Bab2 Pengali Lagrange

8/13/2019 Bab2 Pengali Lagrange

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TUGASKALKULUS LANJUT

KELOMPOK 2

BAB2

DOSEN : Drs. SURYO WIDODO,MPD

NAMA KELOMPOK :NPM:

1.ASMAUL SHOLEKAH04.1.03.01.00052.IDA ELISA04.1.03.01.00143.MAY NUKE SURYANA04.1.04.01.00174.PRATIWI04.1.03.01.00265.QOMARIAH04.1.03.01.0027

6.TUTIK SURYANTI04.1.03.01.00417.NUR CHOLIS04.1.03.01.0024

INSTITU KEGURUAN DAN ILMU PENDIDIKAN PGRI KEDIRIJURUSAN PENDIDIKAN MATEMATIKA

2006

1


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BAB 2

PENGALI LAGRANGE2.1.TINJAUAN UMUM PENGALI LAGRANGE

Dalam Bab 1 tela !"t#$%#&&a$ 'a(a #$t#& me$e$t#&a$ $"la")$"la"ma&*"m#m (elat"+ !a$ m"$"m#m (elat"+ !a(" *#at# +#$,*" -a("abel !#a ta$ aa!a$/a *#at# e(*/a(ata$ te(te$t# ata# ' $*t(a"$t te( a!a +#$,*" te(*eb#t

ma# #$ -a("abel)-a("abel$/a.Teta " !alam ba$/a& ma*ala te(a a$ &"ta *e("$, !" a!a &a$ a!a&e$/ataa$ ba a e$e$t#a$ t"t"& ma&*"m#m ata# m"$"m#m *e("$, !""("$,"

!e$,a$ e(*/a(ata$ ata# &e$!ala /a$, a(#* !" e$# ".M"*al$/a *eb#a e(#*a aa$ "$,"$ me$"$,&at&a$ ( !#&*"$/a !a$me$'a(" t"t"& ma&*"m#m !a(" ( !#&*"$/a teta " "a te("&at a!a !a$a /a$,

te(*e!"a #$t#& ma&*#! te(*eb#t.Te(*e!"a *#at# 'a(a !"&e$al *eba,a" met !e Lagrange Multiplier

#$t#& me$a$,a$" ma*ala *e e(t" "$".#at# met !e #$t#& mem e( le *#at# $"la")$"la" ma&*"m#m (elat"+ ata# m"$"m#m (elat"+ !a(" +#$,*" + / /a$, !" e$,a(# " le & $!"*"

e(*/a(ata$ , / 8 0 te(!"(" ata* embe$t#&a$ +#$,*" e$ l $,.9 / 8 + / , /

De$,a$ e(*/a(ata$:

0=

x F

0=

y F

0=

F

;a$, me(# a&a$ */a(at e(l# #$t#& ma&*"m#m (elat"+ ma# #$ m"$"m#m

(elat"+ .Pa(amete( /a$, t"!a& te(,a$t#$, a!a !a$ / !"*eb#t Pengali

Lagrange.

2.2.


3/11

=>NT>? 1:Te$t#&a$ $"la" ma&*"m#m !a(" + / 8 / !e$,a$ */a(at:

G / 8 / @ 16 8 0

Ja ab:9 / 8 9 / , /

8/ / @ 16

016016

0

0

==+=

==+=

==+=

y x F

x x y

F

y y x

F

8 8AT"t"& &("t"* !"'a a" %"&a 8 !a$ / 8 !e$,a$ $"la" + / 8 8 64.

=>NT>? 2:eb#a ab("& mem ( !#&*" !#a ma'am me*"$ !a$ / !a$ +#$,*"

$,& * ,ab#$,a$ a!ala:= / 2 3 / @ 6/

U$t#& m"$"m#m&a$ b"a/a be(a a ba$/a& me*"$ !a(" *et"a %e$"* a(#*!" ( !#&*" %"&a &e*el#(# a$$/a a(#* be(%#mla 42 me*"$.

Ja ab:

Pe(*/a(ata$ /a$, a(#* !" e$# " / 8 42!"t#l"* : , / 8 / @ 42 8 0

+#$,*" e$ l $,$/a:9 / 8 = / , /

82 3 / @ 6/ / @ 42

042

063

032

=+

=+=

=++=

y x F

x y

F

y x x

F

Pe$/ele*a"a$ !a(" *"*tem "$" membe("&a$ 8 30 / 8 12 B6=

ma&a b"a/a ma&*"m#m !" e( le %"&a ab("& mem ( !#&*" 33 me*"$ !a$ me*"$ /.

=>NT>? 3:eb#a & ta& te(b#&a mem #$/a" - l#me te(te$t#. J"&a !""$,"$&a$a,a( ba a$ /a$, !" a&a" l#a*$/a *em"$"mal m#$,&"$ be(a a #(a$ & ta&

te(*eb#t. G#$a&a$ e$,al" la,(a$,e.

Ja ab:C l#me C 8 / 8 & $*ta$

L#a* 8 / 2 2/9 / 8 / C /

3


4/11

9 / 8 / 2 2/ / @ C

0

22022

202

202

==

+==++=

+==++=

+==++=

V xyz F

xy

y x xy y x

z

F

xz

z x xz z x

y

F

yz

z y yz z y

x

F

2222

yz yz xz xy xz

z x yz

z y +=++=+

y x yz xz == 22

222222

yz xz yz xy xy

y x xz

z x=

+=

+

xz xy 2=

82

1 / 8

2

1

C 8 / 8 2

1 8

2

1 3

3 8 2C

83 2V / 8

3 2V 83 221

V

=>NT>? 4:Te$t#&a$ - l#me te(be*a( !a(" & ta& te,a& l#(#* /a$, !a at!"l#&"*&a$ !alam el" * "!a 16 2 4/ 2 2 8 144. "*")*"*" & ta& *e%a%a( b"!a$,

& (!"$at. G#$a&a$ e$,al" la,(a$,e.

Ja ab:J"&a P / a!ala #%#$, & ta& /a$, tela me$/e$t# e(m#&aa$

el" * "!a ma&a - l"me & ta& :C 8 /

T#%#a$ &"ta "ala me$e$t#&a$ $"la" ma&*"m#m !a(" C be(*/a(at&a$:, / 8 16 2 4/ 2 2 @ 144 8 0

9 / 8 / 16 2 4/ 2 2 @ 144

01AA01AA

0AA0AA

032A032A

2

2

2

=+=+=

=+=+=

=+=+=

z xyz z xy z F

y xyz y xz y

F

x xyz x yz x F

0144B416 222 =++=

z y x

F

24/ 2 16 2 4/ 2 2 8 0

24/ 2 144 8 0 ata# /8 12

4

/


5/11

/ 32 2 8 0 12 32 2 80 03 2 =+ x

=+ 03 2

x80ta& be(la=+ 03 2 x 2 8 3 83

/ / 2 8 0 0A12 2 =+ y

3212

0012

012

2

2

2

==

==+

=+

y y

y

y

/ 1 2 8 0 01A122 =+ z

316

163

00316

0316

22

2

2

==

==+=+

z z

z

z

8334

C ma&* 8 / 8 3 2 3 334

8 543

=>NT>? 5:J"&a + / 8 4 2 / 2 5 2 te$t#&a$ t"t"& a!a b"!a$, 2 3/

4 8 12 !"ma$a + / mem #$/a" $"la" te(&e'"l.

Ja ab:


6/11

U$t#& mem e( le $"la")$"la" (elat"+ ma&*"m#m ata# m"$"m#m !a("+#$,*" 9 / !e$,a$ e(*/a(ata$ / 8 0 !"be$t#& +#$,*" emba$t#.

G / 8 9 / /;a$, a(#* meme$# " e(*/a(ata$

xG

8 0 yG

8 0

z G

8 0 !a$

G

8 0Met !e "$" !a at !" e(l#a*. J"&a &"ta "$,"$ me$e$t#&a$ $"la")$"la" ma&*"m#mata# m"$"m#m !a(" *#at# +#$,*" 9 2 2 3 . . $ / $, a(#* meme$# "&e$!ala 1 1 2 . . $ 8 0 2 1 2 . . $ 8 0 . . & 1 2 . . $ 8 0

!"be$t#& +#$,*" e$ l $,:G 1 2 . . $ 1 . . k 8 9k k +++ .....2211

;a$, meme$# " e(*/a(ata$:

1 x

G

80

2 x

G

80 . . .

n x

G

80

1

G 80 . . .

k

G

80

!e$,a$ 1 2 . . . k t"!a& te(,a$t#$, a!a 1 2 . . . $ !a$ !"*eb#t Pengali Lagrange.

=>NT>? 6:Te$t#&a$ $"la")$"la" e&*t("m (elat"+ !a(" +#$,*" + / 8 /!a$ t"t"& / te(leta& a!a e( t $,a$ a$ta(a e(m#&aa$ 2 2 8 2 !a$ /

8 2.

Ja ab:9#$,*" e$ l $,$/a:

9 / 8 / 2 2 @ 2 / @ 2

02 =+=

x z x

F 02 =+++=

y z y x

z

F

0=+=

z z

y F

022 =+=

z x

F

1= 80ta& be(la 02 ==

yz F

x

z

2= .#b*t"t#*" me$, a*"l&a$:

/ 2

x

z

2

1 / 8 0 / x

z 2/ 8 0

!" e( le 2 8 2. #b*t"t#*" &e !alam !#a e(*amaa$ te(a& "( me$, a*"l&a$:2 2 @ 2 8 0 ata# 2 8 1 8 1 81

Da(" ma*"$,)ma*"$, $"la" !" e( le !#a $"la" /a"t# 8 1 !a$ 8 1 .Pe(*amaa$ / @ 2 8 0 membe("&a$ / 8 2 %"&a 8 1 !a$ / 8 2 %"&a 8 1 .

D" e( le em at &el m & e$/ele*a"a$.

8 1 / 8 2 8 121= 1=

8 1 / 8 2

812

1= 1=

8 1 / 8 2 8 1

21= 1=

8 1 / 8 2 81

21= 1=

6


7/11

&el m & e$/ele*a"a$ e(tama !a$ &eem at me$, a*"l&a$ + / 8 3 !a$/a$, &e!#a !a$ &et",a membe("&a$ + / 8 1. Ma&a + / mem #$/a"

$"la" ma&*"m#m (elat"+ 3 !a$ m"$"m#m (elat"+ 1.

=>NT>? 7:Le$,$, = me(# a&a$ e( t $,a$ a$ta(a &e(#'#t 2 8 2 / 2 !a$

b"!a$, 8 1 /. Te$t#&a$ t"t"&)t"t"& a!a le$,$, = /a$, al"$, !e&at!a$ al"$, %a# a!a t"t"& a*al 0 0 0 0.

Ja ab:eba(a$, t"t"& P / mem #$/a" %a(a& 222 z y x ++ te( a!at"t"& a*al 0 0 0 . Ma&a &"ta a&a$ me$e$t#&a$ $"la" e&*t(em !a(" +#$,*"+ / 8 2 / 2 2 *eba,a" a!(at %a(a& t"t"& P te( a!a t"t"& 0. teta " t"t"& P

"$" a(#* te(leta& a!a = be(a(t" te(leta& a!a:, / 8 2 @ 2 @ /2 8 0

/ 8 @ 1 @ @ / 8 09#$,*" e$ l $,:

9 / 8 2 / 2 2 2 @ 2 @ /2

022 == x x

x

F

022 ==

y y

y F

022 =++=

z z

z

F

8 0 membe("&a$ 8 0 !a$ / 8 0 ata# t"t"& 0 0 0 0 . Teta " t"t"& 0 0 0 t"!a& te(leta& a!a b"!am, 8 1 /. Ma&a 8 1 ta& be(la !a$

!a(" e(*amaa$: @ / 8 @ / !"amb"l &e*"m #la$

@ / 8 0 ata# 8 /B"!a$, 8 / mem t $, b"!a$, 8 1 / a! ,a("* l#(#* /a$, mem t $,&e(#'#t !" !#a t"t"&. T"t"&)t"t"& "$" &"ta te$t#&a !e$,a$ me$*#b*t"t#*"&a$ / 8

!a$ !" e( le 8 1 2.2 2 8 1 2 2 2 2 4 1 8 0

8221

1

1 8 2211 + / 1 8 2

211 + 1821 +

2 8 221

1 / 2 8 221

1 2821

T"t"& te(!e&at !e$,a$ 0 a!ala P2122

112

2

11 +++.

T"t"& te(%a# !e$,a$ 0 a!ala 21221

1221

1 .

2.4.PEN;ELE AIAN BERBAGAI MA ALA?MENGGUNA


8/11

= / 8 6 2 10/ 2 @ / 309#$,*" e(*/a(ata$ , / 8 / @ 34 8 0.

9#$,*" e$ l $, a!ala:9 / 8 6 2 10/ 2 @ / 30 / @ 34

=+=

=+=

=+=

034

020

012

y x F

x y y F

y x x F

8 21 / 8 13 823B

>$,& * m"$"mal !"'a a" *ebe*a( = 21 13 8 40 3.

2.=am #(a$ #t #t a a&a a&a$ membe("&a$ &e#$t#$,a$ ma&*"m#m &e a!a e(#*a aa$ %"&a +#$,*" &e#$t#$,$a a!ala 8 0 @ 2 2 @ / @ 3/ 2 100/

!a$ &a a*"ta* #t #t ma&*"m#m "ala / 8 12.

Ja ab:9#$,*" e$ l $,:

9 / 8 0 @ 2 2 @ / @ 3/ 2 100/ / @ 12

=+=

=++=

=+=

012

01006

04A0

y x F

y x y F

y x x

F

8 5 / 8 7 53=


9/11


10/11

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Bab2 Pengali Lagrange

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