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解题方法
1 . 若关于
的二项式
的展开式中各项的系数和为
,则![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4904f9ba87703283e13a44f3823b6758.png)
__________ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2466748f730da3899ef46d3e0410a98f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9c1dfc770fc27e0b9bf602ec2f842ae9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4904f9ba87703283e13a44f3823b6758.png)
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2 . 知识卡片: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.当
,
时,有如下表达式:
,两边同时积分得:
,从而得到如下等式:
请根据以上材料所蕴含的数学思想方法,由二项式定理
计算:![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b7fe8f27f7d0118b645b0577c990cb9f.png)
_______ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4fe7d5809da02c15a43a0e9a898b9086.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4776c85b79df196f606d3ebf3697fbc3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/babadc15694ea4139b1bb919a7d49b93.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2eb0e20a88409f5d7e899876d9d5ef09.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/24a57996290794e082b21d8f1dfc322a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c354be10f49f79ca7fdd3da9837a9b5a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2079e9798470edc75b66126cf06da150.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4a8306906c6ffde45231e08776c0fea7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e4613b9e01d001fab00a2f288d28b782.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e3f303360ac002854ad3d63a5fca122e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b7fe8f27f7d0118b645b0577c990cb9f.png)
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3 . 已知
,则二项式
展开式中的常数项为______ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ecc4e97fd5fcd7d200bcbfb12fa5dfd6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fd7e8844bb80ddb2c97a24934f122545.png)
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4 . 设
,则二项式
的展开式中
项的系数是_______ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/db3bc9597f8bea5a8c9ed020fbf086b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c10145ee48f6e60562a01e954da13cef.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9b0a89e3c30f6e4d4c5db4378b05d987.png)
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5 . 设
,则![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a36e682e4eea73c6357c3b06ff29bbdd.png)
________ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/77a94fd1ee96531561137dd0a48fbde9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a36e682e4eea73c6357c3b06ff29bbdd.png)
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6 . 我们知道通过牛顿莱布尼兹公式,可以求曲线梯形(如图1所示阴影部分)的面积
,其中
,
.如果平面图形由两条曲线围成(如图2所示阴影部分),曲线
可以表示为
,曲线
可以表示为
,那么阴影区域的面积
,其中
.
在区间
与
的图形分别为直径为1的上、下半圆周,在区间
与
的图形分别为直径为2的下、上半圆周,设
.求
的值;
上某一个点处作切线,便之与曲线和x轴所围成的面积为
,求切线方程;
(3)正项数列
是以公差为d(d为常数,
)的等差数列,
,两条抛物线
,
记它们交点的横坐标的绝对值为
,两条抛物线围成的封闭图形的面积为
,求证:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f102cb4c683b01f1cd728f543f703f4c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/924840404483fbdef9af58e844c001ad.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/babadc15694ea4139b1bb919a7d49b93.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b1241216f3c1cb5e73043dd1037f556d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9bc1742d6a5f52c750720b3f099c3fcc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/23f3ffe7abc59e2f65d827c8eab8d36a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cfbd9b2c33ae11831377f140b728ca50.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/38b668edb1a8974b1e882f78178b265c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1a8d35b10249abb8a2d50677f56db638.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0afb80007983e5b99dcdeebf87d18ff4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8ac9833eb592cadc3503eebc041aab21.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/376cd70523564eb2a9b8509ca5fec6ce.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b1bc5c4dc60dacc4d88d0e929574c0f6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/62c2cc4cd1e8bcb4b75b6e799156736e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d7f76d4f2878eb5a4879b74719b8a69c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/12c3b46e63249a782f911667bbd68e9d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/61f3ad8e800d5fde4f5e48567509a074.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09d7abf02717d6e59d8a64a65a87c412.png)
(3)正项数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f329b217e1051b23f0d61023cdc6e69.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c4ce64685821c3e55c07f151996ca8c3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/59dd6c97d2ee3e74ba5730f1cbcc1d43.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9b0c92880b4a88b78d0324564628be0e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a8c5c0257094e981bec043dfa99a6373.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/96abfe2da27a63e6affb19a0c80236d9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/08eb71ecf8d733b6932f4680874dbbf3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9929daf4497eb9657b95b17b212a0886.png)
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解题方法
7 . 已知正数
满足
,则
的最大值为( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/632244ea6931507f8656e1cc3437d392.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f9ae4d4b27c467f928bc36aebd488be5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ff18e4d66fe16878da060b41f17280cd.png)
A.![]() | B.![]() | C.![]() | D.![]() |
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8 . 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
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7卷引用: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)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(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
9 . 设
,
且
,则
的最小值是____________ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/94440d3e4c073f94f2b266ff99d50e74.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/67ca5fd57c2c2fcc3c7a574fdd1467d9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f8e951ae5addb6fee01e110998e1e4a2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c00ec2a6dce11c20d484c9d5eaec419b.png)
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10 . 函数
与函数
的图像围成一个封闭图形,这个封闭图形的面积是______ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fea7001a3b28ef91de34ca1263e1fa22.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c50866229ec5a3640fb250f9bd2192b3.png)
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2023-01-06更新
|
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