kalkulus multivariat-4

7/23/2019 KALKULUS MULTIVARIAT-4

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TURUNAN PARSIAL

TURUNAN PERTAMA DAN

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TURUNAN PARSIAL



TURUNAN PARSIAL

• 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%



N&TASI



TURUNAN PARSIAL



'&NT&(



• )*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#



• )*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#



•

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#



• 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#



• 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#



• 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#



• 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



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



• 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#



• 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



S d & d P ti l



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



De:niti*n

• If > 2 f/0- ,1- ten



TERIMA KASI(

kalkulus multivariat-4

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