設(shè)直線
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824014833739620.png)
是曲線
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824014833755324.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240148337861001.png)
的一條切線,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824014833817815.png)
.
(Ⅰ)求切點(diǎn)坐標(biāo)及
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824014833833337.png)
的值;
(Ⅱ)當(dāng)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824014833864477.png)
時,存在
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824014833880630.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824014833911771.png)
,求實(shí)數(shù)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824014833942283.png)
的取值范圍.
(1)切點(diǎn)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824014833973499.png)
,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824014833989534.png)
,切點(diǎn)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824014834082562.png)
,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824014834114462.png)
.
(2)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824014834145521.png)
試題分析:解:(Ⅰ)設(shè)直線
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824014834176280.png)
與曲線
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824014834192313.png)
相切于點(diǎn)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824014834223642.png)
,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824014834238235.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824014834254718.png)
,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824014834285195.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824014834394591.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824014834426279.png)
, 解得
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824014834441399.png)
或
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824014834472439.png)
, 3分
當(dāng)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824014834441399.png)
時,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824014834504435.png)
,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824014834238235.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824014833973499.png)
在曲線
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824014834192313.png)
上,∴
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824014833989534.png)
,
當(dāng)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824014834472439.png)
時,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824014834738503.png)
,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824014834238235.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824014834082562.png)
在曲線
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824014834192313.png)
上,∴
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824014834114462.png)
,
切點(diǎn)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824014833973499.png)
,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824014833989534.png)
, 5分
切點(diǎn)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824014834082562.png)
,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824014834114462.png)
. 7分
(Ⅱ)解法一:∵
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824014833864477.png)
,∴
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824014834114462.png)
,
設(shè)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240148350031383.png)
,
若存在
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824014833880630.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824014833911771.png)
,則只要
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824014835065692.png)
, 10分
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240148350961107.png)
,
(ⅰ)若
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824014835112450.png)
即
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824014835143381.png)
,令
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824014835159593.png)
,得
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824014835190829.png)
,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824014835221663.png)
,∴
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824014835252484.png)
在
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824014835268708.png)
上是增函數(shù),
令
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824014835299597.png)
,解得
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824014835315696.png)
,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824014834285195.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824014835252484.png)
在
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824014835377614.png)
上是減函數(shù),
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824014834285195.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824014835424939.png)
,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824014835440807.png)
,
解得
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824014835471413.png)
, 12分
(ⅱ)若
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824014835564436.png)
即
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824014835580368.png)
,令
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824014835159593.png)
,解得
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824014835627830.png)
,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824014835221663.png)
, ∴
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824014835252484.png)
在
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824014835689535.png)
上是增函數(shù),
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824014835705219.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824014835767617.png)
,不等式無解,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824014834285195.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824014833942283.png)
不存在, 13分
綜合(ⅰ)(ⅱ)得,實(shí)數(shù)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824014833942283.png)
的取值范圍為
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824014834145521.png)
. 14分
解法二:由
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824014835845598.png)
得
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824014835861899.png)
,
(ⅰ)當(dāng)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824014835892400.png)
時,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824014835908832.png)
,設(shè)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824014835939927.png)
若存在
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824014833880630.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824014833911771.png)
,則只要
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824014835986701.png)
, 10分
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240148360011122.png)
,
令
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824014836017596.png)
解得
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824014836048413.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824014834285195.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824014835252484.png)
在
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824014836095512.png)
上是增函數(shù),
令
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824014836110590.png)
,解得
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824014834285195.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824014835252484.png)
在
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824014836204466.png)
上是減函數(shù),
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824014834285195.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824014836235865.png)
,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824014834285195.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824014835471413.png)
, 12分
(ⅱ)當(dāng)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824014836266367.png)
時,不等式
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824014835861899.png)
不成立,
∴
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824014833942283.png)
不存在, 13分
綜合(ⅰ)(ⅱ)得,實(shí)數(shù)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824014833942283.png)
的取值范圍為
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824014834145521.png)
. 14分
點(diǎn)評:主要是考查了導(dǎo)數(shù)在研究函數(shù)單調(diào)性中的運(yùn)用,屬于基礎(chǔ)題。
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![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824013921311465.png)
中,點(diǎn)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824013921342318.png)
到兩點(diǎn)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824013921358555.png)
,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824013921373546.png)
的距離之和為
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824013921405386.png)
,設(shè)點(diǎn)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824013921342318.png)
的軌跡為曲線
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824013921436313.png)
.
(1)寫出
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824013921436313.png)
的方程;
(2)設(shè)過點(diǎn)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824013921373546.png)
的斜率為
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824013921483312.png)
(
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824013921514418.png)
)的直線
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824013921545280.png)
與曲線
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824013921436313.png)
交于不同的兩點(diǎn)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824013921576399.png)
,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824013921592357.png)
,點(diǎn)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824013921607289.png)
在
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824013921639310.png)
軸上,且
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824013921654676.png)
,求點(diǎn)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824013921607289.png)
縱坐標(biāo)的取值范圍.
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