名校
解题方法
1 . 一般地,设函数
在区间[a,b]上连续,用分点
将区间[a,b]分成
个小区间.每个小区间长度为
.在每个小区间
上任取一点
作和式
.如果
无限接近于0(亦即
)时,上述和式
无限趋于常数
,那么称该常数
为函数
在区间[a,b]上的定积分,记为
.当
时,定积分
的几何意义表示由曲线
,两条直线
与
轴所围成的曲边梯形的面积.如下图所示:
是区间[a,b]上的连续函数,并且
,那么![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b5f992bc847d76afe46f63c83dc32b85.png)
(1)求
;
(2)设函数
.
①若
恒成立,求实数
的取值范围;
②数列
满足
,利用定积分的几何意义,证明:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4fe7d5809da02c15a43a0e9a898b9086.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2779faf49c4c603fdb73ef6f03cc8d82.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2adc9f9006f4b099bcd85a3d3432da15.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/78d74528a8974c188d81391d4f158c28.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/173ccb5cf09e9a104c7d9c969801463b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e827229de8424d07fe1f5d4dfb8b0dd6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f44febc79ae3b32724dcbaf76b835ab5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a107eb946e0fe41629c644b7628d5cba.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/08eb71ecf8d733b6932f4680874dbbf3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cf231f8f86fb922df4ca0c87f044cec3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cf231f8f86fb922df4ca0c87f044cec3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4fe7d5809da02c15a43a0e9a898b9086.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1138c0cc8c4a956a413bd3300337e2b1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/22e38c541dec8fce1d26886e5ef7d21f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/21e26a7c7b8d42a7752b6d3c508d8345.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/51c530f4b7491b95acb8ce3eef9aa09d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/436ff3cf58de28b55f7605675a47d818.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4fe7d5809da02c15a43a0e9a898b9086.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/babadc15694ea4139b1bb919a7d49b93.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b5f992bc847d76afe46f63c83dc32b85.png)
(1)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a0452f7a65f8d088836241db2af124e.png)
(2)设函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f6fd2f18661c82c289ffe94052dac8f7.png)
①若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f9da87f71c121cb80a1120f14aa525a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/294f5ba74cdf695fc9a8a8e52f421328.png)
②数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/76aef4cdcb5af742ce28003b7b6c8c20.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/598da453256520641a243aba79e072d3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b7c9d14bfa527b5ba538cc3960e9396f.png)
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名校
2 . 如果函数
的导数
,可记为
.若
,则
表示曲线
,直线
以及
轴围成的“曲边梯形”的面积.
(1)若
,且
,求
;
(2)已知
,证明:
,并解释其几何意义;
(3)证明:
,
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/46be55c8f2760d6db125f46691a3de48.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83d4d758bac9a7272c1d40a5ea4176c2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/edd8f5b33be6db5be0833f1801bd7a46.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9e9c599e8d420006448905acec2b8234.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8c6a5e6776e205fb09d8a689e1638947.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/942c2141d01bde6b48210c56a17fc75e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/436ff3cf58de28b55f7605675a47d818.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
(1)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c8ed0afb829f4d5c61ce89a556376d93.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4a0dc2a031743126b8b4fabb843a55bf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/46be55c8f2760d6db125f46691a3de48.png)
(2)已知
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1bc282dae4ac9132196ac5d13f63b901.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9c38abf9dbef1c45d9fd8143798fa0ea.png)
(3)证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e59176a49cf2e21c94cf550888de88c5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a37a59558292ad6b3d0978bfd7484990.png)
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2024-02-20更新
|
2426次组卷
|
7卷引用:重庆市第八中学校2023-2024学年高三下学期入学适应性考试数学试题
重庆市第八中学校2023-2024学年高三下学期入学适应性考试数学试题(已下线)压轴题函数与导数新定义题(九省联考第19题模式)练(已下线)微考点2-5 新高考新试卷结构19题压轴题新定义导数试题分类汇编湖北省十一校2024届高三联考考后提升数学模拟训练一湖北省黄冈市浠水县第一中学2024届高三下学期第三次高考模拟数学试题(已下线)第5套 新高考全真模拟卷(二模重组)(已下线)压轴题01集合新定义、函数与导数13题型汇总-2
解题方法
3 . 已知抛物线
的焦点为
,过点F作直线l交抛物线C于A,B两点.椭圆E的中心在原点,焦点在x轴上,点F是它的一个顶点,且其离心率
.
![](https://img.xkw.com/dksih/QBM/2015/7/9/1572177036853248/1572177042956288/STEM/de637f1525654a66b84d3f121f79e1a0.png)
(Ⅰ)分别求抛物线C和椭圆E的方程;
(Ⅱ)经过A,B两点分别作抛物线C的切线
,切线
相交于点M.证明
;
(Ⅲ)椭圆E上是否存在一点
,经过点
作抛物线C的两条切线
(
为切点),使得直线
过点F?若存在,求出抛物线C与切线
所围成图形的面积;若不存在,试说明理由.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/47c66f4296aff5fe8eda04b8dbb1157d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ad43fd96bf0f234fcb7de9e571d99db7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5de85df85401e7e8da683ea4a784963c.png)
![](https://img.xkw.com/dksih/QBM/2015/7/9/1572177036853248/1572177042956288/STEM/de637f1525654a66b84d3f121f79e1a0.png)
(Ⅰ)分别求抛物线C和椭圆E的方程;
(Ⅱ)经过A,B两点分别作抛物线C的切线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/44434b647ec546fe787e2164e0be6cd2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f35c75fa397cd9657012887e09d65695.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4e0e405f595799e1b154cf1655c32945.png)
(Ⅲ)椭圆E上是否存在一点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/26e633f9d68db5891ab79a10d758a113.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/26e633f9d68db5891ab79a10d758a113.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6d0303e4d06db4a49e9b6725b1f4b623.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6ddb94a36a7b710b090f600ad0a1ff24.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6db576bfd903bd4525e6f6568b7d52b9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6d0303e4d06db4a49e9b6725b1f4b623.png)
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