名校
1 . 《见微知著》谈到:从一个简单的经典问题出发,从特殊到一般,由简单到复杂:从部分到整体,由低维到高维,知识与方法上的类比是探索发展的重要途径,是思想阀门发现新问题、新结论的重要方法.
阅读材料一:利用整体思想解题,运用代数式的恒等变形,使不少依照常规思路难以解决的问题找到简便解决方法,常用的途径有:(1)整体观察;(2)整体设元;(3)整体代入;(4)整体求和等.
例如,
,求证:
.
证明:原式
.
波利亚在《怎样解题》中指出:“当你找到第一个藤菇或作出第一个发现后,再四处看看,他们总是成群生长”类似问题,我们有更多的式子满足以上特征.
阅读材料二:基本不等式
,当且仅当
时等号成立,它是解决最值问题的有力工具.
例如:在
的条件下,当x为何值时,
有最小值,最小值是多少?
解:∵
,∴
,即
,∴
,
当且仅当
,即
时,
有最小值,最小值为2.
请根据阅读材料解答下列问题
(1)已知如
,求下列各式的值:
①
___________.
②
___________.
(2)若
,解方程
.
(3)若正数a、b满足
,求
的最小值.
阅读材料一:利用整体思想解题,运用代数式的恒等变形,使不少依照常规思路难以解决的问题找到简便解决方法,常用的途径有:(1)整体观察;(2)整体设元;(3)整体代入;(4)整体求和等.
例如,
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ca27cc54ca0332245f5167488daa3408.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0e2764ccd2cfe6de0c53dce98e45b120.png)
证明:原式
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/87898da3367d13667477a10c9cc47ac2.png)
波利亚在《怎样解题》中指出:“当你找到第一个藤菇或作出第一个发现后,再四处看看,他们总是成群生长”类似问题,我们有更多的式子满足以上特征.
阅读材料二:基本不等式
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a28514741f365301978e922fdca0fcc1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1f22fec5a381ae8aca93d876e54c79de.png)
例如:在
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/08115d6d9f876dea921a4d32260ff1fb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/13f40c24c64bbb0645fcf585f4e66872.png)
解:∵
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1c42b50f6f9e56ea5f222b0a40cb4a3c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/91bb4a7110c19cd10cb915e55438314b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4d32ba3941cef6b1d549f050f0d314e1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/63af71b9e6f71cd26e6e97541154cd8c.png)
当且仅当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1b6a593ef3641dbd11e324dbe78b4dc8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9b384412acba251d87902ab928902f16.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/13f40c24c64bbb0645fcf585f4e66872.png)
请根据阅读材料解答下列问题
(1)已知如
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ca27cc54ca0332245f5167488daa3408.png)
①
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f0dd92f322200ecabfb74ffd7cf3f4a.png)
②
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/af71e37295978173629004816b65791a.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/56667aabbe787eb1c3189d487d203e22.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1d9093a255130a938a4d84595c0c56ce.png)
(3)若正数a、b满足
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ca27cc54ca0332245f5167488daa3408.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2ab1cbf887eca130c254f6e0cf3fdb2f.png)
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2021-10-29更新
|
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3卷引用:江苏省南通中学2020-2021学年高一上学期开学考试数学试题
江苏省南通中学2020-2021学年高一上学期开学考试数学试题江西省南昌市第二中学2023-2024学年高一上学期月考数学试题(一)(已下线)第二章 等式与不等式(压轴题专练)-速记·巧练(沪教版2020必修第一册)
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2 . 已知集合
,若
且
,则称
为集合生成的一个“交错数”,所有“交错数”组成的集合
称为集合
生成的交错集
(1)写出集合
生成的交错集;
(2)若集合
,求证:集合
的交错数各不相同;
(3)无穷数列
的前
项和为
,且对任意
都有
.记
,判断集合
生成的交错集
与正整数集
的关系,并说明理由.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1fd2491dc0189bacbcb09d74ee95e9b0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6f8bb74bfdabb77c25312c1636fcf309.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a73df366305c13505aa32d142f8e96e8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e53a5ea9c349d77bdd3e19dfdcaa6784.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/604f15dc563da9528ee12d1b2bb341e0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f9e8449aad35c5d840a3395ea86df6d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
(1)写出集合
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ef5bc58791e16a37f58c66d95e905ad4.png)
(2)若集合
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bdff394b957c5d6881d041a887758d13.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
(3)无穷数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/08eb71ecf8d733b6932f4680874dbbf3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cea4ac187cbb465180e89f38250b3970.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c3ddd6d99ad32dd7fdb1797d8cf94786.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a66e3cd1e23a2ee92c01c030642d8df4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f9e8449aad35c5d840a3395ea86df6d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/52866a74e4af867ceea0efb1ad06602c.png)
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3 . 设函数
的图象在
处取得极值4.
(1)求函数
的单调区间;
(2)对于函数
,若存在两个不等正数
,
,当
时,函数
的值域是
,则把区间
叫函数
的“正保值区间”.问函数
是否存在“正保值区间”,若存在,求出所有的“正保值区间”;若不存在,请说明理由.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/88deb283438e42fa6d5356a8ccf039e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9b384412acba251d87902ab928902f16.png)
(1)求函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/51c530f4b7491b95acb8ce3eef9aa09d.png)
(2)对于函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6a1cfb60420ff7e72c1b9d64f69ae063.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5873c01192b7d33b7483f444f90b5b0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3388f7a51aab1f949c9d130f7e12910d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/816365eb548ee95772d1bbae268b5cca.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6a1cfb60420ff7e72c1b9d64f69ae063.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9f18667bdb7b35b23e1cdaaaf4f52f9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9f18667bdb7b35b23e1cdaaaf4f52f9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6a1cfb60420ff7e72c1b9d64f69ae063.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/51c530f4b7491b95acb8ce3eef9aa09d.png)
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名校
4 . 设
是定义在
上的函数,若存在
,使得
在
单调递增,在
上单调递减,则称
为
上的单峰函数,
为峰点,包含峰点的区间称为含峰区间,其含峰区间的长度为:
.
(1)判断下列函数中,哪些是“
上的单峰函数”?若是,指出峰点;若不是,说出原因;
;
(2)若函数
是
上的单峰函数,求实数
的取值范围;
(3)若函数
是区间
上的单峰函数,证明:对于任意的
,若
,则
为含峰区间;若
,则
为含峰区间;试问当
满足何种条件时,所确定的含峰区间的长度不大于0.6.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4fe7d5809da02c15a43a0e9a898b9086.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4776c85b79df196f606d3ebf3697fbc3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/85b7e150af2052a1664cde963273d905.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4fe7d5809da02c15a43a0e9a898b9086.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d9a801780561c48c27b05e3894de99a2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/eae40787b884e40c9fbff558491372d7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4fe7d5809da02c15a43a0e9a898b9086.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4776c85b79df196f606d3ebf3697fbc3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/13502d46b8563c54c09b29b20b3006a4.png)
(1)判断下列函数中,哪些是“
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/304226ca50149b49702928e44d565964.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c29d3981fe4fc667bfc4b9ab72a0f938.png)
(2)若函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2d99175f13f12333b9bf574b79cf38e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9210e75c35fb455d0446eb7ddba7d79c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
(3)若函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4fe7d5809da02c15a43a0e9a898b9086.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/304226ca50149b49702928e44d565964.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4aa1825e7e125bba03a5617d0ebe2830.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5ce40da2cbd52723210bbfa98a7f81b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/14283f108568721e6d9ec8d42036be33.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/25ba30f1aa5e75750c67b142fc1d7837.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5f2a570f0086433e604736679f7192c3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8ce7ae90d808f05e86ea063238e4b2f9.png)
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5 . 设
,
,其中
.
(1)当
时,求
的值;
(2)对
,证明:
恒为定值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bfcdfa654b5874be6cff58c317351e2e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f312c9fb6d2b3a42bc721b5214be4df7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9a360c7fcb63fe54cde413f4c1859c31.png)
(1)当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cf0086b054ef120408acac806a1b1318.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6b5f1cb8dd4dc2abe96c68b66d52d5f6.png)
(2)对
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fa7b3e0d7e761be5620effba0e1fc40a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/08585c3f07f917012927a448080e1c7d.png)
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2018-06-16更新
|
1029次组卷
|
5卷引用:【全国百强校】江苏省海安高级中学2017-2018学年高二6月月考数学(理)试题
【全国百强校】江苏省海安高级中学2017-2018学年高二6月月考数学(理)试题2020届江苏省南通市高三下学期3月开学考试数学试题2020届江苏省苏州市吴中区高三高考模拟数学试题江苏省扬州市江都区大桥高级中学2020届高三下学期学情调研(二)数学试题(已下线)热点11 计数原理-2022年高考数学【热点·重点·难点】专练(新高考专用)
名校
6 . 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)
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2017-04-11更新
|
1287次组卷
|
4卷引用:2017届安徽省黄山市高三第二次模拟考试数学(理)试卷
2017届安徽省黄山市高三第二次模拟考试数学(理)试卷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(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)
7 . 已知集合
对于
,
,定义A与B的差为
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e9494aad384d2bbd9f570f12c6fc31ee.png)
A与B之间的距离为![](https://staticzujuan.xkw.com/quesimg/Upload/formula/23b53822fe6093b43b46beae65d6abe3.png)
(Ⅰ)当n=5时,设
,求
,
;
(Ⅱ)证明:
,且
;
(Ⅲ) 证明:
三个数中至少有一个是偶数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8c0062971d409798b8a716209536536f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3615fd277cc1be2d8d8468a1ab9e3e96.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/eddb6f1abafe3023e19e095346474f9d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e9494aad384d2bbd9f570f12c6fc31ee.png)
A与B之间的距离为
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/23b53822fe6093b43b46beae65d6abe3.png)
(Ⅰ)当n=5时,设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4660939da3ac24195b0a7b3773e9fd2d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5e9e460c144f7a2141d2df0308b125f2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ce4a2681390214200443ae07c01a4abe.png)
(Ⅱ)证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4010da33cf43870f86be1bf9bfd6d0e1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8513f18376e4e456b939d0f1cdb6e602.png)
(Ⅲ) 证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f859a0d4fb5579ac99e061da9a8a6de1.png)
您最近一年使用:0次
2016-11-30更新
|
457次组卷
|
4卷引用:2010年普通高等学校招生全国统一考试数学(文)(北京卷)