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1 # 83823堃
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2 # y吃橘子嗎
函數y=sinx-cosx+sinxcosx的值域為[-(1+2√2)/2,1]。
解答過程如下:
y=sinx-cosx+sinxcosx
=√2[(√2/2)sinx-(√2/2)cosx]+½sin2x
=√2sin(x-π/4)+½cos(π/2 -2x)
=√2sin(x-π/4)+½cos[2(x-π/4)]
=√2sin(x-π/4)+½[1-2sin²(x-π/4)]
=-sin²(x-π/4)+√2sin(x-π/4)+½
=-sin²(x-π/4)+√2sin(x-π/4)-½+1
=-[sin(x-π/4) - √2/2]²+1
sin(x-π/4)=√2/2時,y取得最大值,ymax=1
sin(x-π/4)=-1時,y取得最小值,ymin=-(1+2√2)/2
函數的值域為[-(1+2√2)/2,1]
解:原函數為:(1/2)[lnsin(x+π/4)+x]+C
我只能輸入沒幾個字,要點是:cosx+sinx=2^(1/2)sin(x+π/4)
令t=x+π/4 代入:dx=dt 就能得到結果
分子分母同時除以cosx 得 1/(1+tanx)
(-cosx+C)'=sinx,(sinx+C)'=cosx,
sinx原函數是-cosx+C;
cosx原函數是sinx+C.
cosx的原函數是sinx, sinx的原函數是-cosx