Rolle's Theorem - Mean Value Theorem - L'Hôpital's Rule - Taylor Expansion - Lecture 16

Explore advanced calculus theorems in this MIT Real Analysis lecture that demonstrates Rolle's Theorem, the Mean Value Theorem, and their applications. Learn the Cauchy Mean Value Theorem, a more sophisticated version involving two functions, and discover how it serves as the foundation for proving L'Hôpital's rule for evaluating limits of quotients. Master both versions of L'Hôpital's rule and understand the crucial Taylor expansion theorem, gaining insight into how these fundamental results connect and build upon each other in real analysis. Develop a deeper understanding of these essential tools used throughout advanced mathematics and their practical applications in limit evaluation and function approximation.