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Íàâ÷àëüíå âèäàííÿ ÊÅÐÅÊÅØÀ Ïåòðî Âîëîäèìèðîâè÷ ËÅÊÖ²¯ ² ÂÏÐÀÂÈ ç âèùî¿ ìàòåìàòèêè Äëÿ ñòóäåíò³â åêîíîì³÷íèõ ñïåö³àëüíîñòåé Çàâ. ðåäàêö³ºþ Ò. Ì. Çàáàíîâà Ðåäàêòîð Æ. Á. Ìåëüíè÷åíêî Òåõí³÷í³ ðåäàêòîðè Ð. Ì. Êó÷èíñüêà, Ã. Î. Êóêëºâà Êîðåêòîð Í. ². Êðèëîâà Çäàíî ó âèðîáíèöòâî 11.04.2002. Ï³äïèñàíî äî äðóêó 06.05.2003. Ôîðìàò 60×84/16. Ïàï³ð îôñåòíèé. Ãàðí³òóðà SchoolBook. Óì. äðóê. àðê. 30,23. Òèðàæ 500 ïðèì. Çàì. ¹ 308. Âèäàâíèöòâî ³ äðóêàðíÿ «Àñòðîïðèíò» (Ñâ³äîöòâî ÄÊ ¹ 132 â³ä 28.07.2000 ð.) 65026, ì. Îäåñà, âóë. Ïðåîáðàæåíñüêà, 24, Òåë./ôàêñ: (0482) 26-98-82, 26-96-82, 37-14-25 www.astroprint.odessa.ua
Ê36 Êåðåêåøà Ï. Â. Ëåêö³¿ ³ âïðàâè ç âèùî¿ ìàòåìàòèêè: Äëÿ ñòóäåíò³â åêîíîì³÷íèõ ñïåö³àëüíîñòåé: Íàâ÷àëüíèé ïîñ³áíèê. — Îäåñà: Àñòðîïðèíò, 2003. — 520 ñ. ISBN 966-549-871-1. Íàâ÷àëüíèé ïîñ³áíèê íàïèñàíî 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. 160200000—054 Ê 549 — 2003 Áåç îãîëîø. ÁÁÊ 22.1ÿ73 ÓÄÊ 51(075.8)
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Ì²Í²ÑÒÅÐÑÒÂÎ ÎÑÂ²Ò
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2.4. Ðàíã ìàòðèö³ ......
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7.9. Ïîõ³äí³ âèùèõ ïî
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10.2. Ãðàíèöÿ ³ íåïåð
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Ïåðåäìîâà Ç óñ³õ ñî
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äåòüñÿ” âèêîðèñòî
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1 îòðèìàºìî äð³á m = 2
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1.2.7. Ìîäóëü ä³éñíîã
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Òåìà 2 Îñíîâè àëãåá
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Öÿ ñèñòåìà ð³âíÿíü
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2 0 . Ó ÿêîñò³ åëåìåí
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Äîâ³ëüíèé åëåìåíò
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Ïðèêëàä 2.2.2. Çíàéòè
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a a ... a a ... a a a ... a a ... a
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Òîä³ a 12 12 ∆ 1 =∆+ m =∆
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Äâ³ ìàòðèö³ À ³ Â íà
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3 0 . Îá÷èñëèòè àëãåá
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ÒÅÌÀ 3 ÑÈÑÒÅÌÈ Ë²Í²
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Ïðè ðîçâ’ÿçàíí³ ñè
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äå æa11 a12 ... a ö 1n a21 a22
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Ðîçâ’ÿçàííÿ. Øëÿõî
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3.3. Êîðèñòóþ÷èñü ïð
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Äëÿ äàíî¿ òî÷êè Ì â
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Îòæå, êîîðäèíàòè òî
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ìèì îáìåæåííÿì, îáó
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Ïðèêëàä 4.2.6. Ñêëàñò
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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 − =
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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
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6 4 1 1 1 5 1 3 = ∫t − dt+ ∫t
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[ñ, b] ñåãìåíòà [a, b] ¿
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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.
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Òîä³ ìîæíà ïîêàçàò
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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,
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ÂÏÐÀÂÈ Äîñë³äèòè í
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Òàêèì ÷èíîì, ìè âèç
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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
Page 211 and 212: ÂÏÐÀÂÈ Ðîçâ’ÿçàòè
Page 213 and 214: Äàë³ âàð³þºìî ñòàë
Page 215 and 216: 11.6.2. Ë³í³éí³ îäíîð³
Page 217 and 218: Òîä³ ( ) ( ) y2 x ≡λy1 x ,
Page 219 and 220: 2x Çàãàëüíèé ðîçâ’ÿ
Page 221 and 222: 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
Page 233 and 234: íà çàâæäè íå ïîðîæí
Page 235 and 236: ÂÏÐÀÂÈ Âèçíà÷èòè ³
Page 237 and 238: 12.5.6. Äîâåäåííÿ ôîðì
Page 239 and 240: êö³þ, êîòðó íàé÷àñò
Page 241 and 242: Âèðàç âèäó a+bi=z, äå à
Page 243 and 244: 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
Page 249 and 250: 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¯
Page 255 and 256: 14.5. ÃÐÀÔ²ÊÈ ÄÅßÊÈÕ
Page 257 and 258: 14.5.11. y = arctg x (ðèñ. 14.19
Page 259: 14.7. ÍÀÁËÈÆÅÍÅ ÇÍÀ×Å
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