TUGAS KALKULUS LANJUT KELOMPOK 2
B A B 2
DOSEN : Drs. SURYO WIDODO,MPD : NPM 04.1.03.01.0005 04.1.03.01.0014 04.1.04.01.0017 04.1.03.01.0026 04.1.03.01.0027 04.1.03.01.0041 04.1.03.01.0024
NAMA KELOMPOK : ASMAUL SHOLEKAH .1 IDA ELISA .2 MAY NUKE SURYANA .3 PRATIWI .4 QOMARIAH .5 TUTIK SURYANTI.6 NUR CHOLIS .7
INSTITU KEGURUAN DAN ILMU PENDIDIKAN PGRI KEDIRI JURUSAN PENDIDIKAN MATEMATIKA
2006
1
BAB 2 PENGALI LAGRANGE TINJAUAN UMUM PENGALI LAGRANGE .2.1 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$a a!a$/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(aa$ &"ta *e("$, !"a!a&a$ a!a &e$/ataa$ baa 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(#*aaa$ "$,"$ 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 *ee(t" "$" #at# met!e #$t#& meme(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 / λ 8 + / λ , / : De$,a$ e(*/a(ata$ ∂ F
∂ F
∂ F
=0 ∂ y = 0 ∂λ ∂ x ;a$, me(#a&a$ */a(at e(l# #$t#& ma&*"m#m (elat"+ ma##$ m"$"m#m .(elat"+ Pa(amete( λ /a$, t"!a& te(,a$t#$, a!a !a$ / !"*eb#t Pengali . Lagrange =0
=
∂ x ∂ F ∂ y ∂ F ∂λ
∂ f ∂ x
=
∂ F ∂ y
∂ g + λ =0 ∂ x ∂ F + λ =0 ∂ y
= g x y = 0
et"a e$/ele*a"a$ !a(" *"*tem e(*amaa$ "$" a!ala *#at# $"la" .&("t"* !a(" +#$,*" + /
2
: =>NT>? 1 : Te$t#&a$ $"la" ma&*"m#m !a(" + / 8 / !e$,a$ */a(at G / 8 / @ 16 8 0 : Jaab 9 / λ 8 9 / λ , / / λ / @ 16 8
∂ F = y + λ = 0 → y = −λ ∂ x ∂ F = x + λ = 0 → x = −λ ∂ y ∂ F = x + y − 16 = 0 → −λ − λ − 16 = 0 ∂λ
8 λ 8 − λ .T"t"& &("t"* !"'aa" %"&a 8 !a$ / 8 !e$,a$ $"la" + / 8 8 64
−A
: =>NT>? 2 eb#a ab("& mem(!#&*" !#a ma'am me*"$ !a$ / !a$ +#$,*" : $,&* ,ab#$,a$ a!ala 2 = / 3/ @ 6/ U$t#& m"$"m#m&a$ b"a/a be(aa ba$/a& me*"$ !a(" *et"a %e$"* a(#* .!"(!#&*" %"&a &e*el#(#a$$/a a(#* be(%#mla 42 me*"$ : Jaab Pe(*/a(ata$ /a$, a(#* !"e$#" / 8 42 !"t#l"* : , / 8 / @ 42 8 0 : +#$,*" e$l$,$/a 9 / λ 8 = / λ , / 2 3/ @ 6/ λ / @ 42 8
∂ F = 2 x + 3 y + λ = 0 ∂ x ∂ F x = 3 − 6 + λ = 0 ∂ y ∂ F x + y − 42 = 0 ∂λ
Pe$/ele*a"a$ !a(" *"*tem "$" membe("&a$ λ = −B6 8 30 / 8 12 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( baa$ /a$, !"a&a" l#a*$/a *em"$"mal m#$,&"$ be(aa #&#(a$ &ta& .te(*eb#t. G#$a&a$ e$,al" la,(a$,e
: Jaab Cl#me C 8 / 8 &$*ta$ L#a* 8 / 2 2/ 9 / λ 8 / λ C /
3
/
9 / λ 8 / 2 2/ λ / @ C
∂ F y + 2 z = y + 2 z + λ yz = 0 → λ = − ∂ x yz ∂ F x + 2 z = x + 2 z + λ xz = 0 → λ = − xz ∂ y ∂ F 2 x + 2 y = 2 x + 2 y + λ xy = 0 → λ = − ∂ z xy ∂ F = xyz − V = 0 ∂λ
− y + 2 z = − x + 2 z → − xy + 2 xz = − yz + 2 yz
−
yz xz − 2 xz = 2 yz → x = y x + 2 z 2 x + 2 y =−
xz
→ − xy − 2 yz = −2 xz − 2 yz
xy
− xy = −2 xz 8
1 2
/ 8
1 2
C 8 / 8 13 2
1 2
2V 8 3 8 2C
8 →
1 2
3 3
8
2V
3
/8
2V
: =>NT>? 4 Te$t#&a$ -l#me te(be*a( !a(" &ta& te,a& l#(#* /a$, !aat !"l#&"*&a$ !alam el"*"!a 162 4/2 2 8 144. "*")*"*" &ta& *e%a%a( b"!a$, .&(!"$at. G#$a&a$ e$,al" la,(a$,e : Jaab J"&a P / a!ala #%#$, &ta& /a$, tela me$/e$t# e(m#&aa$ : el"*"!a ma&a -l"me &ta& C8/ : T#%#a$ &"ta "ala me$e$t#&a$ $"la" ma&*"m#m !a(" C be(*/a(at&a$ , / 8 162 4/2 2 @ 144 8 0 9 / λ 8 / λ 162 4/2 2 @ 144
∂ F = A yz + 32 xλ = 0 → A xyz + 32 x 2λ = 0 ∂ x ∂ F = A xz + A yλ = 0 → A xyz + A y 2λ = 0 ∂ y ∂ F = A xy + 1A z λ = 0 → A xyz + 1A z 2λ = 0 ∂ z ∂ F ∂λ 2
2
2
2
2
= 16 x + 4 y + B z − 144 = 0
2
/ 2 λ 16 4/ 8 0 24 −12λ 8 / 2 λ 144 8 0 ata# / 24 4
/ 322 λ 8 0 2 2 − 3λ + x λ = 0 → 32 λ 80 −12λ 2 ta& be(la&# 08 λ −3 + x = 0 → λ 8 2 8 3 → − 3 + x 2 = 0 → 2 −12λ + A y λ = 0 / /2 λ 8 0 → 3
−12λ + y
2
λ = 0
λ −12 + y 2
= 0 → λ = 0
y 2 = 12 → y = 2 3 →
− 12λ + 1A z λ = 0 / 12 λ 8 0 2
− 16λ + 3 z λ = 0 2 λ −16 + 3 z = 0 → λ = 0 2
3 z 2
16
= 16 → z 2 = 4 3
C ma&* 8 / 8
3
2
3
3
4 3
3
3 8 3 8 54
: =>NT>? 5 J"&a + / 8 4 / 5 te$t#&a$ t"t"& a!a b"!a$, 2 3/ .4 8 12 !"ma$a + / mem#$/a" $"la" te(&e'"l 2
2
2
: Jaab <"ta "$,"$ me$em#&a$ $"la" m"$"m#m !a(" + / !e$,a$ : e(*/a(ata$ , / 8 2 3/ 4 @ 12 8 0. 9#$,*" e$l$,$/a 9 / λ 8 42 /2 52 λ 2 3/ 4 @ 12 ∂ F ∂ x ∂ F ∂ y ∂ F ∂ z ∂ F ∂λ
= A x + 2λ = 0 → x = −
1 4 3
= 2 y + 3λ = 0 → y = −
2
= 10 z + 4λ = 0 → z = −
λ λ
2 5
= 2 x + 3 y + 4 z −12 = 2−
λ 1 4
− 10 − B0 − 32
20 A 11
A 11
3
2
2
5
λ + 3− λ + 4− λ −12 = 0
λ = 12 → λ = −
8 8 30 11
5 11
5 11
20 11
/ 8
30 11
$"la" m"$"m#m te('aa" !" t"t"t&
5
U$t#& meme(le $"la")$"la" (elat"+ ma&*"m#m ata# m"$"m#m !a(" .+#$,*" 9 / !e$,a$ e(*/a(ata$ / 8 0 !"be$t#& +#$,*" emba$t# G / λ 8 9 / λφ / ;a$, a(#* meme$#" e(*/a(ata$
∂G 8 0 ∂ x
G ∂
8 0 y ∂
∂G ∂G 8 0 !a$ ∂ z ∂λ
80 Met!e "$" !aat !"e(l#a*. J"&a &"ta "$,"$ me$e$t#&a$ $"la")$"la" ma&*"m#m ata# 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$l$, λ 1φ 1 + λ 2φ λ 2 +..... + λ k λ k G 1 2 . . $ λ k 8 9 1 . . : ;a$, meme$#" e(*/a(ata$ ∂G ∂G ∂G ∂G ∂G 0 8 . . . 0 8 0 8 . . . 0 8 0 8
∂λ k
!e$,a$
xn ∂
∂λ 1
λ λ .2. . 1
x2 ∂
x1 ∂
λ t"!a& te(,a$t#$, a!a 1 2 . . . $ !a$ !"*eb#t k
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 : Jaab : 9#$,*" e$l$,$/a 2 2 9 / λ µ 8 / λ @ 2 µ / @ 2 ∂ F ∂ z
∂ F = z + 2λ x = 0 ∂ x
= x + y + 2λ z + µ y = 0 ∂ F ∂λ
∂ F ∂ µ
2
= x + z − 2 =
∂ F
0
∂ y
z = 0 = z + µ
ta& be(la&# 0 8 µ = −1
= yz − 2 = 0
: #b*t"t#*" me$,a*"l&a$ . λ = − / 2 −
z 2 x
−1 / 8 0
→
/
−
2
z
x
z
2 x
−/ 8 0
: !"e(le 2 8 2. #b*t"t#*" &e !alam !#a e(*amaa$ te(a&"( me$,a*"l&a$ −1 8 22 @ 2 8 0 ata# 2 8 1 → 8 1 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 emat &elm& e$/ele*a"a$
µ = −1
λ = −
1 2
81
/82
81
−2 18 8 1 / 8 − 2 1 −1 / 8 2 8 1 µ = −1 λ = 8 2 1 −1 / 8 − 2 µ = −1 λ = − −1 8 8
µ = −1
λ
=
1
2
6
&elm& e$/ele*a"a$ e(tama !a$ &eemat 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$ta(a &e(#'#t 8 / !a$ b"!a$, 8 1 /. Te$t#&a$ t"t"&)t"t"& a!a le$,&#$, = /a$, al"$, !e&at .!a$ al"$, %a# a!a t"t"& a*al 0 000 2
2
2
: Jaab eba(a$, t"t"& P / mem#$/a" %a(a& x 2 + y 2 + z 2 te(a!a t"t"& a*al 000. Ma&a &"ta a&a$ 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 : "$" a(#* te(leta& a!a = be(a(t" te(leta& a!a , / 8 2 @ 2 @ /2 8 0 / 8 @ 1 @ @ / 8 0 : 9#$,*" e$l$, 2 2 2 2 µ 9 / λ 8 / λ @ 2 @ /2
∂ F = 2 x − 2 xλ − µ = 0 ∂ x )/ 8 )/ λ
∂ F ∂ y
µ 1= 0 = 2 y − 2 yλ λ − 8
∂ F
= 2 z + z λ µ = 0λ 28 +/) /
∂ z
8 /t"t"& @ → 8 0 8 0 membe("&a$ 8 0 !a$ / 8 0 ata# t"t"& 0 000./ Teta" 000 t"!a& te(leta& a!a b"!am, 8 1 /. Ma&a λ 8 1 ta& be(la&# !a$ : !a(" e(*amaa$ @ / 8 @ / λ !"amb"l &e*"m#la$ @ / 8 0 ata# 8 / B"!a$, 8 / memt$, b"!a$, 8 1 / a! ,a("* l#(#* /a$, memt$, &e(#'#t !" !#a t"t"&. T"t"&)t"t"& "$" &"ta te$t#&a !e$,a$ me$*#b*t"t#*"&a$ / 8 .!a$ !"e(le 8 1 2 2 2 2 2 8 1 2 → 2 4 1 8 0 −1 ±
−1 +
−1 − . −1 + . −1 −
1 2 1
2
1 2
2 8
2 8 1 8 −1 +
2 8 2 8 −1 −
2 −1 +
2 −1 −
1 2 1
2
1
2 1
2
2 −1 +
2 /1 8 −1 +
2 /2 8 −1 −
1 2 1
2
2 1
2 2
2 T"t"& te(!e&at !e$,a$ 0 a!ala P
2 −1 − 2 T"t"& te(%a# !e$,a$ 0 a!ala
PEN;ELEAIAN BERBAGAI MAALA? .2.4 .MENGGUNA
= / 8 62 10/2 @ / 30 .9#$,*" e(*/a(ata$ , / 8 / @ 34 8 0 : 9#$,*" e$l$, a!ala 2 2 9 / λ 8 6 10/ @ / 30 λ / @ 34
− 23B
8 λ 8 21 / 8 13
∂ F = 12 x − y + = 0 ∂ x ∂ F = 20 y − x + λ = 0 ∂ y ∂ F = x + y − 34 = 0 ∂ λ
→
.>$,&* m"$"mal !"'aa" *ebe*a( = 21 13 8 403 =am#(a$ #t#t aa&a a&a$ membe("&a$ &e#$t#$,a$ ma&*"m#m &ea!a .2 π e(#*aaa$ %"&a +#$,*" &e#$t#$,$a a!ala 8 0 @ 22 @ / @ 3/2 100/ .!a$ &aa*"ta* #t#t ma&*"m#m "ala / 8 12 : Jaab : 9#$,*" e$l$, 2 2 9 / λ 8 0 @ 2 @ / @ 3/ 100/ λ / @ 12
λ
= −53 8 5
/ 87
∂ F = A0 − 4 x − y + λ = 0 ∂ x ∂ F = − x − 6 y + 100 + λ = 0 ∂ y ∂ F = x + y − 12 = 0 ∂ λ
→
.
π 57 8 6
8 Te$t#&a$ $"la")$"la" e&*t("m !a(" +#$,*" + / 8 / !e$,a$ */a(a , /.3 .
8
λ
2/
2
A
+
y
2
2
−1 = 0
: Jaab : 9#$,*" e$l$, 2 λ y −1 9 / 8 /
2
±
x
x λ + A 2
=± 2
→
2 x 1 ∂ F = y + λ = 0 → y + xλ = 0 A 4 ∂ x ∂ F 2 y = x + λ = 0 → x + yλ = 0 ∂ y 2
∂ F ∂λ
=
x
2
+
A
y
2
2
−1 =
2
2
0 → x + 4 y = A
#b*t"t#*" 8 ± 2/ me$,a*"l&a$ 4/ 4/ 8 → /2 8 1 / 8 ± 1 N"la")$"la" e&*t("m !a(" + / a!ala − 2 8 / 8 2 !a$ / 2
2
Te$t#&a$ t"t"& a!a b"!a$, 2 @ 3/ 5 8 1 /a$, al"$, !e&at a!a t"t"& a*al .4 .0 000. G#$a&a$ e$,al" la,(a$,e : Jaab 9#$,*" /a$, !"'a(" m"$"m#m$/a !aat !""l" *eba,a" &#a!(at !a(" %a(a& t"t"& : P / te(a!a t"t"& 0 000 "ala 9 / 8 2 /2 2 .
= 2 x + 2λ = 0 = 2 y − 3λ = 0
= 2 z + 5λ = 0 = 2 x − 3 y + 5 z −1B = 0
− 5 λ 8 8 − λ 2
− 2λ −
B 2
λ −
25 2
λ = 1B → −
5 2
T"t"& P 1
3A 2
/8
3 2
λ
λ = 2B → λ = −1
8 8 1 / 8
3
− 2
− 3 5 te(leta& al"$, !e&at a!a t"t"& a*al 0 000 !a$ P te(leta& 2 2
a!a . b"!a$, 2 @ 3/ 5 8 1 .Te$t#&a$ %a(a& te(!e&at !a(" t"t"& a*al &e "e(bla 2 / 7/2 8 225 8 0 .5
: Jaab <#(-a "$" te(leta& a!a b"!a$, 0 / &a(e$a 8 0. Ja(a& !a(" t"t"& P 2 x 2 + y &"ta / &e t"t"& a*al "ala ! 8 "l" &#a!(at !a(" %a(a&$/a *eba,a" : +#$,*" /a$, a(#* !"te$t#&a$ m"$"m#m$/a. Ma&a 9 / 8 2 /2 : /a(at /a$, a(#* !"e$#" 2 , / 8 / 7/2 @ 225 8 0 : 9#$,*" e$l$, 9 / λ 8 2 / 7/2 @ 225 λ 2 /2
∂ F = 2 x + A y + 2 xλ = 0 → λ + 1 x + 4 y = 0 ∂ x ∂ F = A x + 14 y + 2 yλ = 0 → 4 x + λ + 7 y = 0 ∂ y :
4 λ2
λ ) 8 0 λ 8 1 !a$ 0 8
λ + 7
λ 1 + 4
λ 8
)
2
φ 1 =
x
+
4
y
2
5
2
y
+
4
2
2
+
z
5
2
<"ta a(#* me$e$t#&a$ e&*t("m !a(" 2
x
25
8 1 !a$ 8 / : Jaab !e$,a$ &e$!ala
2
/
2
+
z
25
φ 8 2 @ @ / 8 0. &"ta ,#$a&a$ !#a e$,al"
!a$ −1 =0
.la,(a$,e λ 1 !a$ λ 2 : 9#$,*" e$l$,$/a λ 1φ + λ 2φ G 8 9 G 8 2 /2 2
λ 1
x 2
4
+
y 2
5
z 2
+
− 1 + λ 2 / @
25
∂ F
= 2 x +
∂ x ∂ F
= 2 y +
∂ y ∂ F
2 z +
∂ z
λ 1 x
+ λ 2 = 0
2 2λ 1 y 5
2λ 1 z 25
+ λ 2 = 0
− λ 2 = 0
Da(" t",a e(*amaa$ "$" !"e(le − 5λ 2 − 2λ 2 25λ 2 88 /8 2λ 1 + 10 λ 1 + 4 2λ 1 + 50 #b*"t#*" !alam e(*ama$ b"!a$, / @ 8 0 &em#!"a$ !"ba," λ 2 membe("&a$ 2
λ 1 + 4
Me$,al"&a$ !e$,a$ 2
λ 1
+ 4
λ 1 + 5
+
2λ 1 + 10
+ 25 !a$
λ 1
ata# λ 1 + 10 17 λ 1 + 75 8 0 λ 1
2
λ 1
5
+
25 2λ 1
=0
me$/e!e(a$a&a$ : !"e(le
+ 245λ 1 + 750 = 0 17
=−
75 17
Ma&a
λ 1
λ 1
= −10
= −10
ata#
:
10
x
1
1
3
2
λ 2/ 8
8 2
+
4
y
2
5
λ 2 8
5 6
2
+
z
25
= 1 me$,a*"l&a$
5
1B
!#a t"t"& &("t"* 2
5
3
20 + 45 + 125
1B
= 10.
1B
5
λ 2
λ 2. #b*"t#*"&a$ !alam &$!"*"
=
2
1A0 1B
ata# λ 2
!a$ 1B 5
−2
5
= ±6
5 1B
1B
−3
5 1B
&e$!ala
34
−
/λ 82 17 x
2
+
y
4
2
+
5
17
17
4
2A
λ 2 8
2
z
25
. N"la" 1B 5
−5
!a(" 2 /2 2 be(&a"ta$ !e$,a$ t"t"& &("t"* "ala
λ 1 8
. I$" membe("&a$
75
17
:
λ .2 #b*"t#*"&a$ &e!alam
λ 2
membe("&a$ 1 =
=−
140
=±
17 646
/a$, membe("&a$ : t"t"&)t"t"& &("t"*
− 5 646
35 646
646
−40 !a$ 646
5
2
2
− 35
646
40 646
2
: N"la" !a(" / be(&a"ta$ !e$,a$ t"t"&)t"t"& &("t"* "ala 75 17
.
75 17
8
1600 + 1225 + 25 646
Ma&a $"la" ma&*"m#m "ala 10 !a$ m"$"m#m "ala
11
