Nama : Fitria Rahmadhani

NIM : 8156171045

Kelas : A-2

Mata Kuliah : Kalkulus Lanjut

Dosen : Prof. Dr. Asmin, M.Pd

Tentukan deret Maclaurin untuk f (x) sampai 3 suku pertama!

4. f ( x )=x sec xf (0 )=0 (1 )f (0 )=0

f ( x )=x sec xMisal : u(x )=x v ( x )=sec xu' (x )=1 v ' ( x )=sec x tan xMaka :

f ' ( x )=u' ( x ) v ( x )+u ( x ) v ' ( x )f ' ( x )=1 (sec x )+x ( sec x tan x )f ' ( x )=sec x+x sec x tan x

f ' (0 )=(1 )+(0 ) (1 ) (1 )f ' (0 )=1

f ' ( x )=sec x+x sec x tan xMisal :s= x sec x tan xt=x sec xu=tan xv=xu'=sec2 x

v '=1w=sec xw '=sec x tan x

t '=v 'w+v w'

t '=sec x+x (sec x tan x)s'=t 'u+t u'

s'=((sec x+x (sec x tan x ))( tan x))+((x sec x )(sec2 x ))s'=sec x tan x+x sec x tan2 x+x sec3 x

Maka,

f ' ( x )=sec x+x sec x tan x

f ' ( x )=sec x+sf ' ' ( x )=sec x+s 'f ' ' ( x )=sec x tan x+sec x tan x+x sec x tan2 x+x sec 3 x

f ' ' ( x )=2 sec x tan x+x sec x tan2 x+x sec3 x

f ' ' (0 )=2 (1 ) (0 )+0 (1 ) (0 )+0 (1 )f ' ' (0 )=0

f ' ' ( x )=2 sec x tan x+x sec x tan2 x+x sec3 xMisal,u=2 sec x tan x

u'=2 sec x tan2 x+2 sec3 x

v=x sec x tan2 x

v '=sec x tan2 x+x sec x tan3 x+2 x sec3 x tan x

w=x sec3 x

w '=sec 3 x+3 x sec3 x tan x

Maka,

f ' ' ( x )=2 sec x tan x+x sec x tan2 x+x sec3 x

f ' ' ( x )=u+v+wf ' ' ' ( x )=u '+v '+w 'f ' ' ' ( x )=2 sec x tan2 x+2 sec3 x

+sec x tan 2 x+x sec x tan3 x+2 x sec3 x tan x

+sec3 x+3 x sec3 x tan x

f ' ' ' (x)=3 sec x tan2 x+3 sec 3 x+x sec x tan 3 x+5 x sec3 x tan x

f ' ' ' (0 )=0+3 (1 )+0+0

f ' ' ' (0 )=3

f ' ' ' (x)=3 sec x tan2 x+3 sec 3 x+x sec x tan 3 x+5 x sec3 x tan x

Misal,

u=3 sec x tan2 x

u'=3 sec x tan 3 x+6 sec 3 x tan x

v=3 sec3 x

v '=9 sec3 x tan x

w=x sec x tan3 x

w '=sec x tan3 x+ x sec x tan 4 x+3 x sec3 x tan2 x

z=5 x sec3 x tan x

z '=5 sec3 x tan x+15 x sec3 x tan2 x+5 x sec5 x

Maka,

f ' ' ' (x)=u+v+w+zf ' ' ' ' ( x )=u '+v '+w '+ z 'f ' ' ' ' ( x )=3 sec x tan3 x+6 sec3 x tan x

+9 sec3 x tan x

+sec x tan 3 x+x sec x tan4 x+3x sec3 x tan2 x

+5 sec3 x tan x+15x sec 3 x tan2 x+5 x sec5 x

f ' ' ' ' ( x )=4 sec x tan3 x+20 sec3 x tan x+x sec x tan4 x

+18 x sec3 x tan2 x+5 x sec5 x

f ' ' ' ' ( x )=0

f ' ' ' ' ( x )=4 sec x tan3 x+20 sec3 x tan x+x sec x tan4 x

+18 x sec3 x tan2 x+5 x sec5 xMisal,

t=4 sec x tan3 x

t '=4 sec x tan4 x+12 sec3 x tan 2 x

u=20 sec3 x tan x

u'=60 sec3 x tan 2 x+20 sec5 x

v=x sec x tan4 x

v '=sec x tan4 x+x sec x tan5 x+4 x sec3 x tan3 x

w=18 x sec3 x tan2 x

w=18 sec3 x tan2 x+54 x sec3 x tan3 x+36 x sec5 x tan x

z=5 x sec5 x

z '=5 sec5 x+25x sec 5 x tan x

Maka,

f ' ' ' '(x )=t+u+v+w+zf ' ' ' ' ' (x)=t'+u'+v'+w'+ z'

f ' ' ' ' ' (x)=4 sec x tan4 x+12 sec3 x tan 2 x

+60 sec3 x tan2 x+20 sec5 x

+sec x tan 4 x+x sec x tan5 x+4 x sec3 x tan3 x

+18 sec3 x tan2 x+54 x sec3 x tan3 x+36 x sec5 x tan x

+5 sec5 x+25 x sec5 x tan x

f ' ' ' ' ' (x)=5 sec x tan4 x+90 sec 3 x tan2 x+x sec x tan5 x

+25 sec5 x+58 x sec3 x tan3 x+61 x sec5 x tan x

f ' ' ' ' ' (0)=0+0+25+0+0+0

f ' ' ' ' ' (0)=25

Dengan memasukkan harga-harga turunan ini ke deret Maclaurin diperoleh:

f ( x )=x sec x=f ' (0 ) x+ f ' ' ' (0)3 !

x3+f ' ' ' ' ' (0)

5 !x5+…

¿1 ( x )+ 36x3+ 25

120x5+…

5. f ( x )=e−1sin xf (0 )=0

f ( x )=e−1sin x

f ' ( x )=e−1 cos x

f ' (0 )=e−1 (1 )f ' (0 )=e−1

f ' ( x )=e−1 cos x

f ' ' ( x )=−e−1 sin x

f ' ' (0 )=0

f ' ' ( x )=−e−1 sin x

f ' ' ' ( x )=−e−1cos x

f ' ' ' (0 )=−e−1 (1 )f ' ' ' (0 )=−e−1

f ' ' ' ( x )=−e−1cos x

f ' ' ' ' ( x )=e−1 sin x

f ' ' ' ' ( x )=0

f ' ' ' ' ( x )=¿f ' ' ' ' ' (x )=e−1cos x

f ' ' ' ' ' (0 )=e−1

Dengan memasukkan harga-harga turunan ini ke deret Maclaurin diperoleh:

f ( x )=e−1sin x¿ f ' (0 ) x+ f ' ' ' (0)3 !

x3+f ' ' ' ' ' (0)

5 !x5+…

¿e−1 x+−e−1

6x3+ e

−1

120x5+…

¿ xe− x3

6e+ x5

120e−…

6. f ( x )= 11+sin x

f (0 )= 11+0

f (0 )=1

f ( x )= 11+sin x

f ( x )= (1+sin x )−1

f ' ( x )=−1 (1+sin x )−2(cos x )

f ' ( x )= −cos x

(1+sin x )2

f ' (0 )= −1

(1+0 )2

f ' (0 )=−1

f ' ( x )= −cos x

(1+sin x )2

Misal,u=−cos x v=(1+sin x )2

u'=sin x v '=2 (1+sin x )(cos x )Maka,

f ' ' ( x )=u'v−uv 'v2

f ' ' ( x )=(sin x (1+sin x )2)−¿¿f ' ' ( x )=(1+sin x )¿¿f ' ' ( x )=sin x¿¿

f ' ' ( x )= sin x+sin2 x+cos2 x+cos2 x(1+sin x )3

f ' ' ( x )= sin x+1+cos2 x(1+sin x )3

f ' ' (0 )=0+1+11

f ' ' (0 )=2

Dengan memasukkan harga-harga turunan ini ke deret Maclaurin diperoleh:

f ( x )= 11+sin x

¿ f (0 )+ f '(0)x+f ' '(0)

2 !x2+…

¿1+(−1 ) x+ 22x2+…

¿1−x+x2−…