电大《经济数学基础》考试参考答案（电大参考答案）-中央电大专科考试参考答案.doc

<p>Four short words sum up what has lifted most successful individuals above the crowd: a little bit more.
------------------------------------------author
------------------------------------------date

电大《经济数学基础》考试参考答案(完整版电大参考答案)-中央电大专科考试参考答案

5
经济数学基本积分学
一、单项选择题
1．在切线斜率为2x的积分曲线族中，经由过程点（1, 4）的曲线为（ &nbsp;A &nbsp; &nbsp;）．
&nbsp; &nbsp;A．y = x2 + 3 &nbsp; &nbsp;B．y = x2 + 4 &nbsp; &nbsp; C．y = 2x + 2 &nbsp; &nbsp; D．y = 4x
&nbsp; &nbsp;2. 若= 2，则k =（ &nbsp; A &nbsp; ）．
&nbsp; &nbsp;A．1 &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;B．-1 &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;C．0 &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; D．
&nbsp; &nbsp;3．下列等式不成立的是（ D &nbsp;）． &nbsp; &nbsp;
&nbsp; &nbsp;A． &nbsp; &nbsp; &nbsp;B．
&nbsp; &nbsp;C． &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; D．
&nbsp; &nbsp;4．若，则=（　D &nbsp; &nbsp;）.
　　A. 　 &nbsp;B. &nbsp; &nbsp;　C. &nbsp; &nbsp;D.
&nbsp; &nbsp;5. （ B &nbsp;）． &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;
A． &nbsp; &nbsp;B． C．
D．
&nbsp; &nbsp;6. 若，则f (x) =（ &nbsp; C &nbsp; ）．
&nbsp; &nbsp;A． &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;B．- &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;C． &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;D．-
&nbsp; &nbsp;7. 若是的一个原函数，则下列等式成立的是( &nbsp; B &nbsp; )． &nbsp;
A． &nbsp; &nbsp; &nbsp; &nbsp;B．
C． &nbsp;D．
&nbsp; &nbsp;8．下列定积分中积分值为0的是（ &nbsp;A &nbsp; &nbsp;）．
&nbsp; &nbsp;A． &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;B．
&nbsp; &nbsp;C． &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; D．
&nbsp; &nbsp;9．下列无限积分中收敛的是（ &nbsp;C &nbsp;）． &nbsp; &nbsp;
A． &nbsp;B． &nbsp;C． D．
10．设(q)=100-4q ，若发卖量由10单元削减到5单元，则收入R的转变量是（ B &nbsp;）．
&nbsp; &nbsp;A．-550 &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;B．-350 &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; C．350 &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; D．以上都不合错误
&nbsp; &nbsp;11．下列微分方程中，（ D &nbsp;）是线性微分方程．
&nbsp; &nbsp;A． &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; B．
&nbsp; &nbsp;C． &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;D．
&nbsp; &nbsp;12．微分方程的阶是（　C &nbsp; ）.
A. 4　　 &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; 　B. 3　 &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;C. 2　　 &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;　D. 1
13．在切线斜率为2x的积分曲线族中，经由过程点（1, 3）的曲线为（ C &nbsp; ）．
A． &nbsp;B． &nbsp;C． D．
&nbsp; &nbsp;14．下列函数中，（ &nbsp;C &nbsp;）是的原函数．
A．- &nbsp;B． &nbsp;C． &nbsp;D．
&nbsp; 15．下列等式不成立的是（ D &nbsp;）． &nbsp; &nbsp;
&nbsp; &nbsp;A． &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;B．
C． &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;D．
&nbsp; &nbsp;16．若，则=（　D &nbsp; &nbsp;）.
A. 　 &nbsp; &nbsp; B. &nbsp; &nbsp; &nbsp;　C. 　　 &nbsp; D.
&nbsp; 17. （ B &nbsp;）． &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;
A．B． C． D．
&nbsp; &nbsp;18. 若，则f (x) =（ &nbsp; C &nbsp; ）．
A． &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;B．- &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;C． &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; D．-
&nbsp; &nbsp;19. 若是的一个原函数，则下列等式成立的是( &nbsp;B &nbsp;)． &nbsp;
A． &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; B．
C． &nbsp; &nbsp;D．
&nbsp; &nbsp;20．下列定积分中积分值为0的是（ &nbsp;A &nbsp; &nbsp;）．
&nbsp; &nbsp;A． &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; B．
C． &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; D．
&nbsp; &nbsp;21．下列无限积分中收敛的是（ C &nbsp; ）． &nbsp; &nbsp;
A． &nbsp;B． &nbsp; C． &nbsp;D．
&nbsp; &nbsp;22．下列微分方程中，（ D &nbsp;）是线性微分方程．
&nbsp; &nbsp;A． &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; B．
C． &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;D．
&nbsp; 23．微分方程的阶是（　C &nbsp; ）.
A. 4　　 &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; 　B. 3　 &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;C. 2　　 &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;　D. 1
24.设函数，则该函数是（ A &nbsp;）.
A. 奇函数 &nbsp;B.偶函数 &nbsp; C.非奇非偶函数 &nbsp;D.既奇又偶函数
25. 若，则( A )．
A. 　 &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;B. &nbsp; &nbsp; &nbsp; &nbsp; C. 　　 &nbsp; D.
26. 曲线在处的切线方程为( &nbsp;A &nbsp;).
&nbsp; &nbsp; A． &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;B．
&nbsp; &nbsp; C． &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;D． &nbsp;
27. 若的一个原函数是, 则=（　D）．
　 &nbsp; 　A．　　　　　B． &nbsp; &nbsp; &nbsp;C．　　 &nbsp; 　D．
28. 若, 则（ C &nbsp;）.
&nbsp; &nbsp; &nbsp;A. &nbsp; &nbsp; B. &nbsp;C. &nbsp; &nbsp;D. &nbsp; &nbsp;
二、填空题
1． ．
&nbsp; &nbsp;2．函数的原函数是-cos2x + c (c 是肆意常数) &nbsp;．
　　3．若，则　　.
　　4．若，则= &nbsp; &nbsp; &nbsp;.
　　5．　0　　.
6． 0　 ．
7．无限积分是 收敛的 ．（判别其敛散性）
8．设边际收入函数为(q) = 2 + 3q，且R (0) = 0，则平均收入函数为2 + ．
　　9. 是　 2 &nbsp; &nbsp;阶微分方程.
&nbsp; &nbsp;10．微分方程的通解是 ．
11．
12．。谜底：
13．函数f (x) = sin2x的原函数是 ．
14．若，则　. 谜底：
15．若，则= &nbsp; .
谜底：
16．　　　　　　. 谜底：0
17． ．谜底：0
18．无限积分是 ．谜底：1
19. 是　　 &nbsp; &nbsp; &nbsp;阶微分方程. 谜底：二阶
20．微分方程的通解是 ．谜底：
&nbsp; &nbsp; 21. 函数的界说域是(-2,-1)U(-1,2]．
22. 若，则　4 &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; 　．
23. 已知，则=　27+27 ln3　　　　　　　．
24. 若函数在的邻域内有界说，
且则　1　．.
25. 若, 则　　-1/2　　　．．
&nbsp; &nbsp; &nbsp; &nbsp;(三) 判定题
11. . &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; ( &nbsp;× &nbsp; )
12. 若函数在点持续，则必定在点处可微. &nbsp; &nbsp; &nbsp; &nbsp; ( &nbsp;× &nbsp; ) &nbsp; &nbsp; &nbsp; &nbsp;
13. 已知，则= &nbsp; （ &nbsp;√ &nbsp; ）
14. . &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;（ × &nbsp; &nbsp;）. &nbsp;
15. 无限限积分是发散的. &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;( &nbsp;√ &nbsp; &nbsp;
三、计较题
⒈ &nbsp; &nbsp; &nbsp; ⒈ 解 &nbsp;
2． &nbsp; &nbsp; &nbsp;2．解
3． &nbsp;
3．解 &nbsp;
4． &nbsp;
4．解 &nbsp;= &nbsp;
&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;=
5． &nbsp;
5．解 &nbsp;=
= = &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;
6．
6．解

7． &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;
7．解 &nbsp;=== &nbsp; &nbsp; &nbsp;
8．
8．解 &nbsp;=-==
9． &nbsp;
9．解法一 &nbsp; = &nbsp;
&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;=＝=1 &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;
&nbsp; &nbsp; &nbsp;解法二 &nbsp; &nbsp;令，则
&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;=
10．求微分方程知足初始前提的特解．
10．解 &nbsp;由于 &nbsp;， &nbsp; &nbsp; &nbsp; &nbsp;
&nbsp; &nbsp; &nbsp; 用公式 &nbsp; &nbsp;
&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;
由 &nbsp; ， 得
&nbsp; &nbsp; &nbsp; 所以，特解为 &nbsp; &nbsp; &nbsp; &nbsp;
11．求微分方程知足初始前提的特解．
11．解 &nbsp;将方程分手变量： &nbsp; &nbsp;
等式两头积分得 &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;
将初始前提代入，得 ，c = &nbsp; &nbsp; &nbsp;
所以，特解为： &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;
12．求微分方程知足 的特解. &nbsp;
12．解：方程两头乘以，得

即
&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;
双方求积分，得 &nbsp;
&nbsp; &nbsp; &nbsp; &nbsp; 通解为： &nbsp; &nbsp;
&nbsp; &nbsp; &nbsp; &nbsp; 由，得
&nbsp; &nbsp; 所以，知足初始前提的特解为：
13．求微分方程 的通解．
13．解 &nbsp;将原方程分手变量 &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;
&nbsp; &nbsp; &nbsp; &nbsp; &nbsp;两头积分得 &nbsp;lnlny = lnC sinx &nbsp; &nbsp; &nbsp; &nbsp;
&nbsp; &nbsp; &nbsp; &nbsp; &nbsp;通解为 &nbsp; &nbsp; &nbsp; &nbsp; y = eC sinx &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;
14．求微分方程的通解.
14. 解 &nbsp;将原方程化为：，它是一阶线性微分方程，
&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;，
用公式 &nbsp;
&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;
&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;
15．求微分方程的通解．
15．解 在微分方程中，
由通解公式 &nbsp;

16．求微分方程的通解．
16．解：由于，，由通解公式得
&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;
&nbsp; &nbsp; &nbsp; &nbsp; &nbsp;= =
&nbsp; &nbsp; &nbsp;= &nbsp; &nbsp; &nbsp;
17． &nbsp;
解
= &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; = &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;
18． &nbsp; &nbsp; &nbsp;
解：
&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;
19．
解：
= &nbsp; &nbsp;
20．
解：
&nbsp; &nbsp; &nbsp;=（谜底：
21． &nbsp;
解： &nbsp;
22．
解 &nbsp;=
23．

24.

25．

26．设，求
&nbsp; &nbsp; &nbsp;27. 设，求.
&nbsp; &nbsp; 28．设是由方程确定的隐函数,求.

29．设是由方程确定的隐函数,求.

30.

31.
32.
33.
34.
35.
36.
37.
四、应用题
&nbsp; &nbsp;1．投产某产物的固定成本为36(万元)，且边际成本为=2x + 40(万元/百台). 试求产量由4百台增至6百台时总成本的增量，及产量为若干好多时，可使平均成本达到最低.
&nbsp; &nbsp;1．解 &nbsp;当产量由4百台增至6百台时，总成本的增量为
&nbsp; &nbsp; &nbsp; &nbsp; &nbsp;== 100（万元） &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;
又 &nbsp;= = &nbsp; &nbsp; &nbsp;
令 &nbsp;， 解得.
x = 6是惟一的驻点，而该问题的确存在使平均成本达到最小的值. 所以产量为6百台时可使平均成本达到最小. &nbsp; &nbsp;
2．已知某产物的边际成本(x)=2（元/件），固定成本为0，边际收益(x)=12-0.02x，问产量为若干好多时利润最年夜？在最年夜利润产量的基本上再出产50件，利润将会发生什么转变？
&nbsp; 2．解 &nbsp;由于边际利润
&nbsp; &nbsp; &nbsp; &nbsp;=12-0.02x –2 = 10-0.02x &nbsp; &nbsp; &nbsp;
&nbsp; &nbsp; &nbsp; 令= 0，得x = 500 &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;
x = 500是惟一驻点，而该问题的确存在最年夜值. 所以，当产量为500件时，利润最年夜.
&nbsp; &nbsp;当产量由500件增加至550件时，利润转变量为
&nbsp; &nbsp; &nbsp; &nbsp; =500 - 525 = - 25 （元）
即利润将削减25元. &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;
3．出产某产物的边际成本为(x)=8x(万元/百台)，边际收入为(x)=100-2x（万元/百台），其中x为产量，问产量为若干好多时，利润最年夜？从利润最年夜时的产量再出产2百台，利润有什么转变？
3. 解 &nbsp;(x) =(x) -(x) = (100 – 2x) – 8x =100 – 10x &nbsp; &nbsp; &nbsp;
令(x)=0, 得 x = 10（百台）
又x = 10是L(x)的独一驻点，该问题的确存在最年夜值，故x = 10是L(x)的最年夜值点，即当产量为10（百台）时，利润最年夜. &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;
&nbsp; &nbsp;又
4．已知某产物的边际成本为(万元/百台)，x为产量(百台)，固定成本为18(万元)，求最低平均成本. &nbsp; &nbsp;
4．解：由于总成本函数为
&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; = &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;
当x = 0时，C(0) = 18，得 c =18
即 &nbsp; &nbsp; &nbsp; &nbsp; C(x)= &nbsp; &nbsp; &nbsp;
&nbsp; &nbsp; &nbsp; &nbsp; 又平均成本函数为 &nbsp; &nbsp; &nbsp; &nbsp;
令 ， 解得x = 3 (百台)
该题的确存在使平均成本最低的产量. 所以当x = 3时，平均成本最低. 最底平均成本为
&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;(万元/百台) &nbsp;
5．设出产某产物的总成本函数为 (万元)，其中x为产量，单元：百吨．发卖x百吨时的边际收入为（万元/百吨），求：
&nbsp; &nbsp;(1) 利润最年夜时的产量；
(2) 在利润最年夜时的产量的基本上再出产1百吨，利润会发生什么转变？
&nbsp; 5．解：(1) &nbsp;由于边际成本为 ，边际利润 = 14 – 2x
令，得x = 7 &nbsp; &nbsp; &nbsp;
由该题现实意义可知，x = 7为利润函数L(x)的极年夜值点，也是最年夜值点. 是以，当产量为7百吨时利润最年夜. &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;
&nbsp; (2) &nbsp;当产量由7百吨增加至8百吨时，利润转变量为
&nbsp; &nbsp; &nbsp; =112 – 64 – 98 + 49 = - 1 （万元）
即利润将削减1万元. &nbsp;
6．投产某产物的固定成本为36(万元)，且边际成本为=2x + 40(万元/百台). 试求产量由4百台增至6百台时总成本的增量，及产量为若干好多时，可使平均成本达到最低.
解 &nbsp;当产量由4百台增至6百台时，总成本的增量为
&nbsp; &nbsp; &nbsp; &nbsp; &nbsp;== 100（万元） &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;
又 &nbsp;= = &nbsp; &nbsp; &nbsp;
令 &nbsp;， 解得.
x = 6是惟一的驻点，而该问题的确存在使平均成本达到最小的值. 所以产量为6百台时可使平均成本达到最小.
7．已知某产物的边际成本为(万元/百台)，x为产量(百台)，固定成本为18(万元)，求最低平均成本. &nbsp;
&nbsp; &nbsp; 解：由于总成本函数为
&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; = &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;
当x = 0时，C(0) = 18，得 c =18
即 &nbsp; &nbsp; &nbsp; &nbsp; C(x)= &nbsp; &nbsp; &nbsp;
&nbsp; &nbsp; &nbsp; 又平均成本函数为 &nbsp; &nbsp; &nbsp; &nbsp;
令 ， 解得x = 3 (百台)
该题的确存在使平均成本最低的产量. 所以当x = 3时，平均成本最低. 最底平均成本为
&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;(万元/百台) &nbsp;
8．出产某产物的边际成本为(x)=8x(万元/百台)，边际收入为(x)=100-2x（万元/百台），其中x为产量，问产量为若干好多时，利润最年夜？从利润最年夜时的产量再出产2百台，利润有什么转变？
解：已知(x)=8x(万元/百台)，(x)=100-2x，则
令，解出独一驻点
&nbsp; &nbsp; &nbsp; &nbsp; 由该题现实意义可知，x = 10为利润函数L(x)的极年夜值点，也是最年夜值点. 是以，当产量为10百台时利润最年夜. &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;
从利润最年夜时的产量再出产2百台，利润的转变量为
（万元）
即利润将削减20万元.
&nbsp; 9．设出产某产物的总成本函数为 (万元)，其中x为产量，单元：百吨．发卖x百吨时的边际收入为（万元/百吨），求：
&nbsp; &nbsp;(1) 利润最年夜时的产量；
(2) 在利润最年夜时的产量的基本上再出产1百吨，利润会发生什么转变？
&nbsp; &nbsp;解：(1) &nbsp;由于边际成本为 ，边际利润 = 14 – 2x
令，得x = 7 &nbsp; &nbsp; &nbsp;
由该题现实意义可知，x = 7为利润函数L(x)的极年夜值点，也是最年夜值点. 是以，当产量为7百吨时利润最年夜. &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;
(2) &nbsp;当产量由7百吨增加至8百吨时，利润转变量为
&nbsp; &nbsp; &nbsp; =112 – 64 – 98 + 49 = - 1 （万元）
即利润将削减1万元. &nbsp;
精品电年夜复习资料9</p>