In-class work for 14 June 2012
1.) Find the length of the space curve $\vec{r}(t) = < 2 \sin t, 5t, 2 \cos t >$ between $t=-10$ and $t=10$.
2.) Given $\vec{r}(t) = < \cos t, \sin t, 11t^2 >$, find the unit tangent, unit normal, and curvature. (lots of details for the "hard" part of this are on the solutions page)
3.) Find $\vec{T}$, $\vec{N}$, and $\vec{B}$ of the vector function $\vec{r}(t) = < \cos t, \sin t, \log \cos (t) >$ at the point $(1,0,0)$.
4.) Find the equation of the normal and osculating planes of the curve
$$\left\{ \begin{array}{ll}
x &= 2 \sin (3t) \\
y &= t \\
z &= 2 \cos (3t)
\end{array} \right.$$
at the point $(0, \pi, -2)$.
5.) At what point on the curve
$$\left\{ \begin{array}{ll}
x &= t^2 \\
y &= 3t \\
z &= t^4
\end{array} \right.$$
is the normal plane parallel to the plane $6x+6y-8z=1$?