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R2

(x, y)

n N0 P: R R P(x) =anx

n +an1xn1 +...+a1x+a0,

a0, a1,...,an an= 0

a0, a1,...,an

an

a0

x

R

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a >1 x1 < x2 logax1

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f(x) = arccos x [0, ] arccos: [1, 1][0, ]

D(f) ={x R :|x| 1}= [1, 1]

f(x) = arctgx

2,

2 arctg: R

2,

2

R

f(x) = arcctgx 0, arcctg: R 0,

R

f(x, y) = 14 x2 y2

4 x2 y2 0

4 x2 y2 = 0

4

x2

y2

= 0 x2 +y2

= 4

x2 +y2 4 x2+y2

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f(x, y) = 1

x2

4+ 44

y2

f(x, y) =

(4x2 +y2 16)(9 x2 y2)

f(x, y) = ln(x2 y)

x

x= 0 x2 y >0

y < x2 f

D(f) ={(x, y) R2 :x= 0, y < x2} f y = x2

x= 0

f(x, y) = logx(2 y)

f(x, y) =

8x 6y+x2 +y2

log(xy)

f(x, y) = lny 3x

x 2y

+

4 x2 y2

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f(x, y) =

ln(y+x2) ln(x y2) f(x, y) = ln[x ln(y x)]

f(x, y) = arcsinx

y

y= 0 1 x

y 1

y > 0 y y x y y x yx y 0, yxy) ili (y

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z= f(x, y) T(x0, y0)D(f)

fx(x0, y0) = limx0

f(x0+ x, y0) f(x0, y0)x

,

fy(x0, y0) = limy0

f(x0, y0+ y) f(x0, y0)y

.

z = f(x, y) (x, y) x y

fx(x, y) = limx0

f(x+ x, y) f(x, y)x

,

fy(x, y) = limy0

f(x, y+ y) f(x, y)y

.

z= f(x, y)

fx(x, y) z=f(x, y)

x y

fy(x, y) z= f(x, y) y x

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f: I R x0I y= f(x)

f(x0) = limx0

f(x0+ x) f(x0)x

,

y= f(x) x0 x I y = f(x)

f(x) = limx0

f(x+ x) f(x)x

,

y= f(x)

f, g : I R I

[f(x) g(x)] =f(x) g(x)

[f(x) g(x)] =f(x)g(x) +f(x)g(x)

f(x)

g(x)

=

f(x)g(x) f(x)g(x)g2(x)

[f(g(x))] =f(g(x))g(x)

[f1(x)] = 1f(f1(x))

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f(x) f(x)

c 0x 1xn nxn1

1

x 1

x2ax ax ln aex ex

logax 1

x ln a

ln x 1

x

sin x cos xcos x sin xtgx

1

cos2 x

ctgx 1sin2 x

arcsin x 1

1 x2arccos x 1

1 x2arctgx

1

1 +x2

arcctgx 11 +x2

T(1, 1)

f(x, y) = x2

y 2xy

fx(1, 1) = limx0

f(1 + x, 1) f(1, 1)x

= limx0

(1 + x)21

2(1 + x) 1

121 2 1 1

x

= limx0

1 + 2x+ (x)2 2 2x 1 + 2x

= limx0

(x)2

x = lim

x0x= 0

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f T(1, 1) T fx(x, y) f

y(x, y)

fx(1, 1) =2 11 2 1 = 2 2 = 0, fy(1, 1) =1

2

12 2 1 =1 2 =3.

f(x, y) =2x+y

x y T(1, 3)

f(x, y) =

xy (x y)2 T(2, 1)

f(x, y) = ctg ln(2x

y) + xy+

x

y

f(x, y) = (x2y 3x)ex+y2 + arcsin y2

f: D R D R2

fx(x, y) =

f

x (x, y), fy(x, y) =

f

y (x, y).

fxx(x, y) =

x(fx(x, y)) =

2f

x2(x, y), fxy(x, y) =

y(fx(x, y)) =

2f

yx(x, y),

fyx(x, y) =

x(fy(x, y)) =

2f

xy(x, y), fyy(x, y) =

y(fy(x, y)) =

2f

y2(x, y).

yf

x

xf

y

D R2

y

fx

(x, y) =

x

fy

(x, y).

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f(x, y) = 3x cos y 2xy

fx(x, y) =

x(3x cos y 2xy) = 3 cos y 2y,

fy(x, y) =

y(3x cos y 2xy) =3x sin y 2x.

x y

fxx(x, y) =

x(fx(x, y)) =

x(3cos y 2y) = 0;

fxy(x, y) =

y(fx(x, y)) =

x(3cos y 2y) =3sin y 2;

fyx(x, y) =

x(fy(x, y)) =

x(3x sin y 2x) =3sin y 2;

fyy(x, y) =

y(f

y

(x, y)) =

y(

3x sin y

2x) =

3x cos y;

fxy(x, y) =fyx(x, y)

f(x, y) = ln(x2 y)

f(x, y) =

xy+

x

y

f(x, y) =xy

+yx

f(x, y) = arctgx

y

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z(x, y) =ex

y y

2z

xy =

z

y z

x

2z

xy

z

y

z

x

z

x=

x(e

x

y ) =1

yex

y , z

y =

y(e

x

y ) =xy2

ex

y

2z

xy =

x

zy

=

x

x

y2ex

y

= 1

y2ex

y xy3

ex

y

y

1y2

ex

y xy3

ex

y

=x

y2ex

y 1y

ex

y xy2

ex

y 1y

ex

y =xy2

ex

y 1y

ex

y

z(x, y) = xy

x y

2z

x2+ 2

2z

xy+

2z

y2 =

2

x y .

u(x, t) = arctg(2xt) 2u

x2+2

2u

xt= 0

z = z(x, y) F(x,y,z ) = 0 T(x0, y0, z0) F(x,y,z ) F(x0, y0, z0) = 0 Fz(x0, y0, z0)= 0 z(x0, y0) =z0 z= z(x, y)

zx(x, y) =Fx(x,y,z )

Fz(x,y,z ), zy(x, y) =

Fy(x,y,z )

Fz(x,y,z ).

z= z(x, y) x3z+ yz2 =2 T(1, 1, z0 >0)

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z=f(x, y) T(x, y)

z= f(x+ x, y+ y) f(x, y),

x y x y f

z = f(x, y)

T(x, y)

dz= z

x(x, y)dx+

z

y(x, y)dy.

x y dz = z z = f(x, y) (x+ x, y+ y) (x, y)

f(x+ x, y+ y)f(x, y) + zx

(x, y)x+z

y(x, y)y.

z=f(x, y)

T(x, y)

d2z= d(dz) = 2z

x2(x, y)dx2 + 2

2z

xy(x, y)dxdy+

2z

y2(x, y)dy2.

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z= f(x, y) F(x,y,z ) = 0 F(x,y,z ) = 0 T(x0, y0, z0) z0 = f(x0, y0)

Fx(x0, y0, z0)(x x0) +Fy(x0, y0, z0)(y y0) +Fz(x0, y0, z0)(z z0) = 0, T(x0, y0, z0)

n . . . x x0

Fx(x0, y0, z0)=

y y0Fy(x0, y0, z0)

= z z0

Fz(x0, y0, z0).

z= x2y T(2, 1, 4)

z = x2y (2, 1)

zx(x, y) = 2xyzx(2, 1) = 2 2 1 = 4; zy(x, y) =x2 zy(2, 1) = 22 = 4

t. . . z

4 = 4(x

2) + 4(y

1)

t. . . 4x+ 4y

z

8 = 0

n . . .x 2

4 =

y 14

= z 4

1 .

z= y

x y2 x+ 6y T(4, 1, z0)

T(x0 > 0, 2, 1) xz2 +x2y= 6

x0 T(x0 > 0, 2, 1) xz2 +x2y= 6

x0 12 +x20 2 = 62x20+x0 6 = 0x0 =2, x0 =3

2

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x0 >0 x0 =3

2 T( 3

2, 2, 1)

F(x,y,z ) = xz2 +x2y 6

T(

3

2 , 2, 1)

Fx(x,y,z ) =z2 + 2xyFx(3

2, 2, 1) = 12 + 2 3

2 2 = 7;

Fy(x,y,z ) =x2 Fy(3

2, 2, 1) =

32

2=

9

4

Fz(x,y,z ) = 2xzFz(3

2, 2, 1) = 2 3

2 1 = 3

t. . . 7(x 32

) +9

4(y 2) + 3(z 1) = 0t. . . 28x+ 9y+ 12z 72 = 0,

n . . .x 3

27

= y 2

9

4

= z 1

3 .

x2 + 2y2 + 3z2 = 15 T(1, y0 > 0,

2)

x2y2+ z2 = 1 . . . x y+ 2z= ln 2

F(x,y,z ) =0 F(x,y,z ) =x2 y2 +z2 1.

n={1, 1, 2} nt ={Fx(x0, y0, z0), Fy(x0, y0, z0), Fz(x0, y0, z0)} T(x0, y0, z0)

F(x,y,z )

Fx(x,y,z ) = 2xFx(x0, y0, z0) = 2x0;

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Fy(x,y,z ) =2yFy(x0, y0, z0) =2y0; Fz(x,y,z ) = 2zFz(x0, y0, z0) = 2z0

(Fx(x0, y0, z0), Fy(x0, y0, z0), F

z(x0, y0, z0)) =(1, 1, 2), tj.

(2x0, 2y0, 2z0) =(1, 1, 2)

2x0= x0= 12

2y0=

y0=

1

2

2z0= 2z0= . T(x0, y0, z0) = (

1

2,

1

2, )

x20 y20+ z20 = 11

42 1

42 +2 = 12 = 1

1 = 1 2 =1

1 = 1 T1 = 1

2 ,

1

2 , 1

n1 = (1, 1, 2)

t1. . . 1 (x 1

2) 1 (y 1

2) + 2 (z 1) = 0, tj. t1. . . x y+ 2z= 2;

2 =1 T2 =

12

, 12

, 1

n2 = (1, 1, 2)

t2 1 (x +1

2) + 1 (y +1

2) 2 (z+ 1) = 0, tj.t2. . . x y + 2z=2.

x2 y2 +z2 = 1 . . . 2x+y+ 3z= 1

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fx(x, y) = 3x2 6y= 0

fy(x, y) = 24y

2

6x= 0x = 4y2

3(4y2)2 6y= 048y4 6y= 08y4 y= 0y(8y3 1) = 0,

y1= 0 8y32 1 = 0y32 =

1

8 y2= 1

2

T1(0, 0) T2

1,1

2

fxx(x, y) = 6x

fx,y(x, y) =6fy,y(x, y) = 48y

T1(0, 0) A1 = 0 B1=6 C1= 0

D1=

A1 B1B1 C1 =

0 66 0 =36< 0

T1(0, 0)

T2

1,1

2

A2 = 6 B2=6 C2 = 24

D2=

A2 B2B2 C2 =

6 66 24 = 144 36 = 108> 0

T2

1,

1

2

fxx

1,

1

2

= 6> 0

z= x3 3xy y3

z= 2x4

+y4

x2

2y2

z = x ln(x+ y)

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f(x, y) =

x2 + 1 |y| 2|x|

f(x, y) = ln(x2 y 3) +

4 x2 y2 + x

f(x, y) = lnx

y+ arccos

x 3yx+y

f(x, y) =

log|x|(3 y)

f(x, y) =ln(36 9x2 4y2)

x+y + arccos

x+y

f(x, y) =xy

f(x, y) = (x y)3

f(x, y) =xy T(2, 1)

f(x, y) = (x y)3 T(1, 1)

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f(x, y) = ln(x2 +y2 + 1) T(1, 1)

f(x, y) = arctgx2

y cos2(3x 2y)

f(x, y) = log3(2y 4

x5) +xex

2+y2

y

f(x, y) = 9 x3 y2 +xy 2x+ 3y f(x, y) =

xy

f(x, y) = 4

xx

2

y + 4y3

f(x, y) = sin x sin y+ sin(2x y)

s(x, t) = ln1

x 1

t

2s

x2+2

2s

xt=

1

x2

f(x, y) =ex2

y2

2f

x2+

2f

y2 = 4f(x, y)(x2 +y2 1).

z= z(x, y) z= 1 +yx +zy

z= z(x, y)

xz2 +x2 + 2x+y2 + 5 = 0 T(1, 0, z0

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dz d2z

z= 9 x3 y2 +xy 2x+ 3y T(1, 2) z=

x, y T(2, 2)

z= e2xy

z= x3 ln1 +xy3

5 T(2, 0)

z=

3x5 7x3y z= x2 cos(xy)

arctg 1.02

0.95

5.8 + 3

1 3.12

1.052

3

7.9 4

1.053

z= x2 2xy+y2 x+ 2y T(1, 1, 1)

x

2

2y2

+ z2

= 3

T(1, 1, z0

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