Vector Functions And Space Curves

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  1. Find the domain of the vector function rt=t2t15t .
    15
  2. Find limt0+arctante2tlntt
    The limit does not exist because lntt diverges to infinity as t tends to 0 from above.
  3. Sketch the curve given by equation rt=sintt and indicate the direction in which t increases.
  4. Sketch the curve given by equation rt=1cost2sint and indicate the direction in which t increases.
  5. Sketch the curve given by equation rt=t2i+tj+2k and indicate the direction in which t increases.
  6. Find a vector equation and parametric equations for the line segment joining P101 and Q231 .
    Vector equation: rt=231130t .
    Parametric equations: x=2t , y=33t , z=1 , where 0t1 .
  7. Match parametric equations with the given graphs. Explain.
    1. x=cos4t,y=t,z=sin4t .
    2. x=t,y=t2,z=et .
    3. x=t,y=11+t2,z=t2 .
    4. x=etcos10t,y=etsin10t,z=et .
    5. x=cost,y=sint,z=sin5t .
    6. x=cost,y=sint,z=lnt .

    I. space curve 4 II. space curve 3

    III. space curve 1 IV. space curve 5

    V. space curve 2 VI. space curve 6

    III, V, II, I, IV, VI.
  8. Use a computer to graph the curve given by the equation rt=sintsin2tsin3t .
  9. Two neutrons travel along the space curves r1t=tt2t3 and r2t=1+2t1+6t1+14t . Will they collide? Do their paths intersect?
    No collision; intersections at 111 and 248 .