名校
1 . 银行按规定每经过一定的时间结算存(贷)款的利息一次,结算后将利息并入本金,这种计算利息的方法叫做复利.现在某企业进行技术改造,有两种方案:
甲方案:一次性向银行贷款10万元,技术改造后第一年可获得利润1万元,以后每年比上年增加30%的利润;
乙方案:每年向银行贷款1万元,技术改造后第一年可获得利润1万元,以后每年比前一年多获利5000元.
(1)设技术改造后,甲方案第n年的利润 为
(万元),乙方案第n年的利润 为
(万元),请写出
、
的表达式;
(2)假设两种方案的贷款期限都是10年,到期一次性归还本息.若银行贷款利息均以年息10%的复利计算,试问该企业采用哪种方案获得的扣除本息后的净获利更多?(精确到0.1)(净获利=总利润-本息和)(参考数据
,
甲方案:一次性向银行贷款10万元,技术改造后第一年可获得利润1万元,以后每年比上年增加30%的利润;
乙方案:每年向银行贷款1万元,技术改造后第一年可获得利润1万元,以后每年比前一年多获利5000元.
(1)设技术改造后,甲方案第n年的
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/96abfe2da27a63e6affb19a0c80236d9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/686ece75006ad358f23314dc8a246e11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/96abfe2da27a63e6affb19a0c80236d9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/686ece75006ad358f23314dc8a246e11.png)
(2)假设两种方案的贷款期限都是10年,到期一次性归还本息.若银行贷款利息均以年息10%的复利计算,试问该企业采用哪种方案获得的扣除本息后的净获利更多?(精确到0.1)(净获利=总利润-本息和)(参考数据
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8a178984a657a8fc6b53511319f19fdf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2289b28fcda0a30feec45cc9092a3b66.png)
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2 . 已知幂函数
的图像经过点
.
(1)求此幂函数的表达式和定义域;
(2)若
,求实数
的取值范围.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/942c2141d01bde6b48210c56a17fc75e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6688db018aad27e14e7bba19f324dca0.png)
(1)求此幂函数的表达式和定义域;
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/420467940c723dbc72ea19b39d6e8a8b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
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3 . 在2023年杭州亚运会最后两个竞技项目男女马拉松比赛中,中国选手何杰以2小时13分02秒夺得男子组冠军,这是中国队亚运史上首枚男子马拉松金牌.人类长跑运动一般分为两个阶段,第一阶段为前1小时的稳定阶段,第二阶段为疲劳阶段.小明想通过数学建模的方式研究运动员的运动时长与其剩余体力的关系.通过查找资料,小明得知:一位60kg的复健马拉松运动员进行4小时长跑训练,稳定阶段平均速度为30km/h,该阶段每千克体重消耗体力
(
表示该阶段所用时间),疲劳阶段由于体力消耗过大,在原有基础上随时间变大,速度降低,比例系数为
.同时,疲劳阶段速度降低,体力得到一定恢复,该阶段每千克体重消耗体力
,(
表示该阶段所用时间).同时,根据比赛现场的环境,其他运动员的平均配速,以及比赛策略等各方面因素,产生上下5%~10%的速度浮动,其对于运动员的体力影响也更为复杂.已知该运动员初始体力为
,请帮助小明补充完善数学建模的过程:
(1)对于数学建模,我们需要给出合理假设.
假设一:假设该运动员稳定阶段作速度为
的匀速运动;疲劳阶段做
的减速运动
假设二:_________________
(2)提出问题一:该运动员剩余体力Q关于时间t有何关系?请写出函数
;
提出问题二:该运动员在4小时内何时体力达到最低值,最低值为多少?
(3)总结运用:请根据以上计算结论,给出一定的实际建议.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/85f0ede408390464cffb0308ce938f09.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/87c7eb49a823f757461cd5260757b088.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/eff35e4e3cdc188643c46265591575c6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/166636b1f3567f864d7321534afed858.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5cd84a8f95166367063218ee03ffd5a7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/71dba305f8b63d590f0233275eaa3f10.png)
(1)对于数学建模,我们需要给出合理假设.
假设一:假设该运动员稳定阶段作速度为
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f5ee1c3f44dc7187d75effa7133fa678.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b0f7cb632d784a4c00b291cadab83f8d.png)
假设二:_________________
(2)提出问题一:该运动员剩余体力Q关于时间t有何关系?请写出函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2feeb7462a45a01b9b9530248604063e.png)
提出问题二:该运动员在4小时内何时体力达到最低值,最低值为多少?
(3)总结运用:请根据以上计算结论,给出一定的实际建议.
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解题方法
4 . 已知
是定义在
上的函数,如果存在常数
,使得对区间
的任意划分:
,都有
成立,则称
是
上的“绝对差有界函数”.
(1)分别判断
,
是否是
上的“绝对差有界函数”,若是“绝对差有界函数”,直接写出
的最小值(不需证明);若不是“绝对差有界函数”,直接写出函数的值域(不需证明);
(2)对定义在
上的
,若存在常数
,使得对任意的
,都有
,求证:
是
上的“绝对差有界函数”;
(3)设
是
上的“绝对差有界函数”,满足
,
,且对任意的
,都有
,求实数
的取值范围.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f030c36bb8786df88d401792062a4100.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2480f87a11c4cd450bc9454ea7276722.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f030c36bb8786df88d401792062a4100.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/804319e6cb58f07ee82ee364e334f36b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9cddd157e5a81d11a17daeae7882b85f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f030c36bb8786df88d401792062a4100.png)
(1)分别判断
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ee02fa2349fe9b9dd17c11665352c06e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3a552e0f8ccb78f2eec126ba95d8c399.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/eb87c830a03204a5b783ad4c2ba49c4e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
(2)对定义在
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f030c36bb8786df88d401792062a4100.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2f0d68648b10fce54dfc19c5ee60086d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6f20b947d584a1dc48676c2ae6e2af52.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/16ecdccf4a334ea959a456533c40d53b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f030c36bb8786df88d401792062a4100.png)
(3)设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9e11f4ca0e7ace69f92130d0525bcdb3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b8ac1c23f2a39df0652588ce63221df8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/01bea8bf593f594c51fc7cc547482bee.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f7dbb416ec1ff1984a724a4f48bf692.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4fd80e859f2a7935d7d621e202422621.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f0a532e15e232cb4b99a8d4d07c89575.png)
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解题方法
5 . 若
是定义在
上的增函数,其中
,存在函数
,
,且函数
图像上存在两点
,
图像上存在两点
,其中
两点横坐标相等,
两点横坐标相等,且
,则称
在
上可以对
进行“
型平行追逐”,即
是
在
上的“
型平行追逐函数”. 已知
是定义在
上的奇函数,
是定义在
上的偶函数.
(1)求满足
的
的值;
(2)设函数
,若不等式
对任意的
恒成立,求实数
的取值范围;
(3)若函数
是
在
上的“
型平行追逐函数”,求正数
的取值范围.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f8fd1e808e015f4cb43d2e3a0529ac6a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f030c36bb8786df88d401792062a4100.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b3df718662452f53034b6f702e46dcdb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/958a7330503a10fecf9b6b6f4a30ea8f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cc5ffb9d0b5dc940f53d7370419a08db.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3859cd3de4d6d485df050b0f5321d6c6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/01c74a907dda6bb7d9d56d009d9df253.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3c33c52de7a2c9d148e44283eec3dbb2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/39acab3cfb59bfc9591371721ab01d93.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/098a3e7d1f1890863b7483a98b618119.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fbdb21011ea821b91d539cb763aac649.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cdeb8cfadb41d94aaa1ba534aa040dcd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f030c36bb8786df88d401792062a4100.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4669810732b633b60dbeaf0bf57204f6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/294f5ba74cdf695fc9a8a8e52f421328.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4669810732b633b60dbeaf0bf57204f6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f030c36bb8786df88d401792062a4100.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/294f5ba74cdf695fc9a8a8e52f421328.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9e257d683ceaca6c6c3ee5ada0e447b9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a43b2faa4f81f32d94612dce724e772b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d908675a3ce0661cf6b3d7823143d4b0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a43b2faa4f81f32d94612dce724e772b.png)
(1)求满足
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cfb874a24bc2fa3d232070f16712aa05.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
(2)设函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/86df4f1b31c51da3551a76606d553f69.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e0bb6dbc79a74521af6338b0140b713b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5b8c164755dc2d7cff80fb4c9cffc9be.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
(3)若函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4669810732b633b60dbeaf0bf57204f6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a86644d5b1157b35cf7b825f108d4c39.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/294f5ba74cdf695fc9a8a8e52f421328.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/294f5ba74cdf695fc9a8a8e52f421328.png)
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6 . 如果对于函数
的定义域内任意的
,都有
成立,那么就称函数
是定义域上的“平缓函数”.
(1)判断函数
是否是“平缓函数”;
(2)若函数
是闭区间
上的“平缓函数”,且
,证明:对于任意的
,都有
成立.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8ce7ae90d808f05e86ea063238e4b2f9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9c2f1ca03ade14de6711c85de8fc5df0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
(1)判断函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f1ea19565e4feac073e898ab188fc3f5.png)
(2)若函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9e11f4ca0e7ace69f92130d0525bcdb3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2aeb3ca8cbc4facb2467b1a618f33794.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83a6c0fddb9074dfc96be03b4aa24d2b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/14e9387190a323961884c302798c9e4e.png)
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解题方法
7 . 已知函数
是定义在
上的奇函数,且
.
(1)求函数
的解析式;
(2)判断并用定义法证明
在
上的单调性;
(3)解关于x的不等式
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fbb474dac35d7d9b9b823f5fdb8db266.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4ead3fdcb8fe8f5eb3dbe7d96cabc28b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/34f2ef95d5254995f52a67c732b51243.png)
(1)求函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
(2)判断并用定义法证明
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4ead3fdcb8fe8f5eb3dbe7d96cabc28b.png)
(3)解关于x的不等式
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6e2a0f02510cbf59115751ba5a6e60d7.png)
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名校
解题方法
8 . 已知函数
,
满足以下条件:
①
,
;
②
,
,
,
.
(1)求
,
的值.
(2)判断函数
,
的奇偶性,并说明理由.
(3)若
,
,试判断函数
的周期性,并说明理由.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d275fbb3ee5cd1177ca5a2ceecbbef0f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/028517e8bebe634441e0a5c79828e88a.png)
①
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/306984f0895ba32a7b3bb607065b1eaa.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e83e9f562c10762097469dea27c1e109.png)
②
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/df6593a700bf3e89107556454666b787.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a50ac027c6ebce491ae836524d89901c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9ebeb1c1f2826da8a2e0761f2d2ba87d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/98eb23b8c96a34dd720e00669aa8ed2b.png)
(1)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a078165d75cfb890141845324a6173b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c60cf12a81b11e33356fe7e1c9e3d0b9.png)
(2)判断函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d275fbb3ee5cd1177ca5a2ceecbbef0f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/028517e8bebe634441e0a5c79828e88a.png)
(3)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dfaff6cfe1d15bd64c1fa76af5e52831.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/158fbbd2cbedeb9a6fa1a900630369f4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/028517e8bebe634441e0a5c79828e88a.png)
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9 . 已知函数
.
(1)若
,求
在区间
上的最大值和最小值;
(2)若
在
上恒成立,求
的取值范围.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1e569275b72bb0ccd25959dea844035f.png)
(1)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8e258ab9e600435b37465092243d99f6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d275fbb3ee5cd1177ca5a2ceecbbef0f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9e354f44c5841806fdc363073abdd052.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/58196b9e63ec00aa1119052b6de6ae12.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/825094a0d5c41dcb90880b3a5f27e435.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
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名校
10 . 记集合
.对任意
,
,记
,对于非空集合
,定义集合
.
(1)当
时,写出集合
;对于
,写出
;
(2)当
时,如果
,求
的最小值;
(3)求证:
.
(注:本题中,
表示有限集合A中的元素的个数.)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f0a89f895a14b4f202dfe6b19224857c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/74c86c0b2a71ee538df6ab1eab3c8b2e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bd58ba2338450bd94bc2a1ec0a0a51ed.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/120df58b92e747fc3091f1a3aeff228d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/13342dd73eb34ca37aaca5b521706442.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/19ca324ae5ead82dd03b6cb5afac67a5.png)
(1)当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cc2d3df37e73a8abea815f37dbb3fff5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0cb9ad1e34877b0db02d0219332b0f7b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4469a0542c773e329e8cc42e14a84169.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d0ca0b9e99203ec575c46cdbf2d4ef0d.png)
(2)当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/be604061cf1591f7069472269d4c9719.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8839c83b988c42da1fce4a96787583eb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c045c7a097a2908732932f4c0c170693.png)
(3)求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/25376e139f40d0df5ada2c9ebb1da2e4.png)
(注:本题中,
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c045c7a097a2908732932f4c0c170693.png)
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