已知函數(shù)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824024246595848.png)
的圖象如圖,直線
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824024246610391.png)
在原點處與函數(shù)圖象相切,且此切線與函數(shù)圖象所圍成的區(qū)域(陰影)面積為
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824024246626439.png)
.
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240242466412014.jpg)
(1)求
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824024246641447.png)
的解析式;
(2)若常數(shù)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824024246657457.png)
,求函數(shù)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824024246673463.png)
在區(qū)間
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824024246735445.png)
上的最大值.
(1)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824024246766674.png)
;
(2)當(dāng)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824024246782539.png)
時,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824024246797804.png)
;當(dāng)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824024246813455.png)
時,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240242468291025.png)
.
試題分析:(1)由條件知,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824024246844501.png)
,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824024246860517.png)
,代入可得
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824024246891387.png)
、
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824024246891338.png)
.再用定積分表示出所圍成的區(qū)域(陰影)面積,由面積為
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824024246626439.png)
解得
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824024246922399.png)
,從而得到
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824024246641447.png)
的解析式;(2)由(1)知
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824024246969954.png)
,再列出
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824024246985625.png)
,的取值變化情況,又
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824024247000534.png)
,結(jié)合圖像即可得當(dāng)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824024246782539.png)
時,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824024246797804.png)
;當(dāng)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824024246813455.png)
時,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240242468291025.png)
.
試題解析:(1)由
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824024246844501.png)
得
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824024246891338.png)
, 2分
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824024247109872.png)
.由
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824024246860517.png)
得
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824024246891387.png)
, 4分
∴
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824024247156874.png)
,則易知圖中所圍成的區(qū)域(陰影)面積為
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240242471721019.png)
從而得
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824024246922399.png)
,∴
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824024246766674.png)
. 8分
(2)由(1)知
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824024246969954.png)
.
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824024246985625.png)
的取值變化情況如下:
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824024247250275.png)
| ![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824024247265517.png)
| ![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824024247281262.png)
| ![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824024247281477.png)
| 2
| ![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824024247297545.png)
|
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824024247312485.png)
| ![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824024247343180.png)
| ![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824024247281262.png)
| ![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824024247453165.png)
| ![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824024247281262.png)
| ![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824024247343180.png)
|
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824024246673463.png)
| 單調(diào) 遞增
| 極大值![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824024246844501.png)
| 單調(diào) 遞減
| 極小值![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824024247515572.png)
| 單調(diào) 遞增
|
又
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824024247000534.png)
,
①當(dāng)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824024246782539.png)
時,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824024246797804.png)
; 11分
②當(dāng)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824024246813455.png)
時,
綜上可知當(dāng)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824024246782539.png)
時,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824024246797804.png)
;當(dāng)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824024246813455.png)
時,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240242476091027.png)
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