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Khai trin Taylor Maclaurin (Taylor expansion)

li phn hi Go to comments

Shortlink: http://wp.me/P8gtr-R

Ch dn lch s

1. Cng thc khai trin:

Gi thit hm s y = f(x) c tt c cc o hm n cp n + 1 (k c o hm cp n + 1) trong mt khong no cha im x = a.

Hy xc nh mt a thc bc n m gi tr ca n ti x = a bng gi tr f(a) v gi tr ca cc o hm n hng n ca n bng gi tr ca cc o hm tng ng ca hm s f(x) ti im . Ngha l:

(1)

Ta hy vng s tm c mt a thc nh th trong mt ngha no gn vi hm s f(x).

Ta s xc nh a thc di dng mt a thc theo ly tha (x a) vi cc h s cn xc nh:

3 of 17 4/4/2011 10:37 AM

(2)

Cc h s c xc nh sao cho iu kin (1) c tha mn.

Trc ht, ta tm cc o hm ca :

(3)

Thay x = a vo cc biu thc (2) v (3) ta c:

So snh vi iu kin (1) ta c:

(4)

Thay cc gi tr ca vo cng thc (2) ta c a thc cn tm:

K hiu bng , hiu gia gi tr ca hm s cho f(x) v a thc mi lp (hnh v):

Hay:

4 of 17 4/4/2011 10:37 AM

(6)

gi l s hng d i vi nhng gi tr x lm cho s hng d b, th khi a thc cho biu din gn ng ca hm s f(x).

Do , cng thc (6) cho kh nng thay hm s y = f(x) bng a thc vi chnh xc tng ng bng gi tr ca s hng d

Ta s xc nh nhng gi tr x s hng d kh b .

Vit s hng d di dng: (7)

Trong Q(x) l hm s cn phi xc nh.

Vi x v a c nh, hm s Q(x) c gi tr xc nh, k hiu gi tr bng Q.

Ta xt, hm s ph theo bin t (t l gi tr nm gia a v x) :

(8)

Tm o hm F(t) :

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Rt gn li ta c :

(9)

Vy hm s F(t) c o hm ti mi im t gn im c honh a.

Ngoi ra, t cng thc (8) ta c : F(x) = 0 v F(a) = 0.

V vy, p dng cng thc Rolle cho hm s F(t) , tn ti mt gi tr nm gia a v x sao cho

Th vo (9) ta c :

Suy ra :

Thay biu thc ny vo cng thc (7) ta c :

s hng d Larange

V l gi tr nm gia a v x, nn n c th vit di dng:

Ngha l :

Cng thc:

gi l cng thc khai trin Taylor (Taylor expansion) ca hm s f(x).

Nu trong cng thc Taylor, t a = 0 th n vit di dng:

6 of 17 4/4/2011 10:37 AM

Share this:

l cng thc xp x hm f(x) thnh a thc bc n ti x = 0, vi s d c gi l cng thc khai trin Maclaurin (Maclaurin expansion).

Tm li, ta c nh l sau:

Nu hm s y = f(x) c cc o hm lin tc ti im v c o hm trong ln cn ca th ti ln cn ta c cng thc khai trin:

(c gia v x, )

Cng thc ny gi l cng thc khai trin Taylor cp n, s hng ca cng gi l s hng d ca n. c bit th cng thc Taylor tr thnh cng thc Maclaurin (cng thc khai trin ti ln cn ):

123 bnh chn

Trang: 1 2 3 4

Phn hi (146) Trackbacks (1) li phn hi Trackback

huyphan22.11.2010 lc 22:33 | #1Tr li | Trch dn

Tha thy gi s cho 1 hm F(x)=(1+X^2)cosx ku tnh o hm cp 10 ca F(pi/6), ta c x=pi/6 ch bi khng cp ti x0 vy ta dng cng thc maclorin c dc khng? khi no tabit nn dng maclorin? v trong cc bi tm lim ch cho x tin ti 0 ch u c x0 tin ti 0

1.

7 of 17 4/4/2011 10:37 AM

3 0 B phiu

nguyn vn cng10.12.2010 lc 14:38 | #2Tr li | Trch dn

em cho thy !em mi hc v khai trin Taylor nn em cha bit lm bi tp dng ny nh th no nn em mong thy gii giup em 1 bi lm mu.Bi l:khai trin hm f(x)=e (^2x) vi ly tha x-2.Em xin cm n thy!

2 0 B phiu

2.

2Bo02B10.12.2010 lc 22:20 | #3Tr li | Trch dn

Cch 1: dng cng thc Taylor tng qut. Em c:

(*)

vi: Khi , th vo (*), em s c kt qu. Tuy nhin, vi cch ny em phi tnh o hm n cp n. Vi nhiu hm s, vic tnh o hm cp cao s rt phc tp nht l khng tm cquy lut.Cch 2: bin i a v nhng hm bit cng thc khai trin. Em xem thm pha trn nh. Vi bi ny t t = x 2 Th khai trin ti x = 2 tng ng vi khai trin ti t = 0.Khi : . Em ch cn dng cng thc khai trin Maclaurin cho (u = 2t) s c kt qu.

8 0 B phiu

nguyn vn cng13.12.2010 lc 09:52 | #4Tr li | Trch dn

em xin cm n thy !

6 0 B phiu

3.

ngoc ngoc31.12.2010 lc 16:12 | #5Tr li | Trch dn

4.

8 of 17 4/4/2011 10:37 AM

em chao thay a.thay oi trong khai trien tay-lor khai trien mot ham den bac may thi dc a?

3 0 B phiu

siu nhn bnh rn09.01.2011 lc 20:41 | #6Tr li | Trch dn

cho em hi s hng d lagrange dng lm g .Em cha thy ng dng ca ci ny.Mong thy cho vi v d gip em

0 0 B phiu

5.

2Bo02B09.01.2011 lc 22:31 | #7Tr li | Trch dn

Em xem mc 3.2 tnh gn ng v nh gi sai s trang th 4 ca phn trn nh,

0 0 B phiu

phuonga26.01.2011 lc 00:51 | #8Tr li | Trch dn

tha thy,do ch mi hc s v cng thc ny nn bi gii trn em khng hiu vi iu phn cui.thy ging li cho em c khng ?1.s hng ca cng gi l s hng d ca n ngha l sao ?2.l cng thc xp x hm f(x) thnh a thc bc n ti x = 0, vi s d Rn(x)??3.a v x0 c phi l 2 gi tr khc nhau?4.dng cui,c bit x=0 th cng thc Taylor tr thnh cng thc Maclaurin vy ti sao trong cng thc di vn cn n x ?5.trong sch gio khoa ca em c cng thc taylor: f(x)=(k=0->n)[(f^k)(x0).(x-x0)^k]/k! + [(x-x0).(f (^n+1))(c)]/(n+1)! .cng thc ny em thy khng ging trong nh ngha cng nhtrong nh l ca thy nn em rt thc mc.em cm n thy.

0 0 B phiu

6.

dakqueen05.03.2011 lc 17:58 | #9Tr li | Trch dn

7.

9 of 17 4/4/2011 10:37 AM

Cho e hi l nu bi yu cu khai trin maclaurin ti cp 3 th mnh phi lm o hm ti cp my ?

1 0 B phiu

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01.04.2010 lc 13:56 | #1Mot so them ve Taylor Itoaoaoa's Blog (alpha ver)

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