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Öå ð³âíÿííÿ ïðÿìî¿, ùî º ãðàíèöåþ îáëàñò³ âèçíà÷åííÿ<br />
ôóíêö³¿. Êîæíà òî÷êà ö³º¿ ïðÿìî¿ º òî÷êîþ ðîçðèâó. Òàêèì<br />
÷èíîì, òî÷êè ðîçðèâó óòâîðþþòü ö³ëó ïðÿìó — ë³í³þ<br />
ðîçðèâó äàíî¿ ôóíêö³¿.<br />
Çàóâàæåííÿ 2. Äëÿ ôóíêö³é äåê³ëüêîõ çì³ííèõ ïîíÿòòÿ<br />
ãðàíèö³ ³ íåïåðåðâíîñò³ ââîäÿòüñÿ àíàëîã³÷íî.<br />
10.3. ×ÀÑÒÈÍͲ ÏÎÕ²ÄͲ<br />
10.3.1. ×àñòèíí³ ïîõ³äí³ ïåðøîãî ïîðÿäêó<br />
ßê â³äîìî, ïîõ³äíà f′(x) ôóíêö³¿ îäí³º¿ çì³ííî¿ (çà óìîâè,<br />
ùî âîíà ³ñíóº) õàðàêòåðèçóº øâèäê³ñòü çì³íè ôóíêö³¿ f(x) â<br />
òî÷ö³ x. Ó çâ’ÿçêó ç öèì âèíèêຠïèòàííÿ ïðî ðîçâ’ÿçàííÿ<br />
ïðîáëåìè âèçíà÷åííÿ øâèäêîñò³ çì³íè ôóíêö³¿ äâîõ çì³ííèõ<br />
â çàäàí³é òî÷ö³.<br />
Ðèñ. 10.8<br />
Ïî÷íåìî ç ïðîñòîãî ïðèêëàäà. Ïðîñòèé àíàë³ç ïîêàçóº,<br />
ùî ôóíêö³ÿ z = x + y, ë³í³¿ ð³âíÿ ÿêî¿ çîáðàæåíî íà ðèñóíêó<br />
10.8, â³äïðàâëÿþ÷èñü â³ä òî÷êè Î (0,0), âåäå ñåáå ïî-ð³çíîìó:<br />
à) âçäîâæ á³ñåêòðèñè ïåðøîãî òà òðåòüîãî êîîðäèíàòíèõ<br />
êóò³â âîíà íàéá³ëüøå âñüîãî çðîñòàº; á) âçäîâæ êîîðäèíàòíèõ<br />
â³ñåé âîíà çðîñòຠïîâ³ëüí³øå; â) âçäîâæ á³ñåêòðèñè<br />
äðóãîãî òà ÷åòâåðòîãî êîîðäèíàòíèõ êóò³â âîíà çîâñ³ì íå<br />
çì³íþºòüñÿ.<br />
Íàâåäåíèé ïðèêëàä ïîêàçóº, ùî ãîâîðèòè ïðî øâèäê³ñòü<br />
çì³íè ôóíêö³¿ äâîõ çì³ííèõ â äàí³é òî÷ö³ íå ìຠñìèñëó.<br />
ßñíî, ùî ïðîáëåìà çðîñòàííÿ ôóíêö³¿ äâîõ çì³ííèõ íàáóâàº<br />
çì³ñòó, ÿêùî áóäå çàäàíî íàïðÿìîê, çà ÿêèì çì³íþºòüñÿ<br />
ðîçãëÿäóâàíà ôóíêö³ÿ. ² çíîâó âèíèêຠïðîáëåìà: íàïðÿìê³â<br />
— íåçë³÷åíà ìíîæèíà. ² òîìó íà ïåðøèé ïîãëÿä çäà-<br />
ºòüñÿ, ùî ïîñòàâëåíà ïðîáëåìà íå ìîæå áóòè ðîçâ’ÿçàíà. Çàá³ãàþ÷è<br />
âïåðåä, ñêàæåìî, ùî äëÿ äîñòàòíüî øèðîêîãî êëàñó<br />
ôóíêö³é äâîõ çì³ííèõ âêàçàíà ïðîáëåìà ìîæå áóòè ïîäîëàíà<br />
(äîâåäåííÿ öüîãî ôàêòó áóäå çä³éñíåíî ï³çí³øå), àëå äëÿ<br />
öüîãî òðåáà çíàòè íàïðÿìîê çì³íè ôóíêö³¿ ³ øâèäê³ñòü<br />
çðîñòàííÿ ôóíêö³¿ ó äâîõ âçàºìíî ïåðïåíäèêóëÿðíèõ íàïðÿìàõ.<br />
Ö³ëêîì ïðèðîäíî, ùî â ÿêîñò³ òàêèõ íàïðÿì³â ìè<br />
â³çüìåìî íàïðÿìè, ÿê³ ñï³âïàäàþòü ç íàïðÿìîì êîîðäèíàòíèõ<br />
â³ñåé. Òàê ìè ïðèõîäèìî äî ïîíÿòòÿ ÷àñòèííèõ ïîõ³äíèõ.<br />
Ðîçãëÿíåìî ôóíêö³þ äâîõ çì³ííèõ z = f(x, y). ³çüìåìî<br />
òî÷êó M 0 (x 0 , y 0 ) ³ äàìî ÷èñëó x 0 ïðèð³ñò ∆x ≠ 0. Ïðèïóñòèìî<br />
ïðè öüîìó, ùî òî÷êà Mx ( 0<br />
+∆ xy ,<br />
0)<br />
íå âèõîäèòü ³ç îáëàñò³<br />
âèçíà÷åííÿ ôóíêö³¿ z = f(x, y).  ðåçóëüòàò³ îòðèìàºìî ÷àñòèííèé<br />
ïðèð³ñò<br />
( , ) ( , )<br />
∆ xz = f x0 +∆x y0 − f x0 y0<br />
. (10.3.1)<br />
Ó ð³âíîñò³ (10.3.1) ³íäåêñ x âêàçóº íà òå, ùî ïðèð³ñò ôóíêö³¿<br />
çä³éñíþºòüñÿ çà ðàõóíîê ò³ëüêè çì³íè x.<br />
∆ z<br />
Îçíà÷åííÿ 10.3.1. ßêùî ãðàíèöÿ lim<br />
x<br />
³ñíóº ³ ñê³í÷åííà,<br />
òî âîíà íàçèâàºòüñÿ ÷àñòèííîþ ïîõ³äíîþ ïî çì³íí³é x â<br />
∆x→0<br />
∆x<br />
òî÷ö³ M 0 (x 0 , y 0 ) ³ ñèìâîë³÷íî öå çàïèñóºòüñÿ òàê:<br />
∆ z ∂z ⎛ ∂f ∂f<br />
⎞<br />
= ( x , y ) z′ ( x , y ), ( x , y ), z′<br />
( M ),<br />
( M )<br />
∆x ∂x ⎜<br />
∂x ∂x<br />
⎟<br />
⎝ ⎠ . (10.3.2)<br />
lim x 0 0 x 0 0 0 0 x 0 0<br />
∆x→0<br />
Çàóâàæåííÿ 1.  äóæêàõ âêàçàí³ ³íø³ ñèìâîëè ïîçíà÷åííÿ<br />
÷àñòèííî¿ ïîõ³äíî¿ ïî çì³íí³é õ.<br />
Àíàëîã³÷íî ââîäèòüñÿ ïîíÿòòÿ ÷àñòèííî¿ ïîõ³äíî¿ ôóíêö³¿<br />
z = f(x, y) ïî çì³íí³é y:<br />
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