Calculus I – Lecture 11 Intermediate Value Theorem & Extreme Value Theorem

In this lecture, we study two fundamental results in Calculus: • The Intermediate Value Theorem (IVT) • The Extreme Value Theorem (EVT) We explore: The geometric meaning of the IVT Existence of roots using continuity Applications to polynomial and rational functions Bounded functions Conditions required for the EVT (closed and bounded intervals) Why the hypotheses of EVT are essential Several illustrative sketch problems These theorems are essential tools for existence proofs in calculus and form the bridge toward differential calculus. Instructor: Prof. Shadi Shaqaqha Department of Mathematics – Yarmouk University شرح مبرهنتي القيمة المتوسطة والقيم القصوى مع تطبيقات ورسوم توضيحية. 00:00 Intermediate Value Theorem (Statement & Idea) 02:06 IVT – Polynomial Example 03:54 IVT – Equation with Radical 10:03 IVT – Rational Function Example 12:35 Bounded Functions 15:37 Extreme Value Theorem (Statement) 17:43 Sketch Examples & Counterexamples