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Mathematics LibreTexts

8.3: Arc Length and Curvature

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  1. 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.
     
  2. 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.
     
  3. Determine any values of t at which the vector-valued function r(t)=(13t312t2,0) is not smooth. Graph the function and describe its behavior at the points where it is not smooth.
     
  4. 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.
     
  5. Determine any values of t at which the vector-valued function r(t)=3ti+tj is not smooth. Graph the function and describe its behavior at the points where it is not smooth.
     
  6. Find the arc length of r(t)=(2t2+3,t2+1) for t[1,4].
     
  7. Find the arc length of r(t)=2ti+sintj+costk for t[π,π].
     
  8. Find an arc length parameterization from t=0 for the curve r(t)=12t2i13t3j with the same orientation as the original curve.
     
  9. Find an arc length parameterization from t=0 for the curve r(t)=(cost,t,sint) with the same orientation as the original curve.
     
  10. Given r(t)=13t3i+12t2j, find the values of T(t) and N(t) at t=1.
     
  11. Given r(t)=(2cost,2sint,t), find the values of T(t), N(t), and B(t) at t=0.
     
  12. Given r(t)=ti+etsintj+etcostk, find the values of T(t), N(t), and B(t) at t=0.
     
  13. Find the curvature of r(t)=(sint,4cost).
     
  14. Find the curvature of r(t)=ti+tj+t2k.

8.3: Arc Length and Curvature is shared under a not declared license and was authored, remixed, and/or curated by LibreTexts.

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