Samarqand iqtisodiyot va servis instituti
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funksional va darajali qatorlar
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Teylor qatori, ikkinchisiga Makloren qatori deyiladi. Bu qatorlar х
ning 0
( lim
= ∞ → x R n n bo’ladigan qiymatlarida ) (x f
funksiyaga yaqinlashadi. A nuqtani o’z ichiga oluvchi biror intervalda istalgan n uchun
, ) ( ) (
x f n <
( M
biror musbat son)
tengsizlik bajarilsa, 0 )
lim = ∞ → x R n bo’ladi va ) (x f funksiya Teylor qatoriga yoyiladi. 4. Funksiyalarni darajali qatorlarga yoyish. Ayrim funksiyalarni darajali qatorga yoyyamiz. 1)
= ) ( , istalgan x uchun
...
, 1 ) 0 ( , ... , 1 ) 0 ( , 1 ) 0 ( , 1 ) 0 ( , 0 ,... ) ( ..., , ) ( , ) ( ) ( ) ( = = ′′ = ′ = = = = ′′ = ′
x n x x f f f f dåsàk x e x f e x f e x f
bo’ladi. Bularni Makloren qatoriga qo’yib, ) ( ... ! ... ! 3 ! 2 ! 1 1 3 2 +∞ < < −∞ + + + + + + = x n x x x x e n x
ni hosil qilamiz. Oxirgi tenglikdan 1 =
desak,
.... !
... ! 3 1 ! 2 1 ! 1 1 1 + + + + + + = n e
bo’lib, е soni qator yig’indisi ko’rinishida ifodalanadi. Bundan foydalanib е sonining taqribiy qiymatini istalgan darajadagi aniqlikkacha hisoblash mumkin. 2)
x x f sin
) ( = . Istalgan x uchun,
. ,... sin ) ( , cos
) ( , sin ) ( , cos
) (
x f x x f x x f x x f = ′′′ ′ − = ′′′ − = ′′ = ′ bo’ladi. Bundan
... , 0 ) 0 ( , 1 ) 0 ( , 0 ) 0 ( , 1 ) 0 ( , 0 ) 0 ( = ′′′ ′ − = ′′′
= ′′ = ′ =
f f f f
bo’lib,bularni Makloren qatoriga qo’ysak,
... ! ) 1 2 ( ) 1 ( ... ! 7 ! 5 ! 3 sin
1 2 1 7 5 3 + − − + + − + − = − −
x x x x x x n n
hosil bo’ladi. Bu qator istalgan x uchun yaqinlashuvchi +∞
∞ − x . Oxirgi qatorni hadlab differensiallasak,
... !
2 ( ) 1 ( ! ) 2 2 ( ) 1 ( ...
! 6 ! 4 ! 2 1 cos
2 1 1 2 1 6 4 2 + − + − − + + − + − = − − − n x n x x x x x n n n n
qator hosil bo’ladi, bu x x f cos
) ( = funksiya uchun Makloren qatori bo’ladi. 3) Xuddi yuqoridagidek usul bilan
) 1 ( ) ( + = funksiya uchun
... ....
! 3 ) 2 )( 1 ( ! 2 ) 1 ( ! 1 1 ) 1 ( 3 2 + + − − + − + + = + x m m m x m m x m x m
qatorni hosil qilamiz. Bu qatorga binomial qator deyiladi. U ) 1
1 ( − intervalda absolyut yaqinlashuvchi bo’ladi.
4)
) 1 ln( ) (
x f + = funksiya uchun yuqoridagi usul bilan
) 1 1 ( .... ) 1 ( ...
4 3 2 ) 1 ln( 1 4 3 2 ≤
− +
+ + − + − = + −
n x x x x x x n n
yoyilmani hosil qilish mumkin. 5-misol. x x f cos
) ( = funksiyani x ning darajalari bo’yicha qatorga yoying. Yechish. Yuqoridagi
cos
uchun keltirilgan qatorda x ni
x
bilan almashtirsak, ...
! ) 2 ( ) 1 ( ....
! 4 ! 2 1 cos 2 + − + − + − =
x x x x n n
bo’ladi. Bu qator istalgan x uchun yaqinlashuvchidir, biroq x cos
funksiya 0
da aniqlanmaganligini hisobga olib, hosil qilingan qator x cos
funksiyaga +∞
≤
0 da yaqinlashadi. 5. Qatorlarning taqribiy hisoblashga tatbiqlari. Bir necha misollar qaraymiz. 6-misol.
cos
ning yoyilmasidan foydalanib 0 18 cos ni 001 ,
aniqlikkacha taqribiy hisoblang. Yechish. x cos
funksiyaning qatorga yoyilmasidan foydalanib,
... 10 ! 4 1 10 ! 2 1 1 10 cos 18 cos
4 2 0 − + − = = π π π
qatorni hosil qilamiz.
. 00974
, 0 10 ; 09870
, 0 10 ; 31416
, 0 10 4 2 = = = π π π
va
0001 , 0 10 ! 6 1 6
⋅ π bo’lganligi uchun, taqribiy hisoblashda qatorning birinchi uchta hadi bilan chegaralanamiz, demak
. 9511
, 0 18 cos ; 24 00974 , 0 2 09870
, 0 1 18 cos
0 0 ≈ + − ≈ ёки
7-misol. 0001
, 0 1 , 1 5 ни aniqlikkacha taqribiy hisoblang. Yechish: 5 1 5 ) 1 , 0 1 ( 1 , 1 + = deb, binomial qatordan foydalansak: ....
000048 , 0 0008 , 0 02 , 0 1 ...
001 , 0 ! 3 ) 2 5 1 ( ) 1 5 1 ( 5 1 01 , 0 ! 2 ) 1 5 1 ( 5 1 1 , 0 5 1 1 ) 1 , 0 1 ( 1 , 1 5 1 5 − + − + = + + − ⋅ − ⋅ + − ⋅ + ⋅ + = + =
bo’ladi. To’rtinchi had 0001 . 0 000048 . 0 < bo’lganligi uchun, hisoblashda birinchi uchta hadini olib, hisoblaymiz:
0192 ,
0008 , 0 02 , 0 1 1 , 1 5 = − + ≈ . 8-misol. 001 ,
130 3 ни aniqlikkacha taqribiy hisoblang. Yechish. 130 5
ga eng yaqin butun sonning kubi bo’lganligi uchun 5 5 130 3 + = ko’rinishda ifodalab, binomial qatordan foydalansak,
...
000064 , 0 00018 , 0 0667 , 0 5 ....)
000064 , 0 ! 3 )) 2 3 1 ( ) 1 3 1 ( 3 1 ( 0016 , 0 ! 2 ) 1 3 1 ( 3 1 04 , 0 3 1 1 ( 5 ) 25 1 1 ( 5 ) 25 1 1 ( 5 5 5 130
3 1 3 3 3 3 3 − + − + = + − ⋅ − + + − ⋅ + + = + = + = + =
Bo’ladi. Oxirgi qatorda 3-had 001
, 0 dan kichik bo’lganligi uchun, taqribiy hisoblashda birinchi ikkita had bilan chegaralanamiz: . 0667 , 5 0667 , 0 5 130 3 ≈ + ≈
9-misol. 0001
, 0 04 , 1 ln ни gacha aniqlikda taqribiy hisoblang. Yechish: ) 1 ln( x + funksiyaning darajali qatorga yoyilmasidan foydalanib, , .... 4 04 , 0 3 04 , 0 2 04 , 0 04 , 0 ) 04 , 0 1 ln( 4 3 2 + − + − = + yoki
... 00000064
, 0 000021 , 0 0008 , 0 04 , 0 04 , 1 ln + − + − =
qatorni hosil qilamiz, hamda uchinchi had 0001
, 0 dan kichik bo’lganligi uchun birinchi ikki hadni hisobga olib hisoblaymiz:
. 0392
, 0 04 , 1 ln ≈
38.5-ilova “Funksional va darajali qatorlar” mavzusi bo’yicha ttst topshiriqlari I darajali testlar
1. ( ) x f y = funksiya uchun Teylor qatorini ko’rsating. A) ( ) ( )
( )( ) ( )( ) ( )(
) ( )
( )( ) ... ! 1 ... ! 3 1 ! 2 1 ! 1 1 3 2 + − + + + − ′′′ + − ′′ + − ′ + =
n a x a f n a x a f a x a f a x a f a f x f
В ) ( ) ( )
( )( ) ( )( ) ( )(
) ( )
( )( ) ... ... 3 2 + − + + + − ′′′ + − ′′ + − ′ + = n n a x a f a x a f a x a f a x a f a f x f
D) ( ) ( )
( )( ) ( )( ) ( )(
) ( )
( )( ) ... 0 ...
0 0 0 0 3 2 + − + + + − ′′′ + − ′′ + − ′ + = n n a x f a x f a x f a x f f x f
E) ( ) ( ) ( )(
) ( )(
) ( )(
) ( )
( )( ) ... ... 3 2 + − − − + − ′′′ + − − ′′ + − ′ − = n n a x a f a x a f a x a f a x a f a f x f
2. ...
... 2 2 1 0 + + + + + n n x a x a x a a darajali qator yaqinlashish intervali qanday topiladi? A) yaqinlashish radiusi 1 lim
+ ∞ → = n n n a a R formula bilan topilib, yaqinlashish
intervali ( )
R ; − dan iborat bo’ladi В ) yaqinlashish radiusi n n n a a R 1 lim + ∞ → = formula bilan topiladi D) yaqinlashish radiusi
1 lim + ∞ → = formula bilan topilib,yaqinlashish intervali ( ) R R; − dan iborat bo’ladi E) yaqinlashish intervali ( ) R R; − bo’lib, bunda n n n a a R 1 lim + ∞ → = dan iborat bo’ladi
3.
... ...
2 2 1 0 + + + + + n n x a x a x a a darajali qator yaqinlashish radiusi qanday topiladi? A) hamma koeffisiyentlar 0 dan farqli bo’lganda 1 lim
+ ∞ → = n n n a a R formula bilan topiladi В ) n n n a a R 1 lim + ∞ → = formula bilan topiladi
D)
1 0 lim + → = n n n a a R formula bilan topiladi
E) qator yaqinlashish radiusini topish uchun uning yaqinlashish intervalini topamiz 4. x y l = funksiyaning Makloren qatoriga yoyilmasini ko’rsating. A)
... ! 1 ... ! 3 1 ! 2 1 ! 1 1 1 3 2 + + + + + + = = n x x n x x x y l
В )
... ! 1 ... ! 3 1 ! 2 1 ! 1 1 1 3 2 − + + − + − = = n x x n x x x y l
D) ...
1 ...
3 1 2 1 1 1 1 3 2 − + + − + − = =
x x n x x x y l
E) ...
1 ...
3 1 2 1 1 1 1 3 2 + + + + + + = =
x x n x x x y l
5. x sin
funksiyaning Makloren qatoriga yoyilmasini ko’rsating. A)
... ...
! 7 ! 5 ! 3 sin 7 5 3 + + − + − = x x x x x
В ) ...
... ! 7 ! 5 ! 3 sin
7 5 3 + + + + + = x x x x x
D) ... ...
7 5 3 sin 7 5 3 + + + + + = x x x x x
E) ... 7 5 3 sin
7 5 3 + − + − =
x x x x
6. x cos
funksiyaning Makloren qatoriga yoyilmasini ko’rsating. À)
... ! 6 ! 4 ! 2 1 cos 6 4 2 + − + − =
x x x
В ) ...
! 6 ! 4 ! 2 1 cos
6 4 2 + + + + =
x x x
D) ... 6 4 2 1 cos 6 4 2 + + + + =
x x x
E) ... 6 4 2 1 cos 6 4 2 + − + − =
x x x
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