Consider the plane curve parametrized by where g is a differentiable function....

Consider the plane curve parametrized by

where g is a differentiable function.

(a) Show that the parameter s is the arclength parameter.

(b) Calculate the curvature κ(s).

(c) Use part (b) to explain how you can create a parametrized plane curve with any specified continuous, nonnegative curvature function κ(s).

(d) Give a set of parametric equations for a curve whose curvature κ(s) = |s|. (Your answer should involve integrals.)

(e) Use a computer to graph the curve you found in part (d), known as a clothoid or a spiral of Cornu. (Note: The integrals involved are known as Fresnel integrals and arise in the study of optics. You must evaluate these integrals numerically in order to graph the curve.)