Маслова Т.Н., Суходский А.М. Справочник школьника по математике. 5-11 кл
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136. Ñæàòèå è ðàñòÿæåíèå ãðàôèêîâ
òðèãîíîìåòðè÷åñêèõ ôóíêöèé ................. 179
137. Ãðàôèê ãàðìîíè÷åñêîãî êîëåáàíèÿ
ó = Asin (wx + a) .................................... 180
Ðàçäåë IV. Óðàâíåíèÿ è ñèñòåìû óðàâíåíèé
§ 14. Óðàâíåíèÿ ñ îäíîé ïåðåìåííîé ................. 182
138. Îïðåäåëåíèå óðàâíåíèÿ.
Êîðíè óðàâíåíèÿ .................................... 182
139. Ðàâíîñèëüíîñòü óðàâíåíèé ...................... 182
140. Ëèíåéíûå óðàâíåíèÿ .............................. 183
141. Êâàäðàòíûå óðàâíåíèÿ ........................... 184
142. Íåïîëíûå êâàäðàòíûå óðàâíåíèÿ ............ 186
143. Òåîðåìà Âèåòà ....................................... 187
144. Ñèñòåìû è ñîâîêóïíîñòè óðàâíåíèé ......... 188
145. Óðàâíåíèÿ, ñîäåðæàùèå ïåðåìåííóþ
ïîä çíàêîì ìîäóëÿ ................................ 189
146. Ïîíÿòèå ñëåäñòâèÿ óðàâíåíèÿ.
Ïîñòîðîííèå êîðíè ................................. 190
147. Óðàâíåíèÿ ñ ïåðåìåííîé â çíàìåíàòåëå .... 191
148. Îáëàñòü îïðåäåëåíèÿ óðàâíåíèÿ .............. 192
149. Ðàöèîíàëüíûå óðàâíåíèÿ ........................ 194
150. Ðåøåíèå óðàâíåíèÿ p(x) = 0 ìåòîäîì
ðàçëîæåíèÿ åãî ëåâîé ÷àñòè
íà ìíîæèòåëè ........................................ 195
151. Ðåøåíèå óðàâíåíèé ìåòîäîì ââåäåíèÿ
íîâîé ïåðåìåííîé .................................. 196
152. Áèêâàäðàòíûå óðàâíåíèÿ ........................ 197
153. Óðàâíåíèÿ âûñøèõ ñòåïåíåé ................... 197
154. Ðåøåíèå çàäà÷ ñ ïîìîùüþ óðàâíåíèé ....... 199
155. Èððàöèîíàëüíûå óðàâíåíèÿ. ................... 202
156. Ïîêàçàòåëüíûå óðàâíåíèÿ....................... 204
157. Ëîãàðèôìè÷åñêèå óðàâíåíèÿ................... 205
158. Ïîêàçàòåëüíî-ëîãàðèôìè÷åñêèå
óðàâíåíèÿ .............................................. 207
159. Ïðîñòåéøèå òðèãîíîìåòðè÷åñêèå
óðàâíåíèÿ .............................................. 207
160. Ðåøåíèå òðèãîíîìåòðè÷åñêèõ óðàâíåíèé
ìåòîäîì ðàçëîæåíèÿ íà ìíîæèòåëè ......... 210
161. Ðåøåíèå òðèãîíîìåòðè÷åñêèõ óðàâíåíèé
ìåòîäîì ââåäåíèÿ íîâîé ïåðåìåííîé ....... 212
162. Îäíîðîäíûå òðèãîíîìåòðè÷åñêèå
óðàâíåíèÿ ............................................. 213
163. Óíèâåðñàëüíàÿ ïîäñòàíîâêà. ................... 216
164. Ìåòîä ââåäåíèÿ âñïîìîãàòåëüíîãî
àðãóìåíòà ............................................. 218
165. Ãðàôè÷åñêîå ðåøåíèå óðàâíåíèé ............. 219
166. Óðàâíåíèÿ ñ ïàðàìåòðîì ......................... 221
§ 15. Óðàâíåíèÿ ñ äâóìÿ ïåðåìåííûìè .......... 223
167. Ðåøåíèå óðàâíåíèÿ
ñ äâóìÿ ïåðåìåííûìè ............................ 223
168. Ãðàôèê óðàâíåíèÿ
ñ äâóìÿ ïåðåìåííûìè ............................ 224
169. Ëèíåéíîå óðàâíåíèå ñ äâóìÿ
ïåðåìåííûìè è åãî ãðàôèê ..................... 224
§ 16. Ñèñòåìû óðàâíåíèé ............................... 225
170. Ñèñòåìû äâóõ óðàâíåíèé ñ äâóìÿ
ïåðåìåííûìè. Ðàâíîñèëüíûå ñèñòåìû ..... 225
Ñîäåðæàíèå
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136. Ñæàòèå è ðàñòÿæåíèå ãðàôèêîâ
òðèãîíîìåòðè÷åñêèõ ôóíêöèé ................. 179
137. Ãðàôèê ãàðìîíè÷åñêîãî êîëåáàíèÿ
ó = Asin (wx + a) .................................... 180
Ðàçäåë IV. Óðàâíåíèÿ è ñèñòåìû óðàâíåíèé
§ 14. Óðàâíåíèÿ ñ îäíîé ïåðåìåííîé ................. 182
138. Îïðåäåëåíèå óðàâíåíèÿ.
Êîðíè óðàâíåíèÿ .................................... 182
139. Ðàâíîñèëüíîñòü óðàâíåíèé ...................... 182
140. Ëèíåéíûå óðàâíåíèÿ .............................. 183
141. Êâàäðàòíûå óðàâíåíèÿ ........................... 184
142. Íåïîëíûå êâàäðàòíûå óðàâíåíèÿ ............ 186
143. Òåîðåìà Âèåòà ....................................... 187
144. Ñèñòåìû è ñîâîêóïíîñòè óðàâíåíèé ......... 188
145. Óðàâíåíèÿ, ñîäåðæàùèå ïåðåìåííóþ
ïîä çíàêîì ìîäóëÿ ................................ 189
146. Ïîíÿòèå ñëåäñòâèÿ óðàâíåíèÿ.
Ïîñòîðîííèå êîðíè ................................. 190
147. Óðàâíåíèÿ ñ ïåðåìåííîé â çíàìåíàòåëå .... 191
148. Îáëàñòü îïðåäåëåíèÿ óðàâíåíèÿ .............. 192
149. Ðàöèîíàëüíûå óðàâíåíèÿ ........................ 194
150. Ðåøåíèå óðàâíåíèÿ p(x) = 0 ìåòîäîì
ðàçëîæåíèÿ åãî ëåâîé ÷àñòè
íà ìíîæèòåëè ........................................ 195
151. Ðåøåíèå óðàâíåíèé ìåòîäîì ââåäåíèÿ
íîâîé ïåðåìåííîé .................................. 196
152. Áèêâàäðàòíûå óðàâíåíèÿ ........................ 197
153. Óðàâíåíèÿ âûñøèõ ñòåïåíåé ................... 197
154. Ðåøåíèå çàäà÷ ñ ïîìîùüþ óðàâíåíèé ....... 199
155. Èððàöèîíàëüíûå óðàâíåíèÿ. ................... 202
156. Ïîêàçàòåëüíûå óðàâíåíèÿ....................... 204
157. Ëîãàðèôìè÷åñêèå óðàâíåíèÿ................... 205
158. Ïîêàçàòåëüíî-ëîãàðèôìè÷åñêèå
óðàâíåíèÿ .............................................. 207
159. Ïðîñòåéøèå òðèãîíîìåòðè÷åñêèå
óðàâíåíèÿ .............................................. 207
160. Ðåøåíèå òðèãîíîìåòðè÷åñêèõ óðàâíåíèé
ìåòîäîì ðàçëîæåíèÿ íà ìíîæèòåëè ......... 210
161. Ðåøåíèå òðèãîíîìåòðè÷åñêèõ óðàâíåíèé
ìåòîäîì ââåäåíèÿ íîâîé ïåðåìåííîé ....... 212
162. Îäíîðîäíûå òðèãîíîìåòðè÷åñêèå
óðàâíåíèÿ ............................................. 213
163. Óíèâåðñàëüíàÿ ïîäñòàíîâêà. ................... 216
164. Ìåòîä ââåäåíèÿ âñïîìîãàòåëüíîãî
àðãóìåíòà ............................................. 218
165. Ãðàôè÷åñêîå ðåøåíèå óðàâíåíèé ............. 219
166. Óðàâíåíèÿ ñ ïàðàìåòðîì ......................... 221
§ 15. Óðàâíåíèÿ ñ äâóìÿ ïåðåìåííûìè .......... 223
167. Ðåøåíèå óðàâíåíèÿ
ñ äâóìÿ ïåðåìåííûìè ............................ 223
168. Ãðàôèê óðàâíåíèÿ
ñ äâóìÿ ïåðåìåííûìè ............................ 224
169. Ëèíåéíîå óðàâíåíèå ñ äâóìÿ
ïåðåìåííûìè è åãî ãðàôèê ..................... 224
§ 16. Ñèñòåìû óðàâíåíèé ............................... 225
170. Ñèñòåìû äâóõ óðàâíåíèé ñ äâóìÿ
ïåðåìåííûìè. Ðàâíîñèëüíûå ñèñòåìû ..... 225
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171. Ðåøåíèå ñèñòåì äâóõ óðàâíåíèé
ñ äâóìÿ ïåðåìåííûìè ìåòîäîì
ïîäñòàíîâêè .......................................... 226
172. Ðåøåíèå ñèñòåì äâóõ óðàâíåíèé
ñ äâóìÿ ïåðåìåííûìè ìåòîäîì
ñëîæåíèÿ .............................................. 227
173. Ðåøåíèå ñèñòåì äâóõ óðàâíåíèé
ñ äâóìÿ ïåðåìåííûìè ìåòîäîì ââåäåíèÿ
íîâûõ ïåðåìåííûõ ................................. 228
174. Îïðåäåëèòåëè âòîðîãî ïîðÿäêà.
Èññëåäîâàíèå ñèñòåì äâóõ ëèíåéíûõ
óðàâíåíèé ñ äâóìÿ ïåðåìåííûìè ............ 230
175. Ñèììåòðè÷åñêèå ñèñòåìû ....................... 233
176. Ãðàôè÷åñêîå ðåøåíèå ñèñòåì äâóõ
óðàâíåíèé ñ äâóìÿ ïåðåìåííûìè ............ 234
177. Ñèñòåìû òðåõ óðàâíåíèé ñ òðåìÿ
ïåðåìåííûìè ......................................... 235
178. Îïðåäåëèòåëè òðåòüåãî ïîðÿäêà.
Èññëåäîâàíèå ñèñòåì òðåõ ëèíåéíûõ
óðàâíåíèé ñ òðåìÿ ïåðåìåííûìè ............. 238
179. Ñèñòåìû ïîêàçàòåëüíûõ
è ëîãàðèôìè÷åñêèõ óðàâíåíèé ............... 242
180. Ñèñòåìû òðèãîíîìåòðè÷åñêèõ
óðàâíåíèé ............................................. 243
Ðàçäåë V. Íåðàâåíñòâà
§ 17. Ðåøåíèå íåðàâåíñòâ .............................. 245
181. Îñíîâíûå ïîíÿòèÿ, ñâÿçàííûå
ñ ðåøåíèåì íåðàâåíñòâ ........................... 245
182. Ãðàôè÷åñêîå ðåøåíèå íåðàâåíñòâ
ñ îäíîé ïåðåìåííîé ................................ 246
183. Ëèíåéíûå íåðàâåíñòâà ñ îäíîé
ïåðåìåííîé ............................................ 247
184. Ñèñòåìû íåðàâåíñòâ
ñ îäíîé ïåðåìåííîé ................................ 248
185. Ñîâîêóïíîñòè íåðàâåíñòâ
ñ îäíîé ïåðåìåííîé ................................ 249
186. Äðîáíî-ëèíåéíûå íåðàâåíñòâà ................. 250
187. Íåðàâåíñòâà âòîðîé ñòåïåíè .................... 250
188. Ãðàôè÷åñêîå ðåøåíèå
íåðàâåíñòâ âòîðîé ñòåïåíè ...................... 252
189. Íåðàâåíñòâà ñ ìîäóëÿìè ......................... 254
190. Ðåøåíèå ðàöèîíàëüíûõ íåðàâåíñòâ
ìåòîäîì ïðîìåæóòêîâ ............................ 257
191. Ïîêàçàòåëüíûå íåðàâåíñòâà .................... 260
192. Ëîãàðèôìè÷åñêèå íåðàâåíñòâà ................ 261
193. Èððàöèîíàëüíûå íåðàâåíñòâà. ................. 263
194. Òðèãîíîìåòðè÷åñêèå íåðàâåíñòâà ............. 264
195. Íåðàâåíñòâà è ñèñòåìû íåðàâåíñòâ
ñ äâóìÿ ïåðåìåííûìè............................. 265
§ 18. Äîêàçàòåëüñòâî íåðàâåíñòâ .................... 266
196. Ìåòîä îöåíêè çíàêà ðàçíîñòè .................. 266
197. Ñèíòåòè÷åñêèé ìåòîä äîêàçàòåëüñòâà
íåðàâåíñòâ ............................................. 268
198. Äîêàçàòåëüñòâî íåðàâåíñòâ ìåòîäîì
îò ïðîòèâíîãî ........................................ 269
199. Èñïîëüçîâàíèå íåðàâåíñòâ
ïðè ðåøåíèè óðàâíåíèé.......................... 270
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171. Ðåøåíèå ñèñòåì äâóõ óðàâíåíèé
ñ äâóìÿ ïåðåìåííûìè ìåòîäîì
ïîäñòàíîâêè .......................................... 226
172. Ðåøåíèå ñèñòåì äâóõ óðàâíåíèé
ñ äâóìÿ ïåðåìåííûìè ìåòîäîì
ñëîæåíèÿ .............................................. 227
173. Ðåøåíèå ñèñòåì äâóõ óðàâíåíèé
ñ äâóìÿ ïåðåìåííûìè ìåòîäîì ââåäåíèÿ
íîâûõ ïåðåìåííûõ ................................. 228
174. Îïðåäåëèòåëè âòîðîãî ïîðÿäêà.
Èññëåäîâàíèå ñèñòåì äâóõ ëèíåéíûõ
óðàâíåíèé ñ äâóìÿ ïåðåìåííûìè ............ 230
175. Ñèììåòðè÷åñêèå ñèñòåìû ....................... 233
176. Ãðàôè÷åñêîå ðåøåíèå ñèñòåì äâóõ
óðàâíåíèé ñ äâóìÿ ïåðåìåííûìè ............ 234
177. Ñèñòåìû òðåõ óðàâíåíèé ñ òðåìÿ
ïåðåìåííûìè ......................................... 235
178. Îïðåäåëèòåëè òðåòüåãî ïîðÿäêà.
Èññëåäîâàíèå ñèñòåì òðåõ ëèíåéíûõ
óðàâíåíèé ñ òðåìÿ ïåðåìåííûìè ............. 238
179. Ñèñòåìû ïîêàçàòåëüíûõ
è ëîãàðèôìè÷åñêèõ óðàâíåíèé ............... 242
180. Ñèñòåìû òðèãîíîìåòðè÷åñêèõ
óðàâíåíèé ............................................. 243
Ðàçäåë V. Íåðàâåíñòâà
§ 17. Ðåøåíèå íåðàâåíñòâ .............................. 245
181. Îñíîâíûå ïîíÿòèÿ, ñâÿçàííûå
ñ ðåøåíèåì íåðàâåíñòâ ........................... 245
182. Ãðàôè÷åñêîå ðåøåíèå íåðàâåíñòâ
ñ îäíîé ïåðåìåííîé ................................ 246
183. Ëèíåéíûå íåðàâåíñòâà ñ îäíîé
ïåðåìåííîé ............................................ 247
184. Ñèñòåìû íåðàâåíñòâ
ñ îäíîé ïåðåìåííîé ................................ 248
185. Ñîâîêóïíîñòè íåðàâåíñòâ
ñ îäíîé ïåðåìåííîé ................................ 249
186. Äðîáíî-ëèíåéíûå íåðàâåíñòâà ................. 250
187. Íåðàâåíñòâà âòîðîé ñòåïåíè .................... 250
188. Ãðàôè÷åñêîå ðåøåíèå
íåðàâåíñòâ âòîðîé ñòåïåíè ...................... 252
189. Íåðàâåíñòâà ñ ìîäóëÿìè ......................... 254
190. Ðåøåíèå ðàöèîíàëüíûõ íåðàâåíñòâ
ìåòîäîì ïðîìåæóòêîâ ............................ 257
191. Ïîêàçàòåëüíûå íåðàâåíñòâà .................... 260
192. Ëîãàðèôìè÷åñêèå íåðàâåíñòâà ................ 261
193. Èððàöèîíàëüíûå íåðàâåíñòâà. ................. 263
194. Òðèãîíîìåòðè÷åñêèå íåðàâåíñòâà ............. 264
195. Íåðàâåíñòâà è ñèñòåìû íåðàâåíñòâ
ñ äâóìÿ ïåðåìåííûìè............................. 265
§ 18. Äîêàçàòåëüñòâî íåðàâåíñòâ .................... 266
196. Ìåòîä îöåíêè çíàêà ðàçíîñòè .................. 266
197. Ñèíòåòè÷åñêèé ìåòîä äîêàçàòåëüñòâà
íåðàâåíñòâ ............................................. 268
198. Äîêàçàòåëüñòâî íåðàâåíñòâ ìåòîäîì
îò ïðîòèâíîãî ........................................ 269
199. Èñïîëüçîâàíèå íåðàâåíñòâ
ïðè ðåøåíèè óðàâíåíèé.......................... 270
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Ðàçäåë VI. Ýëåìåíòû êîìáèíàòîðèêè
§ 19. Ðàçìåùåíèÿ, ïåðåñòàíîâêè, ñî÷åòàíèÿ ... 272
200. Ðàçìåùåíèÿ .......................................... 272
201. Ïåðåñòàíîâêè ........................................ 273
202. Ñî÷åòàíèÿ è èõ ñâîéñòâà.
Òðåóãîëüíèê Ïàñêàëÿ ............................. 275
§ 20. Ôîðìóëà áèíîìà Íüþòîíà ..................... 278
203. Áèíîì Íüþòîíà ..................................... 278
204. Ñâîéñòâà ôîðìóëû áèíîìà Íüþòîíà ......... 280
Ðàçäåë VII. Ýëåìåíòû ìàòåìàòè÷åñêîãî àíàëèçà
§ 21. ×èñëîâûå ïîñëåäîâàòåëüíîñòè................ 283
205. Îïðåäåëåíèå ïîñëåäîâàòåëüíîñòè ............. 283
206. Ñïîñîáû çàäàíèÿ ïîñëåäîâàòåëüíîñòè ....... 283
207. Âîçðàñòàþùèå è óáûâàþùèå
ïîñëåäîâàòåëüíîñòè ................................. 284
208. Îïðåäåëåíèå àðèôìåòè÷åñêîé
ïðîãðåññèè ............................................ 285
209. Ñâîéñòâà àðèôìåòè÷åñêîé
ïðîãðåññèè ............................................ 286
210. Îïðåäåëåíèå ãåîìåòðè÷åñêîé
ïðîãðåññèè ............................................ 287
211. Ñâîéñòâà ãåîìåòðè÷åñêîé ïðîãðåññèè ....... 288
212. Ïîíÿòèå î ïðåäåëå ïîñëåäîâàòåëüíîñòè .... 291
213. Âû÷èñëåíèå ïðåäåëîâ
ïîñëåäîâàòåëüíîñòåé ............................... 292
214. Ñóììà áåñêîíå÷íîé ãåîìåòðè÷åñêîé
ïðîãðåññèè ïðè |q| < 1 ............................. 294
§ 22. Ïðåäåë ôóíêöèè ................................... 296
215. Ïðåäåë ôóíêöèè y = f(x) ïðè x ® ×.
Ãîðèçîíòàëüíàÿ àñèìïòîòà ...................... 296
216. Âû÷èñëåíèå ïðåäåëîâ ôóíêöèé
ïðè
.x ¥®
............................................ 298
217. Ïðåäåë ôóíêöèè â òî÷êå.
Íåïðåðûâíûå ôóíêöèè ........................... 299
218. Âåðòèêàëüíàÿ àñèìïòîòà ........................ 301
219. Âû÷èñëåíèå ïðåäåëà ôóíêöèè â òî÷êå...... 302
§ 23. Ïðîèçâîäíàÿ ......................................... 305
220. Ïðèðàùåíèå àðãóìåíòà.
Ïðèðàùåíèå ôóíêöèè ............................. 305
221. Îïðåäåëåíèå ïðîèçâîäíîé ....................... 306
222. Ôîðìóëû äèôôåðåíöèðîâàíèÿ.
Òàáëèöà ïðîèçâîäíûõ ............................. 309
223. Äèôôåðåíöèðîâàíèå ñóììû,
ïðîèçâåäåíèÿ, ÷àñòíîãî ........................... 310
224. Ñëîæíàÿ ôóíêöèÿ
è åå äèôôåðåíöèðîâàíèå ......................... 312
225. Ôèçè÷åñêèé ñìûñë ïðîèçâîäíîé .............. 314
226. Âòîðàÿ ïðîèçâîäíàÿ
è åå ôèçè÷åñêèé ñìûñë ........................... 315
227. Êàñàòåëüíàÿ ê ãðàôèêó ôóíêöèè ............. 316
228. Ôîðìóëà Ëàãðàíæà ................................ 319
§ 24 Ïðèìåíåíèÿ ïðîèçâîäíîé ...................... 319
229. Ïðèáëèæåííûå âû÷èñëåíèÿ ñ ïîìîùüþ
ïðîèçâîäíîé .......................................... 319
230. Äèôôåðåíöèàë ....................................... 321
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Ðàçäåë VI. Ýëåìåíòû êîìáèíàòîðèêè
§ 19. Ðàçìåùåíèÿ, ïåðåñòàíîâêè, ñî÷åòàíèÿ ... 272
200. Ðàçìåùåíèÿ .......................................... 272
201. Ïåðåñòàíîâêè ........................................ 273
202. Ñî÷åòàíèÿ è èõ ñâîéñòâà.
Òðåóãîëüíèê Ïàñêàëÿ ............................. 275
§ 20. Ôîðìóëà áèíîìà Íüþòîíà ..................... 278
203. Áèíîì Íüþòîíà ..................................... 278
204. Ñâîéñòâà ôîðìóëû áèíîìà Íüþòîíà ......... 280
Ðàçäåë VII. Ýëåìåíòû ìàòåìàòè÷åñêîãî àíàëèçà
§ 21. ×èñëîâûå ïîñëåäîâàòåëüíîñòè................ 283
205. Îïðåäåëåíèå ïîñëåäîâàòåëüíîñòè ............. 283
206. Ñïîñîáû çàäàíèÿ ïîñëåäîâàòåëüíîñòè ....... 283
207. Âîçðàñòàþùèå è óáûâàþùèå
ïîñëåäîâàòåëüíîñòè ................................. 284
208. Îïðåäåëåíèå àðèôìåòè÷åñêîé
ïðîãðåññèè ............................................ 285
209. Ñâîéñòâà àðèôìåòè÷åñêîé
ïðîãðåññèè ............................................ 286
210. Îïðåäåëåíèå ãåîìåòðè÷åñêîé
ïðîãðåññèè ............................................ 287
211. Ñâîéñòâà ãåîìåòðè÷åñêîé ïðîãðåññèè ....... 288
212. Ïîíÿòèå î ïðåäåëå ïîñëåäîâàòåëüíîñòè .... 291
213. Âû÷èñëåíèå ïðåäåëîâ
ïîñëåäîâàòåëüíîñòåé ............................... 292
214. Ñóììà áåñêîíå÷íîé ãåîìåòðè÷åñêîé
ïðîãðåññèè ïðè |q| < 1 ............................. 294
§ 22. Ïðåäåë ôóíêöèè ................................... 296
215. Ïðåäåë ôóíêöèè y = f(x) ïðè x ® ×.
Ãîðèçîíòàëüíàÿ àñèìïòîòà ...................... 296
216. Âû÷èñëåíèå ïðåäåëîâ ôóíêöèé
ïðè
.x ¥®
............................................ 298
217. Ïðåäåë ôóíêöèè â òî÷êå.
Íåïðåðûâíûå ôóíêöèè ........................... 299
218. Âåðòèêàëüíàÿ àñèìïòîòà ........................ 301
219. Âû÷èñëåíèå ïðåäåëà ôóíêöèè â òî÷êå...... 302
§ 23. Ïðîèçâîäíàÿ ......................................... 305
220. Ïðèðàùåíèå àðãóìåíòà.
Ïðèðàùåíèå ôóíêöèè ............................. 305
221. Îïðåäåëåíèå ïðîèçâîäíîé ....................... 306
222. Ôîðìóëû äèôôåðåíöèðîâàíèÿ.
Òàáëèöà ïðîèçâîäíûõ ............................. 309
223. Äèôôåðåíöèðîâàíèå ñóììû,
ïðîèçâåäåíèÿ, ÷àñòíîãî ........................... 310
224. Ñëîæíàÿ ôóíêöèÿ
è åå äèôôåðåíöèðîâàíèå ......................... 312
225. Ôèçè÷åñêèé ñìûñë ïðîèçâîäíîé .............. 314
226. Âòîðàÿ ïðîèçâîäíàÿ
è åå ôèçè÷åñêèé ñìûñë ........................... 315
227. Êàñàòåëüíàÿ ê ãðàôèêó ôóíêöèè ............. 316
228. Ôîðìóëà Ëàãðàíæà ................................ 319
§ 24 Ïðèìåíåíèÿ ïðîèçâîäíîé ...................... 319
229. Ïðèáëèæåííûå âû÷èñëåíèÿ ñ ïîìîùüþ
ïðîèçâîäíîé .......................................... 319
230. Äèôôåðåíöèàë ....................................... 321
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231. Ïðèìåíåíèå ïðîèçâîäíîé
ê èññëåäîâàíèþ ôóíêöèé íà âîçðàñòàíèå
(óáûâàíèå) ............................................. 322
232. Ïðèìåíåíèå ïðîèçâîäíîé
ê èññëåäîâàíèþ ôóíêöèé íà ýêñòðåìóì .... 324
233. Îòûñêàíèå íàèáîëüøåãî è íàèìåíüøåãî
çíà÷åíèé íåïðåðûâíîé ôóíêöèè
íà îòðåçêå.............................................. 328
234. Îòûñêàíèå íàèáîëüøåãî
èëè íàèìåíüøåãî çíà÷åíèÿ íåïðåðûâíîé
ôóíêöèè íà íåçàìêíóòîì ïðîìåæóòêå ..... 330
235. Çàäà÷è íà îòûñêàíèå íàèáîëüøèõ
èëè íàèìåíüøèõ çíà÷åíèé âåëè÷èí ......... 331
236. Ïðèìåíåíèå ïðîèçâîäíîé
äëÿ äîêàçàòåëüñòâà òîæäåñòâ ................... 333
237. Ïðèìåíåíèå ïðîèçâîäíîé
äëÿ äîêàçàòåëüñòâà íåðàâåíñòâ ................ 334
238. Îáùàÿ ñõåìà ïîñòðîåíèÿ ãðàôèêà
ôóíêöèè ............................................... 334
§ 25. Ïåðâîîáðàçíàÿ è èíòåãðàë ..................... 338
239. Ïåðâîîáðàçíàÿ ....................................... 338
240. Òàáëèöà ïåðâîîáðàçíûõ ........................... 339
241. Ïðàâèëà âû÷èñëåíèÿ ïåðâîîáðàçíûõ........ 340
242. Èíòåãðàë ............................................... 341
243. Ñâÿçü ìåæäó èíòåãðàëîì è ïåðâîîáðàçíîé
(ôîðìóëà Íüþòîíà Ëåéáíèöà) ............. 344
244. Ïðàâèëà âû÷èñëåíèÿ èíòåãðàëîâ ............. 345
245. Èñïîëüçîâàíèå èíòåãðàëà
äëÿ âû÷èñëåíèÿ ïëîùàäåé ïëîñêèõ ôèãóð 346
246. Âû÷èñëåíèå îáúåìîâ òåë ñ ïîìîùüþ
èíòåãðàëà .............................................. 351
247. Ôèçè÷åñêèå ïðèëîæåíèÿ èíòåãðàëà ......... 353
§ 26. Ïîíÿòèå î äèôôåðåíöèàëüíîì
óðàâíåíèè ............................................. 353
248. Îïðåäåëåíèå äèôôåðåíöèàëüíîãî
óðàâíåíèÿ è åãî ðåøåíèÿ ........................ 353
249. Äèôôåðåíöèàëüíûå óðàâíåíèÿ
ïîêàçàòåëüíîãî ðîñòà è ïîêàçàòåëüíîãî
óáûâàíèÿ .............................................. 355
250. Óðàâíåíèå ãàðìîíè÷åñêèõ êîëåáàíèé....... 357
ÃÅÎÌÅÒÐÈß
Ðàçäåë VIII.
Îñíîâíûå ïîíÿòèÿ ãåîìåòðèè
§ 27. Òî÷êà, ïðÿìàÿ, ïëîñêîñòü.
Ôèãóðû è òåëà ...................................... 359
251. Òî÷êà,ïðÿìàÿ, ëó÷, îòðåçîê.
Óðàâíåíèå ïðÿìîé íà ïëîñêîñòè ............... 359
252. Ïëîñêîñòü. Óðàâíåíèå ïëîñêîñòè
â ïðîñòðàíñòâå. Ôèãóðû è òåëà ................ 362
253. Óãîë ..................................................... 365
254. Ãðàäóñíàÿ è ðàäèàííàÿ ìåðû óãëîâ .......... 367
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231. Ïðèìåíåíèå ïðîèçâîäíîé
ê èññëåäîâàíèþ ôóíêöèé íà âîçðàñòàíèå
(óáûâàíèå) ............................................. 322
232. Ïðèìåíåíèå ïðîèçâîäíîé
ê èññëåäîâàíèþ ôóíêöèé íà ýêñòðåìóì .... 324
233. Îòûñêàíèå íàèáîëüøåãî è íàèìåíüøåãî
çíà÷åíèé íåïðåðûâíîé ôóíêöèè
íà îòðåçêå.............................................. 328
234. Îòûñêàíèå íàèáîëüøåãî
èëè íàèìåíüøåãî çíà÷åíèÿ íåïðåðûâíîé
ôóíêöèè íà íåçàìêíóòîì ïðîìåæóòêå ..... 330
235. Çàäà÷è íà îòûñêàíèå íàèáîëüøèõ
èëè íàèìåíüøèõ çíà÷åíèé âåëè÷èí ......... 331
236. Ïðèìåíåíèå ïðîèçâîäíîé
äëÿ äîêàçàòåëüñòâà òîæäåñòâ ................... 333
237. Ïðèìåíåíèå ïðîèçâîäíîé
äëÿ äîêàçàòåëüñòâà íåðàâåíñòâ ................ 334
238. Îáùàÿ ñõåìà ïîñòðîåíèÿ ãðàôèêà
ôóíêöèè ............................................... 334
§ 25. Ïåðâîîáðàçíàÿ è èíòåãðàë ..................... 338
239. Ïåðâîîáðàçíàÿ ....................................... 338
240. Òàáëèöà ïåðâîîáðàçíûõ ........................... 339
241. Ïðàâèëà âû÷èñëåíèÿ ïåðâîîáðàçíûõ........ 340
242. Èíòåãðàë ............................................... 341
243. Ñâÿçü ìåæäó èíòåãðàëîì è ïåðâîîáðàçíîé
(ôîðìóëà Íüþòîíà Ëåéáíèöà) ............. 344
244. Ïðàâèëà âû÷èñëåíèÿ èíòåãðàëîâ ............. 345
245. Èñïîëüçîâàíèå èíòåãðàëà
äëÿ âû÷èñëåíèÿ ïëîùàäåé ïëîñêèõ ôèãóð 346
246. Âû÷èñëåíèå îáúåìîâ òåë ñ ïîìîùüþ
èíòåãðàëà .............................................. 351
247. Ôèçè÷åñêèå ïðèëîæåíèÿ èíòåãðàëà ......... 353
§ 26. Ïîíÿòèå î äèôôåðåíöèàëüíîì
óðàâíåíèè ............................................. 353
248. Îïðåäåëåíèå äèôôåðåíöèàëüíîãî
óðàâíåíèÿ è åãî ðåøåíèÿ ........................ 353
249. Äèôôåðåíöèàëüíûå óðàâíåíèÿ
ïîêàçàòåëüíîãî ðîñòà è ïîêàçàòåëüíîãî
óáûâàíèÿ .............................................. 355
250. Óðàâíåíèå ãàðìîíè÷åñêèõ êîëåáàíèé....... 357
ÃÅÎÌÅÒÐÈß
Ðàçäåë VIII.
Îñíîâíûå ïîíÿòèÿ ãåîìåòðèè
§ 27. Òî÷êà, ïðÿìàÿ, ïëîñêîñòü.
Ôèãóðû è òåëà ...................................... 359
251. Òî÷êà,ïðÿìàÿ, ëó÷, îòðåçîê.
Óðàâíåíèå ïðÿìîé íà ïëîñêîñòè ............... 359
252. Ïëîñêîñòü. Óðàâíåíèå ïëîñêîñòè
â ïðîñòðàíñòâå. Ôèãóðû è òåëà ................ 362
253. Óãîë ..................................................... 365
254. Ãðàäóñíàÿ è ðàäèàííàÿ ìåðû óãëîâ .......... 367
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255. Ëîìàíàÿ. Ìíîãîóãîëüíèê ........................ 370
256. Ãåîìåòðè÷åñêîå ìåñòî òî÷åê .................... 374
257. Ñèììåòðèÿ ............................................ 375
§ 28. Ïåðïåíäèêóëÿðíûå
è ïàðàëëåëüíûå ïðÿìûå ........................ 378
258. Ïåðïåíäèêóëÿð è íàêëîííàÿ ................... 378
259. Ïàðàëëåëüíûå ïðÿìûå ............................ 380
260. Ïðèçíàêè ïàðàëëåëüíîñòè ïðÿìûõ .......... 381
261. Óãëû ñ ïàðàëëåëüíûìè
è ïåðïåíäèêóëÿðíûìè ñòîðîíàìè ............ 382
§ 29 Ïðîñòåéøèå çàäà÷è íà ïîñòðîåíèå .......... 384
262. Äåëåíèå îòðåçêà ïîïîëàì ........................ 384
263. Ïîñòðîåíèå ïåðïåíäèêóëÿðîâ .................. 384
264. Ïîñòðîåíèå óãëîâ ................................... 385
265. Ïîñòðîåíèå ïðÿìîé, ïàðàëëåëüíîé
äàííîé è ïðîõîäÿùåé ÷åðåç äàííóþ
òî÷êó .................................................... 386
266. Ïîñòðîåíèå ïðîïîðöèîíàëüíûõ
îòðåçêîâ ................................................ 387
267. Ïîñòðîåíèå êàñàòåëüíîé
ê îêðóæíîñòè ........................................ 388
268. Ïîñòðîåíèå âïèñàííîé è îïèñàííîé
îêðóæíîñòåé äëÿ òðåóãîëüíèêà ............... 389
Ðàçäåë IX.
Ãåîìåòðè÷åñêèå ôèãóðû íà ïëîñêîñòè
§ 30. Òðåóãîëüíèê ......................................... 390
269. Ñòîðîíû è óãëû òðåóãîëüíèêà ................. 390
270. Áèññåêòðèñà òðåóãîëüíèêà. Îêðóæíîñòü,
âïèñàííàÿ â òðåóãîëüíèê ........................ 392
271. Ìåäèàíà òðåóãîëüíèêà. Ñðåäíÿÿ ëèíèÿ
òðåóãîëüíèêà ......................................... 395
272. Âûñîòà òðåóãîëüíèêà .............................. 396
273. Îêðóæíîñòü, îïèñàííàÿ
îêîëî òðåóãîëüíèêà. Çàìå÷àòåëüíûå
òî÷êè òðåóãîëüíèêà ............................... 398
274. Ðàâåíñòâî òðåóãîëüíèêîâ ........................ 399
275. Ñâîéñòâà ïðÿìîóãîëüíîãî
òðåóãîëüíèêà ......................................... 401
276. Òåîðåìà êîñèíóñîâ. Òåîðåìà ñèíóñîâ ........ 403
277. Ïëîùàäü òðåóãîëüíèêà ........................... 405
278. Ïðèçíàêè ïîäîáèÿ òðåóãîëüíèêîâ ............ 409
§ 31. ×åòûðåõóãîëüíèêè ................................ 413
279. Ïàðàëëåëîãðàìì, ïðÿìîóãîëüíèê,
ðîìá, êâàäðàò ........................................ 413
280. Òðàïåöèÿ .............................................. 417
281. Ïëîùàäè ÷åòûðåõóãîëüíèêîâ .................. 420
§ 32. Îêðóæíîñòü .......................................... 428
282. Âçàèìíîå ðàñïîëîæåíèå ïðÿìîé
è îêðóæíîñòè. Êàñàòåëüíàÿ è ñåêóùàÿ .... 428
283. Õîðäà è äèàìåòð. Ñåêòîð è ñåãìåíò .......... 429
284. Óðàâíåíèå îêðóæíîñòè ........................... 431
285. Âçàèìíîå ðàñïîëîæåíèå äâóõ
îêðóæíîñòåé ......................................... 431
§ 33. Óãëû è ïðîïîðöèîíàëüíûå
îòðåçêè â êðóãå ..................................... 433
286. Óãëû ñ âåðøèíîé íà îêðóæíîñòè .............. 433
287. Óãëû ñ âåðøèíîé âíóòðè è âíå êðóãà ....... 436
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255. Ëîìàíàÿ. Ìíîãîóãîëüíèê ........................ 370
256. Ãåîìåòðè÷åñêîå ìåñòî òî÷åê .................... 374
257. Ñèììåòðèÿ ............................................ 375
§ 28. Ïåðïåíäèêóëÿðíûå
è ïàðàëëåëüíûå ïðÿìûå ........................ 378
258. Ïåðïåíäèêóëÿð è íàêëîííàÿ ................... 378
259. Ïàðàëëåëüíûå ïðÿìûå ............................ 380
260. Ïðèçíàêè ïàðàëëåëüíîñòè ïðÿìûõ .......... 381
261. Óãëû ñ ïàðàëëåëüíûìè
è ïåðïåíäèêóëÿðíûìè ñòîðîíàìè ............ 382
§ 29 Ïðîñòåéøèå çàäà÷è íà ïîñòðîåíèå .......... 384
262. Äåëåíèå îòðåçêà ïîïîëàì ........................ 384
263. Ïîñòðîåíèå ïåðïåíäèêóëÿðîâ .................. 384
264. Ïîñòðîåíèå óãëîâ ................................... 385
265. Ïîñòðîåíèå ïðÿìîé, ïàðàëëåëüíîé
äàííîé è ïðîõîäÿùåé ÷åðåç äàííóþ
òî÷êó .................................................... 386
266. Ïîñòðîåíèå ïðîïîðöèîíàëüíûõ
îòðåçêîâ ................................................ 387
267. Ïîñòðîåíèå êàñàòåëüíîé
ê îêðóæíîñòè ........................................ 388
268. Ïîñòðîåíèå âïèñàííîé è îïèñàííîé
îêðóæíîñòåé äëÿ òðåóãîëüíèêà ............... 389
Ðàçäåë IX.
Ãåîìåòðè÷åñêèå ôèãóðû íà ïëîñêîñòè
§ 30. Òðåóãîëüíèê ......................................... 390
269. Ñòîðîíû è óãëû òðåóãîëüíèêà ................. 390
270. Áèññåêòðèñà òðåóãîëüíèêà. Îêðóæíîñòü,
âïèñàííàÿ â òðåóãîëüíèê ........................ 392
271. Ìåäèàíà òðåóãîëüíèêà. Ñðåäíÿÿ ëèíèÿ
òðåóãîëüíèêà ......................................... 395
272. Âûñîòà òðåóãîëüíèêà .............................. 396
273. Îêðóæíîñòü, îïèñàííàÿ
îêîëî òðåóãîëüíèêà. Çàìå÷àòåëüíûå
òî÷êè òðåóãîëüíèêà ............................... 398
274. Ðàâåíñòâî òðåóãîëüíèêîâ ........................ 399
275. Ñâîéñòâà ïðÿìîóãîëüíîãî
òðåóãîëüíèêà ......................................... 401
276. Òåîðåìà êîñèíóñîâ. Òåîðåìà ñèíóñîâ ........ 403
277. Ïëîùàäü òðåóãîëüíèêà ........................... 405
278. Ïðèçíàêè ïîäîáèÿ òðåóãîëüíèêîâ ............ 409
§ 31. ×åòûðåõóãîëüíèêè ................................ 413
279. Ïàðàëëåëîãðàìì, ïðÿìîóãîëüíèê,
ðîìá, êâàäðàò ........................................ 413
280. Òðàïåöèÿ .............................................. 417
281. Ïëîùàäè ÷åòûðåõóãîëüíèêîâ .................. 420
§ 32. Îêðóæíîñòü .......................................... 428
282. Âçàèìíîå ðàñïîëîæåíèå ïðÿìîé
è îêðóæíîñòè. Êàñàòåëüíàÿ è ñåêóùàÿ .... 428
283. Õîðäà è äèàìåòð. Ñåêòîð è ñåãìåíò .......... 429
284. Óðàâíåíèå îêðóæíîñòè ........................... 431
285. Âçàèìíîå ðàñïîëîæåíèå äâóõ
îêðóæíîñòåé ......................................... 431
§ 33. Óãëû è ïðîïîðöèîíàëüíûå
îòðåçêè â êðóãå ..................................... 433
286. Óãëû ñ âåðøèíîé íà îêðóæíîñòè .............. 433
287. Óãëû ñ âåðøèíîé âíóòðè è âíå êðóãà ....... 436