名校
解题方法
1 . 英国数学家泰勒发现了如下公式:
其中
为自然对数的底数,
.以上公式称为泰勒公式.设
,根据以上信息,并结合高中所学的数学知识,解决如下问题.
(1)证明:
;
(2)设
,证明:
;
(3)设
,若
是
的极小值点,求实数
的取值范围.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ccf4a87ad1e9742f47b0c5b44b8dfab0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a6696028290bbaddf628d64bad0ed95b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2976d45a26ec77149a05553e8eb13efb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c78478b44ff22e088fd8e6522c5d78a2.png)
(1)证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9d84ae7f43ef85da907d2917ff5f2a80.png)
(2)设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/66692ec49a458f9e48c7315d03dfc37b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8586154d8c4fb5fef893d39a7701f921.png)
(3)设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dde823e2e88ecb6045d66d61962259b8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bb45f673c56a289ea78831c9237e8d20.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/46be55c8f2760d6db125f46691a3de48.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
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2024-03-03更新
|
2355次组卷
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19卷引用:福建省宁德市古田县第一中学2024届高中毕业班高考前适应性测试数学试题
福建省宁德市古田县第一中学2024届高中毕业班高考前适应性测试数学试题贵州省贵阳市2024届高三下学期适应性考试数学试卷(一)贵州省安顺市2024届高三下学期模拟考试(一)数学试卷海南省海南华侨中学2023-2024学年高三下学期第二次模拟考试数学试题江西省宜春市上高二中2024届高三下学期5月月考数学试卷(已下线)专题11 利用泰勒展开式证明不等式【练】云南省玉溪市第一中学2023-2024学年高二下学期3月月考数学试题重庆市礼嘉中学2023-2024学年高二下学期第一次月考数学试题吉林省通化市梅河口市第五中学2023-2024学年高二下学期第一次月考数学试题重庆第十一中学校2023-2024学年高二下学期3月月考数学试题重庆市璧山中学校2023-2024学年高二下学期第一次月考数学试题广东省东莞市光明中学2023-2024学年高二下学期第一次月考数学试题四川省达州外国语学校2023-2024学年高二下学期3月月考数学试题黑龙江省哈尔滨市双城区兆麟中学2023-2024学年高二下学期第一次月考(4月)数学试题重庆市荣昌中学校2023-2024学年高二下学期4月期中考试数学试题广东省广州市广州中学2023-2024学年高二下学期期中考试数学试题河北省石家庄四十一中2023-2024学年高二下学期第一次月考数学试题河北省石家庄二中润德中学2023-2024学年高二下学期第一次月考数学试题四川省南充市白塔中学2023-2024学年高二下学期期中考试数学试题
名校
解题方法
2 . 对于正实数
有基本不等式:
,其中
,为
的算术平均数,
,为
的几何平均数.现定义
的对数平均数:![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a9b454c722316d2e530e935987adcb81.png)
(1)设
,求证:
:
(2)①证明不等式:
:
②若不等式
对于任意的正实数
恒成立,求正实数
的最大值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dd1f53d48a9ad9f88f4b3c14f2637d3b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/12b0bcbf744c3da99e6488f8e66cb8c7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ee128ea692363f9a7b0cf0958e5f74e2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/632244ea6931507f8656e1cc3437d392.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/54b9514b5e245327b05261ac9a946063.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/632244ea6931507f8656e1cc3437d392.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/632244ea6931507f8656e1cc3437d392.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a9b454c722316d2e530e935987adcb81.png)
(1)设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0fde64f4d3c38e43fbdee24eadc4b0dd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/855eaf612ac4e4505948ee0a1c3c080e.png)
(2)①证明不等式:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d8188a2ffd328c07a359ea9be8102a70.png)
②若不等式
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7b0a551c4d6741cae6d513122166db90.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5aff93e03b22c6053550486ea4e911c4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f0a532e15e232cb4b99a8d4d07c89575.png)
您最近一年使用:0次
2022-05-11更新
|
493次组卷
|
6卷引用:福建省永安第九中学2023届高三上学期期中考试数学试题
名校
3 . 形如
的函数称为幂指函数,幂指函数在求导时,可以利用对数法:在函数解析式两边取对数得
,两边对
求导数,得
,于是
.已知
,
.
(1)求曲线
在
处的切线方程;
(2)若
,
恒成立,求
的取值范围.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e71437cb751931577d0f169f71d96556.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/909e553b064501f7f77747bdab7baccd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b68d7dff7daf140a2dce70b6fa4dd89c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2c9286b2a0d32f57c9d9c4c60679b92d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b213cb659a598e3621fe7da9580755ac.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e5a3f1fcae45113e71f8746982f5f9af.png)
(1)求曲线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/51c530f4b7491b95acb8ce3eef9aa09d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9b384412acba251d87902ab928902f16.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d33d41d398944a02f613784ff1ceeaf4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8d59a99dd9aee98cb9b10fa9d972d689.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
您最近一年使用:0次
4 . 已知函数
.
(1)讨论
的单调性;
(2)若
,直线
与曲线
和曲线
都相切,切点分别为
,
,求证:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/53aa300024c371f7f07942d72e0a45df.png)
(1)讨论
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/be1ce3f01e2b6364f9a9fdaf197d5e29.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0b550ee821ee1838384835e81fc34b67.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f85fca60a11e1af2bf50138d0e3fe62.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/51c530f4b7491b95acb8ce3eef9aa09d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6a1cfb60420ff7e72c1b9d64f69ae063.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5d6f5adf13b4214666292dd64b947741.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/af405a054bfe7fb7ce40e48d816467e1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1c059a88d0be56410f74e0820b02f28f.png)
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2020-04-23更新
|
1503次组卷
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6卷引用:福建省漳州市南平市2019-2020学年高三第二次教学质量检测理科数学试题
福建省漳州市南平市2019-2020学年高三第二次教学质量检测理科数学试题福建省漳州市、南平市2020届高三高考数学(理科)二模试题福建省漳州市2020届高三高中毕业班第二次教学质量检测数学(理)试题(已下线)专题11 导数的几何意义应用-学会解题之高三数学万能解题模板【2022版】四川省乐山市十校2019-2020学年高二下学期期中联考数学(理)试题(已下线)第五章 一元函数的导数及其应用-2020-2021学年高二数学同步课堂帮帮帮(人教A版2019选择性必修第二册)
名校
5 . 已知函数
.
(1)讨论函数
的单调性;
(2)若
,过
分别作曲线
与
的切线
,且
与
关于
轴对称,求证:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9a56cbb379b012b2505624beb10237f6.png)
(1)讨论函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d8b6894e8c345a035e89ec672503a01f.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b30ab064746e49ea3dde4d3c2926ddbe.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3bef49f4847f1c47ba40e100d62355c9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7dc6e69ad1a27916fb5c3d5901ded134.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dd3d641761af730cc20b05a79fad66f1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/44434b647ec546fe787e2164e0be6cd2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2e9b0f5f44abbc6544a2f672b025b013.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3f6f17bc385bafb37e8f964e5eb99cd0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/98448dc43a8afaa97bbbc7ed073bd46e.png)
您最近一年使用:0次
2017-04-11更新
|
1287次组卷
|
4卷引用:福建省福州第一中学2020届高三下学期开学质检数学(理)试题
福建省福州第一中学2020届高三下学期开学质检数学(理)试题2017届安徽省黄山市高三第二次模拟考试数学(理)试卷(已下线)强化卷08(3月)-冲刺2020高考数学之拿高分题目强化卷(山东专版)湖北省孝感市八所重点高中教学协作体2016-2017学年高二7月联合考试数学(理)试题
名校
6 . 已知函数
是
的导函数,
为自然对数的底数.
(1)讨论
的单调性;
(2)当
时,证明:
;
(3)当
时,判断函数
零点的个数,并说明理由.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c8efd7ecbec67cd72ab4c4f71b00f09f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d8b6894e8c345a035e89ec672503a01f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/168b3e4b1d6f04226fa2687a72a268b4.png)
(1)讨论
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/93e03ad0c315806342d6cd732a0b91a3.png)
(2)当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2dc56a349930f604e748c531922c4c52.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/df1a5e7f6cf90e6f97b01bb158ceaba6.png)
(3)当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2dc56a349930f604e748c531922c4c52.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d8b6894e8c345a035e89ec672503a01f.png)
您最近一年使用:0次
2017-02-27更新
|
888次组卷
|
2卷引用:2020届福建省长泰县第一中学高三上学期月考 数学(理)试题
解题方法
7 . 设函数
的定义域
,若对任意
,都有
,则称函数
为“storm”函数.已知函数
的图象为曲线
,直线
与曲线
相切于
.
(1)求
的解析式;
(2)设
,若对
,函数
为“storm”函数,求实数
的最小值.
![](https://img.xkw.com/dksih/QBM/2016/12/23/1619431100727296/1619431101325312/STEM/af3c24ca6cc943559cf6ad556da7639f.png)
![](https://img.xkw.com/dksih/QBM/2016/12/23/1619431100727296/1619431101325312/STEM/520e24911c42472d8183906095486991.png)
![](https://img.xkw.com/dksih/QBM/2016/12/23/1619431100727296/1619431101325312/STEM/3536f996daa74b5d898ce81df7838395.png)
![](https://img.xkw.com/dksih/QBM/2016/12/23/1619431100727296/1619431101325312/STEM/2b9b157793114459b6a0f4b4118c6a4f.png)
![](https://img.xkw.com/dksih/QBM/2016/12/23/1619431100727296/1619431101325312/STEM/af3c24ca6cc943559cf6ad556da7639f.png)
![](https://img.xkw.com/dksih/QBM/2016/12/23/1619431100727296/1619431101325312/STEM/45f9c18008664cb1b9e4c577aa3a5865.png)
![](https://img.xkw.com/dksih/QBM/2016/12/23/1619431100727296/1619431101325312/STEM/6b25258472514776b1d799fc2189c594.png)
![](https://img.xkw.com/dksih/QBM/2016/12/23/1619431100727296/1619431101325312/STEM/fdeb9b6dbc694cff91ec3873782d1c42.png)
![](https://img.xkw.com/dksih/QBM/2016/12/23/1619431100727296/1619431101325312/STEM/6b25258472514776b1d799fc2189c594.png)
![](https://img.xkw.com/dksih/QBM/2016/12/23/1619431100727296/1619431101325312/STEM/b085c7bb9e6144a682511a6fb064bb17.png)
(1)求
![](https://img.xkw.com/dksih/QBM/2016/12/23/1619431100727296/1619431101325312/STEM/0a57fef9658e493fb8978086e2e13687.png)
(2)设
![](https://img.xkw.com/dksih/QBM/2016/12/23/1619431100727296/1619431101325312/STEM/913866449dfc4461984dce10b5b00cb4.png)
![](https://img.xkw.com/dksih/QBM/2016/12/23/1619431100727296/1619431101325312/STEM/734a37be29df4aa8a93a3b853b8d904b.png)
![](https://img.xkw.com/dksih/QBM/2016/12/23/1619431100727296/1619431101325312/STEM/8d77ddc56ac045ea8f005c40ef2c0e1a.png)
![](https://img.xkw.com/dksih/QBM/2016/12/23/1619431100727296/1619431101325312/STEM/0dd07423e9a34562958cc924e679eaf3.png)
您最近一年使用:0次
解题方法
8 . 已知函数
.
(1)当
时,求函数
的
极小值;
(2)设定义在
上的函数
在点
处的切线方程为
:
,当
时,若
在
内恒成立,则称
为函数
的“转点”.当
时,试问函数
是否存在“转点”?若存在,求出转点的横坐标;若不存在,请说明理由.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/eca5755c4a6eeb8dac293711af67f9aa.png)
(1)当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0b550ee821ee1838384835e81fc34b67.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4fe7d5809da02c15a43a0e9a898b9086.png)
![](https://img.xkw.com/dksih/QBM/2017/7/19/1733353706102784/1734184396996608/STEM/1179b0e9bc684f1aa7288f327187f5ef.png?resizew=3)
(2)设定义在
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8455657dde27aabe6adb7b188e031c11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/be1ce3f01e2b6364f9a9fdaf197d5e29.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/caf0d139c9810361b4971904a943856b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f85fca60a11e1af2bf50138d0e3fe62.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e915b67f8f747698b8b46d37bc453667.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e6f263b44213ddcbbedf1fcacb84e249.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/93ae849293af974be15d53cd21994a3f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8455657dde27aabe6adb7b188e031c11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6a1cfb60420ff7e72c1b9d64f69ae063.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/36e15cbd7c42d7b15d7ba8d2b28ab8df.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4fe7d5809da02c15a43a0e9a898b9086.png)
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2016-12-03更新
|
446次组卷
|
2卷引用:福建省2016届高三毕业班总复习(导数)单元过关平行性测试卷(理科)数学试题