KALKULUS MULTIVARIAT-4
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Transcript of KALKULUS MULTIVARIAT-4
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TURUNAN PARSIAL
TURUNAN PERTAMA DAN
KEDUA
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TURUNAN PARSIAL
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TURUNAN PARSIAL
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• Berbagai jenis fungsi dari satuvariabel bebas sangat muda untu!mengitung dan diinter"retasi!an !edalam bentu! turunan#
• Bagaimana dengan fungsi"ersamaan dengan $ atau lebivariabel bebas%
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N&TASI
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'&NT&(
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• )*r te +*m"an, "r*du+ing *nl, testandard b*ard- te +*st fun+ti*n .as
'/01 2 344 5 640
• Di7erentiating .it res"e+t t* #r- .e*btain te marginal +*st fun+ti*n
'/01 2 64
• Sin+e te marginal +*st is +*nstant- 864is te +ange in +*st f*r a I9unit in+reasein "r*du+ti*n at an, *ut"ut level#
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• )*r te +*m"an, "r*du+ing t.* t,"es *fb*ards9a standard m*del and a+*m"etiti*n m*del9te +*st fun+ti*n.as
'/0- ,1 2 644 5 640 5 l44
• N*. su""*se tat .e di7erentiate .itres"e+t t* 0- *lding , :0ed- and den*tete resulting fun+ti*n b, '0/0-,1;
•
*r su""*se .e di7erentiate .it res"e+tt* ,- *lding 0 :0ed- and den*te teresulting fun+ti*n b, ',/0-,1#
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•
Di7erentiating in tis .a,-.e *btain
'/0-,1 2 64 ',<0- ,1 2 =44
• Ea+ *f tese fun+ti*ns is +alled a"artial derivative- and- in tis
e0am"le- ea+ re"resents marginal+*st#
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• Te :rst is te +ange in +*st due t*
a I9unit in+rease in "r*du+ti*n *f testandard b*ard .it te "r*du+ti*n*f te +*m"etiti*n m*del eld :0ed#
• Te se+*nd is te +ange in +*st duet* a I9unit in+rease in "r*du+ti*n *f
te +*m"etiti*n b*ard .it te"r*du+ti*n *f te standard b*ardeld :0ed#
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• In general- if > 2 f/0- ,1- ten te
"artial derivative *f?.it res"e+t t*0-den*ted f0 *r f0 /0-,1- isde:ned b,
•
Re+*gni>e tis f*rmula as te*rdinar, derivative *f j .it res"e+tt* 0- *lding, +*nstant#
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• Similarl,- te "artial derivative *f f.it res"e+t t* ,- den*ted - f0*r f0 /0-,1- is de:ned b,
• .i+ is te *rdinar, derivative .itres"e+t t* ,- *lding 0 +*nstant#
•
Parallel de:niti*ns andinter"retati*ns *ld f*r fun+ti*ns .ittree *r m*re inde"endent variables#
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• E0am"le =
)*r > 2 f/0- ,1 2 $0$ 9@0$, 5 3, 5 =-:nd
a# d>?d0 b# f0/$-@1
• S*luti*n
> 2 $0$ 9@0$, 5 3, 5 =
Di7erentiating .it res"e+t t* 0-*lding , +*nstant /tat is- treating,as a +*nstant1- .e *btain
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f/0- ,1 2 $0$ 9@0$, 5 3, 5 =
• )irst- di7erentiate .it res"e+t t* 0#)r*m "art /A1- .e ave
• Ten evaluate tis euati*n at /$-@1
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• In "art I/B1- an alternative a""r*a+
.*uld bet* substitute , 2 @ int* f/0- ,1and gra" te fun+ti*n f/0-@1 2 960$ 5=- .i+ re"resents te +r*ss se+ti*n*f te surfa+e > 2 f/0- ,1 "r*du+ed b,
+utting it .it te "lane , 2 @#
• Ten determine te sl*"e *f te
tangent line .en 0 2 $# Again- .e+*n+lude tat f0/$- @1 2 9$C /see )ig#=1#
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• E0am"le $
• Using te 'ain Rule )*r
> 2 f/0- ,1 2 e0$5,$ - :nd
a# d>?d0 b# f,/$-=1
• S*luti*n
Using te +ain rule tin!ing *f > 2eu- u 2 u/01; is eld +*nstantF- .e*btain
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S d & d P ti l
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Se+*nd9&rder PartialDerivatives
• Te fun+ti*n
• as t.* :rst9*rder "artial derivativesG
•
Ea+ *f tese "artial derivatives- in turn-as t.* "artial derivatives +alled se+*nd9*rder "artial derivatives *f > 2 f<0- ,1#
• Henerali>ing te vari*us n*tati*ns .e
ave f*r :rst9*rder "artial derivatives- .e.rite te f*ur se+*nd9*rder "artialderivatives *f > 2 f/0-,1 2 0,6 as
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De:niti*n
• If > 2 f/0- ,1- ten
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TERIMA KASI(