kuliah kalkulus 1 tatap muka 1
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Kuliah Kalkulus Sekolah Tinggi Ilmu StatistikTRANSCRIPT
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Fungsi Invers, Eksponensial, Logaritma,
dan Trigonometri
Tim Kalkulus I
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JENIS-JENIS FUNGSIFungsi
F.PangkatF. PolinomF. LinierF. KuadratF. KubikF. Bikuadrat
Fungsi rasionalFungsi irrasional
Fungsi non-aljabar (transenden)
Fungsi aljabar
F. EksponensialF. LogaritmikF. TrigonometrikF. Hiperbolik
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FUNGSI TRANSENDEN
• Fungsi invers• Fungsi logaritma dan eksponen• Turunan dan integral fungsi eksponen dan
logaritma• Fungsi invers trigonometri• Turunan dan integral fungsi invers trigonometri
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Fungsi Invers
Definisi
Jika fungsi f dan g memenuhi dua kondisi
untuk setiap x dalam domain g
untuk setiap x dalam domain f
Maka dikatakan bahwa f adalah invers dari g dan g adalah invers dari f,
Atau f dan g adalah fungsi-fungsi invers.
xxgf ))((
xxfg ))((
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Definisi
Jika fungsi f mempunyai invers, maka dikatakan bahwa dapat diselesaikan untuk x sebagai fungsi dari y dan dikatakan
merupakan penyelesaian dari
untuk x sebagai fungsi y.
)(xfy
)(1 yfx )(xfy
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Teorema
Jika f fungsi satu-satu, maka grafik dari
dan adalah pencerminan dari fungsi satu dengan fungsi yang lain terhadap garis
Contoh suatu fungsi dan inversnya:
)(xfy
)(1 xfy xy
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Contoh:
Carilah invers dari
, kemudian x dan y ditukar
Maka
23)( xxf
23 xy
23 yx
232 yx
23
1 2 xy
0,23
1)( 21 xxxf
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f(x) = x2
Syarat apa yang harus dipenuhi agar f mempunyai invers?
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Latihan Tunjukkan bahwa fungsi-fungsi di bawah ini mempunyai invers tentukan fungsi inversnya jika ada.
1.
2.
3.
4.
22)( xxxf
133)( 23 xxxxfxexf /1)(
1)(
2
3
x
xxf
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Turunan fungsi invers
Andaikan dapat diturunkan, monoton murni pada interval I, dan bila f’(x) ≠ 0 pada suatu titik x dalam interval I, maka invers f dapat diturunkan di titik y = f(x) dan berlaku
)('
1)()'( 1
xfyf
dydxdy
dx
/
1
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)!6(' tentukan maka
2)( Misal .2
)!4(' tentukan maka
12)( Jika 1.
1
3
1
5
f
xxf
f
xxxf
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Fungsi Logaritma Natural
dan
Eksponensial Natural
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Fungsi Eksponensial Natural
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FUNGSI LOGARITMA DAN EKSPONENSIAL UMUM
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Fungsi Invers Trigonometri
Definisi
Fungsi invers sinus, dinotasikan , didefinisikan sebagai invers dari fungsi
Fungsi invers cosinus, dinotasikan , didefinisikan sebagai invers dari fungsi
Fungsi invers tangen, dinotasikan , didefinisikan sebagai invers dari fungsi
Fungsi invers secan, dinotasikan , didefinisikan sebagai invers dari fungsi
1sin
2/2/,sin xx1cos
xx 0,cos1tan
2/2/,tan xx
1sec
2/32/0,sec xatauxx
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Teorema
2/3
1atau
2/0
1jikasecsec
2/2/jikatantan
0
11 jikacoscos
2/2/
11jikasinsin
1
1
1
1
y
x
y
xxyxy
y
xxyxy
y
xxyxy
y
xxyxy
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Fungsi Domain Range Hubungan
x1sin 1,1 2/,2/ 2/2/jika)(sinsin 1 xxx
11jikasinsin 1 xxx
x1cos 1,1 ,0 xxx 0jika)(coscos 1
11jika)cos(cos 1 xxx
x1tan , 2/,2/ 2/2/jika)(tantan 1 xxx
xxx jika)tan(tan 1
x1sec ,11, 2/3,2/,0 2/3at2/0jika)(secsec 1 xxxx
1at1jika)sec(sec 1 xxxx
x1cot , ,0
x1csc ,11, 2/,02/,
xxx 0jika)(cotcot 1
xxx jika)cot(cot 1
2/0at2/jika)(csccsc 1 xxxx
1at1jika)csc(csc 1 xxxx
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Turunan & Integral Fungsi Invers Trigonometri
Teorema
1
1csc
1
1sec
1
1cot
1
1tan
1
1cos
1
1sin
2
1
2
1
21
21
2
1
2
1
xxx
dx
d
xxx
dx
dx
xdx
d
xx
dx
dx
xdx
d
xx
dx
d
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dx
du
uuu
dx
d
dx
du
uuu
dx
ddx
du
uu
dx
d
dx
du
uu
dx
d
dx
du
uu
dx
d
dx
du
uu
dx
d
1
1csc
1
1sec
1
1cot
1
1tan
1
1cos
1
1sin
2
1
2
1
21
21
2
1
2
1
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Ca
u
aauu
duCu
uu
du
Ca
u
aua
duCu
u
du
Ca
u
ua
duCu
u
du
1
22
1
2
122
12
1
22
1
2
sec1
sec1
tan1
tan1
sinsin1
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Contoh
Hitunglah
Substitusi
dx
e
ex
x
21
CeCuu
dudx
e
e
dxedueu
x
x
x
xx
)(sinsin11
,
11
22
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Latihan
1. Carilah dy/dx dari
a.
b.
c.
2. a.
b.
)(sin 31 xy
)(sec 1 xey
yxxy 11 cos)(sin
dx
e
ex
x
21
3
1 1xx
dx
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