Math Easy Solutions

In this video I go over the water wave velocity equation and approximate it when the waves are deep and when the waves are shallow. For deep waves, the hyperbolic tan or tanh(x) term in the water wave velocity equation approaches zero, which greatly simplifies the equation. For shallow waves, we can use the Maclaurin series approximation for tanh(x) and show that 3rd order or higher terms approach zero for shallow waves. I also calculate the error of this shallow wave speed approximation by using the alternating series estimation theorem. Timestamps: - Exercise 4: Water waves speed: 0:00 - Solution to (a): Deep water wave speed approximation: 1:43 - Approximation using hyperbolic tanh(x) = 1: 3:38 - Deep wave speed approximation formula: 5:29 - Solution to (b): Shallow water wave speed approximation: 6:04 - Maclaurin series approximation for tanh(x): 7:05 - Determining derivatives of tanh(x) at a = 0: 8:44 - Writing out our Maclaurin series approximation of tanh(x): 17:18 - 3rd order terms approach zero when water is shallow: 20:07 - Shallow water speed approximation: 21:43 - Shallow water speed approximation formula: 23:22 - Solution to (c): Alternating Series Estimation Theorem for the error: 24:25 - Estimation for tanh(x) approximation: 26:53 - tanh(x) remainder approximation if L is greater than 10d: 30:00 - Estimating accuracy of shallow water wave speed: 32:33 - Error is less than 0.0132gL: 34:02 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

In this video, I use the binomial series approximation to demonstrate that the electric field (E) of an electric dipole (two equal but opposite charges separated by a small distance) is proportional to 1 over the distance cubed when measured from a far distance. Normally, the electric field (E) of a single charge is proportional to the charge divided by the distance squared. However, for a dipole, we must subtract the electric field of one charge from the electric field of the other. By applying the binomial series to rewrite E and considering the limit at large distances, higher-order terms (second-order and beyond) approach zero. This simplification yields the approximate result: E ≈ 2dq/D³, where D is the distance from the dipole, d is the separation between the charges, and q is the magnitude of the charge. Timestamps: - Exercise 3: Electric Dipoles: 0:00 - Solution: Rewrite Electric Field equation to get d/D term: 1:38 - Binomial series approximation for d/D term: 4:24 - Plug in binomial series into electric field equation: 8:25 - Electric field of a dipole is proportional to 1/D^3 when D is much greater than d: 10:58 - E ≈ 2dq/D^3: 12:31 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

Adam Eisenberg, a direct witness at the Pentagon, and Norman aka 9/11 Revisionist join the show to discuss the Pentagon attack on 9/11 and what actually happened. September 14, 2024 SATURDAY at 1 PM PST / 3 PM CST / 4 PM EST / 10 PM CAT / 21:00 GMT (UK) Guest links: - Adam's Instagram: https://www.instagram.com/awakenedadam - Norman's Substack: https://substack.com/@911revision Stream notes and playlist: - Stream notes and links: https://peakd.com/hive-113182/@mes/livestream-54-pentagon-911 - MES Livestreams: https://www.youtube.com/@mes/streams - Playlist: https://www.youtube.com/playlist?list=PLai3U8-WIK0FqwyUa_ICwTlqO0S6Y3kAn . ------------------------------------------------------ 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 This stream is created with #PRISMLiveStudio

Join me and CREW in this special 23rd 9/11 anniversary livestream as we do a summary of all the key evidence on 9/11, which point to the use of cold-DEW and holographic projection technology. 9/11 Avengers that may be joining: - Andrew Johnson aka Check The Evidence - Mark Conlon aka 9/11 Planes Research - Adam Eisenberg Pentagon whistleblower - Norman aka 9/11 Revisionist - William aka That's Live But Not As We Know It (TLBNAWKI) - Jon Fortenberry aka American Fathers - Keith aka Conspiracy Realist - Methylminer - K-J French 9/11 Guy - Jeanne Traduction September 11, 2024 WEDNESDAY at 1 PM PST / 3 PM CST / 4 PM EST / 10 PM CAT / 21:00 GMT (UK) Stream notes and playlist: - Stream notes and links: https://peakd.com/hive-113182/@mes/livestream-53-911-summary - MES Livestreams: https://www.youtube.com/@mes/streams - Playlist: https://www.youtube.com/playlist?list=PLai3U8-WIK0FqwyUa_ICwTlqO0S6Y3kAn . ------------------------------------------------------ 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 This stream is created with #PRISMLiveStudio