1 . 1712年英国数学家布鲁克·泰勒提出了著名的泰勒公式,该公式利用了多项式函数曲线来逼近任意一个原函数曲线,该公式在近似计算,函数拟合,计算机科学上有着举足轻重的作用.如下列常见函数的
阶泰勒展开式为:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ccf4a87ad1e9742f47b0c5b44b8dfab0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1b113d94079c4b2138c2325e1141c5bc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b1557366a2ea0c602935e5da8fb495d6.png)
其中
,读作
的阶乘.
1748年瑞士数学家莱昂哈德·欧拉在泰勒公式的灵感下创造了人类数学最美妙的公式,即欧拉公式
,特别的欧拉恒等式
被后世称为“上帝公式”.欧拉公式建立了复数域中指数函数与圆函数(正余弦函数)的关系,利用欧拉公式还可以完成圆的
等分,即棣莫弗定理
的应用.
(1)请写出复数
的三角形式,并利用泰勒展开式估算出
的3阶近似值(精确到0.001);
(2)请根据上述材料证明欧拉公式,并计算
与
;
(3)记
,由棣莫弗定理得
,从而得
,复数
,我们称其为1在复数域内的三次方根. 若
为64在复数域内的6次方根.求
取值构成的集合,其中
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ccf4a87ad1e9742f47b0c5b44b8dfab0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1b113d94079c4b2138c2325e1141c5bc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b1557366a2ea0c602935e5da8fb495d6.png)
其中
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/815fbba8af7b1ecfb112be6b04284191.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
1748年瑞士数学家莱昂哈德·欧拉在泰勒公式的灵感下创造了人类数学最美妙的公式,即欧拉公式
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/26032c72018539ca7aa3ca66ac845260.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8998724d22d1f99493dd285a9e5bfe63.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/419e0831142916b945a1c1004c7cd6c5.png)
(1)请写出复数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bd7a56b5b169d5ecff40690f5def68e0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/041a7c8fc017f596542c5e6ec7d1c40b.png)
(2)请根据上述材料证明欧拉公式,并计算
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4e5bebae7756550f899bbc18ea8bc923.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8dfbd1655b2e4b2c629b2e77fc3e7f06.png)
(3)记
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bd0c30155ec5bc576f72e97afc42abaf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a606f335bfbfabc3362b1faf49add59.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fb0555a4bd63bc674ceca48ba08c4023.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6c88c2ca3f32231770665622da3ba4be.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7bb85abfc312eb4ac4cd1321b033f328.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/78488089f169e8222beb6cdb772af3d7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/01c83f84dad2257eeb8fd3c6c38c671b.png)
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2 . 英国数学家泰勒发现了如下公式:
,
,其中
.这些公式被编入计算工具,计算工具计算足够多的项就可以确保显示值的精确性.
(1)用前三项计算
;
(2)已知
,
,
,试比较
,
,
的大小.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e001efee18e05afab241c12334d98cd5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/63ba15a427babacf319deb9c4dd8d58b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6f7ce15cf27e4aec8abddd8b495a30f2.png)
(1)用前三项计算
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ccb0821260d845231f860baeabd68eeb.png)
(2)已知
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5aa20a3f02e6430d94120e9df5eaaa70.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/695cf64b357d24bc88ea669ad8f1de57.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/aabd3c4a5fd9278fea6546a52a7a6653.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/071a7e733d466949ac935b4b8ee8d183.png)
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3 . 设
是正实数数列.
(1)若
收敛,求证:存在严格递增的无界正实数数列
满足
收敛.
(2)若
收敛,是否一定存在严格递增的正整数数列
,满足
收敛,且
?
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bc7376941fa463c63b1d4d4ea866b78c.png)
(1)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ccecde965d7557d5ee35dea8ae7164a3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/60c988a3683540149b687486af0ed3a7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/120dcd9c3adc5b08ab9d84f228cc4b90.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/97ad99ac2f9cbe69281dcdc7d4195d8a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fba58d775c69de6d132c58581d614792.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/246e5563a2f86de45879b21393d814f6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9c69eef9b8e90f6a153b87738f759bcf.png)
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4 . 得分去掉一个最高分30,一个最低分0,剩下个得分平均数为10,方差为1,则求不去掉时的平均分和方差分别为多少?
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5 . 在平面直角坐标系中,定义
为两点
,
的“切比雪夫距离”.又设点P及l上任意一点Q,称d(P,Q)的最小值为点P到直线l的“切比雪夫距离”,记作d(P,l).给出下列四个命题:①对任意三点A,B,C,都有
;②已知点P(3,1)和直线
,则
;③到原点的“切比雪夫距离”等于1的点的轨迹是正方形.其中正确的序号为______ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5579881e0d1530cc0a17cc5460acecb9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e6ff82ebdfad5e7de1c7487b0b817a7f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a53e311ee0b5085e7e5a45c606daa5d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6efc72de5cceb54c6959af52491ca762.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/94922febcf02e84401ab8631890532df.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3e0104da8625fcd6af63b19a37274a40.png)
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6 . 已知
,
,
则( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1f99a21e96e65120664d35b976f92943.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/93004624161474daab375d4e31e2344e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/54454cb02cb7e5b7fee900090657482e.png)
A.![]() | B.![]() | C.![]() | D.![]() |
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名校
7 . 泰勒公式通俗的讲就是用一个多项式函数去逼近一个给定的函数,也叫泰勒展开式,下面给出两个泰勒展开式
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/479a14f6e8881b7afd15ec7071ca950e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7129e4fdcbdf718eebedb4a69b8dbaa1.png)
由此可以判断下列各式正确的是( ).
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/479a14f6e8881b7afd15ec7071ca950e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7129e4fdcbdf718eebedb4a69b8dbaa1.png)
由此可以判断下列各式正确的是( ).
A.![]() | B.![]() |
C.![]() | D.![]() |
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2023-04-24更新
|
1248次组卷
|
4卷引用:辽宁省部分学校2022-2023学年高三下学期第二次模拟考试数学试题
辽宁省部分学校2022-2023学年高三下学期第二次模拟考试数学试题(已下线)模块六 专题4 易错题目重组卷(辽宁卷)(已下线)黄金卷03广东省梅州市兴宁市第一中学2023-2024学年高二下学期月考一(3月)数学试题
名校
解题方法
8 . 已知数列
满足
,
,设数列
的前
项和为
,则( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0ea8d0e50065114b05ef2dc1ea1129cf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f2297c0f452647db5bda11bbde1562c0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e2d51f9147b8265c0276c1f2c2659197.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3b9a0d7150fb24be3e28ef7f0e18be93.png)
A.![]() | B.![]() |
C.![]() | D.![]() |
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名校
9 . 设A是由
个实数组成的2行n列的矩阵,满足:每个数的绝对值不大于1,且所有数的和为零.记
为所有这样的矩阵构成的集合.记
为A的第一行各数之和,
为A的第二行各数之和,
为A的第i列各数之和
.记
为
、
、
、
、…、
中的最小值.
(1)若矩阵
,求
;
(2)对所有的矩阵
,求
的最大值;
(3)给定
,对所有的矩阵
,求
的最大值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bd9dbdea32a8f7b9fd4c8982eef6dea6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15fc1924d5c54d4f2824f6accc1238b7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/59b68fd1ac04715b65105c0cf40aa84f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/61a2629e9e3b3fcf0c0bdd49c76b95cc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a954ad5b391cfc9440f0444cbbfa889d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1f128d1af43d66e8048295604ef89046.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/12ba5c6f604da3afa5c18d368fb12060.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/30773f6541752c8d133db5662ccee553.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0d137142642163af066957fe19218ed0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/260bcd4709ef67852ef6e2de9841e75d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5af2bb6f225862039961601a07e7d7fd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b9624751c77e7b93a0166bbdc302cdc6.png)
(1)若矩阵
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/63da318b4a47902b2a7979230e997e28.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/12ba5c6f604da3afa5c18d368fb12060.png)
(2)对所有的矩阵
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f0b432f6219d00bd0b2bc483401b9dff.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/12ba5c6f604da3afa5c18d368fb12060.png)
(3)给定
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b18969d9db906a0f002b762113ecf077.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/01aef0b7f72cd41492cade2785ccc6cc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/12ba5c6f604da3afa5c18d368fb12060.png)
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2022-05-28更新
|
452次组卷
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3卷引用:上海市2022届高三高考冲刺卷六数学试题
10 . 记
行
列的数阵中第
行
列的数为
,数阵内所有数之和为
.若
,
,
,
,
,则定义
.根据以上定义,计算
( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2c05b9832b09731a574d4a4adf7448de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7600d2cfbdc6146db96cc545706004f6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b4ba1bbe411bc71bca016d3fd82352f9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/318c9cdd41802fa89a96c1c37d52689c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f05595aa7994885ed05f1de0033da31a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b4c9705e4d8649224c47228f0d1d0f2e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d2aba5fe219fbc5425be7c26f2d33368.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c49cc81bb63745e8267366ef9476086b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3644ba1d482cf220ae8e89c766e19a72.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b56201f5592683db0fa134075eb9f8ec.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/df287293f4a8ccbf0aebebade10f572c.png)
A.![]() | B.![]() | C.![]() | D.![]() |
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