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问题 解答题
已知曲线C1：y=
x2
e
+e（e为自然对数的底数），曲线C2：y=2elnx和直线l：y=2x．
（1）求证：直线l与曲线C1，C2都相切，且切于同一点；
（2）设直线x=t（t＞0）与曲线C1，C2及直线l分别相交于M，N，P，记f（t）=|PM|-|NP|，求f（t）在[e-3，e3]上的最大值；
（3）设直线x=em（m=0，1，2，3┅┅）与曲线C1和C2的交点分别为Am和Bm，问是否存在正整数n，使得A0B0=AnBn？若存在，求出n；若不存在，请说明理由． （本小题参考数据e≈2.7）．
答案

（1）证明：y=

x2
e
+ey′=
2x
e
由y′=
2x
e
=2得x=e（2分）

在C1上点（e，2e）处的切线为y-2e=2（x-e），即y=2x（3分）

又在C2上点（e，2e）处切线可计算得y-2e=2（x-e），即y=2x

∴直线l与C1、C2都相切，且切于同一点（e，2e）（4分）

（2）f(t)=

t2
e
+e-2t-(2t-2elnt)=
t2
e
+2elnt-4t+ef′(t)=
2t
e
+2e
1
t
-4=
2t2+2e2-4et
et
=
2(t-e)2
et
≥0（6分）

∴f（t）在[e-3，e3]上递增

∴当t=e3时f(t)max=

e6
e
+2elne3-4e3+e=e5-4e3+7e（8分）

（3）AnBn=

(en)2
e
+e-2elnen=
(e2)n
e
+e-2ne

设上式为g（n），假设n取正实数，则g′(n)=

(e2)n
e
•lne2-2e=
2(e2n-e2)
e

当n∈（0，1）时，g′（n）＜0，∴g（n）递减；

当n∈（1，+∞），g′（n）＞0，∴g（n）递增．（12分）

g(0)=A0B0=e+

1
e
g（1）=2e-2e=0g(2)=e3+e-4e=e3-3e≈2.72e-3e＞2e＞e+
1
e

∴不存在正整数n，使得g（m）=g（0）

即AnBn=A0B0．（14分）

单项选择题 A1/A2型题

头痛如雷鸣，头面起核，谓之（）

A.厥头痛

B.雷头风

C.首风

D.偏头风

E.以上均不是
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单项选择题

In a country that defines itself" by ideals, not by shared blood, who should be allowed to work and live here In the wake of the Sept. 11 attacks these questions have never seemed more pressing.
On Dec. 11, 2001, as part of the effort to increase homeland security, federal and local authorities in 14 states staged " Operation Safe Travel"—raids on airports to arrest employees with false identification. In Salt Lake City there were 69 arrests. But those captured were anything but terrorists, most of them illegal immigrants from Central or South American. Authorities said the undocumented worker’s illegal status made them open to blank mall by terrorists. Many immigrants in Salt Lake City were angered by the arrests and said they felt as if they were being treated like disposable goods. Mayor Anderson said those feelings were justified to a certain extent. "We’re saying we want you to work in these places. We’re going to look the other way in terms of what our laws are, and then when it’s convenient for us, or when we can try to make a point in terms of national security, especially after Sept. 11, then you’re disposable. There are whole families being uprooted for all of the wrong reasons. " Anderson said.
If Sept. 11 had never happened, the airport workers would not have been arrested and could have gone on quietly living in America, probably indefinitely. Ana Castro, a manager at a Ben & Jerry’s ice cream shop at the airport, had been working 10 years with the same false Social Security card when she was arrested in the December airport raid. Now she and her family are living under the threat of deportation. Castro’s case is currently waiting to be settled. While she awaits the outcome, the government has granted her permission to work here and she has returned to her job at Ben & Jerry’s.

According to the author, the United States claims to be a nation .

A. composed of people having different values
B. encouraging individual pursuits
C. sharing common interests
D. founded on shared ideals
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