求参数方程x=acos3θ y=sin3θ的二阶导数

2025-04-16 12:02:41
推荐回答(3个)
回答1:

消去参数得:x^2/a^2+y^2=1
两边对x求导:2x/a^2+2yy'=0,y'=-x/(ya^2)
y''=[-ya^2+xy'a^2]/(ya^2)^2=-1/(a^2y^3)=-1/[a^2(sin3θ)^3]

回答2:

dx/dθ=-3a*Sin3θ,

dy/dθ=3*Cos3θ

dy/dx=-a*Ctg3θ

回答3:

dx/dθ==-3a*Sin3θ
dy/dθ=3cos3θ
dy/dx=-(cot3θ)/a
d^2y/dx^2=d(dy/dx)/dθ*(dθ/dx)=[3(csc3θ)^2/a]/(-3asin3θ)
=-[(csc3θ)^3]/a^2