Mult. de Lagrange3 Ejercicios

7/23/2019 Mult. de Lagrange3 Ejercicios

1/9

f(x, y) =x2 +y2 g(x, y) =x2 y2 +xy3 (0, 0, 0) z=x sin(y

x)

(a,b,a sin(

b

a))

a= 0

z=xf( yx

).

x + 2y + 3z= 1

x2

8 + y

2

2+ z2 = 1.

t = g(x, y) g C1 F(t) f(x, y) f(x, y) =F[g(x, y)].

f

x=F [g(x, y)]

g

x

f

y =F [g(x, y)]

g

y

F(t) =esin(t), g(x, y) = cos(x2+y2),

fx

,fy

f(x, y) x y f

x,f

y f

u= (x y)

2 v =

(x+y)

2 , f(u, v) F(x, y)

F

x

F

y

f

u,f

v

u= f(x, y), x= X(s, t), y =Y(s, t), u s t u= F(s, t).


2/9

F

s

F

t f, X Y

2f

xy =

2f

yx

2F

s2 =

f

x

2X

s2 +

2f

x2 (

X

s )2

+ 2

X

s

Y

s

2f

xy +

f

y

2Y

s2 +

2f

y2 (

Y

s )2

.

2F

t2

2F

st

X(s, t) =s+t , Y(s, t) =st

X(s, t) =st , Y(s, t) = st

G(s, t) =f(2s t + 1, s + 3t)

f

(1, 0) f D1f= 1; D2f=

1; D11f= 2; D12f=

2; D22f=

3. D12G(0, 0) D22G(0, 0) u= f(x, y), x= r cos(), y =r sin()

u2x+u

2y ur

u

fxx+fyy

f C2

g : R2 R f(x, y) = g( (x

2 +y2)

2 , xy)

D1g(

52

, 2) = 2, D2g(

52

, 2) = 1, f(1, 2) f (1, 2) (0, 1) (1, 0).

r=

x2 +y2 +z2.

1

r

(x,y,z) =

xr3

, y

r3, z

r3

F(t) = f(x+ht, y+ kt) (x, y) (h, k) f F(t), F(t), F(t).

f : R2 R m N

t

R,

(x, y)

R2f(tx,ty) =tmf(x, y). (I)

f m. f

xD1f(x, y) +yD2f(x, y) =mf(x, y),

t t= 1. f

h(x, y) =f(x+cy) +g(x cy),

f, g R R

c h

hxx 1c2

hyy = 0.


3/9

w= f(x, y) x= u+v y= u v, f C2 2w

uv =

2f

x2

2f

y2.

u(x , y, z, t) = f(t+r)

r +

g(t r)r

f, g

R R r2 =x2 + y2 + z2

uxx+uyy+uzz =utt.

f : R2 R fxx+ fyy = 0.

u(x, y) =f(x2 y2, 2xy)

(x, y) =f

x

x2 +y2,

y

x2 +y2

(r, ) R2 u = 2u

x2+

2u

y2 = 0

2u

r2 +

1

r22u

2 +

1

r

u

r = 0

uxx+uyy +uzz

x = r sin() cos(),

y = r sin() sin(),z = r cos().

f : R3 R3 f(x,y,z) =x +y +z Jf(x,y,z)

f : R3 R3

Jf(x,y,z)

f : R2 R2, g: R3 R2

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

g(u,v,w) = (u+ 2v2 + 3w3) + (2v u2) .

Jf(x, y), Jg(u,v,w)

h(u,v,w) =f[g(u,v,w)]

Jh(1, 1, 1)

f g f g


4/9

f(x,y,z) = (z log(x2y4), cos(y) + x, 1), g(x,y,z) = (xz3, 3yz,xyz) (0, 1, 1) f(x,y,z) = ( xy

xy , zxxz

), g(x,y,z) = (x + y+ z, x + y z, x y z) (1, 2, 3) f(x, y) = (2x, arctg(x+y), y7, xy), g(x,y,z) = ( 1

xy, 1z

) (1, 1, 1)

f(x, y) =ex sin y (0, 0)

f(x,y,z) =x2 3xz+z2 4xy+x4y2

(0, 0, 0)

f(x,y,z) =xyz2 (0, 1, 2)

A= 0,97

15,05 + 3

0,98

f(a+h)=f(a) + f(a) h f(x,y,z) = xy+ 3

z

a= (1, 15, 1).

A 0,2421726...

10 25 0,1

f, g U Rn

p U.

f+g

p. f g

f(x, y) =|x| + |y| (0, 0). f

(0, 0)

f(x, y) =x3 + 3x+y3 +y

f(x,y,z) =x5 +y5 +z3 + 4x+ 2y+ 9z+ 2

f(x,y,z) = arctan(x+ 2y+ 3z)

f(x1,...,xn) =a1x1+...+anxn+b, aj= 0

f(x, y) =ax2 + by2, a, b ab


5/9

f(x, y) = 3x4 4x2y+y2. (0, 0)

f(x, y) = (yx2)(y3x2) f

f : R+ R F : R2 R, F(x, y) = f(1 +x2 +y2)

G(x1,...,xn) =f(1 +x2

1+...+x2

n) Rn

.

f(x, y) =x2 (y 1)2; f(x, y) =x3 +y3;

f(x, y) =x3 +y3 3xy; f(x, y) = 2x2 xy 3y2 3x+ 7y;

f(x,y,z) =x2 +y2 + 3z2 1;

f(x,y,z) = (8x2

6xy+ 3y2)exp(2x+ 3y).

f(x, y) =(x21)2(x2y x1)2

9x2 + 36y2 + 4z2 = 36.

32000 cm3.

y x

x3 xy xy2 y3 1 = 0 x= 0 sin x+ cos y+ 2y = 0 x= 0

1 xy log(x2

+y2

) = 0

x= 0

x2y+ 3x2 2y2 2y= 0 x= 1 z+ x + (y + z)4 = 0 z=f(x, y).

fxx fyy .

g(x, y) = 0

y

x

y = w(x)

f : R2 R h(x) =f(x, w(x)).

h(x)

f

g.

2x2 + 3y2 z2 = 25 x2 +y2 = z2 C (

7, 3, 4) x y

z C z


6/9

T C P

C z

sin(x+y) + sin(y+z) = 1 z x y

z=f(x, y).

D1,2f

x

y

z

f C1 f x0 f. x0 f f(x)

f(x0) x

f : R2 R

f(x, y) =y4 ex2 + 2y2

ex +ex2

(0, 0) f f

y = f(x)

x2

+xy+y2

27 = 0 y x,

f(x, y) =xy(1 x2 y2) {(x, y) R2 / 0x1, 0y1} f(x, y) =x3/3 (3/2)x2 + 2x + y2 2y + 1

x= 0, y = 0, x+y = 1.

f(x, y) = (x2 +y2)2

2(x2 +y2)

{(x, y) / x2 +y2

4

}

f(x, y) =x2y3(1 x y) {(x, y) /|x| + |y| 1}. (2/5,3/5) f 216/3125. (1/2, 1/2) f 1/132

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

cos(y) R= [0, 2] [0, 2] f R 2 (

2, 0) (

2, 2) f R 2

( 32 , )

x,y, z

0 1

6/18,

6/6

6/3.

6/18,

6/6

6/3


7/9

x2+y2 = 1 x+y +z= 0

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

f (1/2, 1/

2,2

2) (1/

2,1/

2, 2

2),

3. f (1/

2,1/

2, 0) (1/

2, 1/

2, 0)

(2, 3, 2) x 1 =(y+ 1) =z+ 1

258/3, (1/3, 1/3,7/3).

x2 xy+y2 z2 = 1yx2 +y2 = 1

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

x2 +y2 +z2 = 8

f(x, y) =x2 +xy+y2

D={(x, y) / x2 + y2 1}

f D

x y y= mx + h m h

(x1, y1), (x2, y2), , (xn, yn),

m h y =mx+h

di = yi (mxi+h) (xi, yi) m h

ni=1 d

2i ,

mni=1

xi+hn=ni=1

yi mn

i=1

x2i +hni=1

xi=n

i=1

xiyi

m h.

D(m, h) =ni=1

d2i =n

i=1

[yi (mxi+h)]2;


8/9

m h

D(m, h) =

n

i=1

2[yi

(mxi+h)](

xi),

n

i=1

2[yi

(mxi+h)](

1) = (0, 0) =

=n

i=1[xiyi (mx2i +xih)] = 0ni=1[yi+mxi+h)] = 0

=

=

mn

i=1 x2i +h

ni=1 xi =

ni=1 xiyi

mn

i=1 xi+hn = n

i=1 yi

= ()

HD(m, h) =

2n

i=1 x2i 2

ni=1 xi

2n

i=1 xi hn

detHD = 4

n n

i=1

x2i

ni=1

xi

2

detHD >0

|x, y| x y x, y Rn

|x, y| =ni=1 xiyi y = (1, 1, ..., 1) := 1

n

i=1

xi2

x

2

1

2

12=n

ni=1

xi

2n

ni=1

x2i x Rn

1

4

HD =n

ni=1

x2

i ni=1

xi2

>0 (

xi= 0)

(m, h) ()

Dmm(m, h) = 2n

i=1 x2i >0


9/9

x Rn

D

P(1, 1), Q(4, 2), R(2, 3)

2,9 m

1975 6421976 644

1977 6561978 6671979 6731980 6881981 6961982 6981983 7131984 7171985 7251986 742

1987 757

Mult. de Lagrange3 Ejercicios

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