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解题方法
1 . (1)已知函数
,证明:
,
,
.
(2)已知函数
,定义:若存在
,
,使得曲线
在点
与点
处有相同的切线
,则称切线
为“自公切线”.
①证明:当
时,曲线
不存在“自公切线”;
②讨论曲线
的“自公切线”的条数.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6ff6838d84b68c6f0d3b93b196d9b08d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/673207f6b77b8192d25463d071737b7c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b1bf60c5e8996d138198fe74f30ce520.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9da092efa74406128332df5a053685a8.png)
(2)已知函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1aba730d4e2ff4c9cc155446b3d12e96.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c814128ea2139e33db94ea590e7c2223.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/85a93969738a9bb969f40cf587f1d5d5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1938c093dd2fbcb752d0eb7a18d143b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/878fd4af5b8fff01627f560767e19b73.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4197070db34f0419b6d85eed4cec9fc5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f85fca60a11e1af2bf50138d0e3fe62.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f85fca60a11e1af2bf50138d0e3fe62.png)
①证明:当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/08115d6d9f876dea921a4d32260ff1fb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1938c093dd2fbcb752d0eb7a18d143b2.png)
②讨论曲线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1938c093dd2fbcb752d0eb7a18d143b2.png)
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2 . 对于函数
,下列说法正确的是( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/728f2cc68f8ca8ef2faa681785798259.png)
A.函数![]() ![]() |
B.![]() |
C.若方程![]() ![]() |
D.对任意正实数![]() ![]() ![]() ![]() |
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2卷引用:重庆市开州中学2023-2024学年高三下学期高考模拟考试数学试题(四)
名校
解题方法
3 . 已知函数
.
(1)当
时,
恒成立,求实数
的取值范围;
(2)已知直线
是曲线
的两条切线,且直线
的斜率之积为1.
(i)记
为直线
交点的横坐标,求证:
;
(ii)若
也与曲线
相切,求
的关系式并求出
的取值范围.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6453c2284ab370e0c3817f5e14bafa7d.png)
(1)当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5a3c442579603164f3fc19458677d307.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/43920f5171ed31db2520ef00e4c5fc24.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
(2)已知直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/50297ad9f7256b4d2efc3462289f18b7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6a1cfb60420ff7e72c1b9d64f69ae063.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9849df491461fb04b28fd5fe6017753e.png)
(i)记
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/79b752f0f189e5d8666daea73e145dff.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9849df491461fb04b28fd5fe6017753e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8f90560052fe43871fd3d594c771723c.png)
(ii)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9849df491461fb04b28fd5fe6017753e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/51c530f4b7491b95acb8ce3eef9aa09d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/632244ea6931507f8656e1cc3437d392.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2c94bb12cee76221e13f9ef955b0aab1.png)
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4 . 已知函数![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f28e4867492d6035296db5e28c6ed599.png)
(1)当
时,求
的零点;
(2)若
恰有两个极值点,求
的取值范围.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f28e4867492d6035296db5e28c6ed599.png)
(1)当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0b550ee821ee1838384835e81fc34b67.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
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4卷引用:重庆市第四十九中学校、江津第二中学校等九校2023-2024学年高二下学期5月联考数学试题
名校
解题方法
5 . (1)证明:当
时,
;
(2)已知正项数列
满足
.
(i)证明:数列
为递增数列;
(ii)证明:若
,则对任意正整数
,都有
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/08115d6d9f876dea921a4d32260ff1fb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ca947e8ea00b7a485097ecafd2dfcae9.png)
(2)已知正项数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6f91f8b7476e67db488d85c3a14ffa6d.png)
(i)证明:数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0496f142d8ae5acb06e83526eaa3ef87.png)
(ii)证明:若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/90f2db682457da2d4abd0e7cca1bdf40.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/38d8a310323506a3c2f3626dec8d781f.png)
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6 . 已知函数
.
(1)求证:
;
(2)若
是
的两个相异零点,求证:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8ba20f73926fa882b592848c085f060f.png)
(1)求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/93296bd064e2c6ae84bc4fe7b22f1e4b.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8ce7ae90d808f05e86ea063238e4b2f9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4fe7d5809da02c15a43a0e9a898b9086.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/31bd625540579bf15a6465a2224c9d61.png)
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2卷引用:重庆市第一中学校2024届高三下学期模拟预测数学试题
7 . 对于数列
,定义
,满足
,记
,称
为由数列
生成的“
函数”.
(1)试写出“
函数”
,并求
的值;
(2)若“
函数”
,求n的最大值;
(3)记函数
,其导函数为
,证明:“
函数”
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c214af101c995038e5229277692350e0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5a9dfaa821691b4a0f548d3a57cafb6d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6f3862a345e98a4a34b3101bffe283be.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0ad01b28c2aa9d86daa50de5a9fe698a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bb654dbe976f077495105b21b7963d0f.png)
(1)试写出“
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2837786afdd7b9b8bc37823040d7dd64.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e167bc147c8a470e8060815303b6313a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/abdb56b20415e38b25c6cc9c4319dc53.png)
(2)若“
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e4e0c84de10f0f2186313169c3dc997b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/502cb9fcc607d8622546867ccf60407f.png)
(3)记函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3ab95fa0eb7b774e15babe6854f4d8e4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5840bbbca91e9057cc99a5b3cd85fe7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bb654dbe976f077495105b21b7963d0f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/506f4aa695eb5785abc58cf6c254c492.png)
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8 . 已知函数![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9a0455e23bde86e7e912f9aca84145d0.png)
.
(1)若函数
有两个极值点,求
的取值范围;
(2)若对
,函数
恒成立,求
的取值范围;
(3)证明:对
,
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9a0455e23bde86e7e912f9aca84145d0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ab9cd3690e7aa3debb1ed054a9f622da.png)
(1)若函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
(2)若对
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6f832d9cca2d5c9d76d38374e2a258d9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9e9c599e8d420006448905acec2b8234.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
(3)证明:对
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e72620c113a6fe83273803a9ac24baa5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8bc8c9f6cbca5a13a20ffe4b0c42838e.png)
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9 . 已知函数
,其中e为自然对数的底数,下列选项正确的有( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/99657b7b7f602075bb3bc46fe58ba90f.png)
A.若函数![]() ![]() |
B.当![]() ![]() ![]() |
C.当![]() ![]() ![]() |
D.若![]() ![]() |
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10 . 阅读材料一:“装错信封问题”是由数学家约翰·伯努利(Johann Bernoulli,1667~1748)的儿子丹尼尔·伯努利提出来的,大意如下:一个人写了
封不同的信及相应的
个不同的信封,他把这
封信都装错了信封,问都装错信封的这一情况有多少种?后来瑞士数学家欧拉(Leonhard Euler,1707~1783)给出了解答:记都装错
封信的情况为
种,可以用全排列
减去有装正确的情况种数,结合容斥原理可得公式:
,其中
.
阅读材料二:英国数学家泰勒发现的泰勒公式有如下特殊形式:当
在
处
阶可导,则有:
,注
表示
的
阶导数,该公式也称麦克劳林公式.阅读以上材料后请完成以下问题:
(1)求出
的值;
(2)估算
的大小(保留小数点后2位),并给出用
和
表示
的估计公式;
(3)求证:
,其中
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/66d4e8502106802f1485c3b0f28f2664.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a8412f5256b2b370e421c07f18cc732.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e4403d632f9a81e52c6cd135c6834bc2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a37a59558292ad6b3d0978bfd7484990.png)
阅读材料二:英国数学家泰勒发现的泰勒公式有如下特殊形式:当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bb45f673c56a289ea78831c9237e8d20.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6ce152ca98ac7e21237e00667f005b62.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/35993bd1db970330494665d925c0be7a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
(1)求出
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/395c6efaa63dcd4ee513323d51c6a7eb.png)
(2)估算
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2598975ac1edb754817eada15b9a473e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/041a7c8fc017f596542c5e6ec7d1c40b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/66d4e8502106802f1485c3b0f28f2664.png)
(3)求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ca08ded0d1136421f0a81517f5c2fc9d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a37a59558292ad6b3d0978bfd7484990.png)
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