问题 解答题

已知函数f(x)=x2-alnx(常数a>0).

(Ⅰ)当a=3时,求曲线y=f(x)在点(1,f(x))处的切线方程;

(Ⅱ)讨论函数f(x)在区间(1,ea)上零点的个数(e为自然对数的底数).

答案

(Ⅰ)当a=3时,f(x)=x2-3lnx,

∴f'(x)=2x-

3
x
(1分)

∴fˊ(1)=-1

又∵f(1)=1,

∴曲线y=f(x)在点(1,f(1))处的切线方程为y-1=-(x-1).

即x+y-2=0.--------------------------------3分

(Ⅱ)(1)下面先证明:ea>a(a≥0).

设g(a)=ea-a(a≥0),则g′(a)=ea-1≥e0-1=0(a≥0),且仅当g′(a)=0⇔a=0,

所以g(a)在[0,+∞)上是增函数,故g(a)≥g(0)=1>0.

所以ea-a>0,即ea>a(a≥0).------------------------------5分

(2)因为f(x)=x2-a lnx,

所以f′(x)=2x-

a
x
=
2x2-a
x
=
2(x-
2a
2
)(x+
2a
2
x

因为当0<x<

2a
2
时,fˊ(x)<0,当x>
2a
2
时,1,fˊ(x)>0.

a
2
<a<ea<e2a(a≥0,a<2a)⇒
2a
2
<ea

所以f(x)在(0,

2a
2
]上是减函数,在[
2a
2
,+∞)是增函数.

所以f(x)min=f(

2a
2
)=
a
2
(1-ln
a
2
)
.------------------------------9分

(3)下面讨论函数f(x)的零点情况.

①当

a
2
(1-ln
a
2
)>0,即0<a<2e时,函数f(x)在(1,ea)上无零点;

②当

a
2
(1-ln
a
2
)=0,即a=2e时,
2a
2
=
e
,则1<
2a
2
<ea

而f(1)=1>0,f(

2a
2
)=0,f(ea)>0,

∴f(x)在(1,ea)上有一个零点;

③当

a
2
(1-ln
a
2
)<0,即a>2e时,ea
2a
2
e
>1,

由于f(1)=1>0,f(

2a
2
)=
a
2
(1-ln
a
2
)
<0.

f(ea)=e2a-a lnea=e2a-a2=(ea-a)(ea+a)>0,

所以,函数f(x)在(1,ea)上有两个零点.(13分)

综上所述,f(x)在(1,ea)上有结论:

当0<a<2e时,函数f(x)有、无零点;

a=2e时,函数f(x)有一个零点;

当a>2e时,函数f(x)有两个零点.------------------------------14分.

阅读理解

Water goes around and around Earth in a never ending journey called the water cycle(循环). The sun heats up lakes, oceans, and other wet places on Earth. When the water gets warm enough, it changes into vapour. Plants also give off lots of water vapour. Some of this water vapour cools off high in the sky and becomes clouds. Then it falls back to Earth in a new place as rain or snow. This cycle happens over and over again.

The ground can absorb water like a sponge (海绵). If you could see this groundwater, it wouldn’t look like a lake or river. The groundwater is mixed in with the rocks and sand that lie in layers(层) below Earth’s surface.

Groundwater moves along slowly. How slowly? Maybe 1.5 kilometres in one century. Some of this water has been underground for thousands of years. And once groundwater is pumped out of the ground from a deep well(井) by people, it may take hundreds of years for another water to take its place.

小题1:In which order does water go around Earth?

a .Fall down as rain or snow.  b. Heated up by the sun on lakes, oceans and other wet places.

c. Cool off high in the sky.     d.  Form clouds.     e.  Change into vapour.

A.dacbe

B.becda

C.caebd

D.bceda小题2:The groundwater seems to________.

A.be just on the ground

B.be pure water like that in a lake or river

C.exist in rocks and sand

D.flow along like rivers or streams小题3:The underlined word “pumped” in the passage can be replaced by________.

A.run

B.pushed

C.drawn

D.picked小题4:What conclusion(结论) can we draw from the passage?

A.Groundwater can be quickly replaced by other water once pumped out.

B.Groundwater is very valuable.

C.Groundwater has nothing to do with human beings.

D.Groundwater travel in an unknown way.

填空题