Arc Length and Curvature
1. Calculate the arc length of the following vector-valued function.
solution
\[ {\bf r}(t)= (2t-3){\bf i} + (4t-1){\bf j}, \: 1 \leq t \leq 5\]
2.
Find the length of the following curve segment.
solution
\[ \langle 12t, 8t^{\frac{3}{2}},3t^2 \rangle, \quad -1 \le t \le 2 \]
3.
Find the curvature of \( \quad {\bf r} (t) = t^2\,{\bf i}+ \ln t \, {\bf
j} + t\ln t \, {\bf k} \quad \) at the point \((1,0,0) \)
solution
4.
Find the curvature of the curve \( \quad {\bf r}(t)=\langle \sqrt{6}
t^2, 2t, 2t^3\rangle \)
solution
5.
Find the unit tangent, unit normal and binormal vectors for \( \quad
{\bf r}(t)=\langle t, 2\cos t, 2\sin t \rangle \)
solution