给定积分 [tex=6.929x2.786]u5/riQTd+DtIC9kBnmlD4Fm0B8eeI54pBfoFTNqTUxR5gc2a509fqnbETnF9nMff[/tex][tex=1.286x1.357]VAHhaW1te0xvoqDVN54/dg==[/tex]利用复化梯形公式计算上述积分值,使其截断误差不超过[tex=4.214x2.357]P6uidfEImc5vmG7Z7jgYkLq/7VShnI2Kjyq9nexWJF4=[/tex][tex=1.286x1.357]BEB68bP4vOVk/XYYizw11w==[/tex]取同样的求积节点,改用复化 [tex=3.929x1.214]8J0egaEhuUVWr6XiydqGew==[/tex]公式计算时,截断误差是多少?[tex=1.286x1.357]H6tHfFjOZ3ZWdB4qPQ9Ocg==[/tex]要求截断误差不超过[tex=2.0x1.214]cWZ6JdRNLl/fAGGLvtfcxw==[/tex],如果用复化 [tex=3.929x1.214]8J0egaEhuUVWr6XiydqGew==[/tex]公式,应取多少个函数值?

2022-05-28

给定积分 [tex=6.929x2.786]u5/riQTd+DtIC9kBnmlD4Fm0B8eeI54pBfoFTNqTUxR5gc2a509fqnbETnF9nMff[/tex][tex=1.286x1.357]VAHhaW1te0xvoqDVN54/dg==[/tex]利用复化梯形公式计算上述积分值,使其截断误差不超过[tex=4.214x2.357]P6uidfEImc5vmG7Z7jgYkLq/7VShnI2Kjyq9nexWJF4=[/tex][tex=1.286x1.357]BEB68bP4vOVk/XYYizw11w==[/tex]取同样的求积节点,改用复化 [tex=3.929x1.214]8J0egaEhuUVWr6XiydqGew==[/tex]公式计算时,截断误差是多少?[tex=1.286x1.357]H6tHfFjOZ3ZWdB4qPQ9Ocg==[/tex]要求截断误差不超过[tex=2.0x1.214]cWZ6JdRNLl/fAGGLvtfcxw==[/tex],如果用复化 [tex=3.929x1.214]8J0egaEhuUVWr6XiydqGew==[/tex]公式,应取多少个函数值?

答案：

 解[tex=1.0x0.286]QWpvogsIlU0ILJld1RPl/A==[/tex] 由于[tex=12.286x2.786]cUpWZ5FLuDIWKEpLMrDIGMvy5ilnME81SRLw6Vimo4eb3Ilo7ju/+S7uPbyLSH27L/Insm8h98aYsAhCCywJUA==[/tex]所以[tex=23.0x2.857]mSbprtsq8joRU2WEP6cXTQXeOCqxDkRKFcPN2rxHmEKP/GOEALlZilrFb04+T7kLI7LS/FMCpHrBEohWar+F2DSXHAguHeJyRj7T74lw/H9S67qs0S8SI2RALo9M5nPRlYg1EJ8kJ9vyuxu92XY5AypTc5etmLy9loNHHMFmDo4=[/tex]故[tex=25.0x2.857]rnBs/1qu96CbrROSAy8r/PQu7sX+aFnYnNuMnUc78Y4tuAAwDY/k1aZLRlt6cvyW9DcdpkUieB2nDwhgR8e9eYmGchYnUOKfhRmgncDfw2c7Ul2bFKtZl3Ban+7iwlXL9UuCKzxniqZPazIsWOTd5ffvtXNqfRTMZ3G2IjCuHHDsIaLEa8evawrPkU3ChB98[/tex](1) 为使复化梯形公式满足误差要求,只需[tex=20.5x2.429]MvIJ2EP+s68pFgR5Qt5cHJs8Ut6URBw+O2ngnAMsK5ibaIeRez9+SmZO7maeb50StBUqw0XguXPED5B/pPv/BKLsHGFUiXEFw+uBF2SQwzl4Vkm0IkbFgUxjrYNHnNeqst0+Ext8NXrD72/BBTSmuyA764SJfzeY6B9pot/IW9Av+3n120Z23dSxLI9KQjGj1fumpNWowbLo3XOuV9V0/w==[/tex]即可,这只需[tex=5.0x1.143]x/F82rWu5MKkv4Okryes2jIWQ6eJSL7IUay7SCXoFKI=[/tex][tex=13.929x2.429]GOMsfXfn0PMzBQoZD9mE+QjTKpyBIfP+Zl5FAArQlT3EgWARozrotoDVCNS0vJgUGm9ImMqvWEBdmoSuN9m6tYwFbUbSRKrNf1tS4LY+Mrk=[/tex]故只需[tex=0.5x1.0]hdFTVbNvvzh5T04p00SpZA==[/tex]等分即可 [tex=0.286x0.429]1uLAO+Lu1/udYB+tMUePcQ==[/tex] 此时 [tex=5.786x2.357]Kdqdftsl9HGVy/28s9iwmAkaeQ08abVSEmQ07WQwtPQ=[/tex] 则[tex=59.357x2.643]ifE9NWj3X6IpRVSt3T5ITq+Q6/wtt0XqiriM8wKj2at2vDfmCzWSVvTr2iWebBjOeQRIW001Rr5q69h/uGCB+7xKtEyU3UU/0JQNO9JHQXv4xF035ge248S+ceiB12CLk2c70v1QrDbZ/BhPOcc+AEZUOyM6zh9qIi204hO13ouYYLsAdsRYb1lyfCsVV/lWxlioi76yP6leTSpVvO556675aNpu51qdU0jZmMIaa9hdsQJdfjRJOlBgDwb4gmD4[/tex](2) 对于同样点数用复化[tex=3.929x1.214]F7yCp5gW1sx/ruB8g8A2LQ==[/tex]公式时, [tex=2.714x2.357]GAE3VEQ4FPM0LTgr8kz/2r/2wLgEJZBsS6HGbJDmvP8=[/tex]其截断误差为[tex=19.714x2.929]MvIJ2EP+s68pFgR5Qt5cHMN8gObyhY1fdVIZbaUDEAqrUnPP/tQfQ5GqYO78+JaAlPlxmGQYBzMrvYUrXzunwujoP32oCpa1MxOZTFvFZ099mDqRxuFwBVqFwkec6ekWhSrQbfUmLWPmNWIx/RDMgE660d7fhh2hIhW3v4opWbJQHzYfhFHduBmYMbwZOHAT[/tex]           [tex=12.571x1.357]pY3mV5t5nF/mwtQwfRXJdoPp5RXLgSsjzL8J3AnU4dwsXWyPD8pzVjQi7abeoNnQ[/tex](3)为了在使用复化[tex=3.929x1.214]F7yCp5gW1sx/ruB8g8A2LQ==[/tex]求积公式时误差不超过 [tex=2.0x1.214]cWZ6JdRNLl/fAGGLvtfcxw==[/tex], 只需[tex=17.571x2.429]MvIJ2EP+s68pFgR5Qt5cHMN8gObyhY1fdVIZbaUDEAqrUnPP/tQfQ5GqYO78+JaAlPlxmGQYBzMrvYUrXzunwujoP32oCpa1MxOZTFvFZ09rnBDwBxWolrRo+NMrkzFAkJYOu88XErdv6TbK6hny7A==[/tex]          [tex=11.357x2.929]yohS65Mghe8K0AlsSgyv1MiqNojotPeSF6q05sN1rnPoTzDNljTCmSffGcWbY9KsJngEQppnKkxBCZ+tlbVo0A==[/tex]解得   [tex=13.571x1.429]nEy0uuf3vElufiXJ4U7j1ucpRycDH+V/UbKYo2nVbkC+pv9RK5uj6iWzBun45BbfO4dK5KpCzdWzOGwvHXxp5w==[/tex]故至少需将 [tex=2.5x1.357]3pwgUFtwXn0Y73oZcPn4eg==[/tex]等分[tex=0.286x0.643]eoHgZ4i6w/0ujLXGXx7//g==[/tex]即取 [tex=5.286x1.143]bXa9lo/dIVftGhqdqLkq6AQm1Ljeq0rAp/YF2wM7n5Q=[/tex]个节点处的函数值.

举一反三

内容

0
利用积分[tex=6.357x2.786]y3OnBNwmvoSQwf6DEdQLFS/vQOUzzj8Z8pW7deCZvLxZKBY6P9Eco1MITeDnO4X0[/tex]计算[tex=1.5x1.0]vHPanl9xxtfjpApeUfUVPA==[/tex]时，若采用复化梯形公式，问应取多少节点才能使其误差绝对值不超过[tex=3.929x2.357]KSSvp6yk+g29yNLh3G8dLOPmCzpQ7lROBVIxbiG8fKU=[/tex]
1
计算积分[tex=5.429x2.429]CchnCnJhcZl/OuDxtq4CRYiPiYJFJgjdRvFd9n6+Y/o=[/tex]，若用复化梯形公式，问区间[tex=1.929x1.286]5WiKxiqIs2aMQ1aNQurkGw==[/tex]应分多少等分才能使截断误差不超过[tex=3.929x2.0]CfPmNn4X7uMAwmerwhHuvzjlcaCAJKamkLHZbclCcyQ=[/tex]？若改用复化辛普森公式，要达到同样精度区间[tex=1.929x1.286]5WiKxiqIs2aMQ1aNQurkGw==[/tex]应分多少等分？
2
已知 [tex=9.5x2.357]AgUz88WXmNsIIZtPYh5ZGzJlujm/5vUEg9XT93YZOhE7uuQFPEIRJDApBpMyd2GViTzDvLO2XxIaDdjyv7C9yQ==[/tex][tex=1.286x1.357]VAHhaW1te0xvoqDVN54/dg==[/tex]推导以这三个点作为求积节点在[tex=2.0x1.357]pL+9s9nh77uX8/Gl5SRykA==[/tex]上的插值型求积公式，[tex=1.286x1.357]BEB68bP4vOVk/XYYizw11w==[/tex] 指明求积公式所具有的代数精度，[tex=1.286x1.357]H6tHfFjOZ3ZWdB4qPQ9Ocg==[/tex]用所求公式计算[tex=4.143x2.786]Pdz2GHkIEDDQHxTDEFSF/zghPG0voo2G12sHNdAQbxc=[/tex]
3
计算下列图形的对称性群:[tex=1.286x1.357]VAHhaW1te0xvoqDVN54/dg==[/tex] 正五边形;[tex=1.286x1.357]BEB68bP4vOVk/XYYizw11w==[/tex] 不等边矩形;[tex=1.286x1.357]H6tHfFjOZ3ZWdB4qPQ9Ocg==[/tex] 圆.
4
分别运用梯形公式、[tex=3.929x1.214]8J0egaEhuUVWr6XiydqGew==[/tex]公式、[tex=2.5x1.0]8ZpxQgDGBRSJXlz0pQ/a5g==[/tex]公式计算积分[tex=4.071x2.786]Pl/c5yC7qtagsjcGCe732uar830kWNotJsAa/Gz4Y1o=[/tex]并估计各种方法的误差[tex=0.429x1.357]ljx4OiPNKel/qklZEW5k2A==[/tex]要求小数点后至少保留 5 位 [tex=0.714x1.357]Aogmvz/qbXcwD2jzVC5Lkw==[/tex]