Create successful ePaper yourself
Turn your PDF publications into a flip-book with our unique Google optimized e-Paper software.
Ò å î ð å ì à 10.3.1. ßêùî ïîõ³äí³ 2-ãî ïîðÿäêó<br />
íåïåðåðâí³ â òî÷ö³ M(x, y), òî âîíè îäíàêîâ³.<br />
Öþ òåîðåìó ìè ïîäàºìî áåç äîâåäåííÿ.<br />
z′′<br />
xy ³<br />
10.3.3. Ãåîìåòðè÷íèé òà åêîíîì³÷íèé çì³ñò ÷àñòèííèõ<br />
ïîõ³äíèõ<br />
Ãåîìåòðè÷íèé çì³ñò ÷àñòèííèõ ïîõ³äíèõ ôóíêö³¿<br />
z = f(x, y) ó òî÷ö³ M 0 (x 0 , y 0 ) ïîêàçàíî íà ðèñ. 10.10.<br />
Ðèñ. 10.10<br />
Íåõàé ãðàô³ê ôóíêö³¿ z = f(x, y) ÿâëÿº ñîáîþ äåÿêó ïîâåðõíþ.<br />
Òîä³ ïðè y = y 0 ìè îòðèìàºìî êðèâó à x , ÿêà º ïåðåð³çîì<br />
ö³º¿ ïîâåðõí³ ç â³äïîâ³äíîþ ïëîùèíîþ. Ó öüîìó âèïàäêó<br />
ïîõ³äíà z′ x ( x0,<br />
y0)<br />
âèðàæຠêóòîâèé êîåô³ö³ºíò äîòè÷íî¿<br />
äî êðèâî¿ Ã x â çàäàí³é òî÷ö³ P 0 (x 0 , y 0 , z 0 ) (z 0 = f(x 0 , y 0 )),<br />
z′ x , y = tgα , äå α 0 êóò íàõèëó äîòè÷íî¿ äî â³ñ³ Ox.<br />
òîáòî x ( 0 0)<br />
0<br />
Àíàëîã³÷íî z ( x , y ) tg<br />
′ = β .<br />
y<br />
0 0 0<br />
z′′<br />
yx<br />
Çàãàëüíèé çì³ñò ÷àñòèííèõ ïîõ³äíèõ â òî÷ö³ ïîëÿãຠâ<br />
òîìó, ùî âîíè âèçíà÷àþòü øâèäê³ñòü çðîñòàííÿ ôóíêö³¿ â<br />
íàïðÿìàõ, ïàðàëåëüíèõ â³ñÿì êîîðäèíàò.<br />
z′ x , y ìàº<br />
Çã³äíî ç îçíà÷åííÿì ÷àñòèííèõ ïîõ³äíèõ x ( 0 0)<br />
ì³ñöå ð³âí³ñòü ∆ z= z′<br />
( x0,<br />
y0)<br />
∆ x+α∆ x, äå lim 0<br />
. ßêùî ïðèð³ñò<br />
∆x äîñòàòíüî ìàëèé, òî<br />
Àíàëîã³÷íî<br />
x<br />
x<br />
x<br />
x<br />
( 0,<br />
0)<br />
α=<br />
∆x→<br />
0<br />
∆ z≈z′<br />
x y ∆ x. (10.3.6)<br />
y<br />
y<br />
( , )<br />
∆ z≈z′<br />
x y ∆ y, (10.3.7)<br />
0 0<br />
äå âåëè÷èíà ∆y äîñòàòíüî ìàëà.<br />
Òåïåð ïðèïóñòèìî, ùî ïðîöåñ çì³íè z òàêèé, ùî çì³íí³ x<br />
òà y òàê³, ùî âîíè íàáàãàòî ïåðåâèùóþòü 1. Òîä³ â íàáëèæåíèõ<br />
÷àñòèííèõ ïðèðîñòàõ (10.3.6) – (10.3.7) ôóíêö³¿<br />
z = f(x, y) â òî÷ö³ M 0 (x 0 , y 0 ) ìîæíà ïîêëàñòè ∆x ³ ∆y ð³âíèìè<br />
1. Ó ðåçóëüòàò³ áóäåìî ìàòè òàê³ íàáëèæåí³ ôîðìóëè:<br />
x<br />
x<br />
( 0,<br />
0)<br />
∆ z ≈ z′<br />
x y , (10.3.8)<br />
y<br />
y<br />
( , )<br />
∆ z≈ z′<br />
x y . (10.3.9)<br />
0 0<br />
Òåïåð ïåðåéäåìî äî ç’ÿñóâàííÿ åêîíîì³÷íîãî çì³ñòó ÷àñòèííèõ<br />
ïîõ³äíèõ â ô³êñîâàí³é òî÷ö³ M 0 (x 0 , y 0 ).<br />
Ç ö³ºþ ìåòîþ ðîçãëÿíåìî âèðîáíè÷ó ôóíêö³þ z = f(x, y),<br />
äå çì³íí³ x òà y â³äïîâ³äíî âèçíà÷àþòü îáñÿã ôîíä³â òà<br />
îáñÿã òðóäîâèõ ðåñóðñ³â.<br />
Íåõàé äëÿ êîíêðåòíîñò³ çì³ííà x ÿâëÿº ñîáîþ ê³ëüê³ñòü<br />
âåðñòàò³â, à y ÷èñëî ðîá³òíèê³â íà ï³äïðèºìñòâ³. Çàô³êñóºìî<br />
ïîòî÷íèé ñòàí ï³äïðèºìñòâà, òîáòî ìè ââîäèìî ô³êñîâàí³ âåëè÷èíè<br />
x 0 òà y 0 .<br />
ßêùî ïðè öüîìó ö³ âåëè÷èíè íàáàãàòî ïåðåâèùóþòü 1,<br />
òî çã³äíî ç ôîðìóëàìè (10.3.8) – (10.3.9) ìîæíà äàòè åêîíîì³÷íèé<br />
çì³ñò ÷àñòèííèõ ïîõ³äíèõ â òî÷ö³ M 0 (x 0 , y 0 ).<br />
×àñòèííà ïîõ³äíà â³ä âèðîáíè÷î¿ ôóíêö³¿ z = f(x, y) çà<br />
îáñÿãîì ôîíä³â ó òî÷ö³ M 0 (x 0 , y 0 ) íàáëèæåíî äîð³âíþº äî-<br />
362 363
Change language