解题方法
1 . 费马点是指位于三角形内且到三角形三个顶点距离之和最小的点.当三角形三个内角都小于
时,费马点与三角形三个顶点的连线构成的三个角都为
.已知点
为
的费马点,角
所对的边分别为
,若
,
,
边上的中线长为
,则
的值为_________ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c0211da37e92f915e781691296578ba0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c0211da37e92f915e781691296578ba0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4ec3f79448524d6848be51fdd5c7a150.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bb1b5305cb9b5a90c4f13bceaaee4fc0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4734e4bdd3d2a5d489dd863b17687e61.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fcdb7a488910743dc5c63afb394b87e2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0dc5c9827dfd0be5a9c85962d6ccbfb1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ec85f29c0860b57a8f0cf8098c13a97e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f352a59a635e3f6570e350ca08de6af5.png)
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2 . 利用基本不等式求最值
已知
,
,则:
(1)如果和
等于定值s,那么当
时,积xy有最大值______ ;
(2)如果积xy等于定值p,那么当
时,和
有最小值______ .
已知
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/08115d6d9f876dea921a4d32260ff1fb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/061813f1ec633c5c4c393c4de7938322.png)
(1)如果和
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b88584cf1df43e28d03592c7998b1653.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/07b9d5aaaceaa3ac514d17fcfefbf9b4.png)
(2)如果积xy等于定值p,那么当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/07b9d5aaaceaa3ac514d17fcfefbf9b4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b88584cf1df43e28d03592c7998b1653.png)
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3 . 几个重要不等式
(1)
(a,
)(当且仅当
时取等号).
变形式:______ (a,
)(当且仅当
时取等号).
(2)基本不等式:______ (
,
)(当且仅当
时取等号).
变形式:
(
,
),
(a,
)(当且仅当
时等号成立).
(3)
(a,b,
)(当且仅当
时取等号).
(4)若
,则
,
(当且仅当
时取等号).
(1)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/48fe5d9cbe4f83926f5c21912df67a2e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8a3849dacb9cd91d905551d858ffdd5c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1f22fec5a381ae8aca93d876e54c79de.png)
变形式:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8a3849dacb9cd91d905551d858ffdd5c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1f22fec5a381ae8aca93d876e54c79de.png)
(2)基本不等式:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/94440d3e4c073f94f2b266ff99d50e74.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/67ca5fd57c2c2fcc3c7a574fdd1467d9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1f22fec5a381ae8aca93d876e54c79de.png)
变形式:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/759c09917e5728d75bf5cfdb5b4a807f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/94440d3e4c073f94f2b266ff99d50e74.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/67ca5fd57c2c2fcc3c7a574fdd1467d9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/79b5c6b3c5470bfc6af6eb17e8ee2c07.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8a3849dacb9cd91d905551d858ffdd5c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1f22fec5a381ae8aca93d876e54c79de.png)
(3)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a59101326c029393a18f8285893fcbb4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6aa21fa613a161efcd163fc2cb55fbcf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/44acc0ee22dc4b7750e8be825e7c1355.png)
(4)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/39d5f0d374837655cc286d326305da36.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3aa23408bcc9f6200f22a310e5f2569a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/95dd00785c80a941afa84bf2437182b5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1f22fec5a381ae8aca93d876e54c79de.png)
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4 . 在
中,
,
,
,则![](https://staticzujuan.xkw.com/quesimg/Upload/formula/28e3fc39d2b508cc1cae769a5e7266c0.png)
______ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/60b97bb18e5ca34d22b5e827316a122a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8e258ab9e600435b37465092243d99f6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/226741a02640e1710d93b2e39ec61440.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/28e3fc39d2b508cc1cae769a5e7266c0.png)
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23-24高二下·全国·课前预习
5 . 等差数列的前
项和公式
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
已知量 | 首项、末项与项数 | 首项、公差与项数 |
求和公式 | ![]() | ![]() |
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6 . 某学校数学实践小组为该校一块长方形空地设计种树方案,在坐标纸上设计如下:第
棵树种在点
处,其中
,当
时,
,[
]表示不大于x的最大整数,按此设计方案,第3株树种植点的坐标为___________ ;第2025棵树种植点的坐标为____________ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f0a532e15e232cb4b99a8d4d07c89575.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e6a9ea0c17c1c1576541f981a202701b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/08e93eb05e988d2fd48fac631e479b3e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7c972cbd63decec197aec1bdc306de67.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d7437284d09b06a4e911be5feaf83dd5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
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2024-05-11更新
|
216次组卷
|
2卷引用:山东省潍坊市2023-2024学年高二下学期期中质量监测数学试题
23-24高一下·全国·课前预习
7 . 正弦定理
条件 | 在△ABC中,角A,B,C所对的边分别为a,b,c |
结论 | ![]() ![]() |
文字描述 | 在一个三角形中,各边和它所对角的 |
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23-24高二下·全国·课前预习
8 . 知识点03等比数列的单调性
等比数列
的首项为
,公比为![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9aa8a716a31b0f51b70fdf9bdb257909.png)
(1)当___ 时,数列为递增数列;
(2)当___ 时,数列为递减数列;
(3)当_____ 时,数列为常数列:
(4)当_______ 时,数列为摆动数列.
等比数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/29518f13a1ebc3fff8181c2d7cfba22f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1b42791b77924729f7e31712177b26af.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9aa8a716a31b0f51b70fdf9bdb257909.png)
(1)当
(2)当
(3)当
(4)当
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23-24高二下·全国·课前预习
9 . 知识点01等比数列的概念
1、等比数列的定义
如果一个数列从第2项起,_______ 等于同一常数,那么这个数列叫做等比数列,这个常数叫做等比数列的_______ ,通常用字母_______ 表示
.
2、对等比数列概念的理解
(1)“从第2项起”,是因为首项没有“前一项”,同时注意公比是每一项与前一项的比,前后次序不能颠倒,另外等比数列中至少含有三项;
(2)定义中的“同一常数”是定义的核心之一,一定不能把“同”字省略,这是因为如果一个数列从第2项起,每一项与它的前一项的比都是一个与
无关的常数,但是如果这些常数不相同,那么此数列也不是等比数列,当且仅当这些常数相同时,数列才是等比数列;
(3)若一个数列不是从第2项起,而是从第3项起或第
项起,每一项与它的前一项的比等于同一常数,则此数列不是等比数列;
(4)由定义可知,等比数列的任一项都不为0,且公比
;
(5)不为0的常数列是特殊的等比数列,其公比为1.
1、等比数列的定义
如果一个数列从第2项起,
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/455931a43e525371730e61177c48aec6.png)
2、对等比数列概念的理解
(1)“从第2项起”,是因为首项没有“前一项”,同时注意公比是每一项与前一项的比,前后次序不能颠倒,另外等比数列中至少含有三项;
(2)定义中的“同一常数”是定义的核心之一,一定不能把“同”字省略,这是因为如果一个数列从第2项起,每一项与它的前一项的比都是一个与
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
(3)若一个数列不是从第2项起,而是从第3项起或第
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bccd5c35461ea19c93e24f80e8538f2d.png)
(4)由定义可知,等比数列的任一项都不为0,且公比
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a3ac83c571110d41a396d12d8eea1c6.png)
(5)不为0的常数列是特殊的等比数列,其公比为1.
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23-24高二下·全国·课前预习
10 . 知识点04等比中项
1、等比中项定义:如果在
与
中间插入一个数
,使
成等比数列,那么
叫做
与
的_______ ,即
是
与
的等比中项
成等比数列![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f39c662a3927de39135c3eee4b9cb68f.png)
_______
2、对等比中项概念的理解
(1)
是
与
的等比中项,则
与
的符号相同,符号相反的两个实数不存在等比中项.此时,
,即等比中项有两个,且互为相反数.
(2)
时,![](https://staticzujuan.xkw.com/quesimg/Upload/formula/895dc3dc3a6606ff487a4c4863e18509.png)
_______ 是
与
的等比中项.例如
,但
不是等比数列;
(3)在等比数列
中,从第2项起,每一项是它相邻两项的等比中项;
(4)与等比数列中的任一项“等距离”的两项之积等于该项的平方,即在等比数列
中,![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6a6169cb2b9f2a373abc0cadecadd4f2.png)
3、等差中项与等比中项区别
(1)任意两数都存在等差中项,但并不是任意两数都存在等比中项,当且仅当两数同号且均不为0时才存在等比中项;
(2)任意两数的等差中项是______ 的,而若两数有等比中项,则等比中项______ .
1、等比中项定义:如果在
![](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/895dc3dc3a6606ff487a4c4863e18509.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e6703df340de9d28c32832badbd30f22.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/895dc3dc3a6606ff487a4c4863e18509.png)
![](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/895dc3dc3a6606ff487a4c4863e18509.png)
![](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/83ab5539817e40ffaf20a517e0978b80.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f39c662a3927de39135c3eee4b9cb68f.png)
2、对等比中项概念的理解
(1)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/895dc3dc3a6606ff487a4c4863e18509.png)
![](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/0a6936d370d6a238a608ca56f87198de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2c94bb12cee76221e13f9ef955b0aab1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ede7e2c31ca68ce700cffa87764dc484.png)
(2)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a1c8379fe535e68721fd84be969d257f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/895dc3dc3a6606ff487a4c4863e18509.png)
![](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/9ce438ef49c36ad7b8a27e918137e9ab.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/419abd403ca442c5aadd04165fc9a528.png)
(3)在等比数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
(4)与等比数列中的任一项“等距离”的两项之积等于该项的平方,即在等比数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6a6169cb2b9f2a373abc0cadecadd4f2.png)
3、等差中项与等比中项区别
(1)任意两数都存在等差中项,但并不是任意两数都存在等比中项,当且仅当两数同号且均不为0时才存在等比中项;
(2)任意两数的等差中项是
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