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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 NACIONAL ABIERTA Y A DISTANCIA
ESCUELA DE CIENCIAS B#SICAS$ TECNOLOG%A E INGENIERIA
MANI&ALES
201
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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−VaI
ε r εa
Ve ;100
A5/'@),)-+ .- 5/ ), ! )+/'( , />'$ 5/' '/ @+' / , 60 )+/'($.)'( '( //'/( 5/ / .8-+' -'( .')'(;
V! )+/'(
V 60 )+/'(
ε a I 60!I 2
ε r
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$3444444444444
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P/ +/.-/ (+'( -)/'( 4 ,?>+'( ,)($ (=' '-(,/)'( '( 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$61;
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$. C'-(+/./ .- .,/' ')5/+' , '( )*+','( 5/ ./ /? , .- .=-
+--,' - .-+ -)/' , +/'-($ '-,'-($ 5/'@)'-( '/).$.(+/8-,'' '- )-'( .- <)5';
Solución:
"%todo It&racion&' (or)ula
B(=- T/< ('/ -+/' - . ( /5+ (+ ,+/)-/ '-/>-$ ( (+-+ -+'$
5/' ( ).' )8( (>./' 5/-'-+// '-/>-
f ( xa ) f ( xb )<0
R> ( ')- ' )<'/ , )*+',' ,(=- , )*+',' , (-+$ '>/ ,+/)-/ .- '-/>- 5'/ ),' , .-.=-$ )*+',' ( ).' )8(/85,' . (=-
a¿k ¿¿−f ¿
f (bk )¿
ck =f ( bk ) ak −f (ak ) bk
¿
NK+'-R5('-
N' +/< ('/ -+/'($ (-' ('/.- =/). , 5/'('( +/+'($' . ' )8( /85,'$ (-)/>' 5/(-+ >.-(,.+,( 5'/ <)5' E(-(/' '-'/ ,/, , .-=-$ . ( ( ,? ,'+-/$ S ,/, , .-=-+') .- '/ /-' /'$ )*+',' 5., -' '-/>/$
C.-,' /?( )+5(5'-')'($ )*+',' (;
xn+1= xn− f ( xn )f ´ ( xn )
' +/+' ,5.-+' <'
P/)+ /('/ ((+)( ,.'-( -' -(/)-+-(; E- 5/+./ ( 5.,.+/ 5/ ,+/)-/ /?( ,.- .-=- , '/)$ ( .-
f ( x )=0
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)*+',' ()/ , (=- . ( ( - 5/'@)'-((.(($ 5'/ ' . ('-(,/,' 5'' -+')5/,' '- '+/'( )+','(
*. D)'(+// . @ @3 4@2 10 +- .- /? - 1$ 2 .+-,' M*+',' ,(=- ,+/)- .- 5/'@)=- /? '- .- 5/(=- , )-'( 104
+.
Solución
f ( x )= x3+4 x2−10
f (1 )=13+4(1)2−10=−5<0
f (2 )=23+4(2)0=14>0
x1=a+b
2 =
1+22 =
3
2
3
2¿2−10=
19
8 >0
3
2¿3+4¿
f (32 )=¿
f (1 ) . f ( 32 )=−9518<0, Aquiestarala raiz
f (2 ) . f ( 32 )=33.25>0, No estarala raiz
x2=1+
3
2
2 =
5
4
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5
4 ¿2−10=
−115
64 <0
5
4 ¿3+4¿
f
(5
4
)=¿
x3=
5
4+3
2
2 =
11
8
11
8 ¿2−10=
83
512>0
11
8 ¿3+4¿
f
(118 )=¿
x4=
5
4+11
8
2 =
21
16
21
16 ¿2−10=
−3475
4096 >0
21
16
¿3+4¿
f ( 2116 )=¿
x5=
21
16+11
8
2 =
43
32
f ( 4332 )=−0.35098<0
x6=
43
32+11
8
2 =
87
64
f ( 8764 )=−0.09641<0
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x7=
87
64+11
8
2 =
175
128
f (175128 )=0.03236>0
x8=
87
64+175
1282
=349256
f
(349
256 )=−0.03215<0
x9=
349
256+175
128
2 =
699
512
f (699512 )=0.000072>0
x10=
349
256+699
512
2 =
1397
1024
f (13971024 )=−0.01605<0
x11=
1397
1024+699
512
2 =
2795
2048
f (27952048 )=−0.007989<0
x12=
2795
2048+699
512
2 =1.36499
f (1.36499 )=−0.003963<0
x13=1.36499+
699
512
2 =1.36511
error=2−1
213 =0.00012
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f (1.36511)=−0.00198
x14=1.36511+
699
512
2 =1.36517
error=2−1
214 =0.0000610
,. U(-,' M*+',' , R> ( 5/'@)/ /? , Q 3$2 Q 0$
- -+/' 3$ 4 '- 0$001
Solución
(3)3.2sin(3)−0.5cos¿
f ( xa )=e−3¿
f ( xa )=3.7138
(4)3.2sin(4 )−0.5cos¿
f ( b=4 )=e−4¿
f ( b=4 )=3.8205
(3.7138 ) (3.8205 )=14.19>0 ;
T--,' - .-+ 5/',)-+' -+/'/$ -' /?;
-. S .-=- Q $ 5/'@)/ ),-+ M*+',' , NK+'-R5('- /?@ 0$ +')-,' ')' '/ - @'0;6$ '- .- @++., , 10
Solución
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f ( x )=ln ( x2+1 )−e x2cos πx
xn+1= xn− f ( xn)f ' ( xn)
π .0.6¿
f ( x )= 2 x
x2+1+eπ x
2sin (π x )−2excos ¿
x0=0.6
0.6¿2 sin (π .0.6 )−2e (0.6)cos(π .0.6)2(0.6)
0.62+1 +eπ ¿
ln(0.62+1)¿
x0+1=0.6−¿
x1=0.473315
E=|0.473315−0.6
0.473315 |=0.267
x1¿2sin ( π x1 )−2e ( x1)cos(π x1)
2 x1
x1
2+1+eπ ¿
ln ( x1
2+1)−e1
2cos (π x x1)
¿ x2=0.473315−¿
x2=0.411978
E=|0.411978−0.473315
0.411978 |=0.14888
x3=0.391263
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E=|0.391263−0.411978
0.411978 |=0.05294
x4=0.388727
E=|0.388727−0.391263
0.388727 |=6.52386 x 10−3
x5=0.388691
E=|0.388691−0.388727
0.388691 |=9.2618 x 10−5
. U(/ M*+',' +/+' , 5.-+' <' 5/ 5/'@)/ /? , 2 Q 4 Q$')--,' '- @'0$ '- +/'-(;
Solución
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f ( x )= x2−4 x−e
x
x2−4 x−e
x=0
x2−4 x=e
x
x [ x−4 ]=e x
→ x= e
x
x−4
g ( x )= e x
x−4
x0=0
x1= e0
0−4=−1
4
x2= e
−1 /4
−1
4 −4
=−0.18325
x3= e
−0.18325
(−0.18325 )−4=−0.19902
x4= e−0.19902
(−0.19902 )−4=−0.195173
x5= e
−0.195173
(−0.195173 )−4=−0.196105
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!onclu'ion&'
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