РГР (2 семестр)

300

5.

II= X

2

+ y

i

X·· I

(

1;1;1

)

'B

(

0;1;4

)

+-+-,A

.

: " y

z .

.

6. z =arctg(x + y ), Po (JJ. J3)

7. z =l-(x2 + Y2it3

2 ' 2

BapnanT 2.

I.

z= ~I

. 2.

z=2x2 -2x+4y2 +4y-2

3.

z =x2

- 2xy+% /-2x, Fa (1;2)4.

z = uv, r.n.e u = sin 2 x, v = cos2x·,

	u dz	xyz (	)'·	·. tr
HaHnt- 5.		u=-+-+-, A	1;2;1,	a=fJ=r<-
.	~.	y Z X		2
6.	Z =~i :-X2 , J6(3;-5)		7. z=ex-y(x2 -2y2 ).
BapnaiJT 3.

'.

I.z=)x2 +/~4 2. z=Jx.+y 3. z= J;.Po(4;3)

4•	:r' )'	,	. (	)	u	az	dz
	Z = e		rAe y = (/)	X	, HaHTH -		H -
					8y		dx
5.	u = xY+X, A(1;2;1), 8(2;0;3) 6.					z ;,)x2 +/, J6( -3;4)

7. z=x4 +)'4 -2x2 +4xy-2/

BapnanT4.

I. z=ln~ 2. Z-=ex)' 3. z=-;.., Po(l;l)
y	y
y	.	u	az	az
4. z =arc tg-, rAe	x = usm v, y = vcosu, HaHTH -			H -
X			au	8v

301

5.

u=z

l

3

'tr

, A(l;l;l), cosa=-, cosr=O, /3<-

5

2

6.

z = arcsin-x-, Po{l;l)

7.

z=xyJI-x2 - /

x+y

BapnanT S.

J 2

2

X-)' 3

2

2

. (2 0)

2

X - y

l.z=

x

-y

+I

.z=e

.z= X

2 + y 2

,Po;

"z

.

()

u

au

au

4.u= ~

+y2

,rAex=tsmt,y=vcost,z=rp v ,HaJ1TH-H-

x

2

·

.

ar

av

5.

u=e

Jxz+i+=z

(

)

B

(

4;2;6

)

6.

2

2

n

(3·

2)

, A

1;2;2,

z=x

+ y, •o

,-

7.

z =-x2 +xy- y 2 -9y+ 6x-35 .

BapnanT6.

z=~x2

-/-l 2.

z=~x2

+/

3

I.

3.

z=arctg~. Po(2;l)

8y . .

.J

2

aw.

aw

aw

4.

2

+v

2

rAe

u=ln

(

X

2

-y

2)

X y

u

W=·

U

'

'

'

V=-

HaHTH- -

H-

~

z3

'

ax '

8y

az

5.

u=x

5

I

tr

,A(I;l;8), cosa=-·, cosy=

r;:,/3>-

36

·

3v2

2

6.

z=~x

3

+/+l, P0 (5;1)

7.

r

y

z

-X+ 6y

Z

=Yv X -

BapnanT7.

I.

z =In (x + l) + In (y -I) 2.

z = ~i

+ 2x + /

302

303

3.

z=arctg

·/

, P (3;2)

BapuauT 10.

. I

(x+y) 2 +y

0

z=~x2

-/ 2.

z=x2 +4x+/'-2y 3.

z=Jx~y' Po(I;O)

,J';

. .

I+''

,,

.:.

du

I.

4.

r.ne

, z

u,=arctg-+smz,

x =e

, y = e

=In tgt, HaHTH -·-

y

&

;

.J2

2

u

oz

oz

4.

Z = e

,

f,lle

X=

y =

,

u

U +

V ,

HaHTH -

H -

5.

u = x 2 -

2xy + i

+ _!_, A{1;2;1), B(3;0;2) 6.

z = x3 + /, P0 (1;1)

Jl/ -v2

au

ov

z

'

z

y

1'(

1'(

5.

7.

z = x3 + x/ + 6xy

u=xz+-+- A(1·3·2)

a=fJ<-

r=-

x

z'

' '

'

2'

2

BapuauT 8.

6.

z=~x

2

+/+12, /1(3;2) 7. z=xy+501x+20/y

I . z = ~y 2 -

x -

I

2. z = x2 - y 2 3.

z = eJx'+y' ,

Po ( 3; 4 )

BapuanT 11.

4. z = u

COS\'

, r.ne

,

r2

r::-2

u

dz

2

u =vI + x- , v = v1- x- ,

HaHTH ~

l.

2

dx

z=ln(x

-2'x-y) 2.

z=3x -6x+4/+4y 3.

z=)f, Pa(6;6)

3

·

3

I 2

3

v

(

)

u ou

ou

5.

u=xy+yz,A(2;0;1), a=fl=r>tr 6.

z=

xc,J1(0;1)

4.

u=z

v

·vx

+y, r.n_e x=tcost,y=-, z=qJ

,HaHTH- H-

2

vY

t

at

av

7. z=x3/(2-x- y)

5.

u=ln(ex rteY +e=), A(l;l;l), B(2;2;2) 6. z= J'f,

/1{4;1)

BapuauT 9.

'

y

I +x+ y

2

2

7

·~

I.

z=

x- J!

2

-1

2. z=2x

2

-3y

2

3. z=

X

+ Y

n

{2·0)

. z =~~ + x2 + y2

~·

.r0

,

vx - y

4.	2X	r.ne y = (/J	()	u	oz	dz
	z =In -,		x ,	HaHTH -		H-
	y				ox	dx
5.	u = x3 + 2x + ~. A(l;-1;1), B(3;3;6) 6. i == 3f7, /1 (1;1)
	y	x				v--y
7.	z=xy1n(x2 +/)

BapuauT 12.

l. z=ln(/-2y+x) 2. z=~/+2x 3. z=eg, Po(0;2)

4. w=.Ju+uv+v, r.ne u=arctg ( x 2- y 2) , v= .j;y,z3

u awaw aw

HaHTH- - H-

.	ox' Oy	oz
	'

304

S.u=x				3	-3x		2	+3yz-2,A							(		)			tr						tr
																	1;2;-4,		a=-, f3=r<-
																					2					2
6	.	z =~ , Po (2; 8)											7.			z= y..Jl+; + x$+Y
				X
BapuanT 13.
I.								1									-			4+2.t2				+7.t
		z=arcsinl._ 2.							z=~-3y 3.									z=e>'						, P (0;1)
								X																	0		.
4.											.		X						•					I.			312
		u = In cos z + arcsm JY, r.ne x = t sm t,																				y = v t cost, z ='I r						,
HaHTH dt5. u=x											-3xz+2yz, A								1;1;1, B					1;4~5
		~			du					2								(		)			(		)
	6.	z=x.JxY, P (2;8)											7.				z=e2x(x+/+2y)
								0
	BapuanT 14.
	I.		z=arccos~ 2. z=ln(-x+y) 3. z=arctg fl, P (4;2)
								y																~~			0
					-														.					~
					I .
	4.		z = u'',				r.ne u = xtgx, v = xarctgx,													HaHTH -
																								dx
	5.		u=3x			3					z			(			)							.			tr
							-4yz+-,A									0;1;3, cosa=cosy<O,										{3=-
								.			y																2
	6.		z=y.JxY, P						(1;4)						7. z=vf(1..,.x)(1-y)(x+y-1)
								0
	BapuauT 15.																											(tr )
		l.z=vtor.;:--x-y											2.z=lnx-2y									3.z=cos.JxY-.••- P0
								2				2					( 2				2)						smx	-4;tr

305

. 4.

.

x

f

(

y

)

~

az

dz

z =arcsm

r:. ,

r.ne x =

, HaHTH -

H -

.

"Y

ay

dy

'

2

S.u=_.:_+L, A(1;1;2), B(4;1;6)

6.

z= ~· P0 (4;l)

~

n

"zy

7.

z = 4 - (x2 + /

)213

BapuanT 16.

I.

z= 1x 2 +y2

-16 2.z=x+2-- 3.

z=ln

( x,-+y.,Po)

( 1;2)

4

2

y-3

4. z = In

.

y)

, r.ne x

= ue

,.

,

v

~

az

az

(

arcsm-

y =---;;, HaUTH -

H -·-

x

.

e

&

~

I

I

I

tr

x 2

5. u=-+-+-,A(1;1;1), a=f3=r<- 6.

z=arctg-, P0 (2;4)

X

y

Z

2

y

7.

z =xy + 2/ x + 51 y

BapuanT 17.

I.

z=yJcosx 2.

z=y-l 3. z=~n(x+/'1:2

+/), P0 (3;4)

x+l

4.

u=ln

I

z

.

v

~

au

au

+/

, r.ne x=ttgt,y=cp(v),z=-.-,Hal<ITH- H-

Jx2

smt

ot

ov

5.

u = ~+-;.+ ~. A(2;1;2), B(2;5;5) 6;

z = arctg( x2 -

JY), P0 {2;9)

y

y

X

•

7.

z = (l + eY)cosx- yeY

BapuauT 18.

I.

z=xvfsiny

2.

z=2,x+7y 3.

z=ln{x3 +/), -Pa(1;7)

306

2

I

~

8w

aw

aw

4.

w= ln

11

, r.uc 11 =In

(

x

2

+ y

2 )

,

•

1

v=arcsm-, HaHTH -,-H.-

,..

xyz

ax

ay

az

.

,[;

2

Fz

(

)

.

Jr

J2j

5.

11 =-+

c

+-,A 1;1;1 , a =/3<-, cosy=-

2

vY

2

2

5

~- Jx+y, p (3;4)

7.

z=x

2

+/-32ln(xy)

6... -

xy

0

BapnanT 19.

I.

r.:::-_

2.

'x

I

2

·J

(

1;2

)

z=xvltJ'

z=- 3.

z=vx

+ y, P0

3y

z

e'-"

2

r.

. 2

~

dv

4. v=arctg-+-., r.ue

x=t ,y=-vt,z=sm

t,HaHTH-

zy

z

~

5.

if;

y

z

A( 4;4;4), B(l 0;-1;3)

6. z = J

xy

,P0 (3;4)

u =-+

11

+ 1r-,

Y

v z

vx

·

x2 + y2

7.

z=3(x

2

,

1

+y~)-x· +4y

BapnanT 20.

2. z= jt,·-- 3.

z={Jx,{ 3 +3y2

I. z=x-v~LI!:JIc=-.

+7,.

P0 (2;2)

2y

v

2

•

3

~

dz

4• Z = -, r.ue

II= COS

X, V =Sill

X, HaHTH -

u'·

·

dx

5•

_ zx

+

zy

yx

A ( 4; 4; 4),

a < Jr

f3

=2Jr _ tr

11 -

'--y

r

+ 1 ,

•

vY

vx

vz

2

3

,y--

4

6.

z = ln~x2

+ /, P0 (3;4)

7.

z =x 2 + /

-2 hL~ -181n y

BapuauT 21.	307

1

I

2

.

z=e

~3

. z=x

J

+3xy

2

X

()

. z=

Jx

2 + /-9

+-,Po

1;1

Y

4.

Z = e

Jx1

+2xy

, f,AC

X=

I( .)

~ az

az

dz

)' , H8HTH -, -

H -

ax

ay

dy

5.u = 7x2 +By'+ xy+ ,/x + Jf. A(1;2;1), B(-1;0;0)

6.z=12· JZ¥Z,P. 0 (2;4) 7. z=x2 +/+xy+llx+lly

BapuanT 22.

)2.z=ln H-

l.z=

(

I

X

y

3.z=-+-,Po(l;l)

In

-x- y

x

y

x

4. u = In

.

y)

~

az

az

( sm- , r,Ae x = utgv, y = vctgu , HaHTH

~ H -

X

.

au

av

5.

X

JY

3

I

3

6

1f

u=3x+2xy+ c+-+x, A.(l;2;1),

cosa=-, cosf3=-, r<-

vY

x

.

7

7

2

6.

z=lnff,P0 (1;9)

7. z=x3 +i-9xy

BapuauT ~3.

I.

.

1-x

z=ln

f}x

z= J

y

,

Pa(3;4)

z=arcsm-- 2.

- 3.

I+ y

y

'x2 + /

2

2 ,r,Ae

. ~

4

.

u=

x

•

.

(

t

)

au

au

1

x=vsmt,y=tcost,z=rp

,HaHTH- H-

+z

at

av

3 /

308

5.

u = 2x3 + 3x2 + ~ + ~ +

fi_, A,{l;2;2), B(3;5;8)

y

X

{;

6.

z =

X)'

7.

z = xy(9- x- y)

7.·/ + y' 1'0 (1;9)

- BapuanT 24.

.

~ =

2x+3y-l

2

~·-I

z = x

0·2

Y

2

+

4

,

1o (2·2),

I•

·

z = log -·- 3.

x +

0

"'

• .

2

y-

I

4.

w=}/

t/ + v2

x

xy

~

ow

ow

ow

sin

(

)' rne u=ln-, v=- ,

Hai1TI1 -,-11-

uv

y

z 3

ox

oy

oz

5.

u=FY+ ~+~,A{I;4;-1): a=fJ=r<;r

~xy

z

2

6.z=ln 4 f:1, P0 (4;1) 7.

z=(x-x2 )(y-/)

v~

HapnanT 25.

l.z=

x+2y-l

r;;-

_

. Y

o(I·)

x-y

2.z=y+v2x3.z-xn1 -,r0

,e

x

4.

u=ln

cos JY J

'I'

~

~

~ [J'--::

(

.

+e· +\fze·, rne x=vt+l,y=\fr+l,z="r+2,

SlllX

.

J1ai1TH

dv

5.

u = x + i + .Jx + JY + f;y +if;, A(l;l;8), B( 4;5;8)

dt

6.

z = arctg

~ , P0 ( 4; 4)

7.

z=xy+51x+61y

\fY

309

BapuauT 26.

I.

z=

x- y

h..

.

x

2.x + 2y -

2. z=2x+y7y 3.. z=arctg--, Po{2;1)

I

·

x + y

4.

Z =

,

)COSI'

, r,ne

2

J

~

dz

(Sin U

U = tg X,

V = Ctg X, H311TI1

-

J7

dx

5.

lr::=

A(1;2;32), a= fJ

;r

u = v..~:y;c,

< -, cosy=-

2

3

6.

~=

~+arctg~, P0 (1;1)

7. z=x3 +8/-6xy+'1

'

'JX

X .

BapuauT 27.

I.

z =

~

1

3.

y

2. z =--.;::;:-

z =arc tg--, Po (3;2)

e

x-y

.

oz

oz

dz

4.

Z =2/',ftl.C X= f(y)' Haii"fl1 ax'

oy 11

dy

lr=

X

X

....

. 5. u=-\f.A._Y,A(l;27;1), B(1;23;4) 6.

z=- +arctg-_, P0 (2;2)

z

y 2

y

7.

z=xly+llx+y

BapuauT 28.

1.

z=

f¥

2. z=-1-

3.

z=xcos 2 y,

?,0 ( 1;ir)-

3'-2)'

4

4.

~-2

2

•

2

~

oz

.oz

z=lnvx

-ry, r.ne x=ucos v,y=vsm

u,Hai1TI1- 11-

au

ov

1,.-

.

;r

3

;r

A(2·I·4)

·p-- r

<-

vxz

'

a=-,

cos

-5,

2,

5.u=-.-2'

,,.

2

y

.

x2 . .

7. z =(I+ X- y)(l + x

2

1 2

6. z=(\rcsm-, P0(1;2)

+ /r '

y

310 HapnanT 29.

•

X

.

}

I .

2

•

'

tr

I. z=arcsm-+arcsmL 2. z=

-

2Q 3.

z=y

sm· x,

Po ( -;-1 )

)'

X

1

(

5)

4

4.

II.'+ V 2

r.ne

•

v

OU

OU

u= J

,

x=tcost, y=vsmt, z=q>(v), HaHTH- H-

In z

ot

OV

5.

X 2

Y2

I

[

X

2J

, P0 (3;1)

u=-+-· +-, A(l;l;-1), B(0;3;1),6. z=ln

-

3

)'

X

Z

.

y

7.z=(x-y+l)2

BapnanT 30.

2

I. z=arccos~ 2. z=(.!_)xy 3. z=ln(.J;+if"Y), Po(4;8)

l-2y

7

4. w = J

J

, r.ne

.

(

x

2

- y

2)

J 2

- z

2

,

v

u =arcstn

, v =arccos

y

HaHTH

u 2 + v 2

0\1'

ow

ow

-,-n-

ox

~)'

oz

_

y

x

xy

6

3

r <"

5.

u -

1

+

r..

+

1

, A (I; 1; -I),

cos a = -, cos fJ

= -

\IX

"y

\IZ

.

7

7'

2

6.

.[;

. (

)

7.

z=xy

J

2

-36y

2

z=ln ,/..' P

4;8

l-4x

VY

0

N2

<l>ynKUllll

1.

y=c

2.

y=xa

a=2n neN y = xl•

a=2n+l

neN

y=xln+l

a=.I-

2n

neN

y=lif;.

1

a=--

2n+l

neN

·y = ln+,f;.

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		I

311

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i

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