2022高二·全国·专题练习
1 . 已知曲线
及点
.
(1)求过点P的切线方程;
(2)求证:与曲线S切于点
的切线与S至少有两个交点.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/44259fbade1b1d6582cae0741d2e7749.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a7bd4e5049fa304e4d352bfe6dee455d.png)
(1)求过点P的切线方程;
(2)求证:与曲线S切于点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9e14e64c39c9f39536da2d0fd06f88a8.png)
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解题方法
2 . 已知函数
.
(1)当
时,求曲线
在点
处的切线方程;
(2)若函数
有两个极值点,求实数
的取值范围;
(3)若函数
有两个极值点
,求证:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/61e39d6ccb78063a345d9dc44302223f.png)
(1)当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0b550ee821ee1838384835e81fc34b67.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/942c2141d01bde6b48210c56a17fc75e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5828873f8369183faf71181cda5b61d2.png)
(2)若函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
(3)若函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/aca579894dad67bc82cb715fd48e0d70.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c3b314f6ccb0a3e4fc15685d85e55bf6.png)
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3 . 已知函数
.
(1)
时,求函数
在
处的切线方程;
(2)讨论函数
的单调性;
(3)证明不等式
恒成立.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2f27e242e405cc9cd23b92198e4bbd37.png)
(1)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0b550ee821ee1838384835e81fc34b67.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9b384412acba251d87902ab928902f16.png)
(2)讨论函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
(3)证明不等式
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/da332f19d8ef4ececad083dbfe14b3bd.png)
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2023-05-19更新
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6卷引用:江苏省南通市海安市实验中学2022-2023学年高二下学期6月期末模拟数学试题
解题方法
4 . 已知函数
.
(1)求曲线
在点
处的切线方程;
(2)当
时,求证:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/badaac3e9c8726f0809ff4ce57527e07.png)
(1)求曲线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/942c2141d01bde6b48210c56a17fc75e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7ea9824af71c9da5db5a00ec06063024.png)
(2)当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0fde64f4d3c38e43fbdee24eadc4b0dd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/49b7c51c62b1b77247f2ce5cd5bb7699.png)
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2卷引用:江苏省2024届高三上学期仿真模拟考试(二)数学试题
5 . 已知
是函数
的极值点.
(1)求
的极值;
(2)证明:过点
可以作曲线
的两条切线.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9b384412acba251d87902ab928902f16.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9e0a42b4fc3ea40b5cf11c5b9b603a84.png)
(1)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
(2)证明:过点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/efea404ec4afc504335f713aa6ee5262.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/942c2141d01bde6b48210c56a17fc75e.png)
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6 . 已知函数
.
(1)若
在
处的切线在
轴上的截距为
,求
;
(2)若
不是单调函数,证明:
,且
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/169bc7b2341bfe69972d81274adf7b87.png)
(1)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/942c2141d01bde6b48210c56a17fc75e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9b384412acba251d87902ab928902f16.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d053b14c8588eee2acbbe44fc37a6886.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/acbc6a613224461ade69362d46550474.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1d33da711e50e96568facb18cef27165.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/752de9fa1d49f8acd0b626a4e3dae381.png)
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7 . 已知函数
的导函数为
,且
.
(1)求函数
在
处的切线方程;
(2)证明:
在
上仅有一个零点
,且
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/724340d69477c0ec2418c392b22b1cab.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9e8a101064190710c2287cfe5931f672.png)
(1)求函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bb45f673c56a289ea78831c9237e8d20.png)
(2)证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5085c14cc9d275af1875b7f58575200e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/79b752f0f189e5d8666daea73e145dff.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/33ff090e7db6b070e17df2269e4bd564.png)
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2023-09-12更新
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2卷引用:江苏省南通市海安市实验中学2022-2023学年高二下学期期中数学试题
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解题方法
8 . 我国南北朝时期的数学家祖冲之(公元429年-500年)计算出圆周率的精确度记录在世界保持了千年之久,德国数学家鲁道夫(公元1540年-1610年)用一生精力计算出了圆周率的35位小数,随着科技的进步,一些常数的精确度不断被刷新.例如:我们很容易能利用计算器得出函数
的零点
的近似值,为了实际应用,本题中取
的值为-0.57.哈三中毕业生创办的仓储型物流公司建造了占地面积足够大的仓库,内部建造了一条智能运货总干线
,其在已经建立的直角坐标系中的函数解析式为
,其在
处的切线为
,现计划再建一条总干线
,其中m为待定的常数.
注明:本题中计算的最终结果均用数字表示.
(1)求出
的直线方程,并且证明:在直角坐标系中,智能运货总干线
上的点不在直线
的上方;
(2)在直角坐标系中,设直线
,计划将仓库中直线
与
之间的部分设为隔离区,两条运货总干线
、
分别在各自的区域内,即曲线
上的点不能越过直线
,求实数m的取值范围.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac814388508dc7b9c8540daa5b2f4ed8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/79b752f0f189e5d8666daea73e145dff.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/79b752f0f189e5d8666daea73e145dff.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b1241216f3c1cb5e73043dd1037f556d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/11b4e7dfbde0ae0a87f234a7a762f0b1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/707ea658f3a9359f5740d5aab48f7948.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9f559ab6b3e37fa29cfe0620f9885d49.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fa1971b598c1482b011e71efa3c48a6c.png)
注明:本题中计算的最终结果均用数字表示.
(1)求出
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/172722d11ea7e01411fa06dbb82f46ee.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b1241216f3c1cb5e73043dd1037f556d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/172722d11ea7e01411fa06dbb82f46ee.png)
(2)在直角坐标系中,设直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ec01a64b41c7c6fd705be73fbea4aaa4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/172722d11ea7e01411fa06dbb82f46ee.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9fbd49bf20f987c05b4d36e31549075c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b1241216f3c1cb5e73043dd1037f556d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/23f3ffe7abc59e2f65d827c8eab8d36a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/23f3ffe7abc59e2f65d827c8eab8d36a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9fbd49bf20f987c05b4d36e31549075c.png)
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6卷引用:江苏省扬州中学2023届高三下学期5月适应性考试数学试题
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9 . 已知函数
.
(1)求曲线
在
处切线的斜率;
(2)当
时,比较
与x的大小;
(3)若函数
,且
(
),证明:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/85e4bdada70f9217234b43e8747a855f.png)
(1)求曲线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/942c2141d01bde6b48210c56a17fc75e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9b384412acba251d87902ab928902f16.png)
(2)当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/66692ec49a458f9e48c7315d03dfc37b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
(3)若函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/03f4241a5db19c15cb647bf520a8570e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8e86a442e2b43d732352ea5f44edc4fd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2684b72f9f38f5046c8ecd4280b7b14b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3fc27e66b5547d400351b99194496883.png)
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2023-10-05更新
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8卷引用:江苏省百校大联考2024届高三上学期10月阶段性考试数学试题
10 . 已知函数![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2e9f70ec2e471997fb193188d306cffe.png)
(1)求
在
处的切线方程;
(2)若
在定义域上有两解
,求证:
①
;
②
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2e9f70ec2e471997fb193188d306cffe.png)
(1)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3b2c84e7b41a841a230ed5f8a42309aa.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/338316b0fe50fdea0f2f75aec4c990dd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8ce7ae90d808f05e86ea063238e4b2f9.png)
①
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d7cfada8fd642ddf968bfd4228d48ec3.png)
②
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9ed7932a5ee7f639c53e6eb0a007eb91.png)
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2023-01-09更新
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