Parametric curves:
A curve can be defined parametrically using an independent variable, usually t, as it
often represents time. When t increases, we can regard the component function as
encoding movement in the plane.
For a function, y=f (x ), the component function is given by:
r ( t )=⟨ t , f ( t) ⟩
For some equations that are not functions, such as circles, extra considerations must
be taken. A circle is defined parametrically as:
r ( θ ) =⟨ r cos (θ ) ,r sin( θ) ⟩
For a curve r =⟨ f , g ⟩ , dom ( r )=dom ( f ) ∩dom ( g ).
If r ( t )=⟨ x( t) , y (t ) ⟩ , then lim
t →a ⟨ t→a ⟩
r (t )= lim x ( t ) , lim y (t) . The function r ( t ) is continuous at
t→a
a if lim r (t )=r (a).
t →a
Curves can be defined in a 3D space as well. The component function is given by:
r ( t )=⟨ x ( t ) , y ( t ) , z (t ) ⟩
Tangent lines:
From limits we can build a tangent vector at t on r ( t )=⟨ x ( t ) , y ( t ) ⟩.
The parametric equation of the tangent line of r (t ) at a is:
⟨ x , y , z ⟩ =r ( a ) +t r ' ( a ) :t ϵ R
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