# Arc length of function graphs via Taylor's formula

## TL;DR

This paper proves that functions with bounded second derivatives are rectifiable using Taylor's formula with Lagrange remainder, focusing on equally spaced interval subdivisions, and discusses educational benefits for calculus teaching.

## Contribution

It introduces a novel proof technique for rectifiability of functions with bounded second derivatives using Taylor's formula, with implications for calculus education.

## Key findings

- Functions with bounded second derivatives are rectifiable under equal interval subdivisions.
- Taylor's formula with Lagrange remainder can be used to establish rectifiability.
- Potential educational benefits for calculus courses are discussed.

## Abstract

We use Taylor's formula with Lagrange remainder to prove that functions with bounded second derivative are rectifiable in the case when polygonal paths are defined by interval subdivisions which are equally spaced. We discuss potential benefits for such an approach in introductory calculus courses.