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Íà ïðàêòèö³ êàðòèíà ðåàë³çàö³¿ òîâàðó Q 0 ïðîäàâöåì<br />
á³ëüø ñêëàäíà, ³ ïîâ’ÿçàíå öå ç òèì, ùî ïðîäàâåöü ç ìåòîþ<br />
ïðîäàæó òîâàðó âèùå ð³âíîâàæíî¿ ö³íè ïðîïîíóº ïîêóïöÿì<br />
Q0<br />
òîâàð ïàðò³ÿìè (ïîðö³ÿìè). Íàïðèêëàä, ∆ Qi<br />
= , i = 1, n, òóò<br />
n<br />
n — íàòóðàëüíå ÷èñëî, ÿêå âèçíà÷ຠê³ëüê³ñòü ïàðò³é òîâàðó.<br />
ϳäðàõóºìî òåïåð íàáëèæåíî ñóìó ãðîøåé, ÿêó âèòðà÷åíî<br />
ïîêóïöåì ïðè íàâåäåí³é òàêòèö³ ïðîäàâö³â:<br />
Γ ≈ P∆ Q + P∆ Q + K+ P∆Q<br />
, äå P = f( Q ) (ðèñ. 9.28).<br />
1 1 2 2 n n<br />
n<br />
n<br />
9.11.3. Çàñòîñóâàííÿ âèçíà÷åíîãî ³íòåãðàëà â çàäà-<br />
÷àõ îá÷èñëåííÿ âèòðàò, äîõîä³â òà ïðèáóòê³â<br />
Íàâåäåìî ïðèêëàäè åêîíîì³÷íèõ ïðîöåñ³â, ïîâ’ÿçàíèõ ç<br />
ïîíÿòòÿì ãðàíè÷íèõ âåëè÷èí (äèâ. ï. 7.11).<br />
Íåõàé (x) º ôóíêö³ºþ çàãàëüíèõ âèòðàò íà âèðîáíèöòâî<br />
x îäèíèöü ïðîäóêö³¿, à f(x) =′(x) — ôóíêö³ÿ ãðàíè÷íèõ<br />
âèòðàò. Ó â³äïîâ³äíîñò³ äî ôîðìóëè Íüþòîíà – Ëåéáí³öà ìîæíà<br />
çàïèñàòè, ùî<br />
b<br />
b<br />
∫ f( x) dx = ( x) = ( b) −( a)<br />
. (9.11.6)<br />
a<br />
a<br />
Ðèñ. 9.28<br />
ßêùî, ä³éñíî ïðîäàâö³ ðåàë³çóþòü òîâàð “ìàëèìè” ïîðö³ÿìè,<br />
òî øóêàíó âåëè÷èíó à ìîæíà íàáëèæåíî îá÷èñëèòè<br />
0<br />
÷åðåç âèçíà÷åíèé ³íòåãðàë Γ ≈ ∫ ( )<br />
Q<br />
Q1<br />
f Q dQ, ùî â áàãàòüîõ âèïàäêàõ<br />
ñòàíîâèòü á³ëüø ïðîñòèé àëãîðèòì îá÷èñëåííÿ ñóìè<br />
Ã, í³æ ïîïåðåäí³é.<br />
10<br />
Ïðèêëàä 9.11.4. Íåõàé P = , Q<br />
Q 0 = 10000 (îä. ïð.),<br />
Q 1 =4.<br />
Òîä³<br />
10000<br />
1<br />
10<br />
10000<br />
2<br />
Γ ≈ ∫ dQ= 10⋅ 2Q<br />
= 20( 100 − 2)<br />
= 1960 (ãð. îä.).<br />
4 Q<br />
4<br />
Ïðè öüîìó âåëè÷èíà (b) –(a) âèçíà÷ຠçàãàëüí³ âèòðàòè<br />
ïðè çðîñòàíí³ â³ä a äî b îäèíèöü ïðîäóêö³¿.<br />
2<br />
Ïðèêëàä 9.11.5. Íåõàé ôóíêö³ÿ f( x) 5 2x 0,003x<br />
= + + .<br />
Òðåáà îá÷èñëèòè çàãàëüí³ âèòðàòè ∆ ïðè çðîñòàíí³ âèðîáíèöòâà<br />
â³ä 100 äî 1000 îäèíèöü ïðîäóêö³¿.<br />
Ðîçâ’ÿçàííÿ. Çã³äíî ç ôîðìóëîþ (9.11.6) ìàºìî<br />
1000<br />
2 2 3<br />
1000<br />
∆ F = ∫ ( 5 + 2x+ 0,003x ) dx= ( 5x+ x + 0,001x ) = 1993500 .<br />
100<br />
100<br />
Îòæå, çàãàëüí³ âèòðàòè âèðîáíèöòâà ó çàçíà÷åíèõ ìåæàõ<br />
äîð³âíþþòü 1 993 500 ãð. îä.<br />
Íàâåäåìî òàêîæ â³äïîâ³äí³ ôîðìóëè äëÿ îá÷èñëåííÿ ïðèðîñòó<br />
äîõîäó Ô(x) òà ïðèáóòêó Ψ(x) ïðè ðåàë³çàö³¿ âèðîáëåíî¿<br />
ïðîäóêö³¿ â³ä a äî b îäèíèöü çà óìîâè, ùî íàì â³äîì³<br />
ãðàíè÷íèé äîõîä ϕ(x) òà ãðàíè÷íèé ïðèáóòîê ψ(x):<br />
( x) dx ( b) ( a) , ( x) dx ( b) ( a)<br />
338 339<br />
b<br />
a<br />
b<br />
∆Φ = ∫ϕ = Φ − Φ ∆Ψ = ∫ φ = Ψ − Ψ . (9.11.7)<br />
a<br />
Ïðèêëàä 9.11.6. Íåõàé, ôóíêö³ÿ ϕ(x) = 100 + 0,2 x, à<br />
ψ(x) = 100 – 0,2 x, a = 0, b = 100.<br />
Òðåáà çíàéòè äîõîä òà ïðèáóòîê ï³äïðèºìñòâà â³ä ðåàë³çàö³¿<br />
âèðîáëåíî¿ ïðîäóêö³¿.<br />
Ðîçâ’ÿçàííÿ<br />
100 100<br />
2<br />
⎛ x ⎞100<br />
4 3<br />
∆Φ = ∫ ϕ ( x) dx = ∫ ( 100 + 0, 2x)<br />
dx = ⎜10x + 0, 2 ⎟ = 10 + 10 = 11000<br />
0 0<br />
⎝ 2 ⎠ 0<br />
,
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