1 . 已知
,求
(1)
及
的值;
(2)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9626e5b9db3f22a442f10fef71b76f83.png)
(1)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5a6aad40c02d5217af4535853938065b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6cc9750c313ee972124cb62c4a6fb7ea.png)
(2)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/626cca880ae8dfcda582fa50f9838a70.png)
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2 . 已知
,函数
.
(1)函数
的图象经过点
,且关于
的不等式
的解集为
,求
的解析式;
(2)若
有两个零点
,
,且
的最小值为
,当
时,判断函数
在
上的单调性,并说明理由;
(3)设
,记
为集合
中元素的最大者与最小者之差,若对
,
恒成立,求实数
的取值范围.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/94440d3e4c073f94f2b266ff99d50e74.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3ad9a11b81b7a8643005608eacc7f9d9.png)
(1)函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1803dc3c76fd2b51696647aa18602412.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6acb0f1ac694dd177e99fc385f23318.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/aa66623cf54b42d6d12be4c8edaa7071.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/e170f206fdbbd834aad7580c727e2cc6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dd5452c267983506a7cb3373e19fd5f7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0268e85df43d66b031e0eccb11284452.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/19f88a76f947e7022ef0c5efd6db060c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ee3c6cdb19ac03dc3c28cd63b09dc907.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4802dfb4352b1162b6cda12fa469f91e.png)
(3)设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ebc20d351d51723c9b0a07a20ac14114.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cf0225bca34eaf19544939b29153aac1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5a0e3589ab6dda85eb6dc9cab30878f5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/136da7c9e087330c312e31d1c2083d06.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c8119e4e1c474bc2adbe014628043609.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
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名校
3 . 已知函数
.
(1)求
的最小正周期及单调递增区间;
(2)若
在
上恒成立,求实数
的取值范围.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bcc84fc7a501d15dc107950120af91b7.png)
(1)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e3adbb701b7084868e1eabc8780c939b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/44d51992c05a557cf6058664f1f8961e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/294f5ba74cdf695fc9a8a8e52f421328.png)
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4 . 已知函数
.
(1)当
时,求
,并判断函数
零点的个数;
(2)当
时,
有三个零点
,记
,
,2,3.证明:①
;②
.
参考公式:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/98dc59a1edc821f4cbd77ef908dff3c7.png)
(1)当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3c6fb3c74056a4b4ebbda9d2be78de34.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/80e6f74fdd5c57d4454f8e840563771a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
(2)当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/722f025aeed67dbc2c16f60c3b06061d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7b41b7154250380217e8dc70ff2a7591.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4142405dce1c99e4c58474c00b2bd6e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7c45176df950dfe48b8ca7eac08ee349.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a65021f540ac58a37746758611f2fd76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1c69673c72476d11a283eb5e4d1f7a72.png)
参考公式:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/26aa2c8a73ca9a054ed46e16eb5811e5.png)
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5 . 设
.
(1)当
时,用函数单调性的定义证明:函数
在区间
上是严格增函数.
(2)①根据a的不同取值,讨论函数
在区间
上零点的个数;
②若函数
在区间
(k为正整数)上恰有7个零点,求k的最小值及此时a的取值范围.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/14696ba10834f2d6b8891bf80abd0c79.png)
(1)当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/301605e86e5a5e61a65c91cd3dd8b77e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/51c530f4b7491b95acb8ce3eef9aa09d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d01dc2d99655cf7598837cb0886166ed.png)
(2)①根据a的不同取值,讨论函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/51c530f4b7491b95acb8ce3eef9aa09d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/137d6a66a015ddd2a8076f35ed191927.png)
②若函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/51c530f4b7491b95acb8ce3eef9aa09d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6c7f465e11dcb6a1cf9b4cf111f7b249.png)
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6 . 已知函数
的部分图象如图所示.
的解析式与单调增区间;
(2)若将
的图象向右平移
个单位,再向上平移1个单位得到
的图象,写出
图象的对称中心的坐标,并求当
时,
的最值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6309e18f51f43800c0a16205c6a491b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d275fbb3ee5cd1177ca5a2ceecbbef0f.png)
(2)若将
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d275fbb3ee5cd1177ca5a2ceecbbef0f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac1a63ab608517bb10aa036783dfb51f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/028517e8bebe634441e0a5c79828e88a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/028517e8bebe634441e0a5c79828e88a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c31a8b7f6d3b664b2c9bae09c3a6f884.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/028517e8bebe634441e0a5c79828e88a.png)
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7 . 已知函数
在区间
上的最小值为3.
(1)求常数
的值;
(2)当
时,将函数
的图象上所有点的横坐标缩短到原来的
(纵坐标不变)得到函数
,求函数
的单调递减区间、对称中心.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/265384d62ae17af01c1562581a893ab6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/44d51992c05a557cf6058664f1f8961e.png)
(1)求常数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/294f5ba74cdf695fc9a8a8e52f421328.png)
(2)当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/eb63478132d4c1fef3c17e591919da83.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f89eef3148f2d4d09379767b4af69132.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4669810732b633b60dbeaf0bf57204f6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4669810732b633b60dbeaf0bf57204f6.png)
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8 . 某同学用“五点法”画函数
在某一个周期内的图象时,列表并填入了部分数据,如下表:
(1)请将上表数据补充完整,并求函数
的解析式;
(2)将函数
的图象向左平移
个单位后,得到函数
的图象.若方程
在区间
上有解,求
的取值范围.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6157ef443108dda491868c1a96e3c8e9.png)
![]() | 0 | ![]() | ![]() | ![]() | ![]() |
![]() | ![]() | ![]() | |||
![]() | 0 | 3 | ![]() | 0 |
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
(2)将函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/942c2141d01bde6b48210c56a17fc75e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac1a63ab608517bb10aa036783dfb51f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1938c093dd2fbcb752d0eb7a18d143b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4d873ccbd26127bb543951eff8af9337.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1a908cff3963ed5281930402c853b0b0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/294f5ba74cdf695fc9a8a8e52f421328.png)
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名校
解题方法
9 . 某工厂有甲、乙两个生产车间,其污水瞬时排放量
(单位:
)关于时间
(单位:
)的关系均近似地满足函数
,其图象如图所示.
![](https://img.xkw.com/dksih/QBM/editorImg/2024/6/23/26d08519-def2-4de0-a24a-c37a31e041f7.png?resizew=162)
(1)根据图象求函数解析式;
(2)若甲车间先投产,1小时后乙车间再投产,求该厂两个车间都投产
时刻的污水瞬时排放量;
(3)由于受工厂污水处理能力的影响,环保部门要求该厂的两个车间任意时刻的污水排放量之和不超过
,若甲车间先投产,为满足环保要求,乙车间比甲车间至少需推迟多少小时投产?
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d053b14c8588eee2acbbe44fc37a6886.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c1ff4266515b801477248b32e10f3ac4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/36a1b09c653185842513e24ebba60bb3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f42a97e3ae09c48e1d587f59af3621bf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/795ea8d61f4e4e138231819356f4073b.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2024/6/23/26d08519-def2-4de0-a24a-c37a31e041f7.png?resizew=162)
(1)根据图象求函数解析式;
(2)若甲车间先投产,1小时后乙车间再投产,求该厂两个车间都投产
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/16dca1f69db5d45d7e0195b60f546a40.png)
(3)由于受工厂污水处理能力的影响,环保部门要求该厂的两个车间任意时刻的污水排放量之和不超过
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d18b1b3fa4bc1f8e48a68b9f77e37e3b.png)
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10 . 已知函数
的最小正周期为
.
(1)求
;
(2)求
图象的对称轴方程;
(3)若
在
上有2个零点,求
的取值范围.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/aa55b232cfcb973c206ba521ad33c103.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/70f5389990c3a0c5373f3bd9fb2454c9.png)
(1)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/074c228ffc7b1e306f8410afe7bc4b5c.png)
(2)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
(3)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3ddbc468411a7a08141f24bed62c024a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
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