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ð³àëüíà òî÷êà ïåðåì³ñòèëàñÿ âçäîâæ îñ³ Îõ ç òî÷êè à â òî÷êó b (ðèñ. 9.5). Òðåáà îá÷èñëèòè ðîáîòó ö³º¿ ñèëè íà øëÿõó (ñåãìåíò³) [à, b]. Ðèñ. 9.5 Â³äîìî, ùî ÿêùî ñèëà ñòàëà (Ð(õ) =Ð = const) ³ ä³º ó íàïðÿìó ïåðåì³ùåííÿ, òî ðîáîòà À äîð³âíþº äîáóòêó ñèëè íà âåëè÷èíó ïåðåì³ùåííÿ: À = Ð(b – à). (9.1.5) Àëå ñèëà çì³ííà, ³ ìè íå ìàºìî ïðàâà êîðèñòóâàòèñÿ ôîðìóëîþ (9.1.5). Òîìó ñïî÷àòêó, ÿê öå áóëî ³ ïðè ðîçâ’ÿçàíí³ çàäà÷³ 9.1.1, çä³éñíèìî ðîçáèòòÿ R ñåãìåíòà [à, b] òî- ÷êàìè à = õ 0 < õ 1 …
9.2.1. Îçíà÷åííÿ òà óìîâè ³ñíóâàííÿ âèçíà÷åíîãî ³íòåãðàëà Íåõàé ôóíêö³ÿ ó = f(õ) âèçíà÷åíà íà cåãìåíò³ [à, b] (à < b). Çä³éñíèìî ðîçáèòòÿ R ñåãìåíòà [à, b] íà n ÷àñòèí äîâ³ëüíî îáðàíèìè òî÷êàìè ä³ëåííÿ: R: à = x 0 < õ 1
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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
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
Page 121 and 122: Íàñë³äîê 3. ßêùî íà
Page 123 and 124: 7.14.9. = ln x 1/ x = 1/ x 1 − =
Page 125 and 126: Çàóâàæåííÿ. Ðîçêëà
Page 127 and 128: 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
Page 135 and 136: 8.3. ⎛ ∫ ⎜sin x+ ⎝ 3 ⎞ dx
Page 137 and 138: 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: [ñ, b] ñåãìåíòà [a, b] ¿
Page 149 and 150: 4. ßêùî ôóíêö³ÿ ó = f(
Page 151 and 152: 9.4. ÎÑÍÎÂÍÀ ÔÎÐÌÓËÀ
Page 153 and 154: â³äíîñò³ äî ôîðìóë
Page 155 and 156: Îçíà÷åííÿ 9.6.1. Íåâë
Page 157 and 158: 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
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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).
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ÂÏÐÀÂÈ Ðîçâ’ÿçàòè
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ÂÏÐÀÂÈ Ðîçâ’ÿçàòè
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Äàë³ âàð³þºìî ñòàë
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11.6.2. Ë³í³éí³ îäíîð³
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Òîä³ ( ) ( ) y2 x ≡λy1 x ,
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2x Çàãàëüíèé ðîçâ’ÿ
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y′ ÷.í. = e x (2Ax +Ax 2 ), y
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x = + + + . 2 2 5 y x c1 c2x c3 x
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Ô(t) =7⋅ 10 6 +3⋅ 10 6 e -0,1t
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S n 1 1 1 = + + K + 1⋅2 2⋅ 3 n
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Îñê³ëüêè ðÿä (12.2.7) ç
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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
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13.16. z =− 2+ 2 3i ; 13.17. z =
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14.2.4. Êîðåí³ òà ¿õí³
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y ⎛ π ⎞ x tg α= k , k ; ctg (
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1 2 tg α+ ctg α= = sin αcos α s
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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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Ê36 Êåðåêåøà Ï. Â. Ëå
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