2021年人教A版新课标高中数学必修一教案《基本不等式》.pdf

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1、1/72.2 基本不等式2abab教材分析：“基本不等式”是必修 1 的重点内容，它是在系统学习了不等关系和不等式性质，掌握了不等式性质的基础上对不等式的进一步研究，同时也是为了以后学习选修教材中关于不等式及其证明方法等内容作铺垫，起着承上启下的作用.利用基本不等式求最值在实际问题中应用广泛.同时本节知识又渗透了数形结合、化归等重要数学思想，有利于培养学生良好的思维品质.教学目标【知识与技能】1.学会推导并掌握基本不等式，理解这个基本不等式的几何意义，并掌握定理中的不等号“”取等号的条件是：当且仅当这两个数相等；2.掌握基本不等式2abab；会应用此不等式求某些函数的最值；能够解决一些简单的实

2、际问题【过程与方法】通过实例探究抽象基本不等式；【情感、态度与价值观】通过本节的学习，体会数学来源于生活，提高学习数学的兴趣.教学重难点【教学重点】应用数形结合的思想理解不等式，并从不同角度探索不等式2abab的证明过程；【教学难点】1.基本不等式2abab等号成立条件；2.利用基本不等式2abab求最大值、最小值.精品w o r d 可编辑资料-第 1 页，共 7 页-2/7教学过程1.课题导入前面我们利用完全平方公式得出了一类重要不等式：一般地，?,?，有a2+b22 ab，当且仅当 a=b时，等号成立特别地，如果a0，b0，我们用?,?分别代替上式中的a，b，可得?+?2当且仅当 a=b

3、时，等号成立.通常称不等式（1）为基本不等式（basicinequality）.其中，?+?2叫做正数 a，b的算术平均数，?叫做正数 a，b的几何平均数.基本不等式表明：两个正数的算术平均数不小于它们的几何平均数.思考:上面通过考察a2+b2=2ab 的特殊情形获得了基本不等式，能否直接利用不等式的性质推导出基本不等式呢？下面我们来分析一下.2.讲授新课1）类比弦图几何图形的面积关系认识基本不等式2abab特别的，如果a0,b0,我们用分别代替a、b，可得2abab，通常我们把上式写作：(a0,b0)2abab2）从不等式的性质推导基本不等式2abab用分析法证明：要证2abab（1）只要证

4、a+b（2）要证（2），只要证a+b-0 （3）精品w o r d 可编辑资料-第 2 页，共 7 页-文档编码:CZ2N3X6P9N2 HF5R1J3F4L9 ZZ8Y4M3T6P3文档编码:CZ2N3X6P9N2 HF5R1J3F4L9 ZZ8Y4M3T6P3文档编码:CZ2N3X6P9N2 HF5R1J3F4L9 ZZ8Y4M3T6P3文档编码:CZ2N3X6P9N2 HF5R1J3F4L9 ZZ8Y4M3T6P3文档编码:CZ2N3X6P9N2 HF5R1J3F4L9 ZZ8Y4M3T6P3文档编码:CZ2N3X6P9N2 HF5R1J3F4L9 ZZ8Y4M3T6P3文档编码:CZ2

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10、 ZZ8Y4M3T6P3文档编码:CZ2N3X6P9N2 HF5R1J3F4L9 ZZ8Y4M3T6P3文档编码:CZ2N3X6P9N2 HF5R1J3F4L9 ZZ8Y4M3T6P3文档编码:CZ2N3X6P9N2 HF5R1J3F4L9 ZZ8Y4M3T6P3文档编码:CZ2N3X6P9N2 HF5R1J3F4L9 ZZ8Y4M3T6P33/7要证（3），只要证（-）20 （4）显然，（4）是成立的.当且仅当a=b 时，（4）中的等号成立.探究 1：在右图中，AB 是圆的直径，点C 是 AB 上的一点，AC=a,BC=b.过点 C 作垂直于AB 的弦 DE，连接 AD、BD.你能利用这个图

11、形得出基本不等式2abab的几何解释吗？易证t AD t DB，那么D2AB即Dab.这 个 圆 的 半 径 为2ba，显 然，它 大 于 或 等 于CD，即abba2，其中当且仅当点C 与圆心重合，即ab 时，等号成立.因此：基本不等式2abab几何意义是“半径不小于半弦”评述：1.如果把2ba看作是正数a、b 的等差中项，ab看作是正数a、b 的等比中项，那么该定理可以叙述为：两个正数的等差中项不小于它们的等比中项.2.在数学中，我们称2ba为 a、b 的算术平均数，称ab为 a、b 的几何平均数.本节定理还可叙述为：两个正数的算术平均数不小于它们的几何平均数.【设计意图】老师引导，学生自

12、主探究得到结论并证明，锻炼了学生的自主研究能力和研究问题的逻辑分析能力.例1 已知 x0，求 x1?的最小值.分析：求x1?的最小值，就是要求一个y0（=x01?），使?x0，都有 x1?y.观察 x+1?，发现x?1?=1.联系基本不等式，可以利用正数x和1?的算术平均数与几何平均数的关系得到y0=2.解：因为 x0，所以x1?2?1?=2 精品w o r d 可编辑资料-第 3 页，共 7 页-文档编码:CZ2N3X6P9N2 HF5R1J3F4L9 ZZ8Y4M3T6P3文档编码:CZ2N3X6P9N2 HF5R1J3F4L9 ZZ8Y4M3T6P3文档编码:CZ2N3X6P9N2 HF

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19、5R1J3F4L9 ZZ8Y4M3T6P34/7当且仅当 x=1?，即 x2=1，x=1时，等号成立，因此所求的最小值为2.在本题的解答中，我们不仅明确了?x0，有x1?2，而且给出了“当且仅当 x=1?，即=1，x=1时，等号成立”，这是为了说明2是x1?（x0）的一个取值，想一想，当y00）的最小值吗？例2 已知 x，y都是正数，求证：（1）如果积 xy等于定值 P，那么当 x=y时，和 x+y有最小值 2?；（2）如果和 x+y等于定值 S，那么当 x=y时，积 xy有最大值14?2.证明：因为 x，y都是正数，所以?+?2?.（1）当积 xy等于定值 P时，?+?2?，所以?+?2?，

20、当且仅当 x=y时，上式等号成立.于是，当 x=y时，和 x+y有最小值 2?.（2）当和 x+y等于定值 S 时，?2,所以?14?2，当且仅当 x=y时，上式等号成立.于是，当 x=y时，积 xy有最大值14?2.例3（1）用篱笆围一个面积为100m2的矩形菜园，当这个矩形的边长为多少时，所用篱笆最短？最短篱笆的长度是多少？（2）用一段长为 36m的篱笆围成一个矩形菜园，当这个矩形的边长为多少时，菜园的面积最大？最大面积是多少？分析：（1）矩形菜园的面积是矩形的两邻边之积，于是问题转化为：矩形的邻边之积为定精品w o r d 可编辑资料-第 4 页，共 7 页-文档编码:CZ2N3X6P9

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25、N2 HF5R1J3F4L9 ZZ8Y4M3T6P3文档编码:CZ2N3X6P9N2 HF5R1J3F4L9 ZZ8Y4M3T6P3文档编码:CZ2N3X6P9N2 HF5R1J3F4L9 ZZ8Y4M3T6P3文档编码:CZ2N3X6P9N2 HF5R1J3F4L9 ZZ8Y4M3T6P3文档编码:CZ2N3X6P9N2 HF5R1J3F4L9 ZZ8Y4M3T6P3文档编码:CZ2N3X6P9N2 HF5R1J3F4L9 ZZ8Y4M3T6P3文档编码:CZ2N3X6P9N2 HF5R1J3F4L9 ZZ8Y4M3T6P3文档编码:CZ2N3X6P9N2 HF5R1J3F4L9 ZZ8Y4

26、M3T6P3文档编码:CZ2N3X6P9N2 HF5R1J3F4L9 ZZ8Y4M3T6P3文档编码:CZ2N3X6P9N2 HF5R1J3F4L9 ZZ8Y4M3T6P3文档编码:CZ2N3X6P9N2 HF5R1J3F4L9 ZZ8Y4M3T6P3文档编码:CZ2N3X6P9N2 HF5R1J3F4L9 ZZ8Y4M3T6P3文档编码:CZ2N3X6P9N2 HF5R1J3F4L9 ZZ8Y4M3T6P3文档编码:CZ2N3X6P9N2 HF5R1J3F4L9 ZZ8Y4M3T6P3文档编码:CZ2N3X6P9N2 HF5R1J3F4L9 ZZ8Y4M3T6P3文档编码:CZ2N3X6P9

27、N2 HF5R1J3F4L9 ZZ8Y4M3T6P3文档编码:CZ2N3X6P9N2 HF5R1J3F4L9 ZZ8Y4M3T6P3文档编码:CZ2N3X6P9N2 HF5R1J3F4L9 ZZ8Y4M3T6P35/7值，边长多大时周长最短.（2）矩形菜园的周长是矩形两邻边之和的2倍，于是问题转化为：矩形的邻边之和为定值，边长多大时面积最大.解：设矩形菜园的相邻两条边的长分别为xm，ym，篱笆的长度为2（x+y）m.（1）由已知得 xy=100.由?+?2?，可得x+y2?=20，所以2（x+y）40，当且仅当 x=y=10时，上式等号成立因此，当这个矩形菜园是边长为10m的正方形时，所用篱笆

28、最短，最短篱笆的长度为40m.（2）由已知得 2（x+y）=36，矩形菜园的面积为xy m2.由?+?2=182=9，可得xy 81，当且仅当 x=y=9时，上式等号成立.因此，当这个矩形菜园是边长为9m的正方形时，菜园的面积最大，最大面积是 81m2.例4 某工厂要建造一个长方体形无盖贮水池，其容积为 4800m2，深为3m.如果池底每平方米的造价为150元，池壁每平方米的造价为120元，那么怎样设计水池能使总造价最低？最低总造价是多少？分析：贮水池呈长方体形，它的高是3m，池底的边长没有确定.如果池底的边长确定了，那么水池的总造价也就确定了.因此，应当考察池底的边长取什么值时，水池的总造价

29、最低.精品w o r d 可编辑资料-第 5 页，共 7 页-文档编码:CZ2N3X6P9N2 HF5R1J3F4L9 ZZ8Y4M3T6P3文档编码:CZ2N3X6P9N2 HF5R1J3F4L9 ZZ8Y4M3T6P3文档编码:CZ2N3X6P9N2 HF5R1J3F4L9 ZZ8Y4M3T6P3文档编码:CZ2N3X6P9N2 HF5R1J3F4L9 ZZ8Y4M3T6P3文档编码:CZ2N3X6P9N2 HF5R1J3F4L9 ZZ8Y4M3T6P3文档编码:CZ2N3X6P9N2 HF5R1J3F4L9 ZZ8Y4M3T6P3文档编码:CZ2N3X6P9N2 HF5R1J3F4L9

30、ZZ8Y4M3T6P3文档编码:CZ2N3X6P9N2 HF5R1J3F4L9 ZZ8Y4M3T6P3文档编码:CZ2N3X6P9N2 HF5R1J3F4L9 ZZ8Y4M3T6P3文档编码:CZ2N3X6P9N2 HF5R1J3F4L9 ZZ8Y4M3T6P3文档编码:CZ2N3X6P9N2 HF5R1J3F4L9 ZZ8Y4M3T6P3文档编码:CZ2N3X6P9N2 HF5R1J3F4L9 ZZ8Y4M3T6P3文档编码:CZ2N3X6P9N2 HF5R1J3F4L9 ZZ8Y4M3T6P3文档编码:CZ2N3X6P9N2 HF5R1J3F4L9 ZZ8Y4M3T6P3文档编码:CZ2N

31、3X6P9N2 HF5R1J3F4L9 ZZ8Y4M3T6P3文档编码:CZ2N3X6P9N2 HF5R1J3F4L9 ZZ8Y4M3T6P3文档编码:CZ2N3X6P9N2 HF5R1J3F4L9 ZZ8Y4M3T6P3文档编码:CZ2N3X6P9N2 HF5R1J3F4L9 ZZ8Y4M3T6P3文档编码:CZ2N3X6P9N2 HF5R1J3F4L9 ZZ8Y4M3T6P3文档编码:CZ2N3X6P9N2 HF5R1J3F4L9 ZZ8Y4M3T6P3文档编码:CZ2N3X6P9N2 HF5R1J3F4L9 ZZ8Y4M3T6P3文档编码:CZ2N3X6P9N2 HF5R1J3F4L9

32、ZZ8Y4M3T6P3文档编码:CZ2N3X6P9N2 HF5R1J3F4L9 ZZ8Y4M3T6P3文档编码:CZ2N3X6P9N2 HF5R1J3F4L9 ZZ8Y4M3T6P3文档编码:CZ2N3X6P9N2 HF5R1J3F4L9 ZZ8Y4M3T6P3文档编码:CZ2N3X6P9N2 HF5R1J3F4L9 ZZ8Y4M3T6P3文档编码:CZ2N3X6P9N2 HF5R1J3F4L9 ZZ8Y4M3T6P3文档编码:CZ2N3X6P9N2 HF5R1J3F4L9 ZZ8Y4M3T6P3文档编码:CZ2N3X6P9N2 HF5R1J3F4L9 ZZ8Y4M3T6P3文档编码:CZ2N

33、3X6P9N2 HF5R1J3F4L9 ZZ8Y4M3T6P3文档编码:CZ2N3X6P9N2 HF5R1J3F4L9 ZZ8Y4M3T6P3文档编码:CZ2N3X6P9N2 HF5R1J3F4L9 ZZ8Y4M3T6P3文档编码:CZ2N3X6P9N2 HF5R1J3F4L9 ZZ8Y4M3T6P3文档编码:CZ2N3X6P9N2 HF5R1J3F4L9 ZZ8Y4M3T6P3文档编码:CZ2N3X6P9N2 HF5R1J3F4L9 ZZ8Y4M3T6P3文档编码:CZ2N3X6P9N2 HF5R1J3F4L9 ZZ8Y4M3T6P3文档编码:CZ2N3X6P9N2 HF5R1J3F4L9

34、ZZ8Y4M3T6P3文档编码:CZ2N3X6P9N2 HF5R1J3F4L9 ZZ8Y4M3T6P3文档编码:CZ2N3X6P9N2 HF5R1J3F4L9 ZZ8Y4M3T6P3文档编码:CZ2N3X6P9N2 HF5R1J3F4L9 ZZ8Y4M3T6P3文档编码:CZ2N3X6P9N2 HF5R1J3F4L9 ZZ8Y4M3T6P3文档编码:CZ2N3X6P9N2 HF5R1J3F4L9 ZZ8Y4M3T6P3文档编码:CZ2N3X6P9N2 HF5R1J3F4L9 ZZ8Y4M3T6P3文档编码:CZ2N3X6P9N2 HF5R1J3F4L9 ZZ8Y4M3T6P3文档编码:CZ2N

35、3X6P9N2 HF5R1J3F4L9 ZZ8Y4M3T6P3文档编码:CZ2N3X6P9N2 HF5R1J3F4L9 ZZ8Y4M3T6P3文档编码:CZ2N3X6P9N2 HF5R1J3F4L9 ZZ8Y4M3T6P3文档编码:CZ2N3X6P9N2 HF5R1J3F4L9 ZZ8Y4M3T6P36/7解：设贮水池池底的相邻两条边的边长分别为xm，ym，水池的总造价为2元.根据题意，有z=15048003+120（2 3x+2 3y）=240000+720（x+y）.由容积为 4800m3，可得3xy=4800，因此xy=1600.所以z240000+720 2?，当x=y=40时，上式等

36、号成立，此时z=297600.所以，将贮水池的池底设计成边长为40m 的正方形时总造价最低，最低总造价是297600 元.【设计意图】例题讲解，学以致用.3.随堂练习1.已知 a、b、c 都是正数，求证：（ab）（b c）（ca）abc分析：对于此类题目，选择定理：abba2（a0，b0）灵活变形，可求得结果.解：a，b，c 都是正数ab2?0 bc2?0 ca2?0（a b）（bc）（c a）2?2?2?abc即（ab）（bc）（ca）abc.【设计意图】讲练结合，熟悉新知.4.课时小结本节课，我们学习了重要不等式a2b22ab；两正数a、b 的算术平均数（?+?2），几何平均数（?）及它们

37、的关系（?+?2?）.它们成立的条件不同，前者只要求a、b 都是实数，而后者要求a、b 都是正数.它们既是不等式变形的基本工具，又是求函数最值的重要工具(下一节我们将学习它们的应用).我们还可以用它们下面的等价变形来解决问题：ab精品w o r d 可编辑资料-第 6 页，共 7 页-文档编码:CZ2N3X6P9N2 HF5R1J3F4L9 ZZ8Y4M3T6P3文档编码:CZ2N3X6P9N2 HF5R1J3F4L9 ZZ8Y4M3T6P3文档编码:CZ2N3X6P9N2 HF5R1J3F4L9 ZZ8Y4M3T6P3文档编码:CZ2N3X6P9N2 HF5R1J3F4L9 ZZ8Y4M3T

38、6P3文档编码:CZ2N3X6P9N2 HF5R1J3F4L9 ZZ8Y4M3T6P3文档编码:CZ2N3X6P9N2 HF5R1J3F4L9 ZZ8Y4M3T6P3文档编码:CZ2N3X6P9N2 HF5R1J3F4L9 ZZ8Y4M3T6P3文档编码:CZ2N3X6P9N2 HF5R1J3F4L9 ZZ8Y4M3T6P3文档编码:CZ2N3X6P9N2 HF5R1J3F4L9 ZZ8Y4M3T6P3文档编码:CZ2N3X6P9N2 HF5R1J3F4L9 ZZ8Y4M3T6P3文档编码:CZ2N3X6P9N2 HF5R1J3F4L9 ZZ8Y4M3T6P3文档编码:CZ2N3X6P9N2

39、HF5R1J3F4L9 ZZ8Y4M3T6P3文档编码:CZ2N3X6P9N2 HF5R1J3F4L9 ZZ8Y4M3T6P3文档编码:CZ2N3X6P9N2 HF5R1J3F4L9 ZZ8Y4M3T6P3文档编码:CZ2N3X6P9N2 HF5R1J3F4L9 ZZ8Y4M3T6P3文档编码:CZ2N3X6P9N2 HF5R1J3F4L9 ZZ8Y4M3T6P3文档编码:CZ2N3X6P9N2 HF5R1J3F4L9 ZZ8Y4M3T6P3文档编码:CZ2N3X6P9N2 HF5R1J3F4L9 ZZ8Y4M3T6P3文档编码:CZ2N3X6P9N2 HF5R1J3F4L9 ZZ8Y4M3T

40、6P3文档编码:CZ2N3X6P9N2 HF5R1J3F4L9 ZZ8Y4M3T6P3文档编码:CZ2N3X6P9N2 HF5R1J3F4L9 ZZ8Y4M3T6P3文档编码:CZ2N3X6P9N2 HF5R1J3F4L9 ZZ8Y4M3T6P3文档编码:CZ2N3X6P9N2 HF5R1J3F4L9 ZZ8Y4M3T6P3文档编码:CZ2N3X6P9N2 HF5R1J3F4L9 ZZ8Y4M3T6P3文档编码:CZ2N3X6P9N2 HF5R1J3F4L9 ZZ8Y4M3T6P3文档编码:CZ2N3X6P9N2 HF5R1J3F4L9 ZZ8Y4M3T6P3文档编码:CZ2N3X6P9N2

41、HF5R1J3F4L9 ZZ8Y4M3T6P3文档编码:CZ2N3X6P9N2 HF5R1J3F4L9 ZZ8Y4M3T6P3文档编码:CZ2N3X6P9N2 HF5R1J3F4L9 ZZ8Y4M3T6P3文档编码:CZ2N3X6P9N2 HF5R1J3F4L9 ZZ8Y4M3T6P3文档编码:CZ2N3X6P9N2 HF5R1J3F4L9 ZZ8Y4M3T6P3文档编码:CZ2N3X6P9N2 HF5R1J3F4L9 ZZ8Y4M3T6P3文档编码:CZ2N3X6P9N2 HF5R1J3F4L9 ZZ8Y4M3T6P3文档编码:CZ2N3X6P9N2 HF5R1J3F4L9 ZZ8Y4M3T

42、6P3文档编码:CZ2N3X6P9N2 HF5R1J3F4L9 ZZ8Y4M3T6P3文档编码:CZ2N3X6P9N2 HF5R1J3F4L9 ZZ8Y4M3T6P3文档编码:CZ2N3X6P9N2 HF5R1J3F4L9 ZZ8Y4M3T6P3文档编码:CZ2N3X6P9N2 HF5R1J3F4L9 ZZ8Y4M3T6P3文档编码:CZ2N3X6P9N2 HF5R1J3F4L9 ZZ8Y4M3T6P3文档编码:CZ2N3X6P9N2 HF5R1J3F4L9 ZZ8Y4M3T6P3文档编码:CZ2N3X6P9N2 HF5R1J3F4L9 ZZ8Y4M3T6P3文档编码:CZ2N3X6P9N2

43、HF5R1J3F4L9 ZZ8Y4M3T6P3文档编码:CZ2N3X6P9N2 HF5R1J3F4L9 ZZ8Y4M3T6P3文档编码:CZ2N3X6P9N2 HF5R1J3F4L9 ZZ8Y4M3T6P3文档编码:CZ2N3X6P9N2 HF5R1J3F4L9 ZZ8Y4M3T6P3文档编码:CZ2N3X6P9N2 HF5R1J3F4L9 ZZ8Y4M3T6P3文档编码:CZ2N3X6P9N2 HF5R1J3F4L9 ZZ8Y4M3T6P3文档编码:CZ2N3X6P9N2 HF5R1J3F4L9 ZZ8Y4M3T6P37/7?2+?22，ab（?+?2）2.我们用两个正数的算术平均数与几何平

44、均数的关系顺利解决了函数的一些最值问题.在用均值不等式求函数的最值，是值得重视的一种方法，但在具体求解时，应注意考查下列三个条件：(1)函数的解析式中，各项均为正数；(2)函数的解析式中，含变数的各项的和或积必须有一个为定值；(3)函数的解析式中，含变数的各项均相等，取得最值即用均值不等式求某些函数的最值时，应具备三个条件：一正二定三取等.教学反思：略精品w o r d 可编辑资料-第 7 页，共 7 页-文档编码:CZ2N3X6P9N2 HF5R1J3F4L9 ZZ8Y4M3T6P3文档编码:CZ2N3X6P9N2 HF5R1J3F4L9 ZZ8Y4M3T6P3文档编码:CZ2N3X6P9N

45、2 HF5R1J3F4L9 ZZ8Y4M3T6P3文档编码:CZ2N3X6P9N2 HF5R1J3F4L9 ZZ8Y4M3T6P3文档编码:CZ2N3X6P9N2 HF5R1J3F4L9 ZZ8Y4M3T6P3文档编码:CZ2N3X6P9N2 HF5R1J3F4L9 ZZ8Y4M3T6P3文档编码:CZ2N3X6P9N2 HF5R1J3F4L9 ZZ8Y4M3T6P3文档编码:CZ2N3X6P9N2 HF5R1J3F4L9 ZZ8Y4M3T6P3文档编码:CZ2N3X6P9N2 HF5R1J3F4L9 ZZ8Y4M3T6P3文档编码:CZ2N3X6P9N2 HF5R1J3F4L9 ZZ8Y4M

46、3T6P3文档编码:CZ2N3X6P9N2 HF5R1J3F4L9 ZZ8Y4M3T6P3文档编码:CZ2N3X6P9N2 HF5R1J3F4L9 ZZ8Y4M3T6P3文档编码:CZ2N3X6P9N2 HF5R1J3F4L9 ZZ8Y4M3T6P3文档编码:CZ2N3X6P9N2 HF5R1J3F4L9 ZZ8Y4M3T6P3文档编码:CZ2N3X6P9N2 HF5R1J3F4L9 ZZ8Y4M3T6P3文档编码:CZ2N3X6P9N2 HF5R1J3F4L9 ZZ8Y4M3T6P3文档编码:CZ2N3X6P9N2 HF5R1J3F4L9 ZZ8Y4M3T6P3文档编码:CZ2N3X6P9N

47、2 HF5R1J3F4L9 ZZ8Y4M3T6P3文档编码:CZ2N3X6P9N2 HF5R1J3F4L9 ZZ8Y4M3T6P3文档编码:CZ2N3X6P9N2 HF5R1J3F4L9 ZZ8Y4M3T6P3文档编码:CZ2N3X6P9N2 HF5R1J3F4L9 ZZ8Y4M3T6P3文档编码:CZ2N3X6P9N2 HF5R1J3F4L9 ZZ8Y4M3T6P3文档编码:CZ2N3X6P9N2 HF5R1J3F4L9 ZZ8Y4M3T6P3文档编码:CZ2N3X6P9N2 HF5R1J3F4L9 ZZ8Y4M3T6P3文档编码:CZ2N3X6P9N2 HF5R1J3F4L9 ZZ8Y4M

48、3T6P3文档编码:CZ2N3X6P9N2 HF5R1J3F4L9 ZZ8Y4M3T6P3文档编码:CZ2N3X6P9N2 HF5R1J3F4L9 ZZ8Y4M3T6P3文档编码:CZ2N3X6P9N2 HF5R1J3F4L9 ZZ8Y4M3T6P3文档编码:CZ2N3X6P9N2 HF5R1J3F4L9 ZZ8Y4M3T6P3文档编码:CZ2N3X6P9N2 HF5R1J3F4L9 ZZ8Y4M3T6P3文档编码:CZ2N3X6P9N2 HF5R1J3F4L9 ZZ8Y4M3T6P3文档编码:CZ2N3X6P9N2 HF5R1J3F4L9 ZZ8Y4M3T6P3文档编码:CZ2N3X6P9N

49、2 HF5R1J3F4L9 ZZ8Y4M3T6P3文档编码:CZ2N3X6P9N2 HF5R1J3F4L9 ZZ8Y4M3T6P3文档编码:CZ2N3X6P9N2 HF5R1J3F4L9 ZZ8Y4M3T6P3文档编码:CZ2N3X6P9N2 HF5R1J3F4L9 ZZ8Y4M3T6P3文档编码:CZ2N3X6P9N2 HF5R1J3F4L9 ZZ8Y4M3T6P3文档编码:CZ2N3X6P9N2 HF5R1J3F4L9 ZZ8Y4M3T6P3文档编码:CZ2N3X6P9N2 HF5R1J3F4L9 ZZ8Y4M3T6P3文档编码:CZ2N3X6P9N2 HF5R1J3F4L9 ZZ8Y4M

50、3T6P3文档编码:CZ2N3X6P9N2 HF5R1J3F4L9 ZZ8Y4M3T6P3文档编码:CZ2N3X6P9N2 HF5R1J3F4L9 ZZ8Y4M3T6P3文档编码:CZ2N3X6P9N2 HF5R1J3F4L9 ZZ8Y4M3T6P3文档编码:CZ2N3X6P9N2 HF5R1J3F4L9 ZZ8Y4M3T6P3文档编码:CZ2N3X6P9N2 HF5R1J3F4L9 ZZ8Y4M3T6P3文档编码:CZ2N3X6P9N2 HF5R1J3F4L9 ZZ8Y4M3T6P3文档编码:CZ2N3X6P9N2 HF5R1J3F4L9 ZZ8Y4M3T6P3文档编码:CZ2N3X6P9N