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Math Help - Sketching vector-valued functions

  1. #1
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    Sketching vector-valued functions

    Sketch r(t) = t^2i + t^4j + t^6k

    Attempt:

    I tried to find a surface that this curve lies on so I said:

    x = t^2 therefore:

    y = x^2 and z = x^4

    I then drew these surfaces, then tried to see what happens when t = large positive number and then a large negative number. I got that the graph lies in the first octant, but I am not sure where on the surfaces y = x^2 and z = x^4 the curve lies on (I learnt that you use these surfaces as a guide to draw the curve).
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  2. #2
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  3. #3
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    How do you know that the curve, curves a bit (the part which looks like a little turning point) near the origin?
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  4. #4
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    Well, it has to produce the projections drawn in dashed lines. Actually, this graph curves everywhere, the curvature is just higher near the origin because z does not grow as fast yet.
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  5. #5
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    If \vec{r}(t)= t^2\vec{i}+ t^4\vec{j}+ t^6\vec{k}, then
    x= t^2, y= t^4, and z= t^6.

    Those certainly satisfy xy= t^2t^4= t^6= z for all t.

    That is, this curve lies on the surface xy= z.

    (Note that any curve lies on an infinite number of surfaces.)
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