高等数学上册习题答案

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高等数学上册习题答案


【篇一:大学高等数学上考试题库(附答案
>一.选择题(将答案代号填入括号内,每题3分,共30分).1.下列各组函数中,是相同的函数的是().
af?x??lnxg?x??2lnxbf?x??|x|g?x??2
cf?x??xg?x??2
df?x??|x|x
g?x??122.函数f?x???ln?1?x??a?
x?0x?0
x?0处连续,则a?().a0b14
c1d2
3.曲线y?xlnx的平行于直线x?y?1?0的切线方程为().
ay?x?1by??(x?1cy??lnx?1??x?1?dy?x4.设函数f?x??|x|,则函数在点x?0处().
a)连续且可导b)连续且可微(c)连续不可导(d)不连续不可微
5.点x?0是函数y?x4的().
a)驻点但非极值点b)拐点c)驻点且是拐点d)驻点且是极值点6.曲线y?1|x|
的渐近线情况是().
a)只有水平渐近线b)只有垂直渐近线c)既有水平渐近线又有垂直渐近线d)既无水平渐近线又无垂直渐近线7?f??

?1?1
?2dx的结果是(.?x?x??1??1??1
bc?c?f??cf????x??x??x?x
af??8?dxe?ex??1
d?c?f????x???c?
的结果是().x?x
aarctane?cbarctane?cx?x
?cdln(e?ex?x?c
9.下列定积分为零的是(.?
a?4?
arctanx1?x2??4
dxb?4??4
xarcsinxdxc?11?1e?e2x?x
ce?e
d?1?1?x2
?x?sinxdx
10.设f?x?为连续函数,则?f??2x?dx等于().af?2??f?0?b12
??f?11??f?0???c12
??f?2??f?0???df?1??f?0?二.填空题(每题4分,共20分)?e?2x?1?
1.设函数f?x???x?a?
x?0x?056
x?0处连续,则a?.
2.已知曲线y?f?x?x?2处的切线的倾斜角为?,则f??2??3y?4?xx?12.
的垂直渐近线有条.dxx?1?lnx?2?.?
5?2??xsinx?cosx?dx?4?2.
三.计算(每小题5分,共30分)1.求极限limx??2x
?1?x????x?limx?0

x?sinxxe?x2?1?
2.求曲线y?ln?x?y?所确定的隐函数的导数y?.x3.求不定积分?
四.应用题(每题10分,共20分)1作出函数y?x?3x的图像.232dx
?x?1??x?3??
?a?0??xe?xdx
2.求曲线y?2x和直线y?x?4所围图形的面积.《高数》试卷1参考答案一.选择题
1b2b3a4c5d6c7d8a9a10c二.填空题1?22
?三.计算题1①e21163
3.4.arctanlnx?c5.22.y??x1x?y?13.ln|2
x?1x?3|?c
ln|x|?c?e?x
?x?1??c四.应用题
1.略2.s?18

《高数》试卷2(上)
.选择题(将答案代号填入括号内,每题3,301.下列各组函数中,是相同函数的是(.(af?x??xg?x??
(bf?x??22
x?1x?122
y?x?1
(cf?x??xg?x??x(sinx?cosx(df?x??lnxg?x??2lnx?sin2?x?1??x?1??
2.设函数f?x???2?2
x?1???x?1
x?1,则limfx?1
?x??().x?1
(a0(b1(c2(d不存在
3.设函数y?f?x?在点x0处可导,且f??x?0,曲线则y?f?x??x0,f?x0??处的切线的倾斜角为{}.(a0(b?2
(c锐角(d钝角
4.曲线y?lnx上某点的切线平行于直线y?2x?3,则该点坐标是(.??
1?1??(b2,?ln???2?2??2?x
(a?2,ln(c??1??1?
,ln2?(d?,?ln2??2??2?5.函数y?xe

及图象在?1,2?内是(.
(a单调减少且是凸的(b单调增加且是凸的(c单调减少且是凹的(d单调增加且是凹的
6.以下结论正确的是(.
(ax0为函数y?f?x?的驻点,x0必为函数y?f?x?的极值点.(b函数y?f?x?导数不存在的点,一定不是函数y?f?x?的极值点.(c函数y?f?x?x0处取得极值,f??x0?存在,则必有f??x0?=0.(d若函数y?f?x?x0处连续,f??x0?一定存在.1
7.设函数y?f?x?的一个原函数为xex,f?x?=(.2
1111
(a?2x?1?ex(b2x?ex(c?2x?1?ex(d2xex8.?f?x?dx?f?x??c,?sinxf?cosx?dx?(.
(af?sinx??c(b?f?sinx??c(cf?cosx??c(d?f?cosx??c9.f?x?为连续函数,?f??1?x?
?dx=(.?2???1??
(af?1??f?0?(b2??f?1??f?0???(c2??f?2??f?0???(d2?f?2??f?0??
????10.定积分?dx?a?b?在几何上的表示(.ab
(a线段长b?a(b线段长a?b(c矩形面积?a?b??1(d矩形面?b?a??1.填空题(每题4,20?ln?1?x2??
1.f?x???1?cosx?a?
x?0x?0
,x?0连续,a=________.
2.y?sin2x,dy?_________________dsinx.3.函数y?xx?12
?1的水平和垂直渐近线共有_______.
4.不定积分?xlnxdx?______________________.5.定积分?

1?1
xsinx?11?x22
?___________.
.计算题(每小题5,301.求下列极限:?
lim?1?2x?xlimx?01
?arctanx1xx???
2.求由方程y?1?xe所确定的隐函数的导数y?x.3.求下列不定积分:?tanxsec3xdx?ya
?0??xedx2x
.应用题(每题10,201.作出函数y?13
x?x的图象.(要求列出表格3
【篇二:上海交大版高等数学课后习题解答】

txt>第一章函数
1.设f(x?x2?1,求f(x2?f(x?
解答:f(x2?(x22?1?x4?1,?f(x??[x2?1]2?x4?2x2?1所属章节:第一章第一节难度:一级aex?be?x
2.设f(x?,求f(x?f(?xa?baex?be?xae?x?be?(?xae?x?bex?解答:f(x?f(?x?a?ba?ba?baex?be?xae?x?be?(?x
f(x?f(?x???ex?e?xa?ba?b22所属章节:第一章第一节难度:一级

?2x?1?x?0,1?3.设?(x??20?x?1,?(3,?(2,?(0,?(?
2?x?11?x?3,?1解答:?(3?2,?(2?1,?(0?1,?(?2所属章节:第一章第一节难度:一级
4.求下列函数的定义域:
1y?2x11?xy?log;(2),(a?0,a?1a2x?3x?221?x3
y?3?2x14
y?arcsin.5lg(1?x
解答:(1)由x2?3x?2?0解得定义域为???,1??1,2??2,???2)由1?x?0,1?x?0解得定义域为??1,1?1?x3)由2?x?0,1?x?0,1?x?1解得定义域为??2,0?
4)由3?x?0,3?2x?1解得定义域为[?1,3]5?0,1?所属章节:第一章第一节难度:一级
5.下列各题中,函数f(xg(x是否相同?x1f(x?lgx2g(x?2lg2f(x?xg(x
3f(x?elnxg(x?x.
解答:(1f(x中的x可为一切实数,g(x中的x要求大于零,即定义域不同,故函数不同;
2f(x将负数对应负数,而g(x把负数对应正数,对应法则不同,故函数不同;
3f(x中的x要求大于零,g(x中的x可为一切实数,即定义域不同,故函数不同。所属章节:第一章第一节难度:一级
6.下列函数中哪些是偶函数,哪些是奇函数,哪些是非奇非偶函数?1y?x2(1?x22y?3x2?x31?xex?e?x
(a?0,a?14y?3y?loga1?x25y?x2cosx?16y?ln(x?.
解答:(1)偶;2)非奇非偶;3)奇;4)偶;5偶;6)奇所属章节:第一章第一节难度:一级

7.下列函数中哪些是周期函数?对于周期函数指出其周期t1y?1?tanx;(2y?cos(3x?13y?xsinx4y?sin2x.所属章节:第一章第一节难度:一级
8.求下列函数的反函数:1y?x2?2x;(2y2x3
y?ln(x;(4y?x.2?1解答:(1)由y?x2?2x,得y?(x?12?1,解得x?1
所以当y??1时,反函数y?1y??1时,反函数y?12
)当y?1时,y?y?1时,y?3
)由y?ln(x
,得ey?x(ey?x2?1?x2e2y?1ey?e?yex?e?x?解得x?,所以反函数为y?2ey22xyy2x
4)由y?x解得2x?,即x?log2,所以反函数为y?log21?x1?y1?y2?1
所属章节:第一章第一节难度:二级
9.下列初等函数由哪些简单函数复合而成?21
y?2y?cosx3
3y?ex;(4y?lnsin2x
5y?sin(3x?16y?arctane解答:(1
y?u?2?x22y?cosu,u?22?1x.2x3y?eu,u?x23
4y?lnu,u?sinv,v?2x5y?u2,u?sinv,v?3x?16y?arctanu,u?ev,v??所属章节:第一章第二节1x2难度:一级
10.设f(x?ex,证明:

1f(x?f(y?f(x?y2f(x?f(x?y.f(y解答:(1f(x?f(y?ex?ey?ex?y?f(x?yf(xex
2?y?ex?y?f(x?yf(ye所属章节:第一章第二节难度:一级
11.设f(x?ln(x?1,证明:f(x2?2?f(x?2?f(x.
解答:f(x2?2?f(x?2?ln((x2?2?1?ln((x?2?1?ln(x2?1?ln(x?1x2?1?ln?ln(x?1?f(xx?1所属章节:第一章第二节难度:一级
12.设f(x具有性质:f(x?y?f(x?f(y,证明:必有f(0?0pf(x?f(pxp为任意正整数)
解答:在f(x?y?f(x?f(y中,令x?0,即得f(0?0f(x?y?f(x?f(y中,令y?x,即得f(2x?2f(x
f(x?y?f(x?f(y中,令y?2x,结合上式,即得f(3x?3f(x
设对正整数k,有f(kx?kf(x,则在f(x?y?f(x?f(y中,令y?kx结合假设有f((k?1x?(k?1f(x,由数学归纳法得证。所属章节:第一章第二节难度:二级
13.设fn(x?f(f(???f(x,若f(x?a?bx,证明:n
a(bn?1nfn(x??bx(b?1.b?1
a(b2?12?bx,即等式成立;解答:当n?2时,f2(x?f(f(x?a?b(a?bx?a?ab?bx?b?12
n?k时等式成立,即a(bk?1kfk(x??bx,则当n?k?1时,b?1fn(x?ff???(fx(
na(bk?1kabk?1?(1?f(fkx?a?b?bx??bk?1x]即等式也成立,得证。b?1b?1
所属章节:第一章第二节难度:二级
14.验证下列恒等式:
1sinh(x?y?sinhxcoshy?coshxsinhy2cosh(x?y?coshxcoshy?sinhxsinhyx?yx?ycosh22

x?yx?ysinh4coshx?coshy?2sinh.223sinhx?sinhy?2sinhex?e?xex?e?x
,coshx?解答:由定义sinhx?,从右往左证明22ex?e?xey?e?yex?e?xey?e?yex?y?e?(x?y
sinhxcoshy?coshxsinhy????sinh(x?y,即证22222得(1)式;类似可证其他三式。所属章节:第一章第二节难度:二级
第二章极限与连续
1.用“??n”定义验证下列极限:
【篇三:大一上学期(第一学期高数期末考试题及答案】

class=txt>1.x?x0时,??x?,??x?都是无穷小,则当x?x0d)不一定是无穷小.(a(c?x????x?(b(d?2
?x???2?x??(x
ln?1??(x??(x?12?(x
?sinx?x?alim??x?asina??2.极限的值是(c.a1bex?0x?0cecota
detana
?sinx?e2ax?1?f(x??x?a?3.
x?0处连续,则a=d.celim

h
a14.
b0d?1?
f(a?h?f(a?2h
f(x在点x?a处可导,那么h?0a.
a3f?(af?(a
b2f?(a13f?(a(cd
二、填空题(本大题有4小题,每小题4分,共16分)5.极限x?06.exylim
ln(x?a?lnax
(a?01
的值是a.
y(x,则导函数y???ylnx?cos2x确定函数x.xyxe?lnx
7.直线l过点m(1,2,3且与两平面x?2y?z?0,2x?3y?5z?6都平行,则直?x?1
2sin2x?y?yexy
线l的方程为8.求函数1

2?
y?2?1?
z?3?1.
)和(1+?.
三、解答题(本大题有4小题,每小题8分,共32分)1
y?2x?ln(4x
9.计算极限x?0lim
(1?xx?ex.11
解:x?0lim
(1?xx?ex
?elimex
ln(1?x?1?1x?0x
?elimx
ln(1?x?xx2x?0??e2f(x?
10.f(x[ab]上连续,且xx
?(x?tf(tdt

a
x?[a,b]
,试求出f??(x解:
f(x?x?f(tdt??tf(tdtaxax
f?(x??a
f(tdt?xf(x?xf(x?cosxsinx3?a
f(tdt
f??(x?f(x?11.x.s?2??xcs123xx?2????oi?idx

12i
四、解答题(本大题有4小题,每小题8分,共32分)2xs?2?2
dxxx?12
12.1x3.
11t1t2?t
原式?1232(??11t2dt33??212dt1?ty?2
?arcsint2122x2??

6
1?x的极值与拐点.13.求函数解:函数的定义域(-?+?y??
2(1?x(1?x(1?x22
y???
?4x(3?x(1?x232
y??0x1=1,x2=-1
y??(1?0x=1是极大值点,y??(?1?0x=-1是极小值点12
极大值y(1?1,极小值y(?1??1y???0333323
故拐点(-3-2),(00)(3214.求由曲线解:x3y?2x
4y?3x?x所围成的平面图形的面积.324
?3x?x,x?12x?4x?0,
x(x?6(x?2?0,x1??6,x2?0,x3?2.33
02xx22
s??(?3x?xdx??(3x?x?dx?6044?(x4

16?3213x??472x133
0?6?(32x?2x33?x41620
?45?2
15.设抛物线32
y?4?x上有两点a(?1,3ab上,求一点p(x,y使?abp的面积最大.?x?2x?32
ab连线方程:y?2x?1?02x?y?1
,在弧ab?45pab的距离?abp的面积b(3,?5
2
?(?1?x?312?45??x?2x?35
?2(?x?2x?32
s(x?
s?(x??4x?4x?1s?(x?0s??(x??4?0x?1s(x取得极大值也是最大值此时y?3所求点为(13
另解:由于?abc的底ab一定,故只要高最大而过c点的抛物线的切线与ab平行时,高可达到最大值,问题转为求c(x04?x0,使f?(x0??2x0??5?2
六、证明题(本大题4分)16.x?0,试证e2x3?1
??2,解得x0?1,所求c点为(1,3(1?x?1?x.
f??(x?0,因此f?(x在(02x
证明:设f(x?e2x
(1?x?(1?x,x?02x2x
f?(x?e(1?2x?1f??(x??4xex?0,
+?)内递减。在(0+?)内,f?(x?f?(0?0,f(x在(0+?)内递减,在(0+?)内,f(x?f(0,ee2x2x
(1?x?(1?x?0亦即当x0时,e(1?x?1?x试证
(1?x?1?x.

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