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Îçíà÷åííÿ 10.4.2 Äèôåðåíö³àëîì äèôåðåíö³éîâíî¿ ôóíêö³¿<br />
z = f(x, y) â òî÷ö³ M 0 (x 0 , y 0 ) íàçèâàºòüñÿ ë³í³éíà â³äíîñíî<br />
ïðèðîñò³â ∆x i ∆y ÷àñòèíà ïîâíîãî ïðèðîñòó ö³º¿ ôóíêö³¿ â<br />
òî÷ö³ M 0 (x 0 , y 0 ). Äèôåðåíö³àë ôóíêö³¿ z = f(x, y) â òî÷ö³<br />
M 0 (x 0 , y 0 ) ïîçíà÷àºòüñÿ òàê: dz(x 0 , y 0 ). Òîä³ çà îçíà÷åííÿì<br />
³ òåîðåìîþ 10.4.2 äèôåðåíö³àë ìîæíà çàïèñàòè ó âèãëÿä³:<br />
( 0, ) ′<br />
0 ( 0, 0) ′( 0,<br />
0)<br />
dz x y = z x y ∆ x + z x y ∆ y . (10.4.5)<br />
x<br />
Äèôåðåíö³àëàìè íåçàëåæíèõ çì³ííèõ õ ³ ó íàçâåìî ïðèðîñòè<br />
öèõ çì³ííèõ: dx = ∆x, dy = ∆y. Òîä³ äèôåðåíö³àë íàáóâàº<br />
ñèìåòðè÷íîãî âèãëÿäó:<br />
y<br />
( , ) ′ ( , ) ′( , )<br />
0 0 x 0 0 y 0 0<br />
y<br />
dz x y = z x y dx + z x y dy . (10.4.6)<br />
z′ x y = A,<br />
Íåõàé ïðèíàéìí³ îäíà ³ç êîíñòàíò x ( 0,<br />
0)<br />
( 0,<br />
0)<br />
z′ x y = B â³äì³ííà â³ä íóëÿ. Òîä³ ó ð³âíîñò³ (10.4.4) òðåò³é<br />
³ ÷åòâåðòèé äîäàíîê ÿâëÿþòü ñîáîþ íåñê³í÷åííî ìàë³<br />
á³ëüø âèñîêîãî ïîðÿäêó ìàëèçíè, í³æ ïåðø³ äâà. ² â öüîìó<br />
âèïàäêó ìîæíà çàïèñàòè íàáëèæåíó ôîðìóëó<br />
∆z ≈ dz,<br />
ÿêó âåëüìè ÷àñòî âèêîðèñòîâóþòü â íàáëèæåíèõ îá÷èñëåííÿõ.<br />
Çàóâàæåííÿ. ßêùî ôóíêö³ÿ z = f(x, y) äèôåðåíö³éîâíà<br />
â äåÿê³é îáëàñò³ D, òî â äîâ³ëüí³é òî÷ö³ ö³º¿ îáëàñò³ äèôåðåíö³àë<br />
ìຠòàêèé âèãëÿä:<br />
x<br />
( , ) ( , )<br />
10.4.3. Ïîõ³äíà çà íàïðÿìîì<br />
dz = z′ x y dx + z′<br />
x y dy . (10.4.7)<br />
×àñòèííà ïîõ³äíà z′ ( x , y ) ( zy<br />
( x0,<br />
y0)<br />
)<br />
′ ó òî÷ö³ M<br />
x 0 0<br />
0 (x 0 , y 0 )<br />
âèðàæຠøâèäê³ñòü çðîñòàííÿ ôóíêö³¿ z = f(x, y) â òî÷ö³ â<br />
äîäàòíîìó íàïðÿìó îñ³ Ox (Oy), îäíàê äëÿ ôóíêö³¿<br />
z = f(x, y) ìîæíà ïîñòàâèòè ïèòàííÿ ïðî øâèäê³ñòü ¿¿ çðîñòàííÿ<br />
â òî÷ö³ M 0 (x 0 , y 0 ) â äîâ³ëüíîìó íàïðÿì³.<br />
y<br />
Íåõàé íàïðÿì ó òî÷ö³ M 0 (x 0 , y 0 ) çàäàíî îäèíè÷íèì âåêòîðîì<br />
e r (ðèñ. 10.11), ÿêèé óòâîðþº ç äîäàòíèì íàïðÿìîì<br />
îñ³ Ox êóò α. Ðîçãëÿíåìî ïðÿìó, ÿêà ïàðàëåëüíà âåêòîðó e r .<br />
Ðèñ. 10.11<br />
Íà í³é â³çüìåìî äîâ³ëüíó òî÷êó M 0 (x 0 +∆x, y 0 + ∆y), ÿêà<br />
â³äì³ííà â³ä òî÷êè M 0 , ³ ââåäåìî â³äñòàíü<br />
( ) ( )<br />
2 2<br />
ρ=ρ<br />
MM ,<br />
= ∆ x + ∆ y . Ðîçãëÿíåìî òåïåð ôóíêö³þ z = f(x, y),<br />
0<br />
ÿêà äèôåðåíö³éîâíà â òî÷ö³ M 0 (x 0 , y 0 ), ïðè öüîìó òî÷êà<br />
M 0 (x 0 +∆x, y 0 + ∆y) íàëåæ³òü δ-îêîëó ö³º¿ òî÷êè.<br />
∆z<br />
Òîä³ â³äíîøåííÿ âèðàæຠñåðåäíþ øâèäê³ñòü çðîñòàííÿ<br />
ôóíêö³¿ z = f(x, y) íà â³äð³çêó M 0<br />
ρ<br />
M.<br />
∆<br />
Îçíà÷åííÿ 10.4.3. ßêùî ãðàíèöÿ lim z<br />
³ñíóº, òî âîíà<br />
ρ→0<br />
ρ<br />
íàçèâàºòüñÿ ïîõ³äíîþ ôóíêö³¿ z = f(x, y) â òî÷ö³ M 0 çà íàïðÿìîì<br />
âåêòîðà e r<br />
∂zx<br />
( 0,<br />
y0)<br />
³ ïîçíà÷àºòüñÿ . Ñèìâîë³÷íî öå<br />
∂e<br />
çàïèñóºòüñÿ òàê:<br />
∆<br />
lim z ∂<br />
=<br />
z x y<br />
ρ→0<br />
ρ ∂e<br />
( , )<br />
0 0<br />
. (10.4.8)<br />
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