1 . 固定项链的两端,在重力的作用下项链所形成的曲线是悬链线.1691年,莱布尼茨等得出“悬链线”方程
,其中
为参数.当
时,就是双曲余弦函数
,类似地我们可以定义双曲正弦函数
.它们与正、余弦函数有许多类似的性质.
(1)类比正弦函数的二倍角公式,请写出双曲正弦函数的一个正确的结论:
_____________.(只写出即可,不要求证明);
(2)
,不等式
恒成立,求实数
的取值范围;
(3)若
,试比较
与
的大小关系,并证明你的结论.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/852665ec9c3a65b758898059361f11a7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/071a7e733d466949ac935b4b8ee8d183.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4580cc037c0c760c728cdbb74a8154c6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f0a7c1d3681898e25187a896aeb0c8c0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e0718c04bdf70989bcc90b902671a692.png)
(1)类比正弦函数的二倍角公式,请写出双曲正弦函数的一个正确的结论:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3d8fe1e65b09697538d4dee0746846f4.png)
(2)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2fe9f3099ed9429dc5b4e38a350e524a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/343e7c30c2a5d166819b28e23fad2203.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/294f5ba74cdf695fc9a8a8e52f421328.png)
(3)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/563f464c94feac28033f6f3a271fbe8f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9a2cebaab3423dfb2f2c944dfc43df8c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bb966b7b2dd6581640bcee2d97dacf77.png)
您最近一年使用:0次
2024-01-27更新
|
941次组卷
|
9卷引用:福建省宁德市2023-2024学年高一上学期1月期末质量检测数学试题
福建省宁德市2023-2024学年高一上学期1月期末质量检测数学试题(已下线)专题04 三角函数恒等变形综合大题归类 -期末考点大串讲(苏教版(2019))(已下线)专题08 期末必刷解答题专题训练的7种常考题型归类-期末真题分类汇编(北师大版2019必修第二册)河南省名校联盟2023-2024学年高一下学期3月测试数学试题(已下线)第八章:向量的数量积与三角恒等变换章末重点题型复习(2)-同步精品课堂(人教B版2019必修第三册)河南省信阳市信阳高级中学2023-2024学年高一下学期3月月考(一)数学试题(已下线)第8章:向量的数量积与三角恒等变换章末综合检测卷(新题型)-【帮课堂】(人教B版2019必修第三册)重庆市缙云教育联盟2024届高三下学期2月月度质量检测数学试题(已下线)压轴题函数与导数新定义题(九省联考第19题模式)讲
名校
解题方法
2 . 设函数
定义在区间
上,若对任意的
、
、
、![](https://staticzujuan.xkw.com/quesimg/Upload/formula/177e54e8deea5da9dc6bc82eb3de0c2c.png)
,当
,且
时,不等式
成立,就称函数
具有M性质.
(1)判断函数
,
是否具有M性质,并说明理由;
(2)已知函数
在区间
上恒正,且函数
,
具有M性质,求证:对任意的
、![](https://staticzujuan.xkw.com/quesimg/Upload/formula/aec19b68e3add9d5bfcc6269a1855b87.png)
,且
,有
;
(3)①已知函数
,
具有M性质,证明:对任意的
、
、![](https://staticzujuan.xkw.com/quesimg/Upload/formula/291c25fc6a69d6d0ccfb8d839b9b4462.png)
,有
,其中等号当且仅当
时成立;
②已知函数
,
具有M性质,若
、
、
为三角形
的内角,求
的最大值.
(可参考:对于任意给定实数
、
,有
,且等号当且仅当
时成立.)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/51c530f4b7491b95acb8ce3eef9aa09d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ea9c587f6257331045c362ef25677c20.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/770cf3716f1e9dc8023a898df7f33783.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/177e54e8deea5da9dc6bc82eb3de0c2c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9d589f18d16b1a6bbd5108409c53fd05.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a49c641617f38855f6abc7baf36af8e3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3f05279fb93940ea0741b64227cc58c0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8a70644524df044d4a24b998a81d44c0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/51c530f4b7491b95acb8ce3eef9aa09d.png)
(1)判断函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bee6881a170f6ef9ed5c133b95c2f448.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/475a20b276768b190ac15c9aa5c352ef.png)
(2)已知函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/51c530f4b7491b95acb8ce3eef9aa09d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ea9c587f6257331045c362ef25677c20.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/80fcd5a1ca4f9abf76c88db3a3542b38.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fb5348b540c0b2e012191ae95351aaac.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/9d589f18d16b1a6bbd5108409c53fd05.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/33bd24e647a626899a243a3f3984f90a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/450fb41cf5543a06035606ff29a9e934.png)
(3)①已知函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/51c530f4b7491b95acb8ce3eef9aa09d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fb5348b540c0b2e012191ae95351aaac.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/9d589f18d16b1a6bbd5108409c53fd05.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f183be2a65b185fd240990dffdec3ba7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b62e63003be4ad8c4c51e36e71df2ac3.png)
②已知函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b923078510697d5f7f9ea392eb76dd9a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/089e6e44271b4c08be46dda1e7403741.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/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7bef5239ddbb0972700ce01daf9ee7cf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5a8080fef9bdfa92ae70f3e314eef3e3.png)
(可参考:对于任意给定实数
![](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/205ca5a7d5bede14db0175445bb6d508.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2f6b79d363c080275b93b8cc4b279653.png)
您最近一年使用:0次
2021-12-27更新
|
698次组卷
|
5卷引用:专题06 期末解答压轴题-《期末真题分类汇编》(上海专用)
(已下线)专题06 期末解答压轴题-《期末真题分类汇编》(上海专用)上海市黄浦区2022届高三一模数学试题(已下线)上海市黄浦区2022届高三上学期一模数学试题(已下线)第04讲 函数最值与性质-3上海市文来高中2023届高三上学期期中数学试题
名校
解题方法
3 . 设函数
的定义域为
.若存在常数
,
,使得对于任意
,
成立,则称函数
具有性质
.
(1)判断函数
和
具有性质
?(结论不要求证明)
(2)若函数
具有性质
,且其对应的
,
.已知当
时,
,求函数
在区间
上的最大值;
(3)若函数
具有性质
,且直线
为其图像的一条对称轴,证明:
为周期函数.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4aa0df7f1e45f9de29e802c7f19a4f64.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0b68df477b3ee45ac0f725db00d465a1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/809ea2eff71a0de3db640313ad25b7a7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c4166972dec0aa3e8694a44eeb941a08.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/404d068b60dd901194f1684d023212ed.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
(1)判断函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d77f5191798242b7b9b88a75e17e4425.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/58c65d71e57e6e7697e2f627dcd58583.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
(2)若函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8073fa685bc10cf01a0128220feac940.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e5e661ad31aa4c6d8684923cf904bf1b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/45d0588454ec8b64bf86578fb90b39e2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/aa7f9b35017daa8b524c5717a355834a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c55351494cd96fed31976fdc5d9c7292.png)
(3)若函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4669810732b633b60dbeaf0bf57204f6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/36d71f015144ffaf1faec94a259b4a06.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4669810732b633b60dbeaf0bf57204f6.png)
您最近一年使用:0次
2021-08-01更新
|
580次组卷
|
3卷引用:北京市西城区2020-2021学年高一下学期期末数学试题
解题方法
4 . 已知函数
.
(1)试用周期函数的定义证明函数
是周期函数,并指出该函数的一个周期;
(2)若函数
在
上取最大值、最小值时,所对应的x的值按从小到大依次记为
,试求
关于
的函数关系式;
(3)在满足(2)的条件下,记
,求证:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/04a19ed84596e39625983668dee15dd8.png)
(1)试用周期函数的定义证明函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
(2)若函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8938db94f49dcbe0c383fba0241bb0da.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/79ae6558e11384a40f3a338b73385ee1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3282e5fde4ae53fcb1bb072a685304c9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/223ed9652852ca4d996fd1f20808df9a.png)
(3)在满足(2)的条件下,记
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/25c0bf9a26970107ec9ad726dc4dbd7c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/04ce595c87542ef504dae056509d008a.png)
您最近一年使用:0次
名校
解题方法
5 . 在
中,内角
的对边分别为
的面积为
,且
.
(1)证明:
;
(2)若
,求
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/24e0c10fb103930eabd5fa18e8f9bb06.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/718b5b48053888ab3b234b8cb56a0fc1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cf231f8f86fb922df4ca0c87f044cec3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/078e30318231eb60cd787c7b595d3b6b.png)
(1)证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/da8607bde1fa6cde631a46e921d959a0.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/56ee49209b78441c35512d86ad426275.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0fa6361e919ac07ee6ed642556e1d1ae.png)
您最近一年使用:0次
7日内更新
|
665次组卷
|
3卷引用:专题05 解三角形大题常考题型归类-期期末考点大串讲(人教B版2019必修第四册)
(已下线)专题05 解三角形大题常考题型归类-期期末考点大串讲(人教B版2019必修第四册)江西省多校联考2023-2024学年高一下学期5月教学质量检测数学试卷江苏省扬州中学2024届高三下学期全真模拟数学试卷
名校
6 . 在
中,AD是
的角平分线,AE是边BC上的中线,点D、E在边BC上.
(1)用正弦定理证明
;
(2)若
,求DE的长.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cc3322a2ad9a95bdc9fc576a7a158d4d.png)
(1)用正弦定理证明
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a9c6ba1dce1e32a78ea2f3a85a3c8962.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c95d54823e4a9895941d5b88b802670c.png)
您最近一年使用:0次
名校
解题方法
7 . 现定义“
维形态复数
”:
,其中
为虚数单位,
,
.
(1)当
时,证明:“2维形态复数”与“1维形态复数”之间存在平方关系;
(2)若“2维形态复数”与“3维形态复数”相等,求
的值;
(3)若正整数
,
,满足
,
,证明:存在有理数
,使得
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d9dc4e868a310c371ff88075d8a966a6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ef9d830212489b316bb052455098108e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3a7035cd4adda5d72a9fc9f9fda75995.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/48f093c61867ee4ce75f951d46b9b123.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/edc8299790d98621b87e73212a2ebb91.png)
(1)当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/905dd10639c9fef5ef8d66a124756140.png)
(2)若“2维形态复数”与“3维形态复数”相等,求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c136aaf9b5dedec254a92ce302f4a70c.png)
(3)若正整数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/294f5ba74cdf695fc9a8a8e52f421328.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/94742ebbb028c50d7a58e3e8f4ab329c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/35490c12e57ecd91af9934cb17b5c927.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4ed110fbfeb14003270a1039ba174d0e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9aa8a716a31b0f51b70fdf9bdb257909.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2f02f2606180ffeda602ff9ae747af6f.png)
您最近一年使用:0次
2024-05-11更新
|
590次组卷
|
3卷引用:湖北省黄冈市浠水县第一中学2023-2024学年高一下学期期末质量检测数学试题
名校
解题方法
8 . 在
中,角A,B,C所对的边分别为a,b,c,且满足
,
.
(1)求证:
;
(2)求
的值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/988b7e964e313579ab8869d67d5be007.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7c6fdc45e193a71f67399d7a9f3320c0.png)
(1)求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/de371aef17ea71040f165f9b7f653799.png)
(2)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c1311f32edf13f8caee5edb03f24a7ba.png)
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名校
9 . 设
,函数
.
(1)讨论函数
的零点个数;
(2)若函数
恰有两个零点
,求证:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/10bbdef421c976962a270a2beabbad91.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/65e36c4a0587c78c0d17e90b20b422f2.png)
(1)讨论函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
(2)若函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8ce7ae90d808f05e86ea063238e4b2f9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c07bbaa783c21744c573ce71de07b92a.png)
您最近一年使用:0次
10 . 定义:
为实数
对
的“正弦方差”.
(1)若
,则实数
对
的“正弦方差”
的值是否是与
无关的定值,并证明你的结论
(2)若
,若实数
对
的“正弦方差”
的值是与
无关的定值,求
值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/772833a69a6fac02a9a7047af3759912.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e30a14046b3202838a0dd1481153515a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1a4438bae1705c0f26beddf41322c087.png)
(1)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5e7f35fd3cea28f5ddda1786d10d6dd5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cef1543ed59105b12a3afb1d356f9d61.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1a4438bae1705c0f26beddf41322c087.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1100379a4385b9ce064847bc21760adc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1a4438bae1705c0f26beddf41322c087.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0b40d00a9164afda82e9560a2f05c1ab.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cef1543ed59105b12a3afb1d356f9d61.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1a4438bae1705c0f26beddf41322c087.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1100379a4385b9ce064847bc21760adc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1a4438bae1705c0f26beddf41322c087.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c4e288596fa3811dd2c17bded60e82e7.png)
您最近一年使用:0次