解题方法
1 . 已知双曲线
的一个焦点为
为坐标原点,点
在双曲线上运动,以
为直径的圆过点
,且
恒成立,则
的离心率的取值范围为______ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/844a1c77cdc51fb57f2fc55d791ea64f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09224287f31db61f4335560cdaf28ae3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/01c74a907dda6bb7d9d56d009d9df253.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/01c74a907dda6bb7d9d56d009d9df253.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1dde8112e8eb968fd042418dd632759e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4ed568f7b6490ebbb104d105dbc07f6e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
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解题方法
2 . 一个完美均匀且灵活的项链的两端被悬挂, 并只受重力的影响,这个项链形成的曲 线形状被称为悬链线.1691年,莱布尼茨、惠根斯和约翰・伯努利等得到“悬链线”方程
,其中
为参数.当
时,就是双曲余弦函数
,类似地双曲正弦函数
,它们与正、余弦函数有许多类似的性质.
(1)类比三角函数的三个性质:
①倍角公式
;
②平方关系
;
③求导公式![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fad7736e047d89385512f5715c4434a4.png)
写出双曲正弦和双曲余弦函数的一个正确的性质并证明;
(2)当
时,双曲正弦函数
图象总在直线
的上方,求实数
的取值范围;
(3)若
,证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f07f8015f0a035e80a166092be0b7318.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/7ddb06bbda9da4a045750637f4215593.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6ee7a18d65bcc8b5a94292365009462e.png)
(1)类比三角函数的三个性质:
①倍角公式
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/190a6011b263200d13f62e636398e26d.png)
②平方关系
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9da8f21743a3a14ce326eaeecb86a417.png)
③求导公式
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fad7736e047d89385512f5715c4434a4.png)
写出双曲正弦和双曲余弦函数的一个正确的性质并证明;
(2)当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/08115d6d9f876dea921a4d32260ff1fb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0b9213864ba0aa83b0f11be6ad6ed6bc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac02a054bd0771a56987af33454baaea.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f0a532e15e232cb4b99a8d4d07c89575.png)
(3)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/db1c01b5cfd9630ca3e7d8f48ada6ef7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a2a602db560a460408aae63f5cde96d6.png)
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2024-06-10更新
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2卷引用:2024届山东省潍坊市高考三模数学试题
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3 . 已知函数
,
为
的导数
(1)讨论
的单调性;
(2)若
是
的极大值点,求
的取值范围;
(3)若
,证明:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0405779583ded3b24cfa5479851dbf20.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6a4b04824a308519a61318a82aa97a05.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4fe7d5809da02c15a43a0e9a898b9086.png)
(1)讨论
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6a4b04824a308519a61318a82aa97a05.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bb45f673c56a289ea78831c9237e8d20.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4fe7d5809da02c15a43a0e9a898b9086.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
(3)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5caabda288fc01cc168938846eec5d2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6a901b3cb6a4b5201add46eb26a0d8c2.png)
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2024-06-08更新
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6卷引用:山东省枣庄市2024届高三三调数学试题
山东省枣庄市2024届高三三调数学试题山东省青岛市2024届高三下学期第二次适应性检测数学试题(已下线)山东省济南市2024届高三下学期5月适应性考试(三模)数学试题(已下线)专题9 利用放缩法证明不等式【练】湖北省武汉市汉铁高级中学2024届高考数学考前临门一脚试卷江苏省扬州市扬州中学2023-2024学年高二下学期5月月考数学试题
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4 . 已知函数
,则下列说法正确的是( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1e06c3168a8206fa6a62e05e3d61eb3c.png)
A.存在实数![]() ![]() |
B.存在实数![]() ![]() |
C.存在实数![]() ![]() |
D.存在实数![]() ![]() |
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解题方法
5 . 定义:函数
满足对于任意不同的
,都有
,则称
为
上的“
类函数”.
(1)若
,判断
是否为
上的“2类函数”;
(2)若
为
上的“3类函数”,求实数a的取值范围;
(3)若
为
上的“2类函数”,且
,证明:
,
,
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4fe7d5809da02c15a43a0e9a898b9086.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e9d6fe21d6ed78bfc1d2b9cc41a766c1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5f2a699f43d6836c18eaced5758a37a3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4fe7d5809da02c15a43a0e9a898b9086.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f030c36bb8786df88d401792062a4100.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f0a532e15e232cb4b99a8d4d07c89575.png)
(1)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6ece875296333d786d8a671b2749255.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4fe7d5809da02c15a43a0e9a898b9086.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a2ec965488c7e1cea085463c7731285.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c93ed0f3a123e0e3b1c08db887fa1697.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d47e734b17201fe992be7775714e9558.png)
(3)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4fe7d5809da02c15a43a0e9a898b9086.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9210e75c35fb455d0446eb7ddba7d79c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a2db4bd08a64c7ceefac83e2fce50b90.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/673207f6b77b8192d25463d071737b7c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4ea6e1bfbda5c4f6421ed18e802aba04.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9b1ab2e5e3dd3a1c768a88eb182b44d9.png)
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2卷引用:山东省滨州市2024届高三下学期二模数学试题
6 . 已知函数
,其中
.
(1)当
时,判断
的单调性;
(2)若
存在两个极值点
.
(ⅰ)证明:
;
(ⅱ)证明:
时,
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c83a7dd3632df5886f0018e85890140c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/10bbdef421c976962a270a2beabbad91.png)
(1)当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/10ede78fd7ac619ea597856254bb5d75.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/39c975f44ddb3ba2a07bdaa4f1461fca.png)
(ⅰ)证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/861c839ef4c6fafe3b59a10c6c7ac969.png)
(ⅱ)证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4921923069c4f38a0af1ff8637e35b3c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f12cf6946e5f7b7d2e5d5d79dcda9b43.png)
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7 . 已知函数
.
(1)当
时,求证:
存在唯一的极大值点
,且
;
(2)若
存在两个零点,记较小的零点为
,t是关于x的方程
的根,证明:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9250ac88895db27b0ccb5869b0e8bf19.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/79b752f0f189e5d8666daea73e145dff.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/84ffe0afc6fa9e62ff75d13f656e7cc4.png)
(2)若
![](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/17f7f01bcb51cd8fd65827c26b065a99.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7867f9fdfa7811958bf68b7ef10dd792.png)
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8 . 数学家加斯帕尔·蒙日在研究圆锥曲线时发现:椭圆
任意两条互相垂直的切线的交点都在以原点O为圆心,
为半径的圆上,这个圆被称为该椭圆的蒙日圆.已知椭圆C:
可以与边长为
的正方形的四条边均相切,它的左、右顶点分别为A,B,则( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7dd54b9df3402ad91e2d34c40efe0c7a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/da001dad7941e6c9858637d7b62cec59.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dec92768b052f6ceb3efc112b7e48625.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b10e8abf8690e4b129466ddb918bcc94.png)
A.![]() |
B.若矩形的四条边均与椭圆C相切,则该矩形面积的最大值为12 |
C.椭圆C的蒙日圆上存在两个点M满足![]() |
D.若椭圆C的切线与C的蒙日圆交于E,F两点,且直线OE,OF的斜率都存在,记为![]() ![]() ![]() |
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9 . 已知函数
,
.
(1)判断函数
在区间
上的零点个数,并说明理由;
(2)函数
在区间
上的所有极值之和为
,证明:对于
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b1044dcf4fba551e1b7fbfeb895ea08c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/df3e4d7bd045ca908e0d291df76e02d3.png)
(1)判断函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4669810732b633b60dbeaf0bf57204f6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ad14579830d0293b1390911cb603eb02.png)
(2)函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/88d035309c953cd9ea43094ffbd888b5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/daba0444595815c4aa0bcbca50b2f46c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8fa2ed9c260f345b10e8da2e80545655.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f1245cd8a6cf18939a889c1e64782e4f.png)
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10 . 已知函数
.
(1)讨论
的单调性;
(2)若
恒成立,求实数
的取值范围.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b0fd16778910dccc8bb0bcd690f9d57a.png)
(1)讨论
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d275fbb3ee5cd1177ca5a2ceecbbef0f.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/44f490ff297bcfeb77f1d40b50f26411.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/294f5ba74cdf695fc9a8a8e52f421328.png)
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2024-05-15更新
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764次组卷
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2卷引用:山东省烟台市2024年高考适应性练习(二模)数学试题