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1 . 阅读知识卡片,结合所学知识完成以下问题:知识卡片1:一般地,如果函数
在区间
上连续,用分点
将区间
等分成
个小区间,在每个小区间
上任取一点
,作和式
(其中
为小区间长度),当
时,上述和式无限接近某个常数,这个常数叫做函数
在区间
上的定积分,记作
即
.这里,
与
分别叫做积分下限与积分上限,区间
叫做积分区间,函数
叫做被积函数,
叫做积分变量,
叫做被积式.从几何上看,如果在区间
上函数
连续且恒有
,那么定积分
表示由直线
和曲线
所围成的曲边梯形的面积.知识卡片2:一般地;如果
是区间
上的连续函数,并且
,那么
.这个结论叫做微积分基本定理,又叫做牛顿-莱布尼茨公式.
(1)用定积分表示曲线
及
所围成的图形的面积,并确定
取何值时,使所围图形的面积最小;
(2)一列火车在平直的铁轨上行驶,由于遇到紧急情况,火车以速度
(单位:
)紧急刹车至停止.求:
①求火车在刹车4秒时速度的瞬时变化率(即4秒时的瞬时加速度);
②紧急刹车后至停止火车运行的路程.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d275fbb3ee5cd1177ca5a2ceecbbef0f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8ca6d68f1de3e70696f1d5d60affe6ef.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2779faf49c4c603fdb73ef6f03cc8d82.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8ca6d68f1de3e70696f1d5d60affe6ef.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/78d74528a8974c188d81391d4f158c28.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3d3b6ed0cf8753da31759fcdec5e2f4b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6b41a531cbf0efb7c364d373dd278df1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f44febc79ae3b32724dcbaf76b835ab5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/19893fdb67307d35a9115ef4f3f1202a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d275fbb3ee5cd1177ca5a2ceecbbef0f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8ca6d68f1de3e70696f1d5d60affe6ef.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/024c4cff71b59cd710d9e69618ed0428.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/357985155afb3e4fb692de8426162ea2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2c94bb12cee76221e13f9ef955b0aab1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8ca6d68f1de3e70696f1d5d60affe6ef.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d275fbb3ee5cd1177ca5a2ceecbbef0f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6b5f852626c9a09019c20fba7b388183.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8ca6d68f1de3e70696f1d5d60affe6ef.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d275fbb3ee5cd1177ca5a2ceecbbef0f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/58196b9e63ec00aa1119052b6de6ae12.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/024c4cff71b59cd710d9e69618ed0428.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c0b37d2ec9e139b8043263ec01024cd8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0afb80007983e5b99dcdeebf87d18ff4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d275fbb3ee5cd1177ca5a2ceecbbef0f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8ca6d68f1de3e70696f1d5d60affe6ef.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/31fb89dedbd069ce7f90dc1781c801b1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c2bc109bbee91a3a9e3206575324cd24.png)
(1)用定积分表示曲线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b4f0bf1a1e12f085c845efd618ba0434.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e2a7df955fc17e92fd86302f8c34664a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f0a532e15e232cb4b99a8d4d07c89575.png)
(2)一列火车在平直的铁轨上行驶,由于遇到紧急情况,火车以速度
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/aa8c0428e9b02facf1a503377e3e68fc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/53afeaf21f93091b71608d21540be239.png)
①求火车在刹车4秒时速度的瞬时变化率(即4秒时的瞬时加速度);
②紧急刹车后至停止火车运行的路程.
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解题方法
2 . 帕德近似(Pade approximation)是有理函数逼近的一种方法.已知函数
在
处的
阶帕德近似定义为:
,且满足:
,
,
,….又函数
,其中
.
(1)求实数
,
,
的值;
(2)若函数
的图象与
轴交于
,
两点,
,且
恒成立,求实数
的取值范围.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/abfb2c1441a7d94cf142af07fa69c062.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bb45f673c56a289ea78831c9237e8d20.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b2158b65e10dbd08c2cb1e265c55f578.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a7ab02976c65cd2523a875b23afbff91.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/67a9428f7efe344ff19d910626bc7b3c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c378a9dead44c9e42f438191dc80032d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e6b5cdadafa6454202069ffa98507aff.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ea0753b6f262da7b99776ae7a403d777.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7577b18ba31abfe26b6677f191a2e512.png)
(1)求实数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2c94bb12cee76221e13f9ef955b0aab1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/071a7e733d466949ac935b4b8ee8d183.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5c02bc0c74292b1e8f395f90935d3174.png)
(2)若函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d275fbb3ee5cd1177ca5a2ceecbbef0f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0fe622d63eb6d0d9568e4ef85deff47e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b34779af6b2c2b139c32c94104f01088.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/26d8dafc71b106f39f4e15442220897b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6bb0f7f3ff2c266a03d45a368ddacd7f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/294f5ba74cdf695fc9a8a8e52f421328.png)
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