Horizontal and Vertical Tangents on Parametric Curves

A parametric curve is traced by a point moving with two coordinates, x(t) and y(t), each changing with time. The slope of the curve is dy/dx = (dy/dt)/(dx/dt), a ratio of two separate rates. The curve momentarily stops moving sideways when dx/dt = 0 — the x-coordinate freezes while y keeps changing, so the path runs straight up or down: a vertical tangent. It stops moving up or down when dy/dt = 0 — y freezes while x keeps changing, giving a horizontal tangent. The curve isn't actually standing still unless both derivatives vanish at once; one of them hitting zero just means motion in that one direction has paused.

The method: compute dx/dt and dy/dt separately. Set dx/dt = 0 and solve for t to get candidate vertical tangents; set dy/dt = 0 to get candidate horizontal tangents. Always check that the other derivative is nonzero at that same t — if both are zero together, the slope is 0/0, and the point needs extra care (it may be a cusp or a point with no tangent at all, not a guaranteed horizontal or vertical line). Once you have valid t values, plug back into x(t) and y(t) to locate the actual points on the curve if asked.

This sits right next to the general slope formula dy/dx = (dy/dt)/(dx/dt) from the previous topic — tangent lines are just the special cases where the top or bottom of that fraction vanishes. It also sets up later work on curvature and motion, where knowing exactly when a particle's horizontal or vertical velocity component vanishes tells you about turning points and direction changes along the path.