解题方法
1 . 已知函数
.
(1)当
时,求函数
在区间
上的最大值;
(2)函数
在区间
上存在最小值,求
的取值范围.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8aff9b5571a0c86c788bbf159b461643.png)
(1)当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3b4d795709b0abcf47bceec2250f2f9b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4fe7d5809da02c15a43a0e9a898b9086.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/03db4ea1dcb63b22cf4e917df5db581e.png)
(2)函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4fe7d5809da02c15a43a0e9a898b9086.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/02e1c9c97de9198d47306216e9961b80.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
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2 . 已知
的二项式系数之和是64.
(1)求展开式中含
的项的系数;
(2)写出展开式二项式系数最大的项.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0e97a55fecff2e9affe29c5e30ec86c1.png)
(1)求展开式中含
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9b0a89e3c30f6e4d4c5db4378b05d987.png)
(2)写出展开式二项式系数最大的项.
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3 . 已知函数
在
时取得极值.
(1)求函数
的单调区间;
(2)求函数
在区间
上的最小值;
(3)若
有两个零点,求
的值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/23e896d105cce0a6ea3003a78dae6346.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9b384412acba251d87902ab928902f16.png)
(1)求函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4fe7d5809da02c15a43a0e9a898b9086.png)
(2)求函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4fe7d5809da02c15a43a0e9a898b9086.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30a5498bb0236a2bb04ae38329b408.png)
(3)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ed118174009b84dad78216de99db52e5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/294f5ba74cdf695fc9a8a8e52f421328.png)
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4 . 在校运动会上,只有甲、乙、丙三名同学参加铅球比赛,比赛成绩达到
以上(含
)的同学将获得优秀奖.为预测获得优秀奖的人数及冠军得主,收集了甲、乙、丙以往的比赛成绩,并整理得到如下数据(单位:
):
甲:
;
乙:
;
丙:
.
假设用频率估计概率,且甲、乙、丙的比赛成绩相互独立.
(1)估计甲在校运动会铅球比赛中获得优秀奖的概率;
(2)设
是甲、乙、丙在校运动会铅球比赛中获得优秀奖的总人数,求
的分布列和数学期望
;
(3)在校运动会铅球比赛中,甲、乙、丙谁获得冠军的概率估计值最大?(结论不要求证明)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/90bbeb88466f25da82ee060774501475.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/90bbeb88466f25da82ee060774501475.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e15e00f40396e914d1d9955bd7785f1f.png)
甲:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1b5c2ebff997804ce8da464e9003f904.png)
乙:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3bfe3af7c01ccb01341a765db6aa6631.png)
丙:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9e72a33281f69adb17001e92ed19f1f5.png)
假设用频率估计概率,且甲、乙、丙的比赛成绩相互独立.
(1)估计甲在校运动会铅球比赛中获得优秀奖的概率;
(2)设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f022950e0faa45b617d497b01b5292b9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f022950e0faa45b617d497b01b5292b9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5bf3baba074e8aeb6f3ea117865bbd1b.png)
(3)在校运动会铅球比赛中,甲、乙、丙谁获得冠军的概率估计值最大?(结论不要求证明)
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解题方法
5 . 在数列
中,已知
,且![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8c0eddd71d4fa6c78b1a410784bafb16.png)
(1)求
,
的值;
(2)是否存在实数
,使得数列
为等差数列?若存在,求出
的值;若不存在,请说明理由.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f6065aaa8f3f103d1bc960da8318ce35.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8c0eddd71d4fa6c78b1a410784bafb16.png)
(1)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3e88093a749c0d46e0ee931ecfaff925.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6c1ccc6c74b8754e9bcbb3f39a11b6f1.png)
(2)是否存在实数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/df64046e91b047037f19e4032e3b6de3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9e6224c0adf74d78a27ba6d4222840b9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/df64046e91b047037f19e4032e3b6de3.png)
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6 . 已知等差数列
中,
,
.
(1)求这个数列的第
项;
(2)
和
是不是这个数列中的项?如果是,求出是第几项;如果不是,说明理由.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6df1038200f2d97a52c716aab6c3bcb6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/919f8ca81f48d968e0b4d88ec7a61eb3.png)
(1)求这个数列的第
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d07ae0b4264da6a8812454ffd2f20d94.png)
(2)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/54e60696d1c681628c57bb2c7e9d1814.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2ab66291ef17f0d1664cefd6f6bffd31.png)
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解题方法
7 . 已知某种药物对某种疾病地治愈率为
,现有甲、乙、丙、丁4个患有该病的患者服用了这种药物,观察其中有多少患者会被这种药物治愈.
(1)求出甲、乙、丙都被治愈而丁没被治愈的概率;
(2)设有
人被治愈,求
的分布列和数学期望.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8b2a698891d42c70b597f0da4f215f09.png)
(1)求出甲、乙、丙都被治愈而丁没被治愈的概率;
(2)设有
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f022950e0faa45b617d497b01b5292b9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f022950e0faa45b617d497b01b5292b9.png)
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名校
解题方法
8 . 对给定的实数a,b,q,其中
,
.如果函数
,
:满足(1)对任意的
,
且
;(2)对任意的
,
.则称
为在区间
上的一个“q-压缩函数”.区间
上所有“q-压缩函数”构成的集合记作
.
(1)判断下列函数,是否属于集合
?(直接写出结论)
①
②
③![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f80e8d2d66b432fd2b8a2830ba21e5dd.png)
(2)设
,若
求实数a的取值范围.
(3)设
.若对任意的
,
,均有
,求M的最小值,并说明理由.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c6a46e678bf9d2df5ad4c782b3dc22f5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1ae99e050d0f1cfc0447304f06424d17.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/942c2141d01bde6b48210c56a17fc75e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e02cab1add26335b3cb43d5b54c7c853.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e207cf62e3a7e282eac4c4a3455bbf9a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e02cab1add26335b3cb43d5b54c7c853.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/48318c1b3f9dffc48e3ce0626b277e4d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/664c92988d61280ed12488f270151940.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e1a8b4f05d2ba1a5514ebf390dbbd325.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/942c2141d01bde6b48210c56a17fc75e.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/a293e0b4f751f4102c795b567a633194.png)
(1)判断下列函数,是否属于集合
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f3c564b7a711c6ac5945a0d41b2564c8.png)
①
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a740cdaca0afce73f68c890f86acc6f1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cef72cf8297d91045117d50b4e9d3476.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f80e8d2d66b432fd2b8a2830ba21e5dd.png)
(2)设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/db0e64d809acabedccf8a8516ff8b6a7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5c109bacd687055d3a6a79978dfa7c0e.png)
(3)设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2e39b23fb1290ed449f1700c42c3efc6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cfcd04b625189228b6d697edf095f7c3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3c97169936ca9785dfca39ca368c65c8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2b0aa28b363cb79e6f8988b9b2bad7dc.png)
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名校
9 . 已知函数
.
(1)求函数
的图象在点
处的切线方程;
(2)求函数
在区间
上的最大值与最小值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/eedaa6aad781cb80241b78ae9d4c97cc.png)
(1)求函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d275fbb3ee5cd1177ca5a2ceecbbef0f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9770b18d9984172e6c08bf0921ec4200.png)
(2)求函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d275fbb3ee5cd1177ca5a2ceecbbef0f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/17b63fb475fbdacbbb8eb2bf6a6716d2.png)
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名校
10 . 在等差数列
中,
,
.
(1)求数列
的首项
和公差
;
(2)设数列
的前n项和为
,求
的最小值及取最小值时n的值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ff52aa657973485243c460aad1502267.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/00d6bd814515a297f39e59b3ffa2cffa.png)
(1)求数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e72adb45c60c2f63b46e65ff787302bf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5c02bc0c74292b1e8f395f90935d3174.png)
(2)设数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/08eb71ecf8d733b6932f4680874dbbf3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/08eb71ecf8d733b6932f4680874dbbf3.png)
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