Question

高数之曲率对数曲线y=lnx上哪一点处的曲率半径最小?求出该点处的曲率半径.答案是（根号2/2,-ln2/2)处曲率半径

Answer 1

y=lnx

y'=1/x

y''=-x^(-2)

曲率半径公式ρ=[(1+y'^2)^(3/2)]/∣y"∣

= (1+ (1/x)^2)^(3/2) / (x^(-2))

=x^2 * (1+ x^(-2) )^(3/2)

对它求导

=2x*(1+x^(-2))^(3/2) +x^2 * 3/2*(1+x^(-2))^(1/2) *(-2)x^(-3)

=2x*(1+x^(-2))*(1+x^(-2))^(1/2)-3*x^(-1)*(1+x^(-2))^(1/2)

=(2x-1/x)*(1+x^(-2))^(1/2)

=(2x^2-1)*(1+x^2))^(1/2)/x

2x^2-1=0

x=2^(1/2)时,曲率半径最小

代入曲率半径公式ρ=2*(1+1/2)^(3/2)=2*(3/2)^(3/2)