Function

[ x(t) = t cos(t), y(t) = t sin(t), z(t) = t ]

Domain

t is an element of the set of all Real numbers

Graphs

Derivative

With respect to t: [cos(t) - t sin(t), sin(t) - t cos(t), 1]

Critical points are formed where all the derivatives equal 0. These critical points may be minimums, maximums, or saddle points. Since the final value is 1, and 1 will never equal 0, there are no critical points.

Integral

With respect to t: [cos(t) + t sin(t), sin(t) - t cos(t), ( ¹/₂ )t²]

Interesting Features

This is a modified helix curve. While the function looks very similar, the results are very different. This is caused by the first t coefficient. This scales the level outward as the function raises. The final graph shows how the function flips and mirrors the upward spiral downward.