Calculus1-Lecture 24|Asymptotes, Vertical Tangents and Cusps Explained скачать в хорошем качестве

Calculus1-Lecture 24|Asymptotes, Vertical Tangents and Cusps Explained

In this lecture, we study vertical and horizontal asymptotes and analyze the behavior of functions near these asymptotes. Topics covered: Definition of vertical asymptotes Definition of horizontal asymptotes Using limits to determine asymptotes Sign analysis near vertical asymptotes Examples involving rational and trigonometric functions Vertical tangents Vertical cusps and how they differ from vertical tangents Several examples illustrate how limits and derivatives describe the behavior of graphs near asymptotes and special points. Instructor: Prof. Shadi Shaqaqha Department of Mathematics – Yarmouk University شرح المقاربـات الرأسية والأفقية (Asymptotes) وكذلك المماس الرأسي والنقطة الحادة (Vertical Cusp) مع تطبيقات على دوال كسرية ومثلثية. 00:00 Vertical asymptotes – definition 01:58 Horizontal asymptotes – definition 02:47 Example – asymptotes of 2x/(x²−9) 07:49 Example – asymptotes of cos(x)/x 12:10 Example – asymptotes of (x/(1−2x))³ 13:41 Example – asymptotes of √x /(2√x − x − 1) 18:45 Vertical tangents – definition 19:49 Vertical cusps – definition 20:47 Examples – tangent vs cusp classification