网考数学模拟试题

试点高校网络教育部分公共基础课全国统一考试
高等数学（A）试卷
2008年4月
注意：是非题、选择题、填空题及解答题的解答均必须写在答题纸上，写在试卷上的任何解答一律无效.

一、是非题（满分18分）本大题共6个小题，每小题3分.对每小题给出的命题，认为正确的在答题纸相应的空格内填“对”，否则填“错”.
1．函数的定义域为.(??? )
2．极限.(??? )
3．函数在点处可导，则该函数在点处必连续.(??? )
4．函数在其定义域上的极小值可能大于它的一个极大值.(??? )
5．设为连续函数，则.(??? )
6．设为上连续奇函数，则.(??? )

二、选择题（满分20分）本大题共5个小题，每小题4分. 在每小题给出的四个选项中，只有一项符合题目要求，把所选项前的字母写在答题纸相应的空格内.
7．极限(??? ).
A．????????????????????? B．0????????????????????????? C．1????????????????????????? D．2

8．函数在内是(??? )．
A．偶函数?????????????? B．奇函数??????????????? C．无界函数??????????? D．单调函数

9．设函数，则(??? ).
A．????????????????? B. ??????????????????????????? C. ??????????? D.
10．设二元函数，则(??? )．
A．??????????????????????? B．??????????????????? C．????????????????????????? D．?
11．设函数在内可导，且，则在内(??? ).
A. 单调增加????????????????????????????????????????????? B. 单调减少
C. 是常数????????????????????????????????????????????????? D. 依条件不能确定单调性

三、填空题（满分20分）本大题共5个小题，每小题4分. 把答案写在答题纸相应的空格内.
12．极限(??? )．
13．设函数在处可导，且，则(??? )．
14．设为的一个原函数，则函数=(??? )．
15．设区域，则二重积分(??? )．
16．若级数收敛，则常数满足条件(??? )．

四、解答题（满分42分）本大题共6个小题，每小题满分7分. 解答应写出推理、演算步骤，将解答写在答题纸相应的位置上.
17．设函数，求及 ．
18．设函数，求（1）定义域；（2）；（3）函数图形的凹区间．
19．设函数由方程所确定，求及．
20．计算不定积分．
21．设二元函数，求（1）；（2）；（3）．
22．求幂级数的收敛半径与收敛区间（不需考虑区间的端点）．

试点高校网络教育部分公共基础课全国统一考试

高等数学（A）试卷参考解答与评分标准

2008年4月

1. 是非题（满分18分，每小题3分）

1. 错?????????? 2. 错??????????? 3. 对??????????? 4. 对??????????? 5. 错??????????? 6. 对
二、选择题（满分20分，每小题4分）
7. D??????????????????? 8. A???????????????????? 9. B???????????????????? 10. C??????????? 11. B
三、填空题（满分20分，每小题4分）
12. e2??? 13. 1???????????? 14. 1???????????? 15. ??????? 16. >0或填p>0
四、解答题（满分42分，每小题满分7分）
17．解：
··························································································· 5分
····························································································· 7分
18．解：
（1），（写为也正确）······················································· 2分
（2）··························································································· 4分
··································································································· 5分
（3）函数图形的凹区间为··························································· 7分
19．解：
方程两边对x求导数，得··················································· 3分
整理得························································································ 5分
所以? ························································································· 7分
20．解法一：
················································································· 3分
··········································································· 7分
解法二：
设···························································································· 1分
······························································································ 2分
························································································· 3分
················································································ 5分
········································································· 7分
注：①丢掉常数C扣1分；②对数的真数不加绝对值符号不扣分．
21．解：
(1) ································································································ 2分
(2) ································································································ 4分
(3) ··················································································· 6分
············································································· 7分
22．解：
····································································· 3分
所以···································································································· 5分
收敛区间为····················································································· 7分

试点高校网络教育部分公共基础课全国统一考试
高等数学（B）试卷
2008年4月
注意：是非题、选择题、填空题及解答题的解答均必须写在答题纸上，写在试卷上的任何解答一律无效.

一、是非题（满分18分）本大题共6个小题，每小题3分.对每小题给出的命题，认为正确的在答题纸相应的空格内填“对”，否则填“错”.
1．函数的定义域为.(??? )
2．极限.(??? )
3．函数在点处可导，则该函数在点处必连续.(??? )
4．函数在其定义域上的极小值可能大于它的一个极大值.(??? )
5．设为连续函数，则.(??? )
6．设为上连续奇函数，则.(??? )

二、选择题（满分20分）本大题共5个小题，每小题4分. 在每小题给出的四个选项中，只有一项符合题目要求，把所选项前的字母写在答题纸相应的空格内.
7．极限(??? ).
A．????????????????????? B．0????????????????????????? C．1????????????????????????? D．2

8．函数在内是(??? )．
A．偶函数?????????????? B．奇函数??????????????? C．无界函数??????????? D．单调函数

9．设函数，则(??? ).
A．????????????????? B. ??????????????????????????? C. ??????????? D.
10．不定积分(?? ?)．
A．???????????????????? B．???????????????? C．???????????? D．?
11．设函数在内可导，且，则在内(??? ).
A. 单调增加?????????????????????? B. 单调减少
C. 是常数???????????????????????? D. 依条件不能确定单调性

三、填空题（满分20分）本大题共5个小题，每小题4分. 把答案写在答题纸相应的空格内.
12．极限（??? ）．
13．设函数在处可导，且，则(??? )．
14．设为的一个原函数，则函数=(??? )．
15．设，若函数在处连续，则常数(??? )．
16．微分方程的通解是(??? )．

四、解答题（满分42分）本大题共6个小题，每小题满分7分. 解答应写出推理、演算步骤，将解答写在答题纸相应的位置上.
17．求极限．
18．设函数，求及 ．
19．设函数，求（1）定义域；（2）；（3）函数图形的凹区间．
20．计算不定积分．
21．求由曲线及直线和所围成的平面图形的面积（如下图）．

22．求微分方程的通解．

试点高校网络教育部分公共基础课全国统一考试

高等数学（B）试卷参考解答与评分标准

2008年4月

1. 是非题（满分18分，每小题3分）

1. 错?????????? 2. 错??????????? 3. 对??????????? 4. 对??????????? 5. 错??????????? 6. 对
二、选择题（满分20分，每小题4分）
7. D??????????????????? 8. A???????????????????? 9. B???????????????????? 10. D??????????? 11. B
三、填空题（满分20分，每小题4分）
12．??? 13．1?? ?? 14．?? ????? 15．??? 16．?
四、解答题（满分42分，每小题满分7分）
17．解法一：
······························································ 3分
······································································ 6分
················································································ 7分
解法二：
······························································· 3分
??????????????????????????????? 6分
················································································ 7分
18．解：
····························································································· 5分
······························································································· 7分
19．解：
（1），（写为也正确）······················································· 2分
（2）···························································································· 4分
···································································································· 5分
（3）函数图形的凹区间为··························································· 7分
20．解法一：
············································································· 3分
······································································· 7分
解法二：
设·························································································· 1分
····························································································· 2分
························································································· 3分
················································································ 5分
········································································· 7分
注：①丢掉常数C扣1分；②对数的真数不加绝对值符号不扣分．
21．解：
······················································································· 3分
····························································································· 6分
···································································································· 7分
22．解：
，················································································· 1分
························································ 2分
···································································· 3分
················································································ 5分
?或 ································································ 7分

试点高校网络教育部分公共基础课全国统一考试
高等数学（B）试卷
2007年4月
注意：选择题、填空题及解答题的解答均必须写在答题纸上，写在试卷上的任何解答一律无效。

一、选择题（满分20分）本大题共5个小题，每小题4分。在每小题给出的四个选项中，只有一项符合题目要求，把所选项前的字母写在答题纸上。
1．的定义域为 (1) .  
A．??? B．????? C．?????? D．

2．= (2) .

A． ??????? -1B．0????????? C．1??????????? D．2

3．在内是 (3) ．

A．偶函数??????????? B．奇函数??????? C．无界函数? D．单调函数

4．设，则?(4) .

A．???????? B．??????? C．??????? D．
5．设二元函数，则?(5) ．

A．????????????? B．???? C．?????????? D．1

二、填空题（满分28分）本大题共7个小题，每小题4分。把答案写在答题纸上。
6．?(6) ．

7．函数的间断点为?(7) ．

8．函数在闭区间上的最小值为 (8) ．

9．= (9) ．
10． ?(10) ．

11．设区域D可以表示为，，则?(11) ．

12．微分方程 的一个解为 = (12) ． 

三、解答题（满分52分）本大题共7个小题。解答应写出推理、演算步骤，将解答写在答题纸上。
13.（本题满分7分）求（1），（2）．

14.（本题满分8分）设，求及．

15.（本题满分7分）计算不定积分．

16.（本题满分7分）设，求的最小值点和最小值．

17.（本题满分9分）设，求．

18.（本题满分9分）设二元函数，求（1）；（2）；（3）．

19.（本题满分5分）求微分方程 的通解．

试点高校网络教育部分公共基础课全国统一考试

高等数学（B）试卷参考解答与评分标准

2007年4月
一、选择题（满分20分，每小题4分）
1．D??? 2．C??????? 3．A??????? 4．B??????? 5．D

二、填空题（满分28分，每小题4分）
6．???? 7．????? 8．1??????? 9．?（不加不扣分） 10．0???? 11．2?????? 12． 
三、解答题（满分52分）
13．（本题满分7分）
解：(1)?????????????????? ………………………………… 3分

??? (2)??????????? ………………………………………… 7分

14．（本题满分8分）
解： 两边对求导数
??????????????????? ……………………………………… 5分
???????????????????? …………………………………… 8分
15．（本题满分7分）
解：????????????? ……………………………… 2分

????????????? ……………………………… 3分

???????????? ……………………………… 7分

注： (1) 丢掉常数 ?只扣1分，丢掉绝对值符号不扣分.
(2) 与的计算各2分.

16．（本题满分7分）
解法1：
……………………………………………………? 2分

???? 令 得驻点??? …………………………………………… 3分

?????? ………………………………………………………? 4分

??????

可知为的极小值点，由于驻点唯一，因此为的最小值点?????? …………………………………………………………………… 6分

最小值为????????? ………………………………………… 7分

解法2：

???? …………………………………………………… 2分

令?得驻点 ????………………………………………… 3分

?????? 当时 ；

? ?????当时 ；

可知为的极小值点，由于驻点唯一，因此为的最小值点???????? ………………………………………………………………… 6分

最小值为 ?????????………………………………………… 7分

解法3：????? ………………………………? 3分

时，
时，????? ………………………………… 5分
可知为最小值点?? ……………………………………………… 6分

最小值为? ????……………………………………………… 7分

17．（本题满分9分）
解法1：????????????????? ………………………………? 7分

??????????????????? ……………………………… 9分

解法2： = ????????????? ……………………………? 4分

????????????????? ………………………………? 7分

??? ?????????????????…………………………… 9分

18．（本题满分9分）
解：(1)  ??????????????????????………………………………… 3分
(2) ?????????????????? ………………………………? 6分
(3) ???????????? …………………………………… 8分

????? ……………………………………? 9分
19．（本题满分5分）
解：该方程为一阶线性微分方程，通解公式为
?……………………………………… 1分
其中，，??????????? ……………………………… 2分
因此通解为??? ……………………………? 3分

?????? ……………………………? 4分

?
???????? …………………………………? 5分
注：（1）丢掉常数只扣1分，(2)只得出齐次方程通解，给3分.