电大微积分初步专科期末复习题及答案资料小抄精华打印版.doc

电大微积分初步期末复习资料小抄
一、填空题
⒈函数的定义域是　　．
答案：
⒉函数的间断点是=　　　．答案：
⒊曲线在点的斜率是　　．答案：
⒋若，则　　　　　．
答案：
⒌微分方程的阶数是　　2　　．
6.函数，　　　．答案：
7．函数在处连续，则=　　2　．
8.曲线在点的斜率是　　．答案：
9.　　　．答案：4
10.微分方程的阶数是　．答案：2
11.函数的定义域是　　．答案：
12.若，则　　　　．答案：2
13.已知，则=　　　　．答案：
14.若　　　　．答案：
15.微分方程的阶数是　3　．
16.函数的定义域是(-2,-1)∪(-1，4】．
17.若，则　2．
18.曲线在点处的切线方程是_y=x+1__．
19. 0 ．
20.微分方程的特解为 y=e的x次方 ．
21.函数的定义域是　 　．
22.若函数,在处连续，则　2　 ．
23.曲线在点处的斜率是 ．
24. ．
25.微分方程满足初始条件的特解为 ．
26．函数的定义域是　　　．
答案：
27．函数的定义域是　　　　　．
答案：
28．函数的定义域是　　．
答案：
29．函数，则 ．
答案：
30．函数，则　　 ．
答案：
31.函数，则　　 ．
答案：
32．函数的间断点是　　　．
答案：
33．　　　　　．
答案： 1
34．若，则　　　　　．
答案： 2
35．若，则　　　　　．
答案：
36．曲线在点的斜率是．
37．曲线在点的切线方程是．
38．曲线在点处的切线方程是．
39．．
40．若y = x (x – 1)(x – 2)(x – 3)，则(0) = －6 ．
41．已知，则=．
42．已知，则=．
43．若，则－2 ．
44．函数在区间内单调增加，则a应满足 大于零
45．若的一个原函数为，则 。
答案：` （c为任意常数）
46若的一个原函数为，则 。
答案：
47若，则　　　　　．
答案：
48若，则　　　　　．
答案：
49．若，则　　　　　．
答案：
50．若，则　　　　　．
答案：
51． ．
答案：
52． ．
答案：
53．若，则　　．
答案：
10．若，则　　．
答案：
54．
答案：
55．
答案：2
56．已知曲线在任意点处切线的斜率为，且曲线过，则该曲线的方程是 。
答案：
57．若　　　　　 答案：4
58．由定积分的几何意义知，= 。答案：，它是1/4半径为a的圆的面积。
59．　　　 答案：0
60．=　　　　． 答案：
61．微分方程的特解为 . 答案：1
62．微分方程的通解为 .答案：
63．微分方程的阶数为 ．
答案：2
64．函数的定义域是___且 。
65．函数＋的定义域是_ _。
66．设＝，则＝___0____。
67．函数，则___ 。
68．_______ 。
69．设，则_______。
70．曲线在点的切线方程是______ 。
71．函数在区间__________内是单调减少的。
72．函数的单调增加区间是　　　　．
73．若，则　　　．
74．_______。
75． ．
76．　　0　.
77． 2 .
78．微分方程的阶数是　二阶　．

二、单项选择题
⒈设函数，则该函数是（偶函数）．
⒉若函数，则（）.
⒊函数在区间是（先减后增）　
⒋下列无穷积分收敛的是（）．
⒌微分方程的通解是（）
6.函数的定义域（且）．
7.若函数，则（ 1 ）.
8.函数在区间是（先减后增）　
9.下列无穷积分收敛的是（）．
10.下列微分方程中为一阶线性微分方程的是（）
11.设函数，则该函数是（偶函数）．
12.当=（ 2 ）时，函数，在处连续.
13.微分方程的通解是（　）
14.设函数，则该函数是（偶函数）．
15.当（2）时，函数，在处连续.
16.下列结论中（在处不连续，则一定在处不可导. ）正确．
17.下列等式中正确的是（）．
18.微分方程的阶数为（3）
19.设，则（）
20.若函数f (x)在点x0处可导，则(，但 )是错误的．
21.函数在区间是（先减后增）　22.若，则（）.
23.微分方程的阶数为（3）
24．设函数，则该函数是（偶函数）．
25．设函数，则该函数是（奇函数）．
26．函数的图形是关于（坐标原点）对称．
27．下列函数中为奇函数是（ ）．
28．函数的定义域为（且）．
29．函数的定义域是（　）．
30．设，则（ ）
31．下列各函数对中，（，）中的两个函数相等．
32．当时，下列变量中为无穷小量的是（ ）. 答案：C
33．当（ 1 ）时，函数，在处连续.
34．当（ 3 ）时，函数在处连续.
35．函数的间断点是（ ）
36．函数在区间是（先减后增）　
37．满足方程的点一定是函数的（驻点）.
38．若，则=（-1　）．
39．设是可微函数，则（ ）．
40．曲线在处切线的斜率是（ ）．
41．若，则（ ）．
42．若，其中是常数，则（ ）．
43．下列结论中（在处连续，则一定在处可微.）不正确．
44．若函数f (x)在点x0处可导，则(，但)是错误的．
45.下列结论正确的有（x0是f (x)的极值点，且(x0)存在，则必有(x0) = 0）．
46．下列等式成立的是（）．
47．若，则（）.
48．若，则（ ）.
49．以下计算正确的是（ ）
50．（ ）
51．=（ ）．
52．如果等式，则（ ）
53．在切线斜率为2x的积分曲线族中，通过点（1, 4）的曲线为（ y = x2 + 3 ）．
54．若= 2，则k =（ 1 ）．
55．下列定积分中积分值为0的是（ ）．
56．设是连续的奇函数，则定积分（ 0 ）　
57．（ ）．
58．下列无穷积分收敛的是（　）．
59．下列无穷积分收敛的是（　）．
60．下列微分方程中，（ ）是线性微分方程．
61．微分方程的通解为（ ）．
62．下列微分方程中为可分离变量方程的是（　）
63.函数y＝的定义域是（(－2，2]）。
64．设，则（ ）。
65．函数的图形关于（轴）对称．
66、当时，变量（ ）是无穷小量．
67．函数 在x = 0处连续，则k = (-1)．
68．曲线在点(1,0)处的切线方程是（ ）。
69．若，则（ ）。
70．函数在区间内满足（单调上升　）．
71．函数y＝x2－2x＋5在区间 (0，1) 上是（单调减少 ）。
72．下列式子中正确的是（ ）。
73．以下计算正确的是（ ）
74．若，则（ 　）．
75．（ ）。
76．下列定积分中积分值为0的是（ ）．
77．微分方程的通解是（ ）。

三、计算题（本题共44分，每小题11分）
⒈ 计算极限．
解
　⒉ 设，求.
解
3．计算不定积分
解
4．计算定积分
解
5．计算极限．
解
　 6． 设，求.
解
7．计算不定积分
解 =
8．计算定积分
解

9.计算极限．
解

10.设，求.
解

11.计算不定积分
解 =
12.计算定积分
解

12.计算极限．
解：原式
13.设，求.
解：

14.计算不定积分
解：=
15.计算定积分
解：
16.计算极限．
解：原式
17.设，求.
解：

18.计算不定积分
解：=
19.计算定积分
解：
20.计算极限．
解
21．计算极限
解
22．
解
23．计算极限
解
24．计算极限．
解
25．计算极限．
解
26．计算极限．
解
27.设，求．
解
28．设，求.
解
29．设，求.
解
30．设，求.
解，
31．设，求.
解

32．设是由方程确定的隐函数，求.
解
33．设是由方程确定的隐函数，求.
解
34．设，求．
解
35．
解 利用分部积分法
设，，则，

36．
解 利用分部积分法
设，，则，

四、应用题
1．用钢板焊接一个容积为4的底为正方形的无盖水箱，已知钢板每平方米10元，焊接费40元，问水箱的尺寸如何选择，可使总费最低？最低总费是多少？
解：设水箱的底边长为，高为，表面积为，且有
所以

令，得，
因为本问题存在最小值，且函数的驻点唯一，所以，当时水箱的表面积最小. 此时的费用为 （元）.

2．欲做一个底为正方形，容积为62.5立方米的长方体开口容器，怎样做法用料最省？
解：设底边的边长为，高为，容器的表面积为，由已知，

令，解得是唯一驻点，易知是函数的极小值点，此时有，所以当，时用料最省．

3. 欲用围墙围成面积为216平方米的一成矩形的土地，并在正中用一堵墙将其隔成两块，问这块土地的长和宽选取多大尺寸，才能使所用建筑材料最省？
解：设土地一边长为，另一边长为，共用材料为
于是 =3

令得唯一驻点（舍去）
因为本问题存在最小值，且函数的驻点唯一，所以，当土地一边长为，另一边长为18时，所用材料最省.

4. 欲做一个底为正方形，容积为108立方米的长方体开口容器，怎样做法用料最省？
解：设底边的边长为，高为，用材料为，由已知

令，解得是唯一驻点，
且，
说明是函数的极小值点，所以当，时用料最省。

5．欲用围墙围成面积为216平方米的一成矩形的土地，并在正中用一堵墙将其隔成两块，问这块土地的长和宽选取多大尺寸，才能使所用建筑材料最省？
解 设土地一边长为，另一边长为，共用材料为
于是 =3

令得唯一驻点（舍去）

五、证明题（本题5分）
1、函数在（是单调增加的．
证 只需证明当时，有
因为
当时，，即有
所以，当时，是单调增加的。
1、证明等式。
证明：显然是偶函数，是奇函数，

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