Question

Find the curvature of the parabola y=x^2 at A(1,1)

Answer 1

Answer:

Step-by-step explanation:

The curvature of the curve y=f(x) is

$y=f(x)=x^2\\\\y'=f'(x)=2x\\\\y''=f''(x)=2\\\\K(x)=courbure(in\ French)=\Bigg|\dfrac{y''(x)}{(1+y'(x)^2)^{\dfrac{3}{2} } }\Bigg|\\\\At\ A=(1,1),\\K(1)=\Bigg|\dfrac{2}{(1+2^2)^{\dfrac{3}{2} } }\Bigg|\\\\=\Bigg|\dfrac{2}{(5)^{\dfrac{3}{2} } }\Bigg|\\\\=\Bigg|\dfrac{2}{5*\sqrt{5} }\Bigg|\\\\=\Bigg|\dfrac{2*\sqrt{5}}{25 }\Bigg|\\\\\\\boxed{=\dfrac{2*\sqrt{5}}{25 }}$