Infinite Sequences and Series: Applications of Taylor Polynomials

In this video I go over further into infinite sequences and series, and this time look at the many applications of Taylor polynomials. When I was still a student in university, it was this particular concept of Taylor polynomials and their applications to all aspects of physics that got me interested in creating my Math Easy Solutions channel. And in this video I explore Taylor polynomials in approximating functions which can then be extended to most functions in physics.

The accuracy of our approximations can be determined using the Alternating Series Estimation theorem, Taylor's Inequality, and through graphical or even brute force calculation methods. The particular science and physics applications that I cover in this video include optics, electromagnetism, water waves, pendulums, and surveying.

The topics covered in this video are listed below with their time stamps.

- @ 0:00 - Intro
- @ 1:04 - Topics to Cover
1. @ 2:18 - Introduction
- @ 2:48 - Recap on Power Series and Taylor Polynomials
2. @ 8:22 - Approximating Functions by Polynomials
- @ 30:37 - Example 1: f(x) = x^(1/3)
- Example 2: f(x) = sin x
- @ 53:48 - Using Alternating Series Estimation Theorem
- @ 1:14:22 - Using Taylor's Inequality
- @ 1:21:11 - Using Graphical Methods
3. @ 1:32:04 - Applications to Physics
- @ 1:32:38 - Example 3: Einstein's Theory of Special Relativity
- @ 2:02:16 - Optical Physics
4. @ 2:21:25 - Exercises
- @ 2:21:42 - Exercise 1: Taylor Derivatives
- @ 2:30:50 - Exercise 2: Optical Physics
- @ 3:19:35 - Exercise 3: Electric Dipoles
- @ 3:33:36 - Exercise 4: Water Waves
- @ 4:09:52 - Exercise 5: Pendulum Period
- @ 4:52:04 - Exercise 6: Surveying

Download Video Notes: https://1drv.ms/b/s!As32ynv0LoaIiMQFpSc5JN-fpBUyHw

Video notes and playlists:

- PDF notes: https://1drv.ms/b/s!As32ynv0LoaIiMQFpSc5JN-fpBUyHw
- HIVE notes: https://peakd.com/hive-128780/@mes/infinite-sequences-and-series-applications-of-taylor-polynomials
- Video sections playlist: https://www.youtube.com/playlist?list=PLai3U8-WIK0F76sIU8xm09oqBTq1mlry3
- Vectors and the Geometry of Space playlist: https://www.youtube.com/playlist?list=PLai3U8-WIK0FjJpwnxwdrOR7L8Ul8VZoZ .
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