Finding Tangent to Cycloid

Calculus 2 · Axiom Academy

EXAMPLE Finding Tangent to Cycloid Find the slope and the special tangent lines of a parametric curve using the chain rule. A point on the rim of a rolling wheel of radius r traces a cycloid Find the slope at a general parameter value t , evaluate it at , and locate every horizontal and vertical tangent. (We take r = 1 for the figure and the numerical answer.) One arch, with the tangent at t = π/2 The dashed orange line is the tangent at t = π/2 , where the slope is exactly 1. At the apex (t = π) the tangent is horizontal; at the cusps (t = 0, 2π) the curve comes to a sharp point and has no tangent line. Nice work — you found the slope of a cycloid and located its special tangents. The whole method rests on one parametric idea: Parametric slope: differentiate x and y separately, then . Here that gives . A concrete value: at the point is and the slope is . Horizontal tangents: at , but only the odd multiples also have — the apex of each arch. No vertical tangents — cusps instead: only at , where dy/dt = 0 too. Both derivatives vanish, so those ground-touching points are sharp cusps, not vertical tangents. The takeaway: when dx/dt = 0 , always check dy/dt — a true vertical tangent needs ; if both are zero you have a cusp. This same recipe works for any parametric curve: dy/dx = (dy/dt)/(dx/dt) , and the zeros of dy/dt and dx/dt are where the interesting tangents live.

This is the written version of the interactive lesson above. See the full Calculus 2 course.