求参数方程x=acos^3t,y=asin^3t所描绘的曲线在t=pi/4处的曲率,其中a>0

Question

求参数方程x=acos^3t,y=asin^3t所描绘的曲线在t=pi/4处的曲率,其中a>0

Answer 1

x=acos^3t,y=asin^3t
dx=-3asintcos^2tdt dy=3asin^2tcostdt
dy/dx=-sint/cost=-tant
y'(π/4)=-1
d(dy/dx)=-sec^2tdt
d^2y/dx^2=-sec^2tdt/(-3asintcos^2tdt)
=1/(3asintcos^4t)
y''(π/4)=1/(3a√2/2*1/4)
=4√2/(3a)
K=4√2/(3a)/(1+1)^(3/2)
=2/(3a)