Derivatives in Parametric Equations 2D Rectangular Coordinates  Contents

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Critical Points in Parametric Equations 2D Rectangular Coordinates  Top of Page  Contents

A critical point of a parametric curve (x(t), y(t)) in the plane is a value t0 of t such that x’(t0) = 0 and y’(t0) = 0.

Figure 3

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Exercises


Find the critical points of the following parametric curves:
  •     1. (t2,t3),
  •     2. (cos(t),cos(2t))
  •     3. ((1 + cos(t))cos(t), (1+cos(t))sin(t))


Tangent Lines and Normal Vectors in Parametric Equations 2D Rectangular Coordinates  Top of Page  Contents

If the coordinate functions of a parametric curve (x(t), y(t)) are differentiable at t = t0, then (x’(t0),y’(t0)) is called the velocity vector.  At a point that is not critical, the velocity vector will be non-zero.  The “tangent line to the parametric curve (x(t),y(t)) at (x(t0),y(x0)) is then given by (x(t0) + x’(t0)(t-(t0)), y(t0) + y’(t0)(t-t0) = (x(t0),y(t0)) + (x’(t0),y’(t0))(t-t0).
 
The normal vector at t0, (-y '(t0), x'(t0)), is perpendicular to the tangent line, and the normal line is then given by by (x(t0) -y’(t0)(t-(t0)), y(t0) + x’(t0)(t-t0) = (x(t0),y(t0)) + (-y’(t0),x’(t0))(t-t0).

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Differentiability in Parametric Equations  2D Rectangular Coordinates  Top of Page  Contents

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