математика_2семестр

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25

Вариант 1.

1:

z = 2x3y 4xy3:

:

z

p

2:

z = x2 + y2 + 3; x = ln t; y = p3 t:

yp

2

y2

x

y:

3

=

x

+ 14

3x + y xy;

4:

z =

D : y = x; y = 4; x = 0:

z = x2e y

x = cos(u v); y = sin

u

2:

;

:

v

3:

z = x3 + 8y3 6xy + 5:

4:

z = xy x 2y; D : x = 3; y = x; y = 0:

Вариант 3.

1:

z = x2y sin x 3y:

2:

z = ln ex

+ e y

; x = t3; y = t2:

3

:

z

= 1

xy

2y2:

+ 15x

2x2

4:

z = x2 + 2xy 4x + 8y;

D

: x = 0; y = 0; x = 1; y = 2:

Вариант 4.

1:

z = arcsin xy 3xy2:

y2

p

3:

z = 1 + 6x x2

xy

:

2:

z = sin x cos y; x = ln

u + v2

; y =

v

u2:

4:

z = 5x2 3xy + y2;

D

: x = 0; x = 1; y = 0; y = 1:

Вариант 5.

1:

z = 5xy4 + 2x2y7:

2:

z =

x

; x = eu v; y = sin(u + v):

x y

3:

z = 2x3 + 2y3 6xy + 5:

4:

z = x2 + 2xy y2 4x;

D

: x y + 1 = 0; x = 3; y = 0:

Вариант 6.

1:

cos

x2 y2

+ x3:

y

v

2:

z = x ;

u

;

y = ln(v u):

x = e

3:

z = 3x3 + 3y3 9xy + 10:

4:

z = x2 + y2 2x 2y + 8;

D

: x + y = 1; y = 0; x = 0:

Вариант 7.

:

z

= ln2

3x2

2y2

:

1

y

;

y = cos uv:

2:

z = x e

x = sin(u v);

3:

z = x2 + xy + y2 + x y + 1:

4:

z = 2x3 xy2 + y2;

D

: x = 0; x = 1; y = 0; y = 6:

Вариант 8.

1:

z = 5xy2 3x3y4:

p

3:

z = 4(x y) x2

y2

:

1 ; y =

v2:

2:

z = x sin xy;

x = ln

u2

1

z = 3x + 6y x2 xy y2;

: x = 0; x = 1; y = 0; y = 1:

4:

D

1:

z = arcsin(x + y):

2:

z = xy;

x = ln(u v);

u

y = ev :

3:

z = 6(x y) 3x2 3y2:

4:

z = x2 2y2 + 4xy 6x 1;

D

: x = 0; y = 0; x + y = 3:

Вариант 10.

1:

z = arctg(2x y):

x

:

z

arcsin

;

y = x2

+ 1:

2

=

2

y

2

p

3:

z = x

+ xy + y

6x 9y:

4:

z = x2 + 2xy 10;

D

: y = 0; y = x2 4:

Вариант 11.

1:

z = 7x3y p

xy:

u

2:

z = x2 ln y; x =

; y = 3u 2v:

v

3:

z = (x 2)2 + 2y2 10:

4:

z = xy 2x y; D : x = 0; x = 3; y = 0; y = 4:

Вариант 12.

1

= py

:

z

x2x+ y

+ 2 3

+ 1

2

2

xy

:

2:

z = e

;

x = u ;

y = u sin v:

3:

z = (x 5)2 + y2 + 1:

4:

z = 0; 5x2 xy;

D

: y = 8; y = 2x2:

Вариант 13.

1:

z = ex+y 4:

2x

y = cos2 t:

2:

z = arccos

;

x = sin t;

y

3:

z = x3 + y3 3xy:

4:

z = 3x2 + 3y2 2x 2y + 2;

D

: x = 0; y = 0; x + y = 1:

Вариант 14.

1:

z = cos(3x + y) x2:

2:

z = arcsin

x

;

x = sin t;

y = cos2 t:

2y

3:

z = 2xy 2x2 4y2:

4:

z = 2x2 + 3y2 + 1; D : y = r

; y = 0:

9 4x2

9

2:

z = ey 2x 1; x = cos t; y = sin t:

3: z = xp

x2 y + 6x + 3:

y

z = x2 2xy y2 + 4x + 1;

4:

D : x = 3; y = 0; x + y = 1:

2:

z = ln e x + ey

; x = t2; y = t3:

3

:

z = 2

5

x2

3

+ 2

xy

y2

:

z = 3x2 + 3y2 x y + 1;

: x = 5; y = 0; x y = 1:

4:

D

Вариант 17.

1:

z = xy4 3x2y + 1:

2:

z = x2e y;

x = sin t;

y = sin2 t:

3:

z = xy (12 x y) :

4:

z = 2x2 + 2xy 0; 5y2 4x;

D

: y = 2x; y = 2; x = 0:

Вариант 18.

1:

z

ln

x + xy

y2

:

2:

=

y

2x

;

3

:

z = e

x = sin t;

y = t

3:

z = xy x2 y2 + 9:

5

4:

z = x2 2xy +

y2 2x; D : x = 0; x = 2; y = 0; y = 2:

2

Вариант 19.

1:

z = 2x2y2 + x3 y3:

2:

z = xy; x = et; y = ln t:

3:

z = 2xy 3x2 2y2 + 10:

4:

z = xy 3x 2y; D : x = 0; x = 4; y = 0; y = 4:

Вариант 20.

z = p

:

1:

3xx2 2yy2 + 5

2

3

2:

z = ln (e + e ) ; x = t ; y = t :

3:

z = x3 + 8y3 6xy + 1:

4:

z = x2 + xy 2;

D

: y = 4x2 4; y = 0:

1:

z = arcsin

x + y

:

x

2:

z = x2ey; x = cos t; y = sin t:

Вариант 24.

1:

z = y2 + 3xy x4:

2:

z = ln e2x + e y ; x = t4; y = t3:

3

:

z

= 2

xy2

+

+ y2:

x3

5x2

z = 6xy 9x2 9y2 + 4x + 4y;

: x = 0; y = 0; x = 1; y = 2:

4:

D

Вариант 25.

:

z

arcsin

x2

+ y3

:

1

=

2

2

t

; y = cos t:

2:

z = x

+ (x + y) ;

x = e

3:

z = 3x2 x3 + 3y2 + 4y:

4:

z = x4 y4;

D

: x2 + y2 = 1:

1:

dx

f(x; y) dy:

1

p

1 x2

2:

ZZ

12x2y2 + 16x3y3 dxdy; D : x = 1; y = x2; y = p

x:

D

3:

x = 4y y2; x + y = 6:

p

p

4:

y = 16 2x; y = 2x; z = 0; x + z = 2: