Derivadas parciales

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∆& ' → ,

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∆& ' → ,

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! ! ! % &! ! ! ! &! ( ) ! ! ! ! . /#% ( > restart: f:=(x,y)->x^4-3*x^2*y+3*y-y^3; print(`DERIVADAS PARCIALES`); fx:=D[1](f); fy:=D[2](f); *+ *+ *+ 0 ! ! ) →
,

! !

& ! !

! &

! ! ' &

!! !

−→,

.

+- − −/
.-

.

→ −! !

+-−-

# + $ 1$

> fx_p:=D[1](f)(1,-1); fy_p:=D[2](f)(1,-1); *+ $2 *+ 1. /#% 0( > restart: f:=(x,y)->exp(x-y)+sin(x+2*y); print(`DERIVADAS PARCIALES y su valor en (0,0)`); fx:=D[1](f); fy:=D[2](f); fx_p:=D[1](f)(0,0); fy_p:=D[2](f)(0,0);

*+ *+ *+

→ → →−

−

+ + +.

+. +. +.

− −

*+ .
#

*+ $
.

. /#%

1(

3

&!

)

!

> restart:f:=(x,y,z)->x^2*z-2*z^2*y+x^2*y^3; print(`DERIVADAS PARCIALES y su valor en (1,2,3)`); fx:=D[1](f); fy:=D[2](f); fz:=D[3](f); fx_p:=D[1](f)(1,2,3); fy_p:=D[2](f)(1,2,3); fz_p:=D[3](f)(1,2,3);

*+ *+ *+ *+

→

.

−.

.

+

.

-

! →. → −. → *+ 1/ *+ 1.. 2 & ')
0 . .

+. +.

.

−,

*+ ..

−

0

+

3 & ' > restart: f:=(x,y)->x^2*y-y^2*x;fx:=Limit((f(x+h,y)-f(x,y))/h,h=0)=limit((f(x+h,y)-f(x,y))/h,h=0 ); fy:=Limit((f(x,y+h)-f(x,y))/h,h=0)=limit((f(x,y+h)-f(x,y))/h,h=0 ); fx_p:=subs(x=1,y=1,fx); fy_p:=subs(x=1,y=1,fy); *+ *+ !
"→2 .

→
.

.

−
.

.

+"

.

−

+" − " +" "
.

+ +

.

=
.

. − −.

*+ !
"→2

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−
.

.

= =$ = 1$

*+ !
"→2

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−$−"
.

" "− $+" "
# -

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"→2

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!

%

&!

!! . 2 0( * +*

& ' > restart: f:=(x,y)->PIECEWISE([m*(abs(x)+abs(y))/sqrt(x^2+y^2),(x,y)(0,0 )],[n,(x,y)=(0,0)]); # *+ +∆ ∆ ∆ ∆ #= ! ! &
.

+
.

→

+

.

≠ 2 2 2+∆ = 2 2 2 − 2 2 ∆

+ !
∆ →2

−

+4

22 + !
∆ →2

+

# ∆

− + !
∆ →2

!
∆ →2

#− ∆ #≠

+4 5 & ! #= ! ! ! 3 ! !

+2

=∞ !

0

&

! ! ! !

! !

!

" !

' '

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0

! !
0

! & ! !

! 3 !* &

! &! ')

4 − & − 0' − & − 1'

> restart: with(plots): with(plottools): > f:=(x,y)->6-((x-2)^2+(y-3)^2); *+ →/− −.
.

−

−-

.

% # ! & ) & ! ' ' 6 7% 1. 6 6 / ! ! ! & ! ! > Superficie:=plot3d([x,y,f(x,y)],x=-2..5,y=-2..6, grid=[30,30],shading=ZHUE, axes=none,linestyle=2, , #

! !

1. 6 3

orientation=[60,80]): display(Superficie);

' !

!

! ' 8 # !8 !! '

!"

!

!

& )

!

!

!

> Superficie:=plot3d([x,y,f(x,y)],x=-1..5,y=-sqrt(9-(x-2)^2)+3..sq rt(9-(x-2)^2)+3, grid=[40,40],shading=ZHUE, axes=normal,linestyle=2,color=aquamarine, orientation=[-45,80]): Vista_1:=display(Superficie): display(Vista_1);

#

7

9

&! ! ! ! ! ! ! $%" & '( ! 8 ) 8 & ! > Otramanera:=view=[-1..5,0..6,0..12],axes=none,labels=[x,y,z], tickmarks=[5,5,5],orientation=[-45,80]: Superficie:=plot3d([x,y,f(x,y)],x=-1..5,y=-sqrt(9-(x-2)^2)+3..sq rt(9-(x-2)^2)+3, grid=[40,40], style=PATCHNOGRID): display(Superficie,Otramanera,axes=normal);

#

/

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!

& )

&

!

! ! ! 5' " ! ! !

)

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! '

+ -.,

(

" ! ! ! # ! ! ! ! ! =2 ' ! =. > PX:=plot3d([x,2,z],x=-1..5,z=0..9,linestyle=2,style=wireframe,color=blue):Etiqueta1:=textplot3d([5,2,0,` plano Y = constante`],color =black): display({Superficie,PX,Hx,Etiqueta1},Otramanera,axes=normal); Curva_x:=[x,2,f(x,2)]: Etiqueta:=textplot3d([3,2,4,` A`],color =black): Tra_x:=spacecurve(Curva_x,x=-1..5,axes=none,thickness=2,color=or ange,axes=normal): Hx:=spacecurve([x,2,0],x=-2..6,thickness=2,color=red,axes=normal ): display({Superficie,Tra_x,Hx,Etiqueta},Otramanera);

#

:& ' ! " ! !

! ! !

=. !

! !

! ! ;

" ! ! ' > plot([x,f(x,2),x=-1..5],view=[-1..5,0..6],labels=[x,z],color=red );

#

<

> diff(Curva_x,x); Vector_directorx:=subs(x=3,%); Recta_tangente_x:=[3,2,f(3,2)]+t*Vector_directorx; Tangente_x:=spacecurve(evalm(Recta_tangente_x),t=-5..5,axes=norm al,thickness=2,color=navy): display({Tra_x,Hx,Tangente_x},Otramanera,Etiqueta,axes=normal);...