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

For a parametric curve [math] [math] one can derive the following formula for curvature: [math] where the dots indicate derivatives with respect to [math], so [math]

One can derive a second formula for curvature by regarding the curve [math] as the parametric curve [math] with parameter [math]. Then one can prove the curvature formula [math]

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 [math] at the point (1, 1).

[math]


(b) At what point on the parabola [math] does it have maximum curvature?

[math] ( , )