2023考研大纲：大连理工大学2023年考研单独考试 数学 考试大纲

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大连理工大学2023年单独考试硕士研究生入学考试大纲

数学

单考“数学”试题分为客观题型和主观题型，具体复习大纲如下：

一、函数、极限、连续

理解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。

类型<Object: word/embeddings/oleObject1.bin>

理解并掌握无穷小和无穷大的概念及其关系、无穷小的性质及无穷小的比较。

当<Object: word/embeddings/oleObject2.bin>时 <Object: word/embeddings/oleObject3.bin>

求极限的方法：熟练理解并掌握极限的四则运算、极限存在的单调有界准则和夹逼准则、两个重要极限

利用连续性

两个重要极限<Object: word/embeddings/oleObject4.bin>

无穷小等价代换

当<Object: word/embeddings/oleObject5.bin>时 <Object: word/embeddings/oleObject6.bin> <Object: word/embeddings/oleObject7.bin>

“<Object: word/embeddings/oleObject8.bin>”型 <Object: word/embeddings/oleObject9.bin> 利用重要极限式指数化

<Object: word/embeddings/oleObject10.bin>

有理函数<Object: word/embeddings/oleObject11.bin>极限（<Object: word/embeddings/oleObject12.bin>）

4.理解函数的连续性（含左连续与右连续）、会求函数间断点的类型。

类型<Object: word/embeddings/oleObject13.bin>

理解续函数的性质和初等函数的连续性，能判断分段函数的连续性。

1．定义：如果<Object: word/embeddings/oleObject14.bin>那么就称函数<Object: word/embeddings/oleObject15.bin>在点<Object: word/embeddings/oleObject16.bin>连续。<Object: word/embeddings/oleObject17.bin>

2．主要条件：<Object: word/embeddings/oleObject18.bin>（由此可求两个参数）

熟练理解并掌握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、零点定理)。

二、一元函数微分学

1. 理解导数和微分的概念、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、掌握平面曲线的切线和法线方程的计算方法。

导数定义：<Object: word/embeddings/oleObject19.bin>=<Object: word/embeddings/oleObject20.bin>，

<Object: word/embeddings/oleObject21.bin> 和 <Object: word/embeddings/oleObject22.bin>

<Object: word/embeddings/oleObject23.bin> <Object: word/embeddings/oleObject24.bin>

可导必连续，连续未必可导

2. 掌握基本初等函数的导数、导数和微分的四则运算、一阶微分形式的不变性。

初等函数求导公式（16个求导公式，5个求导法则）

（1）<Object: word/embeddings/oleObject55.bin>

（2）<Object: word/embeddings/oleObject56.bin> ， <Object: word/embeddings/oleObject57.bin>

（3）。<Object: word/embeddings/oleObject58.bin> <Object: word/embeddings/oleObject59.bin>

(4) 复合函数导数<Object: word/embeddings/oleObject60.bin>，<Object: word/embeddings/oleObject61.bin>称为中间变量，<Object: word/embeddings/oleObject62.bin> （5）<Object: word/embeddings/oleObject63.bin>；参数方程求二阶导数<Object: word/embeddings/oleObject64.bin> <Object: word/embeddings/oleObject65.bin>，

<Object: word/embeddings/oleObject66.bin> <Object: word/embeddings/oleObject67.bin>

3. 熟练掌握复合函数、反函数、隐函数以及参数方程所确定的函数的微分法。

例如：隐函数求二阶导数：F(x,y)=0 y=y(x),方程两边对x求导，y的函数看成x的复合函数

4. 理解高阶导数的概念并会计算分段函数的二阶导数、某些简单函数的n阶导数。

5. 熟练理解并掌握微分中值定理，包括罗尔定理、拉格朗日中值定理。

6. 熟练理解并掌握利用洛必达(L’Hospital)法则与求未定式极限。

例如：洛必达法则：“<Object: word/embeddings/oleObject68.bin>，<Object: word/embeddings/oleObject69.bin>”型 <Object: word/embeddings/oleObject70.bin>

7. 理解函数的极值并会利用导数判别函数单调性、函数图形的凹凸性、拐点及渐近线(水平、铅直和斜渐近线)。

1．方法：利用最值，单调性证不等式

单调性：单调升：<Object: word/embeddings/oleObject71.bin>，当<Object: word/embeddings/oleObject72.bin>时

单调降：<Object: word/embeddings/oleObject73.bin>，当<Object: word/embeddings/oleObject74.bin>时

<Object: word/embeddings/oleObject75.bin>，<Object: word/embeddings/oleObject76.bin>单调升，<Object: word/embeddings/oleObject77.bin>，<Object: word/embeddings/oleObject78.bin>单调降

利用单调性证不等式，证<Object: word/embeddings/oleObject79.bin>，<Object: word/embeddings/oleObject80.bin>，<Object: word/embeddings/oleObject81.bin>

2．求导时最多到二阶

8. 理解函数最大值和最小值并掌握其简单应用。

三、一元函数积分学

理解原函数和不定积分的概念.

1．原函数：在区间上，若<Object: word/embeddings/oleObject82.bin>，称为的一个原函数。

2．不定积分：在区间<Object: word/embeddings/oleObject83.bin>上，<Object: word/embeddings/oleObject84.bin>的原函数的全体称为<Object: word/embeddings/oleObject85.bin>的不定积分，记为<Object: word/embeddings/oleObject86.bin>

理解不定积分的基本性质、基本积分公式.

① <Object: word/embeddings/oleObject87.bin>是常数) ② <Object: word/embeddings/oleObject88.bin>

③ <Object: word/embeddings/oleObject89.bin>, ④ <Object: word/embeddings/oleObject90.bin>

⑤ <Object: word/embeddings/oleObject91.bin> ⑥ <Object: word/embeddings/oleObject92.bin>

⑦ <Object: word/embeddings/oleObject93.bin> ⑧ <Object: word/embeddings/oleObject94.bin>

⑨ <Object: word/embeddings/oleObject95.bin> ⑩ <Object: word/embeddings/oleObject96.bin>

(11)<Object: word/embeddings/oleObject97.bin> (12)<Object: word/embeddings/oleObject98.bin>

(13)<Object: word/embeddings/oleObject99.bin>

理解定积分的概念和基本性质，掌握定积分中值定理、理解变上限定积分确定的函数并会求其导数、掌握牛顿-莱布尼茨(Newton-Leibniz)公式.

例如：<Object: word/embeddings/oleObject100.bin>

<Object: word/embeddings/oleObject101.bin>，<Object: word/embeddings/oleObject102.bin>

掌握不定积分和定积分的换元积分法与分部积分法.

凑分法：<Object: word/embeddings/oleObject103.bin>

掌握下列常用凑分法

（1）<Object: word/embeddings/oleObject104.bin>

（2）<Object: word/embeddings/oleObject105.bin>

（3）<Object: word/embeddings/oleObject106.bin>

分布积分法：<Object: word/embeddings/oleObject107.bin> <Object: word/embeddings/oleObject108.bin>

掌握（1）<Object: word/embeddings/oleObject109.bin>

（2）<Object: word/embeddings/oleObject110.bin>

（3）<Object: word/embeddings/oleObject111.bin>

（4）<Object: word/embeddings/oleObject112.bin>

<Object: word/embeddings/oleObject113.bin>

简化计算的技巧

例如：（1）1）若<Object: word/embeddings/oleObject114.bin>在<Object: word/embeddings/oleObject115.bin>上连续且为偶函数，则 <Object: word/embeddings/oleObject116.bin>

2）若<Object: word/embeddings/oleObject117.bin>在<Object: word/embeddings/oleObject118.bin>上连续且为奇函数，则 <Object: word/embeddings/oleObject119.bin>

2）<Object: word/embeddings/oleObject120.bin> <Object: word/embeddings/oleObject121.bin>

3）换元法（结合凑微分法）

掌握有理函数、三角函数的有理式和简单无理函数的积分.

熟练掌握利用定积分计算平面图形的面积、平面曲线的弧长、旋转体的体积．

四.常微分方程

理解常微分方程的基本概念：微分方程及其解、阶、通解、初始条件和特解等。

熟练掌握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)方程的计算方法。

例如：形式<Object: word/embeddings/oleObject122.bin> 通解：<Object: word/embeddings/oleObject123.bin>

理解线性微分方程解的性质及解的结构定理.

掌握二阶常系数齐次线性的计算方法。

例如：二阶常系数线性齐次方程通解。标准型<Object: word/embeddings/oleObject124.bin>，其中常数。

解法：特征方程：<Object: word/embeddings/oleObject125.bin>，特征根<Object: word/embeddings/oleObject126.bin>

通解<Object: word/embeddings/oleObject127.bin>

熟练理解并掌握简单的二阶常系数非齐次线性微分方程：自由项为多项式、指数函数，以及它们的和与积的计算方法。

例如：二阶常系数线性非齐次方程通解。标准型<Object: word/embeddings/oleObject128.bin>，其中<Object: word/embeddings/oleObject129.bin>常数 <Object: word/embeddings/oleObject130.bin>，<Object: word/embeddings/oleObject131.bin>

解法：通解<Object: word/embeddings/oleObject132.bin>，其中<Object: word/embeddings/oleObject133.bin>为对应齐次方程通解，<Object: word/embeddings/oleObject134.bin>为本身的特解。

<Object: word/embeddings/oleObject135.bin>，其中<Object: word/embeddings/oleObject136.bin>，<Object: word/embeddings/oleObject137.bin>

会用微分方程解决一些简单的应用问题。

五、多元函数微分学

了解二元函数的极限和连续的概念、有界闭区域上多元连续函数的性质。

理解并掌握多元函数偏导数和全微分、全微分存在的必要条件和充分条件。

熟练理解并掌握多元复合函数求二阶偏导数，会求隐函数的导数。

例如：多元复合函数求偏导数

设函数<Object: word/embeddings/oleObject138.bin>和<Object: word/embeddings/oleObject139.bin>在<Object: word/embeddings/oleObject140.bin>点分别具有对<Object: word/embeddings/oleObject141.bin>和<Object: word/embeddings/oleObject142.bin>的偏导数，而对应的函数<Object: word/embeddings/oleObject143.bin>在相应的<Object: word/embeddings/oleObject144.bin>点具有对<Object: word/embeddings/oleObject145.bin>和<Object: word/embeddings/oleObject146.bin>的连续偏导数，则复合函数<Object: word/embeddings/oleObject147.bin>在<Object: word/embeddings/oleObject148.bin>点具有对<Object: word/embeddings/oleObject149.bin>的偏导数，且

<Object: word/embeddings/oleObject150.bin> <Object: word/embeddings/oleObject151.bin>

同链相乘，分链相加

若<Object: word/embeddings/oleObject152.bin>和<Object: word/embeddings/oleObject153.bin>二阶可偏导，<Object: word/embeddings/oleObject154.bin>具有二阶连续偏导数，则

<Object: word/embeddings/oleObject155.bin> <Object: word/embeddings/oleObject156.bin>

理解方向导数和梯度的概念，并掌握其计算方法。

例如：(1) 方向导数：函数<Object: word/embeddings/oleObject157.bin>f (x , y , z)在<Object: word/embeddings/oleObject158.bin>(<Object: word/embeddings/oleObject159.bin>)点沿方向e<Object: word/embeddings/oleObject160.bin><Object: word/embeddings/oleObject161.bin>的方向导数

<Object: word/embeddings/oleObject162.bin>=<Object: word/embeddings/oleObject163.bin>

（2）梯度：函数<Object: word/embeddings/oleObject164.bin>f (x , y , z)在<Object: word/embeddings/oleObject165.bin>(<Object: word/embeddings/oleObject166.bin>)点的梯度<Object: word/embeddings/oleObject167.bin>

会求空间曲线的切线和法平面、曲面的切平面和法线。

例如：1）空间曲线切线与法平面方程

设空间曲线<Object: word/embeddings/oleObject168.bin>在<Object: word/embeddings/oleObject169.bin>参数<Object: word/embeddings/oleObject170.bin>，

切向量<Object: word/embeddings/oleObject171.bin>，切线方程：<Object: word/embeddings/oleObject172.bin>

法平面方程：<Object: word/embeddings/oleObject173.bin>

2）空间曲面的切平面与法线方程

设空间曲面：<Object: word/embeddings/oleObject174.bin>在切点<Object: word/embeddings/oleObject175.bin>，法向量<Object: word/embeddings/oleObject176.bin>

切平面方程：<Object: word/embeddings/oleObject177.bin>，

法线方程：<Object: word/embeddings/oleObject178.bin>

熟练理解并掌握多元函数极值和条件极值、拉格朗日乘数法、多元函数的最大值、最小值及其简单应用。

例如：条件极值问题可表述为：求函数<Object: word/embeddings/oleObject179.bin>在条件<Object: word/embeddings/oleObject180.bin>下的极值。

方法：构造拉格朗日函数<Object: word/embeddings/oleObject181.bin>，令<Object: word/embeddings/oleObject182.bin>，<Object: word/embeddings/oleObject183.bin>，<Object: word/embeddings/oleObject184.bin>，<Object: word/embeddings/oleObject185.bin>，解出<Object: word/embeddings/oleObject186.bin>，代入<Object: word/embeddings/oleObject187.bin>，其中最大（小）者为最大（小）值。

六、多元函数积分学　

理解二重积分和三重积分的概念及性质、熟练掌握二重积分的计算(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)。

例如：1）积分区域D为X－型区域 <Object: word/embeddings/oleObject188.bin>，<Object: word/embeddings/oleObject189.bin>

<Object: word/embeddings/oleObject190.bin>=<Object: word/embeddings/oleObject191.bin>，

积分区域D为Y－型区域<Object: word/embeddings/oleObject192.bin>，<Object: word/embeddings/oleObject193.bin>

<Object: word/embeddings/oleObject194.bin>=<Object: word/embeddings/oleObject195.bin>，

2）对于二重积分,如果区域<Object: word/embeddings/oleObject196.bin>关于<Object: word/embeddings/oleObject197.bin>轴对称,函数<Object: word/embeddings/oleObject198.bin>是关于<Object: word/embeddings/oleObject199.bin>的奇函数(既<Object: word/embeddings/oleObject200.bin>), 则<Object: word/embeddings/oleObject201.bin>;

若是偶函数(既<Object: word/embeddings/oleObject202.bin>),则<Object: word/embeddings/oleObject203.bin>

其中<Object: word/embeddings/oleObject204.bin>是<Object: word/embeddings/oleObject205.bin>在<Object: word/embeddings/oleObject206.bin>轴的上半部分

对于二重积分, 如果区域<Object: word/embeddings/oleObject207.bin>关于<Object: word/embeddings/oleObject208.bin>轴对称,函数<Object: word/embeddings/oleObject209.bin>是关于<Object: word/embeddings/oleObject210.bin>的奇函数(既<Object: word/embeddings/oleObject211.bin>), 则<Object: word/embeddings/oleObject212.bin>;

若是偶函数(既<Object: word/embeddings/oleObject213.bin>),则<Object: word/embeddings/oleObject214.bin>

其中<Object: word/embeddings/oleObject215.bin>是<Object: word/embeddings/oleObject216.bin>在<Object: word/embeddings/oleObject217.bin>轴的右半部分

理解两类曲线积分的概念、性质及两类曲线积分的关系，掌握两类曲线积分的计算方法。

熟练掌握格林(Green)公式和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。

例如：1）第二型曲线积分（平面曲线）

积分形式：<Object: word/embeddings/oleObject218.bin>＋<Object: word/embeddings/oleObject219.bin>=<Object: word/embeddings/oleObject220.bin>

曲线积分与路径无关的充要条件之一是：<Object: word/embeddings/oleObject221.bin>在<Object: word/embeddings/oleObject222.bin>内恒成立；

2）格林(Green)公式

<Object: word/embeddings/oleObject223.bin><Object: word/embeddings/oleObject224.bin>

其中L是D的正方向边界曲线。

了解两类曲面积分的概念、性质，掌握两类曲面积分的计算方法，熟练掌握用高斯公式计算曲面积分的方法。

例如：高斯（Gauss）公式

<Object: word/embeddings/oleObject225.bin><Object: word/embeddings/oleObject226.bin>

七、无穷级数

了解常数项级数的收敛与发散的概念、收敛级数的和的概念。

例如：两种级数

（1）<Object: word/embeddings/oleObject227.bin>级数<Object: word/embeddings/oleObject228.bin>当<Object: word/embeddings/oleObject229.bin>时收敛，当<Object: word/embeddings/oleObject230.bin>时发散

（2）等比级数<Object: word/embeddings/oleObject231.bin> 当<Object: word/embeddings/oleObject232.bin>收敛,且其和为<Object: word/embeddings/oleObject233.bin>;当<Object: word/embeddings/oleObject234.bin>时,等比级数发散

掌握级数的基本性质，掌握级数收敛的必要条件，掌握几何级数与p级数及其收敛性，掌握正项级数收敛性的比较判别法，掌握交错级数并会用莱布尼茨(Leibniz)判别法。

例如：1）正项级数的比值判别法：

正项级数<Object: word/embeddings/oleObject235.bin>,<Object: word/embeddings/oleObject236.bin>

2）比值审敛法，（达朗贝尔（D’Alembert）判别法）设<Object: word/embeddings/oleObject237.bin>为正项级数，如果<Object: word/embeddings/oleObject238.bin>则当<Object: word/embeddings/oleObject239.bin>时级数收敛；<Object: word/embeddings/oleObject240.bin>(或<Object: word/embeddings/oleObject241.bin>)时级数发散；<Object: word/embeddings/oleObject242.bin>时级数可能收敛也可能发散.

3．交错级数<Object: word/embeddings/oleObject243.bin><Object: word/embeddings/oleObject244.bin>的莱布尼茨定理判别法：若（1）<Object: word/embeddings/oleObject245.bin> （2）<Object: word/embeddings/oleObject246.bin>则级数收敛

4（比较审敛法的极限形式） 设<Object: word/embeddings/oleObject247.bin>和<Object: word/embeddings/oleObject248.bin>都是正项级数，(其中<Object: word/embeddings/oleObject249.bin>), 如果 <Object: word/embeddings/oleObject250.bin>则

当<Object: word/embeddings/oleObject251.bin>时，两个级数同时敛散；

当l=0时,若<Object: word/embeddings/oleObject252.bin>收敛,则<Object: word/embeddings/oleObject253.bin>也收敛,若<Object: word/embeddings/oleObject254.bin>发散，则<Object: word/embeddings/oleObject255.bin>也发散.

当<Object: word/embeddings/oleObject256.bin>时,若<Object: word/embeddings/oleObject257.bin>发散,则<Object: word/embeddings/oleObject258.bin>也发散;若<Object: word/embeddings/oleObject259.bin>收敛,则<Object: word/embeddings/oleObject260.bin>也收敛.

了解任意项级数的绝对收敛与条件收敛。

了解函数项级数的收敛域与和函数的概念。

会求幂级数的收敛半径、收敛区间（指开区间）、收敛域。

例如：幂级数<Object: word/embeddings/oleObject261.bin>

（1）收敛半径，收敛域：如果<Object: word/embeddings/oleObject262.bin>其中<Object: word/embeddings/oleObject263.bin>是幂级数<Object: word/embeddings/oleObject264.bin>的相邻两项的系数，则这幂级数的收敛半径<Object: word/embeddings/oleObject265.bin>开区间<Object: word/embeddings/oleObject266.bin>叫做幂级数的收敛区间。再由幂级数在<Object: word/embeddings/oleObject267.bin>处的收敛性就可以决定它的收敛域是<Object: word/embeddings/oleObject268.bin>或<Object: word/embeddings/oleObject269.bin>这四个区间之一。

（1）<Object: word/embeddings/oleObject270.bin>，<Object: word/embeddings/oleObject271.bin>

（2）<Object: word/embeddings/oleObject272.bin>，<Object: word/embeddings/oleObject273.bin>

理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)，熟练掌握简单幂级数的和函数的求法。

理解初等函数的幂级数展开式，熟练掌握应用它们将简单函数间接展开成幂级数。

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