The curvature $\kappa$ at a point $P$ of a curve is defined as $$\kappa=\left|\frac{d\phi}{ds}\right|,$$ 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: $$\kappa = \frac{|\dot{x}\ddot{y}-\ddot{x}\dot{y}|}{[\dot{x}^2+\dot{y}^2]^{3/2}}$$ 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 $$\kappa=\frac{|d^2y/dx^2|}{[1+(dy/dx)^2]^{3/2}}$$

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?

$(x, y) =$ (, )