名校
1 . 已知函数
.
(1)当
时,讨论函数
的单调性;
(2)若不等式
恒成立,求
的取值范围;
(3)在(1)的条件下,设
,
,且
.求证:当
,且
时,不等式
成立.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b7e7bafcc08b76256e0ec491fb36f712.png)
(1)当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cf0086b054ef120408acac806a1b1318.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9d6ea20aa7804cb6e41322bc3d8dc99b.png)
(2)若不等式
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/810e8fa5cf17ae94a41803772c488726.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/294f5ba74cdf695fc9a8a8e52f421328.png)
(3)在(1)的条件下,设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bbf2a234b8102356b2c13a3c0b75a00e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/31c0c165afc6ed30f0d41808f8442f86.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e9645bd4d2002993b90ec6d48f9c04f7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0704f453b2de48d36911f7db496bbf82.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/48f093c61867ee4ce75f951d46b9b123.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5922c1719da2520a49b75db30ab0c276.png)
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2 . 定义在
上的函数
,若对任意
,存在常数
,都有
成立,则称
是
上的有界函数,其中
称为函数
的上界.已知函数
.
(1)若
是奇函数,判断函数
是否为有界函数,并说明理由;
(2)若
在
上是以
为上界的函数,求
的取值范围.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8455657dde27aabe6adb7b188e031c11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e02cab1add26335b3cb43d5b54c7c853.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2480f87a11c4cd450bc9454ea7276722.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/aea232de27d21a2646fd4520ea0726bf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8455657dde27aabe6adb7b188e031c11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/65a45e285f154675b459b2247f3682fb.png)
(1)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f2d2a706da87c1775d9e89799e45b4df.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a2ec965488c7e1cea085463c7731285.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/56d266a04f3dc7483eddbc26c5e487db.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/294f5ba74cdf695fc9a8a8e52f421328.png)
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3 . 用篱笆在一块靠墙的空地围一个面积为
的等腰梯形菜园,如图所示,用墙的一部分做下底
,用篱笆做两腰及上底,且腰与墙成
,当等腰梯形的腰长为多少时,所用篱笆的长度最小?并求出所用篱笆长度的最小值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7c8fbf64cd29a08ae703e9fa2035f3c0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/03902478df1a55bc99703210bccab910.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/be6a6301878fed2a01413020b27310a5.png)
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解题方法
4 . 已知函数
.
(1)若
,求不等式
的解集;
(2)若
时,
的最小值为
,求
的值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a4b23fabe4b8c9810a3e043dfc27ffc7.png)
(1)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b9a81e971f501d78f5560c0c3d42f0f6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9f1a722a2595fdc71462654b764ee49d.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/24aa16b780156e18f12baa2b8ee0f9a5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/acbc6a613224461ade69362d46550474.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
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解题方法
5 . 已知函数
.
(1)求不等式
的解集;
(2)若
,不等式
恒成立,求
的取值范围.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/276246854dd33691f6798db87faeea0b.png)
(1)求不等式
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1c73a98c1b3504e09bfbe0db849b0d24.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/08115d6d9f876dea921a4d32260ff1fb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d37dafb12565279285111a5948d835b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
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解题方法
6 . 记全集
,已知集合
,
.
(1)若
,求
;
(2)若
,求
的取值范围.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/442be389c6471b36eff2652b75beb114.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/126129b13dc3e8817eb0eb2de6dd1cf3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/672388bcf46a971284cd2f4763bd37e6.png)
(1)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8e258ab9e600435b37465092243d99f6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5a69c58edb641bdf02f3685d4f36d793.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1d7d6537709d1ab4bec43bd14c9ba696.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
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解题方法
7 . 已知数列
的前n项和为
,且
.
(1)求数列
的通项公式;
(2)从数列
中剔除第1项,第4项,第7项,
,第
项后,将剩下的项保持顺序不变组成一个新数列
,求数列
的前
项和
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/76aef4cdcb5af742ce28003b7b6c8c20.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/08eb71ecf8d733b6932f4680874dbbf3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f6ae2a1d6845f05120f0201da06e1467.png)
(1)求数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/76aef4cdcb5af742ce28003b7b6c8c20.png)
(2)从数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/76aef4cdcb5af742ce28003b7b6c8c20.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/daa5e9bd516f6282483b92cfe6074623.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/803446057e38b5376b31ebf8fc78d2f4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f329b217e1051b23f0d61023cdc6e69.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f329b217e1051b23f0d61023cdc6e69.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e2d51f9147b8265c0276c1f2c2659197.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fbd85b79372dc6e596d465f738c3c300.png)
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解题方法
8 . 在棱长均为2的正三棱柱
中, E为
的中点.过AE的截面与棱
分别交于点F, G.
(1)若F为
的中点,求三棱柱被截面AGEF分成上下两部分的体积比 ![](https://staticzujuan.xkw.com/quesimg/Upload/formula/301709b97852b4c2b951cfa2b2cfa2d2.png)
(2)若四棱锥
的体积为
求截面 AGEF 与底面ABC所成二面角的正弦值;
(3)设截面AFEG的面积为
面积为S₁,△AEF面积为
当点F在棱
上变动时,求
的取值范围.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/770abdd10660689c605577f9cb6d9db1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fa717be2a43918b9eb13655f86042a93.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b828011b4a0be31af4f30d829d7e30b1.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2024/6/18/c1fc03ae-c978-409e-8fa2-9e23147b9838.png?resizew=123)
(1)若F为
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5a5fb47a5caaa26007a7606acf05c52c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/301709b97852b4c2b951cfa2b2cfa2d2.png)
(2)若四棱锥
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/04eb4a437f9aee581cb2d0fff0d9d588.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4c181e33261a6c8df1e5291b0a119dc9.png)
(3)设截面AFEG的面积为
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/431bceebfd7dbb3dcbd0348e486a97fd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/71c9821bd2db0d34acbd7acd0fb2bb8a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5a5fb47a5caaa26007a7606acf05c52c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d17646ce1c1499a928584904d416a2a2.png)
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9 . 已知桶
中盛有3升水,桶
中盛有1升水.现将桶
中的水的
和桶
中的水的
倒入桶
中,再将桶
与桶
中剩余的水倒入桶
中;然后将桶
中的水的
和桶
中的水的
倒入桶
中,再将桶
与桶
中剩余的水倒入桶
中;如此继续操作下去.
(1)求操作1次后桶
中的水量;
(2)求操作
次后桶
中的水量;
(3)至少操作多少次,桶
中的水量与桶
中的水量之差小于
升?(参考数据:
,
)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7356ec98b600ece41f3a6b4bc26a7d59.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c41d793c851a2f72f787913ba23e459c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7356ec98b600ece41f3a6b4bc26a7d59.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bf31876698721a199c7c53c6b320aa86.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c41d793c851a2f72f787913ba23e459c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4dac452fbb5ef6dd653e7fbbef639484.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a18722354086c42e62334983fc50eb6a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7356ec98b600ece41f3a6b4bc26a7d59.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c41d793c851a2f72f787913ba23e459c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/97c01fdc7bc471af0b264a04aef0823e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a18722354086c42e62334983fc50eb6a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bf31876698721a199c7c53c6b320aa86.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/97c01fdc7bc471af0b264a04aef0823e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4dac452fbb5ef6dd653e7fbbef639484.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cd3b9e816b14051f785aa5aae72b8eed.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a18722354086c42e62334983fc50eb6a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/97c01fdc7bc471af0b264a04aef0823e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a6c89fcb4633216e492ed5a04cf45eed.png)
(1)求操作1次后桶
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/97c01fdc7bc471af0b264a04aef0823e.png)
(2)求操作
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/63f5c583c98a1fd516c6ceaa60b55dec.png)
(3)至少操作多少次,桶
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/20f32bf06b351a6f046912c035898364.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/38067f3a30fe018bccae0508a38d4e57.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/18ebb51ff653764e0f964cf4ecda0199.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2bfb5a9ba77ae3ff13997225d5ba02f8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/70345587c2d90c50abb161cd7e158a67.png)
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解题方法
10 . 已知A,B分别是椭圆
的左、右顶点,R为椭圆C上异于A,B的一点,且满足
.
(1)求椭圆C的标准方程;
(2)已知点
,过点
的直线交椭圆C于D,E两点,直线
,
分别交直线
于
两点,探究
是否为定值,若为定值,求出该定值;若不为定值,请说明理由.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/19d5b538dbc0d6d7aca647794be954b0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/27117e63de9412bd33ac53f6a1971d3d.png)
(1)求椭圆C的标准方程;
(2)已知点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0bea681006f614f8a070e9c6a942c04.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7c699a757a7ee06c063150def9c5dffa.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2ef9d0784478511736255074b9395fae.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/80f30cb81835298cb612a828f355060a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f23d29646155e27b172ecdf263e2d702.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/748e0ea9f0f974ea69fef4d04a0933d4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1e74181b941bc0f0cae3cc6a91309ce3.png)
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