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

www.cyberofcampus.co.cc

BUKU ACUAN

1. ”CALCULUS”

Dale varberg

Edwin J.Purcell

.2.”Calculus dan Geometri Analitik”

Thomas

3. .”Calculus dan Geometri Analitik”

Howard Anton


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


• 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 ∞


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


• 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


• 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


• 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


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


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


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


06)5)(x-(x

4)-2)(x1)(x-(x3.

0 65x - 2.2

01.x

2

23

3

xx

x

solusiTentukan

soal


NILAI MUTLAK

BABABABA

B

ABA

A

.4.3

B

A.2AB1.

mutlak nilai dari 2Sifat

0 A jika A,-

0A jika, A

mutlak Nilai


///////_a////////a_________//////////

axandaxaX

a_________/////////////////a_._________

..axaaX

///////_a////////a_________//////////

axandaxaX

a_________/////////////////a_._________

..axaaX


62

1224

8428

84-2x

1Contoh

x

x

x


54/6

20464

13741374

137-4x

2Contoh

xorx

xx

xorx


#########1________5#######

IIIsolutionThe

22x

2-xIIor2

2x

2-xI

22x

2-x2

22x

2-x

3Contoh


####)#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


62.6

2.5

1.4

123

42.2

34x1.

inidibawah solusiTentukan

2

2

xx

xx

xx

xx

xx

x


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


130034

0130034

)1()3(

)3)(1(34

0034)34(

0034,34

034x

2

2

2

22

22

2

xxifandxxIIbif

xorxxifandxxifIIa

xxxx

xifandxxifxxIIb

xifandxxifxxIIa

xifxI


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


),(,

42

4242

,0

02

14242

,02

xxsolutionThe

xxxx

xcheck

solutionasx

xxxxx

xcheck


11

10),1(1

0,1

1,0,,11

:3

xifxx

xifandxifxxxx

ifxxx

xifandxifxxxx

solusi


Documents

The Real Number Sistem