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Белорусский национальный технический университет

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ɨɩɪɟɞɟɥɟɧɢɹ

ɮɭɧɤɰɢɢ

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

ɢɥɢ

f xk c(x1 , x2

,..., xn ) .

wxk

ɉɪɢɦɟɪ 3.2. ɇɚɣɬɢ

, wu

, ɝɞɟ u

x2 yz3 x y2 .

Ɋɟɲɟɧɢɟ.

Ⱦɥɹ ɧɚɯɨɠɞɟɧɢɹ

ɚ ɮɭɧɤɰɢɸ

ɫɱɢɬɚɟɦ y,

z Ȼɤɨɧɫɬɚɧɬɚɦɢ,

u x2 yz3 x y2 – ɮɭɧɤɰɢɟɣ ɨɞɧɨɣ ɩɟɪɟɦɟɧɧɨɣ x. Ɍɨɝɞɚ

2 c

(x yz x y )

(x yz )

(xɢ) ( y )

2xyz3 1

2xyz3 1.

x2 z3

3z2 x2 y .

Ⱥɧɚɥɨɝɢɱɧɨ ɦɨɠɧɨ ɩɨɥɭɱɢɬɶ: wu

2 y,

ɑɚɫɬɧɵɦɢ

ɩɪɨɢɡɜɨɞɧɵɦɢ

2-ɝɨ

ɩɨɪɹɞɤɚ

ɮɭɧɤɰɢɢ u f (x1, x2 ,..., xn )

ɨɬ ɟɟ ɱɚɫɬɧɵɯ ɩɪɨɢɡɜɨɞɧɵɯ ɩɟɪɜɨɝɨ ɩɨɪɹɞɤɚ.

ɧɚɡɵɜɚɸɬɫɹ ɱɚɫɬɧɵɟ ɩɪɨɢɡɜɨɞɧɵɟɢ

ɉɪɨɢɡɜɨɞɧɵɟ ɜɬɨɪɨɝɨ ɩɨɪɹɞɤɚ ɨɛɨɡɧɚɱɚɸɬɫɹ ɫɥɟɞɭɸɳɢɦ ɨɛɪɚɡɨɦ:

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

ɨ fx

wxk

©wxk

wxk

wx wx

ɢ ɬ. ɞ.

Ⱥɧɚɥɨɝɢɱɧɨ ɨɩɪɟɞɟɥɹɸɬɫɹ ɢ ɨɛɨɡɧɚɱɚɸɬɫɹ ɱɚɫɬɧɵɟ ɩɪɨɢɡɜɨɞɧɵɟ ɩɨɪɹɞɤɚ ɜɵɲɟ ɜɬɨɪɨɝɨ.

ɉɪɢɦɟɪ 3.3. ɇɚɣɬɢ ɱɚɫɬɧɵɟ ɩɪɨɢɡɜɨɞɧɵɟ ɜɬɨɪɨɝɨ ɩɨɪɹɞɤɚ ɞɥɹ ɮɭɧɤɰɢɢ

z	x	.

	y2

Ɋɟɲɟɧɢɟ.

; wz

;

w2 z

·c

w2 z

·c

;

wywx

¹ x

w2 z

1 ·c

w2 z

2x ·c

;

wxwy

2 ¸

¹ y

3.3.3. ɉɨɥɧɵɣ ɞɢɮɮɟɪɟɧɰɢɚɥ ɮɭɧɤɰɢɢ

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

ɉɨɥɧɵɦ

ɩɪɢɪɚɳɟɧɢɟɦ

ɮɭɧɤɰɢɢ

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

ɬɨɱɤɟ

ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɦ ɩɪɢɪɚɳɟɧɢɹɦ ɚɪɝɭɦɟɧɬɨɜ

'x1, 'x2 ,..., 'xn , ɧɚɡɵɜɚɟɬɫɹ ɪɚɡɧɨɫɬɶ

f (x1 'x1, x2 'x2 ,..., xn

'xn ) f (x1, x2 ,..., xn ) . Ɏɭɧɤɰɢɹ u=f(P) ɧɚɡɵɜɚɟɬɫɹ

ɞɢɮɮɟɪɟɧɰɢɪɭɟɦɨɣ

) ,

ɟɫɥɢ

ɧɟɤɨɬɨɪɨɣ ɨɤɪɟɫɬɧɨɫɬɢ

ɷɬɨɣ

ɜ ɬɨɱɤɟ

(xɨ, x ,..., x

ɬɨɱɤɢ

ɩɨɥɧɨɟ

ɩɪɢɪɚɳɟɧɢɟ

ɮɭɧɤɰɢɢ

ɦɨɠɟɬ

ɛɵɬɶ

ɩɪɟɞɫɬɚɜɥɟɧɨ

ɜɢɞɟ

A 'x

o(U) ,

A 'x

...

ɢ2

ɝɞɟ

;

A1, A2 ,.., An

–

ɱɢɫɥɚ,

ɧɟ

ɡɚɜɢɫɹɳɢɟ

ɨɬ

'x1

'x2

... 'xn

'x , 'x

,..., 'x .

ɨn

f (x , x

,..., x

) ɜ

ɉɨɥɧɵɦ ɞɢɮɮɟɪɟɧɰɢɚɥɨɦ du 1-ɝɨ ɩɨɪɹɞɤɚ ɮɭɧɤɰɢɢ u

ɬɨɱɤɟ (x1, x2 ,..., xn ) ɧɚɡɵɜɚɟɬɫɹ ɝɥɚɜɧɚɹ ɱɚɫɬɶ ɩɨɥɧɨɝɨ ɩɪɢɪɚɳɟɧɢɹ ɷɬɨɣ ɮɭɧɤɰɢɢ ɜ

ɪɚɫɫɦɚɬɪɢɜɚɟɦɨɣ ɬɨɱɤɟ, ɥɢɧɟɣɧɚɹ ɨɬɧɨɫɢɬɟɥɶɧɨ 'x1, 'x2 ,..., 'xn , ɬɨ ɟɫɬɶ

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

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

ɫɩɪɚɜɟɞɥɢɜɚ

ɮɨɪɦɭɥɚ

...

ɉɪɢɦɟɪ 3.4. ɇɚɣɬɢ ɩɨɥɧɵɣ ɞɢɮɮɟɪɟɧɰɢɚɥ ɮɭɧɤɰɢɢ

ln(y

x2 y2 ) .

Ɋɟɲɟɧɢɟ.

y x2 y 2

2 x2

y 2

x2 y 2 y x2 y 2

1ɣ,

¨1

y x

xdx

ɪx y

ɉɨɥɧɵɣ

ɞɢɮɮɟɪɟɧɰɢɚɥ ɢɫɩɨɥɶɡɭɟɬɫɹ ɞɥɹ ɩɪɢɛɥɢɠɟɧɧɵɯ ɜɵɱɢɫɥɟɧɢɣ

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

ɡɧɚɱɟɧɢɣ ɮɭɧɤɰɢɢ. Ɍɚɤ,

ɧɚɩɪɢɦɟɪ,

ɡɚɦɟɧɹɹ 'z | dz , ɩɨɥɭɱɢɦ

, y0 ) df ( x0 , y0 ) .

f ( x

0 'x, y0

'y)

| f ( x0

ɉɪɢɦɟɪ 3.5. ȼɵɱɢɫɥɢɬɶ ɩɪɢɛɥɢɠɟɧɧɨ ɫ ɩɨɦɨɳɶɸ ɩɨɥɧɨɝɨ ɞɢɮɮɟɪɟɧɰɢɚɥɚ

ɩ¹

1,97

arctg¨

1,02

1¸.

Ɋ
ɟɊɟɲɟɧɢɟ. Ɋɚɫɫɦɨɬɪɢɦ ɮɭɧɤɰɢɸ	§	x	·
ɟɊɟɲɟɧɢɟ. 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¸

¨arctg

1¸

¹ x

¹ y

ɢɥɢ, ɩɨɫɥɟ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɯ ɩɪɟɨɛɪɚɡɨɜɚɧɢɣ,

§ x 'x

§ x

'y .

arctg¨

1¸

| arctg¨

1¸

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)

0,02Ɍ

arctg¨

1¸ | arctg¨

1¸

1)2

12 (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),ɣɧɚɡɵɜɚɟɬɫɹ ɫɥɨɠɧɨɣ ɮɭɧɤɰɢɟɣ

ɩɟɪɟɦɟɧɧɵɯ x ɢ y. Ⱦɥɹ ɧɚɯɨɠɞɟɧɢɹ ɱɚɫɬɧɵɯɢɩɪɨɢɡɜɨɞɧɵɯ ɫɥɨɠɧɵɯ ɮɭɧɤɰɢɣ

ɢɫɩɨɥɶɡɭɸɬɫɹ ɫɥɟɞɭɸɳɢɟ ɮɨɪɦɭɥɵ:

;

.ɬ

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

– ɮɭɧɤɰɢɹ ɨɞɧɨɣ

ȼ ɫɥɭɱɚɟ, ɤɨɝɞɚ u=M(x), v=\(x), ɛɭɞɟɬ:

ɩɟɪɟɦɟɧɧɨɣ ɢ, ɫɨɨɬɜɟɬɫɬɜɟɧɧɨ,

wɨz du wz

ɉɪɢɦɟɪ 3.6.

ɇɚɣɬɢ ɱɚɫɬɧɵɟ ɩɪɨɢɡɜɨɞɧɵɟ ɮɭɧɤɰɢɢ

arctg u , ɝɞɟ u=x+y,

v=x–y.

Ɋɟɲɟɧɢɟ. ɉɨ ɮɨɪɦɭɥɟ

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

Fxc(x, y)

Fyc(x, y)

ȿɫɥɢ ɭɪɚɜɧɟɧɢɟ F (x, y, z)

ɞɜɭɯ ɩɟɪɟɦɟɧɧɵɯ z(x, y) (ɜ

ɡɚɞɚɟɬ ɮɭɧɤɰɢɸɢ

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

Fc(x, y, z)

Fyc(x, y, z)

;

Fzc(x, y, z)

ɉɪɢɦɟɪ 3.7.

ɇɚɣɬɢ ɱɚɫɬɧɵɟ ɩɪɨɢɡɜɨɞɧɵɟ ɮɭɧɤɰɢɢ z, ɡɚɞɚɧɧɨɣ ɧɟɹɜɧɨ

ɭɪɚɜɧɟɧɢɟɦ

xyz x

ɢ3 3

0 .

z 5

Ɋɟɲɟɧɢɟ.

ɨyz 3x

xy 3z

;

xz 3y2

xy 3z2

3z2 xy .

ɟwy

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

M 0 ( x0 , y0 , z0 ) ɩɨɜɟɪɯɧɨɫɬɢ,

ɬɚɤɨɜɨ:

x x0

y y0

z z0

f xc(x0

, y0 )

f yc(x0 , y0 )

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 x

y ɢy

z z

Fxc(x0 , y0 , z0 )

Fyc(x0 , y0 , z0 )

Fzc(x0 , y0 , z0 )

ɉɪɢɦɟɪ

3.8. Ɂɚɩɢɫɚɬɶ

ɭɪɚɜɧɟɧɢɹ ɤɚɫɚɬɟɥɶɧɨɣ ɩɥɨɫɤɨɫɬɢ ɢ ɧɨɪɦɚɥɢ ɤ

ɨ2 2

0 ɜ ɬɨɱɤɟ P0(2;–1;1).

ɨɞɧɨɩɨɥɨɫɬɧɨɦɭ ɝɢɩɟɪɛɨɥɨɢɞɭ x

2 y

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 ɟɜɢɞɟ 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

f ( p)

ɢɦɟɟɬ ɦɚɤɫɢɦɭɦ (ɦɢɧɢɦɭɦ)

ɜ ɬɨɱɤɟ P ( x0

, x0 ,..., x0 ) ,

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 ɩɪɨɢɡɜɨɞɧɵɟɌ

1-ɝɨ

ɩɨɪɹɞɤɚ

f xck (P0 )

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 )

f (x, y) , ɩɪɢɱɟɦ ɷɬɚ ɮɭɧɤɰɢɹ

– ɫɬɚɰɢɨɧɚɪɧɚɹ ɬɨɱɤɚ ɮɭɧɤɰɢɢ z

ɞɜɚɠɞɵ ɞɢɮɮɟɪɟɧɰɢɪɭɟɦɚ ɜ ɧɟɤɨɬɨɪɨɣ ɨɤɪɟɫɬɧɨɫɬɢ ɬɨɱɤɢ P0 ɢ ɜɫɟ ɟɟ ɜɬɨɪɵɟ

ɱɚɫɬɧɵɟ ɩɪɨɢɡɜɨɞɧɵɟ ɧɟɩɪɟɪɵɜɧɵ ɜ ɬɨɱɤɟ P0. Ɉɛɨɡɧɚɱɢɦ

, y0 ), D AC B

A fxx (x0

, y0 ), B fxy (x0

, y0 ), C f yy (x0

Ɍɨɝɞɚ:

( x0 ,

y0 ) ɮɭɧɤɰɢɹ z

f (x, y) ɢɦɟɟɬ ɷɤɫɬɪɟɦɭɦ, ɚ

1) ɟɫɥɢ D>0, ɬɨ ɜ ɬɨɱɤɟ

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

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

3)ɩɟɫɥɢ 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.

	wz	3(x2 y) 0;	wz	3( y2 x) 0.
	wx		wy
ɉɨɥɭɱɚɟɦ ɫɢɫɬɟɦɭ:
x2 y	0;
®	0.
¯y2 x	0.

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

(1,1)

. ɇɚɣɞɟɦ

ɱɚɫɬɧɵɟ ɩɪɨɢɡɜɨɞɧɵɟ 2–ɝɨ ɩɨɪɹɞɤɚ:

w2 z

6x;

6 y .

wx2

wxwy

wy2

Ɂɚɬɟɦ ɫɨɫɬɚɜɢɦ ɞɢɫɤɪɢɦɢɧɚɧɬ D

AC B2 ɞɥɹ ɤɚɠɞɨɣ ɫɬɚɰɢɨɧɚɪɧɨɣ ɬɨɱɤɢ.

w2 z

Ⱦɥɹ ɬɨɱɤɢ P :

2 z

;

3 ; C

w2 z

0 ; D

9 0 .

wx2

wxwy

wy 2

ɋɥɟɞɨɜɚɬɟɥɶɧɨ, ɷɤɫɬɪɟɦɭɦɚ ɜ ɬɨɱɤɟ P1 ɧɟɬ.

w2 z

Ⱦɥɹ

ɬɨɱɤɢ

P :

6 ;

ɪ wxwy

36 9 ! 0;

A ! 0 . ɋɥɟɞɨɜɚɬɟɥɶɧɨ, ɜ ɬɨɱɤɟ P 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

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 ɮɭɧɤɰɢɢɍz ɜ

ɤɪɭɝɟ.

3x2 6 y 2

(3x

6x)e

6x 0;

°z

12 yex

6 y

° zc

¯y 0.

Ɇ2(-2;0)

-1

ɢɊɢɫ. 3.2

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

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

ɇɚɣɞɟɦ ɧɚɢɛɨɥɶɲɟɟ ɢ ɧɚɢɦɟɧɶɲɟɟ ɡɧɚɱɟɧɢɟ ɮɭɧɤɰɢɢ z ɧɚ ɨɤɪɭɠɧɨɫɬɢ

x3 3x 2 6

ɇɚ

ɧɟɣ

;

x [ 1;1];

z(x)

ɂɦɟɟɦ

x3 3x2

z( 1)

; z(1)

Ⱦɚɥɟɟ,

ɪɟɲɚɹ

ɭɪɚɜɧɟɧɢɟ

z (x)

(3x

6x)e

0 ,

ɧɚɯɨɞɢɦ ɫɬɚɰɢɨɧɚɪɧɭɸ ɬɨɱɤɭ:

x1 0 ( 1;1);

z(x1 )

z(0)

Ɋɟɩɨɡɢɬɨɪɢɣ

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
	Ɍ
ɡɧɚɱɟɧɢɣ ɮɭɧɤɰɢɢ ɧɚ ɤɚɠɞɨɣ ɢɡ ɱɚɫɬɟɣ ɝɪɚɧɢɰɵ.	ɍ
	ɇ
	Ȼ

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