2012考研数学一大纲

Pushing ahead, East, North, West development, facilitating shanty, complete xincheng West Road, stability and reconstruction of the street, Star Ding garden built century Garden District, subject public housing construction, heating the gas expansion accelerated implementation, Jing Rui River and bridge style line solid shaft, further expanded the town development. Beautiful country adhere to the advance and typical for a group to lead. According to \cal lay based, and industry increased income, and live upgrade civilization, and culture rich connotation\ht, caught built beautiful village, and whole village advance, and easily relocation, and \d three built\oject near 30 a; depth into ecological, and industry, and culture, and tourism, elements, high standards caught party original city Liu, and jade are Xia AO, model village, in province City set has benchmark. Four is equal benefits to promote sharing, coordinated development of various social undertakings. In the case of rigid spending slowing revenue growth, increase, financial 75% to the people's livelihood, the masses have more access in the sharing. Cuoban something good and benefit. Adhere to the affordable, best, carefully cuoban village oil, water and heat, cultural, city and County columns practical, developments benefit families, bring more welfare to the masses. Development of social convenience. Acceptance of balanced development of compulsory education through the national assessment, the County vocational education center was named \onal education in China\batch of national reform and development model through provincial ... Five is to deepen reform and innovation to break bottlenecks and endogenous development of effective fire. As an engine of dynamism, adding reform key to adopting reform measures stress energy and break bottlenecks and promoting development. Pay close attention to the key areas of reform. Conscientiously implement the provincial deployment and cancel, adjustment, to undertake the administrative approval items according to Regulation 347 to further improve administrative efficiency continuing deepening reform for poverty alleviation, speeding up rural land right registration certification pilot project, encouraging the orderly transfer of land management, utilize the land resources, in order to promote scale management laid the Foundation. Actively cultivate the market. Firmly establish a market-oriented, nurturing family farms, industrial users, professional cooperation organization new market management main body 456, effectively improve the industry, intensive and marketization of encouraging support social capital into the small micro-credit, urban developmentculture and tourism, public services and other areas, easing development shortfall. To promote mass innovation. To support the rural sector to use new thinking and new ideas, creative work incentives such as actively promoting financial intermediation, bring added 159 in non public enterprises, individually-1540, created a2012考研数学一大纲

考试科目：高等数学、线性代数、概率论与数理统计 考试形式和试卷结构

一、试卷满分及考试时间

试卷满分为150 分，考试时间为180 分钟. 二、答题方式

答题方式为闭卷、笔试. 三、试卷内容结构 高等教学56% 线性代数22%

概率论与数理统计22% 四、试卷题型结构 试卷题型结构为：

单选题8 小题，每题4 分，共32 分 填空题6 小题，每题4 分，共24 分 解答题(包括证明题) 9 小题，共94 分 高等数学

一、函数、极限、连续 考试内容

函数的概念及表示法函数的有界性、单调性、周期性和奇

偶性复合函数、反函数、分段函数和隐函数基本初等函数的性质 及其图形初等函数函数关系的建立

数列极限与函数极限的定义及其性质函数的左极限与右

极限无穷小量和无穷大量的概念及其关系无穷小量的性质及无 穷小量的比较极限的四则运算极限存在的两个准则：单调有界准

函数连续的概念函数间断点的类型初等函数的连续性闭区间上连续函数的性质 考试要求

1.理解函数的概念，掌握函数的表示法，会建立应用问题的函数关系. 2.了解函数的有界性、单调性、周期性和奇偶性.

3.理解复合函数及分段函数的概念，了解反函数及隐函数的概念. 4.掌握基本初等函数的性质及其图形，了解初等函数的概念.

5.理解极限的概念，理解函数左极限与右极限的概念以及函数极限存在与左、右极限之间的关系。.

6.掌握极限的性质及四则运算法则。

7.掌握极限存在的两个准则，并会利用它们求极限，掌握利用两个重要极限求极限的方法。 8.理解无穷小量、无穷大量的概念，掌握无穷小量的比较方法，会用等价无穷小量求极限。 9.理解函数连续性的概念(含左连续与右连续)，会判别函数间断点的类型。 10.了解连续函数的性质和初等函数的连续性，理解闭区间

上连续函数的性质(有界性、最大值和最小值定理、介值定理)，并会应用这些性质。 二、一元函数微分学 考试内容

导数和微分的概念导数的几何意义和物理意义函数的

可导性与连续性之间的关系平面曲线的切线和法线导数和微分的四则运算基本初等函数的

Pushing ahead, East, North, West development, facilitating shanty, complete xincheng West Road, stability and reconstruction of the street, Star Ding garden built century Garden District, subject public housing construction, heating the gas expansion accelerated implementation, Jing Rui River and bridge style line solid shaft, further expanded the town development. Beautiful country adhere to the advance and typical for a group to lead. According to \cal lay based, and industry increased income, and live upgrade civilization, and culture rich connotation\ht, caught built beautiful village, and whole village advance, and easily relocation, and \d three built\oject near 30 a; depth into ecological, and industry, and culture, and tourism, elements, high standards caught party original city Liu, and jade are Xia AO, model village, in province City set has benchmark. Four is equal benefits to promote sharing, coordinated development of various social undertakings. In the case of rigid spending slowing revenue growth, increase, financial 75% to the people's livelihood, the masses have more access in the sharing. Cuoban something good and benefit. Adhere to the affordable, best, carefully cuoban village oil, water and heat, cultural, city and County columns practical, developments benefit families, bring more welfare to the masses. Development of social convenience. Acceptance of balanced development of compulsory education through the national assessment, the County vocational education center was named \onal education in China\batch of national reform and development model through provincial ... Five is to deepen reform and innovation to break bottlenecks and endogenous development of effective fire. As an engine of dynamism, adding reform key to adopting reform measures stress energy and break bottlenecks and promoting development. Pay close attention to the key areas of reform. Conscientiously implement the provincial deployment and cancel, adjustment, to undertake the administrative approval items according to Regulation 347 to further improve administrative efficiency continuing deepening reform for poverty alleviation, speeding up rural land right registration certification pilot project, encouraging the orderly transfer of land management, utilize the land resources, in order to promote scale management laid the Foundation. Actively cultivate the market. Firmly establish a market-oriented, nurturing family farms, industrial users, professional cooperation organization new market management main body 456, effectively improve the industry, intensive and marketization of encouraging support social capital into the small micro-credit, urban developmentculture and tourism, public services and other areas, easing development shortfall. To promote mass innovation. To support the rural sector to use new thinking and new ideas, creative work incentives such as actively promoting financial intermediation, bring added 159 in non public enterprises, individually-1540, created a导数复合函数、反函数、隐函数以及数的极值函数图形的凹凸性、拐点及渐近线函数图形的描绘函数的最大值和最小值弧微分曲率的概念曲率圆与曲率半径。 考试要求

1.理解导数和微分的概念，理解导数与微分的关系，理解导

数的几何意义，会求平面曲线的切线方程和法线方程，了解导数的物理意义，会用导数描述一些物理量，理解函数的可导性与连续性之间的关系.

2.掌握导数的四则运算法则和复合函数的求导法则，掌握基

本初等函数的导数公式.了解微分的四则运算法则和一阶微分形式的不变性，会求函数的微分.

3.了解高阶导数的概念，会求简单函数的高阶导数.

4.会求分段函数的导数，会求隐函数和由参数方程所确定的函数以及反函数的导数.

5.理解并会用罗尔(Rolle)定理、拉格朗日(Lagrange)中值定理和泰勒(Taylor)定理，了解并会用柯西(Cauchy)中值定理.

6.掌握用洛必达法则求未定式极限的方法.

7.理解函数的极值概念，掌握用导数判断函数的单调性和求函数极值的方法，掌握函数最大值和最小值的求法及其应用.

8.会用导数判断函数图形的凹凸性(注：在区间内，设函数具有二阶导数。当时， 的图形是凹的;当时， 的图形是凸的)，会求函数图形的拐点以及水平、铅直和斜渐近线，会描绘函数的图形.

9.了解曲率、曲率圆与曲率半径的概念，会计算曲率和曲率半径. 三、一元函数积分学 考试内容

原函数和不定积分的概念不定积分的基本性质基本积分公式定积分的概念和基本性质定积分中值定理积分上限的和定积分的换元积分法与分部积分法有理函数、三角函数的有理式 和简单无理函数的积分反常(广义)积分定积分的应用 考试要求

1.理解原函数的概念，理解不定积分和定积分的概念.

2.掌握不定积分的基本公式，掌握不定积分和定积分的性质及定积分中值定理，掌握换元积分法与分部积分法.

3.会求有理函数、三角函数有理式和简单无理函数的积分.

4.理解积分上限的函数，会求它的导数，掌握牛顿-莱布尼茨公式. 5.了解反常积分的概念，会计算反常积分.

6.掌握用定积分表达和计算一些几何量与物理量(平面图形的面积、平面曲线的弧长、旋转体的体积及侧面积、平行截面面积为已知的立体体积、功、引力、压力、质心、形心等)及函数的平均值四、向量代数和空间解析几何 考试内容

向量的概念向量的线性运算向量的数量积和向量积向量的混合积两向量垂直、平行的条件两向量的夹角向量的坐标表达式及其运算单位向量方向数与方向余弦曲面方程和空间曲线方程的概念平面方程、直线方程平面与平面、平面与直线直线与直线的夹角以及平行、垂直的条件点到平面和点到直线的距离球面柱面旋转曲面常用的二次曲面方程及其图形空间曲线的参数方程和一般方程空间曲线在坐标面上的投影曲线方程 考试要求

1.理解空间直角坐标系，理解向量的概念及其表示.

2.掌握向量的运算(线性运算、数量积、向量积、混合积)，了解两个向量垂直、平行的条件.

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Pushing ahead, East, North, West development, facilitating shanty, complete xincheng West Road, stability and reconstruction of the street, Star Ding garden built century Garden District, subject public housing construction, heating the gas expansion accelerated implementation, Jing Rui River and bridge style line solid shaft, further expanded the town development. Beautiful country adhere to the advance and typical for a group to lead. According to \cal lay based, and industry increased income, and live upgrade civilization, and culture rich connotation\ht, caught built beautiful village, and whole village dvance, and easily relocation, and \d three built\oject near 30 a; depth into ecological, and industry, and culture, and tourism, elements, high standards caught party original city Liu, and jade are Xia AO, model village, in province City set has benchmark. Four is equal benefits to promote sharing, coordinated development of various social undertakings. In the case of rigid spending slowing revenue growth, increase, financial 75% to the people's livelihood, the masses have more access in the sharing. Cuoban something good and benefit. Adhere to the affordable, best, carefully cuoban village oil, water and heat, cultural, city and County columns practical, developments benefit families, bring more welfare to the masses. Development of social convenience. Acceptance of balanced development of compulsory education through the national assessment, the County vocational education center was named \onal education in China\batch of national reform and development model through provincial ... Five is to deepen reform and innovation to break bottlenecks and endogenous development of effective fire. As an engine of dynamism, adding reform key to adopting reform measures stress energy and break bottlenecks and promoting development. Pay close attention to the key areas of reform. Conscientiously implement the provincial deployment and cancel, adjustment, to undertake the administrative approval items according to Regulation 347 to further improve administrative efficiency continuing deepening reform for poverty alleviation, speeding up rural land right registration certification pilot project, encouraging the orderly transfer of land management, utilize the land resources, in order to promote scale management laid the Foundation. Actively cultivate the market. Firmly establish a market-oriented, nurturing family farms, industrial users, professional cooperation organization new market management main body 456, effectively improve the industry, intensive and marketization of encouraging support social capital into the small micro-credit, urban developmentculture and tourism, public services and other areas, easing development shortfall. To promote mass innovation. To support the rural sector to use new thinking and new ideas, creative work incentives such as actively promoting financial intermediation, bring added 159 in non public enterprises, individually-1540, created a3.理解单位向量、方向数与方向余弦、向量的坐标表达式，

5.会求平面与平面、平面与直线、直线与直线之间的夹角，并会利用平面、直线的相互关系(平行、垂直、相交等)解决有关问题6.会求点到直线以及点到平面的距离. 7.了解曲面方程和空间曲线方程的概念.

8.了解常用二次曲面的方程及其图形，会求简单的柱面和旋转曲面的方程. 9.了解空间曲线的参数方程和一般方程.了解空间曲线在坐标平面上的投影，并会求该投影曲线的方程.

五、多元函数微分学 考试内容

多元函数的概念二元函数的几何意义二元函数的极限与连续的概念有界闭区域上多元连续函数的性质多元函数的偏导数和全微分全微分存在的必要条件和充分条件多元复合函数、隐 函数的求导法二阶偏导数方向导数和梯度空间曲线的切线和法平面曲面的切平面和法线二元函数的二阶泰勒公式多元函数的极值和条件极值多元函数的最大值、最小值及其简单应用 考试要求

1.理解多元函数的概念，理解二元函数的几何意义.

2.了解二元函数的极限与连续的概念以及有界闭区域上连续函数的性质.

3.理解多元函数偏导数和全微分的概念，会求全微分，了解全微分存在的必要条件和充分条件，了解全微分形式的不变性.

4.理解方向导数与梯度的概念，并掌握其计算方法. 5.掌握多元复合函数一阶、二阶偏导数的求法.

6.了解隐函数存在定理，会求多元隐函数的偏导数.

7.了解空间曲线的切线和法平面及曲面的切平面和法线的概念，会求它们的方程.

9.理解多元函数极值和条件极值的概念，掌握多元函数极值存在的必要条件，了解二元函数极值存在的充分条件，会求二元函数的极值，会用拉格朗日乘数法求条件极值，会求简单多元函数的最大值和最小值，并会解决一些简单的应用问题. 六、多元函数积分学 考试内容

二重积分与三重积分的概念、性质、计算和应用两类曲线

积分的概念、性质及计算两类曲线积分的关系格林(Green)公式 平面曲线积分与路径无关的条件二元函数全微分的原函数两类 曲面积分的概念、性质及计算两类曲面积分的关系高斯(Gauss) 公式斯托克斯(Stokes)公式散度、旋度的概念及计算曲线积分 和曲面积分的应用 考试要求

1.理解二重积分、三重积分的概念，了解重积分的性质，了解二重积分的中值定理.

2.掌握二重积分的计算方法(直角坐标、极坐标)，会计算三重积分(直角坐标、柱面坐标、球面坐标).

3.理解两类曲线积分的概念，了解两类曲线积分的性质及两类曲线积分的关系. 4.掌握计算两类曲线积分的方法.

5.掌握格林公式并会运用平面曲线积分与路径无关的条件，会求二元函数全微分的原函数. 6.了解两类曲面积分的概念、性质及两类曲面积分的关系， 掌握计算两类曲面积分的方法，掌握用高斯公式计算曲面积分的方法，并会用斯托克斯公式计算曲线积分.

7.了解散度与旋度的概念，并会计算.

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Pushing ahead, East, North, West development, facilitating shanty, complete xincheng West Road, stability and reconstruction of the street, Star Ding garden built century Garden District, subject public housing construction, heating the gas expansion accelerated implementation, Jing Rui River and bridge style line solid shaft, further expanded the town development. Beautiful country adhere to the advance and typical for a group to lead. According to \cal lay based, and industry increased income, and live upgrade civilization, and culture rich connotation\ht, caught built beautiful village, and whole village advance, and easily relocation, and \d three built\oject near 30 a; depth into ecological, and industry, and culture, and tourism, elements, high standards caught party original city Liu, and jade are Xia AO, model village, in province City set has benchmark. Four is equal benefits to promote sharing, coordinated development of various social undertakings. In the case of rigid spending slowing revenue growth, increase, financial 75% to the people's livelihood, the masses have more access in the sharing. Cuoban something good and benefit. Adhere to the affordable, best, carefully cuoban village oil, water and heat, cultural, city and County columns practical, developments benefit families, bring more welfare to the masses. Development of social convenience. Acceptance of balanced development of compulsory education through the national assessment, the County vocational education center was named \onal education in China\batch of national reform and development model through provincial ... Five is to deepen reform and innovation to break bottlenecks and endogenous development of effective fire. As an engine of dynamism, adding reform key to adopting reform measures stress energy and break bottlenecks and promoting development. Pay close attention to the key areas of reform. Conscientiously implement the provincial deployment and cancel, adjustment, to undertake the administrative approval items according to Regulation 347 to further improve administrative efficiency continuing deepening reform for poverty alleviation, speeding up rural land right registration certification pilot project, encouraging the orderly transfer of land management, utilize the land resources, in order to promote scale management laid the Foundation. Actively cultivate the market. Firmly establish a market-oriented, nurturing family farms, industrial users, professional cooperation organization new market management main body 456, effectively improve the industry, intensive and marketization of encouraging support social capital into the small micro-credit, urban developmentculture and tourism, public services and other areas, easing development shortfall. To promote mass innovation. To support the rural sector to use new thinking and new ideas, creative work incentives such as actively promoting financial intermediation, bring added 159 in non public enterprises, individually-1540, created a8.会用重积分、曲线积分及曲面积分求一些几何量与物理量 七、无穷级数 考试内容

常数项级数的收敛与发散的概念收敛级数的和的概念

级数的基本性质与收敛的必要条件几何级数与级数及其收敛性 正项级数收敛性的判别法交错级数与莱布尼茨定理任意项级数 的绝对收敛与条件收敛函数项级数的收敛域与和函数的概念幂 级数及其收敛半径、收敛区间(指开区间)和收敛域幂级数的和函数 幂级数在其收敛区间内的基本性质简单幂级数的和函数的求法初 等函数的幂级数展开式函数的傅里叶(Fourier)系数与傅里叶级数 狄利克雷(Dirichlet)定理函数在上的傅里叶级数函数在上的 正弦级数和余弦级数 考试要求

1.理解常数项级数收敛、发散以及收敛级数的和的概念，掌握级数的基本性质及收敛的必要条件.

2.掌握几何级数与级数的收敛与发散的条件.

3.掌握正项级数收敛性的比较判别法和比值判别法，会用根值判别法. 4.掌握交错级数的莱布尼茨判别法.

5. 了解任意项级数绝对收敛与条件收敛的概念以及绝对收敛与收敛的关系. 6.了解函数项级数的收敛域及和函数的概念.

7.理解幂级数收敛半径的概念、并掌握幂级数的收敛半径、收敛区间及收敛域的求法. 8.了解幂级数在其收敛区间内的基本性质(和函数的连续

性、逐项求导和逐项积分)，会求一些幂级数在收敛区间内的和函数并会由此求出某些数项级数的和.

9.了解函数展开为泰勒级数的充分必要条件.

11.了解傅里叶级数的概念和狄利克雷收敛定理，会将定义在上的函数展开为傅里叶级数，会将定义在上的函数展开为正弦级数与余弦级数，会写出傅里叶级数的和函数的表达式. 八、常微分方程 考试内容

常微分方程的基本概念变量可分离的微分方程齐次微分方程一阶线性微分方程伯努利(Bernoulli)方程全微分方程可用简单的变量代换求解的某些微分方程可降阶的高阶微分方程 线性微分方程解的性质及解的结构定理二阶常系数齐次线性微分方程高于二阶的某些常系数齐次线性微分方程简单的二阶常系数非齐次线性微分方程欧拉(Euler)方程微分方程的简单应用 考试要求

1.了解微分方程及其阶、解、通解、初始条件和特解等概念2.掌握变量可分离的微分方程及一阶线性微分方程的解法.

3.会解齐次微分方程、伯努利方程和全微分方程，会用简单的变量代换解某些微分方程. 4.会用降阶法解下列形式的微分方程： . 5.理解线性微分方程解的性质及解的结构. 6.掌握二阶常系数齐次线性微分方程的解法，并会解某些高于二阶的常系数齐次线性微分方程.

7.会解自由项为多项式、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非齐次线性微分方程.

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Pushing ahead, East, North, West development, facilitating shanty, complete xincheng West Road, stability and reconstruction of the street, Star Ding garden built century Garden District, subject public housing construction, heating the gas expansion accelerated implementation, Jing Rui River and bridge style line solid shaft, further expanded the town development. Beautiful country adhere to the advance and typical for a group to lead. According to \cal lay based, and industry increased income, and live upgrade civilization, and culture rich connotation\ht, caught built beautiful village, and whole village dvance, and easily relocation, and \d three built\oject near 30 a; depth into ecological, and industry, and culture, and tourism, elements, high standards caught party original city Liu, and jade are Xia AO, model village, in province City set has benchmark. Four is equal benefits to promote sharing, coordinated development of various social undertakings. In the case of rigid spending slowing revenue growth, increase, financial 75% to the people's livelihood, the masses have more access in the sharing. Cuoban something good and benefit. Adhere to the affordable, best, carefully cuoban village oil, water and heat, cultural, city and County columns practical, developments benefit families, bring more welfare to the masses. Development of social convenience. Acceptance of balanced development of compulsory education through the national assessment, the County vocational education center was named \onal education in China\batch of national reform and development model through provincial ... Five is to deepen reform and innovation to break bottlenecks and endogenous development of effective fire. As an engine of dynamism, adding reform key to adopting reform measures stress energy and break bottlenecks and promoting development. Pay close attention to the key areas of reform. Conscientiously implement the provincial deployment and cancel, adjustment, to undertake the administrative approval items according to Regulation 347 to further improve administrative efficiency continuing deepening reform for poverty alleviation, speeding up rural land right registration certification pilot project, encouraging the orderly transfer of land management, utilize the land resources, in order to promote scale management laid the Foundation. Actively cultivate the market. Firmly establish a market-oriented, nurturing family farms, industrial users, professional cooperation organization new market management main body 456, effectively improve the industry, intensive and marketization of encouraging support social capital into the small micro-credit, urban developmentculture and tourism, public services and other areas, easing development shortfall. To promote mass innovation. To support the rural sector to use new thinking and new ideas, creative work incentives such as actively promoting financial intermediation, bring added 159 in non public enterprises, individually-1540, created a8.会解欧拉方程.

9.会用微分方程解决一些简单的应用问题.线性代数 一、行列式 考试要求：

1.了解行列式的概念，掌握行列式的性质.

2.会应用行列式的性质和行列式按行(列)展开定理计算行列式. 二、矩阵 考试内容

矩阵的概念矩阵的线性运算矩阵的乘法方阵的幂方阵乘积的行列式矩阵的转置逆矩阵的概念和性质矩阵可逆的充分必要条件伴随矩阵矩阵的初等变换初等矩阵矩阵的秩矩阵的等价分块矩阵及其运算 考试要求

1.理解矩阵的概念，了解单位矩阵、数量矩阵、对角矩阵、三角矩阵、对称矩阵和反对称矩阵，以及它们的性质.

2.掌握矩阵的线性运算、乘法、转置以及它们的运算规律，了解方阵的幂与方阵乘积的行列式的性质.

3.理解逆矩阵的概念，掌握逆矩阵的性质，以及矩阵可逆的充分必要条件，理解伴随矩阵的概念，会用伴随矩阵求逆矩阵. 4.理解矩阵初等变换的概念，了解初等矩阵的性质和矩阵等价的概念，理解矩阵的秩的概念，掌握用初等变换求矩阵的秩和逆矩阵的方法. 5.了解分块矩阵及其运算. 三、向量 考试内容

向量的概念向量的线性组合与线性表示向量组的线性相关与线性无关向量组的极大线性无关组等价向量组向量组的秩向量组的秩与矩阵的秩之间的关系向量空间及其相关概念维向量空间的基变换和坐标变换过渡矩阵向量的内积线性无关向量组的 2011考研数学一大纲

1.理解维向量、向量的线性组合与线性表示的概念.

2.理解向量组线性相关、线性无关的概念，掌握向量组线性相关、线性无关的有关性质及判别法.

3.理解向量组的极大线性无关组和向量组的秩的概念，会求向量组的极大线性无关组及秩. 4.理解向量组等价的概念，理解矩阵的秩与其行(列)向量组的秩之间的关系.

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