Hi.
I am asked to graph . How do I go about this?
I'll assume that , and point along the x-, y- and z-axes, respectively. You see that along the path, , so the graph lies on the cylinder with radius 1 going along the x-axis. The x-coordinate grows linearly. If x(t) were constant, then y(t), z(t) would follow a circle. Therefore, this is a spiral. Further, (y(t), z(t)) make 2 revolutions when t ranges from 0 to , so when x(t) = 2t, there will be two revolutions when x ranges from 0 to .
This is a spiral with the unit radius and the period , whose center is the x-axis.
See this picture (red: , blue: )
For maple
with(plots):
A1 := spacecurve([2*t, sin(4*t), cos(4*t)], t=0..(1/2)*Pi, color=red, thickness=3, numpoints=1000):
A2 := spacecurve([2*t, sin(4*t), cos(4*t)], t=(1/2)*Pi..Pi, color=blue, thickness=3, numpoints=1000):
display(A1, A2, axes=normal);