Smooth

Definition

A smooth curve is one that is continuously differentiable, meaning it has a continuous first derivative. Such curves have no sharp corners or cusps.

5 Must Know Facts For Your Next Test

1. A curve is considered smooth if it has a continuous first derivative over its entire length.
2. Smooth curves are essential for calculating arc lengths because they ensure the integrals involved are well-defined.
3. The parametric equations of a smooth curve must be continuously differentiable.
4. If a function $f(x)$ is smooth, then both $f'(x)$ and $f''(x)$ exist and are continuous.
5. In surface area calculations involving rotation, the curve being rotated must be smooth to apply standard formulas.

Review Questions

• What condition must a curve meet to be classified as smooth?
• How does the concept of smoothness affect the calculation of arc length?
• Why is it important for parametric equations to be continuously differentiable?

Related terms

Arc Length: The distance measured along the path of a curve.

Parametric Equations: Equations that express the coordinates of the points on a curve as functions of a variable, typically denoted as $t$.

$f'(x)$: The first derivative of a function $f$, representing the rate at which $f$ changes with respect to its variable.