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Parametric Representation of a Curve

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See the attached file.

1. Find a parametric representation of the curve:
x^2 + y^2 = 36 and z = 1/π arctan (x/y)

i.e find a representation in the form: x = x(t); y=y(t), z = z(t)

2. Find what kind of curves are given by the following representations and draw (schematically) the curves:
i) r(t) = (2t - 5, -3t + 1,4):
ii) r(t) = (0, -cos(t), 3sin(t))

3. Find the equations of the tangent (straight line) and the normal plane to the curve given by the followig parametric representation:

x = 2 + 3t^4; y = 2t + t^3, z = t at the point t = 1

4. The equations of motion of a point particle in space are given by:

i) x(t) = 2t^3 - 3; y(t) = -3t^3; z(t) = 4t^3 - 1
ii) x(t) = asin(wt); y(t) = acos(wt); z(t) = t

In each case, calculate the velocity vector v and the acceleration vector a. Also calculate the absolute values of these vectors. What are the curves along which the particle moves? (find these values for an arbitrary t, with a and w being fixed parameters).

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The solution discusses the parametric representation of a curve.

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1. The equation x^2 + y^2 = 36 is the equation of a circle, centered at ...

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