1 . 帕德近似(Pade approximation)是法国数学家帕德(Pade)于l9世纪末提出的,其基本思想是将一个给定的函数表示成两个多项式之比的形式,具体是:给定两个正整数m,n,函数
在
处的
帕德近似为
,其中
,
,
,…,
(
为
的导数).已知函数
在
处的
阶帕德近似为
.
(1)求实数a,b的值;
(2)证明:当
时,
;并比较
与
的大小.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4fe7d5809da02c15a43a0e9a898b9086.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bb45f673c56a289ea78831c9237e8d20.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/db527571cfd256c515424c6f9d114284.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4b73a1e9e6afa355710753d576ea991a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a8cd76f42911e8c8e57ce761b4541137.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1c52140f46c02b2bde412f89d0977bbb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c85cd03aea30c3ee0093afad048b75c3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7b63504a4bf0d4861de7909bbc2e4878.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0bd370c3b127fbdb77b6e5c40318328d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f2db1e56c92e2ebdc5d2cae336a01b63.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5e96546b3259afe4add331673fb835c3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bb45f673c56a289ea78831c9237e8d20.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0d307aa65d930bc8e51835eb147de513.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/96d128f7851b7771f95bffbdbf3ced02.png)
(1)求实数a,b的值;
(2)证明:当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/08115d6d9f876dea921a4d32260ff1fb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c67d0af421900f7a55f52dd805064f6b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4548db97a06a6f27db6af1dd9b063645.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/786a32019158f0d2ec126cf189ccf572.png)
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解题方法
2 . 已知正实数
,
满足
,则下列结论正确的是( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d053b14c8588eee2acbbe44fc37a6886.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/854c7f3b0b8661beceac1cb4d657c702.png)
A.若![]() ![]() |
B.若![]() ![]() |
C.若![]() ![]() |
D.若![]() ![]() |
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2024-06-06更新
|
115次组卷
|
2卷引用:贵州省部分学校2024届高三下学期联考数学试卷
名校
3 . 设
是函数
的导函数,若
可导,则称函数
的导函数为
的二阶导函数,记为
.若
有变号零点
,则称点
为曲线
的“拐点”.
(1)研究发现,任意三次函数
,曲线
都有“拐点”,且该“拐点”也是函数
的图象的对称中心.已知函数
的图象的对称中心为
,求函数
的解析式,并讨论
的单调性;
(2)已知函数
.
(i)求曲线
的“拐点”;
(ii)若
,求证:
.
![](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/724340d69477c0ec2418c392b22b1cab.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/10acd6d864583617dd3e71240bf0c857.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/11abb76da45ffa52b47c3a6b9a03ac7e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/43db00e106c7d08a76a7ba71ca5e63d1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/942c2141d01bde6b48210c56a17fc75e.png)
(1)研究发现,任意三次函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/012429b7101ba0f84e7b45598ed12db9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/942c2141d01bde6b48210c56a17fc75e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/942c2141d01bde6b48210c56a17fc75e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/211aed3f74a18399b2adbcb74420037e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c2e88ebfb5c0d6cce558b515be06404d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
(2)已知函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1ea257a36c48ae67291bb79295085a5d.png)
(i)求曲线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1938c093dd2fbcb752d0eb7a18d143b2.png)
(ii)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2342ead0be84f52b93d85f167fdbb9a7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8c1bb642f5f896ed02ecd76d9a15e500.png)
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4 . 已知函数
和
.
(1)若函数
在点
处的切线与直线
垂直,求
的单调区间和极值;
(2)当
时,证明:
的图象恒在
的图象的下方.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bf88f7ec79fc9e89f1806e1d027d69a4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1610bcd07b02c4ed7184ad586b88f373.png)
(1)若函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5828873f8369183faf71181cda5b61d2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a9e493a2a0d0c1c4cd3c334454419d42.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
(2)当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/561800aa679a45da4dbe0e323de1fd59.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4669810732b633b60dbeaf0bf57204f6.png)
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解题方法
5 . 在平面直角坐标系
中,已知直线
与抛物线C:
相切.
(1)求m的值;
(2)已知点
,
在抛物线C上,A,B分别位于第一象限和第四象限,且
,过A,B分别作直线
的垂线,垂足分别为
,
,当四边形
面积取最小值时,求直线
的方程.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7ee31829d0d4d5f779a957d7df8058ab.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5ab466aedd6e176088d8dee7bc3e3aaa.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ec68c680d1b8c8cfa98b48a27d2c46a1.png)
(1)求m的值;
(2)已知点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/12a3efb79f35db8448f3391252ab7d4e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8df332f01628130c084fd46aaca0a4b7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/82f8ec3a4456209b619d083b73f1218d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ef00713e73b8357cc7900144f5505bc6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a18722354086c42e62334983fc50eb6a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/97c01fdc7bc471af0b264a04aef0823e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e9edc50f7febbc2d5d8dcdc23a3630a7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
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解题方法
6 . 已知函数
图象上的点
与方程
的解
一一对应,则下列选项中正确的是( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/942c2141d01bde6b48210c56a17fc75e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/82a79a33a83a7ba57a34b5093d1d1d02.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8c0ec8aa466248a663cc603fdc06eeef.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/82a79a33a83a7ba57a34b5093d1d1d02.png)
A.![]() | B.0是![]() |
C.![]() ![]() | D.![]() |
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名校
解题方法
7 . 已知函数
.
(1)若
在定义域内不单调,求a的取值范围;
(2)证明:若
,且
,则
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b14dee98f762932a2b717636a20306b2.png)
(1)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
(2)证明:若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4e4fb399cd59f3c65462df72b179a628.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/24f967a3906eff362ae1748b5a49f204.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3ed1fdfd6d053610f476731689209d32.png)
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8 . 已知
,则下列说法正确的是( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/070be6b8b911597bceaa222710aaf321.png)
A.若![]() ![]() ![]() ![]() |
B.若![]() ![]() ![]() ![]() |
C.若![]() ![]() |
D.若![]() ![]() ![]() ![]() |
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9 . 已知函数
.
(1)若函数
有两个零点,求实数a的取值范围;
(2)已知
,
,
(其中
且
,
,
成等比数列)是曲线
上三个不同的点,判断直线AC与曲线
在点B处的切线能否平行?请说明理由.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4cc1b193aa193153eb402df8560778e6.png)
(1)若函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0384a0466920e5bf00231a5c5bf77969.png)
(2)已知
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/12a3efb79f35db8448f3391252ab7d4e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8df332f01628130c084fd46aaca0a4b7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4c16dac1e9bf5804c8907cbc59014d04.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e1310a7a80d1f8751a3f8cafe7f8c8b4.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/291c25fc6a69d6d0ccfb8d839b9b4462.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/942c2141d01bde6b48210c56a17fc75e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/942c2141d01bde6b48210c56a17fc75e.png)
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10 . 布劳威尔不动点定理是拓扑学里一个非常重要的不动点定理,它可运用到有限维空间并构成了一般不动点定理的基石,得名于荷兰数学家鲁伊兹·布劳威尔(L.E.J.Brouwer).简单地讲就是:对于满足一定条件的连续函数
,存在实数
,使得
,我们就称该函数为“不动点”函数,实数
为该函数的不动点.
(1)求函数
的不动点;
(2)若函数
有两个不动点
,且
,若
,求实数
的取值范围.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4fe7d5809da02c15a43a0e9a898b9086.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/79b752f0f189e5d8666daea73e145dff.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d468b616235df122370cf58f03bb678f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/79b752f0f189e5d8666daea73e145dff.png)
(1)求函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/249ae140f4c699e463b914aa0a25a260.png)
(2)若函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/493db98b5e52328db4dd952e589b3cb6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8ce7ae90d808f05e86ea063238e4b2f9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/26d8dafc71b106f39f4e15442220897b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b0cfc6ebbd5ab3be6a65a553e9da0f85.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2c94bb12cee76221e13f9ef955b0aab1.png)
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