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The Real Number Sistem
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Transcript of The Real Number Sistem
CALCULUS IMATERI
1Sistem bilangan reel
2.Pertaksamaan
3.Fungsi dan limit
4.Turunan dan aplikasi turunan
5.Integral dan aplikasi Integral
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BUKU ACUAN
1. ”CALCULUS”
Dale varberg
Edwin J.Purcell
.2.”Calculus dan Geometri Analitik”
Thomas
3. .”Calculus dan Geometri Analitik”
Howard Anton
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BilanganBilangan
ReelReel khayalkhayal
irasionalirasionalrationalrational
bulatbulat asliasliwww.cyberofcampus.co.cc
THE REAL NUMBER SYSTEM
The Integers and The Rational Numbers 1. N =Natural Numbers : 1,2,3,4,5........ 2. Z= Integers .........-3,-2,-1,0,1,2,3...... 3. Q = Rational numbers p/q, ( p,q are integers and q ≠ 0 ) ½,-3/4,3/3,7/6
4 Ir = The Irational Numbers ...... ,51,3 5. R = The real Number IrQZN RQZN
THE REAL NUMBER SYSTEM
The Integers and The Rational Numbers 1. N =Natural Numbers : 1,2,3,4,5........ 2. Z= Integers .........-3,-2,-1,0,1,2,3...... 3. Q = Rational numbers p/q, ( p,q are integers and q ≠ 0 ) ½,-3/4,3/3,7/6
4 Ir = The Irational Numbers ...... ,51,3 5. R = The real Number IrQZN RQZN
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• The Integers and The Rational Numbers
• 1. N =Natural Numbers : 1,2,3,4,5........
• 2. Z= Integers .........-3,-2,-1,0,1,2,3......
• 3. Q = Rational numbers p/q, ( p,q are integers and q ≠ 0 ) ½,-3/4,3/3,7/6
• 4 Ir = The Irational Numbers ......5. R = The real Number
• Bilangan reel
-∞ 0 ∞
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Sifat-sifat
0)(1 x.x5.Invers
x x x.1 0 xElements 4.Identity
xzxy z) x(y Law tive3.Distribu
z )y x z)(y xLaw eAssociativ 2.
xy y x Law f1.Komulati
1-
xx
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• The Order Properties
yz xzy x negative z
yz xzy x positive zen cation wh4.Multipli
z y z xy x 3.Addition
z x z y andy x vity 2.Transiti
yxory xoryx
holds following theof oneexactly numbers, are x ifmy 1.Trichoto
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• Bagaimana mencari penyelesaian
2x 1- x solution The *
positif,daerah pilih If
negatif,daerah pilih If *
negative 1)(2)( 0 x*
2 --------1- bilangan garis kedalamGambar *
1- , 2 ;yaitu nol titik Cari *
0 1)2)(x-(x :Solution
0 2- x - x
darian penyelesaihimpunan 1.Tentukan
NKETAKSAMAA
2
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• Contoh 2
///2///////////////(1)//1/////////- 1 dan x ) 2 x 1- / (
-2------- 1- 0 ).(-)( 0 untuk x
2)________)________( (-1_____,2 1-adalah nol titik
0 )2)(1(
diabaikandapat maka
anpertaksama terhadap
hberpengarutidak , 1 xif 0 1)-(x :Solution
0 )2)(1(1)-(x
T.2
0
2
2
x
xx
xx
darianpenyelesaihimpunanentukan
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2or x 1x1- area. positip theissolution the
)2()1()1(
0 )(-)(-)( 0 xintervaleach Check
____________2__ ______1_____ 1- _____
,2 ,1 1- :point Zero
0 )2)(1)(1(
0 )2)(1)(1(1)-(x 0 )2)(1(1)-(x 4.
1 x and 2x 1-solution the 0 )2)(1(1)-(x 3.
0
23
2
xxx
xxxxx
xx
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2x 1or 2- 3
)2(12)3(
0)(-)(
)(-)( 0
0 p
(2), ,1 ,-2 ) (-3adalah nol titik
pos x neg neg/pos pos x pospos/pos :Solution
0 )3)(2(
)21-x inequality the5.Solve
x
x
enyebut
xx
x
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64
64
0 )/(-)( 0 x
4 , 6point zero
0 4
6 0
4-x
82x-2)-(x
04)-(x
4)-2(x
4-x
2-x 0 2
4-x
2-x :Solution
2 4-x
2-x 6.
xorxnthesolutio
x
x
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06)5)(x-(x
4)-2)(x1)(x-(x3.
0 65x - 2.2
01.x
2
23
3
xx
x
solusiTentukan
soal
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NILAI MUTLAK
BABABABA
B
ABA
A
.4.3
B
A.2AB1.
mutlak nilai dari 2Sifat
0 A jika A,-
0A jika, A
mutlak Nilai
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///////_a////////a_________//////////
axandaxaX
a_________/////////////////a_._________
..axaaX
///////_a////////a_________//////////
axandaxaX
a_________/////////////////a_._________
..axaaX
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62
1224
8428
84-2x
1Contoh
x
x
x
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54/6
20464
13741374
137-4x
2Contoh
xorx
xx
xorx
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#########1________5#######
IIIsolutionThe
22x
2-xIIor2
2x
2-xI
22x
2-x2
22x
2-x
3Contoh
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####)#1__(____________________)5(#######
####)#1__(____________________)5(#########
####)#1_(__________)2(###############
)2()5(
02
50
2x
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)1(30)
2x
2x(2
2x
2-x
III
II
I
x
xII
x
xI
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62.6
2.5
1.4
123
42.2
34x1.
inidibawah solusiTentukan
2
2
xx
xx
xx
xx
xx
x
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13)34(
310,34
//////////////////////////////////0_____
////////31//////////////////
0,0)3)(1(
0034)34(
0034,34
034x
034x
34x.:1
2
2
22
22
2
2
2
xifxxIb
xorxifxxIa
xxxIa
xifandxxifxxIb
xifandxxifxxIa
xifxII
xifxI
xSolusi
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130034
0130034
)1()3(
)3)(1(34
0034)34(
0034,34
034x
2
2
2
22
22
2
xxifandxxIIbif
xorxxifandxxifIIa
xxxx
xifandxxifxxIIb
xifandxxifxxIIa
xifxI
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nextsolutiontheisx
xxxx
xcheck
c
xx
xxx
xifx
xifxxb
ispozerotheaFind
xx
Solusi
..........................2
424242
,2
_______0_______2________.
0,
0,
2)2(
2)2(2.
0,2:int,
42
2
nextsolutiontheisx
xxxx
xcheck
c
xx
xxx
xifx
xifxxb
ispozerotheaFind
xx
Solusi
..........................2
424242
,2
_______0_______2________.
0,
0,
2)2(
2)2(2.
0,2:int,
42
2
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),(,
42
4242
,0
02
14242
,02
xxsolutionThe
xxxx
xcheck
solutionasx
xxxxx
xcheck
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11
10),1(1
0,1
1,0,,11
:3
xifxx
xifandxifxxxx
ifxxx
xifandxifxxxx
solusi
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