Problem :
Consider the parametric curve (x(t), y(t)) = (2t, sin(t)). What function has a graph that
coincides with this curve?
Substituting
t = x(t)/2 in the expression for
y(t) yields
y(t) = sin(x(t)/2). Thus the
desired function is
f (x) = sin(x/2).
Problem :
What is the velocity vector of the parametric curve (2t, sin(t)) at time t = 0? Π/2?
Π?
The derivatives of the parametric functions are
x'(t) = 2,
y'(t) = cos(t), so the velocity
vectors given by
(x'(0), y'(0))  =  (2, 1) 

(x'(Π/2), y'(Π/2))  =  (2, 0) 

(x'(Π), y'(Π))  =  (2,  1) 

Problem :
What is the velocity vector at time t = 2Πk (k≥ 0 an integer)
of the spiral (t cos(t), t sin(t))? Suppose a particle is tracing
out the spiral, so that it is at the point (t cos(t), t sin(t)) at
time t. How fast is the particle moving at time t?
Since
(x'(t), y'(t)) = (cos(t)  t sin(t), sin(t) + t cos(t)), the desired velocity vectors are given by
(x'(2Πk), y'(2Πk)) = (1, 2Πk) 

The speed of the particle at time
t is equal to
= 

This makes sense because the speed of the particle must continue to rise if it is to
continue to complete larger and larger turns of the spiral in the same amount of time (
2Π).