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Ðîçâ’ÿçàííÿ. Íåõàé íà ðèñ. 6.5 çîáðàæåíî ãðàô³ê<br />
ôóíêö³¿ ïîïèòó (ÿê â³äîìî, öÿ ôóíêö³ÿ ñïàäíà, ³ òîìó ãðàô³ê<br />
îáåðíåíî¿ ôóíêö³¿ äëÿ íå¿ ³ñíóº).<br />
1) Ãðàô³ê îáåðíåíî¿ äî íå¿ ôóíêö³¿ ð = ϕ(D) çà ãðàô³êîì<br />
ïðÿìî¿ ôóíêö³¿ D = f(p) òðåáà çíàõîäèòè òàê: à) ÷åðåç<br />
ïî÷àòîê êîîðäèíàò ïðîâåäåìî á³ñåêòðèñó ïåðøîãî ³ òðåòüîãî<br />
êîîðäèíàòíèõ êóò³â; á) â³äíîñíî ïðîâåäåíî¿ á³ñåêòðèñè ïîáóäóºìî<br />
êðèâó, ÿêà ñèìåòðè÷íà ãðàô³êó ôóíêö³¿ D = f(p).<br />
Ïîáóäîâàíà êðèâà ³ áóäå øóêàíèì ãðàô³êîì îáåðíåíî¿<br />
ôóíêö³¿ ïîïèòó.<br />
2) Íåõàé âåëè÷èíà D ô³êñîâàíà (D = w). Ó â³äïîâ³äíîñò³<br />
äî ìàñøòàáó íà â³ñ³ OD çîáðàçèìî òî÷êó, ÿêó ïîçíà÷èìî<br />
áóêâîþ w. ϳñëÿ ÷îãî â³ä òî÷êè w ïðîâîäèìî âïðàâî ïðÿìó<br />
äî ïåðåòèíàííÿ ç ãðàô³êîì ôóíêö³¿ D = f(p). Ïîò³ì ñïðîåêòóºìî<br />
öþ òî÷êó ïåðåòèíó íà â³ñü Op. Îòðèìàíà òî÷êà v ³<br />
º çíà÷åííÿ îáåðíåíî¿ ôóíêö³¿ ïðè D = w.<br />
ÂÏÐÀÂÈ<br />
6.10. Íåõàé ôóíêö³ÿ ïîïèòó íà òîâàð ìຠâèãëÿä<br />
7<br />
D( p)<br />
=<br />
p . Ïîêàæ³òü, ùî îáåðíåíà ôóíêö³ÿ ìຠòîé ñàìèé<br />
âèãëÿä.<br />
7<br />
6.11. Íåõàé D( p)<br />
= , S(p) =p. Òðåáà çíàéòè ð³âíîâàæíó<br />
p<br />
ö³íó ç òî÷í³ñòþ äî ï’ÿòîãî çíàêà ï³ñëÿ êîìè.<br />
2 2<br />
6.12. Íåõàé Dp ( ) =- p+ 50, Sp ( ) = p . Òðåáà çíàéòè ïðèðîäí³<br />
îáëàñò³ âèçíà÷åííÿ ôóíêö³é D(p), S (p) ³ ð³âíîâàæíó ö³íó.<br />
6.13. Íåõàé ôóíêö³¿ ïîïèòó òà ïðîïîçèö³¿ ìàþòü â³äïîâ³äíî<br />
âèãëÿä:<br />
ap + b<br />
Dp ( ) = , Sp ( ) = ep+<br />
f.<br />
cp + d<br />
Òðåáà äîâåñòè, ùî ïðè äîäàòíèõ a, b, c, d, e, f ð³âíÿííÿ<br />
D(p) = S(p) ìຠºäèíèé äîäàòíèé êîð³íü. Ùî öå îçíà÷ຠç<br />
åêîíîì³÷íî¿ òî÷êè çîðó?<br />
6.2. ÏÎÍßÒÒß ÏÐÎ ÃÐÀÍÈÖÞ ÔÓÍÊÖ²¯<br />
6.2.1. Ïðîáëåìí³ ïðèêëàäè ³ îçíà÷åííÿ<br />
 ïîïåðåäí³õ ëåêö³ÿõ áóëî ðîçãëÿíóòî ïèòàííÿ ïðî ãðàíèöþ<br />
÷èñëîâî¿ ïîñë³äîâíîñò³, àáî ôóíêö³¿ íàòóðàëüíîãî àðãóìåíòó.<br />
Ó ö³º¿ ôóíêö³¿ àðãóìåíò çì³íþºòüñÿ äèñêðåòíî,<br />
íàáóâàþ÷è çíà÷åíü 1, 2, 3, …, n, … . Ó çàãàëüíîìó âèïàäêó<br />
àðãóìåíò õ ôóíêö³¿ f(x) íàëåæèòü äåÿê³é ìíîæèí³ Õ. Íàïðèêëàä,<br />
X = ( a, x0) È ( x0,<br />
b)<br />
. Ïðè äîñë³äæåíí³ ôóíêö³¿ f(x) íà<br />
òàê³é ìíîæèí³ çâè÷àéíî âèíèêຠïèòàííÿ ïðî ïîâåä³íêó<br />
ôóíêö³¿ f(x) ïðè íàáëèæåíí³ àðãóìåíòó õ äî õ 0<br />
(õ 0 º ô³êñîâàíå çíà÷åííÿ õ). Ïðè öüîìó ìîæëèâ³ ð³çí³ âèïàäêè.<br />
Äëÿ á³ëüøîãî ðîçóì³ííÿ öüîãî ôàêòó ðîçãëÿíåìî<br />
òàê³ ïðèêëàäè.<br />
Ïðèêëàä 6.2.1. y= x; a=- 1, b= 1, x0<br />
= 0. Ö³ëêîì ïðèðîäíî,<br />
ùî äëÿ ðîçâ’ÿçàííÿ ïðîáëåìè ïðî ïîâåä³íêó ôóíêö³¿ ó = õ<br />
ïðè ïðÿìóâàíí³ àðãóìåíòó õ äî íóëÿ òðåáà áðàòè çíà÷åííÿ<br />
õ, ÿê³ áëèçüê³ äî ÷èñëà íóëü (ôðàçà “çíà÷åííÿ õ, ÿê³ áëèçüê³<br />
äî ÷èñëà íóëü” îçíà÷àº, ùî çíà÷åííÿ õ äîñèòü ìàë³ â³äíîñíî<br />
îäèíèö³ ìàñøòàáó ä³éñíî¿ â³ñ³ 0õ), ³ ïðè öüîìó çíàõîäèòè<br />
â³äïîâ³äí³ çíà÷åííÿ ôóíêö³¿. Ïðîöåñ ïðÿìóâàííÿ<br />
àðãóìåíòó õ äî íóëÿ ìîæíà çàïèñàòè ó âèãëÿä³ ïîñë³äîâíîñò³<br />
{x n }, äå lim x n<br />
= 0 .<br />
n→∞<br />
³çüìåìî, íàïðèêëàä,<br />
1<br />
xn<br />
= , n∈ N . ³äïîâ³äí³ çíà÷åííÿ y n<br />
n<br />
ôóíêö³¿ ó = õ â òî÷êàõ x n äîð³âíþþòü 1 . Íåâàæêî ïîáà÷èòè,<br />
ùî êîëè n →∞, òî xn<br />
→ 0 i yn<br />
→ 0 . ² âçàãàë³, ÿêùî ãðà-<br />
n<br />
íèöÿ ïîñë³äîâíîñò³ {x n } äîð³âíþº íóëþ (ïîñë³äîâí³ñòü {x n } º<br />
íåñê³í÷åííî ìàëîþ), òî ³ ãðàíèöÿ ïîñë³äîâíîñò³ {y n } òåæ<br />
äîð³âíþº íóëþ. ijéñíî, îñê³ëüêè x n = y n , òî ãðàíèö³ ïîñë³äîâíîñòåé<br />
{x n }, {y n } ñï³âïàäàþòü ³ äîð³âíþþòü íóëþ. Îòæå, ìè ñ<br />
ïåâí³ñòþ ìîæåìî ñêàçàòè, ùî ïðè ïðÿìóâàíí³ àðãóìåíòó õ<br />
ôóíêö³¿ ó = õ äî íóëÿ â³äïîâ³äí³ çíà÷åííÿ ôóíêö³¿ òåæ<br />
ïðÿìóþòü äî íóëÿ.<br />
168 169
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