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METODOS NUMERICOS TRABAJO COLABORATIVO 1
JUAN PABLO ECHEVERRY RIVERA CODIGO: 1060646331 EDWIN GERARDO CASTILLO CODIGO: 1112619137 CRISTIAN CAMILO ESCUDERO CODIGO: 10377!96
CURSO 100401"!3 TUTOR JOSE ADEL BARRERA
UNIVERSIDAD UNIVERSIDA D NACIONAL NACIO NAL ABIERTA Y A DISTANCIA DISTANCIA ESCUELA DE CIENCIAS B#SICAS$ TECNOLOG%A E INGENIERIA
MANI&ALES 201
2
Introducción
L'( )*+','( -.)*/'( ('- +*-( ),-+ ( .( ( 5'( '/)./ 5/')( )+)8+'( , + '/) . 5.,- /('/( .(-,' '5/'-( /+)*+($ (+'( )*+','( -'( .- 5+'( 5/ -+-,/ (.)( -.)*/'( , /('/ 5/')( )+)8+'($ /,./ (.)( -.)*/'( 8('(; E 5/(-+ +/<' ''/+' +- ')' - ,/ ('.=- '( 5/')( 5-+,'( - >.? , +,,($ 5/ .-,, 1$ - . ( ,(//'/'- '-+-,'( ')' )*+',' , /> ($ )*+',' , (=-$ @++.,$ 5/(=-$ /,'-,'$ -+/ '+/'(;
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Trabajo colaborativo 1
1. D(, (. )5' , '/)=- 5-+ , ('.=- ,'( <)5'( ('/ '( +5'( ,
//'/( //'/ ('.+'$ /+'$ //'/ /+' 5/'@),'$ //'/ 5'/ +/.-)-+' 5'/ /,'-,'$ +--,' - .-+ 5/(=- @++., , '( )()'(; Solución: VALOR ABSOLUTO Y VALOR RELATIVO
ε
a IVe IVe −VaI
ε r
εa Ve ;100
A5/'@),)-+ .- 5/ ), ! )+/'( , />'$ 5/' '/ @+' / , 60 )+/'($ .)'( '( //'/( 5/ / .8-+' -'( .')'(;
V! )+/'( V 60 )+/'( ε a I 60!I 2 εr
2 60
;100 3$3 F
ERROR POR TRU!A"IETO
E- (.)5' )+)8+' , -8(( -.)*/'$ +/.-)-+' ( +*/)-' .(,' 5/ /,./ -)/' , ,?>+'( ,/ , (5/,'/ ,)$ ,(/+-,' '( )-'( (>-+'(; P'/ <)5' ,,'( '( -)/'( /(: 3$1419263!979;;; 32$43!1912!! 6$3444444444444
4
P/ +/.-/ (+'( -)/'( 4 ,?>+'( ,)($ (=' '-(,/)'( '( 4 ,?>+'( ,/ , ') ,); E /(.+,' (: 3$141 32$43!1 6$3444 ERROR POR REO#EO
L( />( , /,'-,' ( 5- ,) (+.,' - (>.-+ 5'(=- -)/' , ,)( . ( ./ +/-('/)/$ ( ,/$ ( +-)'( .- -)/' , 3 ,)( ./)'( /,'-,/ 2$ ( 5/8 ( />( , /,'-,': D?>+' )-'/ . : S (>.-+ ,) ( )-'/ . $ -+/'/ -' ( )',; E<)5': 12$612; R,'-,-,' 2 ,)( ,/)'( +-/ - .-+ +// ,): 12$612 12$61;
5
$. C'-(+/./ .- .,/' ')5/+' , '( )*+','( 5/ ./ /? , .- .=-
+--,' - .-+ -)/' , +/'-($ '-,'-($ 5/'@)'-( '/).$ .(+/8-,'' '- )-'( .- <)5';
Solución:
"%todo
It&racion&'
B(=-
T/< (' ('/ -+/' - . ( /5+ (+ ,+/)-/ '-/>-$ ( (+-+ -+'$ 5/' ( ).' )8( (>./' 5/ -'-+// '-/>-
R> (
')- ' ' ) )<'/ <'/ , )*+',' , (=- , )*+',' , (-+$ '>/ ,+/)-/ .- '-/>- 5'/ ),' , .- .=-$ )*+',' ( ).' )8( /85,' . (=-
NK+'- R5('-
' +/+' , 5.-+' <'
N' +/< ('/ -+/'($ (-' ('/ .- =/). =/). , 5/'('( 5/'('( +/+'($ +/+'($ ' . . ' )8( )8( /85 /85,' ,'$$ (()/>' 5/(-+ >.-( ,.+,( 5'/ <)5' E( -(/' '-'/ ,/, , .-=-$ . ( ( ,? , '+-/$ S ,/, , .-=+') .- '/ /-' /'$ )*+',' 5., , -' '-/>/$ C.-,' /?( )+5( 5' 5'-') ')'( '($ $ )*+' )*+',' ,' (( ; P/)+ /('/ ((+)( , .'-( -' -(/)-+ -(; E- 5/+./ ( 5., .+/ 5/ ,+/)-/ /?( , .- .-=- , '/)$ ( .-
(or)ula
f ( ( x a ) f ( ( x b ) < 0
a
¿
k
¿¿ −f ¿ ¿ f ( ( b k ) ¿ f ( ( bk ) ak −f ( ( a k ) bk c k = ¿ x n+1= x n−
f ( ( x )=0
f ( ( x x n ) f ´ ( x n )
6
)*+',' ()/ , (=- . ( ( - 5/'@)'-( (.(($ 5'/ ' . ( '-(,/,' 5'' -+ ')5/,' '- '+/'( )+','(
*. D)'(+// . @ @3 4@2 10 +- .- /? - 1$ 2 .+-,' M*+',' ,
(=- ,+/)- .- 5/'@)=- /? '- .- 5/(=- , )-'( 104 +.
Solución
f ( ( x )= x + 4 x −10 3
2
f ( ( 1 )=1 + 4 ( 1 ) −10 =−5 < 0 3
2
f ( ( 2 )= 2 + 4 ( 2) 0= 14 > 0 3
x 1=
a +b 2
=
1 +2 3 = 2 2
3 2 19 ¿ −10 = > 0 2 8 3 3 ¿ +4¿ 2 3 f =¿ 2
()
f ( ( 1 ) . f
( )=−
f ( ( 2 ) . f
( )=
3 2
1+
x 2=
2
3 2 3 2
9518 < 0, Aquiestara Aqui estarala la raiz
33.25 > 0, No estarala estara laraiz raiz
=
5 4
7
5 2 −115 ¿ −10 = <0 4 64 5 3 ¿ +4 ¿ 4 5 f =¿ 4
()
5 3 + 4 2 x 3= 2
=
11 8
11 2 83 ¿ −10 = >0 8 512 11 3 ¿ +4¿ 8 11 =¿ f 8
( )
x 4=
5 11 + 4 8 21 = 2 16
21 2 −3475 ¿ −10 = >0 16 4096 21 3 ¿ +4¿ 16 21 f =¿ 16
( )
21 11 + 16 8 43 x 5= = 2 32
f
( )=− 43 32
x 6=
f
0.35098 < 0
43 11 + 32 8 87 = 2 64
( )=− 87 64
0.09641 < 0
8
87 11 + 64 8 175 x 7= = 2 128
f
87 175 + 64 128 349 x 8= = 2 256
( )= 175 128
f
( )=− 349 256
349 175 + 256 128 699 x 9= = 2 512
f
( )= 699 512
0.000072 > 0
349 699 + 256 512 1397 x 10 = = 2 1024
f
( )=− 1397 1024
0.01605 < 0
1397 699 + 1024 512 2795 x 11= = 2 2048
f
( )=− 2795 2048
0.007989 < 0
2795 699 + 2048 512 x 12= =1.36499 2
f ( ( 1.36499 )=−0.003963 < 0 1.36499 +
x 13 =
error =
2 2−1 2
13
699 512
=1.36511
= 0.00012
0.03236 > 0
0.03215 < 0
9
f ( ( 1.36511 ) =−0.00198 1.36511+
x 14 =
699 512
2
error =
2−1 2
14
=1.36517
= 0.0000610
,. U(-,' M*+',' , R> ( 5/'@)/ /? , Q 3$2 Q 0$
- -+/' 3$ 4 '- 0$001
Solución
( 3) 3.2 sin ( 3 )−0.5cos ¿ f ( ( x x a ) =e −3 ¿ f ( ( x a ) =3.7138
( 4) 3.2 sin ( 4 )−0.5 cos ¿ f ( ( b =4 )= e− 4 ¿ f ( ( b =4 )=3.8205
( 3.7138 ) (3.8205 )=14.19 > 0 ; T--,' T--,' - .-+ 5/',)-+' -+/'/$ -' /?;
-. S .-=- Q $ 5/'@)/ ),-+ M*+',' , NK+'-
R5('- /?@ 0$ +')-,' ')' '/ - @'0;6$ '- .- @++., , 10
Solución
10
f ( ( x )= ln ( x x + 1 )− e x cos πx 2
x n+ 1= x n−
2
f ( x n) f ' ( x n) π . 0.6
¿
f ( ( x )=
2 x
x + 1 2
+eπ x 2 sin (π x )−2 ex cos ¿
x 0=0.6 0.6 ¿ sin ( π .0.6 ) − 2 e ( 0.6 ) cos ( π .0.6 ) 2 ( 0.6 ) 2
0.6 0. 6
2
+ eπ ¿
+1
2
ln ( 0. 6
+1 )
¿ x 0+1=0.6 −¿ x 1=0.473315
|
E=
0.473315 −0.6 0.473315
|=
0.267
x 1 ¿ sin ( π x 1 )−2 e ( x 1) cos ( π x 1) 2
2 x 1 2
x 1+ 1
(
2
ln x1 + 1
+ eπ ¿
)−e cos ( π x ) 2 1
x
1
¿
x 2= 0.473315−¿
x 2=0.411978
|
E=
0.411978−0.473315 0.411978
x 3=0.391263
|=
0.14888
11
|
E=
0.391263 −0.411978 0.411978
|=
0.05294
x 4= 0.388727
|
E=
|
0.388727 −0.391263 0.388727
= 6.52386 x 10−3
x 5=0.388691
|
E=
|=
0.388691 −0.388727 0.388691
−5
9.2618 x 10
. U(/ M*+',' +/+' , 5.-+' <' 5/ 5/'@)/ /? , 2 Q 4 Q$
')--,' '- @'0$ '- +/'-(; Solución
12
f ( ( x )= x − 4 x −e
x
2
2
x
2
x
x −4 x −e =0 x −4 x =e
x
e
x [ x − 4 ] = e → x = x − 4 x
x e ( ) g x x = x − 4
x 0=0 0
x 1=
e
=
0− 4
−1 4
−1 /4
x 2=
e
−1 4
0.1832 5 =−0.1832
−4 −0.18325
x 3=
e
(−0.18325 ) −4
=−0.19902
−0.19902
x 4=
e
(−0.19902 )−4
0.195173 =−0.195173
−0.195173
x 5=
e
(−0.195173 ) −4
=−0.196105
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!onclu'ion&'
+5'( , //'/($ (.( /+/?(+( /+/?(+( A5/-,/ ,/- -+/ '( ,/-+( +5'( 5/-5( .('( . ( +--$ ,)8( , =)' ( ,/-- )')-+' , // '5/'-(
S -' '( '-5+'( (+'( , .-,, 1$ ),-+ ('.=- , '( </'( 5-+,'( - >.? , +,,(;
E( -+/(-+ / ')' '( )*+','( -.)*/'( - ),' ).' '/) , ()5/ </'(;
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RE(ERE!IAS
M=,.' , C./(' M*+','( -.)*/'(; R.5/,' 3 , A>'(+' , 201$ , A. /+.: 100401 C./(' M*+','( -.)*/'( , U-/(,, N'- A/+ D(+-: ++5:,++;.-,;,.;''-+-,'(100401MODULO"20132