函數(shù)f(x)=sinx+cosx(x∈R)的圖象按向量(m,0)平移后,得到函數(shù)y=f′(x)的圖象,其中:m∈(-π,π)則m的值是 .
【答案】
分析:分別將f(x)與f'(x)用輔助角公式合并,可得將f(x)的圖象左移
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131025122941621434070/SYS201310251229416214340011_DA/0.png)
單位,可得y=f′(x)的圖象,結(jié)合題意不難得到m的值.
解答:解:函數(shù)f(x)=sinx+cosx=
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131025122941621434070/SYS201310251229416214340011_DA/1.png)
sin(x+
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131025122941621434070/SYS201310251229416214340011_DA/2.png)
)
而f'(x)=cosx-sinx=
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131025122941621434070/SYS201310251229416214340011_DA/3.png)
cos(x+
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131025122941621434070/SYS201310251229416214340011_DA/4.png)
)=
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131025122941621434070/SYS201310251229416214340011_DA/5.png)
sin(x+
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131025122941621434070/SYS201310251229416214340011_DA/6.png)
)
∵f(x-m)=
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131025122941621434070/SYS201310251229416214340011_DA/7.png)
sin(x-m+
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131025122941621434070/SYS201310251229416214340011_DA/8.png)
)=f'(x),
∴-m+
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131025122941621434070/SYS201310251229416214340011_DA/9.png)
+2kπ=
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131025122941621434070/SYS201310251229416214340011_DA/10.png)
(k∈Z)
∵m∈(-π,π)
∴取k=0,得m=-
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131025122941621434070/SYS201310251229416214340011_DA/11.png)
故答案為:-
點評:本題給出一個三角函數(shù)圖象,平移后與它導數(shù)的圖象重合,求平移的距離.著重考查了三角函數(shù)的圖象與性質(zhì)、輔助角公式和三角函數(shù)的導數(shù)等知識,屬于基礎題.