100401 30 Act Colaborativa 3

METODOS NUMERICOS_100401_30 NUMERICOS_100401_30

TRABAJO COLABORATIVO 3 METODOS NUMÉRICOS

Grupo:

Tutor:

UNIVERSIDAD UNIVERSI DAD NACIONAL N ACIONAL ABIERTA Y A DISTANCIA DISTANCIA ESCUELA DE CIENCIAS BÁSICAS, TECNOLOGÍAS E INGENIERÍAS -ECBTIPROGRAMA QUÍMICA CEAD IBAGUÉ MAYO


INTRODUCCIÓN El pr!"# #r$%$&o 'ol$%or$#()o *l 'ur!o * +#o*o! "u+r('o! #(" l$ ("$l(*$* * $%or*$r u"$ !r( * &r'('(o! pr.'#('o!, '#u$"*o &+plo! * l$ U"(*$* /: D(r"'($'(0", I"#1r$'(0" Nu+r('$ 2 E'u$'(o"! D(r"'($l!3 Por llo " !#$ u"(*$* ! *!$rroll$r$" *!$rroll$r$" lo! 'o"#"(*o! *: D(r"'($'( D(r"'($'(0" 0" Nu+r('$, I"#1r$'(0" I"#1r$'(0" Nu+r('$, R1l$ *l #r$p'(o, R1l$ * S(+p!o", I"#1r$'(0" * Ro+%r1, M#o*o * Eulr, M#o*o * Ru"1-4u##$ 2 M#o*o Mul#(p$!o!3 Lo! 'o"'p#o! 5u ! *!$rroll$" " l$ pr!"# u"(*$* !o" (+por#$"#! por5u 'orr!po"*" $ l$! 6rr$+("#$! u"'(o"$l! * #o*o pro'!o +$#+.#('o 2 !r." * )(#$l (+por#$"'($ " l *!$rrollo * l$ #+.#('$ *l 'ur!o +#o*o! "u+r('o!3 Al +o+"#o * $pl('$r l$! M$#+.#('$! $ !(#u$'(o"! *l +u"*o r$l "o! "'o"# o"#r$+ r$+o! $ +"u*o 'o" pro%l+ %l+$ $! 5u "o pu* *" !r r! r!ul#o! $"$l7#('$+"# o * +$"r$ 8$'#$ 2 'u2$ !olu'(0" *% !r $%or*$*$ 'o" $2u*$ * $l19" pro'*(+("#o "u+r('o3 Lo! +#o*o! "u+r('o! 'o"!#(#u2" pro'*(+("#o! $l#r"$#()o! pro)'6o!o! p$r$ r!ol)r pro%l+$! +$#+.#('o!, p$r$ lo! 'u$l! ! *(('ul#$ l$ u#(l($'(0" * +#o*o! $"$l7#('o! #r$*('(o"$l! 2, o'$!(o"$l+"#, !o" l$ 9"('$ op'(0" po!(%l * !olu'( !olu'(0"3 0"3 So" #'"('$ #'"('$!! !(!#+ !(!#+.#( .#('$! '$! 'u2o! 'u2o! r!ul#$ r!ul#$*o! *o! !o" $pro8( $pro8(+$' +$'(o" (o"! ! *l )r*$*ro )$lor 5u $!u+ l$ )$r($%l * ("#r!; l$ rp#('(0" 'o"!(!#"# * l$ #'"('$, $ lo 'u$l ! l *"o+("$ (#r$'(o"!, ! lo 5u pr+(# $'r'$r! '$*$ ) +.! $l )$lor %u!'$*o3 E! por "* 5u por +*(o *l pr!"# #r$%$&o ! pr#"* $pl('$r l$! #+.#('$! *l 'ur!o 'orr!po"*("#! $ l$ U"(*$* / 2 $'r'$r"o! u" po'o +.! $ lo! +#o*o! propu!#o! p$r$ !olu'(o"$r pro%l+$!3


OBJETIVOS OBJETIVO GENERAL M*($"# l *!$rrollo * lo! &r'('(o! propu!#o! p$r$ l #r$%$&o 'ol$%or$#()o $pl('$r lo! 'o"'p#o! $pr"*(*o! " l$ u"(*$* /3

OBJETIVOS ESPECIFICOS D!$rro D!$rroll$ ll$rr lo! &r'( &r'('(o '(o!! u#(l( u#(l($"* $"*o o lo! +#o*o! +#o*o! !#u*( !#u*($*o! $*o! " l$ u"(*$* u"(*$* /, D(r"'($'(0" Nu+r('$, R1l$ *l Tr$p'(o, R1l$ R1l$ * S(+p!o", I"#1r$'(0" * Ro+%r1, I"#1r$l! Mul#(pl!, M#o*o * Eulr, M#o*o! * T$2lor 2 M#o*o * Ru"1 4u##$3


Guía Int!ra"a " A#t$%$"a"& ' Pa&o 3

() P*an P*ant t + &o*u &o*u#$ #$on on "o& "o& , ,r r#$ #$#$ #$o& o& &o-r &o-r D$. D$.r rn# n#$a $a#$ #$/n /n Nu01 Nu01r$ r$#a #a 2*$#an"o a&o a a&o * ro#"$0$nto ut$*$4a"o) Ejemplo 1:

E,r#$#$o "$.rn#$a 5a#$a atr6& S$ l$ u"'(0" <8= > L"<8= #$" < 18= '$l'ul$r l$ *r()$*$ por +#o*o! "u+r('o! " l pu"#o 8>? 'o" 6> @3 $pl('$"*o l$ or+ul$ *(r"'($ 6$'($ $#r.!

() S r$l($ #$%l$ 8 F<8=

/3 @3/

/3 3@/

/3 3?

?3@ 3@H@

?3 3@@H

?3 3HH

?3/ /3//?

7) S $pl('$ l$ pr(+r$ 0r+ul$ * l$ *(r"'($ 6$'($ $#r.!
( x 0−1 ) f ( ( x0 ) − f ( h

se remplazala remplazala formula formula

f ’ ( x x 0 ) =

1.6050 −1.2849 =3.201 0.1

S '$l'ul$ l rror

|

Er =

E=

V v −V a V v

|

3.534 −3.201 X 100 =9.4 3.534

NOTA: NOTA: /3H/? )$lor )r*$*ro )r *$*ro * l$ *r()$*$ *r ()$*$

3) C$l'ul$+o! l$ !1u"*$ *r()$*$ S $pl('$ l$ !1u"*$ 0r+ul$ * l$ *(r"'($ 6$'($ $#r.!

f ´ ( x x 0 ) =

f ( ( x0 ) −2 f ( ( x 0−1 ) + f ( ( x 0−2 ) 2

h

se remplaz emplazala ala formul formula a

f ’ ( x x 0 ) =

1.6050 −2 ( 1.2894 ) + 1.0326 2

( 0.1)

S '$l'ul$ l rror For+ul$

|

Er =

E=

V v −V a V v

=5.88

|

8.611 −5.88 X 100 =31.71 8.611

NOTA: NOTA: 3 )$lor )r*$*ro * l$ *r()$*$

8) S $pl('$ l$ !1u"*$ *(r"'($ p$r$ l$ pr(+r$ *r()$*$ For+ul$ f ' ' ( x x 0 ) =

3 f ( ( x0 ) −4 f ( ( x 0−1 ) + f ( ( x 0−2 ) 2h

reempla reemplazamosla zamosla formula formula x

( ¿¿ 0 )=

3 ( 1.6050 ) − 4 ( 1.2894 ) + 1.0326 2 ( 0.1 ) ' '

f ¿

S '$l'ul$ l rror For+ul$

|

Er = E=

V v −V a V v

|

3.534 −3.45 x 100=2.3 3.534

= 3.45


9) S $pl('$ l$ !1u"*$ *r()$*$ * l$ !1u"*$ *(r"'($ For+ul$ f ' ' ( x x 0 ) =

2 f ( ( x 0−3 ) x 0 ) −5 f ( ( x 0−1 ) + 4 f ( ( x x 0 −2 )− f ( ( x

h2

S r+pl$$ l$ or+ul$ 2 ( 1.6050 )−5 ( 1.2894 ) + 4 ( 1.0326 ) − 0.8173 =7.61 f ' ' ( ( x x )= ( 0.1 )2 0

S '$l'ul$ rror For+ul$

|

Er = E=

V v −V a V v

|

8.611 −7.61 x 100 =11.6 8.611

Ejemplo 2:

E" !# !(1u("# &+plo, "'o"#r$r+o! u"$ #$%l$, " l 'u$l ! +u!#r$ l$ po!('(0" p$r$ *#r+("$*o #(+po * u"$ p$r#7'ul$ +o)("*o! " l !p$'(o3 D%+o! '$l'ul$r l$ po!(%l $'lr$'(0" * l$ +(!+$3 S( l$ p$r#7'ul$ *!'r(% u" +o)( +o)(+( +(" "#o #o p$r$% p$r$%0l 0l(' ('o o *$*o *$*o por f ( ( t )=r 2 '$l'ul l rror * l$ $'lr$'(0" $pro8(+$*$3

, T, 2, @ @ @  @3@ @3@@@  @3@ @3@@@? / @3@/ @3@@@ ? @3@? @3@@ H @3@H @3@@H P$r$ !# '$!o po*+o! '$l'ul$r l$ $'lr$'(0" " #>@3@, @3@@, @3@@/ 2 @3@?3 f ´ ´ ( 0.01) ≈

1

[ 0.0004 −2 ( 0.0001 ) + 0 ] ≈ 2

2

0.01


f ´ ´ ( 0.01) ≈ f ´ ´ ( 0.02) ≈ f ´ ´ ( 0.01) ≈ f ´ ´ ( 0.03) ≈ f ´ ´ ( 0.03) ≈

f ´ ´ ( 0.04 ) ≈

f ´ ´ ( 0.04 ) ≈

1

[ 2

0.01 1

1 ]≈ 2 5000

[ 0.0009−2 ( 0.0004 )+ 0.0001 ]≈ 2

2

0.01

1

[ 2

0.01 1

0.01 1 0.01

1 0.01

1 0.01

2

1 ]≈ 2 5000

[ 0.0016 −2 ( 0.0009 )+ 0.0004 ] ≈ 2

[ 2

1 ]≈ 2 5000

[0.0025 −2 ( 0.0016 ) + 0.0009 ] ≈ 2

2

[ 2

1 ]≈ 2 5000

A6or$, #"("*o " 'u"#$ lo! r!ul#$*o! $"#r(or!, "'o"#r$+o! 5u l$ !1u"*$ *r()$*$ *  ! (1u$l $  <*o!= " #o*o l ("#r)$lo, " l 'u$l ! pu* *'(r 5u " !# '$!o ! o%#u)o u" rror * 'ro p$r$ !# &r'('(o3

7) So*u#$on * &$!u$nt ,r#$#$o ut$*$4an"o *a R!*a "* Tra#$o) n; 8< 8< 2

a)

2

x3 x 3 ∫ 1+ x1 /2 dx →∫ 1 +√ x dx 0 0

L7+(# ("r(or: $ >   3@ L7+(# !upr(or: % > @  @3@ N9+ro * !u%("#r)$lo!: ">?


L$ lo"1(#u* * u" !u%("#r)$lo:

h=

(b− a ) n

→ h=−0,5

x

x

xi

(¿¿ i ) f ¿ ¿

@

,@@@

/,//@



/,//@



,H@@

,H@



/,@/?@H



,@@@

@,H@@@@@



,@@@@@@

/

@,H@@

@,@//



@,???

?

@,@@@

@,@@@@@@



@,@@@@@@

i

(¿¿ i ) mf ¿ ¿

m

∑ ¿ 7,4942108722 2

x 3 ∫ 1+ √ x dx ≈ h2 Σ=−1,8735527180 0

2

-)

∫ √ x x (e )dx 3

x

1

L7+(# ("r(or: $ >   3@ L7+(# !upr(or: % >   3@ N9+ro * !u%("#r)$lo!: ">? L$ lo"1(#u* * u" !u%("#r)$lo:

h=

( b− a )

x

n

→ h=−0,25

x

xi

(¿¿ i ) f ¿ ¿

@

,@@@

,?H/



,?H/



,H@

,H?@



?,/



,H@@

,/H



/,?

i

m

(¿¿ i ) mf ¿ ¿


/

,H@

,/?@



/,@HH

?

,@@@

,/H



,/H

∑ ¿ 15,2112674010 2

∫ √ x x (e )dx ≈ h2 Σ =−1,9014084251 3

x

1

3) So*u#$on& So*u#$on& *o& *o& &$!u$nt& &$!u$nt& ,r#$#$o ,r#$#$o& & ut$*$4an"o ut$*$4an"o *a R!*a R!*a " S$0&on S$0&on (=3 + 3=>) n; 8<

3

4

e x a .∫ dxb. ∫ e x ∈( x ) dx 1 x 2 3

e x Intera!i"n Intera!i"n 1 : ∫ dx 1 x

Regla de Simpson 1/3

Pr(+ro ! *()(* l ("#r)$lo J$,%K " " !u%("#r)$lo! * (1u$l $ lo"1(#u*3 S '$l'ul$ 6 h=

b −a n

D0"*: %>/; $>; ">? # x=

b − a 3− 1 2 = = =# x = 0,50 n 4 4

S '$l'ul$" l$! 'oor*"$*$! X i= a + i∗# x


x 0=1 + 0 ( 0,50 ) =1 x 1=1 + 1 ( 0,50 )=1.5 x 2=1 + 2 ( 0,50 )=2,0 x 3=1 + 3 ( 0,50 )= 2.5 x 4=1 + 4 ( 0,50 ) =3,0

C$l'ul$r l$! $l#ur$! * l$! u"'(o"!, r+pl$$+o! e1

e 1.5

1

1.5

f ( ( x 0 ) = =2.718 f ( ( x 1 )= e

=2.987

2

f ( ( x x 2 )= = 3.694 f ( ( x x 3 )= 2

f ( ( x x 4 ) =

e 3,0 3,0

=6.695

R+pl$$+o! f ( ( x x 4 ) f ( ( x x 3 ) + ¿ f ( ( x 2 ) + 4 ¿ f ( ( x 1) + 2 ¿ f ( ( x 0 ) + 4 ¿ h f ( ( x ) ≈ ¿ 3

b

∫¿ a

f ( ( 6.695 ) f ( ( 4.873 ) + ¿ f ( ( 3.694 )+ 4 ¿ f ( ( 2.987 ) + 2 ¿ f ( ( 2.718 )+ 4 ¿

( x ) ≈ f (

0,5 3

¿

b

∫¿ a

b

∫ f ( ( x x ) ≈ 0,167∗48.248 a

3

e x ∫1 x dx= 8.041

e

2.5

2.5

= 4.873


E* 6ra -a,o *a #ur%a & >)?8(


# x=

b − a 3− 1 2 = = =# x = 0,50 n 4 4

S '$l'ul$" l$! 'oor*"$*$! X i= a + i∗# x x 0=1 + 0 ( 0,50 ) =1 x 1=1 + 1 ( 0,50 )=1.5 x 2=1 + 2 ( 0,50 )=2,0 x 3=1 + 3 ( 0,50 )= 2.5 x 4=1 + 4 ( 0,50 ) =3,0

C$l'ul$r l$! $l#ur$! * l$! u"'(o"!, r+pl$$+o! e1

e 1.5

1

1.5

f ( ( x 0 ) = =2.718 f ( ( x 1 )= e

=2.987

2

f ( ( x x 2 )= = 3.694 f ( ( x x 3 )= 2

f ( ( x x 4 ) =

e 3,0 3,0

=6.695

R+pl$$+o! f ( ( x 4 ) f ( ( x x 3 ) +¿ f ( ( x2 ) + 3 ¿ f ( ( x1 ) + 3 ¿ f ( ( x x 0 ) + 3 ¿ 3 h f ( ( x ) ≈ ¿ 8

b

∫¿ a

e

2.5

2.5

= 4.873


R+pl$$+o! f ( ( 6.695 ) f ( ( 4.873 ) + ¿ f ( ( 3.694 )+ 3 ¿ f ( ( 2.987 ) + 3 ¿ f ( ( 2.718 ) + 3 ¿ b ∫ f ( ( x ) ≈ 0,5∗3 ¿ 8

a b

∫ f ( ( x x ) ≈ 0,188∗44.075 a

3

e x ∫1 x dx = 8.286

E* 6ra -a,o *a #ur%a & >)7>@ 4

Intera!i"n 2 : ∫ e x ∈( x ) dx 2


Pr(+ro ! *()(* l ("#r)$lo J$,%K " " !u%("#r)$lo! * (1u$l $ lo"1(#u*3 S '$l'ul$ 6 h=

b −a n

D0"*: %>?; $>; ">? # x=

b −a 4 −2 2 = = =# x =0,5 n 4 4

S '$l'ul$" l$! 'oor*"$*$! X i= a + i∗# x x 0=2 + 0 ( 0.5 ) =2 x 1=2 + 1 ( 0.5 ) =2.5 x 2=2 + 2 ( 0.5 )=3.0 x 3=2 + 3 ( 0.5 ) =3.5 x 4=2 + 4 ( 0.5 ) =4.0


C$l'ul$r l$! $l#ur$! * l$! u"'(o"!, r+pl$$+o! f ( ( x 0 ) =e ∈ ( 2 )=5.122 f ( ( x x 1 )= e ∈ ( 2.5 ) =11.162 2

2.5

f ( ( x x 2 )= e ∈( 3.0 ) =22.067 f ( ( x x 3 )= e ∈ ( 3.5 )= 41.485 3.0

3.5

f ( ( x x 4 ) =e 4.0 ∈ ( 4.0 )=75.689

R+pl$$+o! f ( ( x x 4 ) f ( ( x x 3 ) + ¿ f ( ( x 2 ) + 4 ¿ f ( ( x x 1) + 2 ¿ f ( ( x 0 ) + 4 ¿ h f ( ( x ) ≈ ¿ 3

b

∫¿ a

f ( ( 75.689 ) f ( ( 41.485 ) + ¿ f ( ( 22.067 )+ 4 ¿ f ( ( 11.162 )+ 2 ¿ f ( ( 5.122 ) + 4 ¿ f ( x ) ≈

0,5 3

¿

b

∫¿ a

b

∫ f ( ( x x ) ≈ 0,167∗335.533 a

2

∫ e ∈( x ) dx=55.922 x

1

E* 6ra -a,o *a #ur%a & 99)77



X i= a + i∗# x

x 0=2 + 0 ( 0.5 ) =2 x 1=2 + 1 ( 0.5 ) =2.5 x 2=2 + 2 ( 0.5 )=3.0 x 3=2 + 3 ( 0.5 ) =3.5 x 4=2 + 4 ( 0.5 ) =4.0

C$l'ul$r l$! $l#ur$! * l$! u"'(o"!, r+pl$$+o! f ( ( x 0 ) =e 2∈ ( 2 )=5.122 f ( ( x x 1 )= e2.5 ∈ ( 2.5 ) =11.162 f ( ( x x 2 )= e3.0 ∈( 3.0 ) =22.067 f ( ( x x 3 )= e3.5 ∈ ( 3.5 )= 41.485

f ( ( x x 4 ) =e 4.0 ∈( 4.0 )=75.689

R+pl$$+o! f ( ( x 4 ) f ( ( x x 3 ) + ¿ f ( ( x2 ) + 3 ¿ f ( ( x1 ) + 3 ¿ f ( ( x x 0 ) + 3 ¿ 3 f ( ( x ) ≈ h ¿ 8

b

∫¿ a

f ( ( 75.689 ) f ( ( 41.485 ) + ¿ f ( ( 22.067 ) + 3 ¿ f ( ( 11.162 )+3 ¿ f ( ( 5.122 ) + 3 ¿

( x ) ≈ f (

3∗0,5

b

∫¿ a

8

¿

METODOS NUMERICOS_100401_30 NUMERICOS_100401_30 b

∫ f ( ( x x ) ≈ 0,188∗304.953 a

2

∫ e ∈( x ) dx=57.179 x

1

E* 6ra -a,o *a #ur%a & 9)( 8) So*u#$on So*u#$on *o& *o& &$!u$nt& &$!u$nt& ,r#$#$ ,r#$#$o& o& ut$*$4an"o ut$*$4an"o *a *a Int!ra#$/n Int!ra#$/n " Ro0-r!) U&an"o &!0nto& " *on!$tu" ( (=7 + (=8)

So*u#$/n 2

∫ e x dx 3

1

S1+"#o!: , ,  $ll+o! I : 3

f ( ( x )= e x

E" !# '$!o ">: f ( ( a ) + f ( ( b )

I 1=

2

3

=

3

e1 +e 2

I : "> L$ 0r+ul$ '$+%($

2

≈ 1491.8


[

b

n− 1

(

b −a f ( ( a ) + f ( ( b ) b −a I = f ( ( x ) dx + 2∗ f a + $ n n 2 $ =1 a

∫

∑

)]

E" !# '$!o ">

[

(

1 b −a f ( ( a )+ f ( ( b ) b − a + 2∗∑ f a +$ I 2 = n n 2 $ =1

)]

R+pl$$"*o 1

I 2 =0.5

3

e +e

2

3

2

3 2

3

+ 2∗e =775.14

I/ : ">? E!#o! )$lor! !o" *%(*o! $ lo! !1+"#o! *$*o! L$ 0r+ul$ ! l$ +(!+$ 5u p$r$ I

[

b

n− 1 b −a f ( ( a ) + f ( ( b ) b −a I =∫ f ( ( x ) dx + 2∗∑ f a + $ n n 2 $ =1 a

(

[

(

3

b − a f ( ( a )+ f ( ( b ) b− a I 3 = + 2∗ f a + $ 2 n n $ = 1

∑

)]

R+pl$$"*o )$lor! I 3 =0.25

[

3

1

e + e 2

2

3

(

5 4

3

1 2

3

+ 2∗ e + e + e

Y$ #"+o! l pr(+r "()l:

3 4

3

)]= 374.82

)]


I 3 =374.82

I 2 =775.14 I 1 =1491.8

$ll+o! l !1u"*o "()l 'o" l$! or+ul$! 4 1 I m − I l 3 3

I m=interal interalmas masexa exa!ta !ta I l =interal interal menos exa!ta exa!ta

$ll$+o! *o! )$lor!: El pr(+r '$!o ! p$r$ lo! )$lor! * ">, 2 ">: 4 1 ∗775.14 − 1491 3 3

¿ 536.25333 P$r$ l '$!o *o"* lo! )$lor! * ">, 2 ">?: 4 1 ∗374.82 − 775.14 3 3

¿ 241.38

PARA EL NIVEL /: 16 1 I m− I l 15 15

A6or$ lo! )$lor! * I lo! o%#"+o! *l "()l $"#r(or: 16 1 ∗241.38− ∗536.25 =221.722 15 15


Por !# +#o*o l )$lor * l$ ("#1r$l ! *

221.722

?3% 4

∫e

x

ln ( x ) dx

2

I1u$l 5u " l &r'('(o $"#r(or u!$+o! lo! +(!+o! !1+"#o! 2 lo! +(!+o! )$lor! * ": <">, , ?=, $*+.! $>, %>?

$ll+o! I : x f ( ( x )= e ln ( x x )

E" !# '$!o ">: I 1=

f ( ( a ) + f ( ( b ) 2

e ln ( 2 )+ e ln ( 4 ) 2

=

4

2

≈ 32.0333

I : "> L$ 0r+ul$ '$+%($

[

b

n− 1 b −a f ( ( a ) + f ( ( b ) b a ( x ) dx + 2∗∑ f a + $ − I =∫ f ( 2 n n $ =1 a

(

E" !# '$!o ">

[

1

(

b −a f ( ( a )+ f ( ( b ) b− a I 2 = + 2∗ f a + $ n n 2 $ =1

∑

)]

R+pl$$"*o I 2 =1

[

e 2 ln (2 )+ e 4 ln ( 4 ) 2

]

+ 2∗e3 ln ( 3 ) =48.2

)]


I/ : ">? E!#o! )$lor! !o" *%(*o! $ lo! !1+"#o! *$*o! L$ 0r+ul$ ! l$ +(!+$ 5u p$r$ I

[

b

n− 1

(

b −a f ( ( a ) + f ( ( b ) b −a I = f ( ( x ) dx + 2∗ f a + $ n n 2 $ =1 a

∫

[

∑

3

(

b − a f ( ( a )+ f ( ( b ) b − a I 3 = + 2∗∑ f a +$ 2 n n $ = 1

)]

R+pl$$"*o )$lor!

( 2.5 ) +e 3 ln ( 3 ) + e 3.5 ln (3.5 ) e2.5 ln ¿ e2 ln ( 2 ) + e 4 ln ( 4 ) + 2∗¿ =181.5 2

I 3 =0.25 ¿

Y$ #"+o! l pr(+r "()l: I 3 =181.5

I 2 =48.2 I 1 =32.03

$ll+o! l !1u"*o "()l 'o" l$! or+ul$! 4 1 I m − I l 3 3

I m=interal interalmas masexa exa!ta !ta I l =interal interal menos exa!ta exa!ta

)]


$ll$+o! *o! )$lor!: El pr(+r '$!o ! p$r$ lo! )$lor! * ">, 2 ">: 4 1 ∗48.2− ∗32.03 3 3

¿ 53.59

P$r$ l '$!o *o"* lo! )$lor! * ">, 2 ">?: 4 1 ∗181.5 − ∗48.2 3 3

¿ 225.93 PARA EL NIVEL /: 16 1 I m− I l 15 15

A6or$ lo! )$lor! * I lo! o%#"+o! *l "()l $"#r(or: 16 1 ∗53.59− ∗225.93 15 15

¿ 42.1

El )$lor * l$ ("#1r$l ! $pro8(+$*$+"# ?3

9) So*u#$on So*u#$on *o& &$!u$nt &$!u$nt& & ,r#$#$o& ,r#$#$o& " Int!ra*& Int!ra*& M*t$* M*t$*& & #o0ru- #o0ru- u: 2

0.5 x

∫∫ e /

% x

3

0.1 x

d% .dx ≈ 0.0333054


{ } x

0.5

2

∫ ¿ ∫ e / d% % x

0.1

x

3

0.5

dx →∫ ¿ 0.1

{ } x

2

∫ e / d% % x

x

dx

3

1 2 x

∫∫ ( x x + % ) d% . dx≈ 1.000122 2

0

{

1

x

(

2

x % + lim % → x =+ x

% 4

}

2 x

∫ ¿ ∫ x + % 0

3

2

3

d% dx →

x

{

1

%

4

∫ ¿ x % + 4 0

2

)

}

dx

(

4

1 % x + 4 ) → lim % →2 x =− x x 2 % + = x 3 ( x 4 4 4

1 4

¿ 2 x 3 ( 2 x + 1 )− x 3

15 x ( x x + 4 )= 4

4

+ x 3

S '$l'ul$ l$ ("#1r$l ("*("(*$

∫

(

15 x 4

4

)

5

3 x + x dx = 4 3

x 4

+

(

3 x → lim 4 4 x →0

5

+

x 4 4

)=(

3 (0) 4

5

+

)=

( 0) 4 4

0

)

=2 x 3 ( 2 x +1 )


(

3 x lim 4 x→ 1

5

+

x

4

4

)=(

3 (1)

5

4

+

)=

( 1 )4 4

3 1 12 + 4 16 + = = =1 4 4 16 16

→=1− 0 → =1

@) D0o&trar D0o&trar u * %a*or %a*or aro2$0 aro2$0a"o a"o n 2 ; ?7 " *a &o*u#$/n &o*u#$/n "* "* ro-*0a ro-*0a " %a*or $n$#$a* +

;

  ?<; ? u&an"o * M1to"o " Eu*r #on 5 ; ?)?9

& 0 =0  & ??(99?@79

So*u#$/n S 6$ll$ l )$lor * x 1 % % 1 x 1= x 0+ h=0 + 0.05=0.05 % 1= % 0 + hf ( ( x 0  % 0) =0 + 0.05 f ( ( 0,0 )

¿ 0 +0.05∗0 =0 A6or$ por x 2 % % 2 x 2= x 1+ h=0.05 + 0.05

¿ 0.1 % 2= % 1 + hf ( ( x 1  % 1 )

¿ 0 + 0.05 f ( ( 0.05,0 )

¿ 0 +0.05∗0.05 =0.0025


A6or$ por x 3 % % 3 x 3= x 2+ h=0.1 + 0.05

¿ 0.15 % 3= % 2 + hf ( ( x 2  % 2 )

¿ 0.0025 + 0.05 f ( ( 0.1,0 .0025)

¿ 0.0025 +0.05∗0.1025 ¿ 0.007625

A6or$ por x 4 % % 4 x 4= x 3 + h =0.15 + 0.05 =0.2

% 4 = % 3+ hf ( ( x3  % 3 )=0.007625 + 0.05 f ( ( 0.15,0 .007625 )

¿ 0.007625 + 0.05∗0.157625

¿ 0.01550625 x n

f ( ( x n  % n )

%n

@ @

@

@

 @3@H

@

@3@H

 @3

@3@@H

@3@H


/ @3H

@3@@H

@3HH

? @3

?)?(99?@79

Co+o Co+o ! pu* pu*  o%! o%!r) r)$r $r " l$ #$%l$ $%l$ l )$lor $lor $pro $pro8( 8(+ +$*o $*o "

x =0,2

!

0,01550625

)

A*$#ar * * 01t 01to"o " " Ta Ta+*or " or or"n "o& "o& a *a #u #uaa#$/n % ´ =cos ( x% )  #on *a #on"$#$/n $n$#$a*: % ( 0 )=1

%´ 0= % ( 0) 2

´ ( + hf ( ( x i  % i ) + %´ ( 11= %

h

[ f x ( x x  % ) +f % ( x x  % ) f ( ( x x  % ) ] 2 '

'

i

i

i

i

% ' =cos ( x% x% ) =f ( ( x  % )

% ( 0 )=1 h= 0.1 [ 0,1 ]

f ' x ( x x  % )=− %sen ( x% x% )

f ' x ( x x  % )=− xsen ( x% x% )

% 0=1 Itera!i"n Itera!i"n 1 x =0 % 0=2

2

0.1 %´ i=1 +0.1cos ( o ( 1 ) )+ [−1 sen ( 0 (1 )) −0 sen ( 0 ( 1 ) ) cos ( 0 ( 1 ) ) ] 2

i

i


%´ 1=1 + 0.1 +

0.01 [ −0 − 0 ] 2

¿ 1 +0.1=1.1 Itera!i"n Itera!i"n 2 x =0.1 % 1=1.1 2

0.1 [− 1.1 sen ( 0.1 ( 1.1 ) )−0.15 sen ( 0.1 ( 1.1 ) ) cos ( 0.1 (1.1 )) ] %´ 2=1.1 +0.2cos ( 0.1 ( 1.1 ) ) + 2

%´ 2=1.1 +0.0993956 +

0.01 [ −0.120756 −0.0109115 ] 2

%´ 2=1.198703

Itera!i"n 3 x 2=0.2 %´ 2=1.1987 2

0.1 %´ 3=1,1987 + 0.1cos ( 0.2 ( 1.1987 ) )+ 2

[ −1.1987 sen ( 0.2 ( 1.1987 ) ) cos ( 0.2 ( 1.1987 ) ) ]

%´ 3=1.294186 ≈ 1.2942

Itera!i"n 4 x 3 =0.3 %´ 3=2.2942

2

0.1 %´ 4 =1.2942 + 0.2cos ( 0.3 ( 1,2942 ) )+ [ −1,2942 sen ( 0.3 ( 1.2942 ) ) −0.3 sen ( 0.3 ( 1.2942 ) ) cos ( 0.2( 1.2942 ) ) ] 2

%´ 4 =1.38395673 ≈ 1.3840


Itera!i"n 5 x 4=0.4 %´ 4=1.3840

%´ 5=1.3840 + 0.1 cos ( 0.4 ( 1.3840 ) ) +

2

0.1 2

[−1.304 sen ( 0.4 ( 1,3840 ) )−0.4 sen ( 0.4 ( 1.3840 ) ) cos ( 0.4 ( 1.3840 ) ) ]

%´ 5=1.465

Itera!i"n 6 x 5=0.5 %´ 5=1.465

2

0.1 %´ 6=1.465 +0.1cos ( 0.5 (1.465 ) )+ 2

[ −1.465 sen ( 0.5 ( 1.465 ) )−0.5 ( 1.465 ) ) cos (0.5 ( 1.465 ) )¿

%´ 6=1.533

Itera!i"n Itera!i"n 7 x 6=0.6 %´ 6= 1.533 2

0.1 %´ 7=1.533 +0.1cos ( 0.6 ( 1.533 ) ) + 2

[−1.533 sen ( 0.6 ( 1.533 ) )− 0.6 sen ( 0.6 (1.533 ) ) cos ( 0.6 ( 1.533 ) ) ]

%´ 7=1.5860 =1.586

Itera!i"n 8 x 7=0.7 %´ 7= 1.586

0.1 %´ 8=1.586 + 0.1cos ( 0.7 ( 1.586 ) ) + 2

2

[−1.586 sen ( 0.7 (1.586 ) )−0.6 sen ( 0.6 ( 1.586 ) ) cos ( 0.6 ( 1.586 ) ) ]


%´ 8=1.622

Itera!i"n 9 x 8=0.8 %´ 8=1.622

2

0.1 1.622 sen ( 0.8 (1.622 ) ) −0.8 sen ( 0.8 ( 1.622 ) ) cos ( 0.8 ( 1.622 ) ) %´ 9=1.622 +0.1cos ( 0.8 ( 1.622 ) ) + 2

[

%´ 9=1.64

Itera!i"n Itera!i"n 10 x 9=0.9 %´ 9=1.64

2

0.1 %´10=1.64 + 0.1cos ( 0.9 (1.64 ) ) + [−1.64 sen ( 0.9 (1.64 ) )− 0.9 sen ( 0.9 ( 1.64 ) ) cos ( 0.9 ( 1.64 ) ) ] 2

%´10=1.64

>) P*ant P*ant + &o*u#$on &o*u#$on a&o a&o a a&o un un ,r#$#$o ,r#$#$o or or * M1to"o M1to"o " Run!H Run!H utta " #uarto or"n) Pro-*0a !nra* d% x 0 ) = % 0 =f ( ( x  % )  % ( x dx

For0u*a x i+ 1= x i + h x 1=0 + 0.5 =0.5

]


% i+ 1= % i +

h 6

( $ 1+ 2 $ 2+2 $ 3 + $ 4 )

Don" $ 1= f ( ( x i  % i )

(

1 2

1 2

)

(

1 2

1 2

)

$ 2= f x x i + h  % i+ $ 1 h

$ 3= f x x i + h  % i+ $ 2 h

$ 4= f ( ( x x i + h  % i + $ 3 h )

E,r#$#$o) R&o*%r * ro-*0a "* %a*or $n$#$a* 5; ?)9 d% x + 1 )∗cos ( x x2 + 2 x )  % ( 0 ) =4 =( % + 1 )∗( x dx

% 2= % 1 +

h 6

( $ 1+ 2 $ 2+ 2 $ 3 + $ 4 )

Pr$0r a&o $ 1= f ( ( 0, 4 )=( 4 + 1 )∗ ( 0 + 1 )∗cos ( 02 + 2∗0 ) $ 1= f ( ( 0, 4 )=( 5 )∗( 1 )∗cos ( 0 )

¿5


(

1 2

1 2

$ 2= f 0 + ∗0.5,4 + 5∗0.5

)

$ 2= f ( ( 0.25,5.25 )=( 5.25 + 1 )∗( 0.25 + 1 )∗cos ( 0.252 + 2∗0.25 ) $ 2= f ( ( 0.25,5.25 )=( 5.25 + 1 )∗ ( 0.25 + 1 )∗cos ( 0.25 + 2∗0.25 ) 2

$ 2= f ( ( 0.25,5.25 )=( 6.25 )∗( 1.25 )∗cos ( 0.5625 )

¿ 6.609

(

1 2

1 2

$ 3= f 0 + 0.5,4 + 6.609∗0.5

)

$ 3= f ( ( 0.25, 0.25, 4 + 5.652 ) $ 3= f ( ( 0.25, 5.652)=( 5.652 + 1 )∗( 0.25 + 1 )∗cos ( 0.5625 )

¿ 7.034

$ 4= f ( ( 0 + 0.5,4 + 7.034∗0.5 ) $ 4= f ( ( 0.5,7.534 )=( 7.534 + 1 )∗( 0.5 + 1 )∗cos ( 0.52 + 2∗0.5 ) $ 4= f ( ( 0.5,7.534 )=( 8.534 )∗( 1.5 )∗cos ( 1.25 )

¿ 4.036


% 2= 4 +

0.5 ( 5 + 2∗6.609 +2∗7.034 +4.036 ) 6

% 2= 4 +

0.5 ( 5 + 2∗6.609 +2∗7.034 +4.036 ) 6

¿ 7.027

S!un"o a&o x =0.5 % =7.027

$ 1= f ( ( 0.5,7.027 )=( 7.027 + 1 )∗( 0.5 + 1 )∗cos ( 0.52 + 2∗0.5 )

$ 1= f ( ( 0.5,7.027 )=( 8.027 )∗( 1.5 )∗cos ( 0.25 + 1 )

¿ 3.797

(

1 2

1 2

$ 2= f 0.5 + ∗0.5,7.027 + 3.797∗0.5

)

2 $ 2= f ( ( 0.75, 0.75, 8 )=( 8 + 1 )∗( 0.75 + 1 )∗cos ( 0.75 + 2∗0.75 )

¿− 7.436

(

1 2

1 2

$ 3= f 0.5 + 0.5,7.027 + (−7.436 )∗0.5

)

$ 3= f ( ( 0.75, 5.168) $ 3= f ( ( 0.75, 5.168 )=( 5.168 + 1 )∗( 0.75 + 1 )∗cos ( 0.752 + 2∗0.75 )


$ 3= f ( ( 0.75, 5.168)=( 6.168 )∗( 1.75 )∗cos ( 0.752 + 2∗0.75 )

¿− 5.1 $ 4= f ( ( 0.5 +0.5,7.027 +(−5.1 )∗0.5 ) $ 4= f ( ( 1,4.477 )=( 4.477 + 1 )∗( 1 + 1 )∗cos ( 12 + 2∗1 ) $ 4= f ( ( 1,4.477 )=( 5.477 )∗( 2 )∗cos ( 3 )

¿− 10.844 x 1=0..5 + 0.5=1

% 1=7.027 +

0.5 ( 3.797 +2∗(−7.436 ) + 2∗(−5.1 )+(−10.844 ) ) 6

¿ 4.35

CONCLUSIONES 

E! (+por#$"# $"#! * ("('($r ("('($r u" #r$%$&o #r$%$&o 'ol$%or$#() 'ol$%or$#()o, o, 'o"o'r  (*"#(('$r (*"#(('$r l$ #+. #+.#(#('$ '$ pl$" pl$"# #$* $*$, $, lo! lo! o%& o%&#(#()o )o!! !p !pr$ r$*o *o!! 2 l$! l$! $'#( $'#()( )(*$ *$* *!! $ *!$rroll$r; !#o 'o" l (" * prou"*($r  ("*$1$r " l 'o"#"(*o 2 !#$%l'r u" 'ro"o1r$+$ * #r$%$&o 5u $!1ur l 'u+pl(+("#o * l$! +#$! !#(pul$*$!3



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A%$lo!, C3 C3 <@@H=3 Revisión acerca de las normas para la presentación de referencias bibliográficas según el estilo de la American Psychological Association (APA)3 E8#r$7*o l / * M$ro * @ *!* l$ %$! * *$#o! E-L(%ro " l$

orl* (* %: 6##p:!(#3%r$r23'o+l(%u"$*!p

Bu'6l( C6$)!, C$rlo! I)." <@/=3 Modulo métodos numéricos !nidad " 3 U"()r!(*$* N$'(o"$l A%(r#$ 2 $ D(!#$"'($3 D(!po"(%l ": 6##p:*$#$#'$3u"$*3*u3'o'o"#"(*o!@@?@Mo*uloU"(*$*/3p*


C$"$l!, T3 T3 <@@, ? * M$2o=3 For+$#o For +$#o APA APA  Qu("#$ E*('(0"3 E8#r$7*o l / * M$ro * @ *!* 6##p:3+(!#$r$!3'o+3)l$!-"or+$!-$p$3p6p

C6$pr$, S#)" C3  C$"$l, R$2+o"* P3 <=3 Métodos numéricos para ingenieros3 <$p$#$

S, C$rlo!  Cor#! A"$2$, Alr*o, #r$*3=M8('o3 M'Gr$-

(ll3 D(!po"(%l ": 6##p:?3@3H/3?#!(u$+(L(%ro!L/?3p*

E!#u*($"#! * l$ Fu"*$'(o" U"()r!(#$r($ *l Ar$ A"*("$3 < * No)(+%r * @/=3 Tu#or($l3 Método de #uler y #uler Me$orado 3 O%#"(*o * 6##p!:32ou#u%3'o+$#'6)>PW/2T%E F7!('$ C("'($, ? * D('(+%r * @3Método de %impson &'"  ntegración *umérica  Regla de %impson &'"+ R'upr$*o * l$ p.1("$ %

6##p:32ou#u%3'o+ R2! , Lu'll2 <"3*=3 Método numérico de Runge,ut #$3 #$3 R'upr$*o * l$ p.1("$ %: 6##p:(!('$3u*$3*u3'o Vr$ C6.), I!$$' <@/=3 Método de Romberg 3 V7*o #u#or($l3 D(!po"(%l ": 6##p!:32ou#u%3'o+$#'6)>l??6SGuWY4


100401 30 Act Colaborativa 3

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