2017年电大2017电大【经济数学基础】形成性考核册参考答案

∴AA是对称矩阵。 ∵ATATT??T?AT?AT??T?ATA

∴AA是对称矩阵.

3.设A,B均为n阶对称矩阵,则AB对称的充分必要条件是:AB?BA。 证: 必要性:

∵A?A , B?B 若AB是对称矩阵,即?AB??AB

TTT而?AB??BA?BA 因此AB?BA

TT充分性:

若AB?BA,则?AB??BTAT?BA?AB

T∴AB是对称矩阵.

4.设A为n阶对称矩阵,B为n阶可逆矩阵,且B 证:∵A?A BT?1?1?BT,证明B?1AB是对称矩阵。

?BT

?1T?B

?1AB?1???AB???B?TT?BT?AT?BT??T?B?1AB

∴BAB是对称矩阵. 证毕.

经济数学基础作业4

(一)填空题 1.函数f(x)?4?x?21的定义域为___________________。答案:(1,2)??2,4?.

ln(x?1)2. 函数y?3(x?1)的驻点是________,极值点是 ,它是极 值点。答案:x=1;(1,0);小。 3.设某商品的需求函数为q(p)?10e111?p2,则需求弹性Ep? .答案:Ep=?.答案:4.

p 24.行列式D??1?111?____________?11?1A???0??015. 设线性方程组AX?b,且

6?,则t__________时,方程组有唯一解. 答案:t?132??0t?10??13

1??1.

(二)单项选择题

1. 下列函数在指定区间(??,??)上单调增加的是( B ).

A.sinx B.e x C.x 2 D.3 – x 2. 设f(x)?1,则f(f(x))?( C ). x112A. B.2 C.x D.x

xxx?x1e?e11ex?e?xdx?0 B.?dx?0 C.?xsinxdx?0 D.?(x2?x3)dx?0 A.??1?1-1-12213. 下列积分计算正确的是( A ).

4. 设线性方程组Am?nX?b有无穷多解的充分必要条件是( D ).

A.r(A)?r(A)?m B.r(A)?n C.m?n D.r(A)?r(A)?n

?x1?x2?a1?5. 设线性方程组?x2?x3?a2,则方程组有解的充分必要条件是( C ).

?x?2x?x?a233?1A.a1?a2?a3?0 B.a1?a2?a3?0 C.a1?a2?a3?0 D.?a1?a2?a3?0

三、解答题

1.求解下列可分离变量的微分方程: (1) y??ex?y 解:

dy?ex?ey , e?ydy?exdx ?e?ydy??exdx , ?e?y?ex?c dxdyxex(2)?2

dx3y解: 3ydy?xedx

2x?3y2x3xx3xxdy??xde y?xe??edx y?xe?e?c

2. 求解下列一阶线性微分方程: (1)y??解:y?e2y?(x?1)3 x?1?2??3?x?1dx???x?1?e??e2ln?x?1?dx?c?????2?????dx?x?1?????x?1?e3?2ln?x?1?dx?c??x?1??2???x?1?dx?c?

??x?1??2?1?x?1?2?c?? ?2? 14

(2)y??解:y?ey?2xsin2x x??1??????dx??2xsin2x?e?x?dx?c??elnx??????1?????dx?x???2xsin2x?e?lnxdx?c

?1???x??2xsin2x?dx?c??x?sin2xd2x?c ?x??cos2x?c?

x??3.求解下列微分方程的初值问题: (1)y??e2x?y,y(0)?0

??dye2x?解: dxey

?eyydy??e2xdx

e?12xe?c 2 用x?0,y?0代入上式得:

101e?c, 解得c? 2212x1y ∴特解为:e?e?

22 e?0 (2)xy??y?e?0,y(1)?0 解:y??x11y?ex xx?11xdx?e?xdx??x?dx?c? y?e??x?e?

?? ?e?lnx?1xlnx????e?edx?c? ?x?xx ?1x??edx?c??1?ex?c

? 用x?1,y?0代入上式得: 0?e?c 解得:c??e ∴特解为:y?1xe?c x??(注意:因为符号输入方面的原因,在题4—题7的矩阵初等行变换中,书写时应把(1)写成①;(2)写成②;(3)写成③;…)

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4.求解下列线性方程组的一般解:

?2x3?x4?0?x1?(1)??x1?x2?3x3?2x4?0

?2x?x?5x?3x?0234?12?1?2?1??10?10?2???1??1???01?11??3???2??1?3???1????2?解:A=?11?32??????????????????2?15?3???0?11?1???102?1??01?11? ???0??000?所以一般解为

??x1??2x3?x4 其中?xx3,x4是自由未知量。2?x3?x 4

?2x1?x2?x3?x4?1(2)??x1?2x2?x3?4x4?2

??x1?7x2?4x3?11x4?5??解:A??2?1111?12?14??1?,?2???12?142??2???1????2??12?2??????2?1111??3???1????1??7?4115??1??17?4115??????????0?5????05??12???3???2??1??2?140???2?????1??5???12?142?1??????53?7?3??????01?373?????1????2?????2???????00000???00055?0?05?0???0?????x?416?秩?A?=2,所以方程组有解,一般解为?1?5?5x3?因为秩A5x4?337

?x2?5?5x3?5x4其中x3,x4是自由未知量。

5.当?为何值时,线性方程组

??x1?x2?5x3?4x4?2??2x1?x2?3x3?x4?1xx ?31?2x2?23?3x4?3??7x1?5x2?9x3?10x4??有解,并求一般解。

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?142?3?7?3?

?373???0164?5355?1?73?005505?0????

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