FR-2012_123224006 (Faidah Dwi Yunita Sari)

7/25/2019 FR-2012_123224006 (Faidah Dwi Yunita Sari)

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NAMA : FAIDAH DWI YUNITA SARI

NIM : 123224006

KELAS : FR D/2012

CHAPTER 4

SECTION 1

1. u= x

2

x2+y2

u x

=2x (x2+y2 )x2 (2x )

(x2+

y2 )2

u y

=0 (x2+y2 )x2 (0+

(x2+

y2 )2

2x3+2x y22x3

(x2+y2 )2

2x2y(x2+y2 )

2

2xy (x2+y2)

2. s=tu

s t=1

s

u

=ut

3. z=ln u2+v2+w2

zu

= u

u +v +w

z

v=

v

u +v +w

zw

= w

u +v +w

4. w=x3y32xy+6

w x=3x22y

2 w x2

=6x

w y

=3y22x

2 w

y2=6y

Saatw x

=w y

=0

w x

=3x22y=0

3x2

=2y

y=32

x2

x=0 ataux=23

w y

=3y 2x=0

3y2=2x

3

2

x=32

y =) x

Saat (x,y) = (0,0)

2 w x2

=6x=6.0=0

2 w y

=6y=6.0=0

Saat (x,y) = (-2/3, 2/3)

2 w

x2=6x=6(23)=4

2 w y

=6y=6( 23 )=4

5. w=8x4+y42x y2

w x=32x

3

2y2


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2 w x2

=96x

w y

=4y3

4xy

2

w y

2=12y 4x

Saatw x

=w y

=0

w x

=32x 2y =0

32x =2y

x= 116 y w y

=4y 4xy=0

4y =4xy

y=y x

y4=116

x=14, y=1

2

Saat (x,y) = (0,0)

2 w

x2=96x2=96.0=0

2 w y

=12y 4x=12.04.0=0

Saat (x,y) = (1/4, 1/2)

2

w

x2=96x2=96 ( 14 )

2

=6

2 w y

=12y 4x=12( 12 )2

4 (1

4)=2

6. u=ex cosy

2 ux y

= 2u y x

ux

=x ex ,

u y

=siny

x(u y)= x(siny )=0

y( u x )= y(x ex

)=0

Jadi

2 ux y

= 2u y x

Diketahui :

z=x +2y , x=rcos, y=rsin

7.

z x

.

zx

z=x2+2 (rsin)2

x2+2 r2 si n2

x2+2 r2 (1co s2 )

x2+2 r2(1x2

r2 )

x2+2 r22x2

x2+2 r2


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z x

!.

z x

z=x2+y2

x2+2 r2 si n2

x2+2( xcos )2

sin2

x +2x sin cos

z x

2x (1+2 tan2)

10.

z y

11. ( z y )r=

Ja"a# :

z=x2+2y2x=r cos y=r sin

$aka %i&ai da'i ,

z=r2 cos2 +2y2

z=r2 (1sin2 )+2y2

z=r2(1y2

r2 )+2y2

z=r2y2+2y2 z=r2+y2

(

z

y )r

=2y

12. ( z y )=

Ja"a# :


$aka %i&ai da'i ,

z=r2 cos2 +2y2

z= y

2

sin2(1sin2)+2y2

z= y2

sin2y2+2y2

( z y )=2y cosec2+2y 2y (cosec2+1)

13. ( z )x=

Ja"a# :

z=x2

+2y2

x=r cos y=r sin

$aka %i&ai da'i ,

z=x2+2r2 sin2

z=x2+2 x2

cos2

sin2

z=x2

+2x2

tan2

( z y )=2x22 sec2 tan 4x2 sec2 tan

4 x2

cos2

tan

4 r2 tan

14. ( z )y=

Ja"a# :


$aka %i&ai da'i ,

z=r2 cos2 +2y2


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z= y2

sin2(1sin2)+2y2

z= y2

sin2y2+2y2

z=y2 cosec2+y2

1+cot( 2 )+y2

z=y2

z=y2+y2 cot2 +y2

z=y2+y2 cot2 +y2

z=2y2+y2 cot2

( z )y=2y2cosec

2cot 2

y2

sin2

cot

2 r2 cot

15.

(

z

)r

=

Ja"a# :


$aka %i&ai da'i ,

z=r2 cos2 +2 r2sin2

z=r2(1sin2)+2 r2sin2

z=r2r2sin2 +2 r2sin2

z=r2+r2 sin2

( z )r=r22sin cos r2sin2

16. ( zr )=

Ja"a# :


$aka %i&ai da'i ,

z=r2 cos2 +2 r2sin2

z=r2(1sin2)+2 r2sin2

z=r2r2sin2 +2 r2sin2

z=r2+r2 sin2

( z r )=2 r+2r sin2 2 r (1+sin2 )

17. ( z r )x=

Ja"a# :


$aka %i&ai da'i ,

z=x2+2r2 sin2

z=x2+2r2(1cos2 )

z=x2+2r22r2 cos2

z=x2+2r22r2x2

r2

z=x2+2r22x2

( z r )x=4 r

1. ( zr )y=

Ja"a# :


$aka %i&ai da'i ,

z=r2 cos2 +2y2

z=r2 (1sin2 )+2y2

z=r2r2sin2+2y2

z=r

2

r

2 y2

r2+2y

2

z=r2+y2


5/38

( zr )y=2 r

1!.2z

r y=

Ja"a# :


$aka %i&ai da'i ,

z=r2 cos2 +2y2

z=r2 (1sin2 )+2y2

z=r2r2sin2+2y2

z=r2r2y2

r2+2y2

z=r2+y2

o

r z y

=

o

z y

=2y

o

r

(2y )=0

20. 2z

x =

Ja"a# :


$aka %i&ai da'i ,

z=r2 cos2 +2 r2sin2

x

z

=

z

=r2 2sin cos +2 r22sin cos r2 sin2+2r2

r 2sin 2

x(r2sin2 )=0

21. ( 2z

y )=

( z )y=2y cotcsc

( 2z

y )=4y cotcsc

y cot=x

4xcsc

22. ( 2z

r x )=z=x2+2 (rsin )2

zr

=2 r , zx

=0

x2+2 r2 si n2

x2+r 2 2 si n2

r(

z x )=

r

.0=0

x2+r 2 (1co s2 )

x2+r 2r2 co s2

x2+r 2r2(x2

r 2 )

r

23. ( 2z

r )=

z=(rcos )2+2 (rsin )2

z

r( z

)=

r2r cossin

r2 (co s2+2 si n2 )=4 r sincos


6/38

r2 (co s2 +2(1co s2 ))

2r .2 sincos

r2 (cos2+22co s2 )

2r sin2

r2 (2co s2 )

zr

=2 r (2co s2 )

( zr)= (2r (2co s2 ))

z

=r2 (2co s2 )

2r (2 sincos )

r2

(0+2cossin )

2r sin2

2r cossin

24. ( 2

zx y )=

z=x +2y

zx

=2x z y

=4y

x(z y)= x.0=0

SECTION 2

1. cosxsinhy=(1x2

2!+

x4

4 !+)(y+y

3

3!+)

(y+y3

3 !

x2y2!

x2y3

2!3 !+

x4y4 ! +

x4y3

4 !3 !+)

y+ 16

y312x2y1

12x2y3+ 1

24x4y+ 1

144x 4y3+

2. cos (x+y )=1(x+y )2

2! +

(x+y )4

4 ! +

1 (x2

+2xy+y3

)2 + (x4

+4x3

y+6x2

y2

+4x24

11

2x2xy1

2y2+ 1

24x4+ 1

6x3y+ 1

4x 2y

3.ln (1+x )

1+y =ln (1+x ) (1+y ) 1

(xx2

2+

x3

3+)(1y+ 2y

2

2 !+)

xxy+2x y2

2 !

x2

2+

x2y2

2x2y2

2.2! +

x3

3

x3y3 +

2x3y2

3.2 !

xxy+x y212x2+12x

2y12x2y2+ 13x

313x3y+ 13

4. exy=1+xy+

(xy )2

2 ! +

(xy )3

3 ! +

1+xy+ 12x2y2+ 1

6x3y3+

5. 1+xy=(1+xy )

1+12

xy18

(xy )2+116

x3y3+

1+1

2xy1

8x2y + 1

16x3y3+

6. ex+y=1+(x+y )+

(x+y )2

2! +

1+x+y+ (x2+2xy+y2 )

2 +

1+x+y+ 12

x2+xy+12

y2+

SECTION 4

. #*ut h*" +uh (i% e'e%t) d*e a% e''*' *

1 i% aa%d baet a2b3

Ja"a# :


7/38

De%a%daa=1=0,01 da%

dba=1=0,01

a2#3= 2 &% a 3 &% #

= 2daa 3

dbb

= 2 (0,01) 3 (0,01)= 0,02 0,03= 0,05= 5

!. Ja"a# :f = gh&%f = &%g.h&%f = &%g &% h di i%te'a&ka%

dff =

dgg +

dhh

11. Ja"a# : (4,!)2 +e%dekati (5)2

$aka :

( x ) = x 2 ( x ) = 2x , De%a%

x= 5 da%

x= (4,! 5) = - 0,02

Sehi%a : (4,!)2

= ( x

x)

( x

) ( x

) x

=x 2

2x x

= (5)2 2(5)(- 0,02)= 25 0,2= 24,

(3,03)2 +e%dekati (3)2

$aka : ( x ) = x 2 ( x ) = 2x ,

De%a%x

= 3 da% x

= (3,03 3) = 0,03

Sehi%a : (3,03)2 = ( x

x

)

( x

) ( x

) x

=

x 2

2x x

= (3)2 2(3)(0,03)= ! 0,1= !,1, Sehi%a(4,98)

2+(3,03)2 = (24,8 )+9,18 =

15,62

8e%dekata% %i&ai 15,62 ada&ah . . .

15,62 +e%dekati 16

( x ) = x ( x ) =1

2x ,

De%a%x

= 16 da% x

= (15,62 16) = -

0,3di+a%a (- 0,3)

(- 0,4). Sehi%a : (15,62)2 = (

x

x)

( x

) ( x

) x

= x 1

2x x

= 16 1

216 (-0,4)

= 4 0,05

= 3,!5

12. Ja"a# : (2,05)2 +e%dekati (2)2

$aka :

( x ) = x 2 ( x ) = 2x ,

De%a%x

= 2 da% x

= (2,05 2) = 0,05

Sehi%a : (2,05)2 = ( x

x

)

( x

) ( x

) x

=x 2

2x x

= (2)2 2(2)(0,05)= 4 0,2= 4,2

(1,!)2 +e%dekati (2)2

$aka :

( x ) = x 2 ( x ) = 2x ,

De%a%x

= 2 da% x

= (1,! 2) = - 0,02

Sehi%a : (1,!)2 = ( x

x

)

( x

) ( x

) x

=x

22x x

= (2)2 2(2)(-0,02)

= 4 0,02= 3,!2, Sehi%a3

(2,05)2+(1,98)2 =

34,2+3,92


8/38

=38,12

8e%dekata% %i&ai38,12 ada&ah . . .

38,12 +e%dekati

38

( x ) = 3x ( x ) =1

33x2 ,

De%a%x

= da% x

= (,12 ) = 0,12

Sehi%a :38,12 = (

x

x)

( x

) ( x

) x

=3

x

1

3 3x2

x

=38

1

3382 (0,12)

= 20,01= 2,01

14. Ja"a# :

Di+e%i*% * a #*x = 200 x 200 x 100 *' 2 x 2

x 19he%, di+e%i*% * a #*x ha%e t* 201 x 202 x

!! *' 2,01 x 2,02 x 0,!!e%th * a ae dia*%a& * a #*x a% "'ite :

d' = p2+l2+t2

$aka :8a%;a% dia*%a& 'ua% de%a% di+e%i = 2 x 2 x

1d' = p

2+l2+t2

= 22+22+12

= 9

= 3

Sete&ah di+e%i da'i #a&*k di'u#ah +e%;adi 2,01

x 2,02 x 0,!!. $aka : = 2,01< ehi%a e%dekata% %i&ai e%dekata%

(2,01)2ada&ah(2,01)2 +e%dekati (2)2

$aka :

( x ) = x 2 ( x ) = 2x ,

De%a%x

= 2 da% x

= (2,01 2) = 0,01.

Sehi%a : (2,05)2 = ( x

x )

( x

) ( x

) x

=x 2

2x x

= (2)2 2(2)(0,01)= 4 0,04= 4,04

& = 2,02< ehi%a e%dekata% %i&ai e%dekata%

(2,02)2ada&ah (2,02)2 +e%dekati (2)2$aka :

( x ) = x 2 ( x ) = 2x ,

De%a%x

= 2 da% x

= (2,02 2) = 0,02.

Sehi%a : (2,02)2 = ( x

x

)

( x ) ( x )

x

=x 2

2x x

= (2)2 2(2)(0,02)= 4 0,0= 4,0

t = 0,!!< ehi%a e%dekata% %i&ai e%dekata%

(0,!!)2ada&ah (0,!!)2 +e%dekati (1)2

$aka : ( x ) = x 2 ( x ) = 2x ,

De%a%x

= 1 da% x

= (0,!! 1) = -

0,01. Sehi%a : (0,!!)2 = ( x

x

)

( x

) ( x

) x

=x 2

2x x

= (1)2 2(1)(-0,01)= 1- 0,02= 0,!

8e'u#aha% a%;a% dia*%a& 'ua% ada #a&*k :

d' = p2+l2+t2

= 4,04+4,08+0,98

= 9,1

8e%dekata% %i&ai 9,1 ada&ah . . .

9,1 +e%dekati 9


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( x ) = x ( x ) =1

2x , De%a%

x= ! da%

x= (!,1 !) = 0,1. Sehi%a :

9,1

= (

x

x

)

( x

) ( x

) x

= x 1

29 (0,1)

= 9 1

2x (0,1)

= 3,0167

15. >ti+ate the ha%e i%

f( x ) = 0

xet

t2+0,51 dt

i x ha%e '*+ 0,7 t* 0,71

Ja"a# :

?etika x dia%ti da'i 0,7 ke 0,71

FR-2012_123224006 (Faidah Dwi Yunita Sari)

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