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7.3. ÇÀËÅÆͲÑÒÜ Ì²Æ ÍÅÏÅÐÅÐÂͲÑÒÞ<br />
² ÄÈÔÅÐÅÍÖ²ÉÎÂͲÑÒÞ ÔÓÍÊÖ²¯<br />
ßêùî ôóíêö³ÿ â òî÷ö³ õ 0 ìຠñê³í÷åííó ïîõ³äíó, òî êàæóòü,<br />
ùî â ö³é òî÷ö³ ôóíêö³ÿ äèôåðåíö³éîâíà. ßêùî æ<br />
ôóíêö³ÿ äèôåðåíö³éîâíà â óñ³õ òî÷êàõ äåÿêî¿ ìíîæèíè Õ,<br />
òî âîíà íàçèâàºòüñÿ äèôåðåíö³éîâíîþ íà ö³é ìíîæèí³ Õ.<br />
Äëÿ äèôåðåíö³éîâíî¿ â òî÷ö³ ôóíêö³¿ ìຠì³ñöå òàêà òåîðåìà.<br />
Òåîðåìà 7.3.1. ßêùî ôóíêö³ÿ f(õ) â òî÷ö³ õ 0 äèôåðåíö³éîâíà,<br />
òî âîíà â ö³é òî÷ö³ íåïåðåðâíà.<br />
Äîâåäåííÿ. Íåõàé ôóíêö³ÿ f(õ) â òî÷ö³ õ 0 äèôåðåíö³éîâíà.<br />
Öå îçíà÷àº, ùî ôóíêö³ÿ â òî÷ö³ õ 0 ìຠñê³í÷åííó<br />
ïîõ³äíó f′(õ 0 ), òîáòî ³ñíóº<br />
∆y<br />
lim = f′<br />
( x0<br />
). (7.3.1)<br />
∆x<br />
∆x→0<br />
³çüìåìî òåïåð ∆õ ≠ 0 ³ çîáðàçèìî ïðèð³ñò ôóíêö³¿ ∆ó<br />
â òî÷ö³ õ 0 ó âèãëÿä³:<br />
Dy<br />
D y= ×Dx. (7.3.2)<br />
Dx<br />
Òîä³ ³ç (7.3.2) ç óðàõóâàííÿì (7.3.1) ìàòèìåìî<br />
⎛∆y<br />
⎞ ∆y<br />
lim∆ y = lim ⎜ ⋅∆ x = lim ⋅lim∆ x = f′<br />
( x0) ⋅ 0 = 0<br />
x<br />
⎟<br />
.<br />
⎝∆<br />
⎠ ∆x<br />
∆x→0 ∆x→0 ∆x→0 ∆x→0<br />
Îòæå, ãðàíèöÿ ïðèðîñòó ôóíêö³¿ f(õ) â òî÷ö³ õ 0 äîð³âíþº<br />
íóëþ, êîëè ∆x → 0. Òîìó çà îçíà÷åííÿì 6.3.5 ôóíêö³ÿ f(õ)<br />
â òî÷ö³ õ 0 º íåïåðåðâíîþ.<br />
Òåîðåìó äîâåäåíî.<br />
Îáåðíåíà òåîðåìà, âçàãàë³ êàæó÷è, íåâ³ðíà. Äëÿ á³ëüøîãî<br />
ðîçóì³ííÿ öüîãî ôàêòó ðîçãëÿíåìî ïðèêëàäè.<br />
Ïðèêëàä 7.3.1. Äîñë³äèòè íà äèôåðåíö³éîâí³ñòü â òî÷ö³<br />
õ = 0 íåïåðåðâíó ôóíêö³þ f(x) =x α , α >0.<br />
Ðîçâ’ÿçàííÿ. Ñïî÷àòêó â³äçíà÷èìî, ùî îáëàñòü âèçíà-<br />
÷åííÿ ôóíêö³¿ f(x) =x α , α > 0 çàëåæèòü â³ä ïàðàìåòðà α (äèâ.<br />
ï. 6.1.1). ßêùî îáëàñòü âèçíà÷åííÿ äàíî¿ ôóíêö³¿ º [0; ∞), òî<br />
çã³äíî ç îçíà÷åííÿì äèôåðåö³éîâíîñò³ çðàçó ñêàæåìî, ùî<br />
âîíà â òî÷ö³ õ = 0 íå º äèôåðåö³éîâíîþ.  öüîìó âèïàäêó<br />
ìîæíà ãîâîðèòè ò³ëüêè ïðî îäíîñòîðîííþ ïîõ³äíó (â îçíà-<br />
÷åíí³ òðåáà ðîçãëÿäàòè ãðàíèöþ (7.2.1), êîëè ∆x → +0).<br />
Ó çâ’ÿçêó ç öèì ìè éîãî íå áóäåìî ðîçãëÿäàòè, òîáòî áóäåìî<br />
ââàæàòè, ùî ïàðàìåòð α ãàðàíòóº âèçíà÷åííÿ ôóíêö³¿<br />
f(x) =x α , α > 0 â îêîë³ òî÷êè õ = 0. Òîä³ ïîõ³äíà ôóíêö³¿ f(õ)<br />
â òî÷ö³ õ = 0 ó â³äïîâ³äíîñò³ äî îçíà÷åííÿ 7.2.1 îá÷èñëþºòüñÿ<br />
òàê:<br />
α<br />
( ∆x)<br />
f′ (0) = lim = lim( ∆x)<br />
∆x→0 ∆x<br />
∆x→0<br />
α−1<br />
. (7.3.3)<br />
Ïðîñòèé àíàë³ç ñï³ââ³äíîøåííÿ (7.3.3) ïîêàçóº, ùî ïðè<br />
α≥1 ç óðàõóâàííÿì îáìåæåíîñò³ çíà÷åíü ïàðàìåòðà α ³ñíóº<br />
â òî÷ö³ õ = 0 ñê³í÷åííà ïîõ³äíà ôóíêö³¿ f(õ), ïðè÷îìó<br />
ì 0, ÿêùî α > 1<br />
f ¢ (0) = ï<br />
í ï .<br />
ïî 1, ÿêùî α = 1<br />
ßêùî æ 0 < α < 1, òî ïîõ³äíà ôóíêö³¿ f(õ) â òî÷ö³ õ =0<br />
íå ³ñíóº. Îòæå, ìîæíà çðîáèòè òàêèé âèñíîâîê: ó çàëåæíîñò³<br />
â³ä ïàðàìåòðà α íåïåðåðâíà ôóíêö³ÿ f(x) =x α , α >0 â<br />
òî÷ö³ õ =0 ìîæå áóòè ÿê äèôåðåíö³éîâíîþ, òàê ³ íåäèôåðåíö³éîâíîþ.<br />
Çàóâàæåííÿ 1. Ó âèïàäêó 0 < α < 1 ïîõ³äíà ôóíêö³¿<br />
f(x) =x α , α > 0 â òî÷ö³ õ = 0 íå ³ñíóº. Àëå â ö³é òî÷ö³ ìîæíà<br />
ââåñòè íåñê³í÷åííó ïîõ³äíó, àáî îäíîñòîðîííþ íåñê³í÷åííó<br />
2<br />
ïîõ³äíó. Íàïðèêëàä, äëÿ ôóíêö³é fx ( ) = x 3 ìîæíà ââåñòè<br />
îäíîñòîðîíí³ íåñê³í÷åíí³ ïîõ³äí³: f′(+0) = +∞, f′(−0) = −∞, à<br />
1<br />
2<br />
äëÿ ôóíêö³¿ fx ( ) = x ìîæíà ââåñòè ò³ëüêè îäíó îäíîñòîðîííþ<br />
íåñê³í÷åííó ïîõ³äíó: f′(+0) = +∞.<br />
×èòà÷åâ³ ïðîïîíóºìî äàòè ãåîìåòðè÷íèé çì³ñò íàâåäåíèõ<br />
ó çàóâàæåíí³ 1 ïîíÿòü.<br />
Çàóâàæåííÿ 2. Ó ïðèêëàä³ 7.3.1 ðîçãëÿíóòî ò³ëüêè<br />
äîäàòí³ çíà÷åííÿ ïàðàìåòðà α. Öå ïîâ’ÿçàíî ç òèì, ùî äëÿ<br />
â³ä’ºìíèõ çíà÷åíü ïàðàìåòðà α f(õ) â òî÷ö³ õ = 0 íå âèçíà-<br />
÷åíà.  öüîìó âèïàäêó ãîâîðèòè ïðî ³ñíóâàííÿ ïîõ³äíî¿<br />
f(õ) â òî÷ö³ õ = 0 íå ìຠñåíñó, òèì ïà÷å ãîâîðèòè ïðî äèôåðåíö³éîâí³ñòü.<br />
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