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+ 0.74 + 0.67 + 0.61 + 0.55) = 0.81 .<br />
1<br />
∫<br />
0<br />
dx<br />
1 +<br />
1 (0.99 0.96 0.92 0.86 0.8<br />
2<br />
x ≈ 10<br />
+ + + + +<br />
+ 0.74 + 0.67 + 0.61 + 0.55 + 0.5) = 0.76 .<br />
1<br />
∫<br />
0<br />
dx<br />
1 +<br />
1 (0.75 0.99 0.96 0.92 0.86<br />
2<br />
x ≈ 10<br />
+ + + + +<br />
+ 0.8 + 0.67 + 0.61 + 0.55) = 0.782 .<br />
Òåïåð îá÷èñëèìî òî÷íî äàíèé ³íòåãðàë:<br />
1<br />
∫<br />
0<br />
dx<br />
1 + x<br />
2<br />
1<br />
π<br />
= arctgx<br />
0<br />
= ≈ 0.78539 .<br />
4<br />
Ïîð³âíÿííÿ íàáëèæåíèõ îá÷èñëåíü äàíîãî ³íòåãðàëà ç<br />
éîãî òî÷íèì çíà÷åííÿì äàþòü òàêèé ðåçóëüòàò: îòðèìàíà<br />
îäíà â³ðíà öèôðà çà ôîðìóëàìè ïðÿìîêóòíèê³â, à çà ôîðìóëîþ<br />
òðàïåö³¿ — äâ³ â³ðí³ öèôðè.<br />
Çàóâàæåííÿ 1. Î÷åâèäíî, ùî ç³ çá³ëüøåííÿì n òî÷í³ñòü<br />
ôîðìóë (9.10.1 – 9.10.3) ïîêðàùóºòüñÿ.<br />
Çàóâàæåííÿ 2. Êð³ì ôîðìóë (9.10.1 – 9.10.3), ³ñíóþòü<br />
é ³íø³, íàïðèêëàä ïàðàáîë³÷íà ôîðìóëà ѳìïñîíà 1 .<br />
1<br />
ѳìïñîí Òîìàñ (1710 – 1761) — àíãë³éñüêèé ìàòåìàòèê.<br />
ÒÅÌÀ 10<br />
ÔÓÍÊÖ²¯ ÁÀÃÀÒÜÎÕ Ç̲ÍÍÈÕ<br />
10.1. ÎÑÍÎÂͲ ÎÇÍÀ×ÅÍÍß ² ÏÎÍßÒÒß<br />
10.1.1. Ïðîáëåìí³ ïðèêëàäè<br />
Ïðè âèð³øåíí³ áàãàòüîõ ïèòàíü ãåîìåòð³¿, ïðèðîäîçíàâñòâà<br />
³, çîêðåìà, åêîíîì³êè ìàþòü ì³ñöå âèïàäêè, êîëè îäíà<br />
âåëè÷èíà çàëåæèòü â³ä äåê³ëüêîõ.<br />
Ïîÿñíèìî öå íà ïðèêëàäàõ.<br />
Ïðèêëàä 10.1.1. Ïëîùà S äîâ³ëüíîãî ïðÿìîêóòíèêà<br />
òàêà: S = xy. Ç ö³º¿ ôîðìóëè âèäíî, ùî âåëè÷èíà S çàëåæèòü<br />
â³ä äâîõ çì³ííèõ âåëè÷èí x ³ y (x — äîâæèíà, y —<br />
øèðèíà).<br />
Ïðèêëàä 10.1.2.  òåî𳿠åëåêòðèêè â³äîìèé çàêîí Îìà,<br />
ÿêèé ó â³äïîâ³äíèõ îäèíèöÿõ ïîâ’ÿçóº ñèëó ñòðóìó (I), íàïðóãó<br />
(U) ³ îï³ð (R). Öåé çàêîí ìîæíà îïèñàòè òàêîþ ôîðìóëîþ<br />
U<br />
I = .<br />
R<br />
Ïðèêëàä 10.1.3. Îá’ºì V äîâ³ëüíîãî ïðÿìîãî ïàðàëåëåï³ïåäà<br />
òàêèé: V = xyz. Öÿ ôîðìóëà ïîêàçóº, ùî âåëè÷èíà V<br />
áóäü-ÿêîãî ïðÿìîãî ïàðàëåëåï³ïåäà çàëåæèòü â³ä òðüîõ âåëè÷èí<br />
(x — äîâæèíà, y — øèðèíà, z — âèñîòà).<br />
Ïðèêëàä 10.1.4.  êëàñè÷í³é òåî𳿠åêîíîì³êè â³äîìî<br />
òàêå ñï³ââ³äíîøåííÿ<br />
MV = PY. (10.1.1)<br />
Âîíî íàçèâàºòüñÿ ð³âíÿííÿì îáì³íó Ô³øåðà, àáî îñíîâíèì<br />
ð³âíÿííÿì êëàñè÷íî¿ ê³ëüê³ñíî¿ òåî𳿠ãðîøåé.  öüîìó<br />
ð³âíÿíí³ áóêâîþ Ì ïîçíà÷åíî çàãàëüíó ê³ëüê³ñòü ãðîøåé;<br />
áóêâîþ V ïîçíà÷åíî øâèäê³ñòü ¿õ îáîðîòó (ñê³ëüêè ðàç<br />
êîæíà ãðèâíÿ, äîëàð ïðèéìຠó÷àñòü ó ðîçðàõóíêàõ ó ñåðåäíüîìó<br />
çà ð³ê); áóêâîþ Y ïîçíà÷åíî íàö³îíàëüíèé ïðîäóêò<br />
àáî äîõîä (íàö³îíàëüíèé ïðîäóêò, âèðàæåíèé âàðò³ñòþ, ñï³â-<br />
342 343
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