The vector function is, r(t) = [tex]\bold{ < t,2t^2,9t^2+4t^4 > }[/tex]
Given two surfaces for which the vector function corresponding to the intersection of the two need to be found.
First surface is the paraboloid, [tex]z=9x^2+y^2[/tex]
Second equation is of the parabolic cylinder, [tex]y=2x^2[/tex]
Now to find the intersection of these surfaces, we change these equations into its parametrical representations.
Let x = t
Then, from the equation of parabolic cylinder, [tex]y=2t^2[/tex].
Now substituting x and y into the equation of the paraboloid, we get,
[tex]z=9t^2+(2t^2)^2 = 9t^2+4t^4[/tex]
Now the vector function, r(t) = <x, y, z>
So r(t) = [tex]\bold{ < t,2t^2,9t^2+4t^4 > }[/tex]
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