AP Calculus BC glossary

Smooth curve

Also called: Regular curve

A smooth curve is one that is continuously differentiable with no corners or cusps. For a parametric curve x of t, y of t, this means x prime and y prime are continuous and never both zero, so the curve has a tangent direction everywhere. Arc length only needs x prime and y prime to be continuous.

For a function $y = f(x)$ , smooth means $f'$ is continuous, so the graph has no corner or cusp. For a parametric curve $\langle x(t), y(t)\rangle$ it means $x'(t)$ and $y'(t)$ are continuous and never vanish together, so the velocity vector is never the zero vector.

The arc-length integral $\int_a^b \sqrt{\left(x'(t)\right)^2 + \left(y'(t)\right)^2}\,dt$ needs $x'$ and $y'$ to exist and be continuous. At a corner the derivative does not exist, so a curve with a corner is not smooth and must be split into smooth pieces before the formula applies.

The mistake

Treating continuous and smooth as the same thing. The graph of $|x|$ is continuous everywhere but has a corner at the origin, where the derivative does not exist, so it is not smooth there. Smoothness requires a continuous derivative, which is stronger than continuity of the curve itself.

Appears in: Unit 9: Parametric, Polar, and Vector (BC)