɸɞɚ	ɜɢɞɧɨ,	ɱɬɨ
zɧɚɢɛ.	z(0;1)	e6 ,	zɧɚɢɦ.	z(0;0)	1.

ȿɫɥɢ ɝɪɚɧɢɰɚ Ƚ ɫɨɫɬɨɢɬ ɢɡ ɧɟɫɤɨɥɶɤɢɯ ɱɚɫɬɟɣ, ɬɨ ɧɚɢɦɟɧɶɲɟɟ ɢ ɧɚɢɛɨɥɶɲɟɟ ɡɧɚɱɟɧɢɟ ɮɭɧɤɰɢɢ z ɧɚ ɝɪɚɧɢɰɟ Ƚ ɫɥɟɞɭɟɬ ɢɫɤɚɬɶ ɫɪɟɞɢ ɧɚɢɛɨɥɶɲɢɯ ɢ ɧɚɢɦɟɧɶɲɢɯ ɡɧɚɱɟɧɢɣ ɮɭɧɤɰɢɢ ɧɚ ɤɚɠɞɨɣ ɢɡ ɱɚɫɬɟɣ ɝɪɚɧɢɰɵ.

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