Derivatives and Functions: Definitions and Interpretations

This lecture covers the definition of derivative functions in the context of functions from R to RM, focusing on the concept of differentiability. It also explores the usual definition of the derivative at a point, along with propositions related to functions defined on closed intervals. The lecture delves into the interpretation of functions as velocities in the context of time and position, emphasizing the instantaneous velocity of an object. Additionally, it discusses the length of continuously differentiable curves, providing examples and insights into the interpretation of functions as traces of paths. The lecture concludes with a discussion on equivalence classes of continuously differentiable functions and their relevance in defining curves.