ÐÐÐÐ¦ÐÐ Â² ÐÐÐ ÐÐÐ

More documents

Recommendations

Info

$C:\Documents and Settings\ÐÐ¸ÐºÑÐ¾$

Ðàö³îíàëüí³ ³ ³ððàö³îíàëüí³ ôóíêö³¿ âõîäÿòü äî á³ëüø çàãàëüíîãî êëàñó — àëãåáðà¿÷íèõ ôóíêö³é. Âñ³ ³íø³ åëåìåíòàðí³ ôóíêö³¿ íàçèâàþòüñÿ òðàíñöåíäåíòíèìè. 6.1.4. Äåÿê³ âàæëèâ³ êëàñè ôóíêö³é Ôóíêö³ÿ f(x) (x∈D(f)), ùî ìàº âëàñòèâ³ñòü f(x) =f(−x), íàçèâàºòüñÿ ïàðíîþ, íàïðèêëàä õ 2 , cos x, à ùî ìàº âëàñòèâ³ñòü f(x) =−f(−x) — íåïàðíîþ, íàïðèêëàä õ 3 , sin x. Áàãàòî ôóíêö³é º í³ ïàðíèìè, í³ íåïàðíèìè, íàïðèêëàä à õ , x . ßêùî ³ñíóº ä³éñíå ÷èñëî Ò > 0, ùî ïðè áóäü-ÿêèõ çíà÷åííÿõ àðãóìåíòó õ∈D(f) ìàº ì³ñöå ð³âí³ñòü f(x) =f(x + kT), (6.1.1) äå k — ö³ëå ÷èñëî, òî ó = f(x) íàçèâàºòüñÿ ïåð³îäè÷íîþ ôóíêö³ºþ, à Ò — ïåð³îäîì. Ðîçãëÿíåìî ôóíêö³þ y = f(x), îáëàñòþ âèçíà÷åííÿ ³ çíà- ÷åíü ÿêî¿ º â³äïîâ³äíî D(f) i E(f). Òîä³ áóäü-ÿêîìó çíà÷åííþ x0 Î Df () â³äïîâ³äàòèìå îäíå çíà÷åííÿ y0 Î Ef (), ÿêå äîð³âíþº f(x 0 ). Òàêèì ÷èíîì, ð³âíÿííÿ y = f(x) ïðè y0 Î Ef () ìàº ïðèíàéìí³ îäèí êîð³íü. ²íøèìè ñëîâàìè, ð³âíÿííÿ (6.1.1) êîæíîìó yÎ Ef () ñòàâèòü ó â³äïîâ³äí³ñòü îäíå àáî ê³ëüêà çíà÷åíü x0 Î Df (). ßêùî ïðè öüîìó êîæíîìó yÎ Ef () â³äïîâ³äàº ò³ëüêè îäíå çíà÷åííÿ x0 Î Df (), òî êàæóòü, ùî íà ìíîæèí³ E(f) çàäàíî ôóíêö³þ x=j ( y) (áóêâà ϕ îçíà÷àº, ùî õàðàêòåðèñòèêà íîâî¿ ôóíêö³¿ ³íøà). Íîâó ôóíêö³þ x = ϕ(y) íàçèâàþòü îáåðíåíîþ ôóíêö³ºþ äî ôóíêö³¿ y = f(x), à y = f(x) ïðÿìîþ ôóíêö³ºþ. Ïðè öüîìó ôóíêö³¿ x = ϕ(y) ³ y = f(x) íàçèâàþòü âçàºìíî îáåðíåíèìè. Íàâåäåí³ âèùå ì³ðêóâàííÿ ïîêàçóþòü, ùî îáëàñòü âèçíà÷åííÿ òà îáëàñòü çíà÷åíü âçàºìíî îáåðíåíèõ ôóíêö³é ì³íÿþòüñÿ ì³æ ñîáîþ. Íàïðèêëàä, f = à õ , D(f) =R, E(f) =R + ; ϕ = log a y, D(ϕ )=R + , E(ϕ )=R. Ãðàô³êè âçàºìíî îáåðíåíèõ ôóíêö³é çá³ãàþòüñÿ. ßêùî äëÿ çðó÷íîñò³ ìè ðîçãëÿíåìî ôóíêö³þ y = ϕ(x), òî ìîæíà ïîêàçàòè, ùî ¿¿ ãðàô³ê ñèìåòðè÷íèé äî ãðàô³êà ïðÿìî¿ ôóíêö³¿ â³äíîñíî á³ñåêòðèñè ïåðøîãî ³ òðåòüîãî êîîðäèíàòíèõ êóò³â. Öå òâåðäæåííÿ íàâîäèòüñÿ â êóðñ³ åëåìåíòàðíî¿ ìàòåìàòèêè. ßêùî äëÿ äâîõ áóäü-ÿêèõ ð³çíèõ çíà÷åíü àðãóìåíòó õ 1 ³ õ 2 , ÿê³ íàëåæàòü îáëàñò³ âèçíà÷åííÿ D(f), ³ç íåð³âíîñò³ x 1 < õ 2 âèïëèâàº: 1) f(x 1 )f(x 2 ) — ñïàäíîþ; 4) f(x 1 ) ≥ f(x 2 ) — íåçðîñòàþ÷îþ. Òàê³ ôóíêö³¿ íàçèâàþòüñÿ ìîíîòîííèìè. Ïðè öüîìó ôóíêö³¿, ÿê³ çàäîâîëüíÿþòü âèìîãàì 1) àáî 3), íàçèâàþòüñÿ ñòðîãî ìîíîòîííèìè. 6.1.5. Ïîáóäîâà ãðàô³ê³â ôóíêö³é ìåòîäîì ïåðåòâîðåíü Âèâ÷åííÿ âèùî¿ ìàòåìàòèêè ïðèïóñêàº äîáð³ çíàííÿ âëàñòèâîñòåé îñíîâíèõ åëåìåíòàðíèõ ôóíêö³é. ßê áóëî âæå ñêàçàíî, äî íèõ â³äíîñÿòüñÿ ñòàëà, ñòåïåíåâà, ïîêàçíèêîâà, ëîãàðèôì³÷íà, òðèãîíîìåòðè÷í³ ³ îáåðíåí³ òðèãîíîìåòðè÷í³ ôóíêö³¿. Ïîâòîðèòè ¿õ âëàñòèâîñò³ ç â³äïîâ³äíîþ ïîáóäîâîþ ãðàô³ê³â ìîæíà ïî øê³ëüíèì ï³äðó÷íèêàì. Ïðè ïîäàëüøîìó âèâ÷åíí³ òåìè, ÿêà ïîâ’ÿçàíà ç ïîáóäîâîþ ãðàô³ê³â ôóíêö³é, âåëüìè êîðèñíèìè áóäóòü áåçïîñåðåäíüî ïåðåâ³ðÿºì³ òàê³ òâåðäæåííÿ: 1. Ãðàô³ê ôóíêö³¿ y = −f(x) ñèìåòðè÷íèé ãðàô³êó ôóíêö³¿ y = f(x) â³äíîñíî â³ñ³ Îõ. 2. Ãðàô³ê ôóíêö³¿ y = f( x) ñï³âïàäàº ç ãðàô³êîì ôóíêö³¿ y = f(x) â òèõ òî÷êàõ, äå f(x) ≥ 0, ³ ñèìåòðè÷íèé éîìó â³äíîñíî â³ñ³ Îx â òî÷êàõ, äå f(x) 0, àáî íà b îäèíèöü ìàñøòàáó âíèç, ÿêùî b
4. Äëÿ ïîáóäîâè ãðàô³êà ôóíêö³¿ y = f(x − a) òðåáà ãðàô³ê ôóíêö³¿ y = f(x) çñóíóòè ïàðàëåëüíî â³ñ³ Îx íà à îäèíèöü 6.1.6. Äåÿê³ ôóíêö³îíàëüí³ çàëåæíîñò³, ÿê³ âèêîðèñòîâóþòüñÿ â åêîíîì³ö³ ìàñøòàáó âïðàâî, ÿêùî à > 0, àáî íà a îäèíèöü ìàñøòàáó âë³âî, ÿêùî à
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 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
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: òâ³, ïðè òåõí³÷íèõ
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

Create successful ePaper yourself

Delete template?

Save as template?

ÐÐÐÐ¦ÐÐ Â² ÐÐÐ ÐÐÐ