The curvature $\kappa$ at a point $P$ of a curve is defined as where $\phi$ is the angle of inclination of the tangent line at $P$, as shown in the figure below. Thus, the curvature is the absolute value of the rate of change of $\phi$ with respect to arc length. It can be regarded as a measure of the rate of change of direction of the curve at $P$.

For a parametric curve $x=x(t),$ $y=y(t),$ one can derive the following formula for curvature: where the dots indicate derivatives with respect to $t$, so $\dot{x}=dx/dt.$

One can derive a second formula for curvature by regarding the curve $y=f(x)$ as the parametric curve $x=x,\quad y=f(x),$ with parameter $x$. Then one can prove the curvature formula

You should make sure that you can prove that these formulas are true!

(a) Use the second curvature formula given above to find the curvature of the parabola $y=x^2$ at the point (1, 1).

$\kappa =$

(b) At what point on the parabola $y=x^2$ does it have maximum curvature?

$(x, y) =$ ( , )