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经济数学基础形成性考核册及参考答案
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在(1,1)的切线方程是 .答案:y?211x? 224.设函数f(x?1)?x?2x?5,则f?(x)?____________.答案:2x 5.设f(x)?xsinx,则f??()?__________.答案:?(二)单项选择题 1. 函数y?π2π 2x?1的连续区间是( )答案:D
x2?x?2A.(??,1)?(1,??) B.(??,?2)?(?2,??)
C.(??,?2)?(?2,1)?(1,??) D.(??,?2)?(?2,??)或(??,1)?(1,??) 2. 下列极限计算正确的是( )答案:B A.limxxx?0?1 B.lim?x?0xx?1
C.limxsinx?01sinx?1 D.lim?1
x??xx3. 设y?lg2x,则dy?( ).答案:B A.
11ln101dx B.dx C.dx D.dx 2xxln10xx4. 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.当x?0时,下列变量是无穷小量的是( ). 答案:C
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A.2x B.(三)解答题 1.计算极限
sinx C.ln(1?x) D.cosx xx2?3x?21(x?2)(x?1)x?2??(1)lim = = limlimx?1x?1(x?1)(x?1)x?1(x?1)x2?12x2?5x?61(x?2)(x?3)x?3(2)lim2=lim = lim =
x?2x?6x?8x?2(x?2)(x?4)x?2(x?4)2(3)limx?0(1?x?1)(1?x?1)1?x?1=lim x?0xx(1?x?1) =limx?0?x?11=lim??
2x(1?x?1)x?0(1?x?1)1?35?2x?3x?51xxlim?lim?(4) x??3x2?2x?4x??2433??2xx2(5)limsin3x5xsin3x33?lim=
x?0sin5xx?03xsin5x55x2?4(x?2)(x?2)(6)lim?lim?4
x?2sin(x?2)x?2sin(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:
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(1)y?x?2?log2x?2,求y?
2x2答案:y??2x?2ln2?(2)y?x1 xln2ax?b,求y?
cx?d答案:y?=
a(cx?d)?c(ax?b)ad?cb ?(cx?d)2(cx?d)213x?513x?5,求y?
12(3)y?答案:y?=(3x?5)? y???32(3x?5)3
(4)y?答案:y??(5)y?ex?xex,求y?
12xax?(x?1)ex
sinbx,求dy
axax答案:y??(e)?sinbx?e(sinbx)?
?aeaxsinbx?eaxcosbx?b
?eax(asinbx?bcosbx) dy?eax(asinbx?bcosbx)dx
(6)y?e?xx,求dy
1x311x?2ex)dx 答案:dy?(2x(7)y?cosx?e?x,求dy 答案:dy?(2xen2?x2?sinx2x)dx
(8)y?sinx?sinnx,求y? 答案:y?=nsinn?1xcosx+cosnxn=n(sinn?1xcosx?cosnx)
(9)y?ln(x?1?x2),求y?
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答案:
1?11x22??(1?(1?x)2x)y??(x?1?x)?(1?)22222x?1?xx?1?xx?1?x1?x121?11?x2
(10)y?2cot1x?1?3x2?2xx3,求y?
ln21?21?6??x?x 答案:y?126x2sinx4.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
222cot1x5答案:解:方程两边关于X求导:2x?2yy??y?xy??3?0
y?3?2xdx
2y?x(2y?x)y??y?2x?3 , dy?(2)sin(x?y)?exy?4x,求y?
答案:解:方程两边关于X求导cos(x?y)(1?y?)?exy(y?xy?)?4
(cos(x?y)?exyx)y??4?yexy?cos(x?y)
4?yexy?cos(x?y) y??xyxe?cos(x?y)5.求下列函数的二阶导数: (1)y?ln(1?x),求y??
22?2x2答案:y??? 22(1?x)(2)y?1?xx,求y??及y??(1)
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3?1?答案:y???x2?x2,y??(1)?1
4453作业(二)
(一)填空题 1.若2.
?f(x)dx?2x?2x?c,则f(x)?___________________.答案:2xln2?2
?(sinx)?dx?________.答案:sinx?c ?f(x)dx?F(x)?c,则?xf(1?x2)dx? .答案:?1F(1?x2)?c 23. 若
de4.设函数ln(1?x2)dx?___________.答案:0 ?dx15. 若P(x)??0x11?t2dt,则P?(x)?__________.答案:?11?x2
(二)单项选择题
1. 下列函数中,( )是xsinx2的原函数. A.
11cosx2 B.2cosx2 C.-2cosx2 D.-cosx2 22答案:D
2. 下列等式成立的是( ).
A.sinxdx?d(cosx) B.lnxdx?d()
C.2dx?x1x1d(2x) ln2 D.
1xdx?dx
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.x1?xdx C.xsin2xdx D.
??2?x?1?x2dx
答案:C
4. 下列定积分计算正确的是( ).
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