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) |
= |