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Òîä³ ìîæíà ïîêàçàòè ó â³äïîâ³äíîñò³ äî åêîíîì³÷íîãî<br />
çì³ñòó âèðîáíè÷î¿ ôóíêö³¿ Êîááà – Äóãëàñà, ùî îáñÿã ïðîäóêö³¿<br />
çà T ðîê³â ñêëàäàº:<br />
T<br />
0<br />
()<br />
Q= ∫ q t dt, (9.11.2)<br />
äå q(t) — ôóíêö³ÿ, ÿêà âèçíà÷åíà çà ôîðìóëîþ (9.11.1).<br />
Ïðèêëàä 9.11.2. Òðåáà çíàéòè îáñÿã ïðîäóêö³¿, âèðîáëåíî¿<br />
ï³äïðèºìñòâîì çà 5 ðîê³â, ÿêùî âèðîáíè÷à ôóíêö³ÿ<br />
Êîááà – Äóãëàñà ïî âèãîòîâëåííþ ïðîäóêö³¿ ìຠâèãëÿä:<br />
() 49( 1 ) 7 t<br />
te<br />
qt<br />
= + . (9.11.3)<br />
Ð î ç â ’ ÿ ç à í í ÿ. Çàñòîñîâóºìî ôîðìóëó (9.11.2) ç óðàõóâàííÿì<br />
(9.11.3).<br />
Òîä³ ìàòèìåìî<br />
7t<br />
⎡u = 1 + t , dv = e dt⎤<br />
5<br />
7t<br />
Q= 49 ( 1 t)<br />
e dt ⎢<br />
⎥<br />
∫ + = 1 =<br />
7t<br />
0<br />
⎢du = dt , v = e ⎥<br />
⎢⎣<br />
7 ⎥⎦<br />
Ïðèêëàä 9.11.3. Íåõàé I(t) = 5000 t.<br />
Òðåáà: à) îá÷èñëèòè ïðèð³ñò êàï³òàëó çà 5 ðîê³â;<br />
á) çà ñê³ëüêè ðîê³â ïðèð³ñò êàï³òàëó áóäå ñêëàäàòè<br />
250 000 (ãð. îä.).<br />
Ðîçâ’ÿçàííÿ.<br />
à) Çã³äíî ç ôîðìóëîþ (9.11.4) ìàºìî<br />
5<br />
∆ K = tdt = t =<br />
0<br />
á)<br />
5<br />
∫ 5000<br />
2<br />
2500 62500<br />
0<br />
T<br />
2<br />
250000 = ∫ 5000tdt<br />
= 2500T<br />
,<br />
0<br />
(ãð. îä.);<br />
çâ³äñè T = 10 (ðîê³â).<br />
9.11.2. Çàñòîñóâàííÿ âèçíà÷åíîãî ³íòåãðàëà â çàäà-<br />
÷àõ ðåàë³çàö³¿ òîâàð³â<br />
Ðîçãëÿíåìî ôóíêö³þ ïîïèòó äåÿêîãî òîâàðó ó âèãëÿä³<br />
P = f(Q) (ðèñ. 9.27).<br />
( ) 7 5<br />
t<br />
5<br />
7 t<br />
= 71+ t e − 7∫ e dt = ( 41e<br />
35 −6)<br />
(îä. ïð.).<br />
0 0<br />
3. Ïðèïóñòèìî, ùî çà îäèíèöþ ÷àñó êàï³òàë K(t) ï³äïðè-<br />
ºìñòâà çá³ëüøóºòüñÿ íà âåëè÷èíó ÷èñòèõ ³íâåñòèö³é I(t)<br />
(êàï³òàëîâêëàäåííÿ ç óðàõóâàííÿì ïîâíèõ çàòðàò íà âèðîáíèöòâî<br />
ïðîäóêö³¿).<br />
Òîä³ ìàºìî<br />
() ′()<br />
I t = K t .<br />
×àñòî â åêîíîì³ö³ ñòàâèòüñÿ ïèòàííÿ ïðî ïðèð³ñò êàï³òàëó<br />
çà ïåð³îä T, ÿêùî â³äîìà ³íâåñòèö³ÿ I(t).<br />
Î÷åâèäíî, ùî<br />
( ) ( 0) ( )<br />
T<br />
K T − K =∆ K = ∫ I t dt . (9.11.4)<br />
0<br />
Ðèñ. 9.27<br />
Òóò P 0 — ð³âíîâàæíà ö³íà, à Q 0 — ê³ëüê³ñòü òîâàðó, ÿêèé<br />
ðåàë³çóºòüñÿ ïî ö³é ö³í³.<br />
ßñíî, ùî çàãàëüíà ñóìà ãðîøåé, ÿêà âèòðà÷åíà ïîêóïöÿìè,<br />
ñòàíîâèòü âåëè÷èíó<br />
à = P 0 Q 0 . (9.11.5)<br />
336 337