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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|
8/9/2019 Taylor Series[1]
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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| ' ' '
8/9/2019 Taylor Series[1]
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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
8/9/2019 Taylor Series[1]
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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¤ +
8/9/2019 Taylor Series[1]
http://slidepdf.com/reader/full/taylor-series1 5/21
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
8/9/2019 Taylor Series[1]
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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'
8/9/2019 Taylor Series[1]
http://slidepdf.com/reader/full/taylor-series1 7/21
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 +
8/9/2019 Taylor Series[1]
http://slidepdf.com/reader/full/taylor-series1 8/21
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+
8/9/2019 Taylor Series[1]
http://slidepdf.com/reader/full/taylor-series1 9/21
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 !
8/9/2019 Taylor Series[1]
http://slidepdf.com/reader/full/taylor-series1 10/21
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
8/9/2019 Taylor Series[1]
http://slidepdf.com/reader/full/taylor-series1 11/21
\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'
8/9/2019 Taylor Series[1]
http://slidepdf.com/reader/full/taylor-series1 12/21
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
8/9/2019 Taylor Series[1]
http://slidepdf.com/reader/full/taylor-series1 13/21
]`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|'¶
8/9/2019 Taylor Series[1]
http://slidepdf.com/reader/full/taylor-series1 14/21
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|'
8/9/2019 Taylor Series[1]
http://slidepdf.com/reader/full/taylor-series1 15/21
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'
8/9/2019 Taylor Series[1]
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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
J
8/9/2019 Taylor Series[1]
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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
V
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
V
i
8/9/2019 Taylor Series[1]
http://slidepdf.com/reader/full/taylor-series1 18/21
]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
V
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'.
8/9/2019 Taylor Series[1]
http://slidepdf.com/reader/full/taylor-series1 19/21
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'
8/9/2019 Taylor Series[1]
http://slidepdf.com/reader/full/taylor-series1 20/21
;' 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
8/9/2019 Taylor Series[1]
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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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