高中数学三角函数章节已知一个函数为fx=Asin(ax+b),然后题目会给你一个该函数的单调增或减区间, 【高中数学-三角函数】已知函数f(x)=sin(ωx+φ),...
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例:已知函数y=sin(wx)+1 在(-兀/3,兀/4)内是减函数,则w的取值范围?
解此题主要有以下三点考虑:
(1) T/2>=π/4+π/3=7π/12
(2)函数最高点的横坐标<=-兀/3
(3)正向相邻最低点的横坐标>=兀/4
解析:∵函数f(x)=sin(wx)+1 在(-兀/3,兀/4)内是减函数
T/2>=π/4+π/3=7π/12==>T>=7π/6==>|w|<=12/7(w≠0) *
f(x)=sinwx+1=2==>sinwx=1==>x=π/(2ω)
令π/(2ω)<=- π/3==>ω>=-3/2 **
f(x)=sinωx+1=0==>sinωx=-1==>x=-π/(2ω)
令-π/(2ω)>=π/4==>ω>=-2 ***
取*,**,***三者交
∴ω取值范围为-3/2<=ω<0
函数的单调区间的大小,只与a的值相关。换言之,即三角的周期函数的周期(和区间相关),是用a来表示的。T=2π/a,或者说,a=2π/T
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