91Ó°ÊÓ

$$\begin{gathered} \mathrm{Give}\mathrm{precise}\mathrm{mathematical}\mathrm{definitions}\mathrm{or}\mathrm{descriptions}\mathrm{of}\mathrm{each}\mathrm{of}\mathrm{the}\mathrm{concepts}\mathrm{that}\mathrm{follow}. \\ \mathrm{Then}\mathrm{illustrate}\mathrm{the}\mathrm{definition}\mathrm{with}\mathrm{a}\mathrm{graph}\mathrm{or}\mathrm{an}\mathrm{algebraic}\mathrm{example}. \\ \mathrm{an}\mathrm{arc}\mathrm{length}\mathrm{parametrization}\mathrm{for}\mathrm{the}\mathrm{graph}\mathrm{of}\mathrm{a}\mathrm{differentiable}\mathrm{function}\mathrm{r}\left(\mathrm{t}\right) \end{gathered}$$

Give a step-by-step procedure for finding the equation of the osculating circle at a point on a planar curve where the curvature is nonzero.

Given a twice-differentiable vector-valued function $r\left(t\right)$, what is the definition of the binormal vector $B\left(t\right)$?

Let f(t) be a non-constant continuous function.

(a) Explain why a particle moving along the vector function r(t) = f(t),( f(t))² is on a parabolic path.

(b) What condition must f(t) satisfy so that r(t) · a(t) = 0?

As we saw in Example 1, the graph of the vector-valued function $r\left(t\right)=\left(\cos t,\sin t,t\right),\mathrm{for}t∈[0,2\mathrm{π}]$ is a circular helix that spirals counterclockwise around the z-axis and climbs ast increases. Find another parametrization for this helix so that the motion along the helix is faster for a given change in the parameter.

Unit tangent vectors: Find the unit tangent vector for the given function at the specified value of t.

As we saw in Example 1, the graph of the vector-valued function $r\left(t\right)=\left(\cos t,\sin t,t\right),\mathrm{for}t∈[0,2\mathrm{π}]$ is a circular helix that spirals counterclockwise around the z-axis and climbs as t increases. Find another parametrization for this helix so that the motion is downwards.

Unit tangent vectors: Find the unit tangent vector for the given function at the specified value of t.

$\mathbf{r}\left(t\right)=⟨3\sin 2t,3\cos 2tâŸ\copyright ,t=\frac{\mathrm{Ï€}}{3}$