1 BILANGAN

8/18/2019 1 BILANGAN

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MATEMATIKA TEKNIK 1.BILANGAN

2. MATRIK dan DETERMINASI

3. FUNGSI

4. LIMIT

5. DERET

6. DIFERENSIAL danAPLIKASINYA

7. INTEGRASI danAPLIKASINYA


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I. BILANGAN 1. Bian!an Pada "ian!an a#i $1% 2% 3% .........& da'a( diada)an*'+,a#i-*'+,a#i da#a, ai(/0

a. P+n/aan dan '+n!/,an!an

" P+,)aian dan '+"a!ian . P+n!)a( dan +na,i) a)a,

i)a a dan " "ian!an a#i a)a a " dan a 8 " d%di ana dan d /!a

+,/'a)an "ian!an a#i.

S)+a i'/nan "ian!an


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S)+a i'/nan "ian!an

Bian!anK*'+)#

Bian!an Na(a

$ R+a &

Bian!an Kaa

$ Ia!in+, &

Bian!anRa#i*na

Bian!anI,a#i*na

Bian!an B/a( Bian!anP+aan

Bian!an B/a(N+!a(i9

Bian!an B/a(P*#i(i9 $ A#i &

N*


4/41

• Bian!an R+a

:i'/nan "ian!an ,+a (+,di,i da,i0

a. Bian!an Ra#i*na% ai(/ "ian!an-"ian!an "/a('*#i(i9% "/a( n+!a(i9% n*

dan '+aan a;" $di ana a dan " "ian!an "/a( &". Bian!an i,,a#i*na% ai(/ "ian!an-"ian!an an! (ida)da'a( dina(a)an

#+"a!ai "+n(/) '+aan.


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. Si9a( a#*#ia(i9

Un(/) n1% n2% n3 = N "+,a)/0

• $ n1 n2 & n3 n1 $ n2 n3 &

•

$ n1

8 n2

& n3

n1

$n2

8 n3

&

d. Si9a( *d//#

i)a n = N% a)a n > > n n

Si9a( *d//# '+n/aan $ > "ian!an )+#a(/an /n(/)

'+n/aan > = N & Un(/) n 8 1 1 8 n

Si9a( *d//# '+,)aian $ 1 adaa "ian!an )+#a(/an/n(/) '+,)aian 1 = N

+. Si9a( di#(,i"/(i9

Un(/) n1% n2% n3 = N "+,a)/0

• $n1 n2& n3 n1 n3 n2 n1 #i9a( di#(,i"/(i9 )i,i

• n1 $n2 n3& n1 n2 n1 n3 #i9a( di#(,i"/(i9 )i,i


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9. Si9a( in?+#;)+"ai)an /n(/) '+n/aan

• i)a a = I% (+,da'a( "ian!an @ a #+in!!a a $ -a & >

• i)a a = I dan a >% (+,da'a( "ian!an$)+"ai)an;in?+,# da,i a& #+d+i)ian

#+in!!a0 a. 1. Bian!an di(/i# /!a a-1.

• i)a a% " = I dan " > a)a di#+"/( a#i "a!ida,i a *+ "

a di#+"/( '+"ian!% " di#+"/( '+n+"/(% )aa/"/)an #/a(/ "ian!an

"/a( a)a di#+"/( '+aan.

•Bian!an 8 di#+"/( 'an!)a( n da,i "ian!an a "ia "+,a)/8 a.a.a n "/a% di(/i# /!a 8 an.

•Bian!an 8 di#+"/( a)a, n da,i "ian!an a "ia "+,a)/ a 8n.

a 8n di(/i# /!a

2

a

1

a

1

ba

b

a 1=

b

a

na x =

2442 === x


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2. P+,(ida)#aaan

a di)a(a)an +"i "+#a, da,i'ada " di(/i# a " i)a "+"i )+i da,i'ada a a(a/ " C a% adi a " i)a dan ana" C a.

Tanda % C% $+"i "+#a, a(a/ #aa&% dan $+"i )+ia(a/ #aa& di#+"/( )+(ida)#aaan.

Si9a(-#i9a(0

• a > i)a dan ana i)a a '*#i(i9

• a C > i)a dan ana i)a a n+!a(i9

• a > i)a dan ana i)a @a C >

• a C > i)a dan ana i)a @a >

• i)a a C " dan " C % a)a a C

• i)a a C "% a)a a C " /n(/) #+(ia' • i)a a C " dan C d% a)a a C " d

• i)a a C " dan '*#i(i9% a)a a C "

• i)a a C " dan n+!a(i9% a)a a "

• i)a > C a C " dan > C C d% a)a a C "d

(anda C "*+ di!an(i d+n!an (anda dan (anda


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3. In(+,?a $S+an!&

• In(+,?a "/)a

In(+,?a (+,"/)a In(+,?a (+,"/)a da,i a )+ " di(/i# $a% "&%a)a i'/nan "a!ian da,i R adaa0 $a% "& 8 a C 8

C "H% da'a( di!a"a,)an0

• In(+,?a (/(/' "/)a $#+(+n!a (+,"/)a& In(+,?a (/(/'"/)a da,i a )+ " di(/i# a%"&% a)a i'/nan "a!ian da,i

R adaa0 a% "& 8 a 8 C "H% da'a( di!a"a,)an0

• In(+,?a (/(/'

In(+,?a (+,(/(/' da,i a )+ " di(/i# a% "J% a)a i'/nan"a!ian da,i R adaa0

a% "J 8 a 8 "H% da'a( di!a"a,)an0

• In(+,?a "/)a (/(/' $in(+,?a #+(+n!a (+,(/(/'&

In(+,?a #+(+n!a (+,(/(/' da,i a )+ " di(/i# $a% "J% a)a

i'/nan da,i R adaa0 $a% "J 8 a C 8 "H% da'a(di!a"a,)an0

a "

a "

a "

a "

I ( i ( ( ) ( i


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In(+,?a-in(+,?a (a) (+,in!!a0

• A 8 8 C aH 8 - C 8 C aH -% aH

• B 8 8 aH 8 a C 8 C H a% H

• < 8 8 aH 8 - C 8 aH -% aH

• D 8 8 aH 8 a 8 C H a% H

• E 8 = R H 8 - H - % H

aa-

a

- a

a

-


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11/41


12/41

S*a La(ian;PR I1. T+n(/)an "+#a,an a,!a 8 #+"a!ai "ian!an

"/a( "ia0

a&. 84 256 "&. 27 83

2. T+n(/)an '+,(ida)#aaan i)a0

a&. 4 a C " di)ai)an d+n!an 2 dan -2"&. -2 8 5 d+n!an +n!!an(i (anda d+n!an

3. T+n(/)an a,!a /(a) da,i0

a&. 2 $ 6 @ &

"&. 2 8 1>>


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5. Bian!an +

Bian!an + "ia#a di!/na)an daa R//#-,//# I/Fi#i)a dan I/ T+)ni)

! = fakulteit/factorail/fakultas 5 ! = 5.4...".= "#

Bila$%a$ e = & '"(("((45) *iasa +itulis +e$%a$ 5 +esi,al&-aitu & '"((

+a$ +ise*ut Bila$%a$ Tra$se$+e$tal

• ika $ a+ala *ila$%a$ asli ,aka $ faktorial 0 $! 1 = $ 0$ 2 "1 0$3 1.........."

Berlaku ula $ ! = $ 0$ 3 "1 !

o$to6 " ! = "7 ! = ."7 ! = .."7 4 ! = 4..." = 4

•

....597182818284,2!

1...............

!3

1

!2

1

!1

11)1(

)1

1(

1

0

=++++++=+=

+=

→

∞→

nk mil

h

mil e

k

h

h

h


14/41

6. P+,/(a#i dan K*"ina#i

• i)a da,i i'/nan n /n#/, $*"+)& an! "+,ainan% a)a"ana)na #/#/nan $a,a '+n!/,/(an& /n#/, i(/ di#+"/(

'+,/(a#i i'/nanO (+,#+"/(. i)a da,i i'/nan n /n#/, an! "+,ainan di"/a(#/#/nan an! (+,di,i da,i

/n#/, $ ) C n &% a)a "ana) #/#/nan i(/ di#+"/("ana)na '+,/(a#i

(+,#+"/( d+n!an a"ian )% di,//#)an0

di ana0 n "ana)na /n#/, $*"+)& dan ) "ana)na an!!*(a

daa )+*'*).


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". Mi#a)an : A% B% < H

B+,a'a "ana) #/#/nan i'/nan (+,#+"/( da'a( di"/a(Q

aa"0 "ana)na /n#/, 3% "ana)na an!!*(a daa)+*'*) 3

n 3 dan ) 3

adi0 ada 6 #/#/nan

• i)a da,i i'/nan n "/a /n#/, an! "+,ainan a)andi#/#/n d+n!an a#in!-a#in! #/#/nan an! (+,di,i da,i )/n#/, $ ) n & (an'a +'+,a(i)an /,/(anna% a)a"ana) #/#/nan i(/ di#+"/( "ana)na )*"ina#ii'/nan (+,#+"/( d+n!an a"i)an ) di,//#)an0


16/41

aa"an0 n % ) 6

ada 4 ,+!/

i)a #+)+*'*) (+,#+"/( (+,di,i da,i 6 *,an! a)a"ana) ,+!/ da'a( di#/#/n0

1 ,+!/

) #/)/ )+ $i @ 1&% i#0 #/)/ )+ 4 ) 4 -1 3% a)a)*+#i+n "in*i/O

#/)/ )+ i dina(a)an d+n!an0

#/)/ )+ 4 adaa0

846

504

1.2.3

7.8.9

!3!.6

1.2.3.4.5.6.7.8.9

!)69(!6

!9

!)(!

!====

−=

−=

k nk

n

C k

n

1!6

!6

!)66(!6

!6

!)(!

!==

−=

−=

k nk

n

C k

n

)1().....2)(1(!!)(!

!

−−−−=⇒−=

k nnnnnk nk

nC n

k

33

!3

)2)(1(

!)(!

!ba

nnnba

k nk

nU

nk k nn

k

−− −−=−

=


17/41


18/41

aa"0 a. "ana)na /n#/, 3% "ana)na an!!*(a daa)+*'*) 3

$


19/41

7. Bin*i/ N+(*n

Bin*i/ N+(*n adaa '+n++#aian "in*i/ $#/)/d/a& d+n!an ,//#0

) #/)/ )+ $i @ 1&

#/)/ )+...1% ...........2%..............d#(............n% di ana adan "% #+,(a n adaa

"ian!an a#i

K*+#i+n "in*i/O da'a( di,//#)an d+n!an0

S/)/ )+...$S& % diana ) S @ 1% a)a0

Pandan! "+n(/)-"+n(/) di "aa ini0

• $ a " &n

• $ a " &2 a2 2 a " "2

• $ a " &3 a3 3 a2 " 3 a "2 "3

nn

n

nn

n

n

n

n

n

nbC baC baC aC ba ++++=+ −

−− 1

11

10

......)(

C k

n

)!(!

!

k nk

nC k

n −=

k k nn

k

bak nk

nU

−

−

=

!)(!

!

n

k

U


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Mi#a)an /n(/) #/)/ )+ 4 da,i n 5% a)a i 4 dan ) 4@ 1 3% )*+#i+n #/)/ )+ 4 adaa0

K*+#i+n-)*+#i+n "in*i/ (+,#+"/( dida'a( da,i "+n(/)#+!i(i!a 'a#aO d+n!an 'an!)a( "ian!an $ n & "/a('*#i(i9 dan )+i.

1

1 11 2 1

1 3 3 1

1 4 6 4 1

1 5 1> 1> 5 1

Un(/) 'an!)a( "ian!an '+aan a/'/n "/a( n+!a(i9a(a/ '*#i(i9 di!/na)an "+n(/)0

102

20

!2!.3

!3.4.5

)!35(!3

!5

!)(!

!===

−=

−=

k nk

nC

k

n

nk k n

nnnnnn

bbak

k nnnn

bannnn

bannn

bann

ban

aba

++−−−−

++

−−−+

−−+

−++=+

−

−−−−

...............!

)1(..().........2)(1(........................

!4

)3)(2)(1(

!3

)2)(1(

!2

)1(

!1)(

4433221


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Un(/) $ a " &1;2 .............. $'a)ai a,!a '+nd+)a(an&

a3 "2 1> a2 "3 5 a "4 "5

Ma)a )*+#i+n #/)/ )+ 4 adaa0 ) 4 @ 1 3

". T+n(/)an #/)/ )+ 4 da,i $3a @ 2"&

aa"0 n ) 4 @ 1 3

3535

3388

3

108864).8.(243.56

)2()3()38(!3

!8

!)(!

!

baba

babak nk

nU

k k n

−=−=

−−

=−

=

−−

nk k n

n

bbak

k nnnn

bannnn

bababaaba

++−−−−++

−−−+

−−+

−++=+

−

−−−−

...............!

)1(..().........2)(1(........................

!4

)3)(2)(1(

!3

)2)(1(

!2

)1(

!1)(

443321

21

21

2221

21

121

2

1

21

21

21

21

32335 10!)35(1.2.3

1.2.3.4.5

!)(!

!bababa

k nk

nU

k k nn

k

=−

=−

=

−−


22/41

S*a La(ian;PR III1. T+n(/)an )*+#i+n a7"5 da,i "+n(/) $a "&12

2. :i(/n! )*+#i+n 86 da,i "+n(/) $8 @ 2&11

3. T+n(/)an )*+#in da,i 86 'ada "+n(/) $28 @ 3&


23/41

. Ind/)#i L+n!)a'Ind/)#i +n!)a' di!/na)an /n(/) +"/)(i)an #/a(/,//# "+,a)/ /n(/) #+(ia' "ian!an a#i n $ n 1% 2%3% .........&


24/41

". $ a " &2 a2 2 a " "2

B/)(i0

Mi#a)an a 2 dan " 3 $ 2 3 &2 22 2.2.3 32

T+,"/)(i "+,a)/0 a. 1 3 5 .................. $2n @ 1& n2

". $ a " &2 a2 2 a " "2

. M+na,i a,!a P+nd+)a(anDa,i $ 1 8 &n% i)a n )+i +nd+)a(i n* $ n > &% a)a$ 1 8 &n 1 n8

i)a $ 1 8 &a 1 a 8


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In!a( n $'+aan& Bia V8 )+i di"andin!)an 8% a)a #/)/ )+ 3 d#( da'a(dia"ai)an% #+in!!a0


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1>.Bian!an K*'+)#:i'/nan "ian!an-"ian!an an! "+,"+n(/) a i"% diana a dan " "ian!an ,+a% #+dan! a(a/ i2 - 1% di#+"/(i'/nan "ian!an )*'+)#.


27/41

i)a 82 @ 68 34 > 8 3 5i a(a/ 8 3 @ 5i

Daa '+,na(aan 8 3 5 i% diana0

• 3 di#+"/( "a!ian ,+a da,i 8

• 5 di#+"/( "a!ian iain+, da,i 8

Ga"/n!an )+d/ana di#+"/( "ian!an )*'+)# Bian!an )*'+)# $"a!ian ,+a& $"a!ian iain+,&

W'+,a#i 'ada "ian!an )*'+)#

a. P+n/aan dan '+n!/,an!an "ian!an)*'+)# Ba!ian ,+a dan "a!ian iain+, a,/# di!a,a' #+a,a

(+,'i#a


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i2& $1-2i& $6 - 7i 2>& $1 @ 2 i&

$ 26 @ 7 i& $1 @ 2 i& 26 @ 7 i @ 52 i

14 i2

26 @ 5 i @ 14 12 @ 5 i

P+,)aian 2 9a)(*, "ian!an )*'+)# a#i '+,i(/n!an'ada //na +,/'a)an "ian!an )*'+)# /!a.

S+)a,an! i(/n!0 $ 5 i& $ 5 @ i& 25 4> i @ 4> i @ 64i2

25 64

:a#ina #+'+n/na ,+a% )+na'aQQQQQQ

K+d/a 9a)(*, id+n(i)% )+/ai (anda '/# $& dan in/#$-&

Pa#an!an "ian!an )*'+)# #+aa ini di#+"/(

"ian!an )*'+)# )*n/!a(O:a#i )ai d/a "ian an )*n / a( #+a/ +,/ a)an


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. P+"a!ian "ian!an )*'+)#• M+"a!i "ian!an )*'+)# d+n!an "ian!an ,+a

Mi#a)an0

• M+"a!i "ian!an )*'+)# d+n!an "ian!an )*'+)#

Mi#a)an0


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11. Bian!an )*'+)# #+a,a !,a#Da'a( dina(a)an #+a,a dia!,aa(i#% ai(/ d+n!an#+"/a !a,i# da,i (i(i) a#a $ > & )+ (i(i) (+,(+n(/ an!

+na(a)an '/# $ & a(a/ in/# $ - & 'ada #)aa an!di(+n(/)an. Ga,i# an! +na(a)an "+#a, $+a('anan!na& dan a,a $+a( +n/n/))an )+'aa ana)'ana& di#+"/( ?+)(*,O.

• i)a $3& 8 $-1& $-3& 9a)(*, $-1& +n+"a")an ?+)(*,"+,"ai) a,a 1>>.

3-3

-3 -2 -1 > 12 3

-3 -2 -1 > 12 3

1>>-3 3

Fa)(*, $ 1& #+(a,a d+n!an i2 ai(/ i 8 i adi +n!ai)an


33/41

Fa)(*, $-1& #+(a,a d+n!an i2 ai(/ i 8 i% adi +n!ai)and+n!an i #+)ai a)an +/(a, ?+)(*, #+"+#a, >> daaa,a '*#i(i9 '+n!/)/,an #/d/(% ai(/ daa a,a"+,aanan d+n!an a,/ a.

• i)a 3 8 i 3i% 3i 8 i 3 i2 - 3% a)a #+a,a !,a# da'a(di!a"a,)an #""0

-3 -2 -1 >1 2 3

3

3i

>>

-3 -2 -1 > 12 3

3

3i

X i-3

X i

• i)a )+d/a "a!ian ,+a dan iain+, d+n!an #/"/ XX


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• i)a )+d/a "a!ian ,+a dan iain+, d+n!an #/"/ XX1

dan YY 1 #+'+,(i "ia#a% a)a0

• i)a in!in +na(a)an 3 2 5 #+"a!ai /a d/a?+)(*,% a)a )+d/a ?+)(*, di !a"a,)an #+a,a "+,an(ai#+"a!ai "+,i)/(0

X X1

Y

Y 1

-3 3

3i

-3i

>8 i

•S)aa #/"/ X+na(a)an "ian!an

,+aXX1 adaa #/"/

"ian!an ,+a•S)aa #/"/ Y+na(a)an "ian!an

iain+, YY 1 adaa #/"/

iain+,

> 1 2 3

4 5

3 2

3 2 5


35/41

• i)a "ian!an )*'+)# 3 2i% a)a di/a)an?+)(*, an! +na(a)an 3 d+n!an ?+)(*, an!+na(a)an 2i

P+,na(aan !,a# ini di)+na #+"a!ai dia!,a A,!andO• A(a/ d+n!an +n!!/na)an #)aa #/"/ XY di ana

#/"/ X #+"a!ai #/"/ ,+a dan #/"/ Y #+"a!ai #/"/iain+,.

> 1 2 34 5

32i32i

X

2>

>

P+,a(i)an ?+)(*, 2iadaa 2 8 i an! +/(a,

?+)(*,na >>.Z+)(*, an! +na(a)an $3 2i& adaa #+"/a?+)(*, da,i 'an!)a ?+)(*,'+,(aa $(i(i) a#a& )+//n! ?+)(*, )+ d/a

-6 -4 -2 > 24 6

4 2 3i2

X

X

Y

• P+n/aan dan '+n!/,an!an "ian!an )*'+)# #+a,a


36/41

• P+n/aan dan '+n!/,an!an "ian!an )*'+)# #+a,a!,a#

Mi#a)an /a an(a,a 1 5 2i dan 2 2 3i

> 1 2 3 4 56 7

2 2

3i

5

4

3

2

1

1 5

2i

1

2

adi 1 2 $5 2i& $ 2 3i& 7 5i /a d/a ?+)(*, daa

dia!,a A,!and+,/'a)an dia!*na da,i aa,an !+nan! an!

di"+n(/) *+ )+d/a?+)(*, (+,#+"/(.

• Un(/) '+n!/,an!an an(a,a 5 2i dan 2 3i


37/41

• Un(/) '+n!/,an!an an(a,a 1 5 2i dan 2 2 3i

da'a( dia)/)an d+n!an #+di)i( #ia#a(% di ana 1 - 2

1 $- 2& $- 2& ?+)(*, n+!a(i9 da,i 2 $"+,aanan

a,a&.

> 1 2 3 4 56 7

2 2

3i

5

4

3

2

1

1 5

2i

adi 1 -

2 $5 2i& - $ 2

3i& 3 - i

'+n!/,an!and/a ?+)(*, daadia!,a A,!and

+,/'a)andia!*na da,i aa,an

!+nan! an!

di"+n(/) *+ )+d/a?+)(*,(+,#+"/(% ai(/0

1 -

2 $5 2i& - $ 2 3i&

1 $-

2&

$5 2i& $ -2

- 3i&

1 @

2 3

- i

-2 - $2

3i&

TUGAS IZ


38/41

TUGAS IZ

K+,a)an #*a di "aa ini0

1. a&. $6 5i& @ $4 @ 3i& $2 @ 7i&

"&. $3 5i& @ $5 @ 4i& @ $-2 @ 3i&2. a&. $4 @ 5i& $3 2i&

"&. $3 4i& $2 @ 5i& $1 @ 2i&

&. $2 7i& $2 @ 7i&

3.

4. Ga"a,)an dia!,a A,!and /n(/) +na(a)an

?+)(*, da,i #*a n*. 1

i

iic

i

ib

i

ia

43

)21)(32().

35

53).

2

54).

+

−+

−

+

−

−


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n


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n

n


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