kupri-ov_chisl_methods


Чтобы посмотреть этот PDF файл с форматированием и разметкой, скачайте файл и откройте на своем компьютере.
ºÐäÕÔàÐØÝäÞàÜÐæØÞÝÝëåáØáâÕÜØâÕåÝÞÛÞÓØÙ http://chair36.msiu.ru ´.Î.ºãßàØïÝÞÒ ÇØáÛÕÝÝëÕÜÕâÞÔë 1.ÇØáÛÕÝÝëÕÜÕâÞÔë ²ÔÐÝÝÞÜàÐ×ÔÕÛÕÞßØáëÒÐîâáïÜÕâÞÔëçØáÛÕÝÝÞÓÞØÝâÕÓàØàÞÒÐÝØïØ çØáÛÕÝÝÞÓÞàÕèÕÝØïãàÐÒÝÕÝØÙ,ÝÕÞÑåÞÔØÜëÕÔÛïãáßÕèÝÞÓÞÝÐßØáÐÝØï ÚãàáÞÒÞÙàÐÑÞâë. ÇØáÛÕÝÝÞÕØÝâÕÓàØàÞÒÐÝØÕ. ÇØáÛÕÝÝÞÕØÝâÕÓàØàÞÒÐÝØÕ  íâÞßàÞ  æÕááÒëçØáÛÕÝØïßàØÑÛØÖñÝÝÞÓÞ×ÝÐçÕÝØïÞßàÕÔÕÛñÝÝÞÓÞØÝâÕÓàÐÛÐ,ÚÞâÞ  àÞÕÞáÝÞÒÐÝÞÝÐâÞÜ,çâÞÒÕÛØçØÝÐØÝâÕÓàÐÛÐçØáÛÕÝÝÞàÐÒÝÐßÛÞéÐÔØ ÚàØÒÞÛØÝÕÙÝÞÙâàÐßÕæØØ,ÞÓàÐÝØçÕÝÝÞÙÓàÐäØÚÞÜØÝâÕÓàØàãÕÜÞÙäãÝÚæØØ f ( x ),ÞáìîÐÑáæØáá,ÐâÐÚÖÕÞâàÕ×ÚÐÜØßàïÜëå x = a Ø x = b (àØá.1),ÓÔÕ a Ø b  ßàÕÔÕÛëØÝâÕÓàØàÞÒÐÝØï I = b Z a f ( x ) dx = S .(1.1) ÀØá.1. ·ÝÐçÕÝØÕØÝâÕÓàÐÛÐÚÐÚßÛÞéÐÔìÚàØÒÞÛØÝÕÙÝÞÙâàÐßÕæØØ ½ÕÞÑåÞÔØÜÞáâìßàØÜÕÝÕÝØïçØáÛÕÝÝÞÓÞØÝâÕÓàØàÞÒÐÝØïçÐéÕÒáÕÓÞÜÞ  ÖÕâÑëâìÒë×ÒÐÝÐÞâáãâáâÒØÕÜãßÕàÒÞÞÑàÐ×ÝÞÙäãÝÚæØØßàÕÔáâÐÒÛÕÝØïÒ íÛÕÜÕÝâÐàÝëåäãÝÚæØïåØ,áÛÕÔÞÒÐâÕÛìÝÞ,ÝÕÒÞ×ÜÞÖÝÞáâìîÐÝÐÛØâØçÕáÚÞ  ÓÞÒëçØáÛÕÝØï×ÝÐçÕÝØïÞßàÕÔÕÛñÝÝÞÓÞØÝâÕÓàÐÛÐßÞäÞàÜãÛÕ½ìîâÞÝÐ  »ÕÙÑÝØæÐ.ÂÐÚÖÕÒÞ×ÜÞÖÝÐáØâãÐæØï,ÚÞÓÔÐÒØÔßÕàÒÞÞÑàÐ×ÝÞÙÝÐáâÞÛìÚÞ áÛÞÖÕÝ,çâÞÑëáâàÕÕÒëçØáÛØâì×ÝÐçÕÝØÕØÝâÕÓàÐÛÐçØáÛÕÝÝëÜÜÕâÞÔÞÜ. ÇØáÛÕÝÝëÕÜÕâÞÔë3 ¾áÝÞÒÝÐïØÔÕïÑÞÛìèØÝáâÒÐÜÕâÞÔÞÒçØáÛÕÝÝÞÓÞØÝâÕÓàØàÞÒÐÝØïáÞ  áâÞØâÒÛÞÚÐÛìÝÞÙ×ÐÜÕÝÕßÞÔëÝâÕÓàÐÛìÝÞÙäãÝÚæØØÝÐÑÞÛÕÕßàÞáâãî, ØÝâÕÓàÐÛÞâÚÞâÞàÞÙÛÕÓÚÞÒëçØáÛïÕâáïÐÝÐÛØâØçÕáÚØ.¿àØíâÞÜÞâàÕ×ÞÚ, ÝÐÚÞâÞàÞÜßàÞÒÞÔØâáïØÝâÕÓàØàÞÒÐÝØÕ,àÐ×ÑØÒÐÕâáïÝÐÞßàÕÔÕÛñÝÝÞÕÚÞÛØ  çÕáâÒÞÞâàÕ×ÚÞÒ,ßÞÔëÝâÕÓàÐÛìÝÐïäãÝÚæØïÝÐÚÐÖÔÞÜÞâàÕ×ÚÕ×ÐÜÕÝïÕâáï ÑÞÛÕÕßàÞáâÞÙäãÝÚæØÕÙ,ÐØáÚÞÜÐïßÛÞéÐÔìÚàØÒÞÛØÝÕÙÝÞÙâàÐßÕæØØÒë  çØáÛïÕâáïÚÐÚáãÜÜÐßÛÞéÐÔÕÙÑÞÛÕÕßàÞáâëåäØÓãà,×ÝÐçÕÝØïÚÞâÞàëå ÜÞÓãâÑëâìßÞÛãçÕÝëÐÝÐÛØâØçÕáÚØ. ¼ÕâÞÔßàïÜÞãÓÞÛìÝØÚÞÒ ¼ÕâÞÔßàïÜÞãÓÞÛìÝØÚÞÒ  íâÞÜÕâÞÔçØáÛÕÝÝÞÓÞØÝâÕÓàØàÞÒÐÝØï,ÚÞ  âÞàëÙßÞÛãçÐÕâáïßàØ×ÐÜÕÝÕßÞÔëÝâÕÓàÐÛìÝÞÙäãÝÚæØØÝÐÚãáÞçÝÞ - ßÞáâÞ  ïÝÝãî.¿ÞÛãçØÜÞáÝÞÒÝëÕáÞÞâÝÞèÕÝØïÔÛïÒëçØáÛÕÝØï×ÝÐçÕÝØïÞßàÕÔÕ  ÛñÝÝÞÓÞØÝâÕÓàÐÛÐ.´ÛïíâÞÓÞàÐ×ÞÑêÕÜÞâàÕ×ÞÚ [ a , b ] ÝÐ n àÐÒÝëåçÐáâÕÙ ØÒëçØáÛØÜÔÛØÝãÚÐÖÔÞÓÞØ×ÞâàÕ×ÚÞÒàÐ×ÑØÕÝØï = ( b � a ) / n .½ÐÚÐÖ  ÔÞÜØ×ßÞÛãçØÒèØåáïÞâàÕ×ÚÞÒ×ÐÜÕÝØÜ×ÐÔÐÝÝãîäãÝÚæØîÚÞÝáâÐÝâÞÙ.² ÚÐçÕáâÒÕÚÞÝáâÐÝâëÜÞÖÝÞÒ×ïâì×ÝÐçÕÝØÕäãÝÚæØØ f ( x )ÒÛîÑÞÙâÞçÚÕÞâ  àÕ×ÚÐ [ x i , x i + 1 ] .¾ÔÝÐÚÞÝÐØÑÞÛÕÕçÐáâÞØáßÞÛì×ãîâáï×ÝÐçÕÝØïäãÝÚæØØ ÝÐÚÞÝæÐåÞâàÕ×ÚÐØÛØÒÕÓÞáÕàÕÔØÝÕ.ÁÞÞâÒÕâáâÒãîéØÕÜÞÔØäØÚÐæØØÝÞ  áïâÝÐ×ÒÐÝØïÜÕâÞÔÐÛÕÒëåßàïÜÞãÓÞÛìÝØÚÞÒ,ßàÐÒëåßàïÜÞãÓÞÛìÝØÚÞÒØ áàÕÔÝØåßàïÜÞãÓÞÛìÝØÚÞÒ(àØá.2  4). ÀØá.2. ÇØáÛÕÝÝÞÕØÝâÕÓàØàÞÒÐÝØÕÜÕâÞÔÞÜÛÕÒëåßàïÜÞãÓÞÛìÝØÚÞÒ 4ÇØáÛÕÝÝëÕÜÕâÞÔë ÀØá.3. ÇØáÛÕÝÝÞÕØÝâÕÓàØàÞÒÐÝØÕÜÕâÞÔÞÜßàÐÒëåßàïÜÞãÓÞÛìÝØÚÞÒ ÀØá.4. ÇØáÛÕÝÝÞÕØÝâÕÓàØàÞÒÐÝØÕÜÕâÞÔÞÜáàÕÔÝØåßàïÜÞãÓÞÛìÝØÚÞÒ ÇØáÛÕÝÝëÕÜÕâÞÔë5 ¸×ÓÕÞÜÕâàØçÕáÚÞÓÞßÞáâÞàÕÝØïÒØÔÝÞ,çâÞ×ÝÐçÕÝØÕÞßàÕÔÕÛñÝÝÞÓÞØÝ  âÕÓàÐÛÐäãÝÚæØØ f ( x )ÝÐÞâàÕ×ÚÕ [ a , b ] ÜÞÖÝÞáçØâÐâìßàØÜÕàÝÞàÐÒÝëÜ áãÜÜÕßÛÞéÐÔÕÙ(áãçÕâÞÜ×ÝÐÚÐäãÝÚæØØ)ßÞÛãçØÒèØåáïßàïÜÞãÓÞÛìÝØ  ÚÞÒ S  n X i = 1 S i ,(1.2) ÓÔÕ S i  ßÛÞéÐÔì i - ÞÓÞßàïÜÞãÓÞÛìÝØÚÐ. ¸áßÞÛì×ãïáâÐÝÔÐàâÝëÕáÞÞâÝÞèÕÝØïÔÛïÝÐåÞÖÔÕÝØïßÛÞéÐÔØßàïÜÞ  ãÓÞÛìÝØÚÐ,ßÞÛãçØÜÒëàÐÖÕÝØïÔÛïßàØÑÛØÖñÝÝÞÓÞÒëçØáÛÕÝØï×ÝÐçÕÝØï ÞßàÕÔÕÛñÝÝÞÓÞØÝâÕÓàÐÛÐÜÕâÞÔÞÜßàïÜÞãÓÞÛìÝØÚÞÒ: ˆ ÔÛïÜÕâÞÔÐÛÕÒëåßàïÜÞãÓÞÛìÝØÚÞÒ I  n � 1 X i = 0 f ( x i ) = n � 1 X i = 0 f ( x i ) = b � a n n � 1 X i = 0 f ( x i );(1.3) ˆ ÔÛïÜÕâÞÔÐßàÐÒëåßàïÜÞãÓÞÛìÝØÚÞÒ I  n � 1 X i = 0 f ( x i + 1 ) = b � a n n � 1 X i = 0 f ( x i + 1 );(1.4) ˆ ÔÛïÜÕâÞÔÐáàÕÔÝØåßàïÜÞãÓÞÛìÝØÚÞÒ I  n � 1 X i = 0 f  x i + x i + 1 2  = b � a n n � 1 X i = 0 f  x i + x i + 1 2  .(1.5) ½ÕÞÑåÞÔØÜÞÞâÜÕâØâì,çâÞâÞçÝÞáâìßÞÛãçÕÝÝÞÓÞ×ÝÐçÕÝØïÞßàÕÔÕÛñÝ  ÝÞÓÞØÝâÕÓàÐÛÐÝÐßàïÜãî×ÐÒØáØâÞâÚÞÛØçÕáâÒÐÞâàÕ×ÚÞÒ,ÝÐÚÞâÞàëÕàÐ×  ÑØÒÐÕâáïØáåÞÔÝëÙÞâàÕ×ÞÚ [ a , b ] .¿ÞíâÞÜãÔÛïßÞÛãçÕÝØïÑÞÛÕÕâÞçÝÞÓÞ ×ÝÐçÕÝØïØÝâÕÓàÐÛÐÝÕÞÑåÞÔØÜÞãÒÕÛØçØâìçØáÛÞ n . ¼ÕâÞÔâàÐßÕæØÙ ¼ÕâÞÔâàÐßÕæØÙÐÝÐÛÞÓØçÕÝÜÕâÞÔãßàïÜÞãÓÞÛìÝØÚÞÒØßÞÛãçÐÕâáïßàØ ×ÐÜÕÝÕßÞÔëÝâÕÓàÐÛìÝÞÙäãÝÚæØØÚãáÞçÝÞ - ÛØÝÕÙÝÞÙ.¾áÝÞÒÝÞÕÞâÛØçØÕ íâÞÓÞÜÕâÞÔÐ×ÐÚÛîçÐÕâáïÒâÞÜ,çâÞÝÐÚÐÖÔÞÜÞâàÕ×ÚÕàÐ×ÑØÕÝØï×Ð  ÔÐÝÝÐïäãÝÚæØï×ÐÜÕÝïÕâáïÛØÝÕÙÝÞÙäãÝÚæØÕÙ,ßàØÝØÜÐîéÕÙÝÐÚÞÝæÐå ÞâàÕ×ÚÐâÕÖÕ×ÝÐçÕÝØï,çâÞØäãÝÚæØï f ( x ).¿ÞÛãçØÒèØÕáïäØÓãàëïÒÛïîâ  áïâàÐßÕæØïÜØ,ØØåßÛÞéÐÔìÒëçØáÛïÕâáïÚÐÚßÞÛãáãÜÜÐÔÛØÝÞáÝÞÒÐÝØÙ âàÐßÕæØØ,ãÜÝÞÖÕÝÝÐïÝÐÕñÒëáÞâã. 6ÇØáÛÕÝÝëÕÜÕâÞÔë ÀØá.5. ÇØáÛÕÝÝÞÕØÝâÕÓàØàÞÒÐÝØÕÜÕâÞÔÞÜâàÐßÕæØÙ ¸×ÓÕÞÜÕâàØçÕáÚØåáÞÞÑàÐÖÕÝØÙÛÕÓÚÞßÞÛãçØâì,çâÞ×ÝÐçÕÝØÕÞßàÕÔÕ  ÛñÝÝÞÓÞØÝâÕÓàÐÛÐäãÝÚæØØ f ( x )ÝÐÞâàÕ×ÚÕ [ a , b ] ,ÒëçØáÛÕÝÝÞÓÞáßÞÜÞéìî ÜÕâÞÔÐâàÐßÕæØÙ,áãçÕâÞÜßÞÓàÕèÝÞáâØÜÕâÞÔÐÑãÔÕâßàØÑÛØ×ØâÕÛìÝÞàÐÒ  ÝÞ I  n � 1 X i = 0 f ( x i ) + f ( x i + 1 ) 2 = b � a n n � 1 X i = 0 f ( x i ) + f ( x i + 1 ) 2 .(1.6) ¼ÕâÞÔ¼ÞÝâÕ - ºÐàÛÞ ¼ÕâÞÔ¼ÞÝâÕ - ºÐàÛÞçÐáâÞßàØÜÕÝïÕâáïÔÛïÒëçØáÛÕÝØïßàÞáâëåØ ÚàÐâÝëåØÝâÕÓàÐÛÞÒ.¿àÕÔßÞÛÞÖØÜ,çâÞÝÕÞÑåÞÔØÜÞÒ×ïâìØÝâÕÓàÐÛÞâ ÝÕÚÞâÞàÞÙäãÝÚæØØ f ( x ).²ÞáßÞÛì×ãÕÜáïÓÕÞÜÕâàØçÕáÚØÜáÒÞÙáâÒÞÜÞßàÕ  ÔÕÛñÝÝÞÓÞØÝâÕÓàÐÛÐ,×ÝÐçÕÝØÕÚÞâÞàÞÓÞçØáÛÕÝÝÞàÐÒÝÞßÛÞéÐÔØßÞÔÓàÐ  äØÚÞÜäãÝÚæØØ.´ÛïÞßàÕÔÕÛÕÝØïíâÞÙßÛÞéÐÔØÜÞÖÝÞØáßÞÛì×ÞÒÐâìáÛÕ  ÔãîéØÙáâÞåÐáâØçÕáÚØÙÐÛÓÞàØâÜ(àØá.6): ˆ ÞÓàÐÝØçØÜáÒÕàåãäãÝÚæØî f ( x )ßàïÜÞÙ y = h ,âÐÚÞÙçâÞ h  f ( x ) ÔÛïÛîÑÞÓÞ x 2 [ a , b ]; ˆ ØáßÞÛì×ãïáâÐÝÔÐàâÝãîäÞàÜãÛã,ÒëçØáÛØÜßÛÞéÐÔìÞÓàÐÝØçØÒÐ  îéÕÓÞßàïÜÞãÓÞÛìÝØÚÐ S ßà = ( b � a ) h ; ˆ áÛãçÐÙÝëÜÞÑàÐ×ÞÜÒÝãâàìßÞÛãçÕÝÝÞÓÞßàïÜÞãÓÞÛìÝØÚÐ  ÒëÑàÞ  áØÜ  ÝÕÚÞâÞàÞÕÚÞÛØçÕáâÒÞâÞçÕÚ( N èâãÚ); ÇØáÛÕÝÝëÕÜÕâÞÔë7 ˆ ÞßàÕÔÕÛØÜçØáÛÞâÞçÕÚ( K èâãÚ),ÚÞâÞàëÕßÞßÐÔãâÒÝãâàìäØÓã  àë,ÞÓàÐÝØçÕÝÝÞÙÓàÐäØÚÞÜäãÝÚæØØ,Þáìî x ,ÐâÐÚÖÕßàïÜëÜØ x = a Ø x = b ; ˆ ÒëçØáÛØÜßÛÞéÐÔìßÞÔÓàÐäØÚÞÜ×ÐÔÐÝÝÞÙäãÝÚæØØ S  K N S ßà = K N ( b � a ) h .(1.7) ÀØá.6. ÇØáÛÕÝÝÞÕØÝâÕÓàØàÞÒÐÝØÕÜÕâÞÔÞܼÞÝâÕ - ºÐàÛÞ ÁÜëáÛÒëàÐÖÕÝØï(1.7)×ÐÚÛîçÐÕâáïÒâÞÜ,çâÞßàØÔÞáâÐâÞçÝÞÑÞÛì  èÞÜçØáÛÕØáßëâÐÝØÙÞâÝÞèÕÝØÕçØáÛÐâÞçÕÚ,ßÞßÐÒèØåßÞÔÓàÐäØÚäãÝÚ  æØØ,ÚÞÑéÕÜãçØáÛãâÞçÕÚáâàÕÜØâáïÚÞâÝÞèÕÝØîßÛÞéÐÔÕÙÞÓàÐÝØçØÒÐ  îéÕÓÞßàïÜÞãÓÞÛìÝØÚÐØÚàØÒÞÛØÝÕÙÝÞÙâàÐßÕæØØ.ÍâÞÞ×ÝÐçÐÕâ,çâÞßàØ n !1 ØáÚÞÜÐïßÛÞéÐÔì S �! K N S ßà .(1.8) ¿ÞíâÞÜãÔÛïãÒÕÛØçÕÝØïâÞçÝÞáâØÒëçØáÛÕÝØÙ×ÝÐçÕÝØïØÝâÕÓàÐÛÐ,ßàÞØ×  ÒÞÔØÜëåÔÐÝÝëÜÜÕâÞÔÞÜ,ÝÕÞÑåÞÔØÜÞ: ˆ ÜÐÚáØÜÐÛìÝÞßàØÑÛØ×ØâìàÐ×ÜÕàëÞÓàÐÝØçØÒÐîéÕÓÞßàïÜÞãÓÞÛì  ÝØÚÐÚÞÑÛÐáâØ,×ÐÔÐÝÝÞÙÓàÐäØÚÞÜäãÝÚæØØ; ˆ ãÒÕÛØçØâìàÐ×ÜÕàÒëÑÞàÚØ(ÚÞÛØçÕáâÒÐ  ÒëÑàÐáëÒÐÕÜëå  âÞçÕÚ). 8ÇØáÛÕÝÝëÕÜÕâÞÔë ºÞÝÕçÝÞ,ßàØÔÞáâÐâÞçÝÞÜÐÛÞÜÚÞÛØçÕáâÒÕ  ÒëÑàÞèÕÝÝëå  âÞçÕÚâÞç  ÝÞáâìÜÕâÞÔмÞÝâÕ - ºÐàÛÞÓÞàÐ×ÔÞÝØÖÕâÞçÝÞáâØ,ßÞÛãçÐÕÜÞÙßàØØá  ßÞÛì×ÞÒÐÝØØÜÕâÞÔÞÒßàïÜÞãÓÞÛìÝØÚÞÒØâàÐßÕæØÙ.ÂÕÜÝÕÜÕÝÕÕ,ÒÝÕÚÞâÞ  àëåáÛãçÐïå,ÚÞÓÔÐØÝâÕÓàØàãÕÜÐïäãÝÚæØï×ÐÔÐÝÐÝÕïÒÝÞØÛØÖÕÞÑÛÐáâì ØÝâÕÓàØàÞÒÐÝØï×ÐÔÐÝÐÒÒØÔÕáÛÞÖÝëåÝÕàÐÒÕÝáâÒ,áâÞåÐáâØçÕáÚØÙÜÕâÞÔ ÜÞÖÕâÞÚÐ×ÐâìáïÝÐØÑÞÛÕÕßàÕÔßÞçâØâÕÛìÝëÜ. ÇØáÛÕÝÝÞÕàÕèÕÝØÕÝÕÛØÝÕÙÝëåãàÐÒÝÕÝØÙ. ½ÕÛØÝÕÙÝëÜØ ãàÐÒÝÕÝØïÜØÝÐ×ëÒÐîâãàÐÒÝÕÝØï,áÞÔÕàÖÐéØÕÐÛÓÕÑàÐØçÕáÚØÕäãÝÚæØØ: æÕÛëÕ,àÐæØÞÝÐÛìÝëÕ,ØààÐæØÞÝÐÛìÝëÕ( ÐÛÓÕÑàÐØçÕáÚØÕãàÐÒÝÕÝØï ),Ð âÐÚÖÕâàØÓÞÝÞÜÕâàØçÕáÚØÕ,ßÞÚÐ×ÐâÕÛìÝëÕ,ÛÞÓÐàØäÜØçÕáÚØÕØÔàãÓØÕäãÝÚ  æØØ( âàÐÝáæÕÝÔÕÝâÝëÕãàÐÒÝÕÝØï ).¼ÕâÞÔëàÕèÕÝØïÝÕÛØÝÕÙÝëåãàÐÒ  ÝÕÝØÙÔÕÛïâáïÝÐÔÒÕÓàãßßë: ˆ âÞçÝëÕÜÕâÞÔë; ˆ ØâÕàÐæØÞÝÝëÕÜÕâÞÔë. ÂÞçÝëÕÜÕâÞÔë ßÞ×ÒÞÛïîâ×ÐßØáÐâìÚÞàÝØãàÐÒÝÕÝØïÒÒØÔÕÝÕÚÞ  âÞàÞÓÞÚÞÝÕçÝÞÓÞáÞÞâÝÞèÕÝØï(äÞàÜãÛë).¸×èÚÞÛìÝÞÓÞÚãàáÐÐÛÓÕÑàë âÐÚØÕÜÕâÞÔëØ×ÒÕáâÝëÔÛïàÕèÕÝØïÝÕÚÞâÞàëåâàØÓÞÝÞÜÕâàØçÕáÚØå,ÛÞÓÐ  àØäÜØçÕáÚØå,ßÞÚÐ×ÐâÕÛìÝëå,ÐâÐÚÖÕßàÞáâÕÙèØåÐÛÓÕÑàÐØçÕáÚØåãàÐÒÝÕ  ÝØÙ.¾ÔÝÐÚÞÜÝÞÓØÕãàÐÒÝÕÝØïÝÕØÜÕîâÐÝÐÛØâØçÕáÚØåàÕèÕÝØÙ.²ßÕàÒãî ÞçÕàÕÔìíâÞÞâÝÞáØâáïÚÑÞÛìèØÝáâÒãâàÐÝáæÕÝÔÕÝâÝëåãàÐÒÝÕÝØÙØßàÞØ×  ÒÞÛìÝëÜÐÛÓÕÑàÐØçÕáÚØÜãàÐÒÝÕÝØïÜáâÕßÕÝØÒëèÕçÕâÒÕàâÞÙ.ºàÞÜÕâÞÓÞ, ÒÝÕÚÞâÞàëåáÛãçÐïåãàÐÒÝÕÝØÕÜÞÖÕâáÞÔÕàÖÐâìÚÞíääØæØÕÝâë,Ø×ÒÕáâ  ÝëÕÛØèìßàØÑÛØ×ØâÕÛìÝÞ,Ø,áÛÕÔÞÒÐâÕÛìÝÞ,áÐÜÐ×ÐÔÐçÐÞâÞçÝÞÜÞßàÕ  ÔÕÛÕÝØØÚÞàÝÕÙãàÐÒÝÕÝØïâÕàïÕâáÜëáÛ.´ÛïØåàÕèÕÝØïØáßÞÛì×ãîâáï ØâÕàÐæØÞÝÝëÕÜÕâÞÔë . ÀÕèØâìãàÐÒÝÕÝØÕ f ( x ) = 0(1.9) ØâÕàÐæØÞÝÝëÜÜÕâÞÔÞÜ  ×ÝÐçØâãáâÐÝÞÒØâì,ØÜÕÕâÛØÞÝÞÚÞàÝØ,ÞßàÕÔÕ  ÛØâìÚÞÛØçÕáâÒÞÚÞàÝÕÙØÝÐÙâØ×ÝÐçÕÝØïÚÞàÝÕÙá×ÐÔÐÝÝÞÙâÞçÝÞáâìî. ¿ÞíâÞÜã×ÐÔÐçÐÝÐåÞÖÔÕÝØïÚÞàÝÕÙãàÐÒÝÕÝØïØâÕàÐæØÞÝÝëÜÜÕâÞÔÞÜáÞ  áâÞØâØ×ÔÒãåíâÐßÞÒ: ˆ ÞâÔÕÛÕÝØÕÚÞàÝÕÙ  ÝÐåÞÖÔÕÝØÕÞâàÕ×ÚÞÒ,áÞÔÕàÖÐéØåâÞÛìÚÞ ÞÔØÝÚÞàÕÝìãàÐÒÝÕÝØï,Ø,ÕáÛØíâÞÝÕÞÑåÞÔØÜÞ,ÒëÑÞàÝÐçÐÛì  ÝÞÓÞßàØÑÛØÖÕÝØï(ßÕàÒÞÓÞßàØÑÛØÖñÝÝÞÓÞ×ÝÐçÕÝØïÚÞàÝï)ÔÛï ÚÐÖÔÞÓÞØ×íâØåÞâàÕ×ÚÞÒ; ˆ ãâÞçÝÕÝØÕßàØÑÛØÖñÝÝëå×ÝÐçÕÝØÙÚÞàÝÕÙ ÔÞ×ÐÔÐÝÝÞÙáâÕ  ßÕÝØâÞçÝÞáâØ. ¿àÞæÕááÞâÔÕÛÕÝØïÚÞàÝÕÙÝÐçØÝÐÕâáïáãáâÐÝÞÒÛÕÝØï×ÝÐÚÞÒäãÝÚæØØ f ( x )ÒÓàÐÝØçÝëåâÞçÚÐåÞÑÛÐáâØÕÕáãéÕáâÒÞÒÐÝØï x = a Ø x = b .¿àØÑÛØ  ÇØáÛÕÝÝëÕÜÕâÞÔë9 ÖñÝÝëÕ×ÝÐçÕÝØïÚÞàÝÕÙ( ÝÐçÐÛìÝëÕßàØÑÛØÖÕÝØï )ÜÞÓãâÑëâìØ×ÒÕáâÝë Ø×äØ×ØçÕáÚÞÓÞáÜëáÛÐ×ÐÔÐçØ,Ø×àÕèÕÝØïÐÝÐÛÞÓØçÝÞÙ×ÐÔÐçØßàØÔàãÓØå ØáåÞÔÝëåÔÐÝÝëåØÛØÜÞÓãâÑëâìÝÐÙÔÕÝëÓàÐäØçÕáÚØÜáßÞáÞÑÞÜ. ²ØÝÖÕÝÕàÝÞÙßàÐÚâØÚÕÝÐØÑÞÛÕÕàÐáßàÞáâàÐÝÕÝÓàÐäØçÕáÚØÙáßÞáÞÑ ÞßàÕÔÕÛÕÝØïßàØÑÛØÖñÝÝÞÓÞ×ÝÐçÕÝØïÚÞàÝÕÙ.¿àØÝØÜÐïÒÞÒÝØÜÐÝØÕ,çâÞ ÔÕÙáâÒØâÕÛìÝëÕÚÞàÝØãàÐÒÝÕÝØï  íâÞâÞçÚØßÕàÕáÕçÕÝØïÓàÐäØÚÐäãÝÚ  æØØ f ( x )áÞáìîÐÑáæØáá,ÔÞáâÐâÞçÝÞßÞáâàÞØâìÓàÐäØÚíâÞÙäãÝÚæØØØ ÞâÜÕâØâìâÞçÚØÕñßÕàÕáÕçÕÝØïáÞáìî ¾å ØÛØÞâÜÕâØâìÝÐÞáØ ¾å ÞâàÕ×  ÚØ,áÞÔÕàÖÐéØÕßÞÞÔÝÞÜãÚÞàÝî. ¾ßàÕÔÕÛïâìßàØÑÛØÖñÝÝëÕ×ÝÐçÕÝØïÚÞàÝÕÙãàÐÒÝÕÝØïÓàÐäØçÕáÚØÜ áßÞáÞÑÞÜãÔÞÑÝÞáßÞÜÞéìîáØáâÕÜÚÞÜßìîâÕàÝÞÙÐÛÓÕÑàëØÛØÓàÐäÞ  ßÞáâàÞØâÕÛÕÙ.²ÞáßÞÛì×ÞÒÐÒèØáìáÞÞâÒÕâáâÒãîéÕÙäãÝæØÕÙßÞáâàÞÕÝØï ÓàÐäØÚÞÒ,ÜÞÖÝÞÞßàÕÔÕÛØâìÞÑéØÙÒØÔäãÝÚæØØ,ÐâÐÚÖÕßãâñÜßÞáÛÕÔÞ  ÒÐâÕÛìÝëåßàØÑÛØÖÕÝØÙ(ÒëÑØàÐïàÐ×ÝëÕÞâàÕ×ÚØÔÛïßÞáâàÞÕÝØï)ÝÐÙâØ ÝãÖÝëÙÞâàÕ×ÞÚ,áÞÔÕàÖÐéØÙÚÞàÕÝìãàÐÒÝÕÝØïØãÔÞÒÛÕâÒÞàïîéØÙãáÛÞ  ÒØïÜßÞáâÐÝÞÒÚØ×ÐÔÐçØ. ¸âÕàÐæØÞÝÝëÙßàÞæÕááßÞÛãçÕÝØïßàØÑÛØÖñÝÝÞÓÞ×ÝÐçÕÝØïÚÞàÝïãàÐÒ  ÝÕÝØï×ÐÔÐÝÝÞÙâÞçÝÞáâØáÞáâÞØâÒßÞáÛÕÔÞÒÐâÕÛìÝÞÜãâÞçÝÕÝØØÝÐçÐÛìÝÞ  ÓÞßàØÑÛØÖÕÝØï å 0 ØÛØÖÕãÜÕÝìèÕÝØïáÞÔÕàÖÐéÕÓÞÕÓÞÞâàÕ×ÚÐ.ºÐÖ  ÔëÙâÐÚÞÙèÐÓÝÐ×ëÒÐÕâáï ØâÕàÐæØÕÙ .²àÕ×ãÛìâÐâÕÒëßÞÛÝÕÝØïØâÕàÐæØ  ÞÝÝÞÓÞßàÞæÕááÐßÞÛãçÐÕâáïßÞáÛÕÔÞÒÐâÕÛìÝÞáâìßàØÑÛØÖñÝÝëå×ÝÐçÕÝØÙ ÚÞàÝï å 1 , å 2 ,..., å n .µáÛØíâØ×ÝÐçÕÝØïáãÒÕÛØçÕÝØÕÜçØáÛÐØâÕàÐæØÙ n ßàØÑÛØÖÐîâáïÚØáâØÝÝÞÜã×ÝÐçÕÝØîÚÞàÝï,âÞÓÞÒÞàïâ,çâÞØâÕàÐæØÞÝÝëÙ ßàÞæÕáááåÞÔØâáï. ¼ÕâÞÔÔÕÛÕÝØïÞâàÕ×ÚÐßÞßÞÛÐÜ(ÔØåÞâÞÜØØ) ÁÐÜëÜßàÞáâëÜáàÕÔØÜÕâÞÔÞÒãâÞçÝÕÝØïÚÞàÝÕÙïÒÛïÕâáïÜÕâÞÔÔÕÛÕ  ÝØïÞâàÕ×ÚÐßÞßÞÛÐÜ(ÜÕâÞÔÔØåÞâÞÜØØ).¸áßÞÛì×ÞÒÐÝØÕíâÞÓÞÜÕâÞÔÐÔÛï àÕèÕÝØïãàÐÒÝÕÝØï(1.9)ÒÞ×ÜÞÖÝÞÛØèìÒâÞÜáÛãçÐÕ,ÚÞÓÔÐäãÝÚæØï f ( x ) ãÔÞÒÛÕâÒÞàïÕâáÛÕÔãîéØÜãáÛÞÒØïÜ: ˆ äãÝÚæØï f ( x )ÝÕßàÕàëÒÝÐÝÐÞâàÕ×ÚÕ[ a , b ]; ˆ ×ÝÐçÕÝØïäãÝÚæØØ f ( x )ÝÐÚÞÝæÐåÞâàÕ×ÚÐØÜÕîâàÐ×ÝëÕ×ÝÐÚØ ( f ( a )  f ( b ) 0); ˆ ÞâàÕ×ÞÚ[ a , b ]áÞÔÕàÖØââÞÛìÚÞÞÔØÝÚÞàÕÝìãàÐÒÝÕÝØï. ¼ÕâÞÔÔØåÞâÞÜØØ×ÐÚÛîçÐÕâáïÒßÞáÛÕÔÞÒÐâÕÛìÝÞÜãÜÕÝìèÕÝØØÞâàÕ×  ÚÐ,áÞÔÕàÖÐéÕÓÞÚÞàÕÝìãàÐÒÝÕÝØï,ßãâÕÜÕÓÞÔÕÛÕÝØïßÞßÞÛÐÜ.¿ãáâìâÞç  ÚÐ c ïÒÛïÕâáïáÕàÕÔØÝÞÙÞâàÕ×ÚÐ[ a , b ],âÞÓÔÐÕÕ×ÝÐçÕÝØÕÜÞÖÝÞÒëçØáÛØâì ßÞäÞàÜãÛÕ c = a + b 2 .(1.10) 10ÇØáÛÕÝÝëÕÜÕâÞÔë ²ÞÑéÕÜáÛãçÐÕâÞçÚÐ á ÜÞÖÕâáÞÒßÐáâìáÚÞàÝÕÜàÐááÜÐâàØÒÐÕÜÞÓÞ ãàÐÒÝÕÝØï(ßàØíâÞÜ×ÝÐçÕÝØÕäãÝÚæØØ f ( x )ÒâÞçÚÕ c àÐÒÝÞÝãÛî),ØÛØÖÕ ÚÞàÕÝìãàÐÒÝÕÝØïÑãÔÕâßàØÝÐÔÛÕÖÐâìÞÔÝÞÜãØ×ÞâàÕ×ÚÞÒ[ a , c ]ØÛØ[ c , b ]. ÁÔÕÛÐâìÒëÑÞàÞâàÕ×ÚÐÜÞÖÝÞ,àÐááÜÞâàÕÒ×ÝÐÚØäãÝÚæØØ f ( x )ÝÐÚÞÝæÐå ÚÐÖÔÞÓÞØ×ÝØå.½ÕâàãÔÝÞßÞÚÐ×Ðâì,çâÞÚÞàÕÝìãàÐÒÝÕÝØïßàØÝÐÔÛÕÖØââÞ  ÜãÞâàÕ×Úã,ÝÐÚÞâÞàÞÜäãÝÚæØïÜÕÝïÕâ×ÝÐÚ(ßÕàÕáÕÚÐÕâÞáì Ox ).¿ÞíâÞÜã ÒëÑÞàÞâàÕ×ÚÐÞßàÕÔÕÛØÜáÛÕÔãîéØÜÞÑàÐ×ÞÜ.µáÛØ f ( a )  f ( c ) 0,âÞÒëÑØàÐÕâáïÞâàÕ×ÞÚ[ a , c ].²ßàÞâØÒÝÞÜáÛãçÐÕÒëÑØàÐ  ÕâáïÞâàÕ×ÞÚ[ c , b ]. ´ÐÛÕÕØâÕàÐæØÞÝÝëÙßàÞæÕááßàÞÔÞÛÖÐÕâáïßãâÕÜÔÕÛÕÝØïÝÞÒÞÓÞÞâ  àÕ×ÚÐ,ßÞáÛÕçÕÓÞÞÔÝÐØ×ÕÓÞßÞÛÞÒØÝÒëÑØàÐÕâáïÝÐÞáÝÞÒÕÞßØáÐÝÝëå ÒëèÕãáÛÞÒØÙ(àØá.7).ÂÐÚØÜÞÑàÐ×ÞÜ,ßÞÛãçÐÕâáïáØáâÕÜÐÞâàÕ×ÚÞÒ,áåÞ  ÔïéØåáïÚÞÔÝÞÙâÞçÚÕ  âÞçÝÞÜã×ÝÐçÕÝØîÚÞàÝïãàÐÒÝÕÝØï.·ÐÒÕàèØâì ØâÕàÐæØÞÝÝëÙßàÞæÕáááÛÕÔãÕââÞÓÔÐ,ÚÞÓÔÐàÐááâÞïÝØÕÜÕÖÔãÓàÐÝØçÝëÜØ âÞçÚÐÜØÝÞÒÞÓÞÞâàÕ×ÚÐ a Ø b áâÐÝÕâÜÕÝìèÕ×ÐÔÐÝÝÞÙâÞçÝÞáâØ " : ( b � a ) " .(1.11) ¿àØíâÞÜßÞáÛÕÔÝÕÕ×ÝÐçÕÝØÕâÞçÚØ c ÜÞÖÝÞáçØâÐâìßàØÑÛØÖñÝÝëÜ×ÝÐ  çÕÝØÕÜÚÞàÝïãàÐÒÝÕÝØï,ÝÐÙÔÕÝÝëÜáßÞÓàÕèÝÞáâìî " . ÀØá.7. ÇØáÛÕÝÝÞÕàÕèÕÝØÕãàÐÒÝÕÝØïÜÕâÞÔÞÜÔØåÞâÞÜØØ ÇØáÛÕÝÝëÕÜÕâÞÔë11 ¼ÕâÞÔåÞàÔ ¼ÕâÞÔåÞàÔ  ØâÕàÐæØÞÝÝëÙßàÞæÕáá,ÒÚÞâÞàÞÜáâàÞØâáïáØáâÕÜÐ ßÞáÛÕÔÞÒÐâÕÛìÝëåßàØÑÛØÖÕÝØÙ(âÞçÕÚ,áâàÕÜïéØåáïÚÚÞàÝîØáåÞÔÝÞÓÞ ãàÐÒÝÕÝØï),ÚÞâÞàëÕßÞÛãçÐîâáïßàØßÕàÕáÕçÕÝØØáâàÞïéØåáïåÞàÔáÞáìî ÐÑáæØáá.ºÐÖÔÐïÝÞÒÐïåÞàÔÐßàÞåÞÔØâçÕàÕ××ÝÐçÕÝØÕäãÝÚæØØÝÐÞÔÝÞÜ Ø×ÚÞÝæÞÒÞâàÕ×ÚÐ[ a , b ](×ÐÒØáØâÞâÒØÔÐäãÝÚæØØ)Ø×ÝÐçÕÝØÕäãÝÚæØØÒ âÞçÚÕßàÕÔëÔãéÕÓÞßàØÑÛØÖÕÝØï(àØá.8). ¸âÐÚ,ßãáâìÔÛïäãÝÚæØØ f ( x )Ø×ãàÐÒÝÕÝØï(1.9)ÒëßÞÛÝïîâáïáÛÕÔãî  éØÕãáÛÞÒØï: ˆ äãÝÚæØï f ( x )ÝÕßàÕàëÒÝÐÝÐÞâàÕ×ÚÕ[ a , b ]; ˆ ×ÝÐçÕÝØïäãÝÚæØØ f ( x )ÝÐÚÞÝæÐåÞâàÕ×ÚÐØÜÕîâàÐ×ÝëÕ×ÝÐÚØ ( f ( a )  f ( b ) 0); ˆ ÞâàÕ×ÞÚ[ a , b ]áÞÔÕàÖØââÞÛìÚÞÞÔØÝÚÞàÕÝìãàÐÒÝÕÝØï. ²ÜÕâÞÔÕåÞàÔ,âÐÚÖÕ,ÚÐÚØÒÜÕâÞÔÕÔØåÞâÞÜØØ,ØáßÞÛì×ãÕâáïÜÕ  åÐÝØ×ÜÔÕÛÕÝØïÞâàÕ×ÚÐ.½ÞÕáÛØÒÜÕâÞÔÕÔØåÞâÞÜØØàÐ×ÑØÕÝØÕÞâàÕ×ÚÐ ßàÞØ×ÒÞÔØâáïÝÐÔÒÕàÐÒÝëÕçÐáâØ,âÞÒÜÕâÞÔÕåÞàÔØáßÞÛì×ãÕâáïÔÕÛÕÝØÕ, ßàÞßÞàæØÞÝÐÛìÝÞÕ×ÝÐçÕÝØïÜäãÝÚæØØ,ÚÞâÞàëÕÞÝÐßàØÝØÜÐÕâÝÐáÒÞØå ÚÞÝæÐå.¾ÑÞ×ÝÐçØÜçÕàÕ× x 1 , x 2 ,..., x n âÞçÚØàÐ×ÑØÕÝØïÞâàÕ×ÚÐ,çÕàÕ×ÚÞ  âÞàëÕßàÞåÞÔïâáâàÞïéØÕáïåÞàÔë.½Ð×ÞÒÕÜíâØâÞçÚØßÞáÛÕÔÞÒÐâÕÛìÝëÜØ ßàØÑÛØÖÕÝØïÜØÚÞàÝïãàÐÒÝÕÝØïØßÞÛãçØÜáÞÞâÝÞèÕÝØïÔÛïÒëçØáÛÕÝØï Øå×ÝÐçÕÝØÙ.´ÛïíâÞÓÞÝÐßØèÕÜãàÐÒÝÕÝØÕåÞàÔë y � f ( a ) f ( b ) � f ( a ) = x � a b � a .(1.12) ²âÞçÚÕßÕàÕáÕçÕÝØïåÞàÔëáÞáìîÐÑáæØáá x = x 1 ,Ð y = 0.¿ÞÔáâÐ  ÒØÜíâØ×ÝÐçÕÝØÕÒãàÐÒÝÕÝØÕ(1.12)ØßÞÛãçØÜÝÞÒÞÕáÞÞâÝÞèÕÝØÕÔÛï ÒëçØáÛÕÝØï×ÝÐçÕÝØï x 1 : x 1 = a � f ( a ) f ( b ) � f ( a ) ( b � a ).(1.13) ·ÐÜÕâØÜ,çâÞ×ÔÕáì,âÐÚÖÕ,ÚÐÚØÒÜÕâÞÔÕÔØåÞâÞÜØØ,ÒÞ×ÜÞÖÝëÔÒÕ áØâãÐæØØ,ÚÞÓÔÐØáÚÞÜëÙÚÞàÕÝìãàÐÒÝÕÝØïßàØÝÐÔÛÕÖØâÞâàÕ×Úã[ a , x 1 ] ØÛØ[ x 1 , b ].¿ÞíâÞÜãÒëÑÞàÞâàÕ×ÚÐÞßàÕÔÕÛØÜâÐÚØÜÖÕÞÑàÐ×ÞÜ.µáÛØ f ( a )  f ( x 1 ) 0,âÞÒëÑØàÐÕâáïÞâàÕ×ÞÚ[ a , x 1 ].²ßàÞâØÒÝÞÜáÛãçÐÕÒëÑØ  àÐÕâáïÞâàÕ×ÞÚ[ x 1 , b ]. ´ÐÛÕÕØâÕàÐæØÞÝÝëÙßàÞæÕááßàÞÔÞÛÖÐÕâáïßãâÕÜÔÕÛÕÝØïÝÞÒÞÓÞÞâ  àÕ×ÚÐ[ a , b ],ßÞáÛÕçÕÓÞÞÔÝÐØ×ÕÓÞçÐáâÕÙÒëÑØàÐÕâáïÝÐÞáÝÞÒÕÞßØáÐÝÝëå ÒëèÕãáÛÞÒØÙ(àØá.8).·ÝÐçÕÝØÕÝÞÒÞÓÞßàØÑÛØÖÕÝØïÚÞàÝïßÞáÛÕØ×ÜÕ  12ÇØáÛÕÝÝëÕÜÕâÞÔë ÀØá.8. ÇØáÛÕÝÝÞÕàÕèÕÝØÕãàÐÒÝÕÝØïÜÕâÞÔÞÜåÞàÔ ÝÕÝØïÓàÐÝØæÞâàÕ×ÚÐÒëçØáÛïÕâáïÐÝÐÛÞÓØçÝÞ: x i = a � f ( a ) f ( b ) � f ( a ) ( b � a ).(1.14) ÂÐÚØÜÞÑàÐ×ÞÜ,ßÞÛãçÐÕâáïßÞáÛÕÔÞÒÐâÕÛìÝÞáâìßàØÑÛØÖÕÝØÙ x 1 , x 2 ,..., x n , áåÞÔïéØåáïÚØáÚÞÜÞÙâÞçÚÕ  âÞçÝÞÜã×ÝÐçÕÝØîÚÞàÝïãàÐÒÝÕÝØï. ´ÛïäãÝÚæØØÞÑéÕÓÞÒØÔÐÝÐßØáÐÝØÕÚàØâÕàØï×ÐÒÕàèÕÝØïØâÕàÐæØ  ÞÝÝÞÓÞßàÞæÕááÐïÒÛïÕâáïÔÞáâÐâÞçÝÞáÛÞÖÝÞÙØÝÕÞÔÝÞ×ÝÐçÝÞÙ×ÐÔÐçÕÙ, ßÞíâÞÜãÒÔÐÛìÝÕÙèÕÜÑãÔÕÜáçØâÐâì,çâÞÔÛïàÐááÜÐâàØÒÐÕÜëåäãÝÚæØÙ ÔÞáâÐâÞçÝëÜãáÛÞÒØÕÜ,ÞßàÕÔÕÛïîéØÜÑÛØ×ÞáâìßàØÑÛØÖÕÝØïÚâÞçÝÞÜã ×ÝÐçÕÝØîÚÞàÝïãàÐÒÝÕÝØï,ÑãÔÕâãáÛÞÒØÕ j x i + 1 � x i j " ,(1.15) ÓÔÕ "  ×ÐÔÐÝÝÐïßÞÓàÕèÝÞáâìÒëçØáÛÕÝØÙ. ¿àØíâÞÜßÞÛãçÕÝÝÐïÒàÕ×ãÛìâÐâÕÒëçØáÛÕÝØÙßÞáÛÕÔÝïïâÞçÚÐ x n ÝÐ  ×ëÒÐÕâáïßàØÑÛØÖñÝÝëÜ×ÝÐçÕÝØÕÜÚÞàÝïãàÐÒÝÕÝØï,ÝÐÙÔÕÝÝëÜáßÞÓàÕè  ÝÞáâìî " .´ÐÛÕÕ,ÝÐàØáãÝÚÕ9,ßàØÒÕÔÕÝëßàØÜÕàëâÞÓÞ,ÚÐÚØ×ÜÕÝïîâáï ÓàÐÝØæëÞâàÕ×ÚÐ[ a , b ]ÒáÛãçÐïå,ÚÞÓÔÐßÕàÒÐïØÒâÞàÐïßàÞØ×ÒÞÔÝëÕäãÝÚ  æØØ f ( x )ÝÐÒáñÜÞâàÕ×ÚÕáÞåàÐÝïîâßÞáâÞïÝÝëÙ×ÝÐÚ. ÇØáÛÕÝÝëÕÜÕâÞÔë13 ÀØá.9. ¸×ÜÕÝÕÝØÕÓàÐÝØæÞâàÕ×ÚÐÒÜÕâÞÔÕåÞàÔÒáÛãçÐïå,ÚÞÓÔÐßÕàÒÐïØ ÒâÞàÐïßàÞØ×ÒÞÔÝëÕäãÝÚæØØÝÐÒáñÜÞâàÕ×ÚÕáÞåàÐÝïîâßÞáâÞïÝÝëÙ×ÝÐÚ ¼ÕâÞÔÚÐáÐâÕÛìÝëå(½ìîâÞÝÐ) ¼ÕâÞÔÚÐáÐâÕÛìÝëå  ØâÕàÐæØÞÝÝëÙßàÞæÕáá,ÒÚÞâÞàÞÜáâàÞØâáïáØ  áâÕÜÐßÞáÛÕÔÞÒÐâÕÛìÝëåßàØÑÛØÖÕÝØÙ,ßÞÛãçÐîéØåáïßàØßÕàÕáÕçÕÝØØÝÞ  ÒÞÙÚÐáÐâÕÛìÝÞÙáÞáìîÐÑáæØáá.ºÐÖÔÐïÝÞÒÐïÚÐáÐâÕÛìÝÐïÚäãÝÚæØØ áâàÞØâáïÒâÞçÚÕßàÕÔëÔãéÕÓÞßàØÑÛØÖÕÝØï(àØá.10). ²ÞâÛØçØÕÞâÜÕâÞÔÞÒÔØåÞâÞÜØØØåÞàÔ,ÒÜÕâÞÔÕ½ìîâÞÝÐÚäãÝÚæØØ f ( x )ßàÕÔêïÒÛïîâáïÑÞÛÕÕáâàÞÓØÕâàÕÑÞÒÐÝØï,ÐØÜÕÝÝÞ: ˆ äãÝÚæØï f ( x )ÔÞÛÖÝÐÑëâìÝÕßàÕàëÒÝÐÝÐÞâàÕ×ÚÕ[ a , b ]ÒÜÕáâÕáÞ áÒÞØÜØßàÞØ×ÒÞÔÝëÜØ1 - ÓÞØ2 - ÓÞßÞàïÔÚÐ; ˆ ×ÝÐçÕÝØïäãÝÚæØØ f ( x )ÝÐÚÞÝæÐåÞâàÕ×ÚÐÔÞÛÖÝëØÜÕâìàÐ×ÝëÕ ×ÝÐÚØ( f ( a )  f ( b ) 0); ˆ ßÕàÒÐïØÒâÞàÐïßàÞØ×ÒÞÔÝëÕäãÝÚæØØ f 0 ( x )Ø f 00 ( x )ÔÞÛÖÝëáÞ  åàÐÝïâìßÞáâÞïÝÝëÙ×ÝÐÚÝÐÒáñÜÞâàÕ×ÚÕ. ¿ÕàÒëÕÔÒÐãáÛÞÒØïÓÐàÐÝâØàãîâ,çâÞÝÐÞâàÕ×ÚÕ[ a , b ]ÝÐÙÔñâáïåÞâïÑë ÞÔØÝÚÞàÕÝìãàÐÒÝÕÝØï,ÐØ×ÜÞÝÞâÞÝÝÞáâØßÕàÒÞÙßàÞØ×ÒÞÔÝÞÙÒâàÕâìÕÜ ãáÛÞÒØØáÛÕÔãÕâ,çâÞäãÝÚæØï f ( x )ÝÐÔÐÝÝÞÜÞâàÕ×ÚÕÑãÔÕâØÜÕâìÕÔØÝ  áâÒÕÝÝëÙÚÞàÕÝì. ·ÐÜÕçÐÝØÕ .ÂÐÚÚÐÚßÞáâàÞÕÝØÕÚÐáÐâÕÛìÝëåâàÕÑãÕââÞÛìÚÞÞÔÝÞÙâÞç  ÚØ,âÞÒíâÞÜÜÕâÞÔÕÝÕâàÕÑãÕâáïæÕÛØÚÞÜ×ÐÔÐÒÐâìÞâàÕ×ÞÚ,áÞÔÕàÖÐéØÙ ÚÞàÕÝìãàÐÒÝÕÝØï(1.9),ÐÔÞáâÐâÞçÝÞÝÐÙâØÛØèìÝÕÚÞâÞàÞÕÝÐçÐÛìÝÞÕßàØ  ÑÛØÖÕÝØÕÚÞàÝï x = å 0 . 14ÇØáÛÕÝÝëÕÜÕâÞÔë ÀØá.10. ÇØáÛÕÝÝÞÕàÕèÕÝØÕãàÐÒÝÕÝØïÜÕâÞÔÞÜÚÐáÐâÕÛìÝëå ²ëÑÞàÝÐçÐÛìÝÞÓÞßàØÑÛØÖÕÝØï x 0 ×ÐÒØáØâÞâÒØÔÐäãÝÚæØØ f ( x ).½Ð àØáãÝÚÕ11ßàÕÔáâÐÒÛÕÝëçÕâëàÕÒÞ×ÜÞÖÝëåáØâãÐæØØ.²ßÕàÒëåÔÒãåáÛã  çÐïå(aØÑ)ÒÚÐçÕáâÒÕÝÐçÐÛìÝÞÓÞßàØÑÛØÖÕÝØï x 0 ÒëÑØàÐÕâáïâÞçÚÐ Ð ,Ø ßÞáÛÕÔÞÒÐâÕÛìÝëÕßàØÑÛØÖÕÝØï: x 0 = a , x 1 , x 2 ,..., x n ÞÑàÐ×ãîâÞÓàÐÝØ  çÕÝÝãîÜÞÝÞâÞÝÝÞÒÞ×àÐáâÐîéãîßÞáÛÕÔÞÒÐâÕÛìÝÞáâì,ßàØçÕÜ x 0 x 1 ::: x i x i + 1 ::: x  b .(1.16) ²ÞáâÐÛìÝëåáÛãçÐïå(ÒØÓ)ÒÚÐçÕáâÒÕ x 0 ÒëÑØàÐÕâáïâÞçÚÐ b ,ØßÞáÛÕ  ÔÞÒÐâÕÛìÝëÕßàØÑÛØÖÕÝØï: x 0 = b , x 1 , x 2 ,..., x n ÞÑàÐ×ãîâÞÓàÐÝØçÕÝÝãî ÜÞÝÞâÞÝÝÞãÑëÒÐîéãîßÞáÛÕÔÞÒÐâÕÛìÝÞáâì,ßàØçÕÜ a x  ::: x i + 1 x i ::: x 1 x 0 .(1.17) »ÕÓÚÞ×ÐÜÕâØâì,çâÞÒëÑÞàÞÔÝÞÓÞØ×ÞßØáÐÝÝëåáÛãçÐÕÒ×ÐÒØáØâÞâ ×ÝÐÚÞÒßÕàÒÞÙØÒâÞàÞÙßàÞØ×ÒÞÔÝëåäãÝÚæØØ f ( x ).ÁÞáâÐÒØÜâÐÑÛØæãØå ×ÝÐçÕÝØÙÔÛïÚÐÖÔÞÓÞáÛãçÐï. Ð Ñ Ò Ó f 0 ( x ) + � + � f 00 ( x ) � + + � ¸×âÐÑÛØæëÒØÔÝÞ,çâÞÒßÕàÒÞÜáÛãçÐÕ×ÝÐÚØßàÞØ×ÒÞÔÝëåÞâÛØçÝëÔàãÓÞâ ÔàãÓÐ,ÐÒÒÞÒâÞàÞÜáÛãçÐÕ  ÞÔØÝÐÚÞÒë.ÂÐÚØÜÞÑàÐ×ÞÜ,ÕáÛØ f 0 ( x )  f 00 ( x ) 0,âÞÒÚÐçÕáâÒÕ x 0 ÒëÑØàÐÕâáïâÞçÚÐ a ,ØÝÐçÕ  âÞçÚÐ b . ÇØáÛÕÝÝëÕÜÕâÞÔë15 Ð) Ñ) Ò) Ó) ÀØá.11. ²ëÑÞàÝÐçÐÛìÝÞÓÞßàØÑÛØÖÕÝØïÒÜÕâÞÔÕÚÐáÐâÕÛìÝëå ¿ÞÛãçØÜáÞÞâÝÞèÕÝØïÔÛïßÞáÛÕÔÞÒÐâÕÛìÝÞÓÞÒëçØáÛÕÝØïßàØÑÛØÖñÝ  Ýëå×ÝÐçÕÝØÙÚÞàÝïãàÐÒÝÕÝØï x 1 , x 2 ,..., x n .´ÛïíâÞÓÞÝÐßØèÕÜãàÐÒÝÕÝØÕ ÚÐáÐâÕÛìÝÞÙ,ßàÞÒÕÔÕÝÝÞÙÚÚàØÒÞÙ y = f ( x )çÕàÕ×âÞçÚãáÚÞÞàÔØÝÐâÐÜØ å 0 Ø f ( å 0 ): y � f ( x 0 ) = f 0 ( x 0 )( x � x 0 ).(1.18) ²âÞçÚÕßÕàÕáÕçÕÝØïÚÐáÐâÕÛìÝÞÙáÞáìîÐÑáæØáá x = x 1 ,Ð y = 0. ¿ÞÔáâÐÒØÜíâØ×ÝÐçÕÝØïÒãàÐÒÝÕÝØÕ(1.18)ØßÞÛãçØÜÝÞÒÞÕáÞÞâÝÞèÕÝØÕ ÔÛïÒëçØáÛÕÝØï×ÝÐçÕÝØïßÕàÒÞÓÞßàØÑÛØÖÕÝØï x 1 : x 1 = x 0 � f ( x 0 ) f 0 ( x 0 ) .(1.19) °ÝÐÛÞÓØçÝÞÜÞÓãâÑëâìßÞÛãçÕÝëáÞÞâÝÞèÕÝØïÔÛïßÞáÛÕÔãîéØåßàØ  ÑÛØÖÕÝØÙ,ÚÞâÞàëÕßÞÛãçÐîâáïÒàÕ×ãÛìâÐâÕßÕàÕáÕçÕÝØïáÞáìîÐÑáæØáá ÚÐáÐâÕÛìÝëå,ßàÞÒÕÔÕÝÝëåÒâÞçÚÐå( x 1 , f ( x 1 )),( x 2 , f ( x 2 ))Øâ.Ô.ÄÞàÜãÛÐ ÔÛï i + 1ßàØÑÛØÖÕÝØïØÜÕÕâÒØÔ: x i + 1 = x i � f ( x i ) f 0 ( x i ) .(1.20) 16ÇØáÛÕÝÝëÕÜÕâÞÔë ·ÔÕáì,âÐÚÖÕ,ÚÐÚØÒÜÕâÞÔÕåÞàÔ,ÑãÔÕÜàÐááÜÐâàØÒÐâìâÞÛìÚÞâÐÚØÕ äãÝÚæØØ,ÔÛïÚÞâÞàëåØâÕàÐæØÞÝÝëÙßàÞæÕááÝÕÞÑåÞÔØÜÞÒëßÞÛÝïâìÔÞ âÕåßÞà,ßÞÚÐÝÕÑãÔÕâÞÑÝÐàãÖÕÝÞ,çâÞ j x i + 1 � x i j " ,(1.21) ÓÔÕ "  ×ÐÔÐÝÝÐïßÞÓàÕèÝÞáâìÒëçØáÛÕÝØÙ.¿àØíâÞÜßÞÛãçÕÝÝÐïÒàÕ×ãÛì  âÐâÕÒëçØáÛÕÝØÙßÞáÛÕÔÝïïâÞçÚÐ x n ÝÐ×ëÒÐÕâáïßàØÑÛØÖñÝÝëÜ×ÝÐçÕÝØÕÜ ÚÞàÝïãàÐÒÝÕÝØï,ÝÐÙÔÕÝÝëÜáßÞÓàÕèÝÞáâìî " . ´ÐÛÕÕÝÐàØáãÝÚÕ12ßàÕÔáâÐÒÛÕÝßàØÜÕàäãÝÚæØØ f ( x ),ÔÛïÚÞâÞàÞÙÝÕ ÒëßÞÛÝïÕâáïâàÕÑÞÒÐÝØÕ,ÝÐÚÛÐÔëÒÐÕÜÞÕÝÐÕñßàÞØ×ÒÞÔÝëÕ.ºÐÚÒØÔÝÞØ× ßàØÜÕàÐ,ÒâÐÚÞÜáÛãçÐÕÜÕâÞÔ½ìîâÞÝÐÜÞÖÕâßàØÒÕáâØÚÝÕÚÞààÕÚâÝÞ  ÜãàÕèÕÝØî,ÝÐßàØÜÕà,ÞçÕàÕÔÝÞÕßàØÑÛØÖÕÝØÕÜÞÖÕâÒëÙâØ×ÐßàÕÔÕÛë ÞâàÕ×ÚÐ[ a , b ],ÓÔÕäãÝÚæØïÜÞÖÕâÑëâìÝÕÞßàÕÔÕÛÕÝÐØÛØØÜÕâìÞáÞÑÕÝ  ÝÞáâØ. ÀØá.12. ¿àØÜÕàÒëßÞÛÝÕÝØïÜÕâÞÔнìîâÞÝÐÔÛïäãÝÚæØØ,ÜÕÝïîéÕÙ×ÝÐ  ÚØáÒÞØåßàÞØ×ÒÞÔÝëåÝÐ×ÐÔÐÝÝÞÜÞâàÕ×ÚÕ

Приложенные файлы

  • pdf 15337853
    Размер файла: 236 kB Загрузок: 0

Добавить комментарий