ÐÐÐÐ¦ÐÐ Â² ÐÐÐ ÐÐÐ

More documents

Recommendations

Info

$C:\Documents and Settings\ÐÐ¸ÐºÑÐ¾$

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 Â — 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. Ð³çíèöåþ 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
Page 1 and 2: Ì²Í²ÑÒÅÐÑÒÂÎ ÎÑÂ²Ò
Page 3 and 4: 2.4. Ðàíã ìàòðèö³ ......
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: 1.2.7. Ìîäóëü ä³éñíîã
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
Page 71 and 72:
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
Page 91 and 92:
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
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 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
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
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
show all

ÐÐÐÐ¦ÐÐ Â² ÐÐÐ ÐÐÐ

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?

ÐÐÐÐ¦ÐÐ Â² ÐÐÐ ÐÐÐ