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)
您最近一年使用:0次
名校
解题方法
2 . 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,如果两数
在区间
上的图象连续不断,用分点
将区间
等分成
个小区间,在每个小区间
上任取一点
(
,2,…,n),作和式![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b13820d7479810174b76da70f2b0c91a.png)
(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,x叫做积分变量,
叫做被积式.从几何上看,如果在区间
上函数
的图象连续不断且恒有
,那么定积分
表示由直线
,
,
和曲线
所围成的区域(称为曲边梯形)的面积.
知识卡片2:
一般地,如果
在区间
上的图象连续不断,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.例如,如图所示,对于函数
(
),从几何上看,定积分
的值为由直线
,
,
和曲线
所围成的区域即曲边梯形
的面积,根据微积分基本定理可得
.
①
;
②
;
③
;
④
.
(2)已知
,计算:
①
;
②![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d8be82da06bf4bad8efa5c65a4d960a2.png)
(3)当
,
时,有如下表达式:
.计算:
知识卡片1:
一般地,如果两数
![](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/96c9975b7cb327da8634aabab7856095.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7c45176df950dfe48b8ca7eac08ee349.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b13820d7479810174b76da70f2b0c91a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8eb09a0fb843fbc6959ab5688771ee01.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/50b037afdaa0872d5004fe3e8206ee89.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/4471a3b99ea6b431271b76b7931cf272.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7d0112f1dbb9abb6bbffc3ccfb77ba03.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/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/4471a3b99ea6b431271b76b7931cf272.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/54b53b86bd516400d6fa7dabb3603f31.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/82b334dafda377c3db77647c8cf1e95f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e2a7df955fc17e92fd86302f8c34664a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0afb80007983e5b99dcdeebf87d18ff4.png)
知识卡片2:
一般地,如果
![](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/d4dd421af6cde8dad1ef435eb12cc45a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a71da6f8d70a273a75779e1aa818d4f9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/60e712d5093491bf63ab4cdd34e8ca44.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/08115d6d9f876dea921a4d32260ff1fb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/644ba16341e356b57ea153e840555290.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/54b53b86bd516400d6fa7dabb3603f31.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/82b334dafda377c3db77647c8cf1e95f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e2a7df955fc17e92fd86302f8c34664a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f42b2a9736c8943106472a7398d2892.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fe83f1ae7e5f05d8bed6bf6f42db0e7d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8e16dc5c9406c6cdadbca9136ef97b99.png)
①
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cc6bfbb31a1766fea0bf95473044d34b.png)
②
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1bdb735861115bfe3412c93538c4a522.png)
③
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ea107a985b40ed680a43dfb1fb9c2535.png)
④
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8dfd1660244935cb37ca57a08a26b05b.png)
(2)已知
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a2e58d66635ee79693d107c8f3c49c5a.png)
①
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bed11655d89fb2e5363377fc38d53d7c.png)
②
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d8be82da06bf4bad8efa5c65a4d960a2.png)
(3)当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/eb63478132d4c1fef3c17e591919da83.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/71842069f1fda57351db2de2e23a3262.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2079e9798470edc75b66126cf06da150.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b1bdc90b519006d41102a40a47a9fd87.png)
您最近一年使用:0次
名校
3 . 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
|
2437次组卷
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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
名校
4 . 如图,设D是图中边长分别为1和2的矩形区域,E是D内位于函数y=
(x>0)图象下方的区域(阴影部分),从D内随机取一个点M,则点M取自E内的概率为
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cf35027e76f8ea593f82023973d4aba3.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2023/5/29/ccd7d99d-2eea-44b1-8a76-f1636c587d5b.png?resizew=122)
A.![]() | B.![]() |
C.![]() | D.![]() |
您最近一年使用:0次
2016-12-04更新
|
1861次组卷
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4卷引用:2015-2016学年贵州遵义一中高二下联考理科数学试卷
2015-2016学年贵州遵义一中高二下联考理科数学试卷2017届湖南长沙长郡中学高三入学考试数学(理)试卷2020届湖南省岳阳市第一中学高三上学期第一次质量检测数学(理)试题(已下线)专题25 统计类(解答题)+概率(几何概型)-2