《经济数学基础12》作业讲解
来源网站:百味书屋
2016-10-18 16:25:55

篇一:《经济数学基础12》作业
经济数学基础
形 成 性 考 核 册
专业:工商管理
学号: 1513001400168
姓名: 王浩
河北广播电视大学开放教育学院
(请按照顺序打印,并左侧装订)
作业一
(一)填空题 1.limx?0x?sinx?___________________.答案:0 x
?x2?1,x?02.设f(x)??,在x?0处连续,则k?________.答案:1 ?k,x?0?
3.曲线y?x+1在(1,2)的切线方程是答案:y?11x? 22
__.答案:2x 4.设函数f(x?1)?x2?2x?5,则f?(x)?__________
5.设f(x)?xsinx,则f??()?__________.答案:?π
2π 2
(二)单项选择题
1. 当x???时,下列变量为无穷小量的是( )答案:D
x2
A.ln(1?x) B.x?1
C.e?1
xD.sinxx
2. 下列极限计算正确的是()答案:B A.limx?0xx?1B.lim?x?0xx?1 C.limxsinx?01sinx?1 D.lim?1 x??xx
3. 设y?lg2x,则dy?().答案:B
A.11ln101dx B.dx C.dx D.dx 2xxln10xx
4. 若函数f (x)在点x0处可导,则( )是错误的.答案:B
A.函数f (x)在点x0处有定义B.limf(x)?A,但A?f(x0) x?x0
C.函数f (x)在点x0处连续 D.函数f (x)在点x0处可微
5.若f()?x,f?(x)?( ). 答案:B
A.
1x1111??B.C. D. xxx2x2
(三)解答题
1.计算极限
x2?3x?21x2?5x?61?? (2)lim2? (1)limx?1x?2x?6x?822x2?1
2x2?3x?51?x?11? (3)lim??(4)lim2x??x?0x23x?2x?43
sin3x3x2?4? (6)lim(5)lim?4 x?0sin5xx?25sin(x?2)
1?xsin?b,x?0?x?2.设函数f(x)??a,x?0,
?sinxx?0?x?
问:(1)当a,b为何值时,f(x)在x?0处有极限存在?
(2)当a,b为何值时,f(x)在x?0处连续.
答案:(1)当b?1,a任意时,f(x)在x?0处有极限存在;
(2)当a?b?1时,f(x)在x?0处连续。
3.计算下列函数的导数或微分:
(1)y?x?2?log2x?2,求y? 答案:y??2x?2ln2?
(2)y?x2x21 xln2ax?b,求y? cx?d
答案:y??ad?cb 2(cx?d)
1
3x?5,求y? (3)y?
答案:y???3
2(3x?5)3
(4)y?
答案:y??x?xex,求y? 1
2x?(x?1)ex
(5)y?eaxsinbx,求dy
答案:dy?eax(asinbx?bcosbx)dx
(6)y?e?xx,求dy 1
x
11
2ex)dx 答案:dy
?x
(7)y?cosx?e?x,求dy
答案:dy?(2xe?x?22sinx
2x)dx
(8)y?sinnx?sinnx,求y?
答案:y??n(sinn?1xcosx?cosnx)
(9)y?ln(x??x2),求y? 答案:y??1
?x
sin1
x2 (10
)y?2,求y? 1
x
答案:y???2sinln2
x211?31?52cos?x?x6 x26
4.下列各方程中y是x的隐函数,试求y?或dy
(1)x?y?xy?3x?1,求dy 答案:dy?22y?3?2xdx 2y?x
xy(2)sin(x?y)?e?4x,求y? 4?yexy?cos(x?y)答案:y?? xexy?cos(x?y)
5.求下列函数的二阶导数:
(1)y?ln(1?x2),求y?? 2?2x2
答案:y??? 22(1?x)
(2)y?1?x
x,求y??及y??(1) 3?21?2答案:y???x?x,y??(1)?1 44
53
作业2
一、填空题
1、若∫f(x)dx=2x+2x+c ,则x2、∫(sinx)'
3、若∫f(x)dx=F(x)+c,则∫xf(1-x22de2ln(x?1)dx?0. 4、 ?1dx
5、若P?
x??
?01xdt,,则P'?
x??
篇二:《经济数学基础12》作业讲解(一)(1)
经济数学基础作业讲解(一)
一、填空题 1.lim
x?sinx
x
x?0
?___________________.
解:lim
x?sinx
x
x?0
sinx??
?lim?1???1?1?0 x?0x??
答案:0
?x2?1,
2.设f(x)??
?k,?
x?0
x?0
x?0x?0
2
,在x?0处连续,则k?________.
解:limf(x)?lim(x?1)?1?f(0)?k 答案:1 3.曲线y?
x在(1,1)的切线方程是 .
解:切线斜率为k?y?|x?1?
12
12
?1
?
12
,所求切线方程为y?1?
12
(x?1)
答案:y?x?
4.设函数f(x?1)?x2?2x?5,则f?(x)?____________. 解:令x?1?t,则f(t)?t2?4,f?(t)?2t 答案:2x
5.设f(x)?xsinx,则f??()?__________
2π
.
解:f?(x)?sinx?xcosx,f??(x)?2cosx?xsinx,f???答案:?
π2
????
???
2?2?
二、单项选择题
1. 当x???时,下列变量为无穷小量的是( ). A.ln(1?x) B.解:lim
sinxx
?lim
1x
x
2
?1
x?1
C.ex D.
1x
2
sinxx
sinxx
?0
x???x???
?sinx,而lim
x???
?0,|sinx|?1,故lim
x???
答案:D
2. 下列极限计算正确的是().
A.lim
xx
x?0
?1B.lim
x?0
x
?
x
?1
C.limxsin
x?0
1x
?1 D.lim
sinxx
x??
?1
1x
sinxx
解:lim
xx
x?0
不存在,lim?
x?0
xx
?lim?
x?0
xx
?1,limxsin
x?0
?0,lim
x??
?0
答案:B
3. 设y?lg2x,则dy?(). A.
12x
dx B.
22xln10
?
1xln10
1xln10
dx C.
ln10x
dx D.
dx
1x
dx
解:y??答案:B
,dy?y?dx?
1xln10
4. 若函数f (x)在点x0处可导,则( )是错误的.
A.函数f (x)在点x0处有定义B.limf(x)?A,但A?f(x0)
x?x0
C.函数f (x)在点x0处连续 D.函数f (x)在点x0处可微 解:可导等价于可微,可导必连续,但(B)为不连续 答案:B 5.若f?A.
1x
2
?1?
??x,则f?(x)?( ). ?x?
B.?
1x
1x
2
C.
,f?(t)??
1x
2
D.?
1x
解:令
?t,则f?t??
1t
1t
答案:B 三、解答题 1.计算极限 (1)lim
x?3x?2x?1
22
x?1
x?2x?1
12
解:原式?lim
(x?1)(x?2)(x?1)(x?1)
x?1
?lim
x?1
?? (约去零因子)
(2)lim
x?5x?6x?6x?8
2
2
x?2
解:原式?lim
(x?2)(x?3)(x?2)(x?4)
x?2
?lim
x?3x?4
x?2
?
12
(约去零因子)
(3
)lim
1x
12
x?0
解:原式?lim
x?0
?? (分子有理化)
(4)lim
x?3x?53x?2x?4
2
2
5
x??
21解:原式?lim? (抓大头)
x??43
3??2
xx
sin3x
(5)lim
x?0sin5x
3x3
? (等价无穷小) 解:原式?lim
x?05x5
1?
32
?
(6)lim
x?4sin(x?2)
2
x?2
解:原式?lim
x?2sin(x?2)
x?2
(x?2)?4 (重要极限)
1?xsin?b,?x?
2.设函数f(x)??a,
sinx??
x?
x?0x?0, x?0
问:(1)当a,b为何值时,f(x)在x?0处有极限存在? (2)当a,b为何值时,f(x)在x?0处连续.
sinxx
1??
即当b?1,?1,f(0?)?lim??xsin?b??b,f(0?)?f(0?),
x?0x??
解:(1)f(0?)?lim
x?0
?
a任意时,f(x)在x?0处有极限存在;
(2)f(0?)?f(0?)?f(0),即当a?b?1时,f(x)在x?0处连续. 3.计算下列函数的导数或微分: (1)y?x?2?log解:y??2x?2ln2?
x
2x
2
x?2,求y? 1
2
xln2
(注意2为常数)
2
(2)y?
ax?bcx?d
,求y?
a(cx?d)?(ax?b)c
(cx?d)
2
解:y???
(ax?b)?(cx?d)?(ax?b)(cx?d)?
(cx?d)
13x?5
2
??
ad?cb(cx?d)
2
(3)y?,求y?
1?3????12
解:y???(3x?5)???(3x?5)2?3?
2??
x
(4)y?解:y??
x?xe,求y?
(e?xe)?
xx
x
?(x?1)e
(5)y?eaxsinbx,求dy
解:y??(eax)?sinbx?eax(sinbx)??eaxasinbx?eaxcosbx?b
dy?y?dx?e(asinbx?bcosbx)dx
1
ax
(6)y?ex?xx,求dy
?1?解:y??ex??2??
?x?1
dy
?1x
2
1
ex)dx
(7)y?cos解:y???(sin
x?e
?x
2
,求dy
?x
2
?e(?2x),dy?(2xe
?x
2
?
sin2x
x
)dx
(8)y?sin
n
x?sinnx,求y?
n?1
解:y??n(sinx)cosx?(cosnx)?n?n(sin
n?1
xcosx?cosnx)
(9)y?ln(x?1?x2),求y?
解:y??
??1??sin
1x
(10
)y?2?
,求y?
解:y?2
y??2
sin
1
sin
1x
?x
?
12
1
?x6?
3
5
1
1??1?1?1?ln2sin1?x
(ln2)?cos???2??x2?x6??22xcos?x??x?26xx?4.下列各方程中y是x的隐函数,试求y?或dy (1)x2?y2?xy?3x?1,求dy
解:方程两边对x求导,得 2x?2y?y??(y?xy?)?3?0,
y?3?2x2y?x
y?3?2x2y?x
y??,dy?dx
(2)sin(x?y)?exy?4x,求y?
解:方程两边对x求导,得 cos(x?y)(1?y?)?exy(y?xy?)?4,
y??
4?yexe
xy
xy
?cos(x?y)
?cos(x?y)
5.求下列函数的二阶导数: (1)y?ln(1?x2),求y??
2x1?x
2
解:y??,y???
2?2x
2
22
(1?x)
(2)y?
1?xx
?12
,求y??及y??(1)
1
解:y?x
?x2,y???
12
x
?
32
?
12
x
?
12
,y???
34
x
?
52
?
14
x
?
32
,y??(1)?1
篇三:《经济数学基础12》作业讲解(二)
经济数学基础作业讲解(二)
一、填空题
1.若?f(x)dx?2x?2x?c,则f(x)?___________________.
解:f(x)?(2x?2x?c)??2xln2?2 答案:2xln2?2 2.
?(sinx)?dx?
________.
解:因为?F?(x)dx?F(x)?c,所以?(sinx)?dx?sinx?c 答案:sinx?c
3. 若?f(x)dx?F(x)?c,则?e?xf(e?x)dx? . 解:令 u?e?x,du??e?xdx, 则
?e
?x
f(e
?x
)dx??
?
f(u)du??F(u)?c??F(e
?x
)?c
答案:?F(e?x)?c 4.设函数
d
e2
dx
?1
ln(1?x)dx?__________
_.
解:因为?ed
2
1
ln(1?x2)dx为常数,所以edx
?1
ln(1?x)dx?0
答案:0 5. 若P(x)?
?
01x
t,则P?(x)?__________.
?t
2
解:P?(x)?
d?0dx
x
t?
d?dx??x???0????
答案:?1
2
?x
二、单项选择题
1. 下列函数中,()是xsinx2的原函数. A.
1222
2
cosx B.2cosx C.-2cosx 解:因为(cosx2)???2xsinx2
,所以(?
12
2
cosx)??xsinx2
答案:D
D.-12
cosx2
2. 下列等式成立的是( ).A.sinxdx?d(cosx) B.lnxdx?d(C.2xdx?
1ln2
d(2)D.
x
1x
)
1x
dx?d
x
解:d(cosx)??sinxdx,d()??
112
dx,d(2)?
2ln2dx,xx
?
x
x
答案:C
3. 下列不定积分中,常用分部积分法计算的是(). A.?cos(2x?1)dx, B.?x?x2dx C.?xsin2xdx 答案:C
4. 下列定积分计算正确的是().A.?1
2xdx?2 B.16?1?
?1
dx?15
C.?
?
23
D.??
sin??
(x?x)dx?0xdx?0
??
答案:D
5. 下列无穷积分中收敛的是( ). A.?
??1?x
1
x
dxB.?
??11
x
2
dx C.?
? D.0
edx?
??1
sinxdx解:?
??11
x
2
dx??
1??
x
?1
1
答案:B 三、解答题
1.计算下列不定积分 x(1)?
3e
x
dx
x
3
x
解:原式xx
???3???e?dx?e?c?1?3?????ln3ln3?1c?e?e
(2)?
(1?x)
2
x
dx
解:原式???335
x2?42
?dx?x2?x2?c ?
352
(3)?
x?4x?2
dx
D.?x1?x
2
dx
解:原式?(4)?
1
?(x?2)dx?
dx
12
x?2x?c
2
1?2x
1
解:原式??
2
?(1?2x)d(1?2x)??
?1
12
ln?2x?c
(5)?x2?x2dx 解:原式?
1
12
?(2?x2
xdx
)d(2?x)?
2
2
13
3
(2?x)2?c
2
(6)?
sin
x
解:原式?2?sin(7)?xsin
x2dx
??2cos
c
解:原式??2?xdcos(8)?ln(x?1)dx
x2
??2xcos
x2
?2?cos
x2
dx??2xcos
x2
?4sin
x2
?c
解:原式?xln(x?1)?2.计算下列定积分 (1)??xx
?12
?
1??
dx?xln(x?1)???1??dx?(x?1)ln(x?1)?x?c x?1x?1??
x
解:原式?
1
?
1?1
(1?x)dx?
?
2
1
?x2?15(x?1)dx?2???x??2??
22?2?1
2
(2)?
21
exx
2
x
2
1
解:原式=-?exd
1
1x
1
2
=-ex
1
=e?
(3)?
e1
3
1x?lnx
x
解:原式?
?
e1
3
x)?|1?2(2?1)?2
e
3
?
(4)?
20
xcos2xdx
?
20
解:原式?
e
1
?2
xdsin2x?
12
?
xsin2x|02?
1
?
20
?2
sin2xdx?0?
14
?
cos2x|02??
12
(5)?xlnxdx
1
解:原式?
4
?
e
1
lnxd
x
2
2
?
x
2
2
lnx|?
e
1
1
?2
e
1
x
2
1x
dx?
e
2
2
?
14
x|1?
2e
14
(e?1)
2
(6)?(1?xe?x)dx
解:原式?4??xde
4
?x
?4?xe
?x
|??edx?4?4e
40
4
?x?4
?e
?x
|0?5?5e
4?4
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