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Taylor Series[1]

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8/9/2019 Taylor Series[1] http://slidepdf.com/reader/full/taylor-series1 1/21 Vk~abx \ixni| V`i Lbiccnlnigv| bc k Tb}ix \ixni|
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Page 1: Taylor Series[1]

8/9/2019 Taylor Series[1]

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Vk~abx \ixni|

V`i Lbiccnlnigv| bc k Tb}ix \ixni|

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Lbiccnlnigv| bc k Tb}ix \ixni|

\yttb|i v`kv }i `ksi cyglvnbg  c  mnsig h~ k tb}ix |ixni|

8 <

4 ; 4 8 4 < 4( . ( . ( . ( . c p k k p p k p p k p p+ J 

]`kv lkg }i |k~ khbyv v`i xiakvnbg|`nt hiv}iig c kgf

v`i lbiccnlnigv|   ! ! ! !6 ! 9  ! ' '2

Kg|}ix? [ynvi k hnv! kgf v`i xik|bgngm |`byaf abbj 

|bei}`kv ckenankx vb ~by'

@ixi³| `b} nv mbi| ' ' '

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8 <

4 ; 4 8 4 < 4( . ( . ( . ( . c p k k p p k p p k p p+ J 

8 <

4 4 ; 4 4 8 4 4 < 4 4( . ( . ( . ( . c p k k p p k p p k p p+ J 

V`ig

Cyxv`ixebxi! V`ibxie tm' 9> ng B_ ! |k~| v`kv

  c p p p p p p pf + J 

'  c p kf +V`y|

N| v`n| himnggngm vb abbj ckenankx2

Nv |`byaf xiengf ~by bc v`i txbli|| h~ }`nl` }i lbetyvif v`i

lbiccnlnigv| bc v`i Vk~abx tba~gbenka kttxbpnekvnbg|

k+

Nc 

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Lbgvngyngm vb vkji fixnskvnsi| kgf iskaykvi kv p !

}i `ksi ' ' '

( . ( . 6 ( . c  ff + J 

+ c pfff¤ +

4 8( . 8  c p kff¤ +

4

8

( .

8

 c p

k

ff¤ +

8

< 4 9 4( . <+ + ( . 9 < ( .  c p k k p p k p pfff J 

4

<

( .

<+

 c pfff¤ +

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Ng migixka! }i `ksi?

+g

g  c p g k+

Ng bv`ix }bxf|! nc k cyglvnbg  c  n| mnsig h~ k tb}ix |ixni| v`kv n|

ligvixif kv ! v`kv tb}ix |ixni| ey|v  hi v`i Vk~abx |ixni| cbx  c  hk|if kv '

8 <

; 8 <( . ( . ( . ( . c k k k k+ J 

}`nl` viaa| y| v`kv( .

4( .

+

g

g

 c  k

g+

Nc! }i `ksi

V`ig

  c p c p c p c p c p p p p p p p

fff+ J 

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 Gbvnli }`kv v`n| fbi| gbv |k~'

Nv fbi| gbv |k~ v`kv isix~ cyglvnbg n| mnsig h~ nv|

Vk~abx \ixni|'

Nv fbi| gbv isig |k~ v`kv isix~ cyglvnbg v`kv `k| k

Vk~abx |ixni| n| mnsig h~ nv| Vk~abx \ixni|'

Nv n| ik|~ vb |ii v`kv Vk~abx |ixni| kxi oy|v k |tilnka jngf

bc tb}ix |ixni|' Byx fn|lbsix~ viaa| y| v`kv v`i~ kxixikaa~ v`i bga~ jngf bc tb}ix |ixni| v`ixi n|'

Vb xinvixkvi? Nc k cyglvnbg c n| mnsig h~ k tb}ix |ixni|!

v`kv tb}ix |ixni| ey|v hi v`i Vk~abx \ixni| cbx  c kv v`i|kei hk|i tbngv'

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Cbx k Cyglvnbg c ! \bei

[yi|vnbg| Kxn|i

' Nc  c  `k| k Vk~abx \ixni|! fbi| v`i |ixni|

lbgsix mi2

Kg|}ix? Bcvig! hyv gbv ka}k~|! kgf lixvknga~ gbvka}k~| bg v`i }`bai fbekng bc v`i cyglvnbg'

Lbg|nfix v`i ckenankx lk|i bc 

]`kv n| v`i Vk~abx \ixni| cbx v`n| cyglvnbg2 ]`kv lkg

}i |k~ khbyv nv| lbgsix migli2

 c   +

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 Gipv [yi|vnbg ' ' '

' Nc v`i Vk~abx \ixni| cbx  c lbgsix mi|! n| nv i{ykavb v`i c  bg nv| ngvixska bc lbgsix migli2

Kg|}ix? Bcvig! hyv gbv ka}k~|'

Lbg|nfix v`i kh|bayvi skayi cyglvnbg

]i jgb} v`kv }i lkggbv iptkgf nv ng k Vk~abx |ixni| khbyv  p54' (]`~2.

Hyv c  p 5 q  p q `k| fixnskvnsi| bc kaa bxfix| kv kaa

bv`ix tbngv|'

]`kv nc }i lbg|nfix k Vk~abx |ixni| iptkg|nbgkhbyv  p 5 2

' c  p p+

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Vk~abx \ixni| cbx  c   p 5 q pq hk|if

kv  p 5 '

]`kv khbyv v`i fixnskvnsi| bc  c 

kv  p 5 2

]`kv fb }i miv cbx v`i Vk~abx 

|ixni| iptkg|nbg kv  p 5 2

V`i Vk~abx \ixni| iptkg|nbg

cbx  c  p 5 q  p q lbgsix mi| bgv`i igvnxi xika angi! hyv n| i{yka

vb c bga~ bg v`i ngvixska Z !

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V`ngm| lkg miv xikaa~ }inxf

i nc 

nc 

 p  p c  p

 p

§» z¢»´

Cklv|?

 c n| lbgvngyby| kgf `k|

fixnskvnsi| bc kaa bxfix| kv  p 5 4'

 c  g 4 54 cbx kaa g'

]`kv fbi| kaa v`n| viaa y| khbyv v`i Eklakyxng \ixni| cbx  c 2

V`i Eklakyxng \ixni| cbx  c lbgsix mi| isix~}`ixi! hyv n| i{yka

vb c bga~ kv  p 5

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\b }`ixi fbi| v`n| aiksi y|2Vb \yeekxnzi?

Isig nc }i lkg lbetyvi v`i Vk~abx \ixni| cbx k cyglvnbg!

v`i Vk~abx \ixni| fbi| gbv ka}k~| lbgsix mi'

Nc v`i Vk~abx \ixni| lbgsix mi|! v`i Vk~abx \ixni| n| gbv

gili||kxna~ i{yka vb v`i cyglvnbg! isig bg nv| ngvixska bc 

lbgsix migli'

]i jgb} ̀ b} vb fivixengi }`iv`ix kgf }`ixi v`i

Vk~abx |ixni| lbgsix mi|%%% Xkvnb vi|v +

Hyv `b} fb }i jgb} nc v`i Vk~abx \ixni| bc v`i cyglvnbgn| i{yka vb v`i cyglvnbg bg v`i ngvixska }`ixi nv lbgsix mi|2

V`i kg|}ix n| kaxikf~ ckenankx ' ' ' Vk~abx³| V`ibxie'

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X ilkaa Vk~abx³| V`ibxie

\yttb|i v`kv c n| xitikvifa~ fnccixigvnkhai bg kg ngvixska N 

lbgvkngngm  p4

kgf v`kv

n| v`i gv` bxfix Vk~abx tba~gbenka hk|if kv  p4'

\yttb|i v`kv  J g*;

n| k gyehix |yl` v`kv cbx kaa  z  ng N !

V`ig cbx  p ng N !

g

g g T  p k k p p k p p k p p k p p+ J 

( ;.

;q ( . q 'g

g c  z  J 

i

;;

4q ( . ( . q

; +

gg

g

 J  c  p T p p p

g

i

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]`kv fbi| v`n| viaa y|2

Tbngvngm byv v`kv k Vk~abx |ixni| cbx  c  ¼enm`v lbgsix mi kv

kaa  p hyv tix`kt| vb k anenv bv`ix v`kg c !¶ B|vihii kgf

_bxg k||yxi y| v`kv

¼Vk~abx³| v`ibxie mykxkgvii| v`kv v`n|

ygcbxvygkvi isigv |iafbe bllyx|'¶

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Tnggngm v`n| fb}g

X ilkaa v`kv  T g

n| v`i gv` tkxvnka |ye bc v`iVk~abx \ixni| bc  c  hk|if kv  p

4'

Kgf v`y|

Eik|yxi| v`i ixxbx ekfi h~  T g( p. ng kttxbpnekvngm  c ( p.'

Vk~abx³| v`ibxie mnsi| y| kg yttix hbygf bg v`n| ixxbx +

q ( . ( . qg

 c  p T p

V`i Vk~abx |ixni| cbx  c  }naa lbgsix mi vb c nc kgf bga~ nc cbx kaa  p

q c ( p. % T g( p. q

mbi| vb zixb k| g ' Vk~abx³| v`ibxie lkg `iat y| i|vkhan|` v`n|'

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Y|ngm Vk~abx³| V`ibxie

( . |ng( . c  p p+

;' Cngf v`i Vk~abx |ixni| cbx  c  v`kv n| hk|if kv  p 5 V "6'

' \`b} v`kv v`n| Vk~abx |ixni| lbgsix mi| vb c  cbx kaa skayi| bc  p'

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;' Vk~abx \ixni| cbx  c ( p

. 5 |ng( p

.g  c (g.( p.  c (g.(  . k

g5 c (g.(  ."g+

4

;

8

<

6

|ng( . p

lb|( . p

|ng( . p

lb|( . p

|ng( . p

6V 

;

; ; ;

;+8 8k + +

;|ng6 8

V +

;|ng 6 8V +

;lb| 6 8V +

;lb|6 8

V +

;|ng6 8

V +

6V

4

; ; ;

4+8 8k + +

8

; ;

8+8k +

<

; ;

<+8k +

6

; ;

6+8k +

8 < 6 9; ; ; ; ; ; ; ; ; ;

6 6 6 6 68 < 6 98 8 8 8 8 8  p p p p pV V V V V  

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]i |vkxv }nv` v`i migixka |iv%yt cbx Vk~abx³| V`ibxie'

]`kv n|  J g*;2

Nv cbaab}| v`kv

\`b} v`kv v`n| lbgsix mi| vb |ng( p

.

;

6|ng( . ( . cbx kaa '; +

g

g T 

g

i

]`kv `kttig|

vb v`n| {ykgvnv~

K| g2

|ng b k kg kg

g

f   p g p

fp

i

;;

|ng( . ( . 6; +

g

g

g

 J   p T p p

g

i

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]i |vkxv }nv` v`i migixka |iv%yt cbx Vk~abx³| V`ibxie'

]`kv n|  J g*;2

Nv cbaab}| v`kv

\`b} v`kv v`n| lbgsix mi| vb |ng( p

.

;;

|ng( . ( . 6; +

gg

g

 J   p T p p

gV 

i

;

6|ng( . ( . cbx kaa '; +

g

g T 

g

i

 Gbvnli v`kv N fnfg³v `ksi vb

jgb} }`kv  T g }k| ng bxfix vb

mkv`ix v`n| ngcbxekvnbg' (Ng

bv`ix }bxf|! byx |ilbgf

{yi|vnbg n| ngfitigfigv bc byx 

cnx|v'.

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 Gb} nv³| ~byx vyxg

X itikv v`n| ipixln|i }nv` v`i Eklakyxng |ixni| cbx  c ( p. 5 lb|(8p. '

;' Cngf v`i Eklakyxng |ixni| cbx  c ( p. 5 lb|(8 p.'

8' \`b} v`kv v`n| |ixni| lbgsix mi| vb c cbx kaa skayi| bc  p'

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;' Vk~abx \ixni| cbx  c ( p

. 5 lb|(8 p

.g  c (g.( p.  c (g.(4. k

g5 c (g.(4."g+

4

;

8

<

6

8|ng(8 . p88 lb|(8 . p

<8 |ng(8 . p

68 lb|(8 . p

; 4k +

8 88 lb| 4 8 +

6 68 lb|(4. 8+

8|ng 4 4 +

<8 |ng(4. 4 +

lb| 4 ;+ 4 ;k +

8

8

8

8+k

+

< 4k +

6

6 86+

k +

6 3 88 6 3 8

4

8 8 8 ( ;. 8;

8+ 6+ 3+ 8 +

g g

g

g

  p p p pg

m

¤ 

§A

lb|(8 . p

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]i |vkxv }nv` v`i migixka |iv%yt cbx Vk~abx³| V`ibxie'

]`kv n|  J g*;2

Nv cbaab}| v`kv

\`b} v`kv v`n| lbgsix mi| vb

lb|(8 p.

;;

lb(8 . ( . ; +

gg

g

 J   p p p

g

i

;8

lb| 8 . ( .   bx kaa '; +

g

g

 p

 p T  p p

g

i

V`n| {ykgvnv~ mbi|

vb 4 k| g+

lb|( . bx kaa kg kaa g

g

g

f  p g p

f  p

i


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