In this video I go over Question 3 of the Laboratory Project: Taylor Polynomials, and this time revisit the quadratic approximation but instead use a slightly different notation. In Question 1 I illustrated how the quadratic or parabola or 2nd order polynomial approximation P(x) = A + Bx + Cx^2 can be used to approximate a function f(x) at x = a, with the conditions that P(a) = f’(a), P’(a) = f’(a), and P’’(a) = f’’(a). But in this video I show that it is often preferable to use a slightly different notation and instead use P(x) = A + B(x – a) + C(x – a)^2. The only difference in using this form is that the constants A, B, C will not necessarily be the same. This notation has the benefit in that determine the constants with the 3 conditions listed above is fairly easy because when we input x = a into P(x) or its derivative, most terms vanish because a – a = 0. When we solve for the constants, I show that we obtain the function P(x) = f(a) + f’(a)(x – a) + f’’(a)(x – a)^2, which is a very convenient form to determine the constants, just from f(x) and its derivatives at x = a. Furthermore, this is the basis for Taylor Polynomials which I will be illustrating in further parts of this Laboratory Project, so make sure to watch this video and fully understand this concept! --- UPDATE: Error at the end which I wrote f"(a)(x-a) instead the correct term of f'(a)(x-a). --- Download the notes in my video: https://1drv.ms/b/s!As32ynv0LoaIh45zx3rzhpzXg-omew View video notes on the Hive blockchain: https://peakd.com/mathematics/@mes/laboratory-project-taylor-polynomials-question-3-x-a-approximation-form Related Videos: Laboratory Project: Taylor Polynomials: Question 2: Approximation Accuracy: https://youtu.be/MRSs0Qofd_M Laboratory Project: Taylor Polynomials: Question 1: Quadratic Approximation: https://youtu.be/8bpF3vccvEU Taylor Polynomials - Introduction and Derivation: http://youtu.be/p2EkXwkbflk Linear Approximation - Introduction and Examples: http://youtu.be/bXEK8bkWTtM tan(x) = sin(x) = x and cos(x) = 1 near x = 0: Linear Approximation in Physics: http://youtu.be/TPtZIxICa3Q Differentials Notation in Linear Approximation: http://youtu.be/s0adatWiZg4 Newton's Method of Linear Approximation - Introduction: http://youtu.be/aT4b_5l50RI . ------------------------------------------------------ SUBSCRIBE via EMAIL: https://mes.fm/subscribe DONATE! ʕ •ᴥ•ʔ https://mes.fm/donate Like, Subscribe, Favorite, and Comment Below! Follow us on: MES Truth: https://mes.fm/truth Official Website: https://MES.fm Hive: https://peakd.com/@mes MORE Links: https://linktr.ee/matheasy 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

In this video I go over a quick proof of the trigonometric identities tan(x + y) and tan(x – y). The identity is simple to derive because we can use the identities which I derived in my earlier videos for sin(x +/- y) and cos(x +/- y), as well as the fact that the tangent function, tan, is just equal to the ratio of sine over cosine. Although deriving the set of identities for tan(x +/- y) is pretty simple, I show how even that process can be simplified further by utilizing the use of the notations “+/-“ and “-/+”. This allows for the solving of two identities or sets of identities at the same time! I will be utilizing this set of trig identities in my later videos so make sure to watch this video! Download the notes in my video: https://1drv.ms/b/s!As32ynv0LoaIhvhE4tV3srF3zrHhxg View Video Notes on Steemit: https://steemit.com/mathematics/@mes/video-notes-trigonometric-identity-tan-x-y-proof Related Videos: Trigonometry Identities: cos(x +/- y) = cos(x)cos(y) -/+ sin(x)sin(y) : http://youtu.be/VuQczhk7HOs Trigonometry Identities: sin(x +/- y) = sin(x)cos(y) +/- cos(x)sin(y): http://youtu.be/edtk9thfwbM Trigonometry: Sine, Cosine and Tan Functions: http://youtu.be/WKTIlF2oWw8 . ------------------------------------------------------ SUBSCRIBE via EMAIL: https://mes.fm/subscribe DONATE! ʕ •ᴥ•ʔ https://mes.fm/donate Like, Subscribe, Favorite, and Comment Below! Follow us on: Official Website: https://MES.fm Steemit: https://steemit.com/@mes Gab: https://gab.ai/matheasysolutions Minds: https://minds.com/matheasysolutions Twitter: https://twitter.com/MathEasySolns Facebook: https://fb.com/MathEasySolutions LinkedIn: https://mes.fm/linkedin Pinterest: https://pinterest.com/MathEasySolns Instagram: https://instagram.com/MathEasySolutions Email me: contact@mes.fm Try our 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 Try our Free Online Tools: https://mes.fm/tools iPhone and Android Apps: https://mes.fm/mobile-apps ... https://www.youtube.com/watch?v=aUkEuD2Vdms

In this video I go over the sum of an infinite series that involves taking multiple derivatives of the Geometric Series. Starting off with the Geometric Series for x^n, we can apply the derivative to it for its radius of convergence, as derived from my earlier video on Power Series. When taking derivatives, the Radius of Convergence remains the same but we still have to determine the endpoints separately. The series with the terms (n^3)x^n has a radius of convergence equal to 1 and diverges at the endpoints, that is it diverges at x = 1 and x = -1. The timestamps of key parts of the video are listed below: - Problem 9: 0:00 - Solution: Start with a Geometric Series: 0:20 - Differentiating with Quotient Rule: 6:42 - Radius of Convergence is 1: 16:40 - Checking for convergence at the endpoints: 18:18 - Endpoints diverge: 21:09 This video was taken from my earlier video listed below: - Infinite Sequences and Series: Problems Plus: https://youtu.be/zjdkQIIdTbg - HIVE video notes: https://peakd.com/hive-128780/@mes/infinite-sequences-and-series-problems-plus - Video sections playlist: https://www.youtube.com/playlist?list=PLai3U8-WIK0FQ96Egr5R7fZGeDTIUKz8P Related Videos: Sequences and Series playlist: https://www.youtube.com/playlist?list=PLai3U8-WIK0EXHAJ3vRg0T_kKEyPah1Lz . ------------------------------------------------------ SUBSCRIBE via EMAIL: https://mes.fm/subscribe DONATE! ʕ •ᴥ•ʔ https://mes.fm/donate Like, Subscribe, Favorite, and Comment Below! Follow us on: MES Truth: https://mes.fm/truth Official Website: https://MES.fm Hive: https://peakd.com/@mes MORE Links: https://linktr.ee/matheasy 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

In this video I rewrite Euler's Formula to make it in the form of a rotation matrix. This follows from my earlier videos on equating complex numbers as vector rotations. Furthermore, for the case x = π, we get Euler's Identity, which in matrix form is just the negative Identity Matrix. Thus, even Euler's famous formula can be interpreted as simply rotating a vector. Since complex numbers are used in many areas of physics, such as electromagnetism and quantum mechanics, it may be fruitful to start reinterpreting other famous formulas as rotational matrices! In fact, this is exactly what @FractalWoman has attempted to do already: https://www.youtube.com/user/FractalWoman/search?query=complex%20numbers The timestamps of key parts of the video are listed below: - Reinterpreting Euler's Formula: 0:00 - Applications to Other Fields: 10:48 This video was taken from my earlier video listed below: - Complex Numbers as Rotation Matrices: https://youtu.be/Mgp2vrQeLEw - Video notes: https://peakd.com/hive-128780/@mes/complex-numbers-as-rotation-matrices - Playlist: https://www.youtube.com/playlist?list=PLai3U8-WIK0E6x0dZEAx77Kqv7LLxYdbx Related Videos: Vectors and the Geometry of Space Playlist: https://www.youtube.com/playlist?list=PLai3U8-WIK0FjJpwnxwdrOR7L8Ul8VZoZ . ------------------------------------------------------ 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 ... https://www.youtube.com/watch?v=aZkd2HKvCdU