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竞赛选拔试题及参考答案

发布时间:2014-04-21 13:14:42  

高等数学竞赛选拔试题(每题10分)

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因为

xlim1?xf(x)?limx2f(x)?2, ?lim?limx???xx???11?2xx???2所以limf(x)?0,limxf(x)?2.于是, x???

(1)lim?1?f(n)?n??n2?lim?1?f(x)??ex???x2x???limx2f(x)?e2.

11?1?f()f()f()2??xx2(2)lim ?limlim?lim?2???x2x?0?x?ln(1?x)x?0??0?x?0?x?ln(1?x)xx?ln(1?x)x????

2x2x(1?x)?limt2f(t)?lim?2?lim?4. t???x?0?x?0?x1?1?x

1

解 x?nm

12233

?????1?Cmnx?Cm(nx)?Cm(nx)?

11??2nm

(1?x)?

1?

?

2233

?1?C1nmx?Cn(mx)?Cn(mx)???limx?0

??

?n(n?1)2m(m?1)2?22

m?nx?o(x)??22?

?lim?

x?0x2

mn

2

?n(n?1)2m(m?1)2?o(x)mn(n?m)

m?n??2

?m2n222x???lim??.

?x?0m?n2mnmn

f(x)

?limg(x)x?1

11

(x?1)3(3x?5)3

(1)lim

x?1

ln[1?(x?1)]

?lim?1,??x?1k(x?1)k(x?1)

4

2(x?1)3

所以

k?2,??

4. 3

1?x(2)lim?

?k??tanx

?x?0k??sinx??

?(tanx?sinx)

x?0(k??sinx)xlim

??(tanx?sinx)?lim?1??x?0k??sinx??

1

lim

x?0(k??sinx)x1?x

?e

?e

?tanx(1?cosx)x?0(k??sinx)xlim

??x?x2

?

?e?e2k?e

13

2

max(lx解 1?1?lnx,0?x???xlnx?x?C,0?x?e?e?, ?1?1?,?1?)1?x?emax(lnx,1)dx??x?C2,?x?e?ee???lnx,x?e?xlnx?x?C3,x?e????

由x?11处连续得:C1?C2?;由x?e处连续得:C3?C2?e,所以 ee

11??xlnx?x?C?,0?x??ee?

1?max(lnx,1)dx?x?C,?x?e??e??xlnx?x?C?e,x?e??

xarcsinexx?x?xx?xdx??arcsined(e)??earcsine?e?ex??x

x??e?xarcsien??xed(?)?e?xxarcesi?n?e??C1)

或解

arcsinexx?x?xxdx??arcsined(e)??earcsine??ex?x

1?t,x?ln(1?t2), 2

arcsinex111?t?xx?xxdx??earcsine?dt??earcsine?ln?C ?ex?1?t221?t

1en?

??earcsi2?xx?C

3

解 令1?t,当x?

0时,

x

??

t??

t t?1?C2??

?t?t??arcsin1?x?arcsin?C 2x

当x?0

时,

1?x??arcsin?C

2x

解cosxxsinx?cosx?, f(x)?()??2xx

2?tanx[f(2x)tanx?2f(2x)]dx?f(2x)tanxdx?2?f?(2x)tanxdx ??

??f(2x)tan2xdx??tanxd[f(2x)]

??f(2x)tan2xdx?f(2x)tanx??f(2x)sec2xdx

??f(2x)(tan2x?sec2x)dx?f(2x)tanx

1cos2x2xsin2x?cos2x ???f(2x)dx?f(2x)tanx???(?)tanx?C。222x(2x)

4

解:I??1dx1

?1(1?ex)(1?x2x??t?dt?1(1?e?t)(1?t2)??1dx

)?1(1?e?x)(1?x2)

?11?11?

2??1??(1?ex)(1?x2)?(1?e?x)(1?x2)??dx

?11dx?

2??11?x2?4.

解:设u?x?t,

x0x

则I??tf(x?t)dt???(x?u)f(u)du??(x?t)f(t)dt

0x0

?x

??x?tdt?xarctanx?1ln(1?x2),0?x?1

?I??2

?01?t2

?1x?t

?dt??x?1ln2,x

??01?t242?1

证明F(?x)??1

?1?x?sintf(t)dt

令t??u?1

?1?x?sinuf(?u)du

???1

?1x?sinuf(u)du???1?1x?sintf(t)dt??F(x)

?F(x)也是奇函数.

5

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