Exercise 11

Let $y = f(x)$ define a curve in the plane. We can consider this curve as a curve in three-space with $z$-coordinate 0.

1. Find a parameterization of the form $\vr(t) = \langle x(t), y(t), z(t) \rangle$ of the curve $y=f(x)$ in three-space.

2. Use the formula $$\begin{gathered} \kappa = \frac{\vecmag{\vr'(t) \times \vr''(t)}}{\vecmag{\vr'(t) }^3} \end{gathered}$$ to show that $$\begin{gathered} \kappa = \frac{\lvert f''(x) \rvert}{\left[1+(f'(x))^2\right]^{3/2}}. \end{gathered}$$