ssoluciones derivada parcialesoluciones_derivadas parciales

7/25/2019 SSOluciones Derivada Parcialesoluciones_Derivadas parciales

1/10

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

f(x, y) =

xx2 +y2

f(x, y) = log

x +y

x y

f(x, y) = arctan

x+y

x y

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

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

f

x

(x, y) = 2xy3 sin(xy)

f

y

(x, y) = 2y cos(xy)xy2 sin(xy)

f

x(x, y) =

y2

(x2 +y2)3/2

f

y(x, y) = xy

(x2 +y2)3/2

f

x(x, y) =

2y

y2 x2 f

y(x, y) =

2x

x2 y2

f

x(x, y) = y

x2 +y2

f

y(x, y) =

x

x2 +y2

f

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

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

f

x(x, y) = 2x 2xy2 sin(x2 +y2)

f

y(x, y) = 2y cos(x2 +y2) 2y3 sin(x2 +y2)

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

xD1f(x, y) y D2f(x, y) = (x y) cos(x+y).

g(x,y,z) = cos(

x+y

2z )

xD1g(x,y,z) +y D2g(x,y,z) +z D3g(x,y,z) = 0.

D1f(x, y) =yexy + cos(x+y)

D2f(x, y) = xe

xy + cos(x+y),

xD1f(x, y) y D2f(x, y) =xyexy + cos(x+y)

yxexy + cos(x+y)

= (x y)cos(x+y).


2/10

D1g(x,y,z) = sin(x+y2z )1

2z

D2g(x,y,z) = sin(x+y2z )1

2z

D3g(x,y,z) = sin(x+y2z

)x+y

2z ,

xD1g(x,y,z) +y D2g(x,y,z) +z D3g(x,y,z) =

x

2z y

2z+x+y

2z

sin(x+y

2z ) = 0.

f(x, y) = x2

y2

(1, 1)

60

OX

f

f

x(x, y) = 2x

f

y(x, y) = 2y.

(1, 1)

f

(1, 1)

Duf(1, 1) = f(1, 1) u u

60

OX u = (cos60 , sin 60 ) = (12 ,32 )

Duf(1, 1) = (2,2) (12 ,32

) = 1

3.

f(x, y) =x2xy+ y2

(1, 1)

OX

f

f

x(x, y) = 2x y

f

y(x, y) = x+ 2y

(1, 1)

f

(1, 1)

Duf(1, 1) = f(1, 1) u

u

OX

u= (cos, sin)

u

Duf(1, 1) = f(1, 1) u= f(1, 1) cos(f(1, 1), u).


3/10

Duf(1, 1) cos(f(1, 1), u) = 1 f(1, 1) u.

Duf(1, 1) cos(f(1, 1), u) = 0 f(1, 1) u.

Duf(1, 1) cos(f(1, 1), u) = 1 f(1, 1) u.

f(1, 1) = (1, 1)

Duf(1, 1) (1, 1) u u= ( 12 , 12)

Duf(1, 1) (1, 1) u u = ( 12 , 12) u= ( 1

2, 1

2)

Duf(1, 1) (1, 1) u u= ( 12 , 12)

f(x,y,z) =x3 y3 3xy(x y) + ez

(0, 0, 0)

f

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

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

z =ez.

f

(0,

0,

0) =f

x (0,

0,

0), f

y (0,

0,

0), f

z (0,

0,

0) f(0, 0, 0) = (0, 0, 1)

(x, y)

10

T(x, y) = 100 (x2 +y2)

(4, 3)

(4, 3)

T(x, y) = (2x,2y)

u =T(4, 3) =

(8,6)

(4, 3)

T(4, 3)

=

(

8,

6)

= (

8)2 + (

6)2 = 10


4/10

Q

x

SO4H2 y

H2O Q=

aybx+y

a

b

x

y

x

x

Q

y(x, y) =

abx

(bx+y)2.

Q

x

(x, y) =

aby

(bx+y)2.

dQ = Q

x(x0, y0)dx+

Qy

(x0, y0)dy

(x0, y0) dx

dy

dx= 0, 05x0

dy = 0, 1y0

dQ= 0, 05aby0x0(bx0+ y0)2

+ 0, 1abx0y0(bx0+ y0)2

=0, 05aby0x0(bx0+ y0)2

.

x0 = 10y0

dQ=0, 05aby010y0(10by0+ y0)2

= 0, 5aby20y20(10b+ 1)

2 = ab

2(10b+ 1)2.

f(x, t) = f1(x+ at) + f2(x at) f1 f2

a

D22f(x, t) =a2D11f(x, t)


5/10

D2f(x, t) =

t

f1(x+at) +f2(x at)

= f1(x+at) a+f2(x at) (a) =

= D22f(x, t) = t

a f1(x+at) a f2(x at)

=a f1 (x+at) a a f2 (x at) (a)=a2

f1 (x+at) +f

2 (x at)

.

D1f(x, t) =

x

f1(x+at) +f2(x at)

= f1(x+at) 1 +f2(x at) 1 =

= D11f(x, t) = x

f1(x+at) +f

2(x at)

= f1 (x+at) 1 +f2 (x at) 1.

D

22f

(x, t

) =a2D

11f

(x, t

)

f(x, y) = alog(x2 +y2) + b

f :=D11f(x, y) +D22f(x, y) = 0.

D1f(x, y) =

xa log(x2 +y2) +b= a

2x

x2

+y2

=

=D11f(x, y) =

x

a

2x

x2 +y2

= a2(x2 +y2) 2x 2x

(x2 +y2)2 =a

2(y2 x2)(x2 +y2)2

D2f(x, y) =

y

a log(x2 +y2) + b

= a

2y

x2 +y2 =

=D22f(x, y) = y

a

2y

x2 +y2

= a2(x2 +y2) 2y 2y

(x2 +y2)2 = a

2(x2 y2)(x2 +y2)2

,

f :=D11f(x, y) +D22f(x, y) =a 2(y2 x2)

(x2 +y2)2 +a2(x2 y2)

(x2 +y2)2 = 0.

f(x, y) =x3y xy3

g(x, y) =x2 +y2 x

h(x,y,z) =xyz3 4x2y2z.


6/10

f(t x) =f(tx, ty) = (tx)3(ty)

(tx)(ty)3 =t4(x3yxy3) =t4 f(x, y) = t4 f(x),

f

f

R2

4 f(x) = f(x) x

f

x(x, y) = 3x2y y3

f

y(x, y) =x3 3xy2,

f(x, y) = (3x2y y3, x3 3xy2)

f(x)

x= (3x2y

y3, x3

3xy2)

(x, y)

= 3x3y xy3 +x3y 3xy3= 4(x3y xy3) = 4 f(x).

g

R2

g

g

x(x, y) = 2x 1

g

y = 2y,

g(x, y) = (2x 1, 2y)

g(x)

x= (2x

1, 2y)

(x, y)

= 2x2 x+ 2y2=r g(x),

r R

r g(x) = rx2 +r2y rx

g

h(t x) = h(tx, ty, tz) = (tx)(ty)(tz)3 4(tx)2(ty)2(tz) =t5(xyz3 x2y2z)= t5 h(x,y,z) =t5 h(x),

h

h

R3

5 h(x) = h(x) x

h

x(x,y,z) =yz3 8xy2z

h

y(x,y,z) =xz3 8x2yz

h

z(x,y,z) = 3xyz2 4x2y2,


7/10

h(x,y,z ) = (yz3 8xy2z,xz3 8x2yz, 3xyz2 4x2y2)

h(x) x= (yz3 8xy2z,xz3 8x2yz, 3xyz2 4x2y2) (x,y,z)=xyz3

8x2y2z+ xyz3

8x2y2z+ 3xyz3

4x2y2z

= 5xyz3 20x2y2z= 5(xyz3 4x2y2z) = 5h(x).

F(x, y) = x2y xy2

F(x, y) = 0

y

x

(1, 1)

y=y(x)

H1 F R

2

F

F

x(x, y) = 2xy y2 F

y(x, y) = x2 2xy,

H2 F(1, 1) = 1

2 1 1 12 = 0.

H3 Fy

(1, 1) =x2 2xy|(1,1) = 1 = 0

(1, 1) F(x, y) = 0

y

x

y = y(x)

x2 y(x) x y(x)2 = 0.

x

2x y(x) +x2 y(x) [y(x) + x 2y(x) y(x)] = 0.

x= 1

2y(1) +y(1) [y(1) + 2y(1) y(1)] = 0,

y(1) = 1

2 +y(1) 1 2y(1) = 0

y(1) = 1

F(x,y,z) = x3 + 2y2 z2

F(x,y,z) = 0

z

x y

(1, 0, 1)

z=z(x, y)

(1, 0)


8/10

H1 F R

3

F

F

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

F

y(x,y,z ) = 4y,

F

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

H2 F(1, 0, 1) = 13 + 2 02 12 = 0.

H3 F

z(1, 0, 1) = 2z|(1,0,1)= 2 = 0

(1, 0, 1)

F(x,y,z ) = 0

z

x

y

z= z(x, y)

x3 + 2y2 z(x, y)2 = 0.

x

3x2 + 0 2z(x, y) zx

(x, y) = 0.

(x, y) = (1, 0)

3 2z(1, 0) zx

(1, 0) = 0,

z(1, 0) = 1

z

x (1, 0) =

3

2

y

4y 2z(x, y) zy

(x, y) = 0.

(x, y) = (1, 0)

0 2z(1, 0) zx

(1, 0) = 0,

z(1, 0) = 1

z

y(1, 0) = 0

F(x,y,z ) = C

z

x

y

P(x0, y0, z0) C z=z(x, y)

(x0, y0)

F(x,y,z) = log z x

2y

z P(1, 1, 1)

F(x,y,z) = log

10x

y+ z

100

P(10, 2, 200)


9/10

G(x,y,z) :=F(x,y,z )C= log z x2y

zC

G

H1 G G

G

x(x,y,z ) = 2xy

y , G

y(x,y,z) = x

2

z , G

z(x,y,z ) =

1

z+x2y

z2 ,

H2 G(1, 1, 1) = log 1 12 11 C= 0 C= 1

H3 G

z(1, 0, 1) = 1

z+ x

2yz2 (1,0,1)

= 2 = 0

C =1

(1, 1, 1) G(x,y,z ) = 0

F(x,y,z ) =1

z

x y

z=z(x, y)

log z(x, y) x2y

z(x, y)= 1.

x

1

z(x, y)z

x

(x, y)

2xy z(x, y) x2y z

x(x, y)

z(x, y)2 = 0.

(x, y) = (1, 1)

1

z(1, 1) zx

(1, 1) 2 z(1, 1) zx

(1, 1)

z(1, 1)2 = 0,

z(1, 1) = 1

z

x(1, 1) = 1

y

1

z(x, y)z

y (x, y) x2

z(x, y)

x2y

zy

(x, y)

z(x, y)2 = 0.

(x, y) = (1, 1)

1

z(1, 1)zy

(1, 1) z(1, 1) z

y(x, y)

z(1, 1)2 = 0,

z(1, 1) = 1

z

y(1, 0) =

1

2

G(x,y,z ) := F(x,y,z)

C = log 10xy + z100

C

G


10/10

H1 G G

G

x(x,y,z) =

1

x, G

y(x,y,z ) = 1

y, G

z(x,y,z) =

1

100,

H2 G(10, 2, 200) = log

10102

+ 200

100C= 0 C= log(50) + 2

H3 G

z(1, 0, 1) = 1

z+ x

2yz2

(1,0,1)

= 2 = 0

C= log(50) + 2

(1, 2, 200)

G(x,y,z) = 0

F(x,y,z) = log(50) + 2

z

x

y

z=z(x, y)

log

10x

y

+z(x, y)

100 = log(50) + 2.

x

log

10x

y

+z(x, y)

100 = 0.

(x, y) = (1, 1)

1

z(1, 1) zx

(1, 1) 2 z(1, 1) zx

(1, 1)

z(1, 1)2 = 0,

z(1, 1) = 1

z

x(1, 1) = 1

y

1

z(x, y)zy

(x, y) x2 z(x, y) x2y z

y(x, y)

z(x, y)2 = 0.

(x, y) = (1, 1)

1z(1, 1)zy (1, 1)

z(1, 1)

z

y(x, y)

z(1, 1)2 = 0,

z(1, 1) = 1

z

y(1, 0) =

1

2

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