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8.5. ²íòåãðóâàííÿ äåÿêèõ ³ððàö³îíàëüíèõ ôóíêö³é ....................................................... 280 8.5.1. ²íòåãðàëè âèäó Rxx ( , α β ∫ , x, K ) dx ..................... 280 8.5.2. ²íòåãðàëè ç ðàäèêàëàìè ............................... 280 Âïðàâè .................................................................. 281 8.6. ²íòåãðóâàííÿ âèðàç³â ç òðèãîíîìåòðè÷íèìè ôóíêö³ÿìè ............................................. 282 Âïðàâè .................................................................. 284 ÒÅÌÀ 9. ÂÈÇÍÀ×ÅÍÈÉ ²ÍÒÅÃÐÀË 9.1. Çàäà÷³, ÿê³ ïðèâîäÿòü äî ïîíÿòòÿ âèçíà- ÷åíîãî ³íòåãðàëà .......................................... 286 9.1.1. Çàäà÷à ïðî ïëîùó êðèâîë³í³éíî¿ òðàïåö³¿ ....................................................... 286 9.1.2. Çàäà÷à ïðî ðîáîòó çì³ííî¿ ñèëè .................. 289 9.1.3. Çàäà÷à ïðî îáñÿã ïðîäóêö³¿ .......................... 291 9.2. Îçíà÷åííÿ âèçíà÷åíîãî ³íòåãðàëà ................. 291 9.2.1. Îçíà÷åííÿ òà óìîâè ³ñíóâàííÿ âèçíà- ÷åíîãî ³íòåãðàëà .......................................... 292 9.2.2. Ãåîìåòðè÷íèé, ìåõàí³÷íèé ³ åêîíîì³÷íèé çì³ñò âèçíà÷åíîãî ³íòåãðàëà .................. 293 9.3. Âëàñòèâîñò³ âèçíà÷åíîãî ³íòåãðàëà ................ 295 9.4. Îñíîâíà ôîðìóëà ³íòåãðàëüíîãî ÷èñëåííÿ ..... 300 9.4.1. ²íòåãðàë ³ç çì³ííîþ âåðõíüîþ ìåæåþ .......... 300 9.4.2. Ôîðìóëà Íüþòîíà — Ëåéáí³öà .................... 301 9.5. Îñíîâí³ ìåòîäè îá÷èñëåííÿ âèçíà÷åíèõ ³íòåãðàë³â .................................................... 302 9.5.1. Áåïîñåðåäí³é ìåòîä ...................................... 302 9.5.2. Ìåòîä çàì³íè çì³ííî¿, àáî ï³äñòàíîâêè ......... 303 9.5.3. Ìåòîä ³íòåãðóâàííÿ ÷àñòèíàìè ..................... 306 Âïðàâè .................................................................. 307 9.6. Íåâëàñí³ ³íòåãðàëè ....................................... 307 9.6.1. Íåâëàñí³ ³íòåãðàëè ² ðîäó ........................... 307 9.6.2. Íåâëàñí³ ³íòåãðàëè II ðîäó .......................... 312 Âïðàâè .................................................................. 317 9.7. Ãåîìåòðè÷í³ çàñòîñóâàííÿ âèçíà÷åíîãî ³íòåãðàëà ..................................................... 317 9.7.1. Îá÷èñëåííÿ ïëîù ô³ãóð ............................... 317 Âïðàâè .................................................................. 323 9.7.2. Îá÷èñëåííÿ äîâæèíè äóã êðèâèõ ë³í³é .......................................................... 323 Âïðàâè .................................................................. 327 9.8. Îá÷èñëåííÿ îá’ºì³â ò³ë .............................. 327 9.8.1. Îá÷èñëåííÿ îá’ºì³â ò³ë çà â³äîìèìè ïëîùàìè ïåðåð³ç³â ..................................... 327 9.8.2. Îá’ºì ò³ëà îáåðòàííÿ ................................ 329 Âïðàâè .................................................................. 330 9.9. Ïëîùà ïîâåðõí³ ò³ëà îáåðòàííÿ .................. 330 9.9.1. Âèïàäîê îáåðòàííÿ êðèâî¿, ÿêà çàäàíà ð³âíÿííÿì y = f(x), x∈ [a, b] ........................ 330 9.9.2. Âèïàäîê îáåðòàííÿ êðèâî¿, ÿêà çàäàíà ïàðàìåòðè÷íî ............................................. 331 Âïðàâè .................................................................. 332 9.10. Ô³çè÷í³ çàñòîñóâàííÿ âèçíà÷åíîãî ³íòåãðàëà.................................................... 332 9.10.1. Ðîáîòà çì³ííî¿ ñèëè .................................. 332 9.10.2. Îá÷èñëåííÿ ïðîéäåíîãî øëÿõó ................... 334 9.11. Çàñòîñóâàííÿ âèçíà÷åíîãî ³íòåãðàëà â åêîíîì³ö³ ................................................ 335 9.11.1. Çàñòîñóâàííÿ âèçíà÷åíîãî ³íòåãðàëà â äèíàì³÷íèõ ïðîöåñàõ .............................. 335 9.11.2. Çàñòîñóâàííÿ âèçíà÷åíîãî ³íòåãðàëà â çàäà÷àõ ðåàë³çàö³¿ òîâàð³â ....................... 337 9.11.3. Çàñòîñóâàííÿ âèçíà÷åíîãî ³íòåãðàëà â çàäà÷àõ îá÷èñëåííÿ âèòðàò, äîõîä³â òà ïðèáóòê³â .............................................. 339 9.12. Íàáëèæåíå îá÷èñëåííÿ âèçíà÷åíèõ ³íòåãðàë³â .................................................. 340 9.12.1. Ôîðìóëà ïðÿìîêóòíèê³â ............................. 340 9.12.2. Ôîðìóëà òðàïåö³é ...................................... 341 ÒÅÌÀ 10. ÔÓÍÊÖ²¯ ÁÀÃÀÒÜÎÕ ÇÌ²ÍÍÈÕ 10.1. Îñíîâí³ îçíà÷åííÿ ³ ïîíÿòòÿ ...................... 343 10.1.1. Ïðîáëåìí³ ïðèêëàäè .................................. 343 10.1.2. Îçíà÷åííÿ ôóíêö³¿ áàãàòüîõ çì³ííèõ.......... 344 10.1.3. Ôóíêö³ÿ äâîõ çì³ííèõ ³ ¿¿ îáëàñòü âèçíà÷åííÿ................................................. 345 Âïðàâè .................................................................. 351 10 11
10.2. Ãðàíèöÿ ³ íåïåðåðâí³ñòü ôóíêö³¿ äâîõ çì³ííèõ............................................. 352 10.2.1. Ãðàíèöÿ ôóíêö³¿ äâîõ çì³ííèõ .................. 352 10.3. ×àñòèíí³ ïîõ³äí³ ........................................ 356 10.3.1. ×àñòèíí³ ïîõ³äí³ ïåðøîãî ïîðÿäêó............. 356 Âïðàâè .................................................................. 360 10.3.2. Ïîõ³äí³ âèùèõ ïîðÿäê³â ............................ 360 10.3.3. Ãåîìåòðè÷íèé òà åêîíîì³÷íèé çì³ñò ÷àñòèííèõ ïîõ³äíèõ ................................... 362 10.4. Äèôåðåíö³éîâí³ñòü ôóíêö³¿ ......................... 366 10.4.1. Äèôåðåíö³éîâí³ñòü â òî÷ö³ ......................... 366 10.4.2. Äèôåðåíö³àë ôóíêö³¿ .................................. 367 10.4.3. Ïîõ³äíà çà íàïðÿìîì ................................. 368 10.5. Ãðàä³ºíò ôóíêö³¿ ....................................... 372 10.5.1. Ïîíÿòòÿ ïðî ñêàëÿðí³ òà âåêòîðí³ ïîëÿ .......................................................... 372 10.5.2. Ãðàä³ºíò ôóíêö³¿ ........................................ 373 Âïðàâè .................................................................. 378 10.6. Åêñòðåìàëüí³ çíà÷åííÿ ôóíêö³¿ áàãàòüîõ çì³ííèõ ..................................................... 378 10.6.1. Åêñòðåìóì ôóíêö³¿ .................................... 378 10.6.2. Íàéá³ëüø³ òà íàéìåíø³ çíà÷åííÿ ôóíêö³¿ ...................................................... 385 Âïðàâè .................................................................. 388 10.7. Óìîâíèé åêñòðåìóì .................................... 388 10.7.1. Îñíîâí³ ïîíÿòòÿ ³ îçíà÷åííÿ...................... 388 Âïðàâè .................................................................. 392 10.8. Ïîíÿòòÿ ïðî åìï³ðè÷í³ ôîðìóëè ................ 392 10.8.1. Ìåòîä íàéìåíøèõ êâàäðàò³â ó âèïàäêó ìîæëèâî¿ ë³í³éíî¿ çàëåæíîñò³ çì³ííèõ x òà y ....................................................... 396 10.8.2. Ìåòîä íàéìåíøèõ êâàäðàò³â ìîæëèâî¿ ñòåïåíåâî¿ çàëåæíîñò³ x òà y ................... 399 ÒÅÌÀ 11. ÄÈÔÅÐÅÍÖ²ÀËÜÍ² Ð²ÂÍßÍÍß 11.1. Çàäà÷³ ïðèðîäîçíàâñòâà, ùî ïðèâîäÿòü äî äèôåðåíö³àëüíèõ ð³âíÿíü ....................... 402 11.1.1. Íàéïðîñò³øà ìîäåëü íàêîïè÷åííÿ êàï³òàëó..................................................... 402 11.1.2. Ïðî ðîçïàä ðàä³þ ...................................... 403 11.1.3. Ïðî â³ëüíå ïàä³ííÿ ìàòåð³àëüíî¿ òî÷êè ......................................................... 404 11.2. Îcíîâí³ ïîíÿòòÿ òà îçíà÷åííÿ..................... 404 11.2.1. Îçíà÷åííÿ .................................................. 404 11.3. Ïðèêëàäè äèôåðåíö³àëüíèõ ð³âíÿíü, ðîçâ’ÿçêè ÿêèõ çíàõîäÿòüñÿ â ÿâíîìó âèãëÿä³ çà äîïîìîãîþ åëåìåíòàðíèõ ïðèéîì³â ......................................................... 405 11.4. Äèôåðåíö³àëüí³ ð³âíÿííÿ ïåðøîãî ïîðÿäêó. Òåîðåìà Êîø³ ...................................... 409 11.4.1. Ïîëå íàïðÿì³â (³çîêë³íè) ........................... 411 11.4.2. Êëàñè íåë³í³éíèõ äèôåðåíö³àëüíèõ ð³âíÿíü ïåðøîãî ïîðÿäêó, ÿê³ ðîçâ’ÿçóþòüñÿ ó êâàäðàòóðàõ ............................................ 413 Âïðàâè .................................................................. 416 Âïðàâè .................................................................. 420 11.5. Ë³í³éí³ äèôåðåíö³àëüí³ ð³âíÿííÿ ïåðøîãî ïîðÿäêó ..................................................... 420 11.5.1. Îçíà÷åííÿ .................................................. 420 Âïðàâè .................................................................. 426 11.6. Ë³í³éí³ äèôåðåíö³àëüí³ ð³âíÿííÿ äðóãîãî ïîðÿäêó ..................................................... 427 11.6.1. Îñíîâí³ ïîíÿòòÿ ³ îçíà÷åííÿ...................... 427 11.6.2. Ë³í³éí³ îäíîð³äí³ äèôåðåíö³àëüí³ ð³âíÿííÿ äðóãîãî ïîðÿäêó ................................... 428 11.6.3. Ë³í³éíî çàëåæí³ òà íåçàëåæí³ ôóíêö³¿ íà ³íòåðâàë³ (a, b) ...................................... 429 11.7. Ë³í³éí³ äèôåðåíö³àëüí³ ð³âíÿííÿ äðóãîãî ïîðÿäêó ç³ ñòàëèìè êîåô³ö³ºíòàìè ............. 434 11.7.1. Îäíîð³äí³ ð³âíÿííÿ .................................... 434 Âïðàâè .................................................................. 436 11.7.2. Íåîäíîð³äí³ ð³âíÿííÿ ................................. 437 Âïðàâè .................................................................. 442 11.8. Äèôåðåíö³àëüí³ ð³âíÿííÿ âèùèõ ïîðÿäê³â, ùî äîïóñêàþòü çíèæåííÿ ïîðÿäêó ....... 443 11.8.1. Ð³âíÿííÿ n-ãî ïîðÿäêó âèäó y (n) = f(x) ........ 443 11.8.2. Ð³âíÿííÿ äðóãîãî ïîðÿäêó.......................... 443 12 13
Page 1 and 2: Ì²Í²ÑÒÅÐÑÒÂÎ ÎÑÂ²Ò
Page 3 and 4: 2.4. Ðàíã ìàòðèö³ ......
Page 5: 7.9. Ïîõ³äí³ âèùèõ ïî
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
Page 55 and 56: ð³âíÿííÿìè ïåðøîãî
Page 57 and 58:
ßêùî k çàäàíå ÷èñëî,
Page 59 and 60:
3) ∆ =0 ³ ∆ x = 0, ∆ y = 0.
Page 61 and 62:
ÿêèé íàðîäèâñÿ â ëþ
Page 63 and 64:
º àíàë³òè÷íå ð³âíÿ
Page 65 and 66:
4.5. ÏÎÍßÒÒß ÏÐÎ Ð²ÂÍ
Page 67 and 68:
Ç ðîçãëÿíóòîãî ïðè
Page 69 and 70:
⎧1 ⎫ Ïðèêëàä 5.1.10. Ï
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5.1.5. Îñíîâí³ òåîðåì
Page 73 and 74:
Îòðèìàëè, ùî ð³çíèö
Page 75 and 76:
Â òàê³é çàãàëüí³é ï
Page 77 and 78:
n n 1 ⎛⎛1+ 5⎞ ⎛1− 5⎞
Page 79 and 80:
1. Àíàë³òè÷íèé ñïîñ
Page 81 and 82:
òâ³, ïðè òåõí³÷íèõ
Page 83 and 84:
4. Äëÿ ïîáóäîâè ãðàô
Page 85 and 86:
Ðîçâ’ÿçàííÿ. Íåõàé
Page 87 and 88:
Öÿ ôóíêö³ÿ ÿâëÿº ñî
Page 89 and 90:
Çã³äíî ç îçíà÷åííÿ
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sin x- 0 < x- 0
Page 93 and 94:
Îçíà÷åííÿ 6.3.5 Ôóíêö
Page 95 and 96:
6.3.6. Ãëîáàëüí³ âëàñ
Page 97 and 98:
Ñïðàâåäëèâ³ñòü ôîð
Page 99 and 100:
6.23. Îá÷èñëèòè ãðàíè
Page 101 and 102:
Çà ïåð³îä ÷àñó â³ä t
Page 103 and 104:
Ïðèêëàä 7.3.2. Ïîêàçà
Page 105 and 106:
Â ÿêîñò³ ïðèêëàäà ð
Page 107 and 108:
Òåîðåìà 7.6.2 (ïðî äèô
Page 109 and 110:
Òàáëèöÿ ïîõ³äíèõ ñ
Page 111 and 112:
7.13. r(ϕ) =ϕ sin ϕ + cos ϕ; î
Page 113 and 114:
7.9.8. Çíàõîäèìî y′: Ò
Page 115 and 116:
Ôóíêö³ÿ ïðîïîçèö³¿
Page 117 and 118:
Îñê³ëüêè â êðèòè÷í
Page 119 and 120:
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
Page 129 and 130:
Çâ³äñè ó êð - ó äîò
Page 131 and 132:
Ð î ç â ’ ÿ ç à í í ÿ.
Page 133 and 134:
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.
Page 139 and 140:
Ïåðøèé òèï äîäàíê³
Page 141 and 142:
8.5. ²ÍÒÅÃÐÓÂÀÍÍß ÄÅ
Page 143 and 144:
6 4 1 1 1 5 1 3 = ∫t − dt+ ∫t
Page 145 and 146:
[ñ, b] ñåãìåíòà [a, b] ¿
Page 147 and 148:
9.2.1. Îçíà÷åííÿ òà óì
Page 149 and 150:
4. ßêùî ôóíêö³ÿ ó = f(
Page 151 and 152:
9.4. ÎÑÍÎÂÍÀ ÔÎÐÌÓËÀ
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â³äíîñò³ äî ôîðìóë
Page 155 and 156:
Îçíà÷åííÿ 9.6.1. Íåâë
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³ îñê³ëüêè +∞ ∫ 1 ðî
Page 159 and 160:
Ïðèêëàä 9.6.12 (òåîðåò
Page 161 and 162:
Ïðèêëàä 9.7.2. Îá÷èñë
Page 163 and 164:
Îñê³ëüêè ôóíêö³ÿ ó
Page 165 and 166:
Ãðàíèöÿ ö³º¿ ñóìè ï
Page 167 and 168:
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
Page 173 and 174:
ïàäàº ç íàö³îíàëüí
Page 175 and 176:
Çàóâàæåííÿ 2. Àíàëî
Page 177 and 178:
10.2. ÃÐÀÍÈÖß ² ÍÅÏÅÐ
Page 179 and 180:
Öå ð³âíÿííÿ ïðÿìî¿,
Page 181 and 182:
Öåé æàðò³âëèâèé åê
Page 183 and 184:
äàòêîâ³é âàðòîñò³
Page 185 and 186:
Îçíà÷åííÿ 10.4.2 Äèôå
Page 187 and 188:
∂u äå ( i = 1, 2, K , n) ∂x
Page 189 and 190:
Àíàëîã³÷íî ( 0, 0 ) ( 0,
Page 191 and 192:
Çàóâàæèìî, ùî ïîíÿò
Page 193 and 194:
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
Page 201 and 202:
ßñíî, ùî âàð³àíò ë³
Page 203 and 204:
11.1.3. Ïðî â³ëüíå ïàä
Page 205 and 206:
ñòå, òî ìè çìîæåìî é
Page 207 and 208:
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 and 260:
14.7. ÍÀÁËÈÆÅÍÅ ÇÍÀ×Å
Page 261:
Ê36 Êåðåêåøà Ï. Â. Ëå
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