8.3: Arc Length and Curvature
( \newcommand{\kernel}{\mathrm{null}\,}\)
- Determine any values of t at which the vector-valued function r(t)=(t,0) is not smooth. Graph the function and describe its behavior at the points where it is not smooth.
- Determine any values of t at which the vector-valued function r(t)=t3i is not smooth. Graph the function and describe its behavior at the points where it is not smooth.
- Determine any values of t at which the vector-valued function r(t)=(13t3−12t2,0) is not smooth. Graph the function and describe its behavior at the points where it is not smooth.
- Determine any values of t at which the vector-valued function r(t)=(t3,5t2) is not smooth. Graph the function and describe its behavior at the points where it is not smooth.
- Determine any values of t at which the vector-valued function r(t)=3√ti+tj is not smooth. Graph the function and describe its behavior at the points where it is not smooth.
- Find the arc length of r(t)=(2t2+3,t2+1) for t∈[1,4].
- Find the arc length of r(t)=2ti+sintj+costk for t∈[−π,π].
- Find an arc length parameterization from t=0 for the curve r(t)=12t2i−13t3j with the same orientation as the original curve.
- Find an arc length parameterization from t=0 for the curve r(t)=(cost,t,sint) with the same orientation as the original curve.
- Given r(t)=13t3i+12t2j, find the values of T(t) and N(t) at t=1.
- Given r(t)=(2cost,2sint,t), find the values of T(t), N(t), and B(t) at t=0.
- Given r(t)=ti+etsintj+etcostk, find the values of T(t), N(t), and B(t) at t=0.
- Find the curvature of r(t)=(sint,4cost).
- Find the curvature of r(t)=ti+tj+t2k.