Question

$$ \text { State the definition of a smooth curve } $$

Short answer

A smooth curve is one that is continuously differentiable. This means it has a derivative that is a continuous function at every point, indicating no abrupt changes in direction or any 'sharp' points. It transitions smoothly from one point to another.

Step-by-step solution

Define smooth curve

A smooth curve is a mathematical term used to describe a curve that is continuously differentiable, that is, it has a derivative at every point and the derivative is a continuous function. This indicates that there are no abrupt changes in direction, and the curve doesn't have any 'sharp' points. It transitions smoothly from one point to another.

Explain continuous differentiability

Continuous differentiation means that you can take the derivative of the function at any point, and the derived function you get is still a continuous function. The curve has no edges or disconnected points, and you can draw the curve without lifting your pen, so to speak. All smooth curves are continuously differentiable.

Stress on differentiability

To be continuously differentiable, a curve must first be differentiable. This means that it must have a derivative not just at a few select points, but at every single point on the curve. The function that represents the curve must have no discontinuities, jumps, or vertical tangent lines anywhere on its domain. The absence of these features is what allows for the 'smoothness' of the curve.