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ÂÏÐÀÂÈ<br />
10.11. Çíàéòè ïîõ³äíó z = x 2 y – y 2 x ó òî÷ö³ M 0 (2, 1) ó<br />
íàïðÿìêó âåêòîðà e r , ÿêèé ñêëàäຠêóò α = 30° ç äîäàòíèì<br />
íàïðÿìîì îñ³ Ox.<br />
5<br />
10.12. Çàäàíà ôóíêö³ÿ z = . Ïîáóäóâàòè ë³í³¿ ð³âíÿ<br />
2 2<br />
x + y<br />
³ gradz ó òî÷ö³ M 0 (–1, 2) òà çíàéòè grad z( − 1,2) .<br />
10.13. Çíàéòè ïîõ³äíó ôóíêö³¿ z = xy 2 z 3 â òî÷ö³<br />
uuuuuuur<br />
M 0 (1, 2, 3) ó íàïðÿìó âåêòîðà MM<br />
0 , äå M(3,2,1).<br />
10.14. Çíàéòè ïîõ³äíó ôóíêö³¿ z = 5ln( x 2 + y<br />
2<br />
) â òî÷ö³<br />
M 0 (1; 2) ó íàïðÿìó ãðà䳺íòà ôóíêö³¿ z ³ íàéá³ëüøó øâèäê³ñòü<br />
¿¿ çðîñòàííÿ â äàí³é òî÷ö³.<br />
10.15. Çàäàíà ôóíêö³ÿ u =tgx – x + 3siny – sin 3 y + z + ctgz.<br />
Òðåáà çíàéòè ãðà䳺íò ö³º¿ ôóíêö³¿, éîãî äîâæèíó ³ íàïðÿì â<br />
M π 4 π 3 π 2 .<br />
10.16. Çíàéòè íàéá³ëüøó êðóò³ñòü ïîâåðõí³ z 2 = xy â òî÷ö³<br />
M 0 (4, 2).<br />
10.17. Çíàéòè ïîõ³äíó ôóíêö³¿ z = ln(e x +e y ) ó íàïðÿìàõ,<br />
ïàðàëåëüíèõ á³ñåêòðèñàì êîîðäèíàòíèõ êóò³â.<br />
10.18. Çíàéòè ïîõ³äíó ôóíêö³¿ U = x 2 + y 2 + z 2 â òî÷ö³<br />
òî÷ö³ 0 ( , , )<br />
M 0 (1, 1, 1) ó íàïðÿìó r e (cos45°, cos60°, cos60°) ³ çíàéòè ãðàä³-<br />
ºíò â ò³é òî÷ö³. Ïîáóäóâàòè òàêîæ ïîâåðõí³ ð³âí³â çàäàíî¿<br />
ôóíêö³¿.<br />
10.19. Ó ïðèêëàä³ 10.3.3 âèçíà÷åíà ôóíêö³ÿ Êîááà – Äóãëàñà.<br />
Òðåáà çíàéòè â òî÷ö³ M 0 (25 ⋅ 10 8 ,10 3 ) ¿¿ ãðà䳺íò ³ øâèäê³ñòü<br />
íàéá³ëüøîãî çðîñòàííÿ. Ùî áóäå îçíà÷àòè ç åêîíîì³÷íî¿<br />
òî÷êè çîðó îòðèìàíèé ðåçóëüòàò?<br />
10.6. ÅÊÑÒÐÅÌÀËÜͲ ÇÍÀ×ÅÍÍß ÔÓÍÊÖ²¯<br />
ÁÀÃÀÒÜÎÕ Ç̲ÍÍÈÕ<br />
10.6.1. Åêñòðåìóì ôóíêö³¿<br />
ßê ³ äëÿ âèïàäêó îäí³º¿ çì³ííî¿, ôóíêö³ÿ z = f(x, y) ìàº<br />
âóçëîâ³ òî÷êè, ÿê³ âèçíà÷àþòü ñòðóêòóðó ãðàô³êà.  ïåðøó<br />
÷åðãó öå òî÷êè åêñòðåìóìó.<br />
Îçíà÷åííÿ 10.6.1. Êàæóòü, ùî ôóíêö³ÿ z = f(x, y), ÿêà<br />
âèçíà÷åíà â δ-îêîë³ òî÷êè M 0 (x 0 , y 0 ), ìຠìàêñèìóì (ì³í³ìóì),<br />
ÿêùî ³ñíóº òàêèé δ 1 -îê³ë (δ 1 < δ), ùî äëÿ óñ³õ òî÷îê<br />
δ 1 -îêîëó âèêîíóºòüñÿ íåð³âí³ñòü<br />
f(M) ≤ f(M 0 )(f(M) ≥ f(M 0 )).<br />
Ðèñ. 10.13<br />
Ðèñ. 10.14<br />
Íà ðèñ. 10.13 ôóíêö³ÿ z = f(x, y) â òî÷ö³ ìຠìàêñèìóì, à<br />
íà ðèñ. 10.14 — ì³í³ìóì. Ìàêñèìóì (ì³í³ìóì) ôóíêö³¿<br />
z = f(x, y) ïîçíà÷àºòüñÿ òàê:<br />
( ,<br />
o) ( ( ,<br />
o)<br />
)<br />
z = f x y z = f x y .<br />
max 0 min 0<br />
Ïîíÿòòÿ åêñòðåìóìó îá’ºäíóº âæå ââåäåí³ ïîíÿòòÿ ìàêñèìóìó<br />
³ ì³í³ìóìó. ²íøèìè ñëîâàìè, òî÷êè ìàêñèìóìó ³ ì³í³ìóìó<br />
íàçèâàþòüñÿ òî÷êàìè åêñòðåìóìó, à çíà÷åííÿ ôóíêö³¿<br />
z = f(x, y) â íèõ íàçèâàþòü åêñòðåìóìîì ôóíêö³¿.<br />
378 379
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