Derivatives and Convexity

This lecture covers the concept of derivatives, local extrema, and convexity/concavity in functions. It explains Taylor's formula, expansions, and compositions of functions. The lecture also delves into Taylor expansion, limits, and examples of functions like Cos(x) - 1. The instructor demonstrates the properties of convex and concave functions, inflection points, and the behavior of functions around them. Additionally, the lecture explores the conditions for functions to be convex or concave, along with the implications of derivatives in determining the nature of functions.