Weight Measurements for Weighted Added Gyro Rising Experiments @mes

Math Easy Solutions | Weight Measurements for Weighted Added Gyro Rising Experiments @mes | Updated October 2024, 3 hours ago.

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$Exercise 2: Deriving the 1st Order and 3rd Order Optics Formulas In this video, I derive the first-order and third-order optics formulas by approximating the refraction formula for light using first and third-order Taylor polynomials, respectively. Refraction refers to the change in the direction of light as it passes between different mediums due to a change in speed. Light naturally travels in a way that minimizes time, and this principle helps us derive formulas for how light refracts when entering a material. The distances light travels before and after passing through an optical lens can be calculated using the Law of Cosines, which leads to a complex equation. However, we simplify this equation using Taylor polynomial approximations. In the first part of the video, I derive the first-order, or Gaussian optics, formula by approximating the cosine function in the distance equations with its first term, which is simply 1. In the second part, I derive the third-order Gaussian optics formula by using the third-degree Taylor polynomial for cosine, which includes the first two non-zero terms. I also apply a given hint to approximate the inverse of the distances using the first two terms of their respective binomial series. Additionally, I use a second hint to approximate the angle with its sine, which introduces a term involving the height. After working through the algebra, I arrive at the third-order optics formula! Timestamps: - Exercise 2: Derive 1st Order and 3rd Order Optics Equations for refraction: 0:00 - Solution to (a): Approximating cosine with First degree Taylor Polynomial: 0:59 - Equation 3: Gaussian optics or 1st order formula: 8:26 - Solution to (b): Approximating cosine with Third degree Taylor Polynomial: 9:19 - Modifying lengths in form of Binomial series: 17:46 - 1st Hint: Approximate lengths using first 2 terms of Binomial series: 23:20 - Plugging in our approximations into Equation 1: 28:06 - 2nd Hint: Approximate ɸ with sin(ɸ) to include the height term: 43:48 - Equation 4: 3rd order optics formula: 46:37 Full video, notes, and playlists: - Full video and playlist: https://www.youtube.com/playlist?list=PLai3U8-WIK0F76sIU8xm09oqBTq1mlry3 - HIVE notes: https://peakd.com/hive-128780/@mes/infinite-sequences-and-series-applications-of-taylor-polynomials - Infinite Sequences and Series: https://www.youtube.com/playlist?list=PLai3U8-WIK0EXHAJ3vRg0T_kKEyPah1Lz . Become a MES Super Fan! https://www.youtube.com/channel/UCUUBq1GPBvvGNz7dpgO14Ow/join DONATE! ʕ •ᴥ•ʔ https://mes.fm/donate SUBSCRIBE via EMAIL: https://mes.fm/subscribe MES Links: https://mes.fm/links MES Truth: https://mes.fm/truth Official Website: https://MES.fm Hive: https://peakd.com/@mes Email me: contact@mes.fm Free Calculators: https://mes.fm/calculators BMI Calculator: https://bmicalculator.mes.fm Grade Calculator: https://gradecalculator.mes.fm Mortgage Calculator: https://mortgagecalculator.mes.fm Percentage Calculator: https://percentagecalculator.mes.fm Free Online Tools: https://mes.fm/tools iPhone and Android Apps: https://mes.fm/mobile-apps$

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