Файл: Методическое пособие по высшей математике.pdf

z′′2

= 36x 2 + 12xy 4 ;

z′′

= 24x 2 y3 ;

z′′2

= −140 y 3 + 24x3 y 2 .

x

xy

y

2. z = 2x 2

− xy + 3y 2

− 2x − 11y + 1 .

:

z = f (x, y ( ,

) :

1) z = f (x, y

z′

= 0; z′ = 0 .

x

y

, ,

. M 0 (x0 ,

y0 .

z = f (x, y

2)

M 0 (x0 ,

y0 .

A = z′′2

(x

0

, y

0

; ) B = z′′ x

0

, y( ;

C

= )z′′2

x

0

,

y

0

.

x

xy

0

y

A = z¢¢2

(1; 2 )= 4; B = z¢¢

1; 2 =( -1; ) C = z¢¢2 1; 2 = 6 .

x

xy

y

:

D = AC - B 2 = 23 .

M (1, 2 .

D > 0

A > 0 , ,

zmin = z(1, 2 = -11.

3.

U (x, y

du = (4 + xy 2 dx +) x 2 y + 2(y dy .

P(x,

y dx)

+ Q x, y(

:

dy du

¢

(x, y =)

¢

x, (y .

U (x, y , Py

Qx

,

du = P(x, y dx)

+ Q x, y(

dy - U (x, y .

:

U x¢ (x, y =) P x,(y

“ y ” :

¢

òU x (x, y dx) = òP x, y( dx +j) y

U (x, y

=)

òP x, (y dx +)j y ,

(1)

j(y

- “ y ”.

j(y

( P(x, y dx) +j y (¢y =) Q(x, y )

ò

(òP(x, y dx) ¢y +j¢y =)Q(x, y )

j¢y = Q(x, y -) (òP x, (y dx ¢y)

)

j(y (1).

:

P(x, y =) 4 + xy 2 ; Q x, y = x 2 y + 2 y .

Py¢ = 2xy;

Q¢x = 2xy. Py¢ = Q¢x . ,

æ

1

2

2

ö

¢

2

U ¢y = ç4x +

x

y

+j(y )÷

= x

y +j¢y ; U ¢y

= Q(x, y

)

2

è

øy

,

j(y = y 2 + c , c - .

x 2 y + j¢y

= x 2 y + 2 y; j¢y

= 2 y;

,

U (x, y =) 4x +

1

x 2 y 2 + y 2 + c .

2

4.

ò(2xy + y dx + (3x 2

-1 dy ,

L y = 3x 2 + x ,

dy = (6x +1 dx ;

M1 (1, 4 M 2 (2, 4 “ x ”

x1

= 1 x2

= 2 .

,

ò(2xy + y dx)+ (3x 2 -1 dy =) ò2 [2x 3x 2 + x +( 3x 2 + x - 3)x 2 -( 1 (6x +1 ]dx)) = (

)

L

1

2

3

2

æ

4

8

3

5

2

ö

2

1

ò(24x

+ 8x

- 5x -1 dx =ç6x

+)

x

-

x

- x ÷

1

= 90

.

2

6

1

è

3

ø

3

2 x−1

5. òòdxdy = òdx

òdy .

D

1

( x−1 2)+1

D ; 2)

:

1)

XOY

1 3

x

y = 2x -1 x =

1

(y +1 )- .

2

+1 - (x > 0 .

y = (x -1 2 +1 x =

y -1

,

y−1+1

2

2 x

1

5

−

òòdxdy = òdx

òdy = òdy

òdx .

D

1

(x−1 2)+1

1

1

( y+1 )

2

3) D :

3

2 x−1

3

2x -1

3

[2x -1 - (x -1 2)-1 dx =

3

[- x 2 + 4]x - 3 dx =

]

S =

ò

dx

dy =

ò

dxy

=

ò

ò

ò2

(x -

1 2)+

1

æ

1

x

3

( x−1

)+1

1

ö

3

4

1

1

2

ç

+ 2x

÷

=

.

= ç-

3

- 3x ÷

1

3

è

ø

5

y−1+1

5

y -1 +1 5

é

1

1 ù

5

æ

1

1

ö

S = òdy

òdx = òdyx

1

= òê

y

-1 +1 -

y -

údy = òç

y -1 -

y +

÷dy =

1

1

( y+1)

1

(y +1 )

1

ë

2

2 û

1

è

2

2

ø

2

2

2

5

æ

1

1

ö

æ

2

25

5

ö

æ

1

1

ö

4

(y

3

)-

2

.

= ç

-1

y

+

y ÷

=

ç

* 8

-

+

÷

- ç

0

-

+

÷

=

3

4

3

3

è

2 ø

1

è

4 2

ø è

4 2

ø

f1 (x )f 2 y(

dx) +j1 x j(2

y) dy = 0 y¢ = f1 (x )f2 y( .

) f1 (x )f 2 y( dx) +j1 x j(2 y) dy = 0 .

: .

f1 (x )f 2 y( dx) + j1 x j(2 y) dy = 0

: f2 (y )j1 x( Þ

f1 (x )

dx +

j2 (y

dy = 0

Þ ò

f1 (

x )

dx + ò

j2 (y

dy = c ,

c - –

j1 (x )

f 2 (y )

j1 (x )

f 2 (y )

:

dy

) y¢ = f1 (x )f 2 y( ; )Þ

= f1 (x )f 2 y(

*) dx Þ dy = f1 x f 2 (y ()dx) : f 2 y ( Þ)

dy

= f1 (x )dx Þ

dy

dx

f1 (x )dx + c - .

Þ

=

f2 (y )

ò f 2 (y )

ò

y¢ + P(x )y = q x (

(1)

y¢ + P(x )y = q x (yα

(2)

a ¹ 1;

P(x ), q x( - .

y = u *n , u = u(x ),n =n x ( -

y = u *n

; y

¢

¢

¢

(3)

= u n + un

(3) (1),

¢

¢

+ unP(x )= q x(

(4)

u n + un

un ¢ +n (u¢ + uP(x ) = q) x

u(x ) ,

ìu¢ + uP(x ) = 0,

í

î un ¢ = q(x )

.

(4) :

¢

¢

+nP(x ) = q) x

u n + u(n