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Ôóíêö³ÿ ïðîïîçèö³¿ S = S(p) õàðàêòåðèçóº çàëåæí³ñòü<br />
ïðîïîçèö³¿ äåÿêîãî òîâàðó â³ä éîãî ö³íè ð. Öÿ ôóíêö³ÿ º<br />
çðîñòàþ÷îþ, òîìó ïîõ³äíà S′(p) º âåëè÷èíà äîäàòíà. Âîíà<br />
ïîêàçóº, ó ñê³ëüêè ðàç³â çá³ëüøóºòüñÿ ïðîïîçèö³ÿ òîâàðó ç³<br />
ñòîðîíè ïðîäàâö³â (âèðîáíèê³â) ó çâ’ÿçêó ç ï³äâèùåííÿì<br />
ö³í íà îäíó îäèíèöþ.<br />
7.11.2. Ïîíÿòòÿ åëàñòè÷íîñò³ ôóíêö³¿<br />
Íåõàé âåëè÷èíà ó çàëåæèòü â³ä õ ³ öÿ çàëåæí³ñòü îïèñó-<br />
ºòüñÿ ôóíêö³ºþ y = f(x). Çì³íà íåçàëåæíî¿ çì³ííî¿ ïðèâîäèòü<br />
íà ï³äñòàâ³ ôóíêö³îíàëüíî¿ çàëåæíîñò³ äî çì³íè ó. Ïîñòàº<br />
ïèòàííÿ, ÿê âèì³ðèòè ÷óòëèâ³ñòü çàëåæíî¿ çì³ííî¿ â³ä<br />
çì³íè õ. ßê áóëî ïîêàçàíî ó ï. 7.11.1, îäíèì ³ç ïîêàçíèê³â<br />
ðåàãóâàííÿ º ïîõ³äíà f′(x). Âîíà ïîêàçóº øâèäê³ñòü çì³íè<br />
ôóíêö³¿ ó çâ’ÿçêó ç³ çì³íîþ àðãóìåíòó õ. Îäíàê â ïèòàííÿõ<br />
åêîíîì³êè öåé ïîêàçíèê íå çàâæäè çðó÷íèé, òîìó ùî<br />
â³í çàëåæèòü â³ä âèáîðó îäèíèö³ âèì³ðó. Íàïðèêëàä, ÿêùî<br />
ìè ðîçãëÿíåìî ôóíêö³þ ïîïèòó íà ìóêó D = D(p), òî ö³ëêîì<br />
î÷åâèäíî, ùî çíà÷åííÿ ïîõ³äíî¿ D′(p) ïðè êîæí³é ö³í³<br />
çàëåæèòü â³ä òîãî, â ÿêèõ îäèíèöÿõ (â ê³ëîãðàìàõ àáî öåíòíåðàõ)<br />
âèì³ðþºòüñÿ ïîïèò íà ìóêó. Ó ïåðøîìó âèïàäêó<br />
ïîõ³äíà âèì³ðþºòüñÿ â êã/ãðí., à â äðóãîìó — ö/ãðí. Òîìó<br />
â åêîíîì³ö³ äëÿ âèì³ðó ÷óòëèâîñò³ çàëåæíî¿ çì³ííî¿ â³ä<br />
çì³íè àðãóìåíòó âèâ÷àþòü çâ’ÿçîê íå àáñîëþòíèõ çì³í õ ³<br />
ó (àáî ∆õ ³ ∆ó), à ¿õ â³äíîñíèõ àáî ïðîöåíòíèõ çì³í. Òàêèé<br />
ï³äõ³ä ïðèâ³â äî ââåäåííÿ íîâîãî ïîíÿòòÿ.<br />
Îçíà÷åííÿ 7.11.1. Ãðàíèöÿ â³äíîøåííÿ â³äíîñíèõ çì³í<br />
çì³ííèõ ó ³ õ íàçèâàºòüñÿ åëàñòè÷í³ñòþ ôóíêö³¿, àáî â³äíîñíîþ<br />
ïîõ³äíîþ, ³ ñèìâîë³÷íî ïîçíà÷àºòüñÿ òàê: E .<br />
y<br />
x<br />
Ç ââåäåíîãî îçíà÷åííÿ âèïëèâàº, ùî â³äíîñíà ïîõ³äíà<br />
ïîâ’ÿçàíà ³ç çâè÷àéíîþ ïîõ³äíîþ òàêèì ñï³ââ³äíîøåííÿì:<br />
y x<br />
E = ⋅ ′<br />
x<br />
y ( x)<br />
y<br />
. (7.11.1)<br />
ijéñíî, ó â³äïîâ³äíîñò³ äî îçíà÷åííÿ ìàºìî<br />
E<br />
y<br />
x<br />
∆y<br />
⎛ ∆ ⎞ ∆<br />
=<br />
y<br />
x y x y x<br />
lim = ⎜ ⋅ ⎟ = ⋅ = ⋅ ′<br />
∆ → ∆ lim lim y ( x)<br />
x x 0 ∆x→ 0 ⎝y ∆x⎠<br />
y ∆x→<br />
0 ∆x y .<br />
x<br />
Çàóâàæåííÿ 1. Ïðè íåîáõ³äíîñò³ åëàñòè÷í³ñòü ôóíêö³¿<br />
ìîæíà çîáðàçèòè ó âèãëÿä³:<br />
y dln<br />
y<br />
E = x<br />
dln<br />
x<br />
.<br />
Çàóâàæåííÿ 2. Ââåäåíå ïîíÿòòÿ äóæå çðó÷íå äëÿ<br />
åêîíîì³ñò³â-ïðàêòèê³â. Êîðèñòóþ÷èñü ïîíÿòòÿì åëàñòè÷íîñò³<br />
ôóíêö³¿, åêîíîì³ñò-ïðàêòèê äîñòàòíüî ãðàìîòíî ìîæå â³äïîâ³ñòè<br />
íà çàïèòàííÿ òèïó:<br />
1. Íà ñê³ëüêè ïðîöåíò³â çì³íèòüñÿ ïîïèò íà òîâàð, ÿêùî<br />
ö³íà íà íüîãî çá³ëüøèòüñÿ íà 1%?<br />
2. Íà ñê³ëüêè ïðîöåíò³â çì³íèòüñÿ ïðîïîçèö³ÿ, ÿêùî<br />
ö³íà íà íüîãî çá³ëüøèòüñÿ íà 1%?<br />
7.12. ÊËÀÑÈ×Ͳ ÇÀÑÒÎÑÓÂÀÍÍß ÏÎÕ²Ä-<br />
Íί  ÌÀÒÅÌÀÒÈÖ² ÒÀ ÅÊÎÍÎֲ̲<br />
7.12.1. Çàñòîñóâàííÿ ïîõ³äíî¿ â ìàòåìàòèö³<br />
7.12.1.1. гâíÿííÿ äîòè÷íî¿<br />
Âîíî âèçíà÷åíî ð³âí³ñòþ (7.2.3). Äëÿ êðàùîãî ðîçóì³ííÿ<br />
ïðîáëåìè ðîçãëÿíåìî êîíêðåòíèé ïðèêëàä.<br />
Ïðèêëàä 7.12.1. Çàïèñàòè ð³âíÿííÿ äîòè÷íî¿ äî êðèâî¿<br />
y = sin x (ñèíóñî¿äè), ÿêà ïðîâåäåíà ÷åðåç òî÷êó O (0, 0).<br />
Ðîçâ’ÿçàííÿ. Çíàéäåìî ïîõ³äíó ôóíêö³¿ y = sinx:<br />
y′(0) = cos x. Î÷åâèäíî, ùî y(0) = f(0) = 0, y′(0) = cos 0 = 1.<br />
Òîä³ çã³äíî ç ôîðìóëîþ (7.2.3) ìàºìî øóêàíå ð³âíÿííÿ<br />
äîòè÷íî¿<br />
y = x.<br />
Äî ðå÷³, îñòàííÿ ôîðìóëà âêàçóº, ùî äîòè÷íà, ÿêà ïðîâåäåíà<br />
äî ñèíóñî¿äè íà ïî÷àòêó êîîðäèíàò, ñêëàäຠç îññþ Îx<br />
êóò ó 45%.<br />
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