第九节
- 格式:doc
- 大小:90.00 KB
- 文档页数:10
第九节随机变量的数字特征、正态分布知识点预习1.离散型随机变量的数学期望与方差(1)数学期望(2)方差2.二点分布与二项分布、超几何分布的期望、方差3.正态曲线4.正态曲线的性质5.正态变量在三个特定区间内取值的概率值预习练习题1、判断下面结论是否正确(请在括号中打“√”或“×”)(1)随机变量的均值是常数,样本的平均值是随机变量,它不确定.( )(2)随机变量的方差和标准差都反映了随机变量取值偏离均值的平均程度,方差或标准差越小,则偏离变量的平均程度越小.( )(3)正态分布中的参数μ和σ完全确定了正态分布,参数μ是正态分布的均值,σ是正态分布的标准差.( ) (4)一个随机变量如果是众多的、互不相干的、不分主次的偶然因素作用结果之和,它就服从或近似服从正态分布.( )(5)均值是算术平均数概念的推广,与概率无关.( ) 2、 (教材改编)某射手射击所得环数ξ的分布列如下: 已知ξ的均值E (ξ)=8.9,则y 的值为( ) A .0.4 B .0.6 C .0.7 D .0.93、设样本数据x 1,x 2,…,x 10的均值和方差分别为1和4,若y i =x i +a (a 为非零常数,i =1,2,…,10),则y 1,y 2,…,y 10的均值和方差分别为( ) A .1+a,4 B .1+a,4+a C .1,4 D .1,4+a4、设随机变量X 的分布列为P (X =k )=15(k =2,4,6,8,10),则D (X )等于( )A .5B .8C .10D .16 5、设随机变量X ~B (8,p ),且D (X )=1.28,则概率p 的值是( ) A .0.2 B .0.8 C .0.2或0.8 D .0.166、已知X 的分布列为设Y =2X +3,则E (Y )的值为( ) A .73B .4C .-1D .1 7、若X ~B (n ,p ),且E (X )=6,D (X )=3,则P (X =1)的值为( ) A .3×2-2B .2-4C .3×2-10D .2-88、有一批产品,其中有12件正品和4件次品,从中有放回地任取3件,若X 表示取到次品的次数,则D (X )=________.9、某糖厂用自动打包机打包,每包重量X (kg)服从正态分布N (100,1.22),一公司从该糖厂进货1 500包,则重量在(98.8,101.2)的糖包数量为________包.11、抛掷两枚骰子,当至少一枚5点或一枚6点出现时,就说这次试验成功,则在10次试验中成功次数的均值为________.例题选讲例1、某银行规定,一张银行卡若在一天内出现3次密码尝试错误,该银行卡将被锁定.小王到该银行取钱时,发现自己忘记了银行卡的密码,但可以确认该银行卡的正确密码是他常用的6个密码之一,小王决定从中不重复地随机选择1个进行尝试.若密码正确,则结束尝试;否则继续尝试,直至该银行卡被锁定. (1)求当天小王的该银行卡被锁定的概率;(2)设当天小王用该银行卡尝试密码的次数为X ,求X 的分布列和均值.例2、设袋子中装有a 个红球,b 个黄球,c 个蓝球,且规定:取出一个红球得1分,取出一个黄球得2分,取出一个蓝球得3分.(1)当a =3,b =2,c =1时,从该袋子中任取(有放回,且每球取到的机会均等)2个球,记随机变量ξ为取出此2球所得分数之和,求ξ的分布列;(2)从该袋子中任取(每球取到的机会均等)1个球,记随机变量η为取出此球所得分数.若E (η)=53,D (η)=59,求a ∶b ∶c .例3、某居民小区有两个相互独立的安全防范系统(简称系统)A 和B ,系统A 和系统B 在任意时刻发生故障的概率分别为110和p .(1)若在任意时刻至少有一个系统不发生故障的概率为4950,求p 的值;(2)设系统A 在3次相互独立的检测中不发生故障的次数为随机变量ξ,求ξ的分布列及均值E (ξ).例4、计划在某水库建一座至多安装3台发电机的水电站.过去50年的水文资料显示,水库年入流量X (年入流量:一年内上游来水与库区降水之和.单位:亿立方米)都在40以上.其中,不足80的年份有10年,不低于80且不超过120的年份有35年,超过120的年份有5年,将年入流量在以上三段的频率作为相应段的概率,并假设各年的入流量相互独立.(1)求未来4年中,至多有1年的年入流量超过120的概率;(2)水电站希望安装的发电机尽可能运行,但每年发电机最多可运行台数受年入流量X 限制,并有如下关系:800万元.欲使水电站年总利润的均值达到最大,应安装发电机多少台?例5、已知某批零件的长度误差(单位:毫米)服从正态分布N (0,32),从中随机取一件,其长度误差落在区间(3,6)内的概率为(附:若随机变量ξ服从正态分布N (μ,σ2),则P (μ-σ<ξ<μ+σ)=68.26%,P (μ-2σ<ξ<μ+2σ)=95.44%.)( )A .4.56%B .13.59%C .27.18%D .31.74%例6、(12分)甲袋和乙袋中都装有大小相同的红球和白球,已知甲袋中共有m 个球,乙袋中共有2m 个球,从甲袋中摸出1个球为红球的概率为25,从乙袋中摸出1个球为红球的概率为P 2.(1)若m =10,求甲袋中红球的个数;(2)若将甲、乙两袋中的球装在一起后,从中摸出1个红球的概率是13,求P 2的值;(3)设P 2=15,若从甲、乙两袋中各自有放回地摸球,每次摸出1个球,并且从甲袋中摸1次,从乙袋中摸2次.设ξ表示摸出红球的总次数,求ξ的分布列和均值.第九节课堂练习1、若离散型随机变量X 的分布列为 则X 的数学期望E (X )=( )A .2B .2或12C .12D .12、设X ~N (μ1,σ21),Y ~N (μ2,σ22),这两个正态分布密度曲线如图所示,下列结论中正确的是()A .P (Y ≥μ2)≥P (Y ≥μ1)B .P (X ≤σ2)≤P (X ≤σ1)C .对任意正数t ,P (X ≥t )≥P (Y ≥t )D .对任意正数t ,P (X ≤t )≥P (Y ≤t )3、已知某批零件的长度误差(单位:毫米)服从正态分布N (0,32),从中随机取一件,其长度误差落在区间(3,6)内的概率为( )(附:若随机变量ξ服从正态分布N (μ,σ2),则P (μ-σ<ξ<μ+σ)=68.26%,P (μ-2σ<ξ<μ+2σ)=95.44%.) A .4.56% B .13.59%C .27.18% D .31.74%4、某校在一次月考中约有600人参加考试,数学考试的成绩ξ~N (90,a 2)(a >0,试卷满分150分),统计结果显示数学考试成绩在70分到110分之间的人数约为总人数的35,则此次月考中数学考试成绩不低于110分的学生约有________人.5、为了整顿道路交通秩序,某地考虑将对行人闯红灯进行处罚,为了更好地了解市民的态度,在普通行人中随机选取了200人进行调查,得到如下数据:(1)若用表中数据所得频率代替概率,则处罚10元时与处罚20元时,行人会闯红灯的概率的差是多少? (2)若从这5种处罚金额中随机抽取2种不同的金额进行处罚,在两个路口进行试验. ①求这两种金额之和不低于20元的概率;②若用X 表示这两种金额之和,求X 的分布列和数学期望.6、为回馈顾客,某商场拟通过摸球兑奖的方式对1 000位顾客进行奖励,规定:每位顾客从一个装有4个标有面值的球的袋中一次性随机摸出2个球,球上所标的面值之和为该顾客所获的奖励额.(1)若袋中所装的4个球中有1个所标的面值为50元,其余3个均为10元,求:①顾客所获的奖励额为60元的概率;②顾客所获的奖励额的分布列及数学期望;(2)商场对奖励总额的预算是60 000元,并规定袋中的4个球只能由标有面值10元和50元的两种球组成,或标有面值20元和40元的两种球组成.为了使顾客得到的奖励总额尽可能符合商场的预算且每位顾客所获的奖励额相对均衡,请对袋中的4个球的面值给出一个合适的设计,并说明理由.7、有甲、乙两种棉花,从中各抽取等量的样品进行质量检验,结果如下:其中X表示纤维长度(单位:mm),根据纤维长度的均值和方差比较两种棉花的质量.8、乒乓球台面被球网分隔成甲、乙两部分,如图,甲上有两个不相交的区域A ,B ,乙被划分为两个不相交的区域C ,D .某次测试要求队员接到落点在甲上的来球后向乙回球.规定:回球一次,落点在C 上记3分,在D 上记1分,其他情况记0分.对落点在A 上的来球,队员小明回球的落点在C 上的概率为12,在D 上的概率为13;对落点在B 上的来球,小明回球的落点在C 上的概率为15,在D上的概率为35.假设共有两次来球且落在A ,B 上各一次,小明的两次回球互不影响.求:(1)小明两次回球的落点中恰有一次的落点在乙上的概率; (2)两次回球结束后,小明得分之和ξ的分布列与均值.9、某投资公司在2015年年初准备将1 000万元投资到“低碳”项目上,现有两个项目供选择: 项目一:新能源汽车.据市场调研,投资到该项目上,到年底可能获利30%,也可能亏损15%,且这两种情况发生的概率分别为79和29;项目二:通信设备.据市场调研,投资到该项目上,到年底可能获利50%,可能损失30%,也可能不赔不赚,且这三种情况发生的概率分别为35,13和115.针对以上两个投资项目,请你为投资公司选择一个合理的项目,并说明理由.10、在某次大型考试中,某班同学的成绩服从正态分布N (80,52),现已知该班同学中成绩在80~85分的有17人.试计算该班成绩在90分以上的同学有多少人.第九节课后作业1.若X ~B (n ,p ),且E (X )=6,D (X )=3,则P (X =1)的值为( ) A .3×2-2B .2-4C .3×2-10D .2-82.随机变量ξ的分布列如下,其中a 、b 、c 为等差数列,若E (ξ)=13,则D (ξ)的值为( )A.49B.59C.13D.233.设随机变量X ~N (μ,σ2),且X 落在区间(-3,-1)内的概率和落在区间(1,3)内的概率相等,若P (X >2)=p ,则P (0<X <2)等于( ) A.12+p B .1-p C .1-2pD.12-p 4.一个袋子中装有6个红球和4个白球,假设每一个球被摸到的可能性是相等的.从袋子中摸出2个球,其中白球的个数为X ,则X 的均值是________.5.若随机变量X 的概率分布密度函数是f (x )=122π·e -(x +2)28(x ∈R ),则E (2X -1)=________.6.已知随机变量ξ的分布列为P (ξ=k )=12k -1,k =1,2,3,…,n ,则P (2<ξ≤5)=________.7.某超市为了响应环保要求,鼓励顾客自带购物袋到超市购物,采取了如下措施:对不使用超市塑料购物袋的顾客,超市给予9.6折优惠;对需要超市塑料购物袋的顾客,既要付购买费,也不享受折扣优惠.假设该超市在某个时段内购物的人数为36人,其中有12位顾客自己带了购物袋,现从这36人中随机抽取两人.(1)求这两人都享受折扣优惠或都不享受折扣优惠的概率; (2)设这两人中享受折扣优惠的人数为ξ,求ξ的分布列和均值.8.在某次数学考试中,考生的成绩ξ服从正态分布,即ξ~N(100,100),已知满分为150分.(1)试求考试成绩ξ位于区间(80,120]内的概率;(2)若这次考试共有2 000名考生参加,试估计这次考试及格(不小于90分)的人数.9.现有一游戏装置如图,小球从最上方入口处投入,每次遇到黑色障碍物等可能地向左、右两边落下.游戏规则为:若小球最终落入A槽,得10张奖票;若落入B槽,得5张奖票;若落入C槽,得重投一次的机会,但投球的总次数不超过3次.(1)求投球一次,小球落入B槽的概率;(2)设玩一次游戏能获得的奖票数为随机变量X,求X的分布列及均值.10.设随机变量X 服从正态分布N (12,σ2),集合A ={x |x >X },集合B ={x |x >12},则A ⊆B 的概率为( )A.14 B.13 C.12D.2311.袋中装有大小完全相同,标号分别为1,2,3,…,9的九个球.现从袋中随机取出3个球.设ξ为这3个球的标号相邻的组数(例如:若取出球的标号为3,4,5,则有两组相邻的标号3,4和4,5,此时ξ的值是2),则随机变量ξ的均值E (ξ)为( ) A.16 B.13 C.12D.2312.马老师从课本上抄录一个随机变量ξ的分布列如下表:请小牛同学计算ξ的均值.尽管“!”处完全无法看清,且两个“?”处字迹模糊,但能断定这两个“?”处的数值相同.据此,小牛给出了正确答案E (ξ)=________.13.PM2.5是指大气中直径小于或等于2.5微米的颗粒物,也称为可入肺颗粒物,对人体健康和大气环境质量的影响很大.我国PM2.5标准采用世卫组织设定的最宽限值,即PM2.5日均值在35微克/立方米以下空气质量为一级;在35微克/立方米~75微克/立方米之间空气质量为二级;在75微克/立方米以上空气质量为超标.某市环保局从360天的市区PM2.5监测数据中,随机抽取15天的数据作为样本,监测值如茎叶图所示(十位为茎,个位为叶).(1)在这15天的数据中任取3天的数据,记ξ表示空气质量达到一级的天数,求ξ的分布列.(2)以这15天的PM2.5日均值来估计这360天的空气质量情况,则其中大约有多少天的空气质量达到一级.14.气象部门提供了某地区今年六月份(30天)的日最高气温的统计表如下:由于工作疏忽,统计表被墨水污染,Y和Z数据不清楚,但气象部门提供的资料显示,六月份的日最高气温不高于32℃的频率为0.9.某水果商根据多年的销售经验,六月份的日最高气温t(单位:℃)对西瓜的销售影响如下表:(1)求Y,Z的值;(2)若视频率为概率,求六月份西瓜日销售额的均值和方差;(3)在日最高气温不高于32℃时,求日销售额不低于5千元的概率.。
第九节二叠系二叠系主要分布于桂中、桂西、桂北,桂东北和桂东南亦有零星出露。
地层发育完整,为海相、海陆交互相碎屑岩、泥质岩、碳酸盐岩、硅质岩及海底基性喷出岩。
总厚385—7164米。
与下覆石炭系一般为整合关系,局部为平行不整合接触。
上下统之间大都为平行不整合接触,少数为整合关系。
桂南钦州一带,上二叠统以角度不整合覆于下二叠统乃至志留系不同层位之上。
化石丰富,主要有蜒类、菊石、腕足类和植物等。
赋存矿产有煤矿、铁矿、锰矿、铝土矿、黄铁矿等,为广西重要含矿地层之一。
栖霞阶沿用江苏省南京市东郊栖霞山的命名,李希霍芬1882年建立,乐森璕1928年引用于广西,1959年全国地层会议称现名。
其岩性一般为深灰、灰黑色的灰岩,富含泥质、燧石和有机质,击之具硫化氢臭味为其特征。
桂中、桂西为灰白色灰岩夹白云质灰岩、白云岩;百色、田林、巴马、田东、武呜等地为硅质岩、硅质灰岩,整合于马平群之上;南丹—河池—带为浅色石英砂岩、黑色炭质泥岩夹泥灰岩、灰岩,南丹县朝阳、拉弓一带夹薄煤层;荔浦、平乐等县底部有黑色炭质页岩、铝土质泥岩、煤层、铝土矿等。
厚15-600米。
平行不整合于马平群之上。
确定时代的化石有米氏蜒、喀劳德米氏蜒、拟纺锤蜒、希瓦格蜒及腕足类和山西带羊齿(植物)等,属早二叠世早期。
茅口阶沿用贵州省朗岱县茅口灰岩的命名,乐森璕1929年建立,张文佑、李捷1935年引用于广西,1959年全国地层会议称现名。
岩性为浅色灰岩,有时夹白云质灰岩、白云岩或燧石,局部夹玄武玢岩,中性熔岩及火山角砾岩。
厚0-932米。
桂北、桂中的灰岩颜色较深,含大量燧石或硅质岩,厚300—600米;桂西、桂西南含燧石或硅质岩相对减少,白云质灰岩和白云岩增多,角砾状灰岩及火山岩较发育。
宁明县亭亮等地有海绵、藻类生物礁。
厚515-643米。
整合于栖霞阶之上。
确定时代的化石有费伯克蜒、新希瓦格蜒、新米氏蜒、广西杨铨蜒、广西矢部蜒、文采尔珊瑚、卫根珊瑚、新石燕等,属早二叠世晚期。
第九节命题自然推理一、什么是命题自然推理所谓自然推理,就是从给定的前提出发,运用推理的有效式即根据推理规则进行的推理。
自然推理和公理化推理不同,它不预设公理,只是根据规则,从给定的前提出发得出结论。
这似乎更符合人们日常思维的习惯,因此,称之为自然推理。
自然推理是判定推理形式是否有效的又一种方法,也是指导有效推理的一种方法。
自然推理的基本思想是确定一些推理规则,这些规则具有保真性,也就是说,依据这些规则,从真前提只会推出真结论。
因此,当我们要判定一个推理是否有效时,就要看,从所要判定的推理的前提出发,依据推理规则,能否形式地推出预期的结论。
如果能推出,就说明该推理如果前提真,结论就一定真,因而是有效的;反之,如果推不出,则说明该推理即使前提真,结论也不一定真,因而是无效的。
同时,这些具有保真性的推理规则,也可以指导我们从给定的前提进行有效的推导。
自然推理并不仅用于判定和指导命题推理。
当我们运用它来判定和指导复合命题推理时,就称之为命题自然推理。
二、命题自推理的基本规则命题自然推理有三条基本规则:1、前提引入规则。
在推理的任何一步都可以引入一个给定的前提。
我们称这条规则为p规则。
2、重言蕴涵规则。
如果在推理中有一些在先的命题形式,它们的合取重言地蕴涵A,那么,我们在推理中就可以引入命题形式A。
我们称这条规则为T规则。
3、条件证明规则。
如果能从一组前提和A推出B,那么,我们就可以从这组前提推出A→B。
我们称这条规则为C·P规则。
实际上,这三条基本规则涵盖了我们前面学过的所有有效的推理形式。
在下面的实例中我们将进一步说明。
三、命题自然推理的应用[例1] 在一起凶杀案中,侦查人员了解到以下一些情况:(1)凶手是甲或乙或丙;(2)只有是盗窃杀人案,甲才是凶手;(3)如果是盗窃杀人案,那么被害人的财物会丢失;(4)如果乙是凶手,那么案件发生在中午12点以后;(5)案件发生在中午12点以前,并且被害人的财物没有丢失。
第九节 常系数非齐次线性微分方程㈠本课的基本要求会求自由项为多项式,指数函数,正弦函数,余弦函数以及它们的和与积的二阶常系数齐次线性微分方程的特解和通解㈡本课的重点、难点由自由项的特点去找特解的思想是本节的重点,难点是自由项为混合型时特解的求法 ㈢教学内容一.二阶常系数线性非齐次方程的解法)(x f qy y p y =+'+'' ⑶由定理3知线性非齐次方程的通解是对应的线性齐次方程的通解与其自身的一个特解之和。
而求二阶常系数线性齐次方程的通解前已解决,现在的关键在于求线性非齐次方程的一个特解,而特解显然与自由项)(x f 有关。
以下介绍当自由项)(x f 属于某些特殊类型函数时的情况。
1.自由项)(x f 为多项式)(x P e m x λ,其中λ是常数,)(x P m 是x 的一个m 次多项式: m m m m m a x a x a x a x P ++++=--1110)(可令⑶的特解x e x Q y λ)(*=,为此将*y 及x e x Q x Q y λλ)]()([*+'=',x e x Q x Q x Q y λλλ)]()(2)([*2+'+''=''代入方程⑶并消去x e λ,得)()()()()2()(2x P x Q q p x Q p x Q m =+++'++''λλλ ⑻由分析得x m k e x Q x y λ)(*=,其中)(x Q m 是与)(x P m 同次(m 次)的多项式,而k 按λ不是特征方程的根、是特征方程的单根或是特征方程的重根依次取0,1或2。
上述结论可推广到n 阶常系数非齐次线性微分方程,但要注意特解中的k 是特征方程含根λ的重复次数(即若λ不是特征方程的根,k 取0;若λ是特征方程的s 重根,k 取s )。
例1.求方程13452+=+'+''x y y y 的一个特解 327181543*2+-=x x y 例2.求)474322221x x ec c x y y x +-++-='+''-的通解( 例3.求x xe y y y -=+'+''323的通解2.自由项]sin )(cos )([)(wx x P wx x P e x f n l x +λ为型利用欧拉公式,有 ]22[]sin )(cos )([)(i e e P e e P e wx x P wx x P e x f iwxiwx n iwx iwx l xn l x ---++=+=λλ x iw x iw x iw n l x iw n l e x P e x P e i P P e i P P )()()()()()(2222-+-++=⎪⎭⎫ ⎝⎛-+⎪⎭⎫ ⎝⎛+=λλλλ其中i P P i P P x P i P P i P P x P n l n l n l n l 2222)(,2222)(+=-=-=+= 是互成共轭的m 次多项式(即它们对应项的系数是共轭复数),而},max{n l m =。
第九节二阶常系数非齐次线性微分方程二阶常系数非齐次线性微分方程x xf(x) [P(x)cos x Q(x)sin x]ef(x) P(x)emmm教学目的:掌握自由项为和的二阶常系数非齐次线性微分方程特解的方法教学重点:二阶常系数非齐次线性微分方程求特解的待定系数法教学难点:二阶常系数非齐次线性微分方程求特解的待定系数法教学内容:二阶常系数非齐次线性微分方程的形式为:y py qy f(x)根据二阶线性微分方程解的结构,要求解二阶常系数非齐次线性微分方程,只需先求得对应齐次线性微分方程的通解和该非齐次线性微分方程的一个特解即可。
而齐次线性微分方程的通解已在上一目得到解决,因此本节将解决非齐次线性微分方程的特解问题。
为此,针对自由项的特点,采用如下待定系数法:根据二阶非齐次线性微分方程解的结构,要求二阶常系数非齐次线性微分方程的通解,__yy Y就是非齐次方程的通Y只需先求得非齐次方程的特解和对应齐次方程的通解,则解。
而用待定系数法求二阶常系数非齐次线性微分方程y py qy f(x)的特解分两种情形讨论:一、f(x) e xPm(x)型这里是常数,Pm(x)是m次多项式.由于指数函数与多项式之积的导数仍是同类型的函数,而现在微分方程右端正好是这种类型的函数.因此,不妨假设方程y py qy f(x)的特解为y* Q(x)e x 其中Q(x)是x的多项式,将y*代入方程并消去e x得Q (2 p)Q ( 2 p q)Q Pm(x)(1) 若不是y py qy 0的特征方程r2 pr q 0的根,那么2 p q 0这时Q(x)与Pm(x)应同次,于是可令Q(x) Qm(x) a0xm a1xm 1 代入Q (2 p)Q ( 2 p q)Q Pm(x), 比较等式两端x同次幂的系数,就得到含a0,a1,以定出这些系数,并求得特解y* Qm(x)e x(2) 若是特征方程r2 pr q 0的单根,那么2 p q 0,而2 p 0.,am的m+1个方程的联立方程组,从而可am 1x amgood此时,Q 应是m次多项式,再注意到此时,Ce x(C为常数)为y py qy 0 的解,故可令Q(x) xQm(x)(3) 若是特征方程r2 pr q 0的重根,那么2 p q 0且2 p 0这时Q (x)应是m次多项式,再注意到此时C1e x和C2xe x(C1,C2为常数)均为y py qy 0的解.故可设Q(x) x2Qm(x)综上所述,有如下结论:如果f(x) e xPm(x),则方程y py qy f(x)具有形如y* xkQm(x)e x的特解,其中Qm(x)是与Pm(x)同次的特定多项式,而k按不是特征方程的根,是特征方程的单根或者是特征方程的重根依次取0,1或2. 例1 求方程y 2y 3y 3x 1的一个特解解本题0,而特征方程为r2 2r 3 0,0不是特征方程的根,设所求特解为y* b0x b1,代入方程:3b0x 3b1 2b0 3x 13b0 31比较系数, 得所以b0 1,b13 2b0 3b1 1于是所求特解为y* x1. 32x例2 求方程y 5y 6y xe的通解解特征方程为r 5r 6 0,其根为r1 2, 2r2 32,对应齐次方程的通解为Y C1e2x C2e3x设非齐次方程特解为y* x(b0x b1)e2x 代入方程得2b0x b1 2b0 x2b0 11比较系数, 得解得b0 ,b1 12 2b0 b1 0因此特解为y* x( x 1)e2x.1good所求通解为y C1e2x C2e3x (x2 x)e2x. 二f(x) e x Pl(x)cos x Pn(x)sin x 型分析思路:( i )x( i )x 第一步将f (x) 转化为f(x) P Pm(x)em(x)e( i )x第二步求出如下两个方程的特解y py qy P m(x)e( i )x y py qy P (x)em 1第三步利用叠加原理求出原方程的特解第四步分析原方程特解的特点解法:第一步利用欧拉公式将f (x) 变形ei x e i xei x e i xf(x) e Pl(x) Pn(x) 22ixPl(x)Pn(x)22i ( i )x Pl(x)Pn(x)e22i ( i )xe( i )x( i )x令m max n,l ,则f(x) P (x)e P(x)emm( i )x第二步求如下两方程的特解y py qy P m(x)e( i )x y py qy Pm(x)e( i )x设i 是特征方程的k 重根( k = 0, 1), 则y py qy P m(x)e 特解: y1 xkQm(x)e( i )x( i )x故(y1 ) p(y1) qy1 Pm(x)e等式两边取共轭y1py qy1 Pm(x)e( i )x1y1 为方程y py qy Pm(x)e( i )x的特解x第三步求原方程的特解y py qy e Pl(x)cos x Pn(x)sin x利用第二步的结果, 根据叠加原理, 原方程有特解(1)x Rm y* y1 y1 xke x Rmcos(2)si nx第四步分析y的特点y y1 y1 xek x(1)(2) Rcos x Rsin x mmgoody y1 y1 y1 y1 y1 y1 y*(1)(2) 所以y 本质上为实函数,所以Rm均为m 次实多项式,Rm例3 求方程y y xcos2x的一个特解r 1 0 ,l(x) x,Pn(x) 0特征方程解0, 2P,i 2i不是特征方程的根,故设特解为2y* (ax b)cos2x (cx d)sin2x代入方程得( 3ax 3b 4c)cos2x (3cx 3d 4a)sin2x xcos2x4,b c 0 914于是求得一个特解y* xcos2x sin2x.39比较系数, 得a ,d例4 第七节例1中若设物体只受弹性恢复力f和铅直干扰力F Hsinpt 的作用求物体的运动规律解问题归结为求解无阻尼强迫振动方程13d2x2kx hsinp t2dt当p ≠ k 时, 齐次通解X C1sinkt C2coskt Asin(kt ) 非齐次特解形式x asinpt bcospt 代入可得ah,b 0k2 p2hsinpt 22k p因此原方程之解为x Asin(kt )自由振动强迫振动当干扰力的角频率p ≈固有频率k 时振幅hk2 p2将很大当p = k 时非齐次特解形式: x t(asinkt bcoskt) 代入可得:a 0,b h 2kgood方程的解为x Asin(kt )htcoskt 2k自由振动强迫振动随着t 的增大, 强迫振动的振幅ht可无限增大,这时产生共振现象 . 2k若要避免共振现象, 应使p 远离固有频率k ;若要利用共振现象, 应使p 与k 尽量靠近, 或使p = k .对机械来说, 共振可能引起破坏作用, 如桥梁被破坏,电机机座被破坏等,但对电磁振荡来说,共振可能起有利作用,如收音机的调频放大即是利用共振原理. 小结与思考:xf(x) P(x)ef(x)m①自由项为多项式与指数函数的乘积,即的情形,此时非齐次*k xQ(x)是与已知多项式Pm(x)同次的多项式(其系数可y xQ(x)em方程的特解,其中m将特解代入非齐次方程,比较方程两端同类项的系数,联立求解而得到),而k按不是特征方程的根、是特征方程的单根和是特征方程的重根分别取0、1和2;xf(x) [P(x)cos x Q(x)sin x]emm②自由项的情形,此时非齐次方程的特解R(x)和Ts(x)是s次的多项式(其系数可将y* xk[Rs(x)cos x Ts(x)sin x]e x,其中s特解代入非齐次方程,比较方程两端同类项的系数,联立求解而得到),s max m,n ,而k按i不是特征方程的根、是特征方程的单根和是特征方程的重根分别取0和1。
第九节古汉语特有的句式在古代汉语中,有一些现代汉语所没有的、特殊的句式结构。
在本节中,我们就来谈谈这个问题。
一、判断句的表示法判断句是表示某种事物是什么东西、属于某一类或不属于某一类的句子。
在现代汉语中,肯定性的判断句,必须在主语和宾语之间用判断动词“是”表示。
例如:“江苏省是一个经济发达的省份。
”“王老师是苏州人。
”否定性的判断句,则必须在“是”前加上否定副词“不”字,例如:“李主任不是我的邻居。
” 古汉语的判断句,多数不用判断词,少数用判断词“是”和“为”。
1 不用判断词的判断句典型的古汉语判断句,是不用判断词的。
其表现形式主要有六种:(1) “……者,……也”式。
这种格式中的“者”是辅助性代词,它的作用是复指前面的名词主语;“也”是语气词,其作用是帮助判断,“也”前是直接作谓语的名词、名词性词组或代词。
例如:《史记·陈涉世家》:“陈胜者,阳城人也,字涉。
吴广者,阳夏人也,字叔。
”《战国策·赵策》:“彼秦者,弃礼义而尚首功之国也。
”《庄子·逍遥游》:“南冥者,天池也。
”《史记·项羽本纪》:“楚左尹项伯者,项羽季父也。
”有时,判断句的主语是“者”字结构(如《孟子·离娄下》:“追我者,谁也?”),或带“者”的“所”字结构(如《韩非子·五蠹》:“吾所欲者,土地也。
”)。
其形式与上面所讲的“……者,……也”式似乎相同,但实际上“者”的语法作用不是复指名词主语(如“陈胜者,阳城人也。
”),而是分别充当“者”字结构和“所”字结构的组成成分(见本章第一节的有关内容)。
这是必须搞清楚的。
(2) “……,……也”式。
这可以看成“……者,……也”式的省略形式。
例如:《汉书·张骞传》:“张骞,汉中人也。
”《左传·僖公五年》:“虢,虞之表也。
”《左传·宣公二年》:“董狐,古之良史也。
”“赵盾,古之良大夫也。
”《战国策·赵策》:“而母,婢也。
代词练习1.--Which do you prefer, tea or coffee?-- __________is OK.I don't care.A. EitherB. NeitherC. Both2. -Which of the caps will you take?--I'll take _______,one for my father ,the other for my brother.A.neitherB.bothC.all3. -Who helped you clean the classroom yesterday?--_______.I cleaned it all by myself.A.SomebodyB.NobodyC.Everybody4. -Can I have some more ice cream,Mum?--Sorry. There is _______left in the fridge.A.noneB.nothingC.no oneD.anything5-Excuse me .I want to buy some stamps.Where can I find a post office?--I know _______not far from here.You can easily find ________.A.that;itB.it;oneC.one;it6. _________of the hats suited him,so Jack had to try on the third one.A.NeitherB.EitherC.Both7-Is the woman a teacher?-Yes. She teaches ________ English.A.you B.us C.our D.your8-Have you got ________ready for the sports meeting?--Not yet.We still have________to do.A.anything;nothingB.something;everythingC.everything;something9. Don't wear jeans,or______won't be Aiiowed to go to the concert.A. IB.youC.they10. -How many students are there in the classroom?--__________. They are all in the lab.A. SomeB. NoneC. AllD. Neither11.To help Tommy learn better, his parents have done _________they could:cards,tapes,special learning centers,in short, everything they can think of.A bothB allC noneD neither12.-How many students like this song?--___________of us likes it.It sounds terrible.A.NoneB.NobodyC.Every oneD.All13.-I don't know _______to do next?--It's easy .Press the red button in 2 minutes.A.what B.how C.where D.when14. Emma, can you introduce to Alice? I want to meet her.A. himB. hisC. meD. my15.She is new here, so we know about her.A. nothingB. somethingC. anythingD. everything16.-How heavily it rained this early morning.?-Yes. But of the students in our class was late for school.A.someB.noneC. all?17. -Lucy's skirt is black. What about _______?-Mine is white.?A.youB.yourC. yours ?18..-How was your visit to the World Park in Beijing?--Wonderful!Weenjoyed______very much.A.itselfB.myselfC.yourselvesD.ourselves19.-Oh,you're reading a newspaper .Is there _______in it?--I don't know. I read it just now.A.new somethingB.anything newC.something new20.-Would you like to go to the concert with me tonight?--I'd love to ,but ________of us couple has tickets.Do you have some?A.bothB.neitherC.none22.I'm expecting a pet dog for long,but mom has no time to buy_______for me.A.itB.oneC.thisD.that23.-Would you like tea or coffee? --_________is OK.I really don't mind.A.NoneB.EitherC.NeitherD.BothA.myselfB.yourselfC.himselfD.herself25.His name is Thomas ,but usually calls ______ Tom.A. himB. himselfC. herD. Herself26.Maybe bamboo has more uses than ______in the world.A.any plantB.all the plantsC.other plantD.any other plant时态:1. This term over. The summer vacation is coming in two weeks.A. isB. wasC. has beenD. will be2. I a mistake. Please don’t be angry with me.A. makeB. madeC. will makeD. had made3. My brother left school in 2005, and since then he in Beijing.A. livesB. livedC. will liveD. has lived4. I my clothes, and the phone rang.A. washB. washedC. am washingD. was washing5.—Where is Peter?—He volleyball with his friends in the school gym.A. playsB.playedC. is playing6. He_____his umbrella to me yesterday.so I didn’t get wet.A.borrowedB.keptC.lentD.bought7. I’m sorry I left the book at home.I______it here tomorrow,I promise.A.bringB.will bringC.broughtD.have brought8—What are you doing ,Simon?--I have finished my homework ,and now I______the computer games.A.playedB.was playingC.am playingD.play9.My brother came back home while I ______homework.A.am doingB.were doingC.was doing10—Did you watch the football match yesterday?--Yes ,I did.You know ,my brother ________in the match.A.is playingB.was playingC.will play11.—Where are the Greens ,may I ask?--Well,they ______England.They have been there for nearly a week now.A.have been toB.are going toC.have gone to12. A moment ,please.I’m checking if Mr.Smith______free tomorrow.A.isB.beingC.to beD.will be13.— How was your day off ? Pretty good ! I ______ the science museum with my classmates.A. visitB. visitedC. am visitingD. will visit14.There _______ a big cake and many candies at the party yesterday.A. wasB. wereC. isD. are15. I _______ the charity show on TV when the telephone rang.A. watchB. watchedC. am watchingD. was watching16. Tom _______ the USA. He _______ back in two months.A. has gone to; comesB. has gone to; will beC. has been to; comesD. has been to; will be17.Today is Father’s Day .My mother ______a special dinner for my grandpa now.A.prepareB.preparedC.is preparingD.will prepare18.--Hello! May I speak to Kate? --Sorry, she isn’t here now. She __________to Shanghai.A. wentB. has beenC. has gone19.—Where’s her second son? I haven’t seen him for a long time.—He _______ Maine looking for a place to work.A.will go to B.has gone to C.had gone to D.goes to 20.—Must I finish my homework at school? --_____.You can do it at home and hand it in next week.A.Yes,you mustB.No,you needn’tC.Yes,I can21.—Where are the Greens ,may I ask?--Well,they______to England.They have been there for nearly a week now.A.have beenB.are going toC.have goneD.will go22.—Where is Zhang Ming?--Oh,don’t you know he____to Beijing to see his parents and he’ll be back tomorrow.A.has goneB.has beenC.had gone23.—You have a nice watch.--Thank you.I______it since I got married.A.hadB.boughtC.have hadD.have bought24.Al though this village isn’t big,all the other villages I________so far are smaller.A.visitedB.have visitedC.would visit25.—How long _________in Chengdu?--For just the weekend to come.I’ll be back next Monday morning.A.are you stayingB.did you stayC.have you stayed26.—Alice,would you mind not playing the guitar?I___on the phone.--Oh,sorry ,mom.A.talkedB.talkC.have talkedD.am talking27.Linda told the policeman she______for the bus at ten o’clock yesterday morning.A. waitedB.was waitingC.has waited28.Mike is from America. He ________ English.A. spokeB. will speakC. speaksD. had spoken29.–Where are you going for your holiday? --Well, we _______ yet.A. haven’t decidedB. hadn’t decidedC. don’t decideD. didn’t decide30. Joe,my close friend,moved to Beijing years a90,and I ________him since then.A don’t seeB didn’t seeC haven’t seenD won’t see系动词及情态动词:1.—Which T-shirt do you prefer?--The red one .It______more comfortable.A.tastesB.smellsC.feelsD.sounds2.—Must I finish my homework at school?--______.You can do it at home and hand it in next week.A.Yes,you mustB.No,you needn’tC.Yes,I can3.—Must I mop up the follr now? --No,you________.A.needn’tB.can’tC.shouldn’tD.mustn’t4.—The lake is said to be dry.Is that ture?--It _______.Look ,some kids are swimming in it.A.must be trueB.can’t be trueC.may not be true5.—Is the man over there Mr. Brown? --It_____be him.He’s much shorter.A.mustB.can’tC.needn’tD.may6.______your American friend eat with chopsticks? -Yes, but he can’t use them well.A. CanB.ShouldC. Must7.–Look at that girl! Is it Susan?--No, it _______ be her. She has gone back to her hometown.A. mustn’tB. can’tC. needn’tD. wouldn’t8.( )—It’s such a long way! What shall I do?----You take my car if you wantA willB mustC mayD shall9.—Must I finish my homework at school?--No,you_________.You can do it at home.A.can’tB.mustn’tC.needn’tD.won’t10.This is a secret between you and me.You_______tell it to others.A.shouldB.mustn’tC.canD.may11.—Dean,I hear you’ve got a set of Canadian stamps.________I have a look?--Of course .I’ll fetch them for you.A.MayB.MustC.ShouldD.Need12.—Where’s Jeff?--I’m not sure. He ______be playing football on the playground.A.needB.mightC.mustD.can13.Song Zuying’s voice is very sweet and her songs_____very beautiful.A.tasteB.soundC.smellD.feel14.( )—The winner of the race is a tall and thin boy with glasses.--Then it_____be my friend,Mike,who looks like very fat.A.mightB.can’tC.don’tD.should15.--Listen! Is it Linda singing in the next room?--No, it __________ be her. She is still in Shanghai.A.can't B.couldn't C.may not D.might not16.__This pair of shoes ____ belong to Jack. He likes this kind very much.—It ____ be his. It’s too large for him.A. can; can’tB. may; needn’tC. must; mustn’tD. must; can’t17.—Dad,must I do my homework now?--No.You________play games with your friends for a little while.A.wouldB.needn’t D.may18.Teachers think students needn’t do so much homework , but they _______ mal students do a lot in the past.A. have toB. mustC. had toD. Should19.You __ ____ smoke. You are only 14 and it isn’t allowed.A. mustn’tB. don’t have toC. needn’tD. won’t20.The skirt is made of silk. It __ ____so soft, I like it.A. washesB. is washedC. is feltD. feels21.— Can I go to the movie , Mum ?—Certaninly , But you ________ be back by ten o’clock .A. canB. mayC. mightD. must非谓语动词:1.At least 300 million people are using QQ________by Ma Huateng to chat on line.A.createB.createsC.creatingD.created2.—Where’s your brother now,Bob?--I saw him _______in the street a moment ago and I told him_________.A.playing;don’t do soB.playing;not to soC.play;to do so2.The woman made his son_____finally after she told him some jokes.ughedB.to laughughughing4.Father often tells me——too much time on computer games.A don’t spendB not spendC not to spendD not spending5.—How would your family like to travel?--It’s a problem in my family.Mother prefers to take a bu s to travel,while father always sticks ______to travel.A.to driveB.to drivingC.drivingD.drive6.The teachers often tell their pubils ________aeross the road when the traffic lightis red.A.not goB.not to goC.don’t goD.didn’t go7.Don’t forget _________an umbrella _______you.It’s going to rain.A.to take;toB.taking;toC.to take;withD.taking;with8.English is very important,so I practice________it very hard.A.speaksB.to speakC.speakingD.speak9. Most of the young people enjoy ______ Jay Chou's songs.A.sing B.sang C.singing D.to sing10. As teenagers, we’re old enough __________ with housework. We can help set the table, wash the dishes and clean our own rooms.A. to helpB. helpingC. helped11.—Hi,Steve!Our teachers told us________an electric bike.It’s too dangerous.--I’m sorry.I won’t do it again.A.to rideB.not rideC.not to ride12.After several campus(校园) attacks , many schools across the country havealready taken action their campuses .A. making, safeB. made, safeC. to make, saferD. make, safe13.— What would you like for breakfast ?—I like hamburgers . But now I’ d like ________ some cakes.A. eatingB. to eatC. to drinkD. drinking14.-How long does it take your father ______ to work every day?-About half an hour.A. drivesB. drivingC. to drivest weekend, the Greens had great fun ______ at Water World.A. swimmingB. cookingC. hiking16.Granny often tells us _____ water in our daily life.A. saveB. savingC. to saveD. saves17.The boy is looking forward to a good time after the exam.A. to haveB. hasC. havingD. have18.When we came to the gate, he stopped _______ me go in first.A. to letB. to tellC. to allowD. to ask19.We have two rooms ______, but I can’t decide ______.A. to live, to choose which oneB. lived, choose which oneC. to live in, which one to chooseD. live, which one20.Just before the Chinese class, I suddenly realized that I forgot _______myChinese text book.A. bringB. bringingC. and bringD. to bring21.There are so many kinds of Mp3 in the shop. We can’t decide .A. what to buyB. to buy whatC. which to buyD. to buy which22.28. While I was walking along the lake, I saw some fish _____ out of the water.A. jumpedB. to jumpC. jumpingD. are jumping感叹句:1.--________beautiful day it is!Let’s go and have a picnic in the park.--Good idea!A.HowB.How aC.WhatD.What a2.—Kate has won the first prize in the singing competition.--______pleasant surprise this gave her classmates!A.WhatB.HowC.What aD.How a3._______ great man Huang Furong is! He lost his life while saving the students in the Yushu earthquake.A. HowB. How aC. WhatD. What a4._______exciting sport it is to climb the mountains!A.HowB.WhatC.What aD.What an5.______wide the streets are!A.WhatB.HowC.What a6.—Let’s go skating,_________?--Ok,Let’s go.A.do youB.don’t youC.will youD.shall we7. She’s never been to Hong Kong,______?A.isn’t sheB.has sheC.is sheD.hasn’t she8.-- __________ beautiful city!--Yes. Lots of tourists come to visit it every year.A.How B.What C.How a D.What a 9.—There are always many volunteers in great events, ____________?—Yes. Many hands makes light work!A. aren’t thereB. are thereC. aren’t they10.____ good job she does! She is really a clever girl.A. WhatB. HowC. What aD. How a11.—Zhou Weilun could hardly speak English three years ago,________?--No,he couldn’t .But now he is quite good at it.A.couldn’t heB.could heC.didn’t he12.--________interesting talk Mr.Black gave us!--Yes.We all like it very much.A.HowB.What aC.What an13.Let’s discuss it after the meeting, ____ _____?A. wil l youB. don’t we C won’t you D. shall we14. The waiter hardly has a day off , ________ ?A. hasn’t heB. doesn’t heC. will heD. does he15. — _______ do you study study for a test ?— I study by working with a group .A.WhereB. HowC. WhenD. Why16. –______ schoolbag is this? –I guess it’s Lily’s.A. WhatB. WhoC. WhoseD. Which17. ( ) ——are you going to the airportA. WhatB. HowC. WhenD. Why18( ) meeting it is!A. What a importantB. What importantC. How an imporantD. What an importat19.--- I’ve just got a postcard from my pen friend.--- Let me have a look. Wow. _____ nice card!A. WhatB. What aC. HowD. Why20.--_______kind girl Nancy is!--Yes,she is always ready to help others.A.WhatB.What aC.HowD.How a反义疑问句:—She doesn’t like geography,does she?—.A. Yes,she doesB. Yes,she doesn’tC. No,she does— ______ is her daughter? — The girl on the right wearing blue jeans.A. WhoB. WhatC. WhichD. Where24Alice had a wonderful time yesterday, ______?A. hadn’t sheB. wasn’t sheC. didn’t sheD. wouldn’t she25-He didn’t go to the lecture this morning, did he?-______. Though he was not feeling very well.A. No, he didn’t.B. Yes, he did.C. No, he did.D. Yes, he didn’t.状语从句:1. The man can’t get on the bus because there ________ no room on it.A. wasB. isC. areD. were2. I was listening to the music ________ there was a knock at the door.A. whileB. whenC. becauseD. until3. You may leave the classroom when you ________ writing.A. will finishB. are finishingC. have finishedD. had finished4. I don’t know if he ________ tomorrow, if he _________ ,I’ll meet him.A. will come, comesB. comes, comesC. will come, will comeD. come, will come5. ________ I read, the more I understand.A. The moreB. So muchC. How muchD. For all6. Would you give Johnny this letter if you ________ to see him this week.A. will happenB. happenC. are happenedD. happened7. She was busy, ________ she couldn’t go to your birthday party last night.A. andB. soC. orD. but8. English is ____ a useful language ____ it is spoken in many countries of the world.A. so…thatB. such…thatC. so…becauseD. such…because9. You will not succeeded if you ________ harder.A. will workB. won’t workC. workD. don’t work10. Write clearly ________ your teacher can understand you correctly.A. sinceB. forC. becauseD. so that11. ________ he failed , he went on doing the experiment. A. Even B. YetC. AlthoughD. in spite of定语从句:1. There are lots of things I need to prepare before the trip.A. whoB. thatC. whomD. whose2.Jim dislikes people ______talk much but never do anything .A.whoB.whomC.whose3.—I hear Sam has gone to Qiandiao for his holiday.--Oh,how nice!Do you know when he_______?A.leftB.was leavingC.has leftD.had left4. We are trying to help those children _____ lost their parents in Yushu , Qinghai Province.A. whichB. whenC. whoD. whose5. Do you know the girl _______is singing in the classroom?A.whoB.whichC.when6.Jim dislikes people _______talk much but never do anything.A.whomB.whenC.whoseD.who7.—The volunteers are doing a great job in Yushu.--Yes.They are helping the people ______are suffering from the earthquake.A.whichB.whatC./D.who8.Have you found the answer to the question ______I asked you this morning?A.whenB.whatC.that9. Friends are those make you smile,always open their hearts to you andencourage you to succeedA whichB whatC whomD who10I began to work in Shanghai in the year _______HongKong was returned to China.A.thatB.whichC.whereD.when情景交际:1.—The party is a great success.We’ve had a good time.--________.A.It doesn’t matterB.No,thanksC.It’s not very goodD.I’m glad to hear that2. —Mike hurt his arm the day before yesterday .Now he’s in hospital.--________.A.I’m sorry to hear thatB.That’s all rightC.I hope you’ll feel better soon3.—Would you mind my closing the window?--______.A.Certainly notB.Yes,pleaseC.Sorry,I won’t4. —Excuse me.Where is the nearest post office?--Sorry,I’m new here. ______!A.Thank you all the sameB.What a pityC.Bad luck5. —Shall we go to the concert this evening?--________.Let’s buy the tickets first.A.Best wishesB.That’s rightC.Good idea6. —Sorry,sir.I made a mistake again.--________Practice more and you will do better.A.Never mind.B.I’m not sure.C.You’re welcomeD.Don’t mention it7.—Are you confident about this evening’s performance, Katie?--________.I’m well prepared and feel I’ve got everything ready.A.I hope soB.It’s hard to sayC.Sure,I amD.I am afraid not8.—I am very worried about tomorrow’s maths test.I am afraid I can’t pass this time. --________!I’m sure you’ll make it.A.No problemB.That’s rightC.Cheer upD.Don’t mention it9. —Sorry , I can’t go shopping with you tonight.— ______.A. Why don’t you ask BillB. Excuse me . Which is the way to the shop?C. It’s very kind of you .D. That’s OK. Maybe some other time . 10—Would you lie me to take you to the new shopping centre?--______.I’ve been there once.A.No,thanksB.No problemC.Yes,I think soD.That’s very kind of you。