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1

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



2

Introducción

L'( )*+','( -.)*/'( ('- +*-( ),-+ ( .( ( 5'( '/)./5/')( )+)8+'( , + '/) . 5.,- /('/( .(-,' '5/'-(/+)*+($ (+'( )*+','( -'( .- 5+'( 5/ -+-,/ (.)( -.)*/'( -, /('/ 5/')( )+)8+'($ /,./ (.)( -.)*/'( 8('(;

E 5/(-+ +/<' ''/+' +- ')' - ,/ ('.=- '( 5/')( 5-+,'( - >.? , +,,($ 5/ .-,, 1$ - . ( ,(//'/'- '-+-,'( ')' )*+',' , /> ($ )*+',' , (=-$ @++.,$ 5/(=-$ /,'-,'$ -+/ '+/'(;



3

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



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&' (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



6

)*+',' ()/ , (=- . ( ( - 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



7

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



8

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



9

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



10

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



11

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



12

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



13

!onclu'ion&'

A5/-,/ ,/- -+/ '( ,/-+( +5'( , //'/($ (.( /+/?(+( 5/-5( .('( . ( +--$ ,)8( , =)' ( ,/-- )')-+' , //

'5/'-(

S -' '( '-5+'( (+'( , .-,, 1$ ),-+ ('.=- , '( </'(5-+,'( - >.? , +,,(;

E( -+/(-+ / ')' '( )*+','( -.)*/'( - ),' ).' '/)

, ()5/ </'(;



14

RE(ERE!IAS

M=,.' , C./(' M*+','( -.)*/'(; R.5/,' 3 , A>'(+' , 201$ , A. /+.:100401 C./(' M*+','( -.)*/'( , U-/(,, N'- A/+ D(+-:

++5:,++;.-,;,.;''-+-,'(100401MODULO"20132

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