In this video I revisit my earlier video on “Partial Fraction Decomposition: Proof of the Simplest Case”, https://youtu.be/Eq5yLJmC6pw, because a person by the username Heliocentric brought to my attention that I had not fully completed the proof. After a few back and forth exchanges I quickly realized that indeed Heliocentric was right and I had still some part of the proof yet to be explained. The issue at hand is in the Lemma 1 part of the proof which required proving that a polynomial exists with some given characteristics. In that video though, I simply assumed that the polynomial existed and worked to prove the characteristics. This is not fully complete because I still had yet to prove that it could have been in fact a polynomial. But as Heliocentric stated, and as I promised him 6 months ago that I would complete the proof of this video “later” (i.e. finally today!), I show how the proof that the function is in fact a polynomial can be shown by the Factor Theorem. Now recall in my earlier videos that the Factor Theorem is just a special case of the Polynomial Remainder Theorem, in which the division of an univariant polynomial by (x – a) has a remainder that equals zero. This in fact is the case with the formulation in the Lemma 1, and hence the factor theorem proves that in fact there does exist a polynomial P(x) with the given characteristics. This is a very important video illustrating now only the importance of thoroughly proving each step of a derivation, but also the importance to learn from constructive feedback and always make sure to correct yourself, so make sure to watch this video! Video notes and playlist: - PDF notes: https://1drv.ms/b/s!As32ynv0LoaIh44BB2VefNaDHI-HDQ - HIVE notes: https://peakd.com/mathematics/@mes/partial-fraction-decomposition-proof-of-the-simplest-case-revisited - Partial Fractions playlist: https://www.youtube.com/playlist?list=PLai3U8-WIK0HrJh8AbOkh3oYxSnthP997 Related Videos: Partial Fraction Decomposition: Proof of the Simplest Case: http://youtu.be/Eq5yLJmC6pw Polynomial Remainder Theorem: Factor Theorem & Factorization: https://youtu.be/ThX3uSXCD14 Euclidean Division of Integers: Theorem and Proof: https://youtu.be/66juubotzi0 Euclidean Division of Polynomials: Theorem and Proof: https://youtu.be/ONxn17okl5c Polynomial Remainder Theorem: Proof + Factor Theorem: https://youtu.be/q4lwSBObkXc Polynomial Remainder Theorem: Elementary Proof: https://youtu.be/Im0c25WbZBg . ------------------------------------------------------ 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

In this video I look further into example 2 on determining tangents to a cycloid and go over Part 2 which looks at determining the locations where the tangents are horizontal or vertical. From the curve of a cycloid it appears that the horizontal tangents occur at the midpoint of each arch, where as the vertical tangents are at the cusps, or sharp points, at the edges of each arch. I show that by using parametric calculus, this is indeed true. For the vertical tangents the derivative dy/dx encounters a scenario where have 0/0 or zero divided by zero so I show that we can still use L'Hospital's Rule to solve it. This is a very interesting and useful example on determining vertical and horizontal tangents, as well as in developing equations for periodic functions such as trig functions, so make sure to watch this video! Download the notes in my video: https://1drv.ms/b/s!As32ynv0LoaIhuMuJTKXFUnV0ZMgSQ View Video Notes on Steemit: https://steemit.com/mathematics/@mes/parametric-calculus-tangents-example-2-part-2 Related Videos: Parametric Calculus: Tangents: Example 1 Part 2: https://youtu.be/IYfBPigvzNc Parametric Calculus: Tangents: Example 1 Part 1: https://youtu.be/QBD_TvcJuTo Parametric Calculus: Tangents: https://youtu.be/deQwD2o0Sas Parametric Equations and Curves: https://youtu.be/Kd3XF4LZoFE Parametric Curves: Example 7: The Cycloid: Proof Part 1: https://youtu.be/XmHpIZqeRKk L'Hospital's Rule - Indeterminate Differences: http://youtu.be/FNj2EE8hEVs 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=4FnBRIl3gJg