08年全国成人高考专升本高数(二)真题及答案
2008年成人高等学校招生全国统一考试数学(理工农医类)
2008 年成人高等学校招生全国统一考试数 学（理工农医类）一、选择题：本大题共 17 小题，每小题 5 分，共 85分。
在每小题给出的四个选项中， 选出一项符合题目要求的．（1）设集合 { } | ||2 A x x =£ ， { } |1 B x x =³- ，则A B =I （A ）{ } | ||1 x x £ （B ）{ } | ||2 x x £ （C ）{ }|12 x x -££ （D ）{ }|21 x x -££- （2）函数 cos 3xy = 的最小正周期是（A ）6p （B ）3p （C ）2p（D ）3p（3）抛物线 2 4 y x =- 的准线方程为（A ） 2 x =- （B ） 1 x =- （C ） 2 x = （D ） 1 x = （4）设甲： 6 x p=；乙： 1sin 2x = ，则 （A ）甲是乙的必要条件，但不是乙的充分条件 （B ）甲是乙的充分条件，但不是乙的必要条件 （C ）甲不是乙的充分条件，也不是乙的必要条件 （D ）甲是乙的充分必要条件 （5）若向量 ( ) ,2 a x = r ， ( ) 2,3 b =- r ，且a r //b r，则x =（A ） 43-（B ） 3 - （C ）3（D ）43（6）下列函数中，为奇函数的是（A ） 3 log y x = （B ） 3 x y = （C ） 23 y x = （D ） 3sin y x=（7）函数 () y f x = 的图象与函数 2 xy = 的图象关于直线 y x = 对称，则 () f x =（A ）2x（B ） 2 log (0) x x > （C ）2x（D ）lg(2) (0)x x > （8）设二次函数 2 y x bx c =++ 的图象过点( ) 1,2 和( ) 2,4 - ，则该函数的解析式为（A ） 2 2 y x x =++ （B ） 2 21 y x x =+- （C ） 2 1233 y x x =++ （D ） 2 1233y x x =+- （9）若 1 a > ，则（A ） 1 2log 0a < （B ） 2 log 0a < （C ） 1a - < （D ） 210a -< （10）已知复数 1 z i =+ ，i 为虚数单位，则 2 z =（A ）22i+ （B ）2i（C ）22i- （D ） 2i- （11）在 ABC D 中，若 1sin 3A = ， 30 AB += o， 4 BC = ，则 AB = （A ）24（B ）63（C ）23（D ）6（12）过函数 6y x= 图象上一点P 作 x 轴的垂线PQ ，Q 为垂足，O 为坐标原点，则OPQ D 的面积为（A ）6（B ）3（C ）2（D ）1（13）已知正方形ABCD ，以A ,C 为焦点，且过B 点的椭圆的离心率为（A ） 2（B ）212+ （C ）2 2（D ）21 2- （14）已知向量 ( ) 2,3,1 a =- r ， ( ) 2,0,3 b = r ， ( )0,0,2 c = r ，则 ( )a b c += r r r g （A ）8 （B ）9 （C ）13 （D ） 61（15）设某项试验每次成功的概率为 23，则在2次独立重复试验中，都不成功的概率为（A ）4 9 （B ）1 3（C ）2 9 （D ）1 9（16）在空间中，下列四个命题中为真命题的一个是（A ）平行于同一条直线的两条直线平行 （B ）垂直于同一条直线的两条直线平行（C ）若a 与b 是异面直线，b 与c 是异面直线，则a 与c 也是异面直线 （D ）若直线a //平面a ，直线b //平面a ，则a //b（17）某学生从6门课程中选修3门，其中甲、乙两门课程至少选一门，则不同的选课 方案共有（A ）4种（B ）12种（C ）16种（D ）20种二、填空题：本大题共 4 小题，每小题4 分，共16分． （18）曲线 2sin y x = 在点( ) ,0 p 处的切线的斜率为 ．（19）设a 是直线 2 y x =-+ 的倾斜角，则a =．（20）一个三棱锥的三个侧面与底面都是等边三角形，则其侧面和底面所成角的余弦 值是．（21）设随机变量x 的分布列为x1 2 3 4 P161 31 31 6则x 的数学期望E x =．三、解答题：本大题共 4 小题，共 49 分．解答应写出推理、演算步骤． （22）（本小题满分 12 分）已知等差数列{ } n a 中， 1 9 a = ， 38 0 a a += ． （Ⅰ）求数列{ } n a 的通项公式；（Ⅱ）当n 为何值时，数列{ } n a 的前n 项和 n S 取得最大值，并求该最大值．（23）（本小题满分 12 分）如图，塔PO 与地平线AO 垂直，在A 点测得塔顶P 的仰角 45 PAO Ð= o ，沿AO 方向 前进至B 点，测得仰角 60 PBO Ð= o ， , A B 相距44m ，求塔高PO ． （精确到0.1m ）（24）（本小题满分 12 分）已知一个圆的圆心为双曲线 22 1 412x y -= 的右焦点，并且此圆过原点．（Ⅰ）求该圆的方程；（Ⅱ）求直线 3 y x = 被该圆截得的弦长．（25）（本小题满分 13 分） 已知函数 ()2 f x x x =- ．（Ⅰ）求函数 () y f x = 的单调区间，并指出它在各单调区间上是增函数还是减函数； （Ⅱ）求函数 () y f x = 在区间[ ] 0,4 上的最大值和最小值．POB A。
2008年河南专升本高等数学真题+真题解析
2008河南省普通高等学校选拔优秀专科毕业生进入本科阶段学习考试高等数学试卷一、选择题 （每小题2 分，共50 分）1．函数()ln(1)f x x =-+的定义域是（ ）A ．[]2,1--B ．[]2,1-C ．[)2,1-D ．()2,1-【答案】C【解析】由1020x x ->⎧⎨+≥⎩可得21x -≤<，故选C ．2．312cos limsin 3x xx ππ→-=⎛⎫- ⎪⎝⎭（ ）A ．1B ．0CD【答案】D【解析】3312cos 2sin limlim sin cos 33x x x xx x ππππ→→-==⎛⎫⎛⎫-- ⎪ ⎪⎝⎭⎝⎭D ．3．点0x =是函数113131xxy -=+的（ ）A ．连续点B ．跳跃间断点C ．可去间断点D ．第二类间断点【答案】 【解析】11311lim 1131xx x-→--==-+，11031lim 131xx x +→-=+，故选B ．4．下列极限存在的是（ ）A ．lim xx e →+∞B ．0sin 2lim x xx →C ．01lim cosx x+→ D ．22lim 3x x x →+∞+-【答案】B 【解析】0sin 2lim2x xx→=，其他三个都不存在，应选B ．5．当0x →时，2ln(1)x +是比1cos x -的（ ） A ．低阶无穷小 B ．高阶无穷小C ．等价无穷小D ．同阶但不等价无穷小【答案】D【解析】0x →时，22ln(1)~x x +，211cos ~2x x -，故选D ．6．设函数11(1)sin ,11()1,10arctan ,0x x x f x x x x ⎧++<-⎪+⎪=-≤≤⎨⎪>⎪⎩，则()f x （ ）A ．在1x =-处连续，在0x =处不连续B ．在0x =处连续，在1x =-处不连续C ．在1,0x =-处均连续D ．在1,0x =-处均不连续【答案】A【解析】1lim ()1x f x -→-=，1lim ()1x f x +→-=，(1)1()f f x -=⇒在1x =-处连续；0lim ()1x f x -→=，0lim ()0x f x +→=，(0)1()f f x =⇒在0x =处不连续，应选A ．7．过曲线arctan x y x e =+上的点(0,1)处的法线方程为（ ）A ．210x y -+=B ．220x y -+=C ．210x y --=D ．220x y +-=【答案】D 【解析】211x y e x'=++，02x y ='=，法线的斜率12k =-，法线方程为112y x -=-，即220x y +-=，故选D ．8．设函数()f x 在0x =处满足，()(0)3()f x f x x α=-+，且0()lim0x x xα→=，则(0)f '=（ ） A ．1- B ．1 C ．3-D ．3【解析】000()(0)3()()(0)limlim 3lim 30x x x f x f x x x f x x xαα→→→--+'===-+=--，应选C ．9．若函数()(ln )(1)x f x x x =>，则()f x '=（ ） A ．1(ln )x x - B ．1(ln )(ln )ln(ln )x x x x x -+C ．(ln )ln(ln )x x xD ．(ln )x x x【答案】B【解析】ln(ln )()(ln )x x x f x x e ==，[]11()(ln )ln(ln )(ln )ln(ln )ln x x f x x x x x x x x x ⎡⎤''==+⋅⋅⎢⎥⎣⎦1(ln )(ln )ln(ln )x x x x x -=+，故选B ．10．设函数()y y x =由参数方程33cos sin x t y t ⎧=⎨=⎩确定，则224|t d ydx π==（ ） A ．2- B ．1- C．3-D．3【答案】D【解析】223sin cos sin 3cos sin cos dy dy dt t t t dx dx dt t t t ===--，22d y dx =1d dy dx dt dx dt⎛⎫⋅ ⎪⎝⎭2211cos 3cos sin x t t-=⋅- 413cos sin t t =，224|t d y dx π==，故选D ．11．下列函数中，在区间[]1,1-上满足罗尔定理条件的是（ ）A ．x y e =B ．ln ||y x =C ．21y x =-D ．21y x=【答案】C【解析】验证罗尔定理得条件，只有21y x =-满足，应选C ．12．曲线352y x x =+-的拐点是（ ）A ．0x =B ．(0,2)-C ．无拐点D ．0,2x y ==-【解析】235y x '=+，6y x ''=，令0y ''=，得0x =，当0x >时，0y ''>，当0x <时，0y ''<，故拐点为(0,2)-，应选B ．13．曲线1|1|y x =-（ ） A ．只有水平渐进线B ．既有水平渐进线，又有垂直渐近线C ．只有垂直渐近线D ．既无水平渐进线，又无垂直渐近线【答案】B 【解析】1lim 0|1|x x →∞=-，曲线有水平渐近线0y =；1lim |1|x x →∞=∞-，曲线有垂直渐近线1x =，故选B ．14．如果()f x 的一个原函数是ln x x ，那么2()x f x dx ''=⎰（ ）A ．ln x C +B ．2xC +C ．3ln x x C +D ．C x -【答案】D【解析】()(ln )1ln f x x x x '==+，21()f x x''=-，2()x f x dx dx x C ''=-=-+⎰⎰，应选D ． 15．243dxx x =-+⎰（ ）A ．13ln 21x C x -+-B ．1ln3x C x -+-C ．ln(3)ln(1)x x C ---+D ．ln(1)ln(3)x x C ---+【答案】A 【解析】211113ln 43(3)(1)23121dx dx x dx C x x x x x x x -⎛⎫==-=+ ⎪-+-----⎝⎭⎰⎰⎰，应选A ．16．设14011I dx x =+⎰，则I 的取值范围为（ ）A ．01I ≤≤B ．112I ≤≤ C ．04I π≤≤D ．14I π<<【答案】B【解析】因01x ≤≤，411121x ≤≤+，根据定积分的估值性质，有112I ≤≤，故选B ．17．下列广义积分收敛的是（ ）A ．31x dx +∞⎰B ．1ln xdx x+∞⎰C ．1⎰D ．0x e dx +∞-⎰【答案】D【解析】D 项中001x xe dx e +∞--+∞=-=⎰，故收敛．18．331xdx --=⎰（ ）A ．3021x dx -⎰B ．1331(1)(1)x dx x dx --+-⎰⎰C ．1331(1)(1)x dx x dx ----⎰⎰ D ．1331(1)(1)x dx x dx --+-⎰⎰【答案】D【解析】3131333131111(1)(1)xdx xdx xdx x dx x dx ----=-+-=-+-⎰⎰⎰⎰⎰，故选D ．19．若()f x 是可导函数，()0f x >，且满足220()sin ()ln 221cos x f t tf x dt t=-+⎰，则()f x =（ ） A ．ln(1cos )x + B ．ln(1cos )x C -++C ．ln(1cos )x -+D ．ln(1cos )x C ++【答案】A【解析】对220()sin ()ln 221cos x f t t f x dt t =-+⎰两边求导有()sin 2()()21cos f x xf x f x x'=-+，即 sin ()1cos x f x x '=-+，从而sin (1cos )()ln(1cos )1cos 1cos x d x f x dx x C x x+=-==++++⎰⎰．由初始条件(0)ln 2f =，代入得0C =，应选A ．20．若函数()f x 满足111()1()2f x x f x dx -=+-⎰，则()f x =（ ）A ．13x -B ．12x -C ．12x +D ．13x +【答案】C【解析】令11()a f x dx -=⎰，则1()12f x x a =+-，从而11111()122a f x dx x a dx a --⎛⎫==+-=- ⎪⎝⎭⎰⎰，得1a =，故1()2f x x =+，应选C ．21．若320()eI x f x dx =⎰，则I =（ ）A ．2()e xf x dx ⎰B ．0()exf x dx ⎰C ．21()2e xf x dx ⎰D ．1()2exf x dx ⎰ 【答案】C【解析】32222001()()2ee I xf x dx x f x dx ==⎰⎰，令2t x =，则220011()()22e e I tf t dt xf x dx ==⎰⎰，故选C ．22．直线24:591x y zL ++==与平面:4375x y z π-+=的位置关系是（ ）A ．斜交B ．垂直C ．L 在π内D ．L π【答案】D【解析】直线的方向向量(5,9,1)=s ，平面的法向量(4,3,7)=-n ，由0⋅=s n 得⊥s n ，而点(2,4,0)--不在平面内，故平行，应选D ．23．220x y →→=（ ）A ．2B ．3C ．1D ．不存在【答案】A【解析】22000001)2x x x y y y →→→→→→===，故选A ．24．曲面22z x y =+在点(1,2,5)处的切平面方程为（ ）A ．245x y z +-=B ．425x y z +-=C ．245x y z +-=D ．245x y z -+=【答案】A【解析】令22(,,)F x y z x y z =+-，(1,2,5)2x F =，(1,2,5)4y F =，(1,2,5)1z F =-，得切平面方程为2(1)4(2)(5)0x y z -+---=，即245x y z +-=，故选A ．25．设函数33z x y xy =-，则2zy x∂=∂∂（ ）A ．6xyB ．2233x y -C ．6xy -D ．2233y x -【答案】B【解析】323z x xy y ∂=-∂，22233z x y y x∂=-∂∂，应选B ．26．如果区域D 被分成两个子区域12,D D ，且1(,)5D f x y dxdy =⎰⎰，2(,)1D f x y dxdy =⎰⎰，则(,)Df x y dxdy =⎰⎰（ ）A ．5B ．4C ．6D ．1【答案】C【解析】根据二重积分的可加性，(,)6Df x y dxdy =⎰⎰，应选C ．27．如果L 是摆线sin 1cos x t ty t =-⎧⎨=-⎩上从点(2,0)A π到点(0,0)B 的一段弧，则曲线积分231(3)sin 3xLx y xe dx x y y dy ⎛⎫++-= ⎪⎝⎭⎰（ ） A ．2(12)1e ππ--B ．22(12)1e ππ⎡⎤--⎣⎦C ．23(12)1e ππ⎡⎤--⎣⎦D ．24(12)1e ππ⎡⎤--⎣⎦【答案】C 【解析】因2P Qx y x ∂∂==∂∂，从而此积分与路径无关，取直线段0x x y =⎧⎨=⎩，x 从2π变成0，则002302221(3)sin 333()3x xx x x L x y xe dx x y y dy xe dx xde xe e πππ⎛⎫++-===- ⎪⎝⎭⎰⎰⎰23(12)1e ππ⎡⎤=--⎣⎦．28．通解为x Ce （C 为任意常数）的微分方程为 （ ）A ．0y y '+=B ．0y y '-=C ．1y y '-=D ．10y y '-+=【答案】B【解析】x y Ce =，x y Ce '=，从而0y y '-=，故选B ．29．微分方程x y y xe -'''+=的特解形式应设为*y = （ ）A ．()x x ax b e -+B ．ax b +C ．()x ax b e -+D ．2()x x ax b e -+【答案】A【解析】特征方程为20r r +=，特征根为10r =，21r =-，1-是特征方程的单根，应设*y =()x x ax b e -+，应选A ．30．下列四个级数中，发散的是（ ）A ．11!n n ∞=∑B ．1231000n n n ∞=-∑C ．12n n n∞=∑D ．211n n ∞=∑【答案】B【解析】231lim 01000500n n n →∞-=≠，故级数1231000n n n∞=-∑发散，应选B ．二、填空题 （每小题 2分，共 30分）31．0lim ()x x f x A →=的________条件是0lim ()lim ()x x x x f x f x A -+→→==．【答案】充分必要（或充要） 【解析】显然为充分必要（或充要）．32．函数sin y x x =-在区间(0,2)π内单调________，其曲线在区间0,2π⎛⎫⎪⎝⎭内的凸凹性为________的．【答案】增加（或递增），凹【解析】1cos 0y x '=->⇒在(0,2)π内单调增加，sin y x ''=在0,2π⎛⎫⎪⎝⎭内大于零，应为凹的．33．设方程22232x y z a ++=（a 为常数）所确定的隐函数为(,)z f x y =，则zx∂=∂________． 【答案】【解析】222(,,)32F x y z x y z a =++-，则6x F x =，2z F z =，故3x z F z xx F z∂=-=-∂． 34．=________．【答案】2ln(1C -++ 【解析】令t =2dx tdt =，212122ln(1)2ln(121t dt dt t t C C t t ⎛⎫==-=-++=++ ⎪++⎝⎭⎰⎰．35．331cos xdx x ππ-=+⎰________．【答案】0【解析】1cos x y x =+在区间,33ππ⎡⎤-⎢⎥⎣⎦上是奇函数，故3301cos x dx x ππ-=+⎰．36．在空间直角坐标系中以点(0,4,1)A -，(1,3,1)B --，(2,4,0)C -为顶点的ABC ∆面积为________．【解析】(1,1,0)AB =-，(2,0,1)AC =-，110(1,1,2)201AB AC ⨯=-=----i j k，故ABC ∆的面积为1122S AB AC =⨯=37．方程221942x y x ⎧+=⎪⎨⎪=-⎩在直角坐标系下的图形为________．【答案】两条平行直线【解析】椭圆柱面与平面2x =-的交线，为两条平行直线．38.函数33(,)3f x y x y xy =+-的驻点________． 【答案】【解析】由22330330fx y xf y x y∂⎧=-=⎪∂⎪⎨∂⎪=-=⎪∂⎩，可得驻点为(0,0)，(1,1)．39．若21z x y e -=+(1,0)|zx ∂=∂________． 【答案】0【解析】(1,0)(,0)000|z zz x x x ∂∂=⇒=⇒=∂∂．40．440cos xydx dy yππ=⎰⎰________．【解析】44444000cos cos cos sin y xy y dx dy dy dx ydy yy yπππππ====⎰⎰⎰⎰⎰．41．直角坐标系下二重积分(,)Df x y dxdy ⎰⎰（其中D 为环域2219x y ≤+≤）化为极坐标形式为________．【答案】231(cos ,sin )d f r r rdr πθθθ⎰⎰【解析】231(,)(cos ,sin )Df x y dxdy d f r r rdr πθθθ=⎰⎰⎰⎰．42．以3312x x y C e C xe --=+为通解的二阶常系数线性齐次微分方程为________．【答案】690y y y '''++=【解析】由通解3312x x y C e C xe --=+可知，有二重特征根3-，从而微分方程为690y y y '''++=．43．等比级数()00n n aq a ∞=≠∑，当________时级数收敛；当________时级数发散． 【答案】1q <，1q ≥【解析】级数0n n aq ∞=∑是等比级数，当1q <时，级数收敛，当1q ≥时，级数发散．44．函数21()2f x x x =--展开成x 的幂级数________． 【答案】11011(1)32n n n n x ∞++=-⎡⎤+-⎢⎥⎣⎦∑，11x -<< 【解析】211111111()231231612f x xx x x x x ⎛⎫==-+=-⋅-⋅ ⎪--+-+⎝⎭- 110001111(1)(1)36232n n n n n n n n n n x x x ∞∞∞++===-⎡⎤=---=+-⎢⎥⎣⎦∑∑∑，11x -<<．45．12nn n n ∞=-⎛⎫ ⎪⎝⎭∑是敛散性为________的级数． 【答案】发散 【解析】(2)2222lim lim 10n n n n n e n n -⋅--→∞→∞-⎛⎫⎛⎫=-=≠ ⎪ ⎪⎝⎭⎝⎭，级数发散．三、计算题（每小题5 分，共40 分）46．求252222lim 3x x x x +→∞⎛⎫+ ⎪-⎝⎭． 【答案】52e【解析】222225535552225232222255lim lim 1lim 1333x x x x x x x x x e x x x ++-+⋅⋅-→∞→∞→∞⎛⎫+⎛⎫⎛⎫=+=+= ⎪ ⎪ ⎪---⎝⎭⎝⎭⎝⎭．47．2400lim x x x →⎰． 【答案】【解析】24300lim 2x x x x x →→→===⎰．48．已知lnsin(12)y x =-，求dy dx ． 【答案】2cot(12)x -- 【解析】lnsin(12)1cos(12)(2)2cot(12)sin(12)dy d x x x dx dx x -==⋅-⋅-=---．49．计算arctan x xdx ⎰．【答案】 【解析】2221111arctan arctan arctan 12221x xdx xdx x x dx x ⎛⎫==-- ⎪+⎝⎭⎰⎰⎰ 22111arctan (arctan )(arctan arctan )222x x x x C x x x x C =--+=-++．50．求函数cos()x z e x y =+的全微分．【答案】[]cos()sin()sin()x x e x y x y dx e x y dy +-+-+ 【解析】cos()sin()x x z e x y e x y x∂=+-+∂，sin()x z e x y y ∂=-+∂，故 []cos()sin()sin()x x z z dz dx dy e x y x y dx e x y dy x y∂∂=+=+-+-+∂∂．51． 计算2D x d y σ⎰⎰，其中D 为由2y =，y x =，1xy =所围成的区域． 【答案】1724【解析】根据积分区域的特征，应在直角坐标系下计算积分，且积分次序为先积x 后积y ，交点坐标为(2,2)，(1,1)，1,22⎛⎫ ⎪⎝⎭，故222122221111117224y y Dx x d dy dx y dy y y y y σ⎛⎫==-= ⎪⎝⎭⎰⎰⎰⎰⎰．52．求微分方程sin cos x y y x e -'+=满足初始条件(0)1y =-的特解．【答案】sin (1)x y e x -=-【解析】()cos P x x =，sin ()x Q x e -=，则通解为cos cos sin sin ()xdx xdx x x y e e e dx C e x C ---⎛⎫⎰⎰=⋅+=+ ⎪⎝⎭⎰， 又(0)1y =-，所以1C =-，特解为sin (1)x y e x -=-．53．求级数031nn n x n ∞=+∑的收敛半径与收敛区间（考虑端点）． 【答案】11,33⎡⎫-⎪⎢⎣⎭【解析】1131lim lim 323n n n n n na n a n ρ++→∞→∞+==⋅=+，收敛半径113R ρ==． 当13x =时，级数为011n n ∞=+∑，该级数发散；当13x =-时，级数为0(1)1n n n ∞=-+∑，该级数收敛， 故收敛域为11,33⎡⎫-⎪⎢⎣⎭．四、应用题 （每小题7 分，共 14 分）54．过曲线2y x =上一点(1,1)M ，作切线L ，D 是由曲线2y x =，切线L 及x 轴所围成的平面图形．求：（1）平面图形D 的面积；（2）平面图形D 绕x 轴旋转一周所生成的旋转体的体积．【答案】（1）112 （2）30π【解析】（1）曲线2y x =在(1,1)M 处的切线斜率为2，过M 点的切线方程为21y x =-，切线与x 轴的交点为1,02⎛⎫ ⎪⎝⎭，则平面图形D 的面积 123100111111223412A x dx x =-⋅⋅=-=⎰． （2）平面图形D 绕x 轴旋转一周所生成的旋转体的体积为12225100111()1325630V x dx x πππππ=-⋅⋅⋅=⋅-=⎰．55．一块铁皮宽24厘米，把它的两边折上去，做成一个正截面为等腰梯形的槽（图略），要使等腰梯形的面积A 最大，求腰长x 和它对底边的倾斜角α．【答案】【解析】由题意知梯形的上、下底分别为2422cos x x α-+，242(0,0)x x α->>． 故221(2422cos 242)sin 24sin 2sin sin cos 2A x x x x x x x αααααα=-++-⋅=-+， 24sin 4sin 2sin cos A x x xαααα∂=-+∂， 222224cos 2cos (cos sin )A x x x ααααα∂=-+-∂， 令0A x∂=∂，0A α∂=∂，联立解得，在定义域内唯一驻点8x =，3πα=， 故当3πα=，8x cm =时正截面面积A 最大．五、证明题 （6 分）56．证明方程0ln x x e π=-⎰在区间3(,)e e 内仅有一个实根．【解析】令0()ln x f x x e π=-+⎰，显然()f x 在3,e e ⎡⎤⎣⎦上连续，且0()0f e ==⎰，3220()360f e e e π=-+<-<⎰，由零点定理得，在3(,)e e 内至少存在一个ξ，使得()=0f ξ． 又11()f x x e'=-，在3(,)e e 内()<0f x '，所以在内单调减少．综上所述，方程0ln x x e =-⎰在区间3(,)e e 内仅有一个实根．。
2008年成考专升本高等数学
第 1 页 2008年成考专升本高等数学 36一、单项选择题(在每小题的四个备选答案中，选出一个正确答案，并将正确答案的序号填在题干的括号内。
每小题4分，共28分)1.设A 为［0，1］中的有理数全体，B 为［0，1］中的无理数全体，则A （ ）B 。
A.<B.=C.>D.无法比较2.有限个可数集的并集是（ ）。
A.有限集B.可数集C.不可数集D.无法确定3.任意多个闭集的交集是（ ）。
A.开集B.闭集C.F σ集D.G δ集4.设E 是R 中无理数全体，则mE （ ）。
A.0B.1C.＋∞D.-∞5.设f(x )是R 1上的可微函数，则导函数f ′(x )是（ ）。
A.连续函数B. a. e.连续函数C.可测函数D.无法判定6.设⎪⎩⎪⎨⎧-∈-∈=={0,1}Q)|0,1(|0{0,1}Q)|0,1(|1 x p qx p )x (f 且p, q 互素，下述哪个说法不成立（ ）。
A. f(x)=0 a. e. B. f(x)=0 a. e.连续C. f(x)处处不连续D. f(x) Riemann 可积7.设｛ϕn (x)｝是简单函数列，(x)f lim n n ∞→存在，则(x)f lim n n ∞→ 是（ ）。
A.简单函数B.连续函数C.可测函数D. Lebesgue 可积函数二、填空题(每空4分，共40分)1.集簇｛A α|α∈Γ｝的交αΓ∈αA =_______________.2.设A 2n =［-1+n 1, 2-12n 1+］, A 2n+1=［-n 1,1+n 1］,∞→n lim A n =_______________.3.设E 是E 的开核，则E 与E 具有关系E _______________E.4.设f(x)是［0，1］上的单调函数，E 是f(x)的不连续点全体，则mE=_______________.5.可测集与G δ型集有如下关系：G δ型集是可测集，反之_______________.6.设p 是Cantor 集，⎩⎨⎧-∈∈=P |,|x P x )x (f 1010 ,则f(x)在21处的振幅为_______________.第 2 页 7.设f(x)是E ⊂R n 上的非负函数，记R n+1中的点集{（x,z ）|x ∈Z,0≤z<f(x)}为G |E, f|,则当x ∉E 时（G ｜E, f ｜）x =_______________.8.设f(x)=⎩⎨⎧-∈∈Q|,|x |,|Q x 101100 , 则)(10f V =_______________. 9.设f(x)是E 上的可测函数，则E ［|f|=0］是_______________.10.设｛f n (x)｝是E 上的单调下降的非负可测函数列，则⎰∞→E n x )x (f d lim n _______________ ⎰∞→E n x )x (f d lim n .三、完成下列各题(每小题8分，共32分)1.设p 0是E 的聚点，证明存在E 中的互异的点所成的｛p n ｝点列，使p n →p 0(n →∞)。
2008年专升本高数二考试真题及参考答案
2008年成人高考专升本高等数学二考试真题及参考答案
一、选择题：每小题4分，共40分，在每小题给出的四个选项中，只有一项是符合题目要求。
第1题
参考答案：C
第2题
参考答案：C
第3题
参考答案：A
第4题
参考答案：B
第5题
参考答案：D 第6题
参考答案：A 第7题
参考答案：C 第8题
参考答案：B 第9题
参考答案：A 第10题
参考答案：D
二、填空题：本大题共10个小题，每小题4分，共40分，把答案填写在题中横线上。
第11题
参考答案：1
第12题
参考答案：2
第13题
参考答案：cos x-xsin x
第14题
参考答案：20x3
第15题
参考答案：(1,1/3) 第16题
参考答案：
第17题
参考答案：x3+ x 第18题
参考答案：2
第19题
参考答案：x2+y2≤1第20题
参考答案：
三、解答题：共70分。
解答应写出推理、演算步骤。
第21题
第22题
第23题
第24题
第25题
第26题
第27题
第28题。
成人高考 08-12年 真题(专升本)
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.
三、解答题：21~28 小题，共 70 分，解答应写出推理、演算步骤，并将其写在答题卡相应题号后。 21.（本题满分 8 分） 计算 lim x →3
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22.（本题满分 8 分） 设函数 y = x3 + sin x + 3 ，求 y ′
23.（本题满分 8 分） 计算 ∫ sin 5 xdx
24.（本题满分 8 分） 设 z= x( x, y ) 是由方程 x + y − ez =0 所确定的隐函数，求
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4. 已知 f ( x) 在区间（- ∞ ,+ ∞ ）内为单调减函数，且 f ( x) > f (1) ，则 x 的取值范围是（ A. (- ∞ ,-1) B. (- ∞ ,1) C. (1, ∞ ) ） D. (- ∞ ,+ ∞ )
2008年成人高考专升本高等数学真题
2008年成人高考专升本高等数学真题浇钢工题库一、填空题1、钢的生产过程主要分为炼钢和浇注两大环节。
2、钢水铸造有两种方法：一是钢锭浇注法，一是连续铸钢法。
3、将高温钢水直接浇注成钢坯的工艺就是连铸铸钢。
4、连铸机按外形可分为立式连铸机、立弯式连铸机、弧形连铸机、椭圆形连铸机、水平连铸机。
我公司目前的 4 机 4 流连铸机是弧形的。
5、钢包回转台由回转部分、固定部分、润滑系统和电控系统组成。
6、中间包是钢包与结晶器之间的中间贮存容器，它有贮钢、稳流、缓冲、分流和分渣的作用，是实现多炉连浇的基础。
7、我厂中间包容量是27吨。
钢水深度为850mm。
8、连铸耐火材料三大件是指：大包套管、塞棒和浸入式水口。
9、塞棒控制是通过塞棒控制机构控制塞棒上下运动，以达到关闭和开启水口调节钢水流量的目的。
10、管式结晶器由铜管、冷却水套、底脚板和足辊等组成。
11、结晶器内腔纵断面的尺寸做成上大下小，形成一个锥度。
12、钢水在结晶器中冷却，若结晶器没有锥度或锥度偏小，就会在坯壳和结晶器之间形成间隙，称气隙。
由于气隙的存在降低了冷却效果，同时由于坯壳过早地脱离了结晶器内壁，在钢水静压力下坯壳会产生鼓肚变形。
13、结晶器倒锥度过大会增加拉坯阻力，结晶器内壁磨损快，寿命短，同时还会形成坯料的凹陷、角裂等缺陷。
14、结晶器振动的目的是为了防止连铸坯在凝固过程中与铜管粘结而发生粘挂拉裂或拉漏事故，以保证拉坯顺利进行。
15、结晶器振动形式有以下几种：同步式、负滑脱式、正弦振动、非正弦振动。
16、负滑脱是指：当结晶器下振速度大于拉坯速度时，铸坯对结晶器的相对运动向上，即逆着拉坯方向运动，这种运动称负滑脱。
17、连铸坯的表面振痕深度与结晶器振动负滑脱时间有关，负滑脱时间越短，振痕深度就越浅。
18、2012年公司挖潜创效目标，质量异议万元产值损失率为小于等于 4 元/万元19、对于二冷区为弧形的连铸机，连铸坯出二冷区必须矫直，否则铸坯无法进行切割、运输、堆垛、以及轧制等后道工序。
08英语真题专升本试题高数（2）
08英语真题专升本试题高数（2）安徽省普通高等学校专升本考试2007年真题I. (词汇,每题1分,共35题)1.Five miles ______ a long distance to an old lady of her age.A. seem to beingB. seem to beC. seems to beingD. seems to be2.The computer doesn’t work well, so something ______ wrong.A. can have goneB. should have goneC. must have goneD. ought to have gone3. A number of students in this class ______ to Beijing once or twice.A. wereB. have beenC. isD. has been4. A good secretary should always be informed ______ the latest flight schedule.A. toB. inC. withD. of5.I got the good news that they ______ in planting trees on the sandy hills.A. have succeededB. had succeededC. succeedD. were succeeding6.Be sure to ______ your work at the end of the exam.A. look overB. hang upC. set asideD. catch sight of7.The football game will be played on ______.A. June sixB. six JuneC. the sixth of JuneD. the six of June8.The police investigation discovered that three young men were ______ in the robbery.A. caughtB. involvedC. connectedD. tightened9.It was in that small room ______ they worked hard and dreamed of better days to come.A. wha tB. in whichC. whichD. that10.The disk I bought from the store yesterday was found to be ______.A. emptyB. blankC. vacantD. hollow11.Please ______ the water tap when you have finished your washing.A. turn onB. turn offC. turn outD. turn over12.If you wish to accept the invitation, please send an email to ______ you acceptance.A. confirmB. confineC. confrontD. conform13.Don’t associate with bad boys ______ your whole life will be ruined.A. orB. butC. andD. so that14.______ George loves his daughter, he is strict with her.A. EvenB. ForC. AlthoughD. Whether15.In many cultures people insist ______ that the importance of being punctual.A. inB. overC. toD. on16.One difficulty has been solved. But another one will ______.A. ariseB. riseC. arouseD. arose17.The classroom is quite clean ______ some waste paper on the floor.A. except forB. besidesC. exceptD. without18.Hold the book ______ please, for I can’t see the words in it.A. more closeB. closerC. more closelyD. closely19.I shall have a companion in the house after all these ______ years.A. aloneB. lonelyC. singleD. simple20.He walked in the street without ______ to buy anything.A. intentionB. heartC. purposeD. thought21.You ______ if you had had higher score in the examination.A. must have got scholarshipB. would have got scholarshipC. should get scholarshipD. had got scholarship22.The disabled children need many things, but ______ , they need love.A. first of allB. above allC. after allD. all in all23.______ will come and lend us a helping hand on such an awkward occasion?A. Do you suppose whoB. Whom do you supposeC. Who do you thinkD. Whom do you think24.He didn’t tell me where he had spent his holidays. ______ to know.A. Neither I caredB. Not did I careC. I didn’t careD. Neither did I care25.I’m for the su ggestion that a special board ______ to examine the problem.A. be set upB. will be set upC. must be set upD. has to be set up26.While studying he financially depended ______ his wife.A. onB. ofC. toD. from27.______ his sister, Jack is quite and does not easily make friends with others.A. DislikeB. UnlikeC. AlikeD. Liking28.No sooner had he got home ______ it began to rain.A. whenB. thatC. thanD. while29.That T-shirt was so tight that he decided to have it ______.A. be enlargedB. enlargeC. enlargedD. to enlarge30.______ is known to the world, Mark Twain is a great American writer.A. ThatB. WhichC. AsD. It31. A lawyer’s income is usually very high, ______ more than 200,000 dollars a year.A. add up toB. added up toC. amounted toD. amounting to32.______, the students had to go to bed.A. The lights had gone outB. The lights having gone outC. The lights went outD. The lights have gone out33.Only in this way ______ out of the plan.A. you can talk himB. can you talk himC. him you can talkD. him can you talk34.Most parts of the country are ______ drought.A. facing atB. coming acrossC. threatened withD. surrounded by35.“The people are said to have been killed in the accident” could be rewritten as ______.A. It is said that ten people are killed in the accident.B. It was said that ten people were killed in the accident.C. It was said that ten people had been killed in the accident.D. It is said that ten people were killed in the accident.II. (完型填空,每题1分,共10题) (取自1991年6月四级真题)Geography is the study of the relationship between people and the land. Geographers compare and contrast various places on the earth. But they also go beyond the individual places and consider the earth as a 36 . The word geography 37 from two Greek words, ge, the Greek word for "earth" and graphein, 38 means "to write." The English word geography means "to describe the earth." Some geography books focus on a small area like a town or city.Others deal with a state, a region, a nation, or an 39 continent. Many geography books deal with the whole earth. Another way to divide the study of 40 is to distinguish between physical geography and cultural geography. The former focuses on the natural world; the 41 starts with human beings and studies how human beings and their environment act upon each other. But when geography is considered as a single subject, 42 branch can neglect the other.A geographer might be described 43 one who observes, records, and explains thedifferences between places. If places 44 alike, there would be little need for geographers.We know, however, 45 no two places are exactly the same. Geography, then, is a point of view, a special way of looking at places.36. (A) whole (B) unit (C) part (D) total37. (A) falls (B) removes (C) results (D) comes38. (A) what (B) that (C) which (D) it39. (A) extensive (B) enormous (C) overall (D) entire40. (A) world (B) earth (C) globe (D) geography41. (A) second (B) latter (C) next (D) later42. (A) either (B) neither (C) one (D) each43. (A) for (B) to (C) by (D) as44. (A) being (B) are (C) were (D) be45. (A) although (B) whether (C) since (D) thatIII. (阅读理解，前两篇每题2分，后两篇每题3分，共50分)Passage 1. （某些远程教育学校《大学英语》考试曾采用过此篇阅读理解，很简单）I remember quite clearly the first time I was really shaken by my wife’s intuition.We were looking for a second-hand car, and I had been much taken by a rather nice drop-head with a well-cared-for appearance.“Well, she runs nicely,” I told my life after the try-out. “Good start-up, engine sounds sweet, low mileage and the right price. What do you think?”“I don’t like it.”“Why not?”“I don’t know. There’s just something about the salesman that I don’t believe.”A few weeks after we had bought another car elsewhere, I happened to find out the history of thedrop-head. It had been seriously damaged in a crash before we saw it, and cleverly done up. The car was, indeed, a very bad risk.Of all the things about women that their husbands do not understand, perhaps the biggest question is this: How can she often be right about things of which she has absolutely no knowledge? How does a woman’s intuition work? What is this strange power that tells her exactly when her husband isn’t working late on business, however honest he pretends to be? Is it guesswork, or some sixth sense that men don’t have?46. When was the “first time” the husband felt “really shaken” by his wife’s intuition?A.When they bought another car.B.When he learned about the history of the drop-head..C.When he realized she knew so much about cars.D.When his wife spoke about the salesman.47. What does a person’s intuition refer to according to this passage?A.The ability to see through things.B.The ability to make a quick decision.C.The ability to pick out the best one among a number of things.D.The ability to know something by using o ne’s feelings.48. Why didn’t the man know that the car was damaged at first?A.The damage was not serious at all.B.He was not careful enough.C.It had been skillfully repaired.D.The salesman only tried it out smoothly.49. What answer does the writer give to a woman’s intuition?A.It is simply a blind guess. C. It must be the sixth sense.B.It is a special kind of knowledge. D. He doesn’t really answer the question.50. From the passage we learn that the salesman was _____.A. cheatingB. honestC. helpfulD. kindPassage 2. (出自上海教育出版社《大学英语四级阅读200篇》第12篇)Social customs and ways of behaving change. Things which were considered impolite many years ago are now acceptable. Just a few years ago, it was considered impolite behavior for a man to smoke on street. No man who thought of himself by smoking when a lady was in a room.Customs also differ from country to country. Does a man walk on the left or the right of a woman in your country? Or doesn't it matter? Should you use both hands when you areeating? Should leave one in your lap or on the table?The Americans and the British not only speak the same language but also share a large number of social customs. For example, in both America and England people shake hands when they meet each otherfor the first time. Also, most Englishmen will open a door for a woman or offer their seat to a woman, and so will most Americans. Promptness is important both in England and in America. That is , if a dinner invitation is for 7 o'clock , the dinner guest either arrives close to that time or calls up to explain his delay.The important thing to remember about social customs is not to do anything that might make other people feel uncomfortable-----especially if they are your guests. When the food was served, one of the guests started to eat his peas with a knife. The other guests were amused or shocked, but the host calmly picked up his knife and began eating in the same way. It would have been bad manners to make his guest feel foolish or uncomfortable.51.If one has accepted a dinner invitation ,what should he do if he is to be late for the dinner?A. He should find an excuseB. He should ask for excuse.C. He should say sorryD. He should telephone to explain his being late.52."It would have been bad manners to make his guests feel foolish or uncomfortable." "Bad manners " means ________.A. uglyB. dishonestC. impoliteD. shameful53.which of the following do you think is the best title for this passage?A. Social CustomsB. Social LifeC. American and British CustomsD. Promptness Is Important54.According to the text, the best host_______.A. tries his best to make his guests feel comfortableB. makes his guests feel excitedC. tries to avoid being naughty to his guestsD. tries to avoid being foolish55.The author of this article may agree with which of the following?A. The guest who ate his peas with a knifeB. The other guests who were amused or shockedC. The host who picked up his knife and began eating in the same wayD. None of the abovePassage 3. [ 某些高中（比如杭州二中）和某些大学大一英语曾用过的阅读，难度不大]Cancer is among the top killer diseases in our society today and scientists have found out that stress helps to bring it on. It is worthwhile to consider, therefore, what are the causes of stress in our life, and whether we can do anything about it.Are we under-employed or overburdened with too many responsibilities? Do we have a right balance of work and leisure(闲暇)in our lives? Are our relationships with family, friends or fellow workers all they should be?All these things can be a cause of stress, and it is best to facethem honestly, and to bring our frustrations( 失意)into the open.People who have a good row and then forget it are doing their health more good than those who bottle their feelings.If our self-examination has brought any causes of stress to light, let us consider what we can do about them. It is possible to change jobs. We can make more leisure and fill it more happily, if we will accept a different living standard. We can improve our personal relationships by a different attitude. It is we who allow other people to make ourselves unhappy. Often the little things that disturb us are not worth an hour’s anger. The teaching in the Bible “ Let not the sun go down upon your wrath(anger)” is good advice from the health point of view as well as the religious.56. Which of the following statements is true?A.Freedom from responsibilities helps reduce stressB.Stress is the direct cause of cancerC.The cause of stress is worthy of serious studyD.Cancer is the number one killer in our society today57. According to the passage, which of the following types of people is more likely to suffer fromstress?A.People who have cancerB.People who like to quarrel with othersC.People whose living standards is lowD.People who have more responsibilities than they can handle58. Judging from the context, the word “row” in the third paragraph most probably means____.A. a noisy quarrelB. a very loud noiseC. a neat line of things side by sideD. a journey in a boat59. Which of the following is not mentioned as a way to reduce our stress?A.changing our jobsB.changing our attitude to little things that make us unhappyC.speaking out our frustrationsD.reading the Bible60. “Let not the sun go down upon you wrath” probably means______.A. Don’t let your anger last longB. Don’t get angry while the sun is going downC. Don’t get angry easily about little things.D. Don’t watch the sun set while you are angry.Passage 4. [某些远程教育学校(比如四川大学网络教育学院)《大学英语3》考试题]Scientists have developed a slimming（减肥）drug that successfully suppresses appetite and results in a dramatic loss of weight without any apparent ill effects.The drug interferes with appetite control and prevents the build-up of fatty tissue. Laboratory mice given the drug lost up to a third of their total body weight.Within 20 minutes of being given the drug, called C75, the mice lost interest in eating and survived apparently content on just 10 per cent of the food they would normally eat.As a result, mice taking the drug lost 45 per cent more weight than mice fed the same amount of food, which compensate for the lack of food by becoming more lazy.Scientists, from the Johns Hopkins University in Baltimore, said that C75 is likely to produce a similar effect on humansbecause appetite control in brain is thought to be based on the same chemical pathways as those in mice.“We are not claiming to have found a fabled（神奇的）weight-loss drug. What we have found, using C75, is a major pathway in the brain that body uses naturally in regulating appetite at least in mice,” said Francis Kuhajda, a pathologist and senior team member.61.The newly-developed drug _______.A.controls appetite and causes obvious weight lossB.reduces weight with some ill-effectC.interferes with appetite and builds up fatty tissueD.is still going through laboratory tests62.One of the reasons that the drug helps to lose weight is that _______.A.it helps mice sleep wellB.it keeps fatty tissue from growingC.it controls eatingD.it prevents overeating63.The mice taking C75 lose nearly half of their weight on the same amount of food become _______.A.more activeB.more lazyC.more aggressiveD.more noisy64.C75 is likely to produce weight-loss effect on humans because ______.A.mice and humans have similar brain structureB.human’s are likely to be more lazy than miceC.human’s appetite is regulated through the same pathway as mice’sD.mice and human body use different pathways of appetite control65.What is the attitude of the scientists toward the slimming drug?A.indifferentB.excitingC.negativeD.confidentIV. (翻译英译汉前3题每题4分，第4题6分；汉译英前3题每题4分，第4题5分；共35分)66. Good teachers often encourage their students to think for themselves.67. Paper money is obviously easier to handle and much more convenient in the modern world.68. You must also become familiar with your teacher s’teaching styles; adjusting your learning style to theirs will be to your advantage.69. Educators think that the generation growing up with television spend so much of their time in front of the TV that they do not have enough time to study.70. 我们在做出任何决定之前，应该听一听其他人的意见。
2008年成人高考专升本高等数学真题
第 1 页 2008年成考专升本高等数学 33一、单项选择题(在每小题的四个备选答案中，选出一个正确答案，并将正确答案的序号填在题干的括号内。
每小题4分，共28分)1.设N 是自然数全体，则N ＝（ ）.A. nB. QC. Q -[0,1]D. R 2.连续势个可数集的并集是（ ）.A.有限集B.可数集C.连续势集D.无法确定 3.可数个开集的交集是（ ）.A.开集B.闭集C.F σ集D.G δ集4.设｛A n ｝是一集列，则n A ∞→n lim （ ）. A.n n A ∞=1B. n n A ∞=1C. m n m n A ∞=∞= 1D. m n m n A ∞=∞= 15.设f (x )是［0,1］上的连续函数，则下述哪个说法不．成立？（ ） A. f (x ) a.e.有有限导数 B. f (x )是可测函数C. f (x ) Riemann 可积D. f (x ) Lebesgue 可积6.关于Cantor 集P ，下述哪个说法不．成立？（ ） A.P 无内点B.P 的测度为0C.P 由可数个闭区间组成D.P 是完备集 7.设函数列｛f n (x )｝处处收敛于有限函数f (x )，则（ ）.A. f n (x )⇒ f (x )B.{f n (x )}一致收敛于f (x )C. f n (x )→f (x ) a.e.D.无法判定二、填空题(每小题4分，共40分)1.设E 是［0，1］中无理数全体，则mE ＝_____________.2.设A n ＝（－1－n 1,1－n1）,则n n A ∞=1 ＝_____________.3.设A ，B ⊂R ,则具有关系(A ∪B)°_____________A °∪B °.第 2 页 4.设f (x )是［0，1］上的单调函数，E 是f (x )的连续点全体，则mE =_____________.5.可测集与F σ型集有如下关系：F σ型集是可测集，反之_____________.6.设D （x ）=⎩⎨⎧∈∈QQ -[0,1]10x x ，则D （x ）在21处的振幅为_____________. 7.设A ＝R p , B =R q ，则当x ∈A 时(A ×B )x =_____________.8.设f (x )=x 2 -1≤x ≤1，则10V （f ）=_____________. 9.设f (x )是E 上的可测函数，则E ［| f | =+∞］是_____________.10.设｛f n (x )｝是E 上的非负可测函数列，则x x f n E )d (lim n ⎰∞→_____________⎰∞→E n x x f )d (lim n . 三、完成下列各题(每小题8分，共32分)1.证明闭集的余集是开集.2.设E ⊂R n ，存在可测集｛B n ｝，使得E ⊂B n 且m (B n －E )→0(n →∞),试证明E 可测.3.证明对R 上的任意a.e.有限的可测函数f (x )，一定存在R 上的连续函数列｛f n (x )｝使 f n (x )→f (x )a.e.于R .4.设［0，1］上的函数f (x )在Cantor 集P 0上定义为0，在Cantor 集P 0的余集中长度为n 31的构成区间上定义为n(n =1,2,3,…)，试求f (x )在［0，1］上的积分值.。