ppt analisis real

8/18/2019 PPT Analisis Real

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KELOMPOK 1

NAMA KHAIRATUL HUSNA

RAUDAH HAIFA ANGGRAINI RINA KHAIRATI NOR HAMDANAH ERNI YANTI

NPM:30612E3001

:30612E3013:30612E3015:30612E3017:30612E3038

:30612E3018BILANGAN

REAL


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2.1.1. Sif!"#if! $%&' (

* Penjumlahan

)A1* + & , & + -!-/ #- & ∈ (

)A2* ) + &* + , + )& + * -!-/ #- &

∈ (

)A3* !'4! 0 ∈ ( #4i/i i

0 + , 0 4 + 0 , -!-/ #- ∈ (

)A* -!-/ #!i ∈ ( !'4! " ∈ (

#4i/i i + )"* , 0 4 )"* + ,

0


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* Perkalian

)M1* . & , & . -!-/#-&∈ (

)M2* ) . &* . , .)& . * -!-/#-&∈

(

)M3* !'4! 1 ∈ ( #4i/ii 1. ,

4 . 1 , -!-/#- ∈ ()M* ∈ ( ≠ 0 !'4!∈ (#4i/ii

. , 1 4 . , 1• Distributifpenjumlahandanperkalian

(D) a . (b + c) = (a . b) + (a . c) dan


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2.1.2. T')* ;i/ < ∈ ( 4 < + , / <

, 0

)&* ;i/ - 4 & ≠ 0 $ ( 4 - . &

, & / -,1

2.1.3 Terema

)* ;i/ ≠ 0 4 & ∈ ( #4i/ii

. & , 1 / & ,

)4* ;i/ . & , 0 / , 0 !- & , 0


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2.1. T' Ti4/ 4$ ' ∈ = #4i/ii '2 , 2>-/!i :

A4i/ ' ∈ = #4i/ii'2, 2 .

/ ' 4! 4i!-$i#/ #&i ' , ?!i4/i$i/i'#/-!-/-$i 1)#$i'i*

, 2

⇔ P2 , 2?2 )/' 2?2/ 2 /i&!@ %-*


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S&& %i/ %i$ / , 2"1 ∈ !-

2 , )2"1*2

, 2 " + 1

, 2 )22 B 2*+1 )>''!i 2 %i$ %4i '-#$*

K' / ,2/ /∈

2 , )2/*2

, /2 S-!i!-#i / '# #&$-@)2 , 2?2 *

C 2 , 2?2 , 4 ? '-# %i$ )/' ? '-##$i 'i*

⇔

/2

, 2?2

⇔ 2/2 , ?2 &''!i ? .

T'4! /!'4i/#i & q 4$ &i$ #$i @

#/$i-# %i$. ;4i 4i 4i !# #$. ;4i!i4/ 4 ' ∈ ( #4i/i i '2, 2


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2.1.5 Sif!"#if! U'-! 4 (

A4 #-! ( @ 4i#&-! 4i- &i$"&i$ '$ #i!if

!# @ -i #if!"#if! &'i/-! :

i. ;i/ & ∈ / + &

ii. ;i/ & / & ∈

iii. ;i/ ∈ / -i !! #!-

/4i#i &'i/-! :

∈ , 0 " ∈


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2.1.6 4i#i

Di&'i/ & ∈ (

. ;i/ B & ∈ / 4i!-$i# > & !- &


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2.1.7 T'

Di&'i/#&' & ∈ (. ;i/ > & 4 & > / >

&. ;i/ > & / + > & + . ;i/ > & 4 > 0 / > &%i/

> & 4 0

;i/


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2.1.8 T'

*. ;i/ ∈( 4 ≠ 0 / 2J0

& *. 1 J 0 *. ;i/ ∈ / J0

Ti4/4&i$ '$ #i!if @

!'/i$ #&&%i/4i&'i/ J0

4/'J0 /4i'$ 0


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2.1. T'

;i/ ∈ ( #4i/ii 0 ≤ < -!-/#!i

0 / > > 0

Di &il0 , / 0


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2.1.10 T'

;i/ & J 0 / &'$/-)i* J 0 4 & J 0 !-)ii* 0 4 & 0

!ukti: %i/ & J0 / 0 4 &0.&i$ 0 / 0 !- J0.

U!-/ J0 / &'$/- 1.)&*J0 !- &J0. ;4i )i* J0 4 &J0

U!-/ 0 / &'$/- 1 )&* 0 !- &0.

;4i &'$/- )ii* 0 4 &0


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2.1.11 A/i&!

%i/ & 0 / &'$/-

i. 0 4 & J 0 !-

ii. J 0 4 & 0


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K-/i )i* "1J0 4 + 2 J0• "1J0⇔ B 1 +1 J 0 +1⇔ J 1

• + 2 J0⇔ + 2 + )"2* J 0 + )"2*⇔ J "2

U!-//-/i )ii* 4' @#4 )i* /4i'$ 1 4 "2

;4i > , ∪


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2.1.13 "nth

Di&'i/ ≥ 0 4 & ≥ 0 /

. & ⇔ 2 &2 ⇔

&. ≤ & ⇔ 2 ≤&2 ⇔ ≤


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Ketaksamaan bernuli ;i/ J "1 / )1+*. ≥ 1 + Q -!-/ #-

∈

>-/!i : )-/ i4-/#i !!i/*

U , 1 4i'$ 1 + ≥ 1 + )&'*

U , / 4i'$ )1+*/ ≥ 1 + / ) &'*

U , /+1 4i'$ Q )1+*/+1 ,)1+*/ )1+*

A/ 4i&-/!i/ &' -!-/ ,/+1 '

!'&-/!i & )1+*. ≥ 1 + &'$/- -!-/

#- ! &i$ #$i


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#ilai Mutlak dan$aris !ilan%an &eal

2.2.1Di#i

Ni$i-!$/4'i#-!-&i$ '$ a

4i#i&$/4


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2.2.2 Terema

)* -!-/#-&∈ (

)&* , 2-!-/#- ∈ (

)* ;i/ ≥ 0 /≤ %i/4@%i/" ≤ ≤

)4* "≤ ≤ -!-/ #- ∈ (


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2.2.3 Ketaksamaan 'e%iti%a

;i/ & ∈ ( /+

>-/!i :

D'i !' 2.2.2 )4* 4i'$"≤ ≤ 4 "≤ ≤

D%-$//4-/!/#

4i'$

M-/!' 2.2.2 )*

4i'$+


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(kibat

C ;i/&∈ ( /

)* ≤

(β) ≤+

C ;i/ 1 2 .4$#&'&i$

'$ /

++ +


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"nth

T!-/ i- A 4'i&i$ '$ @ -i7

P@$#i :D'i !' 2.2.2 )* /i!$i!& ∈ A⇔) "7 2 + 3 7⇔) "10 2 ⇔) "5 2

;4iH A,


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$aris !ilan%an &eal

P4 'i# '$ i$i -!$/ 4'i -#-' ( 4i #&i %'/ 4'i /

-#! 0.

S' -- %'/ !' -#-' 4 &

4i ( 4$ "&

0 1 2 3-1-2

-3

-2 3| (-2) – 3 | = 5


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De)nisi 2.2.

Di&'i/ ( 4 J 0. P'#/i!' "

4'i 4i4i#i/ #&i i-.

)* , ( Q B V U!-/ ( '@! !'-! 4i

V)* /-iW$ 4 '@!

" B ⇔ " +


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Terema 2.2.+

Di&'i/ (. ;i/&'44$'#/i!'

)* -!-/#!i V J 0 / ,>-/!i :

>i$ -i -!-/#!i 0

/4'i!' 2.1. 4i'$&, 0

#i ,0


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ppt analisis real

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