1 . 已知集合
,其中
,
是函数
定义域内任意不相等的两个实数.
(1)若
,同时
,求证:
;
(2)判断
是否在集合A中,并说明理由;
(3)设函数
的定义域为B,函数
的值域为C.函数
满足以下3个条件:
①
,②
,③
.试确定一个满足以上3个条件的函数
要对满足的条件进行说明).
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/855526c0f0bd164fff3f74274a6bbecf.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/4fe7d5809da02c15a43a0e9a898b9086.png)
(1)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7d5f5f81ea1a02b88ff8491fcf4937db.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bbec485ab7b15f1e09f163fe990577c5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b529f58369b7699fcd20be1497d07b4e.png)
(2)判断
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bee6881a170f6ef9ed5c133b95c2f448.png)
(3)设函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4fe7d5809da02c15a43a0e9a898b9086.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4fe7d5809da02c15a43a0e9a898b9086.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4fe7d5809da02c15a43a0e9a898b9086.png)
①
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7d5f5f81ea1a02b88ff8491fcf4937db.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/64afdc6fcdc4cd326bb11679766c223e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9db98efa3d9f09a65fc4c88a32cdc0e7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4fe7d5809da02c15a43a0e9a898b9086.png)
您最近一年使用:0次
解题方法
2 . (1)在
中,角A,B,C所对的边分别是a,b,c,证明余弦定理:
;
(2)长江某地南北岸平行,如图所示,江面宽度
,一艘游船从南岸码头A出发航行到北岸,假设游船在静水中的航行速度
,水流速度
,设
和
的夹角为θ(
),北岸的点
在点A的正北方向.
![](https://img.xkw.com/dksih/QBM/editorImg/2022/11/22/56ebc301-c65c-422c-99b5-7cea25092097.png?resizew=164)
①当
多大时,游船能到达
处,需要航行多少时间?
②当
时,判断游船航行到达北岸的位置在
的左侧还是右侧,并说明理由.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/34369422d71dd95c61cdd1b8245d7b6c.png)
(2)长江某地南北岸平行,如图所示,江面宽度
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c24d22d19c32fcb0b7d11c6f51adb423.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ee679dc49dc97433f04dd84dbdbeebd9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/af98572c480b7bf2d9c8c864bdf1fcae.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c4764374bd2fb78e59cd0b283637baeb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c63055a5d6916f99d07fede49120753f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0c70f029a9f809e41b169cf19eba2e0a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a18722354086c42e62334983fc50eb6a.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2022/11/22/56ebc301-c65c-422c-99b5-7cea25092097.png?resizew=164)
①当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/aefd06c239145a2b6ae87a955aa51414.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a18722354086c42e62334983fc50eb6a.png)
②当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bac738fd004e27b51aee959473e266ae.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a18722354086c42e62334983fc50eb6a.png)
您最近一年使用:0次
名校
3 . 已知
是满足下列性质的所有函数
组成的集合:对任何
(其中
为函数
的定义域),均有
成立.
(1)已知函数
,
,判断
与集合
的关系,并说明理由;
(2)是否存在实数
,使得
,
属于集合
?若存在,求
的取值范围,若不存在,请说明理由;
(3)对于实数
、![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2c94bb12cee76221e13f9ef955b0aab1.png)
,用
表示集合
中定义域为区间
的函数的集合.
定义:已知
是定义在
上的函数,如果存在常数
,对区间
的任意划分:
,和式
恒成立,则称
为
上的“绝对差有界函数”,其中常数
称为
的“绝对差上界”,
的最小值称为
的“绝对差上确界”,符号
;求证:集合
中的函数
是“绝对差有界函数”,并求
的“绝对差上确界”.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4fe7d5809da02c15a43a0e9a898b9086.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b03e6ac25d3f66b036bee546b28296e7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4b5886cf72ed5a1073263eb9ff485c7c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4fe7d5809da02c15a43a0e9a898b9086.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e9d3190fce8d15207e6f1cc7feeb7318.png)
(1)已知函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3d5ab1f6923041b7f9a4e7bb641ead4e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6106d267314d135e59cc9cf5b1c89e0c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4fe7d5809da02c15a43a0e9a898b9086.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
(2)是否存在实数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/73ce34293d9465076c5e397efee9d078.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/305dde0e78d0aa1fd43a7b09f5f46d8c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
(3)对于实数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2c94bb12cee76221e13f9ef955b0aab1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/691dacaa369a98eb9cbb4cff25017be8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/79775b36e40e07114c57a801e713398d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4776c85b79df196f606d3ebf3697fbc3.png)
定义:已知
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0eb7df298a9364b36e079a61caec815c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/627565d32e529cafcd2744d006ec6de2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/094f977194228bed828f3507f5898934.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/627565d32e529cafcd2744d006ec6de2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/32f3cf02726d681c4346276f9f447975.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4ce02fc7915cf064510155079b26e076.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0eb7df298a9364b36e079a61caec815c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/627565d32e529cafcd2744d006ec6de2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0b68df477b3ee45ac0f725db00d465a1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0eb7df298a9364b36e079a61caec815c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0b68df477b3ee45ac0f725db00d465a1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0eb7df298a9364b36e079a61caec815c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/801322dacef77d328962b550041ff021.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/68965738ce4064cc4d9dbabc86f950a9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0eb7df298a9364b36e079a61caec815c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0eb7df298a9364b36e079a61caec815c.png)
您最近一年使用:0次
2018-10-19更新
|
854次组卷
|
2卷引用:【全国百强校】江苏省扬州中学2018-2019学年高一上学期期中考试数学试题
名校
4 . 如果函数
在定义域内存在区间
,使得该函数在区间
上的值域为
,则称函数
是该定义域上的“和谐函数”.
(1)求证:函数
是“和谐函数”;
(2)若函数
是“和谐函数”,求实数
的取值范围.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f030c36bb8786df88d401792062a4100.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f030c36bb8786df88d401792062a4100.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5f15eb7cd066e13367998a2da2653976.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
(1)求证:函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2f52f6ee8ead43c46f73102b87a2d943.png)
(2)若函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9a375eaef2dd9a63ac141237bcf7a3c1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/36a1b09c653185842513e24ebba60bb3.png)
您最近一年使用:0次
2018-02-03更新
|
396次组卷
|
3卷引用:江苏省南京市2017-2018学年度第一学期高一期末统考数学试卷
名校
5 . 对于函数
,若存在一个实数
使得
,我们就称
关于直线
对称.已知
.
(1)证明
关于
对称,并据此求:
的值;
(2)若
只有一个零点,求
的值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4fe7d5809da02c15a43a0e9a898b9086.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/41fd9a7f875d1e3fe3dac7c0b797c93c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/51c530f4b7491b95acb8ce3eef9aa09d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/54b53b86bd516400d6fa7dabb3603f31.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/613ad900cfb3a171639219a273f88c70.png)
(1)证明
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4fe7d5809da02c15a43a0e9a898b9086.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9b384412acba251d87902ab928902f16.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ae82f7c606a77822fb75559e10860f30.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4fe7d5809da02c15a43a0e9a898b9086.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/294f5ba74cdf695fc9a8a8e52f421328.png)
您最近一年使用:0次
2017-11-16更新
|
1131次组卷
|
3卷引用:江苏省连云港市赣榆区海头高中2019-2020学年高一上学期期中数学试题
解题方法
6 . 在平面直角坐标系
中,将从点
出发沿纵、横方向到达点
的任一路径称为
到
的一条“折线路径”,所有“折线路径”中长度最小的称为
到
的“折线距离”.如图所示的路径
与路径
都是
到
的“折线路径”.某地有三个居民区分别位于平面
内三点
,
,
,现计划在这个平面上某一点
处修建一个超市.
![](https://img.xkw.com/dksih/QBM/2017/5/10/1683970740789248/1715263246712832/STEM/5a37f9ceff464a899e865c12f92b60c1.png?resizew=237)
(1)请写出点
到居民区
的“折线距离”
的表达式(用
表示,不要求证明);
(2)为了方便居民,请确定点
的位置,使其到三个居民区的“折线距离”之和最小.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7ee31829d0d4d5f779a957d7df8058ab.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/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/54a5d7d3b6b63fe5c24c3907b7a8eaa3.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/8cb9227785322ac87fe538f3164f30dc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0d3a701135e9a11b344e2f8bff95410.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/7ee31829d0d4d5f779a957d7df8058ab.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/627f1578beab2071cefd6cb0278b57f7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b79485e42b7546ccfa3cd6d08491460f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7685ae8254f0b1442482c892d039ae0d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8701e0cce437edc830438b4fe6277d89.png)
![](https://img.xkw.com/dksih/QBM/2017/5/10/1683970740789248/1715263246712832/STEM/5a37f9ceff464a899e865c12f92b60c1.png?resizew=237)
(1)请写出点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5c02bc0c74292b1e8f395f90935d3174.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9b0fffbec1fe851795dfdd448bf0d165.png)
(2)为了方便居民,请确定点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
您最近一年使用:0次