fundamentos app scilab - Facultad - FCEIAericah/Soft/fundamentos... · 2013. 3. 9. · .e 0 #...
Transcript of fundamentos app scilab - Facultad - FCEIAericah/Soft/fundamentos... · 2013. 3. 9. · .e 0 #...
©
! " # ! $ % &!' % ()* +()*,&%- ' . !/! ' ! , 0 ,!1 ,*)&23)45 &3 6%7 8 %665)3 * ! 9""000!""
.. 6 6%7 ()*"%6)*: .
. 6 6%7 ;6)&5; <
. ()*"%6)*: =
6%7 26>55,4 ;6)&5; ?
< 6 6%7 ?
<. &' @
<.. , @
<. @
<.< & A
<. ( ;0 A
<.= B A
523)C)&5 5) 6%7 A
. A
A
< 5 7D E
<. ! .
< > .
<< ! .
< &' .
% !' F ex ..
= % ..
? 3 .
@ ,*)65)3 4>6(5)52G4>6 .
A 5)4)43 336%3 .<
E 345>3 .
E. 345> ,6% .=
E 345> 5%*2) .=
E< 4! .@
. 24>63 .@
.. 5%6)5265 .A
. 3%75>6H) &3 (>I,65 7I65 .E
.. %54 .E
. %54& .
.< 2)3J5 &3 ,*)65)3
<. I%(37> %6)3% =
<.. 345>3 =
<... 345> ,6% &3 ) 525)3)43 =
<.. 345> 5%*2) &3 ) 525)3)43 ?
<..< 525)3)43 &3 *) 345> ?
<.. 345> 3>5 ?
<..= 345> *)5 @
<..? &66H) @
<..@ 2*%46%66H) 5> *) 3%> A
<..A >5&*45 *)45 E
<..E >5&*45 >*C E
<... )5>2 &3 *) 345> <
<.... 345> *)64>65 <
<... 566H) &3 *) 525)3)43 <
<...< 2765 &3 *) 525)3)43 <.
<... 4>)*34 &3 *) 345> <.
< 24>63 <
<. 525)3)43 &3 *) 24>6C <
< 53>65)3 7I69 <
<. 4>)*34 &3 *) 24>6C <
< &66H) 8 *4>6H) <
<< 2*%46%66H) 5> *) 3%> <=
< 2*%46%66H) &3 24>63 <=
<= 24>6C 6)3> <?
<? 24>6C 6&3)46&&
<@ 42K5 &3 *) 345> 5 *) 24>6C .
<A 3:4>6H) 8 2765 &3 3%323)45 &3*) 24>6C .
<E (3)3>6H) &3 24>63 5) )L23>5%345>65
<. >)(5 &3 *) 24>6C
<.. &343>26))43 &3 *) 24>6C <
<. 24>6C 6)(*%> <
<.< %5>3 >565 345>3 >565
<< 5%*6H) &3 3*65)3 %6)3%3 =
< 7 $ ' ! ! A
<< 5%6)5265 E
<<. 5)4>*6H) &3 5%6)5265 E
<< >M3 &3 *) 5%6)5265 =
<<< 3%*6H) &3 *) 5%6)5265 =.
<< &666H) &3 5%6)5265 =
<<= &3>6& &3 *) 5%6)5265 =<
< ,*)65)3 ==
<= )L23>5 52%3J5 =@
<=. 3 $! ! $ ?
<= ! $ ?<
<=< ! ' $ ?
<= % !' F $ ?
<? 3*65)3 &6,3>3)6%3 5>&6)>6 +3&5- ?A
. (>I,65 5) @.
D# @
< 52)&5 &3 75>>&5 @@
<. F @@
52)&5 F+- @@
= 3% 52)&5 +- @A
=. 4>7J)&5 5) >(*23)45 A
=.. ! A
=. ! A?
=.< ! AA
= 23J5>)&5 % 6*%6C6H) 5) N+-N AE
=< >67%3 L46%3 *)&5 +-9 E
=<. OPFQ E
=< RS E
=<< FR R E.
= 346T*34 E<
= . F $ D# E<
= ) /! ! E=
? %54& E?
@ %54&< E@
A %54& EA
E .
E. .
E >65 345>3 ..
. (>I,65 3) <& 5) %54<& .<
.. 3 >(*23)45 ONN .=
=. 6)4>5&*6H) .@
=.. 3% 3&645> &3 6%7 .@
=. 3J3*4> *) >64 5) F+- .A
=.. * ()*"%6)*: .A
=. * 0 2 ;6)&5; © ...
= B3>>263)4 &3 >5(>26H) ...
=. 5 %' ...
= 0 ...
=< ..=
= * #' ! ..?
= . 3!! ! !' ..?
?. ,6%4>&5 8 >3>33)46H)&3 ,*)65)3 &3 4>),3>3)6 ..E
?.. % !'9 U0#+-U ! # ..E
?. % !'9 U+-U # ! .
?.< % !'9 US+-U ! ! .
? >' 8 .
?< & 7 .
? >3>33)46H) &3 6432 %6)3%3 .?
?= 6432 5)34&5 .@
?? %%6)5 76&623)65)% .<.
?@ 5)4>5%&5>3 6 & .<
6%7 ! F' D 24%7 © 6)>6V* >/!! W .EE ' !' W D #X 0 (%
! ! ' Y' .EE 6 *>% 9""000V/" 9""000 ! /! V ! 0 ! ! #
* X ! F 9
3 $ $ ! ! D
! ! D
! ' ! ! ! $ /! $!
% #X 6%7 9
$ ! ! F
4 ! ' (% D !
X F 9 W D
% ' /! ' #!' 0 ! /! ! ! %6)*: 00 % ' ' ! /!F .< 2 , ! $!' 4 F ! ' ! '! ! W F 2 ! $!
3 ! $ ! 6%7 ! X /! ! ! 0 % ' Y 6%7 <
! ' ()*"%6)*: D /! !F ;0 #! !:# ! ! ( % ' %43: ! %8: %6)*: ,.
& F ! /! ' 6 6D %! ! 6 %! (! ('Y ! X 6%7 ! ! ! 2! J>Y ! X! ! ! ; W D! /! (F 5&&( ! %) $ ! ! ! D /!X # !
& ' /! ' <
F 6%7 %6)*: /! ! ! # ! ! +-
! "#$%& 3 /! !!
X ' '1 F Z D /! ' 6%7
! '%("% ! ! /! ! /! X /! !
$ ' < /! F 0 6%7 ! 9
V< # %!F
V< ! +Y #-
3 ! V< !FV?A?Y % ' ! ! !!
.
! F Z V F ! /!!' ! ! 9 9
!
! V<
"
! Y $ 6%7 X /! $!' ! ! ! ! !! 3 ! ! ! 9
#$%&#$%'%()*'%()*
"!#$% & ' ( & )
$! 6%7 ! ! * +,- F '.
! )*+(,+-"#.
' < ! $ ! /! ! 9 9
!
! V<V>
!
<
' # ! ! ! ' +>3&23R*F- /! D
! !! ! $!9
"
$ 6%7 ! ()*"%6)*: % ' .F $!
% ' 5 ;6)&5; $! 6%7 ;6)&5; ! ! -.- Z & ! Z ' / /! ! #! ..
,! ..9 ! %
! Z ' 0 ! ' / /! ! #! . /! ! /! 6%7 /!D
,! .9 ' ! '
! ! /! ! ! 12& '23-.- ! Z /
&! /! ! /! /!! ! ' !/ D ! /! /! D $! 6%7 /! 6%7 /!D
% ! ! ! /! ! ' Z / '
! /! ! /! /! *%$% 6%7 ! ' ' ! 6%7 ;6)&5;
Z , ! $! 6%7 ;6)&5; F '
!"#$
&! 6%7 ! $! 9 D# ()523 [&3 * +,- %4 ' ! * + $ ()523 * + [&3 ' ! Z ' %4 ! #! .
,! .9 $!'
$! 6%7 ! F !9
4 ! $! ! D X ! %6)*: ! 9
=
?
5 " ' ' *0/67
! D# 6%7 ! $! F ! 9
+
3 D 8 # NV00N
%& '(
2 ;6)&5; ' 6%7 ! +,!- Z $! 6%7 0/010# 9#:9$3 ! 3
,! 9 6 ;6)&5;
) *#*
&! $! 6%7 ! V Y , X +,- /! 6%7 $!
&&&&&&&&&&
-.
/0! 1/2 34 4- 565*#/
&&&&&&&&&&
@
7 '
!
,
% ! $! % * S Y/! ! ! /! S $ !Y F ! 1 9* $9 ! 1 9*+ 1 9*+ ! ! # 1 9*+ ! ! $! ! 6%7
% 6%7 ! = ! #! <
,! <9 7
Z ' ' , 1 ;< : ='8 '! ! /! ! ! ' ' ! ' /! /! $! '
%/+
V$ , #9 ! 6%7 $! /! !
V>9 ! 6%7
V?9 6%7
(," /
V99 ! ! /! V
V9 ! ! ! ! V
A
V99
V39 !
+0
! /! ! 6%7
"1$%# 2%(3 4
! D# F ! '
5+/1
Z ! ! ! !! /!W ! /!
+ %!,
3 ' ! ! !' !! ! FX ! F $ 6%7 $ $ Y$ /! ! D
3 6%7 ' /! ' /! !1 F /! ! $! ! "" X9
,'
! ! /! + &- ! /! ! @ %
E
, 8
,&8
&
8
3%#"%0%( 067#*/ 6 0%(7#*/
,&8
&
8
,&.
&
.
6%7 ! ! ! /! ! A@ 9
&
8
! /! /! ! ! BCD # /! /! $!
,&89
,
! $ /! ! /! ' %
,
&
.
/! !
.
*0
,:
&
3-
+,
,
&
-
" 3*#,
,;
&
8
%&%%8(
,
&
.3<- 8.
5 ' ! Y 6%7 !!
= -.
0
! ! ! X BE\ BFF\ X9
,>
&
3-8
,;;-
&
..
<4
XY ! ! ! ) ! ! /! ! Y '% /! ! ! $ ! ! ] \ ! ! X9
,&;;-9
XY ! ! )9
,?12
&
.
ex
! F ! ! 9
,12
! ! ! ! Y $
* ! . . !
,!1-.3 - 3 2
&
3
,!3132
&
,!-13-<2
&
3
.
! " # $
! ! 9
3$9
,&3
&
3
,
&! ! /! F 6%7! ! 9
,
= <
'
% &'()("* +,)-(./+,)0*
6%7 ! 4 ! /! ) ! /! !' ! ! ]+-\ ! !'
6%7 ! X /! ! D! $ = D /! V' $9
a = 45o·π180o
! 6%79
,&1<8; @23
&
. 84 -
,12
&
..3 A BA√
22
3 ! ! ! !9
.<
*(#%8( 0(3 9+01/ ,C7&12
,C6&12
,C/&12
,C&12
,C$&12
,C&12
,C/!&!12
! .9 ! 6%7
1 (*+0(+"* "*")0"*
6%7 Y X $+G- ! # 6%7 9G )HIJ XY! V.+
√−1- ! $ /! G +π- + ! <. .=- # ! ! G +! @.A- ! !9
, @
@ &
3<384-.
, @
@ &
-.3 - 3
,?132
&
5 9
G ! ! /!W G9
, @
@ &
---*3
3$9
,D&3: @
D &
3
.
G ! !
, @
@ &
5
X ! /!W ! ! ! ! !! /!W
,&3 @
&
G # N ! N + ! - !/! ! /
,& @: @
&
G G
,7& @
7 &
$
,& @
&
E
2 3"+,"*
X X ! /! !Y '
! ! 6%7 ! ! ! ! ' ! # ! D ! ]P Q\9
) &OP Q
.=
:
/! ! # ! ]\ X9
,CE3&F- < G
CE3 &
= - < =
* 9
,CE-&F8A @A @G
CE- &
= 8 3<384-. -.3 - 3 =
3 ! $ ! ! $ !
5 # # /! ! ! !
,C&3'3
C &
= 3 - < 8 . 4 3 =
X /! ! ! ! V # # X9
,C<&3'-'3
C< &
= 3 8 . 4 =
,C8&3' '3
C8 &
= 3 3 - < <- 8 8 .< - 4 4 =
:
! ! X ! NON ! NPN ! 3 ! !
.?
NQN
,C/&F3
,-
,
,<
,8
,G
C/ &
= 3 =
= - =
= =
= < =
= 8 =
= =
% ! ! ! ! N1N X9
,C/-&F 9 8 ; @9 G
C/- &
= =
= 8 =
= =
5 ! ! ! # !9
,CE&3'8'8
CE &
= 3 38 - -8 8 < <8 8 =
,C/&CEH
C/ & = 3 =
= 38 =
= - =
= -8 =
= =
= 8 =
.@
= < =
= <8 =
= 8 =
"(1*+,
! ! F' ! B K D /! F'9
,*&F @ -; @G
* &
= 3<384-. - 3 8 =
,*H
&
= =
= 3<384-. =
= - 3 8 =
4 .0+,)"*
2Y9 ! Y : ! ! ! ! $ Y 79
B =
4 8 9 6
5 2 1 73 5 7 9
3 6%7 ! Y ! !Y ! D9
Y X ! LAL ! ' #' 2YLM NL
% Y /! ! 9 # D ! ! # !Y ! N1N
3$ Y 7 6%7 ! 9
,D&F< 4 98 - 3 .9 8 . 4G
D &
= < 4 =
= 8 - 3 . =
= 8 . 4 =
.A
9
,D&F<A A4A98A-A3A.9A8A.A4G
D &
= < 4 =
= 8 - 3 . =
= 8 . 4 =
# 9
< 4
! ! Y ! $! ! 9
&
,&F3 - G
&
= 3 - =
& (
,0&F< 4G
0 &
= < 4 =
&
,&F 8 -G
&
= 8 - =
&
,&F909G
&
= 3 - =
= < 4 =
= 8 - =
4 ( Y
)(.)*
3 $ 6%7 # /! !' !' # /! /! ! X
% F !9
.E
0 # +;#%+(,+
X ! LAL ! NN LHL # ! LM NL! L<L ! L<L LL /! # # N-N
3$9
,#3&01F3 8 .GAHHAHH2
#3 &
3 : : 82 : .3
# O I @ .
0 "<=
X ! LAL< ! L(L< LH L< X ! LM NL ! L<L LJL
3$9
,&01F3 8 .GAHH2
&
38 3. : 2 33 : 4
X OI @ . #O I@ I. I I
"0,0)5( " -,6&)* 6*)*
# 0(3 '>%#
1/ ,? 1/ ,3
3 D# ! 9
1A 0A I J AI J 0IA I !KI2
( W /! D# Y 9 (
/# ' ) O O ( P 'O
3$9
,&F3'3'3G9
,1A 12AIL 3 3IAI IAI/(7*(I2
1 2 3 4 5 6 7 8 9 10−1.0
−0.8
−0.6
−0.4
−0.2
0.0
0.2
0.4
0.6
0.8
1.0
COSENO
EJE X
coseno de x
,! 9 (D# !' !
! ! ! !9
,&F3'3'3G9
,1A 12AI*M* NIAI IAI/(7*(I2
.
1 2 3 4 5 6 7 8 9 10−1.0
−0.8
−0.6
−0.4
−0.2
0.0
0.2
0.4
0.6
0.8
1.0
COSENO
EJE X
coseno de x
,! =9 (D# !' D
3 D# $ X ! 9
-1 A A I!I2
3$9
,&F'8'G9 ,-1A1-;2A&F<A33A<A33G2
−6 −4 −2 0 2 4 6−2.0
−1.6
−1.2
−0.8
−0.4
0.0
0.4
0.8
1.2
1.6
2.0
,! ?9 (D#
5 D#9
0 1 2 3 4 5−1.100
−0.733
−0.367
0.000
0.367
0.733
1.100
∆
∆
∆∆ ∆
∆
∆
∆
∆
∆
∆∆ ∆
∆
∆
∆
∆
∆
∆∆ ∆
∆
∆
∆
∆
∆
0 1 2 3 4 5−1.100
−0.733
−0.367
0.000
0.367
0.733
1.100
0 1 2 3 4 5 6 7 8 9 10−0.10−0.08−0.06−0.04−0.020.000.020.040.060.080.10
plot2d−Escala Logaritmica
0 1 2 3 4 5 6 7 8 9 10−1.0
−0.8
−0.6
−0.4
−0.2
0.0
0.2
0.4
0.6
0.8
1.0
+
+
+
+++
+
+
+
+
+++
+
+
+
+
+
+++
+
+
+
+
+
+++
+
+
+
+
+
+++
+
+
+
+
+
+++
+
+
+
+
+
+
,! @9 5 D#
.0("7 " &'()("*
!' ! !' !
<
/! !D !' 1 6%7 !' ! # !' D ! ! $9
!'9
nombrefuncion = sen(x) + cos(y) + xy2 −√48x
! 9
AQ
(A
,1HF6G&1A02HAH6&12:102:;0;;-?1< ;;2H2
,1<A2
&
--4.<
L9L LL L(L L L /! !'
) -!& !
X! ! /! ' ! 6%7 ! ! ! $ ! !Y X!
3 X! I % ! $ 6%7 /! D X! ! /! $ !
3"+,"*
$ X '! D ! ' ! ! F ! ! $ /! D ! ' /! ! ! ! ' !Y Y ! & !
:
# ! # ! $! ! 9
(x1, x2, x3 · · · xn)
=
?
3 6%79
,&F. -8 8 -- < G9
:
* ! ! $! ! 9
x1
x2
x3
xn
:
3 F. F ! ! 3 FZ ZV % ! ! 9 ! $
3 6%79
,&01AHH29
,&F3 G9
,&F3A ;A ;;-G
&
= - =
= 3 = AA b =(
1x , x
1 , 6x1 , x2
1
)= =
= 3 3 3 =
:
3 ! ! XV ! Y 6%7 X9
,13A2
@
&
= =
:
3 Y /! ! !U Y ! 9
,1'82
&
= 3 3 3 3 3 3 =
! !U
,J&F3 < .G9
,1J2
&
= 3 3 3 =
",0)("* ( 3"+,"*
9
a = [a1, a2, · · · , an]
b = [b1, b2, · · · , bn]
! + /! ! 9
a + b = [a1 + b1, a2 + b2, · · · , an + bn]
6%7 ! Y ! 9
&'
&' ' Y9
>!
3$9
A
,&F3'3'8G9
,&F< < . G9
,:
&
= 8 8 . 33 3 =
/! ! W !
,&F< 8 G
&
= < 8 =
,&F3989.G
&
= 3 =
= 8 =
= . =
,:
=
3 ! ! ! # ! !
3 ! ! ! R Z /! ! R9
ka = (ka1, ka2, · · · , kan)
3$9
,&F< 8 G
&
= < 8 =
,8
&
= - -8 =
E
@
3 ! ! F /! ! # ! ! ! ! !
* 6%7
^_OP A ?Q1
^_OP1?1AQ1
^_`
O
.
B /! ! /! ! ! ! ! ! 9
# ` ! = ! ` #
,;
& 3<
,;
&
= 3 3- =
= -< < =
= - < < =
A
3 ! !Y ! !' # ' ! '
3 6%79
,&F< 8 4G9
,&F. 3 -G9
,;
&
= - 8 3 =
<
:
! ‖a‖ ! D F 1 ! #' D 9
a = [a1, a2]‖a‖ =
√a2
1 + a22
!Y ! 9 LH&JL
3$9
,&F< 8 4G9
,12
&
33<83
:
! ! ! ! ! ! ! ! ! 8 ' !
3 6%79
,&F< 3 G9
,&1122
&
= . <<<8 34333 8 < < =
!Y & ! 9
,12
&
3
+,* ",0,"*
! ! ! ! ' ! 4 /! ! /! !
<.
a(jn)
LL ! << V= . ' HI@J !Y 9 &HMP <NJ
,&F8''3G9
,1382
&
/! ! ' ! .=
! ! ' /! /!
3$9
,&F3 . G9
,12&
&
= 3 =
$ ' /! X L.L LL
:
! ! ! # !X ! ! # ! !
! ! ! 9
3 6%79
,J&F -G9
,JH
&
= =
= =
<
= - =
) %(&!
* Y 7 ! ! ! ! 3 X L L '(ai1, ai2, · · · , ain) '
!
a1j
a2j
amj
! 4
.("(+"* " '(0 .0+,)8
% 4 7 aij /! # ! $ 7 !
7 ! Y A 7 Y ! * Y ! Y 3 W ! Y Y
6%7 ! Y ! ! ! # W ! LCL !LM NL
* 6%79
,)&F< .93 8 9. 8 G
) &
= < . =
= 3 8 =
= . 8 =
4 9
,&F<AA.G9 &F3A8AG9 &F.A8AG9
,)&F99G
) &
= < . =
= 3 8 =
<<
= . 8 =
% Y ! F D9
,&01AHH29
,&F<AA.G9 &F3;;-A8A;;-G9 &F.A8AG9
,)&F99G
) &
= -8 . =
= =
= 3 3 =
= =
= - =
= 8 =
= =
= -=
= 3 3 =
= =
= . 8 =
= =
= 3 3 3 =
$ Y9
M =
0,25x1
6x
7x1
x2
15x1
3x2
71
51
31
6! /! ! ! 9
,&F<AA.G9 &F3A8AG9 &F.A8AG9
,)13A2&
) &
= < =
= 3 8 =
= . 8 =
# . ! < N@N NN
<
",0)("* 6*)0*9
A
# ! ! # X Y $AM4N # 4 ! /! F $KAM4N 4 # !
3$ 6%79
,)
) &
= < . =
= 3 8 =
= . 8 =
! Y $ ! ! $K $L
,)H
&
= < 3 . =
= 8 8 =
= . =
B
! ! ! ! Y ! ' ! /! ! ! '
3$9
AM4N
SAM4N
1AMH+J4N
3 6%79
.
,&F<AA.G9 &F3A8AG9
,:
&
= 8 33 3 =
<=
,
&
= 3 < =
.
,&F @ -9 <. 8G9
,D&F- .<9 .4 G9
,D&:D
D &
= 8.<384-. 3 =
= 3- 38 =
% !' ! Y $ ! > ! ! Y, ! 9
Fnxm = kMnxm, donde fij = kmij
3$ 6%79
,)&F< 89 < . 9 3 .G9
,O&9
,E&O;)
E &
= 3- -< 38 =
= 3- -3 4 =
= -3 =
! 7 /! /! ! Y NN ! # ! Y N7N 3 !V ' ! Y ! ! # ! ! ! 7 ! D ! ! F'
<?
A2x3 =[
a11 a12 a13
a21 a22 a23
]B3x2 =
b11 b12
b21 b22
b31 b32
B `7
C2x2 = A∗B =
a11 ∗ b11 + a12 ∗ a21 + a13 ∗ a31 a11 ∗ b12 + a12 ∗ b22 + a13 ∗ b32
a21 ∗ b11 + a22 ∗ b21 + a23 ∗ b31 a21 ∗ b12 + a22 ∗ b22 + a23 ∗ b32
* 6%7
.
,&F< 89 < . G9
,D&F. 9 < 9 < G9
,;D
&
= 48 =
= =
,&F< 8G9
,D&F. 9 < 9 < G9
,;D
&
= 48 =
A :
/! ! Y ! ! ! 9
` T! Y ! W A
` T! 3F ! Y S /! 9
FSASFA0 0 Y 4 Y S
Y S A−1 5 /! ' 1 S S
<@
3$ ! ! 9
A =
4 5 9
7 4 38 6 3
A−1 = 1
det A(Cij)T
& (Cij)T Y ! Cij = (−1)i+j ∗ Mij
Mij Y
! 6%7 Y 9
,&F< 8 49 . < 9 G9
&
= < 8 4 =
= . < =
= =
,&12
&
3
9
,33&F< 9 G
33 &
= < =
= =
,3-&F. 9 G
3- &
= . =
= =
,3&F. <9 G
3 &
= . < =
= =
,-3&F8 49 G
-3 &
= 8 4 =
= =
,--&F< 49 G
<A
-- &
= < 4 =
= =
,-&F< 89 G
- &
= < 8 =
= =
,3&F8 49 < G
3 &
= 8 4 =
= < =
,-&F< 49 . G
- &
= < 4 =
= . =
,&F< 89 . <G
&
= < 8 =
= . < =
4 !
Cij (Cij)T
* 6%7 Cij
,33&1332;132;;-
33 &
,3-&13-2;132;;
3- &
,3&132;132;;<
3 &
3
,-3&1-32;132;;
-3 &
<E
4
,--&1--2;132;;<
-- &
,-&1-2;132;;8
- &
3
,3&132;132;;<
3 &
-3
,-&1-2;132;;8
- &
83
,&12;132;;
&
34
/J&F33 3- 39 -3 -- -9 3 - G
/J &
= 3 =
= 4 3 =
= -3 83 34 =
* 6%7 $ (Cij)T
,/J$&/JH
/J$ &
= 4 -3 =
= 83 =
= 3 3 34 =
* 6%7 (Cij )T
det A MainvA = CijTde 9
,)&/J$
) &
= .<.<3 < 3< 38 -84-84 =
= .. .<.<. -4-4 =
= 3-<8 34.84 -<8.4 =
3 6%7 Y ! ! !Y !' L&L $ 9
L&H ' JL
3$9
,&F< 8 49. < 9 G9
,D&12
D &
= .<.<3 < 3< 38 -84-84 =
= .. .<.<. -4-4 =
= 3-<8 34.84 -<8.4 =
,/&1;D2
&
= 3 =
= 3 =
= 3 =
, &1D;2
&
= 3 =
= 3 =
= 3 =
$ Y A−1 = B ! VY S ! ! Y
&< 4 ) '' ' ' ' < T ' & (
A
% Y 6 /! VY ! ! ! /! D In
3$9
In =
1 0 0
0 1 00 0 1
.
3 6%7 !' L(L 9
L(HT ' < T ' JL
3$9
,5&01-A -2
5 &
= 3 =
= 3 =
,5-&01A 2
5- &
= 3 =
= 3 =
= 3 =
C : A
! # " ! ! ! Y !Y 6%7 !' LHJL X9
LH ' &JL
3$9
,P&F< 4 9. 4 -9< . 93 - < 8G9
,1P2
&
= < < =
@ D B A
3 6%7 ! /! ! ' ! Y # /!
3$9
,P&F< 4 9. 4 -9< . 93 - < 8G9
,P1A-2&
P &
= < 4 =
= . 4 - =
= < =
= 3 - < 8 =
# < ! U L.L LL
E F
3 6%7 !Y !' NN ! N .N /! !
3$ 6%79
,&1A2
&
= .< .-8. --.< =
= 8 < 34 83<< -3--. =
= -84 8<<-8. -3< =
/ Y 'H<J Y # !
4 ! ! Y !' L'L !' LL
3$ 6%79
,%&1;1<A22
% &
= 8 3 =
= 3 - =
= 3 - 3 =
= 8 3 =
L!L Y LL ! LL ? L'HQ<JL Y # < !
A
! Y HJ ! # ! ! X ! Y DF # DF !
<
3 6%7 ! Y !Y !' LRHJL3$ 6%79
Y V
,M&F3 - 39- 9 3 8G
M &
= 3 - 3 =
= - =
= 3 8 =
,"1M2
&
-
A
Y A =[
a11 a12
a21 a22
] H- -J Y
9
det A = |A| = a11a22 − a21a12
|A|
6%7 ! Y !' L'H JL
$ 3 6%79
,&F< 4 .9 3 8 49 . 8 -G9
,12
&
344
* $ 9
,D&F< 4 9 8 -G9
,1D2
&
.
A
/! ! Y ! ! Y !
3 6%7 ! Y ! !'L'HJL ! ! ! ! Y ! !
3$ 6%7
.-
, 7!A 5
, &F3 -9- 89 . 3- <G9
,1 2
&
3--*:3
,1 2
&
44*38
-
, !A 5
,E&F3 -9 3- - 9- 3 - 893 3 G9
,1E2
&
38 333
,1E2
&
3
: 6 :
Y ! H 4J F ! & 1 /!9
&A& 3 & =
6%7 !Y ! LHJL
3$ 6%79
, -- J J!
,&F 893 3G
&
= 8 =
= 3 3 =
,12
=
&
= 3 : =
= 3 =
O +.a-+ .V-
,
,&F3 3 <9 - 39 - 3 3G
&
= 3 3 < =
= - 3 =
= - 3 3 =
,12
&
= =
= 3 =
= - =
*')5( " "'0)("* )("0"*
/%( /&+!
!' & 6%7 !' ! ! ! !' 9
M< RNA&H< M<NJ
Y na # ma ! ! na # !' R Y F R !
4 ! & 9
F + A
& ' ! !' !D !
3$9
9
2x + y − 2z = 103x + 2y + 2z = 15x + 4y + 3z = 4
! LL
2x + y − 2z − 10 = 03x + 2y + 2z − 1 = 05x + 4y + 3z − 4 = 0
?
(Ax + b = 0)
A =2 1 −23 2 25 4 3
x =xyz
c =−10−1−4
3 6%79
,&F- 3 -9 - -98 < GA &F39 39 <G
&
= - 3 - =
= - - =
= 8 < =
&
= 3 =
= 3 =
= < =
,FA"G&1A2
" &
FG
&
= 3 =
= - =
= =
/!X !' 9
AI< (A-< A
LRAL /! ! !'
. !
E
! ! ! 9
@ + -( +Q +8 AI
Q +.( + + -8 A.
. + ( +W +8 A@
+ Q( +WU+-8 A-
@
@ + -( +Q +8 A.
Q +.( + + -8 A.
. + ( +W +8 AW
+ Q( +WU+-8 AQ
@ + -( +Q +8 A
Q +.( + + -8 A.
. + ( +W +8 AW
+ Q( +WU+-8 AIQ
!' A
A =
5 2 4 14 7 1 27 1 9 11 4 9 2
X =
x1 x2 x3
y1 y2 y3
z1 z2 z3
w1 w2 w3
B =
10 7 67 7 75 9 92 4 14
* 6%79
,&F8 - < 39< . 3 -9. 3 4 393 < 4 -GA D&F3 . 9. . .98 4 49- < 3<GA
&
= 8 - < 3 =
= < . 3 - =
= . 3 4 3 =
= 3 < 4 - =
D &
= 3 . =
= . . . =
= 8 4 4 =
= - < 3< =
,N&12;D
N &
= 3333333 34.---- 3. =
= 4-..... <888 =
= 8 3-8 8 =
= <8 - .8 8 =
Y : ! F Y 0 ! !
A
# : ;
B ! ! #!
R5E3
R1 R3
R2
R4
I3
I1 I2E1
E2
−
+
−+
−
+
,! <.9 ! <
* %[ +% $ [b- Y $
−E1aE2 a+>.a>-i1 V>i2 V>.i3O
−E2 V+>-i1 a+>a><a> -i2 V+><-i3O
E3 V+>.-i1 V+><-i2 a+>.a><a>=-i3O
5
+>.a>-i1 V>i2 V>.i3 O E1 − E2
V+>-i1 a+>a><a> -i2 V+><-i3 O E2
V+>.-i1 V+><-i2 a+>.a><a>=-i3 O −E3
! ! 9
R1 = 4ΩR2 = 8ΩR3 = 7ΩR4 = 9Ω
E
R5 = 2Ω
E1 = 9VE2 = 6VE3 = 8V
5Y AS
12 −8 −4
−8 24 74 −7 13
i1
i2i3
O
3
6−8
3 6%79
,6&F3- <9 -< .9< . 3GA C&F99 G
6 &
= 3- < =
= -< . =
= < . 3 =
C &
= =
= =
= =
,5&162;C
5 &
= -<< 3 =
= -<< =
= <- 48<< =
!'9
i1O ?<A. i2O ? < <
i3O V AE=
)) . %
(*+,')5( " )(.)*
3 6%7 ! ! # !' L(L ! 9
=
a = [1 2 3 7 8]
! ! ! /! ! ! # ! !
1 + 2x + 3x2 + 7x3 + 8x4
! 6%79
,
,&F3 - . G
&
= 3 - . =
, A 0
,
,&01AHHAHH2
&
3 : - : 2 : .3 : 4
5 ! ! XY LL L(L X9
,&01AHH2
&
,3&;;-:8;;;8:.;;;
3 &
2 : .3 : 85
! /! ! ! ( ! ! 9
,&F3 -G
& = 3 - =
,-&01AHH2
- &
- : : 2
,0<"* " '( )(.)
X ! 6%7 !Y !'N+-N! $ V /! X Y9
=.
,-
- &
- : : 2
,3&1-2
3 &
= 3 =
= - =
5 $ ! 9
pol3 = 10 + 22s + 4s2 + 26s3 + 15s5 + s7
,
,&F3 -- < - 38 3G
&
= 3 -- < - 38 3 =
, II
,&01AHHAHH2
&
3 : -- : <2 : -3 : 385 : 7
,&12
&
= <-<< =
= <-- : 33-8 - =
= <-- 33-8 - =
= 8<.83-< : 3<-..- =
= 8<.83-< 3<-..- =
= 3...< : -<8 =
= 3...< -<8 =
' /! ]<\ ! XY X $
"30'0)5( " '( )(.)
! ! !' LL ! 9
H<& &J
$ ! ! ! X
=
,3&1A13A322
3 &
,-&1A1-A322
- & 8-4*38 8 8*3
,
,-&11A1-A3222
- &
,&11A1A3222
&
,<&11A1<A3222
< &
)3)*)5( " )(.)*
3 6%7 !Y !' N+-N ! ! ! ! '
3$9
, II
,&01AHH2
&
,&3:;;-:-;;;
&
3 : 2 : -3
,?&3-;
? & 3 -
,1A?2
&
8 2
$ / ! ' ! !'
=<
D ! !
M'< NA'&H'&''<'&J
3$9
, A 0
,&01AHH2
&
,3&-:;:;;-
3 &
- : : 2
,-&;:3
- & 3 :
,FA G&13A-2
&
4 :
&
33333333
8 ! ' L'&L
",)300 " '( )(.)
3 6%7 ! !' ' !Y !' L'&HJL
* 6%7
,
,&01AHH2
&
,&8-;
&
8 -
,&-8;;;-:3;;;:;;<
&
=
-82 : 33 : 4
,&
&
8 -
-82 : 33 : 4
,
, &12
&
-8 : -2 : 3 : 4
-84 : 85 : 386 : -7 : 8
! ;
,
,&3'
& = 3 - < 8 =
, &1 A2
&
= 4 3< 3 < 3< 8.3 - 3883 =
4 X ;
,
, &11 22
&
= =
= -8 :3<<.8. =
= -8 3<<.8. =
= 8 =
' < & ( ' < ) 4 & HJ ( ''< < 4
! ! F' $ F !V
==
,
&
-8 : -2 : 3 : 4
-84 : 85 : 386 : -7 : 8
, 12
, &1 2
&
-8 : -2 : 3 : 4
, 12
, &1 2
&
-84 : 85 : 386 : -7 : 8
,
,&
&
8 : : 2
3-83 : .84 : 385 : 6
! XY ! ! +-
,
,&12
&
8 : : 2
,
,12
&
= -8 : 3<<.8. =
= -8 3<<.8. =
)+ ' !
3 6%7 # ! !' ! /! V # ! !' !Y 'X9
=?
'XHKMI<-<NA PH&I<&-<JK<K' PKJ
I ( - & ' '
&I ( &- & ' '
$ # ! !'9
a = x + 5x2 + x3 + x3 + x7
3 6%79
^_""# ! b+-
^_b+UO!+F-UUOFa=`F``aF``<a+F"<-aF``@U-
! !' ! !V /! /! ! ! ! ! ! !
,
,132
&
,132
&
3.
,1-2
&
33 .
, !
,&3'39
,&12
&
3 8
= 383 <. 8 -4.8<. -8 -.44 =
33
= . 33. 3. -3. 33 . 3. =
3- 3.
= 38 . -- 38 . 3. - < =
3 -3
=@
= -<3< -4.44<. <. <338 3383 =
! . . & ! ! & !
)/ 0%!& %.!1
( ! Y D D! V ! ! ! AE.π (3,1415927) exp (2,7182818) −3
8 X ! ! ! ' $ ! ! ! /! ! V$ ' /! ! !' 9
x2 + 40,96 = 0x2 = −40,96√
x2 =√−40,96
x =√−1 · √40,96
% !' ! ! # ! ! 4 X F D + 6%7 ! -
i =√−1
3 6%79
, !
,?132
&
8 ! /!9
i2 = −1 i3 = −i i4 = 1
3 6%7 ! B GD $
=A
, ! @
, @
@ &
, @;;-
&
3
, @;;
&
, @;;<
&
3
80 #"+" (70+" %0G%("% 6 (70+" # 01/+9 !
! Y ! ! ! NcN
, !
,< ; @
&
<
,!3&-; @
!3 &
-
,!-& @; @
!- &
3<384-.
( ! $ ! ! ! V
z = x + yi
! $ ( (< F ! 9
=E
Re(z) = x
Im(z) = y
% ' Y /!X 9
Re(z)
Im(z)
z=7+3i
,! <9 >+Y- 6+Y-
% $ A 9HJ $ ( A0HJ 3 #! < $ /! !
z = 7 + 3iRe(z) = 7Im(z) = 3
4 ! ! $ ! ! 6%7 FV ! ! X9
, ! J
,3&3-8:; @
3 &
?
3-8 :
,132
&
3-8
,!132
&
, J
,-& @:-; @; @
- &
3<384-. : - 3 8
,1-2
&
3<384-.
,!1-2
&
- 3 8
Y ! !# D! /! $
?.
Re(z)
Im(z)
z=7+3i
r
θ
,! <<9 , Y
r θ
r =√
x2 + y2
θ = tan−1 ba
x = r ∗ cosθ
y = r ∗ senθ
4 r θ ! $ !V 9
A = r∠θ
% ' D# ! #! <<
6%7 9
,! J !
,&<:; @
?
&
< :
, 612 512
,&129
&!129
,
,&?1;;-:;;-2
&
8
,
,&129 &;3 @
&
4 4
, 0
,A
&
8
&
4 4
" : :
Y $+-
,J! D
,D&:- ; @
D &
: -
,D&J1D2
D &
-
X ! ! Y /! $ 6%7 $! X9
, J
,&F-A :-; @9 ; @ G
&
?<
= - : - =
= =
,
,H
&
= - =
= - =
Y $! Y ! ! R+-
, J! 0
,3&J12
3 &
= - - =
= =
,3H
&
= - =
= : - =
,J! Q
,Q12
&
= - =
= : - =
* = :
, J
,&3: @
&
3 :
,-&:; @
- & :
,&:-
&
< : <
?
, J
,&-
&
- : -
= # :
, J
,-&<:; @9 &:<; @9
,&-;
&
3- : <
, J
,3&-
3 &
4-.. : <38
:
3 X ' $ ! V! ' !' F$ ejθ
ejθ = cosθ + jsenθ , ! '! 3!
! 3! 6%7 ! ! ! /! D ! D! ! 5o 85o
?=
Re(z)
Im(z)
r=2
r=5
r=8
θ=14
θ=45ο
ο
θ=74ο
,! < 9 !
/! D ! /! < !9 - @ D# ! < ! < ! D! !
! ! /! D ! /! ! ! D ! ! X /! . # A ! ! .U !
, ! -A 8 0
,&F-;13A-.2A 8;13A-.2A ;13A-2G9
,12
&
= 3 =
! /! ' = ' # A= ' ! . # A ! /! !
, 8 0 8 !
??
,&18; @23
&
.-8
,&1 8; @23
&
3< 8-44
,! !
,&
&
34-<
,&.4
&
3..<-
,! 3
,&''9
,12
&
= 3 =
,! J
,&;12:;12;@9
,!
,-112A!12A&FAA38A4G2
?@
0.00 2.14 4.29 6.43 8.57 10.71 12.86 15.000
1
2
3
4
5
6
7
8
9
,! <=9 (D# 6%7
#! <? = A= ! 9
^_"" = A=
^_FOPY+.A-1 +F-Q1
^_OPY+.A-1 +F-Q1
^_ +FOP.=EQ-
?A
0.00 2.14 4.29 6.43 8.57 10.71 12.86 15.000
1
2
3
4
5
6
7
8
9
,! <?9 (D#
5 ) ' O ' -' ( -'Q :9Y,01#3 %/ 310*S
)2 ! ! '!&!! &&3! 4
6%7 !Y !' L'HJL ! ! ! V 3 ! 9
(A'H(<<< J
&9
( 9 ! ! Y !'
9
9 /! !
9 F' !' !
! ?E
3$9
! /! !' 9
t0 = 0;
1 < t < 81
y0 = 11
dydt = y − ycos(t)
!'9
!Y L'XHJL ! t0ty0 !' L'HJL /! ! ! D# !Y L-'HJL
* 6%79
,1IFG&1A02IAI&00;12I29
,0&39&9&3'8' 9
,0&10AAA29
,-1A0A-29
,1H7B 0&0 0;12H AHHAH012H29
>! ,! <@9
@
1 2 3 4 5 6 7 80
200
400
600
800
1000
1200
Soluciˆ‡n a la ode y=y - y*cos(t)
tiempo
y(t)
,! <@9 !' )
3 X! X !X ! D# -' ! D# # D !Y Y D# $! !
+ &-'
! 6%7 /! Y D# ! F
3 $ KK !9
H<(<KK<K(K<KKJ
< (9 '
9 ! $ UFU
(9 ! $UU
9 ! D#
3$ 6%79
! 8:12 0 <;
,&'8'<; @9
,0&8:129
,1A0AHJ HAHJ 0HAH812H2
3 ! ! (D# .
@.
@
0 2 4 6 8 10 12 144.0
4.2
4.4
4.6
4.8
5.0
5.2
5.4
5.6
5.8
6.0
5sen(x)
eje x
eje y
,! .9 (D# O=a+F-
! D# $9
! 8:12 0 <:12
,&'8'<; @9
,0&8:129
,+&<:129
,1F+90G2
3 ! ! (D#
@<
0 40 80 120 160 200 240 2803.0
3.4
3.8
4.2
4.6
5.0
5.4
5.8
6.2
,! 9 (D# =a+F- a+F-
! ! !' =a+F-! a+F-9
! 8:12 ! <:12
,&'8'<; @9
,0&8:129
,+&<:129
,1F0A+G2
3 ! ! (D# <
@
0 100 200 300 400 500 6003.0
3.4
3.8
4.2
4.6
5.0
5.4
5.8
6.2
,! <9 (D# =a+F- ! a+F-
+ * 567#
! D# 6%7 ! D# N +-N ! #! =
,! 9 4X! D# 6%7 %6)*:
,! =9 4X! D# 6%7 2 ;6)&5;
" " @=
X /! D D# ! /! !VY' D ! D# $! /! ! ! $ '!
, !
,&' @'-; @9
,0&129 03&129 0-&129
,9 1A029 9 1A0329 9 1A0-29
3 ! ! #! ? @ A
,! ?9 ) .
@?
,! @9 )
,! A9 ) <
@@
+) % ! &&
=
&! ! D# N N #! /! ! 6%7 ! F+- !+- * F /! F +- +-
< $ X9
,12
8
,1-2
++ % 85*34
3 !' !Y 6%7 ! ! D# /!Y ! ! /! /! /! ! +- !Y9
)
. Y!< X= $? @ A E Y! !.E · · · · · ·<
! .9 F+-
3$9
, 2 < A J
@A
,&<'3'<9
,1A>-<AHHAH0HAHH29
,!182
3 ! !! $ ! (D# E
-4 -3 -2 -1 0 1 2 3 4-4
-2
0
2
4
6
8
10
12
titulo
x
y
,! E9 D# !! $
+/ ! % 34
3 !Y 6%7 D# ! 9
-'H<(< KKJ
9 ! ! Y ! ! UU
( 9 ! ! Y
KK D 9
! @E
9 !
9 ! D# /! PFFFFQ
9 ! $ 9 ! ! UU ! UU ! UU UU
9 # /! ' !$ $
9 ! Y
3 ! /! ! ' ! ! 9
> 9
(# ! # # ! !
(# ! Y ! # ! ! ! # Y
(# ! Y ! Y S <4 ! <4 S /! ! # ! ! ! # ! S
,
,&'3'9
,1222
,&1>-2>-9
,-1A2
3 ! ! (D# .
A
-3 -2 -1 0 1 2 3-3
1
5
9
13
17
21
25
29
33
,! .9 (D# ! '
,
,L & L
,&'3'9
,&1>-2>-9
,!&-;:-9
,-1AFHA!HG2
3 ! ! (D# ..
! A.
-3 -2 -1 0 1 2 3-4
0
4
8
12
16
20
24
28
32
36
,! ..9 (D# # Y
, /
,0&2
,2-
,:3
,&8'3'89
,0&>-9
,03&>--9
,0-&:39
,-1HAF0HA03HA0-HG2
3 ! ! (D# .
A
-5 -4 -3 -2 -1 0 1 2 3 4 5-30
-20
-10
0
10
20
30
,! .9 ! Y
+,0070( ( 0,-'."(+*
*Y D# #! @ ! ! D#
"G*0+(, ,6/+
!Y ! ! D# % ! D# ! 9
! A<
0 1 2 3 4 5-1
0
1
2
3
4
5
Ο
Ο
ΟΟ Ο
Ο
Ο
Ο
ΟΟ Ο
♣
♣
♣♣ ♣
♣
♣
♣
♣♣ ♣
∇
∇
∇∇ ∇
∇
∇
∇
∇∇ ∇
∆
∆
∆∆ ∆
∆
∆
∆
∆∆ ∆
◊
◊
◊◊ ◊
◊
◊
◊
◊◊ ◊
style -9Ο
style -8♣
style -7∇
style -6∆
style -5◊
,! .<9
0 1 2 3 4 5-1
0
1
2
3
4
5
♦
♦
♦♦ ♦
♦
♦
♦
♦♦ ♦
⊕
⊕
⊕⊕ ⊕
⊕
⊕
⊕
⊕⊕ ⊕
×
×
×× ×
×
×
×
×× ×
+
+
++ +
+
+
+
++ +
.
.
.. .
.
.
.
.. .
style -4♦
style -3⊕
style -2×
style -1+
style 0.
,! . 9
A
0 1 2 3 4 5-1
0
1
2
3
4
5
style 1style 2style 3
style 4style 5
,! .=9
0 1 2 3 4 5-1
0
1
2
3
4
5
style 6style 7style 9
style 10style 11
,! .?9
3$9
! A=
,&8'3'89
,0&;;-9
,03&;;--9
,0-&:39
,-1HAF0HA03HA0-HGAF- < -G2
3 ! ! (D# .@
-5 -4 -3 -2 -1 0 1 2 3 4 5-30
-20
-10
0
10
20
30
×××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××
♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦
,! .@9 !
,-1HAF0HA03HA0-HGAF4 < G2
3 ! ! (D# .A
A?
-5 -4 -3 -2 -1 0 1 2 3 4 5-30
-20
-10
0
10
20
30
ΟΟΟΟΟΟΟΟΟΟΟΟΟΟΟΟΟΟΟΟΟΟΟΟΟΟΟΟΟΟΟΟΟΟΟΟΟΟΟΟΟΟΟΟΟΟΟΟΟΟΟΟΟΟΟΟΟΟΟΟΟΟΟΟΟΟΟΟΟΟΟΟΟΟΟΟΟΟΟΟΟΟΟΟΟΟΟΟΟΟΟΟΟΟΟΟΟΟΟΟΟ
♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦
♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣
,! .A9
"G*0+(, "+#,
AM<(<<(N
3 !Y $ X DF
3$9
, ! ? J 0 0 1A2
0 1<A82
,-1HAF0HA03HA0-HGAF- < 4GA&FA A <A 8G2
3 ! ! (D# .E
! A@
0.0 0.4 0.8 1.2 1.6 2.0 2.4 2.8 3.2 3.6 4.00
1
2
3
4
5
,! .E9 ! !
,J - 0 0 -
,-1HAF0HA03HA0-HGAF- < 4GA&F-A -A A G2
3 ! ! (D#
AA
-2 -1 0 1 2 3-2
-1
0
1
2
3
,! 9 $ F V < $ V <
"G*0+(, /+G
3 ! !Y ! ! ! /!D ! D#
/! ! # ! !V /! Z
0
, ,12
,&8'3'89
,0&1;;-29
,03&1;;-2-9
,0-&:39
,-1HAF0HA03HA0-HGAF- < 4GA&F-A -A A GA!&I1;;-2R11;;-2-2R1:32I2
3 ! ! (D# .
! AE
-2 -1 0 1 2 3-2
-1
0
1
2
3
(-x**2)((x**2)-2)(x+1)
,! .9 !
."7,0( 0 3)*'0)80)5( ( ? @A?
+- ! /! $ ! D# & ! D /! D! D# !
' ! +- 9
&! ! # ! !' ! ! +- X9 &AHJ
3$9
,-1HAF0HA03HA0-HGAF- 8 3GA&F-A-AAGA!&H1;;-2R11;;-2-2R1:32H2
,&!129
*Y !/! ! ! # ! ! ! X9 H& J
E
30,)0"* B+)"* '*0( @A9
G"%3HI6?JK
! ! D# UFU UU $
3$9
, 0 1-2 0 J 182
,-1HAF0HA03HA0-HGAF- 8 3GA&F-A-AAGA!&H1;;-2R11;;-2-2R1:32H2
,&!129
,!&F8A-G9
3 ! ! (D#
-2 -1 0 1 2 3-2
-1
0
1
2
3
(-x**2)((x**2)-2)(x+1)
(-x**2)((x**2)-2)(x+1)
,! 9 OP=Q ! O+-
/ GLMG
X ! D /! D# ! ! $ X 6%7 D# !
! E.
/! ! ! RS X9
,&'3'<9
,0&>-:>9
,-1A0A8A!&H;;-:;;HA&F AA3A G2
,&!129
,!&F3A-G9
,!Q! & II9
3 ! ! (D# <
-110
010
110
0
10
20
30
40
50
60
70
80
x**2+x**3x**2+x**3x**2+x**3x**2+x**3
,! <9 ! RSONN O+-1
JL/ #,% (? 6L/ #,% (
! /! 9
1'AL''L
0)'AL L
; ALL
E
3A LL
ALL
3$9
&8'3'89
0&;;-9
03&;;--9
0-&:39
-1HAF0HA03HA0-HGAF- 8 3GA&FAAA GA!&H1;;-2R11;;-2-2R1:32H
2
&!129
Q&II9
0Q&II9
3 ! ! (D#
-3 -2 -1 0 1 2 3
-3
-2
-1
0
1
2
3
(-x**2)((x**2)-2)(x+1)
,! 9 ! FR R
X ! X ! 6%79
! E<
, !
"+)C'"+0*
J,%,/+ 1" +9+ 6 G">;#
D# ' /! ! X! /! ! ! ! F D# $ X9HLL<L4 L<L4 (LJ
3$9
&8'3'89
0&;;-9
03&;;--9
0-&:39
-1HAF0HA03HA0-HGAF- 8 3GA&FAAA GA!&H1;;-2R11;;-2-2R1:32H
2
&!129
Q&II9
0Q&II9
1I*SD(6/5( * T6E5/7IAI J IAIJ 0I2
Q0&89
Q&<9
3 ! ! (D# =
E
-3 -2 -1 0 1 2 3
-3
-2
-1
0
1
2
3
ELABORACION DE GRAFICAS
eje x
eje y
(-x**2)((x**2)-2)(x+1)
,! =9 F ! R RY +-
/! ! D# !9
D# 9
!&FAG9
3 ! ! (D# ?
! E=
-3 -2 -1 0 1 2 3
-3
-2
-1
0
1
2
3
ELABORACION DE GRAFICAS
eje x
eje y
(-x**2)((x**2)-2)(x+1)
,! ?9 ! !
, 6 +,%N*+, 1" / #*"&
X /! ! /! ! D# !Y HJ ! 9
H<(<LL<<J
& F /! ! /! F /! D! D! /! /! ! /! F ! . /! ! !
3$9
&8'3'89
0&;;-9
03&;;--9
0-&:39
-1HAF0HA03HA0-HGAF- 8 3GA&FAAA GA!&H1;;-2R11;;-2-2R1:32H
2
E?
&!129
Q&II9
0Q&II9
1I*SD(6/5( * T6E5/7IAI J IAIJ 0I2
Q0&89
Q&<9
!&FAG9
!1 A AFI I9 IIGAA329
3 ! ! (D# @
-3 -2 -1 0 1 2 3
-3
-2
-1
0
1
2
3
ELABORACION DE GRAFICAS
eje x
eje y
(-x**2)((x**2)-2)(x+1)
Adicional Nota
,! @9 ! F+-
+2 . (
!' !Y ' D# ' % D ! !
% F ! ! !' +-
3$9
# E@
, >-
,&<'8'<9
,--1A;;-A8A!&I ,;;-I29
,&!129
,0Q & II9
,&II9
,1I#S($- - N;;-IAIIAI0I29
,Q0&89
,Q&<9
3 ! ! (D# A
-4 -3 -2 -1 0 1 2 3 40
2
4
6
8
10
12
14
16
PLOT2D2 X**2
x
y
step --> x**2
,! A9 *
+9 . ()
3 6%7 D# /! !
% F ! ! +-
EA
3$9
,! > :3
,&'8'9
,-1A;;:3AA!&I 1;;:32I29
,&!129
,0Q&II9
,Q&II9
,1I-IAIIAI0I29
,Q0&89
,Q&9
3 ! ! (D# E
-3 -2 -1 0 1 2 3
-30
-20
-10
0
10
20
30
plot2d3
x
y
barras verticales (x**3+1)
,! E9 * <
+: . (+
3 6%7 D# ! ! ! D ! S
$ EE
% F ! ! +-
3$9
,1>-2-
,&'8'9
,-<1A-;:8;;;A!&I 1-:8;;2I29
,&!129
,0Q&II9
,Q0&89
,Q&9
,Q&II9
,1I-<IAIIAI0I29
3 ! ! (D# <
-3 -2 -1 0 1 2 3
-8
-6
-4
-2
0
2
4
6
8
plot2d4
x
y
flechas (-2x+0.5x**3)
,! <9 *
.
+; + < #
2 3
/! # !Y !' /! ! ! $ /! # #! <.9
(3,2)
(0,0)x
y
,! <.9
) /! #! x0 = y0 = 0 x1 = 3 y1 = 2 X9
x =
x0 x1
↓ ↓[0 3]
y =
y0 y1
↓ ↓[0 2]
3 6%79
, ! 12 1A-2
,&FAG9
,0&FA-G9
,-<1A0A-A&F3A3A8A8GA!&I1A2A1A-2I29
,1I! IAIIAI0I29
,!1A-AI1A-2I29
,&!129
,Q&II9
,0Q&II9
,Q0&89
% & " ..
,Q&<9
3 ! ! (D# <
-1 0 1 2 3
-1
0
1
2
3
grafica de un vector
x
y
vector(0,0),(3,2)
(0,0)
(3,2)
,! <9 6%7 !
2 30,)* 3"+,"*
# # ! Y : D # NFN Y8 D # NN ! ! /! /! # $ #! <<9
.
x1
1
2 3 4 5 6 7
3
4
5
6
7
2
y
8 9 10 11
8
A
BC
,! <<9
! 7 Y ! Y 7
* 6%79
,3&F9G9
,D3&F9.G9
,D3&F9<G9
,/3&F3:DG9
,-&F9-G9
,D-&F-9 G9
,D-&F9G9
,/-&F-:D-G9
,&F3AD3A/3G9 0&F-AD-A/-G9
,-<1A0AF-A-A-GA&F8A8A A4G2
,1I AD 0 /IAIIAI0I29
,!1-A AII2
,!18A<8AIDI2
,!1<A<8AI/I2
,&!129
,Q&II9
,0Q&II9
3 ! ! (D# <
' .<
0 1 2 3 4 5 6 7 80
1
2
3
4
5
6
7
8
9
vectores A,B y C
x
y
A
BC
,! < 9 6%7
+= &-' ! ) . ()
6%7 ! D# <& !Y !' 'H<(<<J3 D# ! !# +FY-
Y ! D# !' /! ! !9
NFN NN F '
Y ! Y ' +F-
Y+$- !# ! +F+-+--
3$ 6%79
,!
,&'3'-; @9
,0&129
,&0H ;13A29
,1A-A29
3 ! ! ,! <=
.
-1.0
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1.0
Z
01
23
45
67
Y
0 1 2 3 4 5 6 7X
,! <=9 ! !' <&
3 D# ! # LL ?< !' L(L! LHJL ! Y ! LL
5 $ '9
3 6%79
,0&8'3'89
,&'3'9
,&0H;1;;-:-29
,10AA29
3 ! ! ,! <?
' .=
-40
-30
-20
-10
0
10
20
30
40
Z
-5-4-3-2-1012345X
-3 -2 -1 0 1 2 3Y
,! <?9 3$ '
4 " 0,-'."(+ D??
O D# ! /! +- ! $ d
,0&8'3'89
,&'3'9
,&0H;1;;-:-29
,310AAA!&IJ RJ 0RJ I29
3 ! ! ,! <@
.?
-40
-30
-20
-10
0
10
20
30
40
eje z
-5-4-3-2-1012345eje x
-3 -2 -1 0 1 2 3eje y
,! <@9 (D# <& ! O\\
& ! D# 6%7 !/! ! /! ! ! X D# /!
/ (& >
5 X 6%7 ! ! 3 /!W ! +- Y /!! ! 6%7 ! !$! ! ! 3 ! ! !' ! ! ! 6%7
" ")+, " *)0
3 ! F ! 6%7 V 4%"4[ ! ' %' ! $ 6%7 ! $! 'HJ '
$ ' ! ! V D ! $ /! ! ' ! ' ! ! !9
% !' ,- ! % F' /! ! $! 6% ! 9 !
,@ ! $! 6%7
& Z ' Y/! ' /! ! !
4 Z ! ' ' /! ! ! !
.@
.A
' ' ! '
2 ' ' ! /! $!' 6%7
2 1 ' ' ! /! ! # ! X !Y'
2 1 ! H+J W ! + L+L ! -
2 1 ! ! HJ W ! !
6%7 ! ! !!9 1++* D ! 6%7
"7"'+0, '( *,)+ ( @A
(3 OD
F $! /! 6%7 ! ! ! ' /! ' ! !V 6%7 $!
H M<'NJ
! /! ! /! V ' /! D $! 6%7 ! ! ! 8' ! $! 8' 6%7 $ %6)*: 9
,+
&
! ! ! 9 ! # ! ! 4 /! ! 6%7 $! 'HJ X9
,1H!H2
.E
&
% ! ! /! ! ! $! ! 0
3 ! Y ! $! HJ9
^_+-
& !9
&1A-; @A3 29
+&-9
&129
!&;1+;2:;1+;2; @9
-11!2A!1!229
1I IAIIA II2
! ! Z # /! ! & ! /! ! ' / ' & ! ! F +,! =.- $ ! I ! ! Z $! 6%7X9
, 3
3 ! D# ,! =
) /! $! ! ! ! /! ! I1 /! Y '
..
,! =.9 !
-0.3 -0.1 0.1 0.3 0.5 0.7 0.9 1.1-0.2
-0.1
0.0
0.1
0.2
0.3
0.4
0.5
0.6
plot de mi primer script
real
im
,! =9 >! .
( ...
(3 143 6 #$3%" 22 ©3 $! 0 6%7 ! !9
,+
&
/'U 7!UU*
! ! +- 9 ! 9
,1H'U!H2
&
% ! ! /! ! ! $! ! 0
3 ! Y ! $! HJ9
! ! ' =..
6%7 ! ! &! ! ! 9 ' #' ! ! #' !
/ ?!&&%!( ! .& &%>
6%7 ! ! ! ! ' /! !9 !! ' ! ' ! ! 3 !F ! X ' 6%7
3F Y 6%7 3 ! =. ! X /! ! Y ' 6%7
= EF
3F 6%79 8 $! !! ! ! # !VD !
..
X #
OO ! e /!_ /!eO ! /!_O ! /!e_
! =.9 5 '
'
/# 3 ) 4 ' ( O '' 3'
3$9
&3
&3'<A &;A
3 6%79
,
,
,&3
&
3
, &3'<A &;A
&
3
&
-
&
&
-<
3 ! !/! Y ! Y
( ..<
3$9
&39
&F- 3 <GA &:A 9
3 6%79
,
,
,&39
, &F- 3 <GA &:A 9
&
3
&
&
<
3 8 Y ! ! ! ! ! 9
3$9
+
&39 + V3<A &-;A 9
3 6%79
, +
, +
,&39
,+ V3<A&-;A 9
&
-
&
<
&
&
..
3
3 8 ! #Y ! R X9
"
&9
&3'8'3A&:39 ,3 "A 9
3 6%7
, "
, "
,&9
, &3'8'3A&:39 ,3 "A 9
,
&
3 R 9
"&3'9
J&3'<9
":J,< "9 1"29
9
9
3 6%79
,
,
, "&3'9
, J&3'<9
, ":J,< "9 1"29
, 9
, 9
,
..=
3
3
3
-
-
/) *#*
3 6%7 F !! 9 !! !! % !! ! ! F' ' $! ! + '- ' /! ' $! /! !! ! !
&39
, A 0&A A 0&A
&39
, A 0&A A 0&A
3 6%7
,
,
,&39
, , A 0&A A 0&A
0 &
3
,&39
, , A 0&A A 0&A
0 &
3
% ' $ ! F' F $!!
..?
?&1H H29
?,3 A& @A A& @A
A @A 0&?:8A @A 0&?1?2A
3 6%7
,
,
,?&1H H29
,
, ?,3 A& @A A& @A
&
$
, A @A 0&?:8A @A 0&?1?2A
0 &
33
/+ < 7*#*@ $#*
/! # ! !' 6%7 ! !! ! /! !!' 3 ' !! ! !' 6%7 /! D ! ' ! !
"
% ! !' 6%7 !9
! P.QO+F.F-
!' $! ! !' $! ! !' < ! $ ! !
& ..@
$! !' ! $ ! Z +Zf-9
FG&1"2
"&1"2
"V3 "&3A
&39
J&3'"A&;J9
% !' D & 6%7 !V ! !' D !9
,1HH2
&
% ! /! !' D
, 9
,1HH2
&
3
! !' X9
^_ O.9= FO+-
F O
.
F O
F O
?
F O
F O
.
..A
3 ! ' ! ! ! ! 'F
3 X! ! $ F ! 6%7 Y$ D X /! ! 6X 3' 2D ! ' ! ! ! /! ! Y ! ! ! !! 6%7 /! /! 3 # X! ! /! ! !
2 '(& A &!.&!!(>! ' ! ! (&'!&!
# ! !Y 6%7 ! 8HJ (HJ[HJ
! 9 GEH@AG H
3 !' ! F9
M8 <8 < NA8H (< '< )<8(< J
9
( 9 # /! /! UU UU U U U U
' 9 #
) 9 ! + e/+.- /+-e= -
..E
.
8( 9 + UUUUUUUUUZUUU -
9 D
>!9
8 9 # #
8 9 >! # !
9 !
! 9 G= @AG H
! F 9
MNA(H'<<S<1 M<; M<N NJ
MNA(H'</<;J
MNA(H'<!J
9
' 9 H KK < K'K J # 9 ! KK K'K
<S<1<; 9 '
/< ; 9 '
! 9 Y '
9 +N N - '
! 9 GI@AG
F9
[H<J
9
9 Y
9 !
$ 6%79
,F+A+AG&+1HHA-AF- GA HHAF G29
,&3'39
,3&1 @;329
& ..
,-&1-; @;29
,&3:-9
,&01AHH29
,&01HHA01+AHHAHH2;;-29
,0&1A29
,-1A029
!9
0 10 20 30 40 50 60 70 80 90 100-1.0
-0.8
-0.6
-0.4
-0.2
0.0
0.2
0.4
0.6
0.8
1.0
,! ?.9 W
.
0 10 20 30 40 50 60 70 80 90 100-1.9
-1.5
-1.1
-0.7
-0.3
0.1
0.5
0.9
1.3
1.7
2.1
,! ?9 W 3
2 &#*@ . A
D# ! # !Y !' 6%7 KHJK
F9
HJ
&9
O !
3$ 6%79
, J 0 HH
,12
!9
& .<
-2 -1 0 1 2 3-1.4
-1.0
-0.6
-0.2
0.2
0.6
1.0
1.4
transmission zeros and poles
real axis
imag. axis
Ο
Ο
Ο
Ο
Ο
Ο
Ο
Ο
Ο
Ο
Ο
Ο
Ο
Ο
Ο
Ο
Ο
Ο
Ο
Ο
Ο
Ο
Ο
ZerosΟ
×××
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×××
Poles×
,! ?<9 Y
# 6%7 UYU
,1122
&
= <83- : <<4-4 =
= <83- <<4-4 =
= -- -. : .<<8 =
= -- -. .<<8 =
= 888-488 : 3<4.. =
= 888-488 3<4.. =
= 3.< : 4 < < =
= 3.< 4 < < =
= < .8<- : .443 =
= < .8<- .443 =
= .33833 : .3<< =
= .33833 .3<< =
.
= 4.333 : <4833 =
= 4.333 <4833 =
= ..43.. : 3-- 8 =
= ..43.. 3-- 8 =
= 4..- : -8<84 =
= 4..- -8<84 =
= 3 =
= 3-33-8 : 33<3<3 =
= 3-33-8 33<3<3 =
= 3 -4<38 : <<8<<8 =
= 3 -4<38 <<8<<8 =
2) *5B
D# ! 7 6%7 !Y !'9 K'HJK
F9
'H<M < N M<N M<N J
& 9
9 ! !
< 9 U! BYU
9 '
3$9
& !' ! 7
x∗ = 7πx + u
y = 10πx + u
3 6%79
,! HH
,&.; @9
,&39
,&3; @9
,&39
.=
,&01HHAAAA2
&
132 1 0'2
= D / N =
1-2 & &
-34433<4
12 & D &
3
1<2 & / &
3<384-.
182 & &
3
12 & N 1 2 &
1.2 & $ &
,1A3A329
!9
.?
-110
010
110
210
-8
-7
-6
-5
-4
-3
-2
-1
0Magnitude
Hz
db
.
-110
010
110
210
-180
-160
-140
-120
-100
-80
-60
-40
-20
0Phase
Hz
degrees
.
,! ? 9
2+ &!.&!!(> ! (!% !!
% !' !Y 6%7 # K(HJK !' D /! #
5 !' ! K- HJK /! !
! F 9
MNA- HJ
& 9 ! K(HJK
9 Y
3$9
&# !
A =[
1 −37 −2
] B =
[10
] C =
[4 −2
]1
.@
3 6%79
,&F3 9. -G9
,D&F39G9
,/&F< -G9
,&01HHAADA/2
&
132 1 0'2
= D / N =
1-2 & &
= 3 =
= . - =
12 & D &
= 3 =
= =
1<2 & / &
= < - =
182 & &
12 & N 1 2 &
= =
= =
1.2 & $ &
,
,&-12
&
: <
34 : : 2
2/ (!% !(
* 6%7 !
.A
s1 s2
s1
s1
s1
s1
s2
s2
s2
s2
+
+
+
s1*s2
s1 + s2
[s1 , s2]
[s1; s2 ]
s1 / . s2
,! ?=9 4
! 6%7 !Y ! !9
K -HJK ! !' !
F9
A -H J
&9
9 Y
9 + OP7&+-Q -
K HJK
F9
H M<NJ
&9
.E
9 !
9 !
3$9
S1 = 1s+2 ,
S2 = 13−s
& Y 9
. S1 + S2
S1 ∗ S2
<[
S1S2
]
[S1S2]
= S1S2
3 6%79
,&01AHH2
&
,3&31:-2
3 &
3
- :
,-&312
- &
3
,3&01HHA329
,-&01HHA-29
,-&-1-29
,3&-1329
.<
,1-2
& W W : W 3 W
0 & W3 W
,&-;39
,12
W 3 W W W
& W - W : W 3 W
0 & W3 W
,&-12
&
3
: 2
! ! !/! ! ! 9
,3;-
&
3
: 2
,3:-
&
8
: 2
,F39-G
&
= 3 =
= =
= - : =
= =
= 3 =
.<.
= =
= =
,F3 -G
&
= 3 3 =
= =
= - : =
,3-
&
. : 2
22 . %!
3 ! ' ! ' !' ! 7
&89 0&3'-;9 &3'9
&3'A
1A32&39
1A-;2&39
J &3'-;A
13AJ2&39
1AJ2&9
&39
+ ,3A
&9
&-'3A
J &-'-;3A
&-8;113AJ2:1AJ32:1:3AJ2:1AJ:3229
&11AJ229
,
&9
.<
A
1AJ2&9
1A0A29
3 ! 9
% #' ! /! D Y (nX2n) 5 /! ! ! ! :
% #' (V 1 = 1000, V 0 = 0)
% ' ! + DF .- ! 9 Vi,j = 1
4(V i−1,j + Vi,j−1 + Vi+1,jaVi,j+1) #$ V1 = 1000, V0 = 0
3 D! DF Y ! ! # +! DF .-
% ' D# !
!' D ! D# !'
0
100
200
300
400
500
600
700
800
900
1000
Z
01020304050
X
0 10 20 30 40 50 60 70 80 90 100Y
,! ??9 !' F
.<<
0
100
200
300
400
500
600
700
800
900
1000
Z
050100Y0 10 20 30 40 50X
,! ?@9
0
10
20
30
40
50
X
0 10 20 30 40 50 60 70 80 90 100Y
,! ?A9 ! <.+-
.<
29 (& &! .
3 6%7 ! ! ! 6& ! W ' !Y ! L(HJL LHJL
LHJL ! ! !
! F 9
AH<<J
&9
9 !' +'-
9 /! ! !'
9 ! !'
9 !
3$9
6%79
,# 1# 2
,&01AHH29
,&'3'9
,#
,&31;;-:329
,/
,O&39
, &39
,&O;13: ;29
,&;13:;29
,0&01HHA2
,&1HHAA029
,-1AA-29
,1H# 131;;-:322 O&3 &3HAHHAH012H29
!9
# .<=
0 1 2 3 4 5 60.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
Proporcional.Derivativo (1/(s**2+1)) K=1 D=1
tiempo
y(t)
,! ?E9 &
' Z D# !Y L HJ L
39
,&01AHH29
,&'3'9
,#
,&31;;-:329
,/
, &3'8
,O&9
, &39
,&O;13: ;29
,&;13:;29
,0&01HHA29
,&1HHAA029
,-1AA29
.<?
,
,1H# 131;;-:322 O&F3 - < 8G &3HAHHAH012H29
>!9
0 1 2 3 4 5 60.0
0.2
0.4
0.6
0.8
1.0
1.2
Proporcional.Derivativo (1/(s**2+1)) K=[1 2 3 4 5] D=1
tiempo
y(t)
,! ?.9 [O .< =
&9
,&01AHH29
,&'3'9
,#
,&31;;-:329
,/
,&9
, &8'8'-89
,&:39
,O&9
, &9
,&O;13: ;29
# .<@
,&;13:;29
,0&01HHA29
,&1HHAA029
,-1AA29
,
,1H# 131;;-:322 O& &F8A 3A 38A -A -8GHAH
>!9
0 1 2 3 4 50.0
0.2
0.4
0.6
0.8
1.0
1.2
Proporcional.Derivativo (1/(s**2+1)) K=3 D=[0.5, 1, 1.5, 2, 2.5]
tiempo
y(t)
,! ?..9 &O = . .= =
V
,129
,&01AHH29
,&'3'89
,#
,&31<;:29
,/ 1#52
, &3'89
.<A
,O&9
,5&-9
,&O;13:315;229
,&;13:;29
,0&01HHA29
,&1HHAA029
,-1AA29
,9
,1H# 5! #&31<:2A O&F3 - < 8G 5&-HAHHAH012H29
>!9
0 10 20 30 40 500.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
Proporcional Integrador Planta=1/(4s+6), K=[1 2 3 4 5] I=2
tiempo
y(t)
,! ?.9 6 O . < = 6O
6
,129
,&01AHH29
,&'3'89
,#
# .<E
,&31<;:29
,/ 1#52
,&9
, &<'<'-9
,&:39
,O&9
,5&9
,&O;13:315;229
,&;13:;29
,0&01HHA29
,&1HHAA029
,-1AA29
,9
,1H# 5! #&31<:2A O&F3 - < 8G 5&-HAHHAH01
!9
0 4 8 12 16 20 24 28 320.0
0.2
0.4
0.6
0.8
1.0
1.2
Proporcional Integrador Planta=1/(4s+6), K=3 I=[0.4, 0.8, 1.2, 1.6, 2]
tiempo
y(t)
,! ?.<9 6 [O< 6OP A . .? Q
.
.!& C! ? (& !(!.D(
6 %! (! ('Y %! ' *' ) 7D !'9 %%6)5 76&623)V65)%
!"#
I % =
FS @E
.
.A ...! A . @@# .E A <. < ..=$! ?< A .< 6& .<
& A& =<& <& 7 . &' .
.<3! =3&5 ?A3$ E. ..= .<3/! E<F .AF ..F .. .
. S . ...! !' .<!' .<!' .<!' .<!' .<!' .<!' .<,*)65)3 ==,! ..?,! ..E! .! ..?
AE(>I,65 @.(D# .E
B A
..= .
% .<. @E AA @E =%6)*: . = .A ... .. ..S @E
2 .@ <
. .
. )
2 <2Y 2Y 2Y <?2Y ! <2Y .2Y ! <=2Y ! < 2Y ! < @
) $ =@ .
5 E5 ' ...
. .< .E @. .E @A E?< E@ EA< .< ! =. .A E ' = . .E .! .' .@0 .A
>X .E>XY .EXY ..> => @E A?> .>2 .
@= .@ ..=
/ .. .< .? .A @E A! . . .?
. .A ..=4! .@
. A ? ! .= ? # .= = ! @ ? . = .. < @ ! !Y E ! A ! ! E ! @ ! <. ! <
0# ..E0 ...;0 A;6)&5; < ?
F @@F @@F E<