The term “smooth curve” has many meanings, depending on what field you’re working in. Unfortunately, it isn’t defined very well in calculus, with definitions varying from one text to another.

## Informal Definition of Smooth Curve

**Informally**, a **smooth curve** is another name for a curve drawn smoothly, as opposed to jaggedly [1]. It describes a curve’s appearance, rather than any mathematical properties. For example, the following graph is smooth:

This one is

*not*smooth:

## Smoothing in Statistics

In statistics, smooth curves (obtained through a process called *smoothing*) are produced by reducing noise in a dataset.

## Smooth Curve in Calculus

In general, a smooth curve is a rectifiable curve created on an interval from a differentiable function. Unfortunately, definitions found in many elementary calculus textbooks are often slightly ambiguous [2].

For example, Larson & Edwards [3] definition is relatively simple, as the graph of a continuously differentiable, rectifiable function on an interval. Cochran et al [4] also make the link between rectifiable and smooth (p. 366). But Massani et al. [5] use the term in the informal sense (i.e. “draw a smooth curve”).

Smooth curves are sometimes defined a little more precisely, especially in numerical analysis and complex analysis. For example [6]:

A curve is smooth if every point has a neighbourhood where the curve is the graph of a differentiable function. A curve can fail to be smooth if:

- It intersects itself,
- Has a cusp.

## References

[1] Clark. M. & Anfinson, X. (2012). Beginning and Intermediate Algebra: Connecting Concepts. Cengage Learning.

[2] Russell, E. & Sadek, J. (2005). A Note on the Definition of a Smooth Curve. Mathematics and Computer Education, v39 n1 p53-55

[3] Larson, R. & Edwards, B. (2016). Calculus, 10th Edition. Cengage Learning.

[4] Cochran, J. et al. (1987). Advanced Engineering Mathematics. Brooks/Cole Publishing Company.

[5] Massani, P. et al., (2014). Elementary Calculus. Elsevier Science.

[6] 1. Exercises from 3.2. Retrieved April 26, 2021 from: http://www.math.toronto.edu/~jmracek/Tutorial%2011.pdf

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