名校
1 . 已知函数
,
为函数
的反函数
(1)讨论
在
上的单调性,并用定义证明;
(2)设
,求证:
有且仅有一个零点
,且
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8e4ccec118032fd96e0713b04c3a27a2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/028517e8bebe634441e0a5c79828e88a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/12be206d66e65eb92ef08bad8cd8f71d.png)
(1)讨论
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d275fbb3ee5cd1177ca5a2ceecbbef0f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ae4a2b3998705e51dbade9ada0873b2b.png)
(2)设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/041f581f277a2de1ef41c354b6e6991e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/18c49ca8562b98657ca9c499093f7233.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/79b752f0f189e5d8666daea73e145dff.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b7d126a2ae5babaf18b9082a975cdc52.png)
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解题方法
2 . 已知函数
,函数
与
互为反函数.
(1)若函数
的值域为
,求实数
的取值范围;
(2)求证:函数
仅有1个零点
,且
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6ff6838d84b68c6f0d3b93b196d9b08d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1938c093dd2fbcb752d0eb7a18d143b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/942c2141d01bde6b48210c56a17fc75e.png)
(1)若函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c7f56243e7c102bcea2755b9e5ab8455.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a43b2faa4f81f32d94612dce724e772b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/294f5ba74cdf695fc9a8a8e52f421328.png)
(2)求证:函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5f6655e9e9bb9995d0c7e1dd02eb718d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/79b752f0f189e5d8666daea73e145dff.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a1680e0b88a968543d32bb4ccf820e0d.png)
您最近一年使用:0次
2024-03-01更新
|
311次组卷
|
2卷引用:湖北省部分学校2023-2024学年高一上学期期末考试数学试题
名校
3 . 三角函数的定义是:在单位圆C:
中,作一过圆心的射线与单位圆交于点P,自x轴正半轴开始逆时针旋转到达该射线时转过的角大小为θ,则P点坐标为
,转动中扫过的圆心角为θ的扇形,由圆弧面积公式和弧度角的定义,可知面积
.类似地对于双曲三角函数有这样的定义:在单位双曲线E:
中,过原点作一射线交右支于点P,该射线和x轴及双曲线围成的曲边三角形面积是
,双曲角
,则P的坐标是
.其中,
称为双曲余弦函数,
称为双曲正弦函数同样,有类似定义双曲正切函数
双曲余切函数
且有如下关系式:
,![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4d279cc5e9f902480c9a0ea810cf9d3a.png)
,
的初等函数表达式.
(Ⅰ)双曲三角函数有如下和差公式,请任选其一进行证明:
①
;
②
;
(Ⅱ)①求函数
在R上的值域;
②若对
,关于x的方程
有解,求实数a的取值范围.
类似三角函数的反函数,试研究双曲三角函数的反函数artanhx,arcothx.
(2)①证明:![](https://staticzujuan.xkw.com/quesimg/Upload/formula/35b450e870513b9cf6021b6416959224.png)
②已知
的级数展开式为
,写出
的级数展开式.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f240cccaf24af8a796abb95cb42be52e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5b93ac1e1087ef8a7827e22983ab895f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/33074bee68ff41ba4c6b675578f19957.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4e1fa37c4c826b5dcfebe86ab6177906.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/050c00da6d39ad0fae411836b0a26979.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15cd370bd2337b78fe820b7b61438c93.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/de2dc9ac6460d3c72e915e93b9f16d08.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6e7c627427318b62d977ff7a86c2cb5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a53b8e0108664bf39aa302570457199a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7e1fe4e3a61667cfe81973a300859f97.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6a252d4a56c74a8829afb1fccbe09d81.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0961cbc097652b999cd4106c671e4cd1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4d279cc5e9f902480c9a0ea810cf9d3a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a53b8e0108664bf39aa302570457199a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6e7c627427318b62d977ff7a86c2cb5.png)
(Ⅰ)双曲三角函数有如下和差公式,请任选其一进行证明:
①
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1079114cdde9367a22632b0165f1a1a8.png)
②
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3510bba38a7f232cc4d9e437e78f5b6a.png)
(Ⅱ)①求函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e154c56d574646a2a541a3fe70c6307b.png)
②若对
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/32f2372e3d0c3de8f5f0579312efe38b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/937c47cddd4b31aeacfad8f81705b827.png)
类似三角函数的反函数,试研究双曲三角函数的反函数artanhx,arcothx.
(2)①证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/35b450e870513b9cf6021b6416959224.png)
②已知
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/802ae3e64c0bb802cc83bf3cf81bfe49.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac1bbb717893d3adb6ce58b3a99bc257.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cc0593e23740ebd0cd068a2eadf059e3.png)
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4 . 我们知道,函数
与
互为反函数.一般地,设A,B分别为函数
的定义域和值域,如果由函数
可解得唯一
也是一个函数(即对任意一个
,都有唯一的
与之对应),那么就称函数
是函数
的反函数,记作
.在
中,y是自变量,x是y的函数.习惯上改写成
的形式.反函数具有多种性质,如:①如果
是
的反函数,那么
也是
的反函数;②互为反函数的两个函数的图象关于直线
对称;③一个函数与它的反函数在相应区间上的单调性是一致的.
(1)已知函数
的图象在点
处的切线倾斜角为60°,求其反函数
的图象在
时的切线方程;
(2)若函数
,试求其反函数
并判断单调性;
(3)在(2)的条件下,证明:当
时,
,
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/da53929a8f67b9aa3827fdbd73ebd265.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/eda90af8ba1d6f9e21a49e96b709f16c.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/b7edf0a72070071cbbcd54c9e2f5ce1b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/84ae23cf6a2823451f9676220b32c782.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6ed006b944ea64f970fee46e2f558467.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b7edf0a72070071cbbcd54c9e2f5ce1b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/942c2141d01bde6b48210c56a17fc75e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c3fe53f7586f7cfbc17e2fd1c1a091bf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c3fe53f7586f7cfbc17e2fd1c1a091bf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f3a4f23baf90cbc32cba9f6b9bfea2e1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/135bcf6d7f7c04641823b90f1d038eee.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/135bcf6d7f7c04641823b90f1d038eee.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d77f5191798242b7b9b88a75e17e4425.png)
(1)已知函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/942c2141d01bde6b48210c56a17fc75e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3654254401fc902c3cb4912969f21f88.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/135bcf6d7f7c04641823b90f1d038eee.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e55aa0a20848c37c1892c567b2315e04.png)
(2)若函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c83104d98d6920b19fe2cc3cf097bce2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f786a5701dc1a8a015e8843c3360151b.png)
(3)在(2)的条件下,证明:当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/65cc52aacc31a21a443c8de0374b24f1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/47f8c8e4cfd60c1793cfa4526d1fc853.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/897453f27022194d1f57e8b54960111f.png)
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