ITK-121 KALKULUS I
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![Page 1: ITK-121 KALKULUS I](https://reader035.fdocuments.net/reader035/viewer/2022072016/56813215550346895d98707d/html5/thumbnails/1.jpg)
ITK-121KALKULUS I
3 SKS
Dicky Dermawanwww.dickydermawan.890m.com
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DIFFERENSIAL &
TURUNAN FUNGSI PARAMETER
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Pada koordinat baru:
Gradien
Turunan adalah hasil bagi dua differensial
1. turunan dari terhadap x Jadi, mempunyai arti
2. hasil bagi dy terhadap dx
dxmdy
)( 0' xfm → dxxfdy )( 0
' ↔ dx
dyxf )( 0
'
dx
dy )(xfy
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ATURAN DIFFERENSIAL SAMA DENGAN ATURAN DERIVATIFISASI
Turunan Differensial
dx
dv
dx
duvu
dx
d dvduvud )(
dx
dv
dx
duvu
dx
d )( dvduvud )(
dx
duv
dx
dvuvu
dx
d )( duvdvuvud )(
2vdx
dvu
dx
duv
v
u
dx
d
2v
dvuduv
v
ud
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Contoh
1.
2
3
4
Diantara kegunaan differensial adalah untuk penurunan fungsi implisit.
xyd
)tan( xxd
)tan( yxd
x
xdsin
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FUNGSI IMPLISIT
fungsi eksplisit → y bergantung pada x
F(x, y) = 0 fungsi implisit → → y fungsi x juga x fungsi y
Aturan Bisa dibuat atau
tetapi artinya bisa berbeda.
)(xfy
0)(, xfyyxf
422 yx 24 xy 24 yx
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Contoh
1
2
Seringkali fungsi implisit sukar bahkan kadang mustahil dieksplisitkan
2124 xy
21
24 xy
yyxxyx 3sin 25
12sin 2 xyxy
'y'y
??
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Contoh mudah:
33 7 xyy
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FUNGSI PARAMETER
Contoh
1. Persamaan lingkaran
Dalam bentuk fugnsi parameter dinyatakan sebagai
2.
3
422 yx
tax cos2
tay sin2
ttx sinty cos1
t ≥ 21
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4. A
b
ttx 44 2 t ≥ 2
1
241 ty t ≥2
1
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Soal
Tentukan untuk fungsi-fungsi implisit di bawah ini serta
tentukan nilainya di titik yang diberikan
1
2
3
4
5
'y
1 yxyx ; (3, 1)
2 xyxyx ; (1, 1)
2326 yxyxyx ; (0,0)
xyxy 2cos 2
13cos 22 xxyy ; (0, 0)
0,
2
1
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6.
7
8
9
10
11
12
yxy sin ; (1, 0)
1132 232 yxxyyx ; (-1,1)
4322cos yxxxy
12 yxy
x
2cos2sin xyyy
x
yxxy tantan
yxxy sincos
0,
2
1
2,
7
2
2,0
4,1
4,4
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Tentukan turunan dari y terhadap x dari fungsi parameter:
1.
2.
3
4
ttx1
t 0
tty1
tx 2Rt
522 tty
1sin tx0 ≤ t ≤ 2 π
2cos ty
12 2 txRt
212 ty
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5.
6.
tx sec42
11
ty tan3
tx 4cos4Rt
ty 4sin9