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ÂÏÐÀÂÈ<br />
Çíàéòè ãðàíèö³:<br />
7.36. lim cos3x<br />
x /2 cos x<br />
7.38.<br />
x<br />
→+∞ + ;<br />
; 7.37. lim →π x ln ( 1 x)<br />
lim<br />
x→−∞<br />
7.40. lim cos x tg 5x<br />
x /2<br />
2<br />
x − 1<br />
2 x<br />
; 7.39. lim xe ;<br />
x<br />
x→+<br />
0<br />
; 7.41. lim( ctg 1/ )<br />
→π ϕ→0<br />
x<br />
7.42. lim ( 1 e ) 1/<br />
x→∞<br />
ϕ− ϕ ;<br />
x<br />
x<br />
+ ; 7.43. lim ( cos16 / )<br />
x→∞<br />
x ;<br />
7.44. ( ) 1 ln x<br />
lim ctg x ; 7.45. lim( sin 9 ) sin(16 )<br />
x→+<br />
0<br />
7.15. ÔÎÐÌÓËÀ ÒÅÉËÎÐÀ 1<br />
x→0<br />
x<br />
x .<br />
Íà ïî÷àòêó XVIII ñòîð³÷÷ÿ Òåéëîð îïóáë³êóâàâ ôîðìóëó<br />
äëÿ íàáëèæåíîãî çîáðàæåííÿ ôóíêö³é â îêîë³ äåÿêî¿ òî÷êè<br />
ä³éñíî¿ â³ñ³ ìíîãî÷ëåíàìè. Òåïåð öÿ ôîðìóëà íîñèòü éîãî<br />
³ì’ÿ. Âêàçàíà ôîðìóëà äóæå âàæëèâà ÿê ç òåîðåòè÷íî¿, òàê<br />
³ ç ïðàêòè÷íî¿ òî÷îê çîðó.<br />
7.15.1. Ôîðìóëà Òåéëîðà äëÿ ìíîãî÷ëåíà<br />
Íåõàé çàäàíî ìíîãî÷ëåí<br />
( ) =<br />
0<br />
+<br />
1<br />
+ ... + = ∑ n j<br />
p x b bx b x b x . (7.15.1)<br />
n<br />
n n j<br />
j=<br />
0<br />
Ðîçãëÿíåìî çàäà÷ó: çä³éñíèòè ðîçêëàä äàíîãî ìíîãî÷ëåíà<br />
çà ñòåïåíÿìè õ – õ 0 . Òàêó çàäà÷ó ìîæíà áóëî á ðîçâ’ÿçàòè<br />
“â ëîá”. Äëÿ öüîãî òðåáà ñêîðèñòàòèñÿ òîòîæí³ñòþ<br />
õ =(õ –õ 0 )+õ 0 ³ ï³äñòàâèòè çàì³ñòü õ ïðàâó ÷àñòèíó îñòàííüî¿<br />
ôîðìóëè. Ïðè öüîìó îá÷èñëåííÿ áóäóòü íå ïðîñò³.<br />
ßê æå ïîäîëàòè ö³ òðóäíîù³? Âèÿâëÿºòüñÿ, ùî ¿õ ìîæíà ïîäîëàòè<br />
òàêèì ÷èíîì:<br />
1) ñïî÷àòêó çîáðàçèìî ìíîãî÷ëåí ó âèãëÿä³<br />
p ( x) = a + a ( x− x ) + ... + a ( x− x ) + ... +<br />
n<br />
k<br />
0 1 0 k 0<br />
n n<br />
( ) ( )<br />
+ an<br />
x− x = ∑ a x−x<br />
, (7.15.2)<br />
0 k 0<br />
k=<br />
0<br />
äå êîåô³ö³ºíòè ïîêè ùî íåâ³äîì³;<br />
2) ïîò³ì ó ð³âí³ñòü (7.15.2) ï³äñòàâèìî çàì³ñòü õ çíà÷åííÿ<br />
õ 0 .  ðåçóëüòàò³ îòðèìàºìî, ùî<br />
( )<br />
a = p x . (7.15.3)<br />
0 n 0<br />
Äàë³ ïîñë³äîâíî çíàõîäèìî:<br />
′ ′′<br />
a = p x ,2! ⋅ a =⋅p x ,..., k! ⋅ a =⋅ p ( x ),..., n!<br />
a = p x .<br />
( ) ( )<br />
k<br />
( k) ( n )<br />
( )<br />
1 n 0 2 n 0 k n 0 n n 0<br />
Çâ³äêè ä³ñòàºìî, ùî<br />
( k )<br />
n ( x )<br />
p<br />
a 0 k<br />
= , k= 1,2,..., n.<br />
(7.15.4)<br />
k!<br />
Çà îçíà÷åííÿì 0! = 1. Òîä³ ôîðìóëè (7.15.3) — (7.15.4)<br />
ìîæíà çàïèñàòè îäí³ºþ<br />
( k )<br />
n ( x )<br />
p<br />
a 0 k<br />
= , k = 0,1,2,..., n. (7.15.5)<br />
k!<br />
Îòæå, ïîñòàâëåíà ïðîáëåìà âèð³øåíà: ìíîãî÷ëåí p n (x),<br />
ÿêèé çîáðàæåíî ó âèãëÿä³ (7.15.1), ðîçêëàäåíî çà ñòåïåíÿìè<br />
õ–õ 0 . Ïðè öüîìó êîåô³ö³ºíò³ âèçíà÷àþòüñÿ çà ôîðìóëàìè<br />
(7.15.5).<br />
ϳäñòàâèìî (7.15.5) â ð³âí³ñòü (7.15.2). Ó ðåçóëüòàò³<br />
îòðèìàºìî òàê çâàíó ôîðìóëó Òåéëîðà äëÿ ìíîãî÷ëåíà<br />
( k )<br />
n ( x0<br />
) ( )<br />
n p<br />
k<br />
pn<br />
( x)<br />
= ∑ x−x0<br />
. (7.15.6)<br />
k<br />
k=<br />
0 !<br />
1<br />
Òåéëîð Áðóê (1685 – 1731) — àíãë³éñüêèé ìàòåìàòèê ³ ô³ëîñîô.<br />
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