Implications of the Mean Value Theorem

This lecture discusses the implications of the mean value theorem, starting with remarks and a reformulation of the theorem, followed by immediate implications. The reformulation focuses on analyzing small intervals to compare values at different points within a given interval using the mean value theorem and derivative information. The lecture also explores how to restrict functions to sub-intervals while maintaining the properties of the original function. By introducing a new perspective on the theorem, the instructor demonstrates how to use derivative information to predict function values at specific points. The reformulation sets the stage for analyzing small intervals and introduces the concept of a 'small' increment in function values.