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íàãð³âàòè. Åêñïåðèìåíòàëüí³ ³ òåîðåòè÷í³ äîñë³äæåííÿ äîçâîëÿþòü<br />
çðîáèòè ïðîãíîç ïðî çì³íó îá’ºìó êóáèêà ñâèíöþ.<br />
Ïðîöåñ áóäå â³äáóâàòèñÿ òàê: ïðè íàãð³âàíí³ â³ä 0 0 äî 327 0<br />
îá’ºì ñâèíöþ çá³ëüøóºòüñÿ â³ä 1000 ñì 3 äî 1030 ñì 3 ïîñòóïîâî<br />
áåç ð³çêèõ ñòðèáê³â. Ïîò³ì â³í ïðè òåìïåðàòóð³ 327 0<br />
(òåìïåðàòóðà ïëàâëåííÿ ñâèíöþ) ð³çêî çðîñòຠäî 1067 ñì 3 .<br />
Ïðè ïîäàëüøîìó íàãð³âàíí³ äî 500 0 îá’ºì ñâèíöþ (âæå â<br />
ð³äêîìó ñòàí³) çðîñòຠâ³ä 1067 ñì 3 äî 1088 ñì 3 . Îòæå, íà<br />
ñåãìåíò³ [0, 500] îá’ºì ñâèíöþ ÿê ôóíêö³ÿ òåìïåðàòóðè íàãð³âàííÿ<br />
ÿâëÿº ñîáîþ ðîçðèâíó ôóíêö³þ íà ñåãìåíò³ [0, 500]<br />
³ íåïåðåðâíó ôóíêö³þ, íàïðèêëàä, íà ñåãìåíò³ [0, 250]. Ïðîïîíóºìî<br />
÷èòà÷åâ³ ñõåìàòè÷íî íà ãðàô³êó çîáðàçèòè ðîçãëÿäóâàíèé<br />
ïðîöåñ.<br />
Òåïåð ïåðåéäåìî äî ðîçãëÿäó ôóíêö³é, ÿê³ îïèñóþòü åêîíîì³÷í³<br />
ïðîöåñè. Á³ëüø³ñòü ç íèõ — íåïåðåðâí³ ôóíêö³¿.<br />
Àëå ³ â åêîíîì³÷íèõ ïðîöåñàõ áóâຠòàê, ùî ïðîöåñ çì³íè<br />
äåÿêî¿ åêîíîì³÷íî¿ õàðàêòåðèñòèêè ñïî÷àòêó çì³íþºòüñÿ<br />
íåïåðåðâíî, à ïîò³ì ñòðèáêîïîä³áíî. Íàïðèêëàä, òàêà ôóíêö³ÿ,<br />
ÿê ïîïèò òîâàðó, çì³íþºòüñÿ â îêîë³ ð³âíîâàæíî¿ ö³íè<br />
íåïåðåðâíî. Òåïåð íåõàé ö³íà çðîñòຠäî ÿêîãîñü êðèòè÷íîãî<br />
çíà÷åííÿ. Ìè íàçâàëè éîãî êðèòè÷íèì, òîìó ùî ï³ñëÿ<br />
ââåäåíîãî çíà÷åííÿ ö³íè ïîïèò íà òîâàð ð³çêî ïàäàº. Òàêèì<br />
÷èíîì, â íàâåäåíîìó ïðèêëàä³ ôóíêö³ÿ ïîïèòó ñòàº<br />
ðîçðèâíîþ. Äîñâ³ä÷åí³ ìåíåäæåðè ³ ô³íàíñèñòè öåé ôàêò<br />
çíàþòü ³ íå äîïóñêàþòü çàïðîâàäæåííÿ éîãî â æèòòÿ. À<br />
äåÿê³ ç íèõ (äóæå àçàðòí³), îïèðàþ÷èñü íà ÷èñòî ïñèõîëîã³÷í³<br />
ïðè÷èíè íàâåäåíîãî ôàêòó, ââîäÿòü ö³íè, ÿê³ áëèçüê³<br />
äî êðèòè÷íî¿ àëå âñå æ òàêè ìåíø³ â³ä íå¿. ×èòà÷ ìàáóòü<br />
çäîãàäàâñÿ, ùî çàì³ñòü êðèòè÷íî¿ ö³íè â 10 ãðîøîâèõ îäèíèöü<br />
òîâàð ïðîäàþòü çà 9,99 ãðîøîâî¿ îäèíèö³.<br />
Ñë³ä ñêàçàòè, ùî â åêîíîì³ö³ º ôóíêö³¿, ÿê³ íàïåðåä çàäàþòüñÿ<br />
ÿê ðîçðèâí³. Íàïðèêëàä, ôóíêö³ÿ ïîäàòêîâî¿ ñòàâêè<br />
(äèâ. ï. 6.1.6.).<br />
6.3.8. Äåÿê³ âàæëèâ³ ãðàíèö³<br />
Îá÷èñëåííÿ ãðàíèöü ó áàãàòüîõ âèïàäêàõ çä³éñíþºòüñÿ<br />
çà äîïîìîãîþ äâîõ âàæëèâèõ ôîðìóë:<br />
sin<br />
lim<br />
x →0<br />
x→0<br />
x<br />
x<br />
= 1 , (6.3.6)<br />
1<br />
x<br />
lim(1 + x)<br />
= e . (6.3.7)<br />
×àñòî âèêîðèñòîâóþòü òàêîæ òàê³ ôîðìóëè:<br />
sin kx<br />
lim = k , k ≠ 0, (6.3.8)<br />
x→0<br />
x<br />
arcsin x<br />
lim = 1, (6.3.9)<br />
x →0<br />
x<br />
arctg x<br />
lim = 1 , (6.3.10)<br />
x →0<br />
x<br />
x<br />
⎛ 1 ⎞<br />
lim ⎜1+ ⎟ = e , (6.3.11)<br />
x→∞⎝<br />
x ⎠<br />
log<br />
a<br />
(1 + x) 1<br />
lim = log<br />
a<br />
e = , a > 0, a ≠ 1 , (6.3.12)<br />
x→0<br />
x<br />
ln a<br />
x<br />
a − 1<br />
lim = ln a, a > 0 , (6.3.13)<br />
x→0<br />
x<br />
µ<br />
(1 + x) − 1<br />
lim =µ ,<br />
x →0<br />
x µ∈ R, µ≠ 0 . (6.3.14)<br />
Çîêðåìà, ïðè à = å<br />
ln(1 + x)<br />
lim = 1,<br />
x →0<br />
x<br />
x<br />
e − 1<br />
lim = 1 .<br />
x→0<br />
x<br />
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