1 . 已知椭圆
,
,
分别是椭圆C的左、右焦点,点
为左顶点,椭圆上的点到左焦点距离的最小值是焦距的
.
(1)求椭圆
的离心率;
(2)直线
过椭圆C的右焦点
,与椭圆C交于P,O两点(点P在第一象限).且
面积的最大值为
,
①求椭圆C的方程;
②若直线
,
分别与直线
交于
,
两点,求证:以
为直径的圆恒过右焦点
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/851a5d6ec23256f9b4a9e98aa92945fe.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f5076289823db419f94e9c0c8f4aafd9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a3fb78c5f885034612c0e030b920143d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/56d266a04f3dc7483eddbc26c5e487db.png)
(1)求椭圆
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
(2)直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f85fca60a11e1af2bf50138d0e3fe62.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a3fb78c5f885034612c0e030b920143d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d9763846b1131e1e3e2d741ad95d5bb0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dcd3b3e802743d1c239c192c7fb8a599.png)
①求椭圆C的方程;
②若直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/20a541b81584a032f571159ea152c85a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/84d454c82d9e52747563d47b68099249.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3c716601286ac3ef51a4c3c16b12e260.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/54a5d7d3b6b63fe5c24c3907b7a8eaa3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411461db15ee8086332c531e086c40c7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a3fb78c5f885034612c0e030b920143d.png)
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解题方法
2 . 已知函数
和
.
(1)若曲线数
与
在
处切线的斜率相等,求
的值;
(2)若函数
与
有相同的最小值.
①求
的值;
②证明:存在直线
,其与两条曲线
与
共有三个不同的交点,并且从左到右三个交点的横坐标成等差数列.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9ef96ff936eb415b1f8fe6b9166d8e89.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e3c68ab4181ffc22679c971eed6d8286.png)
(1)若曲线数
![](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/9b384412acba251d87902ab928902f16.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
(2)若函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4fe7d5809da02c15a43a0e9a898b9086.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/be1ce3f01e2b6364f9a9fdaf197d5e29.png)
①求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
②证明:存在直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/af79f45b5880c72a349500da9d8e118d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/51c530f4b7491b95acb8ce3eef9aa09d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6a1cfb60420ff7e72c1b9d64f69ae063.png)
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名校
3 . 已知函数
.
(1)当
时,求
的单调区间;
(2)若
时,
,求a的取值范围;
(3)对于任意
,证明:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f27862c9517dbb4eb17a6725eb142969.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/636289ad84b4a3a51095dd32ca201f94.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9e9c599e8d420006448905acec2b8234.png)
(3)对于任意
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a37a59558292ad6b3d0978bfd7484990.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1af027bd16e380d3be03a9761ca56055.png)
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|
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|
9卷引用:天津市和平区耀华中学2024届高三下学期寒假验收考数学试卷
名校
4 . 已知函数
,
,
.
(1)设
,试讨论函数
的单调性;
(2)若
对于定义域内的任意
恒成立,求实数
的取值范围
(3)设
,对于任意的
,总存在
,使不等式
成立,求实数
的取值范围.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/153a430f99c4004a94325ab49d84dc02.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2e9f049a5f960728c60a909821b2404b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/10bbdef421c976962a270a2beabbad91.png)
(1)设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/568041d6adb0ccc31892181f6206afd8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/46be55c8f2760d6db125f46691a3de48.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/29836991ef5eb74372a6827f5810d7de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
(3)设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9122b0871365ec49002a62401ac7f693.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2fae35542c8e8114f3cfc05b400ba565.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/36ad23d93c0e20425a3a7f3a8605a61d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6319c453a96a1a75d46aaf4b51ede08f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f0a532e15e232cb4b99a8d4d07c89575.png)
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名校
5 . 已知函数
.
(1)当
时,求曲线
在点
处的切线方程;
(2)当
时,求使
恒成立的最大偶数a.
(3)已知当
时,
总成立.令
,若在
的图像上有一点列
,若直线
的斜率为
,求证:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f4767ea62e6d768d139e72f14a324f0.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)当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0fde64f4d3c38e43fbdee24eadc4b0dd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1c73a98c1b3504e09bfbe0db849b0d24.png)
(3)已知当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a6e2e79843faf62dde86bf858d1e0569.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8852262dcc3b39fceedd7ddd34d96e43.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/976581d4a974fe50f9f29d430c1289f2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4669810732b633b60dbeaf0bf57204f6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b5e4aab2b573ce9ae9094490368bfe5b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/425f3ce645095842006c80a509268f85.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/488e9eb082f829bebb465ee78a27d0bd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f55da0df18e4e42fcf44ff489c861be2.png)
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6 . 对于任意的实数
,总存在三个不同的实数y,使得
成立,其中e为自然对数的底数,则实数a的取值范围是( ).
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5fbb01a7f5e9861aa185c6c63fcd58c0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3ab3178c8f2f812d05b4d622c159f5a7.png)
A.![]() | B.![]() |
C.![]() | D.![]() |
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名校
7 . 已知函数
,若函数
与
的图象恰有5个不同公共点,则实数
的取值范围是______ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/43a74d88c4d841156e04411eeae3d30a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d4a448aee939c98ef0044362e52d14b4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/df6e318aeb4f2bfa58d289dbaedf91a8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
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|
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|
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名校
8 . 已知函数
,
且
.
(1)当
时,求曲线
在
处的切线方程;
(2)若
,且
存在三个零点
,
,
.
(i)求实数
的取值范围;
(ii)设
,求证:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cd34ef1116ad10637283e654cacb2b31.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/94440d3e4c073f94f2b266ff99d50e74.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c400a615a16a1662de98dfb4e49d58d3.png)
(1)当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b0ffecb03c47be920254c4ccffa5b222.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/942c2141d01bde6b48210c56a17fc75e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9b384412acba251d87902ab928902f16.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1d33da711e50e96568facb18cef27165.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.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)
(i)求实数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
(ii)设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e1310a7a80d1f8751a3f8cafe7f8c8b4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7803d86067299198e6d14b0c83947f58.png)
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|
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名校
9 . 已知函数![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6f0ebc77b83210330497b573b39c458e.png)
(1)当
时,求
在点
处的切线方程;
(2)若
对
恒成立,求实数
的取值范围
(3)
若有3个零点
,其中
,求实数
的取值范围,并证明
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6f0ebc77b83210330497b573b39c458e.png)
(1)当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0b550ee821ee1838384835e81fc34b67.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/51c530f4b7491b95acb8ce3eef9aa09d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/54b13280d106fe9c3db2069984325b63.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e728e7423960080153f3ef30e2b708d7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5daf486cf2adbd7af8aa26fe66b57ce3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
(3)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4fe7d5809da02c15a43a0e9a898b9086.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/05b8ec9d4206ea66a02de5c4a1e1e911.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e1310a7a80d1f8751a3f8cafe7f8c8b4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c1c06068ac2d0da29abec54df3f84347.png)
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2卷引用:天津市第一百中学2023-2024学年高三上学期期中数学试题
10 . 已知函数
.
(1)当
时,求曲线
在
处切线的斜率;
(2)讨论函数
的单调性;
(3)若
有两个极值点
,且
恒成立,求实数
的取值范围.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/97227e82a644fdaa750975fbab2be46f.png)
(1)当
![](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/707ea658f3a9359f5740d5aab48f7948.png)
(2)讨论函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
(3)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d98919b1335a8b7ca020636d1494ad0d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/72a844b9c0d7ea04428d8ba115709d9e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/294f5ba74cdf695fc9a8a8e52f421328.png)
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