1.2.8. ×èñëîâ³ ìíîæèíè, ÿê³ íàé÷àñò³øå çóñòð³÷àþòüñÿ â ìàòåìàòè÷íîìó àíàë³ç³. Ïîçíà÷åííÿ ×èñëà 1, 2, 3,…,n,… íàçèâàþòüñÿ íàòóðàëüíèìè ÷èñëàìè. Ö³ ÷èñëà óòâîðþþòü ìíîæèíó, ÿêà ïîçíà÷àºòüñÿ ÷åðåç N. ×èñëà 0,±1,±2,… íàçèâàþòüñÿ ö³ëèìè ÷èñëàìè. Âîíè óòâîðþþòü ìíîæèíó, ÿêà ïîçíà÷àºòüñÿ ÷åðåç Z. ×èñëà âèäó m , m ∈Z, n ∈ N íàçèâàþòüñÿ ðàö³îíàëüíèìè. Ìíîæèíà âñ³õ n ðàö³îíàëüíèõ ÷èñåë ïîçíà÷àºòüñÿ ÷åðåç Q. Ìíîæèíó âñ³õ ³ððàö³îíàëüíèõ ÷èñåë ïîçíà÷èìî ÷åðåç I. Îá’ºäíàííÿ ðàö³îíàëüíèõ é ³ððàö³îíàëüíèõ ÷èñåë óòâîðþº ìíîæèíó ä³éñíèõ ÷èñåë. Öÿ ìíîæèíà ïîçíà÷àºòüñÿ ÷åðåç R. Îòæå, R = Q∪I . Ðîçãëÿíåìî òåïåð ìíîæèíè ÷èñåë, ÿê³ º ï³äìíîæèíàìè ìíîæèíè R. ×åðåç { x∈R a≤ x≤b} ïîçíà÷èìî ìíîæèíó ä³éñíèõ ÷èñåë, ÿê³ çàäîâîëüíÿþòü íåð³âíîñò³ a ≤ x ≤ b. Òóò à i b – äâà ä³éñíèõ ÷èñëà (äâ³ òî÷êè ÷èñëîâî¿ îñ³), ïðè÷îìó à
Òåìà 2 Îñíîâè àëãåáðè âåêòîð³â ³ ìàòðèöü 2.1. ÂÅÊÒÎÐÈ 2.1.1. Îñíîâí³ ïîíÿòòÿ Ó øê³ëüíîìó êóðñ³ ìàòåìàòèêè âåêòîð âèçíà÷åíèé ÿê uuur íàïðÿìëåíèé â³äð³çîê AB , ó ÿêîìó òî÷êà À ðîçãëÿäàºòüñÿ ÿê ïî÷àòîê, à òî÷êà  — ê³íåöü. Âåêòîð ìîæå ïîçíà÷àòèñÿ rrrr r r ³ îäí³ºþ ë³òåðîþ, íàïðèêëàä abcx , , , , X, P . Äîâæèíîþ, àáî uuur ìîäóëåì, âåêòîðà AB íàçèâàþòü äîâæèíó â³äð³çêà AB ³ uuur r ïîçíà÷àþòü AB , a . Ó ïðÿìîêóòí³é ñèñòåì³ êîîðäèíàò, â³äïîâ³äíî íà ïëîùèí³ é ó ïðîñòîð³, âåêòîð uur X ìîæíà çàïèñàòè ó âèãëÿä³ óïîðÿäêîâàíî¿ ïàðè ÷èñåë (õ 1 , õ 2 ) òà óïîðÿäêîâàíî¿ òð³éêè ÷èñåë uur uur (õ 1 , õ 2 , õ 3 ), òîáòî X = ( x 1, x 2) (íà ïëîùèí³) ³ X = ( x 1, x 2, x 3) (ó ïðîñòîð³). Óçàãàëüíèìî òåïåð ïîíÿòòÿ âåêòîðà uur X çà äîïîìîãîþ òàêèõ îçíà÷åíü. Îçíà÷åííÿ 2.1.1. Äîâ³ëüíèé óïîðÿäêîâàíèé íàá³ð x 1 , x 2 ,…, x n ³ç n ä³éñíèõ ÷èñåë íàçèâàºòüñÿ n-âèì³ðíèì âåêòîðîì uur X . ×èñëà x 1 , x 2 ,…, x n , ñêëàäîâ³ íàáîðó, íàçèâàþòüñÿ êîìïîíåíòàìè (êîîðäèíàòàìè) âåêòîðà uur X . ×èñëî n íàçèâàþòü ðîçì³ðí³ñòþ âåêòîðà. Îçíà÷åííÿ 2.1.2. Ñóêóïí³ñòü óñ³õ n-âèì³ðíèõ âåêòîð³â íàçèâàºòüñÿ n-âèì³ðíèì âåêòîðíèì ïðîñòîðîì. Êîìïîíåíòè n-âèì³ðíîãî âåêòîðà ìîæíà ðîçòàøîâóâàòè â ðÿäîê àáî â ñòîâïåöü. Ó ïåðøîìó âèïàäêó ãîâîðÿòü ïðî âåêòîð-ðÿäîê uur X = ( x1, x2,..., x n ), â äðóãîìó — ïðî âåêòîð-ñòîâïåöü r X ⎛x1 ⎞ ⎜ ⎟ ⎜ x ⎟. ... ⎜ ⎟ ⎝xn ⎠ 2 = ⎜ ⎟ Äâà âåêòîðè ð³âí³ òîä³ é ò³ëüêè òîä³, êîëè ð³âí³ ¿õ êîìïîíåíòè. ßê âèäíî ç îçíà÷åííÿ, ãîâîðèòè ïðî ð³âí³ñòü àáî íåð³âí³ñòü âåêòîð³â ìîæíà ëèøå äëÿ âåêòîð³â îäí³º¿ ðîçì³ðíîñò³. Äîâæèíà (ìîäóëü) âåêòîðà îá÷èñëþºòüñÿ çà ôîðìóëîþ: n ( ) uur 1/ 2 X = x + x + ... + x = ∑ x . 2 2 2 2 1 2 n i= 1 1 2.1.2. Îïåðàö³¿ íàä âåêòîðàìè r r Ñóìîþ äâîõ âåêòîð³â a= ( a1, a2,..., a n ) ³ b= ( b1, b2,..., b n) íàçèâàºòüñÿ òðåò³é âåêòîð c= ( c1, c2,..., c r n ), êîìïîíåíòè ÿêîãî äîð³âíþþòü ñóì³ êîìïîíåíò äîäàíê³â âåêòîð³â: c1 = a1 + b1, c2 = a2 + b2,..., cn = an + b n. гçíèöåþ a r r ³ b º âåêòîð r r r c = a− b= ( a1−b1, a2 −b2,..., an −bn) . Íåõàé ìàºìî âåêòîð a r ³ ä³éñíå ÷èñëî λ. Äîáóòêîì ÷èñëà λ íà âåêòîð a r º âåêòîð b r , ÿêèé âèçíà÷àºòüñÿ çà ïðàâèëîì λ a = ( λa1, λa2,..., λa n), ïðè÷îìó äîâæèíà éîãî äîð³âíþº r λ a r . Äëÿ áóäü-ÿêèõ âåêòîð³â ³ ÷èñåë ñïðàâåäëèâî: r r r r 1. a+ b = b+ a — çàêîí ïåðåì³ùåííÿ. r r r r r r 2. ( a+ b) + c = a+ ( b+ c ) — çàêîí ñïîëó÷åííÿ (ïîºäíàííÿ). r r r r 3. a+ 0 = a(0 = (0,...,0)). 4. Äëÿ áóäü-ÿêîãî a r r r r ³ñíóº a ' òàêèé, ùî a+ a′ = 0. 32 33
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Ç ðîçãëÿíóòîãî ïðè
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Îòðèìàëè, ùî ð³çíèö
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 òàê³é çàãàëüí³é ï
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Ñïðàâåäëèâ³ñòü ôîð
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Çà ïåð³îä ÷àñó â³ä t
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 ÿêîñò³ ïðèêëàäà ð
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Òàáëèöÿ ïîõ³äíèõ ñ
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Ôóíêö³ÿ ïðîïîçèö³¿
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Îñê³ëüêè â êðèòè÷í
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âè áóäåòå âèâ÷àòè ó
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Ïåðøèé òèï äîäàíê³
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[ñ, b] ñåãìåíòà [a, b] ¿
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4. ßêùî ôóíêö³ÿ ó = f(
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â³äíîñò³ äî ôîðìóë
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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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ïàäຠç íàö³îíàëüí
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Çàóâàæåííÿ 2. Àíàëî
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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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ÂÏÐÀÂÈ Äîñë³äèòè í
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Ðèñ. 10.21 ³äîìèé Îìó
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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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2x Çàãàëüíèé ðîçâ’ÿ
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Îñê³ëüêè ðÿä (12.2.7) ç
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Çàóâàæåííÿ. Ðÿäè ç
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êö³þ, êîòðó íàé÷àñò
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Âèðàç âèäó a+bi=z, äå à
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y ⎛ π ⎞ x tg α= k , k ; ctg (
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14.5. ÃÐÀÔ²ÊÈ ÄÅßÊÈÕ
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Ê36 Êåðåêåøà Ï. Â. Ëå