名校
1 . 已知函数
,其中
且
.
(1)讨论
的单调性;
(2)当
时,证明:
;
(3)求证:对任意的
且
,都有:
…
.(其中
为自然对数的底数)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7d9471f77a4cd41501471bd85c48d34b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e1e69392d21261afd8e5e5f096634669.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/20849c00c47cbdc43f18d53341b6c4e5.png)
(1)讨论
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4fe7d5809da02c15a43a0e9a898b9086.png)
(2)当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0b550ee821ee1838384835e81fc34b67.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1413a67adedc88a492a3f2e21e426961.png)
(3)求证:对任意的
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cea4ac187cbb465180e89f38250b3970.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0704f453b2de48d36911f7db496bbf82.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52daa0cdc945df33fd98a43b930b71f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f663883e5e739184a7fc18c72a7b62ff.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e25da8298b6a96d627f3e8c990e55f0c.png)
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|
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11卷引用:湖北省郧阳中学、恩施高中、随州二中、襄阳三中、沙市中学2022-2023学年高二下学期四月联考数学试题
湖北省郧阳中学、恩施高中、随州二中、襄阳三中、沙市中学2022-2023学年高二下学期四月联考数学试题湖北省部分重点高中2022-2023学年高二下学期4月联考数学试题重庆市西南大学附属中学2019-2020学年高二下学期阶段性测试数学试题苏教版(2019) 选修第一册 选填专练 第5章 微专题十五 函数、导数与不等式的综合应用重庆市实验中学2021-2022学年高二下学期第一次月考数学试题辽宁省沈阳市东北育才学校2021-2022学年高二下学期4月月考数学试题四川省泸州市泸县第一中学2021-2022学年高二下学期期中数学理科试题江苏省南通市通州区金沙中学2022-2023学年高二下学期5月学业水平质量调研数学试题(已下线)第二篇 函数与导数专题4 不等式 微点9 泰勒展开式(已下线)第三章 重点专攻二 不等式的证明问题(讲)(已下线)专题11 利用泰勒展开式证明不等式【讲】
名校
解题方法
2 . 设
,函数
.
(1)证明:当
时,
恒成立
(2)若函数
无零点,求实数a的取值范围
(3)若函数
有两个相异零点
,求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/94440d3e4c073f94f2b266ff99d50e74.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/819684642bf77a426d2738984bf7ac03.png)
(1)证明:当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0fde64f4d3c38e43fbdee24eadc4b0dd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b17af43cc460a6a7010d51a0c9403d67.png)
(2)若函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
(3)若函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8ce7ae90d808f05e86ea063238e4b2f9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1b4900c67f4b57fa430c4bd863f8e896.png)
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|
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名校
3 . 已知函数
,其中
.
(Ⅰ)讨论
的单调性;
(Ⅱ)当
时,证明:
;
(Ⅲ)求证:对任意正整数n,都有
(其中e≈2.7183为自然对数的底数)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fa9f7cb75c5500ad56dfe0f178dedb92.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e1e69392d21261afd8e5e5f096634669.png)
(Ⅰ)讨论
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4fe7d5809da02c15a43a0e9a898b9086.png)
(Ⅱ)当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0b550ee821ee1838384835e81fc34b67.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/257810d08006d4b886331966c99767ea.png)
(Ⅲ)求证:对任意正整数n,都有
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9bf0f4b1e329db4bf6070f993297f9b9.png)
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2019-01-12更新
|
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10卷引用:湖北省武汉二中2019-2020学年高二下学期4月第二次线上测试数学试题
湖北省武汉二中2019-2020学年高二下学期4月第二次线上测试数学试题【全国百强校】四川省成都市成都外国语学校2018-2019学年高二下学期期中考试文科数学试题四川省宜宾市第四中学校2019-2020学年高二下学期期中考试数学(理)试题四川省宜宾市第四中学校2019-2020学年高二下学期期中考试数学(文)试题广东省佛山市三水区三水中学2019-2020学年高二下学期第二次统考数学试题【区级联考】天津市蓟州等部分区2019届高三上学期期末联考数学(文)试题【区级联考】天津市部分区2019届高三(上)期末数学(文科)试题【全国百强校】河北省武邑中学2019届高三下学期第一次模拟考试数学(文)试题江西省五市八校2019-2020学年高三第二次联考文科数学试题黑龙江省大庆实验中学2019届高三普通高等学校招生全国统一考试文科数学模拟试题
4 . 已知函数
,
为自然对数的底数.
(1)求
的单调区间;
(2)证明:
,
;
(3)当
时,求证:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0b47d4c5d3ddd3ce7f949670d36f974.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/041a7c8fc017f596542c5e6ec7d1c40b.png)
(1)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4fe7d5809da02c15a43a0e9a898b9086.png)
(2)证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0cb81489349bfb327ee7d410735cbc2d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/66c14c9d8bdcf15c6868b91ec14e53bc.png)
(3)当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0704f453b2de48d36911f7db496bbf82.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0fd2427b6f87eb47a9377cb133ae4469.png)
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名校
解题方法
5 . 已知![](https://staticzujuan.xkw.com/quesimg/Upload/formula/163de36d71222265637ad0f150f25f1b.png)
(1)当
时,求证:![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2018f19af87c5bc26c72d7ef037cf621.png)
(2)若
恒成立,求
的取值范围.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/163de36d71222265637ad0f150f25f1b.png)
(1)当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0b550ee821ee1838384835e81fc34b67.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2018f19af87c5bc26c72d7ef037cf621.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4eb13306786482b7de08d446f0742ab7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
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名校
解题方法
6 . 曲率是曲线的重要性质,表征了曲线的“弯曲程度”,曲线曲率解释为曲线某点切线方向对弧长的转动率,设曲线
具有连续转动的切线,在点
处的曲率
,其中
为
的导函数,
为
的导函数,已知
.
(1)
时,求
在极值点处的曲率;
(2)
时,
是否存在极值点,如存在,求出其极值点处的曲率;
(3)
,
,当
,
曲率均为0时,自变量最小值分别为
,
,求证:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dd817a1014876a72ad1971548ed6f52c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fe7522a3f232bd0b7a7850ae674db43f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bad7aa241de8ac2738629f7361a7c8bb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/724340d69477c0ec2418c392b22b1cab.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/10acd6d864583617dd3e71240bf0c857.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/724340d69477c0ec2418c392b22b1cab.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ea058d082b5f7517c3b6a6359dbcb44a.png)
(1)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3b4d795709b0abcf47bceec2250f2f9b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
(2)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/94440d3e4c073f94f2b266ff99d50e74.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/724340d69477c0ec2418c392b22b1cab.png)
(3)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c0c51e20ceeca65fe6821130d94b794c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3387f1c69de6c2407212536b35150e5a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4669810732b633b60dbeaf0bf57204f6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c814128ea2139e33db94ea590e7c2223.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/aec19b68e3add9d5bfcc6269a1855b87.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6fb22f6880c74b35a8285cbb51a50fb1.png)
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名校
解题方法
7 . 如图,对于曲线
,若存在圆
满足如下条件:
①圆
与曲线
有公共点
,且圆心在曲线
凹的一侧;
②圆
与曲线
在点
处有相同的切线;
③曲线
的导函数在
处的导数(即曲线
在点
的二阶导数)等于圆
在点
处的二阶导数(已知圆
在点
处的二阶导数等于
);则称圆
为曲线
在
点处的曲率圆,其半径
称为曲率半径.
在原点的曲率圆的方程;
(2)(i)求证:平面曲线
在点
的曲率半径为
(其中
表示
的导函数);
(ii)若圆
为函数
的一个曲率圆,求圆
半径的最小值;
(3)若曲线
在
处有相同的曲率半径,求证:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b94469fd19f40116e2dec334919d6586.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
①圆
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b94469fd19f40116e2dec334919d6586.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5981cd3691a9166a4d714a2a26b29fd2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b94469fd19f40116e2dec334919d6586.png)
②圆
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b94469fd19f40116e2dec334919d6586.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/92637a7e7dab461f173112dfc8fa7390.png)
③曲线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b94469fd19f40116e2dec334919d6586.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/11abb76da45ffa52b47c3a6b9a03ac7e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b94469fd19f40116e2dec334919d6586.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5981cd3691a9166a4d714a2a26b29fd2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/80999542b0b1e42a23e95363667399a4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5981cd3691a9166a4d714a2a26b29fd2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/98b40504aef42ec81163e9581efbd83b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b94469fd19f40116e2dec334919d6586.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/11bc05f41215f9894e11d1df0465751a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/108c18cb76d7d34b05c991a644c8b136.png)
(2)(i)求证:平面曲线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0afb80007983e5b99dcdeebf87d18ff4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0aeb9ff11c38818c2f3906ea7429a7eb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/03a6c9ccde77a428c1255488d1eefa26.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0bac50c92211d6348b056335f6c83ea1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/090a91e4f3c8930674f98a9fa527709b.png)
(ii)若圆
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5db192285632d1991b4ee7a003a52205.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
(3)若曲线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5db192285632d1991b4ee7a003a52205.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ffebab8e8ac2b96518ebf38fc2e36609.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/259945c2a19261f6d9086e916b5b82c8.png)
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8 . 已知函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/143b917df0520097be222accbddf9394.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/62559d143b4a977be9990eebcbec539e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
(2)证明:曲线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/51c530f4b7491b95acb8ce3eef9aa09d.png)
(3)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/79699156efecc21a555e63da6456031a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8a551a88ac426439803f564a3bbee04a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2c94bb12cee76221e13f9ef955b0aab1.png)
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|
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5卷引用:湖北省黄冈市浠水县第一中学2023-2024学年高二下学期期末质量检测数学试题
湖北省黄冈市浠水县第一中学2023-2024学年高二下学期期末质量检测数学试题2024年新课标全国Ⅰ卷数学真题(已下线)2024年高考数学真题完全解读(新高考Ⅰ卷)专题03导数及其应用(已下线)2024年新课标全国Ⅰ卷数学真题变式题16-19
名校
9 . 已知函数
,其导函数为
.
(1)求函数
的极值点;
(2)若直线
是曲线
的切线,求
的最小值;
(3)证明:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2c54484556715e9741e928f4972ba58f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/724340d69477c0ec2418c392b22b1cab.png)
(1)求函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
(2)若直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5e136e7637543c8ae92c8dcd55b31924.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f499a006927ca1e000afc1f62133c449.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/20d6fc9b90f370fbb27552876b650f8f.png)
(3)证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a7399b4bbfcedfc2dba78865a78471e8.png)
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|
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名校
解题方法
10 . 已知函数
,且曲线
在点
处的切线斜率为1.
(1)求
的表达式;
(2)若
恒成立,求
的值.
(3)求证:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4224ac66f06515bb53aad2c7d9a75b5c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0afb80007983e5b99dcdeebf87d18ff4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c3832d863e6cefdfe45cff4319e1fbdb.png)
(1)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d275fbb3ee5cd1177ca5a2ceecbbef0f.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b387739efd7a170870100f783948d60d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
(3)求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/17658ba57a6f979195e76ab36c7d44dd.png)
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3卷引用:湖北省武汉市第十一中学2023-2024学年高二下学期3月考数学试卷
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