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

2x2

2x

2-x

)1()2(

02

)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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