fr-2012_123224006 (faidah dwi yunita sari)
TRANSCRIPT
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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# :
z=x2+2y2x=r cos y=r sin
$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# :
z=x2+2y2x=r cos y=r sin
$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# :
z=x2+2y2x=r cos y=r sin
$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# :
z=x2+2y2x=r cos y=r sin
$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# :
z=x2+2y2x=r cos y=r sin
$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# :
z=x2+2y2x=r cos y=r sin
$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
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( zr )y=2 r
1!.2z
r y=
Ja"a# :
z=x2+2y2x=r cos y=r sin
$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# :
z=x2+2y2x=r cos y=r sin
$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
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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# :
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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
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=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