(完整word版)高中数学导数知识点归纳总结,推荐文档.pdf

14.导 数知识要点1.导数（导函数的简称）的定义：设0 x是函数)(xfy定义域的一点，如果自变量x在0 x处有 增 量x，则 函 数 值y 也 引 起 相 应 的 增 量)()(00 xfxxfy；比 值xxfxxfxy)()(00称为函数)(xfy在点0 x到xx0之间的平均变化率；如果极限xxfxxfxyxx)()(limlim0000存在，则称函数)(xfy在点0 x处可导，并把这个极限叫做)(xfy在0 x处的导数，记作)(0 xf或0|xxy，即)(0 xf=xxfxxfxyxx)()(limlim0000.注：x是增量，我们也称为“改变量”，因为x 可正，可负，但不为零.以知函数)(xfy定义域为A，)(xfy的定义域为B，则 A与 B 关系为BA.2.函数)(xfy在点0 x处连续与点0 x处可导的关系：函数)(xfy在点0 x处连续是)(xfy在点0 x处可导的必要不充分条件.可以证明，如果)(xfy在点0 x处可导，那么)(xfy点0 x处连续.事实上，令xxx0，则0 xx相当于0 x.于是)()()(lim)(lim)(lim0000000 xfxfxxfxxfxfxxxx).()(0)()(limlim)()(lim)()()(lim0000000000000 xfxfxfxfxxfxxfxfxxxfxxfxxxx如果)(xfy点0 x处连续，那么)(xfy在点0 x处可导，是不成立的.例：|)(xxf在点00 x处连续，但在点00 x处不可导，因为xxxy|，当x0 时，1xy；当x 0 时，1xy，故xyx0lim不存在.注：可导的奇函数函数其导函数为偶函数.可导的偶函数函数其导函数为奇函数.导数导数的概念导数的运算导数的应用导数的几何意义、物理意义函数的单调性函数的极值函数的最值常见函数的导数导数的运算法则3.导数的几何意义：函数)(xfy在点0 x处的导数的几何意义就是曲线)(xfy在点)(,(0 xfx处的切线的斜率，也 就 是 说，曲 线)(xfy在 点P)(,(0 xfx处 的 切 线 的 斜 率 是)(0 xf，切 线 方 程 为).)(00 xxxfyy4.求导数的四则运算法则：)(vuvu)(.)()()(.)()(2121xfxfxfyxfxfxfynn)()(cvcvvccvuvvuuv（c为常数）)0(2vvuvvuvu注：vu,必须是可导函数.若两个函数可导，则它们和、差、积、商必可导；若两个函数均不可导，则它们的和、差、积、商不一定不可导.例如：设xxxf2sin2)(，xxxg2cos)(，则)(),(xgxf在0 x处均不可导，但它们和)()(xgxfxxcossin在0 x处均可导.5.复合函数的求导法则：)()()(xufxfx或xuxuyy复合函数的求导法则可推广到多个中间变量的情形.6.函数单调性：函数单调性的判定方法：设函数)(xfy在某个区间内可导，如果)(xf0，则)(xfy为增函数；如果)(xf0，则)(xfy为减函数.常数的判定方法；如果函数)(xfy在区间I内恒有)(xf=0，则)(xfy为常数.注：0)(xf是 f（x）递增的充分条件，但不是必要条件，如32xy在),(上并不是都有0)(xf，有一个点例外即x=0 时 f（x）=0，同样0)(xf是 f（x）递减的充分非必要条件.一般地，如果 f（x）在某区间内有限个点处为零，在其余各点均为正（或负），那么f（x）在该区间上仍旧是单调增加（或单调减少）的.7.极值的判别方法：（极值是在0 x附近所有的点，都有)(xf)(0 xf，则)(0 xf是函数)(xf的极大值，极小值同理）当函数)(xf在点0 x处连续时，如果在0 x附近的左侧)(xf0，右侧)(xf0，那么)(0 xf是极大值；如果在0 x附近的左侧)(xf0，右侧)(xf0，那么)(0 xf是极小值.文档编码:CV9O7B3P5T10 HN6Q1M9U4N1 ZH5C5O8Y6L4文档编码:CV9O7B3P5T10 HN6Q1M9U4N1 ZH5C5O8Y6L4文档编码:CV9O7B3P5T10 HN6Q1M9U4N1 ZH5C5O8Y6L4文档编码:CV9O7B3P5T10 HN6Q1M9U4N1 ZH5C5O8Y6L4文档编码:CV9O7B3P5T10 HN6Q1M9U4N1 ZH5C5O8Y6L4文档编码:CV9O7B3P5T10 HN6Q1M9U4N1 ZH5C5O8Y6L4文档编码:CV9O7B3P5T10 HN6Q1M9U4N1 ZH5C5O8Y6L4文档编码:CV9O7B3P5T10 HN6Q1M9U4N1 ZH5C5O8Y6L4文档编码:CV9O7B3P5T10 HN6Q1M9U4N1 ZH5C5O8Y6L4文档编码:CV9O7B3P5T10 HN6Q1M9U4N1 ZH5C5O8Y6L4文档编码:CV9O7B3P5T10 HN6Q1M9U4N1 ZH5C5O8Y6L4文档编码:CV9O7B3P5T10 HN6Q1M9U4N1 ZH5C5O8Y6L4文档编码:CV9O7B3P5T10 HN6Q1M9U4N1 ZH5C5O8Y6L4文档编码:CV9O7B3P5T10 HN6Q1M9U4N1 ZH5C5O8Y6L4文档编码:CV9O7B3P5T10 HN6Q1M9U4N1 ZH5C5O8Y6L4文档编码:CV9O7B3P5T10 HN6Q1M9U4N1 ZH5C5O8Y6L4文档编码:CV9O7B3P5T10 HN6Q1M9U4N1 ZH5C5O8Y6L4文档编码:CV9O7B3P5T10 HN6Q1M9U4N1 ZH5C5O8Y6L4文档编码:CV9O7B3P5T10 HN6Q1M9U4N1 ZH5C5O8Y6L4文档编码:CV9O7B3P5T10 HN6Q1M9U4N1 ZH5C5O8Y6L4文档编码:CV9O7B3P5T10 HN6Q1M9U4N1 ZH5C5O8Y6L4文档编码:CV9O7B3P5T10 HN6Q1M9U4N1 ZH5C5O8Y6L4文档编码:CV9O7B3P5T10 HN6Q1M9U4N1 ZH5C5O8Y6L4文档编码:CV9O7B3P5T10 HN6Q1M9U4N1 ZH5C5O8Y6L4文档编码:CV9O7B3P5T10 HN6Q1M9U4N1 ZH5C5O8Y6L4文档编码:CV9O7B3P5T10 HN6Q1M9U4N1 ZH5C5O8Y6L4文档编码:CV9O7B3P5T10 HN6Q1M9U4N1 ZH5C5O8Y6L4文档编码:CV9O7B3P5T10 HN6Q1M9U4N1 ZH5C5O8Y6L4文档编码:CV9O7B3P5T10 HN6Q1M9U4N1 ZH5C5O8Y6L4文档编码:CV9O7B3P5T10 HN6Q1M9U4N1 ZH5C5O8Y6L4文档编码:CV9O7B3P5T10 HN6Q1M9U4N1 ZH5C5O8Y6L4文档编码:CV9O7B3P5T10 HN6Q1M9U4N1 ZH5C5O8Y6L4文档编码:CV9O7B3P5T10 HN6Q1M9U4N1 ZH5C5O8Y6L4文档编码:CV9O7B3P5T10 HN6Q1M9U4N1 ZH5C5O8Y6L4文档编码:CV9O7B3P5T10 HN6Q1M9U4N1 ZH5C5O8Y6L4文档编码:CV9O7B3P5T10 HN6Q1M9U4N1 ZH5C5O8Y6L4文档编码:CV9O7B3P5T10 HN6Q1M9U4N1 ZH5C5O8Y6L4文档编码:CV9O7B3P5T10 HN6Q1M9U4N1 ZH5C5O8Y6L4文档编码:CV9O7B3P5T10 HN6Q1M9U4N1 ZH5C5O8Y6L4文档编码:CV9O7B3P5T10 HN6Q1M9U4N1 ZH5C5O8Y6L4文档编码:CV9O7B3P5T10 HN6Q1M9U4N1 ZH5C5O8Y6L4文档编码:CV9O7B3P5T10 HN6Q1M9U4N1 ZH5C5O8Y6L4文档编码:CV9O7B3P5T10 HN6Q1M9U4N1 ZH5C5O8Y6L4文档编码:CV9O7B3P5T10 HN6Q1M9U4N1 ZH5C5O8Y6L4文档编码:CV9O7B3P5T10 HN6Q1M9U4N1 ZH5C5O8Y6L4文档编码:CV9O7B3P5T10 HN6Q1M9U4N1 ZH5C5O8Y6L4文档编码:CV9O7B3P5T10 HN6Q1M9U4N1 ZH5C5O8Y6L4文档编码:CV9O7B3P5T10 HN6Q1M9U4N1 ZH5C5O8Y6L4也就是说0 x是极值点的充分条件是0 x点两侧导数异号，而不是)(xf=0.此外，函数不可导的点也可能是函数的极值点.当然，极值是一个局部概念，极值点的大小关系是不确定的，即有可能极大值比极小值小（函数在某一点附近的点不同）.注：若点0 x是可导函数)(xf的极值点，则)(xf=0.但反过来不一定成立.对于可导函数，其一点0 x是极值点的必要条件是若函数在该点可导，则导数值为零.例如：函数3)(xxfy，0 x使)(xf=0，但0 x不是极值点.例如：函数|)(xxfy，在点0 x处不可导，但点0 x是函数的极小值点.8.极值与最值的区别：极值是在局部对函数值进行比较，最值是在整体区间上对函数值进行比较.注：函数的极值点一定有意义.9.几种常见的函数导数：I.0C（C 为常数）xxcos)(sin211)(arcsinxx1)(nnnxx（Rn）xxsin)(cos211)(arccosxxII.xx1)(lnexxaalog1)(log11)(arctan2xxxxee)(aaaxxln)(11)cot(2xxarcIII.求导的常见方法：常用结论：xx1|)|(ln.形如).()(21naxaxaxy或).()().()(2121nnbxbxbxaxaxaxy两边同取自然对数，可转化求代数和形式.无理函数或形如xxy这类函数，如xxy取自然对数之后可变形为xxylnln，对两边求导可得xxxxxyyxyyxxxyylnln1ln.导数知识点总结复习经典例题剖析考点一：求导公式。例 1.()fx是31()213f xxx的导函数，则(1)f的值是。文档编码:CV9O7B3P5T10 HN6Q1M9U4N1 ZH5C5O8Y6L4文档编码:CV9O7B3P5T10 HN6Q1M9U4N1 ZH5C5O8Y6L4文档编码:CV9O7B3P5T10 HN6Q1M9U4N1 ZH5C5O8Y6L4文档编码:CV9O7B3P5T10 HN6Q1M9U4N1 ZH5C5O8Y6L4文档编码:CV9O7B3P5T10 HN6Q1M9U4N1 ZH5C5O8Y6L4文档编码:CV9O7B3P5T10 HN6Q1M9U4N1 ZH5C5O8Y6L4文档编码:CV9O7B3P5T10 HN6Q1M9U4N1 ZH5C5O8Y6L4文档编码:CV9O7B3P5T10 HN6Q1M9U4N1 ZH5C5O8Y6L4文档编码:CV9O7B3P5T10 HN6Q1M9U4N1 ZH5C5O8Y6L4文档编码:CV9O7B3P5T10 HN6Q1M9U4N1 ZH5C5O8Y6L4文档编码:CV9O7B3P5T10 HN6Q1M9U4N1 ZH5C5O8Y6L4文档编码:CV9O7B3P5T10 HN6Q1M9U4N1 ZH5C5O8Y6L4文档编码:CV9O7B3P5T10 HN6Q1M9U4N1 ZH5C5O8Y6L4文档编码:CV9O7B3P5T10 HN6Q1M9U4N1 ZH5C5O8Y6L4文档编码:CV9O7B3P5T10 HN6Q1M9U4N1 ZH5C5O8Y6L4文档编码:CV9O7B3P5T10 HN6Q1M9U4N1 ZH5C5O8Y6L4文档编码:CV9O7B3P5T10 HN6Q1M9U4N1 ZH5C5O8Y6L4文档编码:CV9O7B3P5T10 HN6Q1M9U4N1 ZH5C5O8Y6L4文档编码:CV9O7B3P5T10 HN6Q1M9U4N1 ZH5C5O8Y6L4文档编码:CV9O7B3P5T10 HN6Q1M9U4N1 ZH5C5O8Y6L4文档编码:CV9O7B3P5T10 HN6Q1M9U4N1 ZH5C5O8Y6L4文档编码:CV9O7B3P5T10 HN6Q1M9U4N1 ZH5C5O8Y6L4文档编码:CV9O7B3P5T10 HN6Q1M9U4N1 ZH5C5O8Y6L4文档编码:CV9O7B3P5T10 HN6Q1M9U4N1 ZH5C5O8Y6L4文档编码:CV9O7B3P5T10 HN6Q1M9U4N1 ZH5C5O8Y6L4文档编码:CV9O7B3P5T10 HN6Q1M9U4N1 ZH5C5O8Y6L4文档编码:CV9O7B3P5T10 HN6Q1M9U4N1 ZH5C5O8Y6L4文档编码:CV9O7B3P5T10 HN6Q1M9U4N1 ZH5C5O8Y6L4文档编码:CV9O7B3P5T10 HN6Q1M9U4N1 ZH5C5O8Y6L4文档编码:CV9O7B3P5T10 HN6Q1M9U4N1 ZH5C5O8Y6L4文档编码:CV9O7B3P5T10 HN6Q1M9U4N1 ZH5C5O8Y6L4文档编码:CV9O7B3P5T10 HN6Q1M9U4N1 ZH5C5O8Y6L4文档编码:CV9O7B3P5T10 HN6Q1M9U4N1 ZH5C5O8Y6L4文档编码:CV9O7B3P5T10 HN6Q1M9U4N1 ZH5C5O8Y6L4文档编码:CV9O7B3P5T10 HN6Q1M9U4N1 ZH5C5O8Y6L4文档编码:CV9O7B3P5T10 HN6Q1M9U4N1 ZH5C5O8Y6L4文档编码:CV9O7B3P5T10 HN6Q1M9U4N1 ZH5C5O8Y6L4文档编码:CV9O7B3P5T10 HN6Q1M9U4N1 ZH5C5O8Y6L4文档编码:CV9O7B3P5T10 HN6Q1M9U4N1 ZH5C5O8Y6L4文档编码:CV9O7B3P5T10 HN6Q1M9U4N1 ZH5C5O8Y6L4文档编码:CV9O7B3P5T10 HN6Q1M9U4N1 ZH5C5O8Y6L4文档编码:CV9O7B3P5T10 HN6Q1M9U4N1 ZH5C5O8Y6L4文档编码:CV9O7B3P5T10 HN6Q1M9U4N1 ZH5C5O8Y6L4文档编码:CV9O7B3P5T10 HN6Q1M9U4N1 ZH5C5O8Y6L4文档编码:CV9O7B3P5T10 HN6Q1M9U4N1 ZH5C5O8Y6L4文档编码:CV9O7B3P5T10 HN6Q1M9U4N1 ZH5C5O8Y6L4文档编码:CV9O7B3P5T10 HN6Q1M9U4N1 ZH5C5O8Y6L4文档编码:CV9O7B3P5T10 HN6Q1M9U4N1 ZH5C5O8Y6L4考点二：导数的几何意义。例2.已 知 函 数()yf x的 图 象 在 点(1(1)Mf，处 的 切 线 方 程 是122yx，则(1)(1)ff。例 3.曲线32242yxxx在点(13)，处的切线方程是。点评：以上两小题均是对导数的几何意义的考查。考点三：导数的几何意义的应用。例4.已知曲线C：xxxy2323，直线kxyl:，且直线l与曲线C 相切于点00,yx00 x，求直线l的方程及切点坐标。点评：本小题考查导数几何意义的应用。解决此类问题时应注意“切点既在曲线上又在切线上”这个条件的应用。函数在某点可导是相应曲线上过该点存在切线的充分条件，而不是必要条件。考点四：函数的单调性。例 5.已知1323xxaxxf在 R 上是减函数，求a的取值范点评：本题考查导数在函数单调性中的应用。对于高次函数单调性问题，要有求导意识。考点五：函数的极值。例 6.设函数32()2338f xxaxbxc在1x及2x时取得极值。（1）求 a、b 的值；（2）若对于任意的0 3x，都有2()f xc成立，求c 的取值范围。点评：本题考查利用导数求函数的极值。求可导函数xf的极值步骤：求导数xf；求0 xf的根；将0 xf的根在数轴上标出，得出单调区间，由xf 在各区间上取值的正负可确定并求出函数xf的极值。文档编码:CV9O7B3P5T10 HN6Q1M9U4N1 ZH5C5O8Y6L4文档编码:CV9O7B3P5T10 HN6Q1M9U4N1 ZH5C5O8Y6L4文档编码:CV9O7B3P5T10 HN6Q1M9U4N1 ZH5C5O8Y6L4文档编码:CV9O7B3P5T10 HN6Q1M9U4N1 ZH5C5O8Y6L4文档编码:CV9O7B3P5T10 HN6Q1M9U4N1 ZH5C5O8Y6L4文档编码:CV9O7B3P5T10 HN6Q1M9U4N1 ZH5C5O8Y6L4文档编码:CV9O7B3P5T10 HN6Q1M9U4N1 ZH5C5O8Y6L4文档编码:CV9O7B3P5T10 HN6Q1M9U4N1 ZH5C5O8Y6L4文档编码:CV9O7B3P5T10 HN6Q1M9U4N1 ZH5C5O8Y6L4文档编码:CV9O7B3P5T10 HN6Q1M9U4N1 ZH5C5O8Y6L4文档编码:CV9O7B3P5T10 HN6Q1M9U4N1 ZH5C5O8Y6L4文档编码:CV9O7B3P5T10 HN6Q1M9U4N1 ZH5C5O8Y6L4文档编码:CV9O7B3P5T10 HN6Q1M9U4N1 ZH5C5O8Y6L4文档编码:CV9O7B3P5T10 HN6Q1M9U4N1 ZH5C5O8Y6L4文档编码:CV9O7B3P5T10 HN6Q1M9U4N1 ZH5C5O8Y6L4文档编码:CV9O7B3P5T10 HN6Q1M9U4N1 ZH5C5O8Y6L4文档编码:CV9O7B3P5T10 HN6Q1M9U4N1 ZH5C5O8Y6L4文档编码:CV9O7B3P5T10 HN6Q1M9U4N1 ZH5C5O8Y6L4文档编码:CV9O7B3P5T10 HN6Q1M9U4N1 ZH5C5O8Y6L4文档编码:CV9O7B3P5T10 HN6Q1M9U4N1 ZH5C5O8Y6L4文档编码:CV9O7B3P5T10 HN6Q1M9U4N1 ZH5C5O8Y6L4文档编码:CV9O7B3P5T10 HN6Q1M9U4N1 ZH5C5O8Y6L4文档编码:CV9O7B3P5T10 HN6Q1M9U4N1 ZH5C5O8Y6L4文档编码:CV9O7B3P5T10 HN6Q1M9U4N1 ZH5C5O8Y6L4文档编码:CV9O7B3P5T10 HN6Q1M9U4N1 ZH5C5O8Y6L4文档编码:CV9O7B3P5T10 HN6Q1M9U4N1 ZH5C5O8Y6L4文档编码:CV9O7B3P5T10 HN6Q1M9U4N1 ZH5C5O8Y6L4文档编码:CV9O7B3P5T10 HN6Q1M9U4N1 ZH5C5O8Y6L4文档编码:CV9O7B3P5T10 HN6Q1M9U4N1 ZH5C5O8Y6L4文档编码:CV9O7B3P5T10 HN6Q1M9U4N1 ZH5C5O8Y6L4文档编码:CV9O7B3P5T10 HN6Q1M9U4N1 ZH5C5O8Y6L4文档编码:CV9O7B3P5T10 HN6Q1M9U4N1 ZH5C5O8Y6L4文档编码:CV9O7B3P5T10 HN6Q1M9U4N1 ZH5C5O8Y6L4文档编码:CV9O7B3P5T10 HN6Q1M9U4N1 ZH5C5O8Y6L4文档编码:CV9O7B3P5T10 HN6Q1M9U4N1 ZH5C5O8Y6L4文档编码:CV9O7B3P5T10 HN6Q1M9U4N1 ZH5C5O8Y6L4文档编码:CV9O7B3P5T10 HN6Q1M9U4N1 ZH5C5O8Y6L4文档编码:CV9O7B3P5T10 HN6Q1M9U4N1 ZH5C5O8Y6L4文档编码:CV9O7B3P5T10 HN6Q1M9U4N1 ZH5C5O8Y6L4文档编码:CV9O7B3P5T10 HN6Q1M9U4N1 ZH5C5O8Y6L4文档编码:CV9O7B3P5T10 HN6Q1M9U4N1 ZH5C5O8Y6L4文档编码:CV9O7B3P5T10 HN6Q1M9U4N1 ZH5C5O8Y6L4文档编码:CV9O7B3P5T10 HN6Q1M9U4N1 ZH5C5O8Y6L4文档编码:CV9O7B3P5T10 HN6Q1M9U4N1 ZH5C5O8Y6L4文档编码:CV9O7B3P5T10 HN6Q1M9U4N1 ZH5C5O8Y6L4文档编码:CV9O7B3P5T10 HN6Q1M9U4N1 ZH5C5O8Y6L4文档编码:CV9O7B3P5T10 HN6Q1M9U4N1 ZH5C5O8Y6L4文档编码:CV9O7B3P5T10 HN6Q1M9U4N1 ZH5C5O8Y6L4考点六：函数的最值。例 7.已知a为实数，axxxf42。求导数xf；（2）若01f，求xf在区间2,2上的最大值和最小值。点评：本题考查可导函数最值的求法。求可导函数xf在区间ba,上的最值，要先求出函数xf在区间ba,上的极值，然后与af和bf进行比较，从而得出函数的最大最小值。考点七：导数的综合性问题。例 8.设函数3()f xaxbxc(0)a为奇函数，其图象在点(1,(1)f处的切线与直线670 xy垂直，导函数()fx的最小值为12。（1）求a，b，c的值；（2）求函数()f x的单调递增区间，并求函数()f x在1,3上的最大值和最小值点评：本题考查函数的奇偶性、单调性、二次函数的最值、导数的应用等基础知识，以及推理能力和运算能力。文档编码:CV9O7B3P5T10 HN6Q1M9U4N1 ZH5C5O8Y6L4文档编码:CV9O7B3P5T10 HN6Q1M9U4N1 ZH5C5O8Y6L4文档编码:CV9O7B3P5T10 HN6Q1M9U4N1 ZH5C5O8Y6L4文档编码:CV9O7B3P5T10 HN6Q1M9U4N1 ZH5C5O8Y6L4文档编码:CV9O7B3P5T10 HN6Q1M9U4N1 ZH5C5O8Y6L4文档编码:CV9O7B3P5T10 HN6Q1M9U4N1 ZH5C5O8Y6L4文档编码:CV9O7B3P5T10 HN6Q1M9U4N1 ZH5C5O8Y6L4文档编码:CV9O7B3P5T10 HN6Q1M9U4N1 ZH5C5O8Y6L4文档编码:CV9O7B3P5T10 HN6Q1M9U4N1 ZH5C5O8Y6L4文档编码:CV9O7B3P5T10 HN6Q1M9U4N1 ZH5C5O8Y6L4文档编码:CV9O7B3P5T10 HN6Q1M9U4N1 ZH5C5O8Y6L4文档编码:CV9O7B3P5T10 HN6Q1M9U4N1 ZH5C5O8Y6L4文档编码:CV9O7B3P5T10 HN6Q1M9U4N1 ZH5C5O8Y6L4文档编码:CV9O7B3P5T10 HN6Q1M9U4N1 ZH5C5O8Y6L4文档编码:CV9O7B3P5T10 HN6Q1M9U4N1 ZH5C5O8Y6L4文档编码:CV9O7B3P5T10 HN6Q1M9U4N1 ZH5C5O8Y6L4文档编码:CV9O7B3P5T10 HN6Q1M9U4N1 ZH5C5O8Y6L4文档编码:CV9O7B3P5T10 HN6Q1M9U4N1 ZH5C5O8Y6L4文档编码:CV9O7B3P5T10 HN6Q1M9U4N1 ZH5C5O8Y6L4文档编码:CV9O7B3P5T10 HN6Q1M9U4N1 ZH5C5O8Y6L4文档编码:CV9O7B3P5T10 HN6Q1M9U4N1 ZH5C5O8Y6L4文档编码:CV9O7B3P5T10 HN6Q1M9U4N1 ZH5C5O8Y6L4文档编码:CV9O7B3P5T10 HN6Q1M9U4N1 ZH5C5O8Y6L4文档编码:CV9O7B3P5T10 HN6Q1M9U4N1 ZH5C5O8Y6L4文档编码:CV9O7B3P5T10 HN6Q1M9U4N1 ZH5C5O8Y6L4文档编码:CV9O7B3P5T10 HN6Q1M9U4N1 ZH5C5O8Y6L4文档编码:CV9O7B3P5T10 HN6Q1M9U4N1 ZH5C5O8Y6L4文档编码:CV9O7B3P5T10 HN6Q1M9U4N1 ZH5C5O8Y6L4文档编码:CV9O7B3P5T10 HN6Q1M9U4N1 ZH5C5O8Y6L4文档编码:CV9O7B3P5T10 HN6Q1M9U4N1 ZH5C5O8Y6L4文档编码:CV9O7B3P5T10 HN6Q1M9U4N1 ZH5C5O8Y6L4文档编码:CV9O7B3P5T10 HN6Q1M9U4N1 ZH5C5O8Y6L4文档编码:CV9O7B3P5T10 HN6Q1M9U4N1 ZH5C5O8Y6L4文档编码:CV9O7B3P5T10 HN6Q1M9U4N1 ZH5C5O8Y6L4文档编码:CV9O7B3P5T10 HN6Q1M9U4N1 ZH5C5O8Y6L4文档编码:CV9O7B3P5T10 HN6Q1M9U4N1 ZH5C5O8Y6L4文档编码:CV9O7B3P5T10 HN6Q1M9U4N1 ZH5C5O8Y6L4文档编码:CV9O7B3P5T10 HN6Q1M9U4N1 ZH5C5O8Y6L4文档编码:CV9O7B3P5T10 HN6Q1M9U4N1 ZH5C5O8Y6L4文档编码:CV9O7B3P5T10 HN6Q1M9U4N1 ZH5C5O8Y6L4文档编码:CV9O7B3P5T10 HN6Q1M9U4N1 ZH5C5O8Y6L4文档编码:CV9O7B3P5T10 HN6Q1M9U4N1 ZH5C5O8Y6L4文档编码:CV9O7B3P5T10 HN6Q1M9U4N1 ZH5C5O8Y6L4文档编码:CV9O7B3P5T10 HN6Q1M9U4N1 ZH5C5O8Y6L4文档编码:CV9O7B3P5T10 HN6Q1M9U4N1 ZH5C5O8Y6L4文档编码:CV9O7B3P5T10 HN6Q1M9U4N1 ZH5C5O8Y6L4文档编码:CV9O7B3P5T10 HN6Q1M9U4N1 ZH5C5O8Y6L4文档编码:CV9O7B3P5T10 HN6Q1M9U4N1 ZH5C5O8Y6L4