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Mathematics LibreTexts

8.1: Vector-Valued Functions and Space Curves

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  1. Given the vector-valued function r(t)=(2t2+3,3t2), find r(2).
     
  2. Given the vector-valued function r(t)=secti+tantj, find r(π4).
     
  3. Sketch the curve of the vector-valued function r(t)=ti+(1t2)j.
     
  4. Sketch the curve of the vector-valued function r(t)=(t,t2,t).
     
  5. Sketch the curve of the vector-valued function r(t)=(cost,sint). Be sure to indicate the orientation of the curve.
     
  6. Sketch the curve of the vector-valued function r(t)=sinti+costj+tk. Be sure to indicate the orientation of the curve.
     
  7. Find the domain of the vector-valued function r(t)=(1t3,t21).
     
  8. Find the domain of the vector-valued function r(t)=2eti+lntj+1t2+1k.
     
  9. Evaluate limt0(2t+1,cost), if possible.
     
  10. Evaluate limt2(t2t24,et), if possible.
     
  11. Evaluate limt(eti+3t2+2t425t2j+arctan(t)k), if possible.
     
  12. Evaluate limt0+(sintt,lnt,4), if possible.
     
  13. Express the vector-valued function r(t)=2ti(3+t)j in Cartesian coordinates by eliminating the parameter t, then sketch the graph of the vector-valued function. Be sure to indicate the orientation of the curve.
     
  14. Express the vector-valued function r(t)=(t3,t2) in Cartesian coordinates by eliminating the parameter t, then sketch the graph of the vector-valued function. Be sure to indicate the orientation of the curve.
     
  15. Express the vector-valued function r(t)=1t(i+j) in Cartesian coordinates by eliminating the parameter t, then sketch the graph of the vector-valued function. Be sure to indicate the orientation of the curve.
     
  16. Express the vector-valued function r(t)=(2sint,3cost) in Cartesian coordinates by eliminating the parameter t, then sketch the graph of the vector-valued function. Be sure to indicate the orientation of the curve.
     
  17. Express the vector-valued function r(t)=3sec2ti+tantj in Cartesian coordinates by eliminating the parameter t, then sketch the graph of the vector-valued function. Be sure to indicate the orientation of the curve.
     
  18. Find a vector-valued function that traces the line from point P(1,0,1) to point Q(0,1,0).
     
  19. Find a vector-valued function that traces the parabola y=x2 from left-to-right.
     
  20. Find a vector-valued function that traces the ellipse 4x2+9y2=36 in the counterclockwise direction.
     
  21. Find a vector-valued function that traces the hyperbola x216y2=1, where the right branch is oriented top-to-bottom.

8.1: Vector-Valued Functions and Space Curves is shared under a not declared license and was authored, remixed, and/or curated by LibreTexts.

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