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ÁÁÊ 22.1ÿ73 Ê36 ÓÄÊ 51(075.8) Íàâ÷àëüíèé ïîñ³áíèê íàïèñàíèé â³äïîâ³äíî äî òèïîâî¿ ïðîãðàìè Ì³í³ñòåðñòâà îñâ³òè ³ íàóêè Óêðà¿íè ç êóðñó âèùî¿ ìàòåìàòèêè äëÿ åêîíîì³ñò³â. Â³í ì³ñòèòü îñíîâí³ òåìè ç ìàòåìàòè÷íîãî àíàë³çó, âèùî¿ àëãåáðè, àíàë³òè÷íî¿ ãåîìåòð³¿ ³ äèôåðåíö³àëüíèõ ð³âíÿíü. Êð³ì öüîãî, ïîñ³áíèê ìàº äâà äîäàòêè, â ÿêèõ íàâåäåí³ â³äîìîñò³ ïðî êîìïëåêñí³ ÷èñëà ³ äîâ³äêîâèé ìàòåð³àë ç åëåìåíòàðíî¿ òà âèùî¿ ìàòåìàòèêè. Íàâ÷àëüíèé ïîñ³áíèê àäðåñîâàíî ñòóäåíòàì óñ³õ åêîíîì³÷íèõ ñïåö³àëüíîñòåé âèùèõ íàâ÷àëüíèõ çàêëàä³â òà êîëåäæ³â åêîíîì³÷íîãî ïðîô³ëþ, ÿê³ âèâ÷àþòü âèùó ìàòåìàòèêó. Çàïðîïîíîâàíèé ïîñ³áíèê ìîæå áóòè êîðèñíèì åêîíîì³ñòàì-àíàë³òèêàì, ô³íàíñèñòàì ³ á³çíåñìåíàì, ÿê³ ó ñâî¿é ðîáîò³ çàñòîñîâóþòü ìàòåìàòè÷í³ ìåòîäè. Ðåöåíçåíòè: Þ. É. ×åðñüêèé, äîêòîð ô³çèêî-ìàòåìàòè÷íèõ íàóê, ïðîôåñîð êàôåäðè âèùî¿ ìàòåìàòèêè Îäåñüêî¿ äåðæàâíî¿ àêàäåì³¿ áóä³âíèöòâà ³ àðõ³òåêòóðè; 1602010000–054 Ê 549—2003 ISBN 966-549-871-1 Â. Î. Àíäð³ºíêî, äîêòîð ô³çèêî-ìàòåìàòè÷íèõ íàóê, ïðîôåñîð êàôåäðè ìàòåìàòè÷íîãî àíàë³çó Ï³âäåííîóêðà¿íñüêîãî äåðæàâíîãî ïåäàãîã³÷íîãî óí³âåðñèòåòó ³ì. Ê. Ä. Óøèíñüêîãî. Ðåêîìåíäîâàíî äî äðóêó â÷åíîþ ðàäîþ ²íñòèòóòó ï³ñëÿäèïëîìíî¿ îñâ³òè ³ â÷åíîþ ðàäîþ Îäåñüêîãî íàö³îíàëüíîãî óí³âåðñèòåòó ³ì. ². ². Ìå÷íèêîâà. Áåç îãîëîø. © Ï. Â. Êåðåêåøà, 2003 Çì³ñò Ïåðåäìîâà ................................................................... 16 ÒÅÌÀ 1. Ä²ÉÑÍ² ×ÈÑËÀ 1.1. Ìíîæèíè òà ñèìâîëè..................................... 17 1.1.1. Ìíîæèíè òà îñíîâí³ ïîçíà÷åííÿ .................... 17 1.1.2. Ëîã³÷í³ ñèìâîëè ............................................ 19 1.2. Ìíîæèíà ä³éñíèõ ÷èñåë ................................. 20 1.2.1. Íàòóðàëüí³ ÷èñëà òà ä³¿ íàä íèìè ................ 20 1.2.2. Ö³ë³ ÷èñëà..................................................... 21 1.2.3. Ðàö³îíàëüí³ ÷èñëà .......................................... 22 1.2.4. ²ððàö³îíàëüí³ ÷èñëà ....................................... 23 1.2.5. Ä³éñí³ ÷èñëà.................................................. 25 1.2.6. ×èñëîâà â³ñü.................................................. 26 1.2.7. Ìîäóëü ä³éñíîãî ÷èñëà .................................. 28 1.2.8. ×èñëîâ³ ìíîæèíè, ÿê³ íàé÷àñò³øå çóñòð³- ÷àþòüñÿ â ìàòåìàòè÷íîìó àíàë³ç³. Ïîçíà- ÷åííÿ ............................................................ 30 Âïðàâè .................................................................... 31 ÒÅÌÀ 2. ÎÑÍÎÂÈ ÀËÃÅÁÐÈ ÂÅÊÒÎÐ²Â ² ÌÀÒÐÈÖÜ 2.1. Âåêòîðè ......................................................... 32 2.1.1. Îñíîâí³ ïîíÿòòÿ ............................................ 32 2.1.2. Îïåðàö³¿ íàä âåêòîðàìè ................................. 33 2.1.3. Ë³í³éíèé âåêòîðíèé ïðîñò³ð .......................... 34 2.1.4. Ë³í³éíà íåçàëåæí³ñòü âåêòîð³â ....................... 34 2.1.5. Áàçèñ ë³í³éíîãî âåêòîðíîãî ïðîñòîðó .............. 36 2.1.6. Ñêàëÿðíèé äîáóòîê äâîõ âåêòîð³â .................. 37 2.1.7. Ïðîñò³ð òîâàð³â, âåêòîð ö³í òà âàðò³ñòü íàáîðó òîâàð³â ............................................... 38 Âïðàâè .................................................................... 40 2.2. Ìàòðèö³ ³ ¿õ âèäè ......................................... 42 2.2.1. Ïðîáëåìí³ çàäà÷³ ........................................... 42 2.2.2. Îçíà÷åííÿ ..................................................... 43 2.2.3. Ä³¿ íàä ìàòðèöÿìè ........................................ 46 2.3. Âèçíà÷íèê, ì³íîðè ³ àëãåáðà¿÷í³ äîïîâíåííÿ ............................................................ 51 2.3.1. Îçíà÷åííÿ ..................................................... 51 2.3.2. Âëàñòèâîñò³ âèçíà÷íèê³â ................................ 54 3
2.4. Ðàíã ìàòðèö³ ................................................. 58 2.4.1. Îçíà÷åííÿ ..................................................... 58 2.5. Îáåðíåíà ìàòðèöÿ .......................................... 62 2.5.1. Îçíà÷åííÿ. Òåîðåìà ïðî îáåðíåíó ìàòðèöþ................................................................. 62 2.5.2. Àëãîðèòì îá÷èñëåííÿ îáåðíåíî¿ ìàòðèö³ ....... 63 2.5.3. Âëàñòèâîñò³ íåâèðîäæåíèõ ìàòðèöü ................ 64 Âïðàâè .................................................................... 65 ÒÅÌÀ 3. ÑÈÑÒÅÌÈ Ë²Í²ÉÍÈÕ ÀËÃÅÁÐÀ¯×ÍÈÕ Ð²ÂÍßÍÜ 3.1. Ïî÷àòêîâ³ çíàííÿ ïðî ñèñòåìè àëãåáðà¿÷íèõ ð³âíÿíü ................................................... 68 3.1.1. Çîáðàæåííÿ ÑËÀÐ ......................................... 68 3.1.2. Îñíîâí³ ïîíÿòòÿ ³ îçíà÷åííÿ ......................... 69 3.2. Ñèñòåìà n ë³í³éíèõ àëãåáðà¿÷íèõ ð³âíÿíü ç n íåâ³äîìèìè ............................................. 71 3.2.1. Ñèñòåìà äâîõ ð³âíÿíü ç äâîìà íåâ³äîìèìè ................................................................. 71 3.3. Ìåòîäè ðîçâ’ÿçóâàííÿ ñèñòåìè n àëãåáðà- ¿÷íèõ ð³âíÿíü ç n íåâ³äîìèìè ....................... 72 3.3.1. Ìàòðè÷íèé ìåòîä........................................... 72 3.3.2. Ìåòîä âèçíà÷íèê³â (ïðàâèëî Êðàìåðà) ........... 73 3.4. Ñèñòåìa m ë³í³éíèõ àëãåáðà¿÷íèõ ð³âíÿíü ç n íåâ³äîìèìè ............................................. 75 3.5. Ìåòîä Ãàóññà ................................................ 76 3.5.1. Ïðîáëåìè ...................................................... 76 3.5.2. Àëãîðèòì ìåòîäó Ãàóññà................................ 77 3.6. Ñèñòåìà ë³í³éíèõ îäíîð³äíèõ àëãåáðà¿÷íèõ ð³âíÿíü ................................................... 82 3.6.1. Îçíà÷åííÿ ..................................................... 82 3.6.2. Íåòðèâ³àëüí³ ðîçâ’ÿçêè îäíîð³äíî¿ ñèñòåìè ë³í³éíèõ àëãåáðà¿÷íèõ ð³âíÿíü .......... 82 Âïðàâè .................................................................... 83 ÒÅÌÀ 4. Ë²Í²¯ ÍÀ ÏËÎÙÈÍ² ² Ó ÏÐÎÑÒÎÐ² 4.1. Ïðÿìîêóòí³ êîîðäèíàòè íà ïëîùèí³ ³ ¿õ íàéïðîñò³ø³ çàñòîñóâàííÿ ........................ 87 4.1.1. Îçíà÷åííÿ ..................................................... 87 4.1.2. Â³äñòàíü ì³æ äâîìà òî÷êàìè íà ïëîùèí³ ...... 89 4.1.3. Ä³ëåííÿ â³äð³çêà â äàíîìó â³äíîøåíí³ ........... 90 4.1.4. Ïëîùà òðèêóòíèêà ........................................ 92 Âïðàâè .................................................................... 95 4.2. Ìåòîä êîîðäèíàò íà ïëîùèí³ ......................... 95 4.2.1. Ë³í³ÿ ÿê ìíîæèíà òî÷îê ............................... 96 4.2.2. Ð³âíÿííÿ ë³í³¿ íà ïëîùèí³ ............................ 97 4.2.3. Ïîáóäîâà ë³í³¿ çà ¿¿ ð³âíÿííÿì ................... 101 4.2.4. Äåÿê³ åëåìåíòàðí³ çàäà÷³ ............................. 102 Âïðàâè .................................................................. 104 4.3. Ð³âíÿííÿ ë³í³¿ ïåðøîãî ïîðÿäêó ................. 105 4.3.1. Ð³âíÿííÿ ïðÿìî¿ ç êóòîâèì êîåô³ö³ºíòîì ... 105 4.3.2. Êóò ì³æ äâîìà ïðÿìèìè ............................. 108 4.3.3. Ð³âíÿííÿ ïðÿìî¿, ùî ïðîõîäèòü ÷åðåç äàíó òî÷êó â çàäàíîìó íàïðÿì³ ................... 111 4.3.4. Ð³âíÿííÿ ïðÿìî¿, ùî ïðîõîäèòü ÷åðåç äâ³ äàí³ òî÷êè ................................................... 113 4.3.5. Âçàºìíå ðîçòàøóâàííÿ äâîõ ïðÿìèõ ............ 115 4.3.6. Â³äñòàíü â³ä òî÷êè äî ïðÿìî¿ ...................... 117 Âïðàâè .................................................................. 119 4.4. Ë³í³¿ äðóãîãî ïîðÿäêó ................................. 122 4.4.1. Åë³ïñ ........................................................... 123 4.4.2. Ã³ïåðáîëà..................................................... 125 4.4.3. Ïàðàáîëà ..................................................... 126 Âïðàâè .................................................................. 127 4.5. Ïîíÿòòÿ ïðî ð³âíÿííÿ ïëîùèíè ³ ïðÿìî¿ ó ïðîñòîð³ ................................................... 128 4.5.1. Çàãàëüíå ð³âíÿííÿ ïëîùèíè ........................ 128 4.5.2. Ïðÿìà ë³í³ÿ ó ïðîñòîð³ ............................... 130 4.6. Áþäæåòíà ìíîæèíà .................................... 130 4.6.1. Ïðîáëåìíèé ïðèêëàä ................................... 130 4.6.2. Ïîíÿòòÿ ³ îçíà÷åííÿ ................................... 132 Âïðàâè .................................................................. 133 ÒÅÌÀ 5. ÃÐÀÍÈÖ² ×ÈÑËÎÂÈÕ ÏÎÑË²ÄÎÂÍÎÑÒÅÉ 5.1. ×èñëîâ³ ïîñë³äîâíîñò³ òà àðèôìåòè÷í³ ä³¿ íàä íèìè..................................................... 134 5.1.1. Îçíà÷åííÿ ³ ïðèêëàäè ................................. 134 5.1.2. Àðèôìåòè÷í³ îïåðàö³¿ íàä ïîñë³äîâíîñòÿìè .......................................................... 135 5.1.3. Îáìåæåí³ òà íåîáìåæåí³ ïîñë³äîâíîñò³ ......... 135 4 5
Page 1: Ì²Í²ÑÒÅÐÑÒÂÎ ÎÑÂ²Ò
Page 5 and 6: 7.9. Ïîõ³äí³ âèùèõ ïî
Page 7 and 8: 10.2. Ãðàíèöÿ ³ íåïåð
Page 9 and 10: Ïåðåäìîâà Ç óñ³õ ñî
Page 11 and 12: äåòüñÿ” âèêîðèñòî
Page 13 and 14: 1 îòðèìàºìî äð³á m = 2
Page 15 and 16: 1.2.7. Ìîäóëü ä³éñíîã
Page 17 and 18: Òåìà 2 Îñíîâè àëãåá
Page 19 and 20: Öÿ ñèñòåìà ð³âíÿíü
Page 21 and 22: 2 0 . Ó ÿêîñò³ åëåìåí
Page 23 and 24: Äîâ³ëüíèé åëåìåíò
Page 25 and 26: Ïðèêëàä 2.2.2. Çíàéòè
Page 27 and 28: a a ... a a ... a a a ... a a ... a
Page 29 and 30: Òîä³ a 12 12 ∆ 1 =∆+ m =∆
Page 31 and 32: Äâ³ ìàòðèö³ À ³ Â íà
Page 33 and 34: 3 0 . Îá÷èñëèòè àëãåá
Page 35 and 36: ÒÅÌÀ 3 ÑÈÑÒÅÌÈ Ë²Í²
Page 37 and 38: Ïðè ðîçâ’ÿçàíí³ ñè
Page 39 and 40: äå æa11 a12 ... a ö 1n a21 a22
Page 41 and 42: Ðîçâ’ÿçàííÿ. Øëÿõî
Page 43 and 44: 3.3. Êîðèñòóþ÷èñü ïð
Page 45 and 46: Äëÿ äàíî¿ òî÷êè Ì â
Page 47 and 48: Îòæå, êîîðäèíàòè òî
Page 49 and 50: ìèì îáìåæåííÿì, îáó
Page 51 and 52: Ïðèêëàä 4.2.6. Ñêëàñò
Page 53 and 54:
Ïðèêëàä 4.2.14. Çíàéòè
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ð³âíÿííÿìè ïåðøîãî
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ßêùî k çàäàíå ÷èñëî,
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3) ∆ =0 ³ ∆ x = 0, ∆ y = 0.
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ÿêèé íàðîäèâñÿ â ëþ
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º àíàë³òè÷íå ð³âíÿ
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4.5. ÏÎÍßÒÒß ÏÐÎ Ð²ÂÍ
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Ç ðîçãëÿíóòîãî ïðè
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⎧1 ⎫ Ïðèêëàä 5.1.10. Ï
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5.1.5. Îñíîâí³ òåîðåì
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Îòðèìàëè, ùî ð³çíèö
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Â òàê³é çàãàëüí³é ï
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n n 1 ⎛⎛1+ 5⎞ ⎛1− 5⎞
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1. Àíàë³òè÷íèé ñïîñ
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òâ³, ïðè òåõí³÷íèõ
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4. Äëÿ ïîáóäîâè ãðàô
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Ðîçâ’ÿçàííÿ. Íåõàé
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Öÿ ôóíêö³ÿ ÿâëÿº ñî
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Çã³äíî ç îçíà÷åííÿ
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sin x- 0 < x- 0
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Îçíà÷åííÿ 6.3.5 Ôóíêö
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6.3.6. Ãëîáàëüí³ âëàñ
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Ñïðàâåäëèâ³ñòü ôîð
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6.23. Îá÷èñëèòè ãðàíè
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Çà ïåð³îä ÷àñó â³ä t
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Ïðèêëàä 7.3.2. Ïîêàçà
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Â ÿêîñò³ ïðèêëàäà ð
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Òåîðåìà 7.6.2 (ïðî äèô
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Òàáëèöÿ ïîõ³äíèõ ñ
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7.13. r(ϕ) =ϕ sin ϕ + cos ϕ; î
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7.9.8. Çíàõîäèìî y′: Ò
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Ôóíêö³ÿ ïðîïîçèö³¿
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Îñê³ëüêè â êðèòè÷í
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ïðè ∆x =1 i x=1000: (∆y)⏐x
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Íàñë³äîê 3. ßêùî íà
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7.14.9. = ln x 1/ x = 1/ x 1 − =
Page 125 and 126:
Çàóâàæåííÿ. Ðîçêëà
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Îö³íèìî n ( ) n+ 1 θx n+ 1
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Çâ³äñè ó êð - ó äîò
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Ð î ç â ’ ÿ ç à í í ÿ.
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âè áóäåòå âèâ÷àòè ó
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8.3. ⎛ ∫ ⎜sin x+ ⎝ 3 ⎞ dx
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8.12. ∫ sin( ax + b) dx ; 8.13.
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Ïåðøèé òèï äîäàíê³
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8.5. ²ÍÒÅÃÐÓÂÀÍÍß ÄÅ
Page 143 and 144:
6 4 1 1 1 5 1 3 = ∫t − dt+ ∫t
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[ñ, b] ñåãìåíòà [a, b] ¿
Page 147 and 148:
9.2.1. Îçíà÷åííÿ òà óì
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4. ßêùî ôóíêö³ÿ ó = f(
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9.4. ÎÑÍÎÂÍÀ ÔÎÐÌÓËÀ
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â³äíîñò³ äî ôîðìóë
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Îçíà÷åííÿ 9.6.1. Íåâë
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³ îñê³ëüêè +∞ ∫ 1 ðî
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Ïðèêëàä 9.6.12 (òåîðåò
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Ïðèêëàä 9.7.2. Îá÷èñë
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Îñê³ëüêè ôóíêö³ÿ ó
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Ãðàíèöÿ ö³º¿ ñóìè ï
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Ð î ç â ’ ÿ ç à í í ÿ.
Page 169 and 170:
Òîä³ ìîæíà ïîêàçàò
Page 171 and 172:
4 3 ∆Ψ = 10 − 10 = 9000 . Îò
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ïàäàº ç íàö³îíàëüí
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Çàóâàæåííÿ 2. Àíàëî
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10.2. ÃÐÀÍÈÖß ² ÍÅÏÅÐ
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Öå ð³âíÿííÿ ïðÿìî¿,
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Öåé æàðò³âëèâèé åê
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äàòêîâ³é âàðòîñò³
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Îçíà÷åííÿ 10.4.2 Äèôå
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∂u äå ( i = 1, 2, K , n) ∂x
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Àíàëîã³÷íî ( 0, 0 ) ( 0,
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Çàóâàæèìî, ùî ïîíÿò
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Òîä³: 1) ÿêùî a a a a > 0,
Page 195 and 196:
ÂÏÐÀÂÈ Äîñë³äèòè í
Page 197 and 198:
Òàêèì ÷èíîì, ìè âèç
Page 199 and 200:
Ðèñ. 10.21 Â³äîìèé Îìó
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ßñíî, ùî âàð³àíò ë³
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11.1.3. Ïðî â³ëüíå ïàä
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ñòå, òî ìè çìîæåìî é
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êîîðäèíàò (ðèñ. 11.4).
Page 209 and 210:
ÂÏÐÀÂÈ Ðîçâ’ÿçàòè
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ÂÏÐÀÂÈ Ðîçâ’ÿçàòè
Page 213 and 214:
Äàë³ âàð³þºìî ñòàë
Page 215 and 216:
11.6.2. Ë³í³éí³ îäíîð³
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Òîä³ ( ) ( ) y2 x ≡λy1 x ,
Page 219 and 220:
2x Çàãàëüíèé ðîçâ’ÿ
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y′ ÷.í. = e x (2Ax +Ax 2 ), y
Page 223 and 224:
x = + + + . 2 2 5 y x c1 c2x c3 x
Page 225 and 226:
Ô(t) =7⋅ 10 6 +3⋅ 10 6 e -0,1t
Page 227 and 228:
S n 1 1 1 = + + K + 1⋅2 2⋅ 3 n
Page 229 and 230:
Îñê³ëüêè ðÿä (12.2.7) ç
Page 231 and 232:
Çàóâàæåííÿ. Ðÿäè ç
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íà çàâæäè íå ïîðîæí
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ÂÏÐÀÂÈ Âèçíà÷èòè ³
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12.5.6. Äîâåäåííÿ ôîðì
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êö³þ, êîòðó íàé÷àñò
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Âèðàç âèäó a+bi=z, äå à
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òî z z r (cos ϕ + isin ϕ ) r (c
Page 245 and 246:
13.16. z =− 2+ 2 3i ; 13.17. z =
Page 247 and 248:
14.2.4. Êîðåí³ òà ¿õí³
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y ⎛ π ⎞ x tg α= k , k ; ctg (
Page 251 and 252:
1 2 tg α+ ctg α= = sin αcos α s
Page 253 and 254:
14.4. ÅËÅÌÅÍÒÈ ÂÈÙÎ¯
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14.5. ÃÐÀÔ²ÊÈ ÄÅßÊÈÕ
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14.5.11. y = arctg x (ðèñ. 14.19
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14.7. ÍÀÁËÈÆÅÍÅ ÇÍÀ×Å
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