Math Easy Solutions

In this video I recap on Taylor polynomials, Taylor series, Maclaurin series, and Taylor's inequality. A Taylor polynomial is a finite sum with terms consisting of the i-th derivative at a divided by the i-th factorial and multiplied by (x - a) to the power of i. A Taylor series is just an infinite Taylor polynomial. A Maclaurin series is a special type of Taylor series where a = 0. A function is equal to its Taylor series if the remainder approaches 0 as n approaches infinity. Taylor's Inequality states that if the (n + 1) derivative of f(x) is less than or equal to a number M then the absolute value of the n-th remainder of a Taylor series is less than M divided by (n + 1) factorial multiplied by the absolute value of (x - a) to the power of (n + 1). Taylor and Maclaurin series are very useful since they can turn complicated functions into a simple sum of many easily calculatable terms, and is how most calculators work behind the scenes! The timestamps of key parts of the video are listed below: - Question 10: 0:00 - (a) Taylor polynomial: 0:42 - (b) Taylor series: 2:20 - (c) Maclaurin series: 3:33 - (d) A function equals its Taylor series if the remainder approaches 0: 4:26 - (e) Taylor's Inequality: 5:45 This video was taken from my earlier video listed below: - Infinite Sequences and Series: Review and True-False Quiz: https://youtu.be/F0dsQLdXXpI - HIVE video notes: https://peakd.com/hive-128780/@mes/infinite-sequences-and-series-review-and-true-false-quiz - Video sections playlist: https://www.youtube.com/playlist?list=PLai3U8-WIK0FCqXVJv1r7eJvrvphfkr6L Related Videos: Sequences and Series playlist: https://www.youtube.com/playlist?list=PLai3U8-WIK0FjJpwnxwdrOR7L8Ul8VZoZ . ------------------------------------------------------ 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 recap on term-by-term differentiation and integration of power series. This theorem, which I proved in my earlier video, states that the derivative of a power series is equivalent to first taking the derivative of each term and then summing it up. Likewise this is true for integrating a power series as well. In both cases, the radius of convergence is the same as that of the original power series. Note that the interval of convergence need not be the same for either case. Differentiating power series term-by-term has an important application in solving differential equations so make sure you fully grasp this concept! The timestamps of key parts of the video are listed below: - Question 9: 0:00 - (a) Derivative of Power Series: 0:32 - (b) Integral of Power Series: 3:59 - Radius of convergence of derivative and integral is R: 6:06 - Rewriting the derivative and integral equations: 6:27 - Note 1: Infinite sum vs finite sums: 7:39 - Note 2: Interval of convergence may change: 8:26 - Note 3: Application in solving differential equations: 8:58 This video was taken from my earlier video listed below: - Infinite Sequences and Series: Review and True-False Quiz: https://youtu.be/F0dsQLdXXpI - HIVE video notes: https://peakd.com/hive-128780/@mes/infinite-sequences-and-series-review-and-true-false-quiz - Video sections playlist: https://www.youtube.com/playlist?list=PLai3U8-WIK0FCqXVJv1r7eJvrvphfkr6L Related Videos: Sequences and Series playlist: https://www.youtube.com/playlist?list=PLai3U8-WIK0FjJpwnxwdrOR7L8Ul8VZoZ . ------------------------------------------------------ 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 recap on power series and the conditions that it converges, that is, its radius and interval of convergence. A power series has terms of the form: constant * (x - a)^n. There are only three possibilities for its convergence: Either it converges only for x = a, or converges for all x, or it converges for |x - a| is less than the radius of convergence R. In the first case, the radius of convergence is considered as R = 0, and the second case it is R = infinity. The interval of convergence has 4 possibilities, since at |x - a| = R, the series can converge or diverge. Thus the interval of convergence is either (a - R, a + R), (a - R, a + R], [a - R, a + R), or [a - R, a + R]. I also graphically show this in the video. The timestamps of key parts of the video are listed below: - Question 8: 0:00 - (a) General form of a power series: 0:20 - (b) Radius of convergence of a power series: 1:11 - Summary for the radius of convergence: 6:28 - (c) Interval of convergence of a power series: 7:40 This video was taken from my earlier video listed below: - Infinite Sequences and Series: Review and True-False Quiz: https://youtu.be/F0dsQLdXXpI - HIVE video notes: https://peakd.com/hive-128780/@mes/infinite-sequences-and-series-review-and-true-false-quiz - Video sections playlist: https://www.youtube.com/playlist?list=PLai3U8-WIK0FCqXVJv1r7eJvrvphfkr6L Related Videos: Sequences and Series playlist: https://www.youtube.com/playlist?list=PLai3U8-WIK0FjJpwnxwdrOR7L8Ul8VZoZ . ------------------------------------------------------ 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 further on the Malaysian Airlines Flight MH370 alleged teleportation videos, and this time double down on the duplicate frame by showing that the UFO orb rotates the WRONG direction at the 45th second frame! In my previous video I showed that the supposed MH370 thermal plane footage likely involved a copy and paste job for one of the frames, which is unlike any typical video compression techniques, and this video further confirms it. Going frame by frame on the orb, it maintains its rotational motion throughout the video except at the 45 second frame, where is rotates abruptly the wrong direction before abruptly rotating back in the next frame. At the 47 second frame, the duplicate frame shows proper orb motion with no abrupt changes. Thus the 47th frame is likely the correct frame while the 45th frame is the edited frame. Note also that the first "stereoscopic satellite 3D" plane footage involves the plane, orbs, and clouds moving every 5th frame while the mouse moves every frame besides duplicating every 5th frame. In the thermal video, it too doubles every 5th frame. If you know why this is or have further information on it, please let me know! Links to all the references and videos used are listed below: - MES Livestream 11: Emergency MH370 Broadcast: https://www.youtube.com/live/C2vv4FjSh6o - MH370 Teleportation Video Fakery: Duplicate Frames (NOT Video Compression): https://youtu.be/tazw6CAcrKo - Duplicate frames MES Links telegram post: https://t.me/meslinks/21496 - Mick West tweet: https://twitter.com/MickWest/status/1726385274534679014 - @ophello tweet: https://twitter.com/ophello/status/1726828364600840534 - NOT video compression: https://twitter.com/528vibes/status/1726567372968898691 - NOT video motion vector compression: https://www.securitylinkindia.com/technology-paper/2019/07/15/video-compression-and-its-significance3/ - Camtasia video editing file: https://1drv.ms/u/s!As32ynv0LoaIirhfhyD_2K1OSVA5vw - GIMP photoshop file: https://1drv.ms/u/s!As32ynv0LoaIirhguJ36QTkDgImkVA - MH370 videos on MES Links telegram: https://t.me/meslinks/21412 - May 19, 2014: Satellite Video: Airliner and UFOs by RegicideAnon: http://web.archive.org/web/20170606182854/https://youtube.com/watch?v=5Ok1A1fSzxY - Download link: https://web.archive.org/web/20140526071334oe_... - MES local download: https://1drv.ms/v/s!As32ynv0LoaIirhd1lFnID1lbw0Yzw - Used the WayBack Machine YouTube downloader: https://www.waybackyt.download/ - June 12, 2014 video: UAV-Captures Airliner and UFOs by RegicideAnon: http://web.archive.org/web/20140827060121/https://youtube.com/watch?v=ShapuD290K0 - Download link: https://web.archive.org/web/20190107190546oe_... - MES local download: https://1drv.ms/v/s!As32ynv0LoaIirheG432fgdwTFwXEA - Ashton Forbes calls me out LOL: https://twitter.com/JustXAshton/status/1727520818182209640 - My Duplicate Frames video got shared on the r/AirlinerAbduction2014 Reddit forum! https://www.reddit.com/r/AirlinerAbduction2014/comments/181uh0y/mh370_teleportation_video_fakery_duplicate_frames/ The timestamps of key parts of the video are listed below: - Links to videos and references: 0:00 - Ashton Forbes calls out MES... LOL: 1:55 - My earlier video shared on Reddit: 3:12 - Notes: Both videos duplicate every 5th frame: 3:28 - "Satellite" Footage moves every 5th frame: 4:09 - Mouse duplicates every 5th frame: 5:41 - "Thermal" footage duplicates every 5th frame: 6:09 - 45th second orb rotates the wrong way: 6:44 - 47th second orb rotates correctly: 8:07 - Checking the orb rotation one more time: 10:11 - Let me know if you have information on the duplicate 5th frames: 11:00 . ------------------------------------------------------ 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 my brother Mark aka MFA goes over how the NBA's new in-season tournament works and how each group is selected. The tournament involves splitting each conference into 3 groups of 5 teams, which are randomly drawn from 5 pots of 3 teams each (listed in order of last year's season rankings). The group stage involves playing each team inside the group twice, one road and one away, while the games count towards the regular season. The top team in each group, and 1 wildcard from each conference make it to the knockout stage. The first knockout stage is the quarter finals, while the losing teams play each other in each conference; once again all games here count towards the regular season. Next is the Semi-finals, which also count towards the regular season. The Finals will be held in Las Vegas and will not count towards the regular season. Every player in the knockout stages gets a financial bonus, while the champion gets $500,000 per player! Note also that every in-season tournament game will have the court painted in custom tournament colors. This is the first year that the National Basketball Association (NBA) has included a tournament within the regular season. The goal is to increase the excitement around the early part of the long NBA season. I think it's a great idea, what do you think? The timestamps of key parts of the video are listed below: - NBA In-Season Tournament: 0:00 - Group Play: 0:36 - Knockout Rounds: Quarter Finals: 1:39 - Semifinals: 2:25 - Finals: 2:44 - Conclusion: 3:12 - Tie Breaker: 3:51 - Choosing the Groups: 4:16 - Example: Eastern Conference: 4:49 - What do you think of the Tournament? 5:37 Download Video Notes: - PDF: https://1drv.ms/b/s!As32ynv0LoaIirpTLmaguLb8DrxC0g - HIVE: https://peakd.com/nba/@mes/nba-in-season-tournament-how-does-it-work Related Videos: NBA basketball tutorials: https://www.youtube.com/playlist?list=PL43B7E7AE41C847FA . ------------------------------------------------------ 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 show that 2 frames 2 seconds apart from the missing Malaysia airlines flight MH370 plane alleged teleportation videos likely involved manually copying, pasting, scaling, and moving a portion of the frame. This is an important piece of evidence that shows that the alleged teleportation videos were faked to at least some degree. These similar frames are not the result of video compression, since the entire frame was not merely duplicated, nor any specific vector tracking was implemented to track motion. The portion of the frame included the entire plane, an entire orb, and a select portion of the background noise. This was, again, cut, scaled up, and moved to a different position. And interestingly, the later frame has the better quality, which suggests that it was the frame that was cropped and scaled down, thus reducing its quality and sharpness. This piece of video fakery falls in line with the other VFX stock image fakery and overall shady Ashton Forbes character, which I explored in my previous livestream: https://www.youtube.com/live/C2vv4FjSh6o Links to all the references and videos used are listed below: - MES Livestream 11: Emergency MH370 Broadcast: https://www.youtube.com/live/C2vv4FjSh6o - Camtasia video editing file: https://1drv.ms/u/s!As32ynv0LoaIirhfhyD_2K1OSVA5vw - GIMP photo editing file: https://1drv.ms/u/s!As32ynv0LoaIirhguJ36QTkDgImkVA - MH370 videos on MES Links telegram: https://t.me/meslinks/21412 - June 12, 2014 video: UAV-Captures Airliner and UFOs by RegicideAnon: http://web.archive.org/web/20140827060121/https://youtube.com/watch?v=ShapuD290K0 - Download link: https://web.archive.org/web/20190107190546oe_... - MES Local download: https://1drv.ms/v/s!As32ynv0LoaIirheG432fgdwTFwXEA - Duplicate frames MES Links telegram post: https://t.me/meslinks/21496 - Mick West tweet: https://twitter.com/MickWest/status/1726385274534679014 - @ophello tweet: https://twitter.com/ophello/status/1726828364600840534 - NOT video compression: https://twitter.com/528vibes/status/1726567372968898691 - NOT video motion vector compression: https://www.securitylinkindia.com/technology-paper/2019/07/15/video-compression-and-its-significance3/ . ------------------------------------------------------ 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 recap on estimating the sum of a series based on whether it converges by the Integral Test, Comparison Test, or the Alternating Series Tests. If a series is convergent by the Integral Test, we can estimate its sum by first finding an estimate of the size of its remainder. Since an integral is formed by an infinite set of rectangles, the remainder can be estimated based on whether the rectangles are above or below the curve. From this, we can estimate the sum of the series for the integral test. If a series is convergent by the Comparison Test, we can estimate its sum by estimating the sum of the series it is being compared to. For example, if a series is less than another series that happens to converge by the integral test, we can use the sum estimate for the integral test. If a series is convergent by the alternating series test, we can apply the alternating series estimation theorem. This theorem states that the absolute value of the remainder of the n-th partial sum of a series is less than the next positive term of the series. This can be seen visually since each subsequent term is larger than the difference between the sum and any given partial sum. The timestamps of key parts of the video are listed below: - Question 7: 0:00 - (a) Integral Test sum estimation: 0:27 - Estimating the size of the remainder: 1:47 - Remainder estimate for the integral test: 6:57 - (b) Estimating sum using the Comparison Test: 8:25 - (c) Estimating sum of alternating series: 15:02 - Visualization of the alternating series estimation theorem: 16:31 This video was taken from my earlier video listed below: - Infinite Sequences and Series: Review and True-False Quiz: https://youtu.be/F0dsQLdXXpI - HIVE video notes: https://peakd.com/hive-128780/@mes/infinite-sequences-and-series-review-and-true-false-quiz - Video sections playlist: https://www.youtube.com/playlist?list=PLai3U8-WIK0FCqXVJv1r7eJvrvphfkr6L Related Videos: Sequences and Series playlist: https://www.youtube.com/playlist?list=PLai3U8-WIK0FjJpwnxwdrOR7L8Ul8VZoZ . ------------------------------------------------------ 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 #911Truth Part 25, I have uploaded the full 9-1-1 phone call from Melissa Doi inside the impact zone at the 83rd floor of the South Tower on 9/11. The phone call shows a frantic Melissa saying "she can't breathe" and she is feeling intense heat but with very little indication of actual conventional thermal heat. She says she is "burning up" but she does not appear burned. She says there is smoke blocking out all the air, yet she hardly coughs and her voice is never hindered. She says she can't see fire but only that she can "feel" fire. The New York Police Department (NYPD) 9-1-1 emergency dispatcher tries to console her for over 20 minutes, and finally Melissa ends up "unconscious and snoring"; literally 18 minutes before the South Tower's fall. The World Trade Center 2 (WTC 2) or the South Tower was struck by an alleged plane 14 minutes before Melissa's phone call and between the 77th and 85th floors. Melissa was quite literally in the middle of the impact zone on the 83rd floor. She was with 5 other people trapped on the floor and even says she heard other people walking around, which may have been the Firefighters that quite literally walked up all the way up the alleged "flaming inferno". From the phone call, it is clear that Melissa did not experience heat from a fire, but rather something more subtle, such as possible microwave directed energy. This is inline with what the "jumpers" may have experienced as they took their clothes off while on the window edge before either throwing themselves or being launched against their wills. Here are some important links for reference: - Melissa Doi Wiki: https://en.wikipedia.org/wiki/Melissa_Doi - I downloaded the phone call from NPR: https://www.npr.org/templates/story/story.php?storyId=5658516 - I found portions of the beeped out audio from Metro.co.uk: https://metro.co.uk/2022/09/11/9-11-the-haunting-last-calls-from-victims-of-the-september-11-attacks-2-17345144/ - I used most of this phone call transcript: https://en.wikisource.org/wiki/9/11_Dispatcher_transcript - Articles about Melissa Doi: - 2006 New York The Sun: https://web.archive.org/web/20080907020542/http://www.nysun.com/editorials/melissa-doi/38055/ - 2006 Daily News: https://www.nydailynews.com/2006/08/17/i-feel-like-im-dying/ - 2006 The Washington Post: https://www.washingtonpost.com/archive/politics/2006/08/17/more-voices-from-911-im-going-to-die-arent-i/d99eba25-43df-4ae6-82c4-f3268a603123/ The timestamps of key parts of the video are listed below: - WTC 2 impact at 9:03:02 AM between the 77th and 85th floors: 0:00 - Melissa Doi calls 911 at 9:17 AM from the 83rd floor, she was with 5 other people: 0:36 - "We can't breathe", "it's very hot": 1:18 - "but no fire, right?" 2:25 - "I think there is fire" 2:44 - "you see no fire": 2:49 - "I don't see any air anymore": 2:58 - "I'm burning up": 3:30 - "We hear voices! HELLO! HELP!: 3:50 - "I feel like I'm dying": 4:39 - Court order removed civilians and Melissa Doi's voice from the remainder of the call: 5:02 - "I feel fire": 7:18 - Melissa Doi requests a note to her mother Evelyne Alderete: 7:59 - "Tell her that she's the best mom in the world... I love her with all my heart and soul": 8:54 - "Tell her I love her so much. I love her. She's the best mother a person could ever have": 9:13 - "Melissa, Melissa, Melissa..." 12:08 - "They sound like they might be unconscious": 22:23 - "Everybody's snoring": 24:15 - It's not dead there. They're snoring": 24:22 - "sounds like they were unconscious and snoring": 24:56 - "The line is now dead.... oh my Lord": 25:06 - MES Note: Melissa Doi experienced a different mechanism other than a conventional fire: 25:31 - WTC 2 South Tower falls at 9:59:00 - WTC 1 North Tower turning to dust (45X speed): 26:21 Stay tuned for #911Truth Part 26... - HIVE video screenshots: https://peakd.com/hive-113182/@mes/911truth-part-25-melissa-doi-911-call-from-the-83rd-floor-of-the-south-tower Related Videos: - #911Truth video series: https://www.youtube.com/playlist?list=PLai3U8-WIK0EzqTamtIXtgX8QudQSxuxh and https://peakd.com/truth/@mes/911 - 9/11 Observable Evidence video sections playlist: https://www.youtube.com/playlist?list=PLai3U8-WIK0G_HHWt33moIqEeUBP3cgCh . ------------------------------------------------------ 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 quickly recap on absolutely and conditionally convergent series. An absolutely convergent series is such that the absolute value of all the terms of a series is convergent. Since the absolute value of all the terms means that the series sums up to its maximum size, if it is absolutely convergent then it means it is also convergent. On the other hand, if a series is convergent but NOT absolutely convergent, then it is said to be conditionally convergent. This arises from positive and negative terms canceling out. The timestamps of key parts of the video are listed below: - Question 6: 0:00 - (a) Absolutely convergent series: 0:14 - (b) Absolutely convergent series are convergent: 0:56 - (c) Conditionally convergent series: 1:12 This video was taken from my earlier video listed below: - Infinite Sequences and Series: Review and True-False Quiz: https://youtu.be/F0dsQLdXXpI - HIVE video notes: https://peakd.com/hive-128780/@mes/infinite-sequences-and-series-review-and-true-false-quiz - Video sections playlist: https://www.youtube.com/playlist?list=PLai3U8-WIK0FCqXVJv1r7eJvrvphfkr6L Related Videos: Sequences and Series playlist: https://www.youtube.com/playlist?list=PLai3U8-WIK0FjJpwnxwdrOR7L8Ul8VZoZ . ------------------------------------------------------ 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 recap on 7 different types of tests for determining if an infinite series converges or diverges. The first is the Test for Divergence which states that if the limit of the sequence of terms does not approach 0, then the series is divergent. The second is the Integral Test, which states that the series is only convergent if its corresponding integral is also convergent. The third is the Comparison Test, which states that a series is convergent if every term of its sequence is smaller than that of a known convergent series. The fourth is the Limit Comparison Test, which states that if the limit of the ratio of the terms of 2 sequences is equal to a finite number c that is greater than 0, then either both series converge or both diverge. The fifth is the Alternating Series Test, which states that if the absolute value of all the terms of an alternating series is less than each subsequent term, and their limit approaches 0, then the alternating series is convergent. The sixth is the Ratio Test, which states that if the limit of the absolute value of the ratio of the n + 1 term divided by the n-th term is equal to a finite number less than 1, then the series is convergent. The seventh test is the Root Test, which states that if the n-th root of the absolute value of the n-th term is equal to a finite number less than 1, then the series is convergent. Note that for all the tests listed, they have similar conditions for when the series is divergent, or in the case of the Ratio and Root Tests, where the tests are inconclusive when the limits equal to 1. The timestamps of key parts of the video are listed below: - Question 5: 0:00 - (a) Test for Divergence: 0:22 - (b) Integral Test: 1:25 - (c) Comparison Test: 3:32 - (d) Limit Comparison Test: 4:43 - (e) Alternating Series Test: 5:28 - (f) Ratio Test: 8:01 - (h) Root Test: 9:23 This video was taken from my earlier video listed below: - Infinite Sequences and Series: Review and True-False Quiz: https://youtu.be/F0dsQLdXXpI - HIVE video notes: https://peakd.com/hive-128780/@mes/infinite-sequences-and-series-review-and-true-false-quiz - Video sections playlist: https://www.youtube.com/playlist?list=PLai3U8-WIK0FCqXVJv1r7eJvrvphfkr6L Related Videos: Sequences and Series playlist: https://www.youtube.com/playlist?list=PLai3U8-WIK0FjJpwnxwdrOR7L8Ul8VZoZ . ------------------------------------------------------ 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 recap on how the sum of a series affects the limit of its terms and its partial sums. If an infinite series sums up to 3, then this must mean that the terms of the series all approach the limit of 0 while the limit of the partial sums approach 3. In other words, the limit of the sequence of partial sums is equal to the sum of the infinite series. The timestamps of key parts of the video are listed below: - Question 4: 0:00 - Solution: 0:47 - Limits of the sequence vs partial sums: 1:06 This video was taken from my earlier video listed below: - Infinite Sequences and Series: Review and True-False Quiz: https://youtu.be/F0dsQLdXXpI - HIVE video notes: https://peakd.com/hive-128780/@mes/infinite-sequences-and-series-review-and-true-false-quiz - Video sections playlist: https://www.youtube.com/playlist?list=PLai3U8-WIK0FCqXVJv1r7eJvrvphfkr6L Related Videos: Sequences and Series playlist: https://www.youtube.com/playlist?list=PLai3U8-WIK0FjJpwnxwdrOR7L8Ul8VZoZ . ------------------------------------------------------ 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 further into Conic Sections and this time go over the definition of a Hyperbola and derive its resulting formula. I had gone over this derivation several years ago but I have decided to revisit it to be better tie into the Conic Sections video series I have recently been making, as well as to go over it in more detail. The Hyperbola is defined as the set of points on a plane with the property that the difference in the distances from each point to two fixed points (known as the foci) is a constant. Using this definition for the foci on the x-axis, I first show that the difference is equal to +/- 2a because this is the difference when the Hyperbola is also on the x-axis, at either of the two points known as the vertices, and is thus defined as constant for Hyperbolas. The derivation for the hyperbola is very similar to that for the ellipse, which I covered in my earlier video, and involves using the Pythagorean Theorem for the distances and combined with a lot of algebra to simplify the resulting formulation. After some careful algebra, I show that we can eventually write the hyperbola as x^2/a^2 – y^2/b^2 = 1 and where b^2 = c^2 – a^2. The resulting graph of the hyperbola consists of two branches that extend outwards approaching the slant asymptote lines, which I covered in my last two videos, and are y = +/ (b/a)x. This is known as a Horizontal Hyperbola, but we can switch up the x and y terms to get a Vertical Hyperbola y^2/a^2 – x^2/b^2 = 1 and with slant asymptotes y = +/- (a/b)x. This is a very extensive video covering the definition and derivation of Hyperbolas so make sure to watch this video! Video notes: - PDF: https://1drv.ms/b/s!As32ynv0LoaIh5FP9bXo4PFSmGkJZw - HIVE: https://peakd.com/mathematics/@mes/conic-sections-hyperbola-definition-and-formula Related Videos: Conic Sections: Parabolas: Definition and Formula: https://youtu.be/kCJjXuuIqbE Conic Sections: Ellipses: Definition and Derivation of Formula (Including Circles): https://youtu.be/9dETsJ2tz_M Slant Asymptote Lines: Example 2: Horizontal Hyperbola: https://youtu.be/XSudAYmWJ-M Slant Asymptote Lines: Example 3: Vertical Hyperbola: https://youtu.be/oFYvWFhcTwI Hyperbola - Definition and derivation of the equation: x^2/a^2 - y^2/b^2 = 1: http://youtu.be/Y6iYC4VEAi0 . ------------------------------------------------------ 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 review of geometric series and p-series and show the circumstances for when they are convergent. A geometric series has all the terms being a constant number multiplied by a common ratio starting from the power of 0 and incrementing by 1 for each successive term. The series is convergent and equal to the (first term) / (1 - common ratio) when the absolute value of the common ratio is less than 1. I also show a geometric interpretation of the geometric series using similar triangles. A p-series is of the form 1/n^p and it is convergent when p is greater than 1 and divergent for all other values. The timestamps of key parts of the video are listed below: - Question 3: 0:00 - Solution to (a): Geometric series: 0:28 - Geometric representation via similar triangles: 2:04 - Solution to (b): p-Series: 4:51 This video was taken from my earlier video listed below: - Infinite Sequences and Series: Review and True-False Quiz: https://youtu.be/F0dsQLdXXpI - HIVE video notes: https://peakd.com/hive-128780/@mes/infinite-sequences-and-series-review-and-true-false-quiz - Video sections playlist: https://www.youtube.com/playlist?list=PLai3U8-WIK0FCqXVJv1r7eJvrvphfkr6L Related Videos: Sequences and Series playlist: https://www.youtube.com/playlist?list=PLai3U8-WIK0FjJpwnxwdrOR7L8Ul8VZoZ . ------------------------------------------------------ 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 review bounded and monotonic sequences as well as the monotonic sequence theorem. A bounded sequence is one in that is bounded above and bounded below. A sequence is bounded above if there is a number that is less than every term in the sequence. Likewise, a sequence bounded below is if there is a number that is larger than every term in the sequence. A monotonic sequence is either always increasing or always decreasing. The monotonic sequence theorem, which I covered in my earlier video, states that a bounded, monotonic sequence is always convergent. The timestamps of key parts of the video are listed below: - Question 2: 0:00 - Solution to (a): Bounded sequence: 0:22 - Solution to (b): Monotonic sequence: 1:33 - Solution to (c): Monotonic sequence theorem: 2:51 This video was taken from my earlier video listed below: - Infinite Sequences and Series: Review and True-False Quiz: https://youtu.be/F0dsQLdXXpI - HIVE video notes: https://peakd.com/hive-128780/@mes/infinite-sequences-and-series-review-and-true-false-quiz - Video sections playlist: https://www.youtube.com/playlist?list=PLai3U8-WIK0FCqXVJv1r7eJvrvphfkr6L Related Videos: Sequences and Series playlist: https://www.youtube.com/playlist?list=PLai3U8-WIK0FjJpwnxwdrOR7L8Ul8VZoZ . ------------------------------------------------------ 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 review of what convergent sequences and convergent series are. A sequence is just a list of numbers, and it is said to be convergent if its limit approaches a specific number as the number of terms gets large. A series is a summation of a sequence, and it is said to be convergent if the limit of the n-th partial sum of the sequence approaches a specific real number. I compare the definitions for when a both the sequence and series converge to the number 3. The timestamps of key parts of the video are listed below: - Question 1: 0:00 - Solution to Part (a): Convergent sequence: 0:39 - Solution to Part (b): Convergent series: 2:29 - Solution to Part (c): Limit of sequence equals 3: 4:31 - Solution to Part (d): Limit of series equals 3: 4:59 This video was taken from my earlier video listed below: - Infinite Sequences and Series: Review and True-False Quiz: https://youtu.be/F0dsQLdXXpI - HIVE video notes: https://peakd.com/hive-128780/@mes/infinite-sequences-and-series-review-and-true-false-quiz - Video sections playlist: https://www.youtube.com/playlist?list=PLai3U8-WIK0FCqXVJv1r7eJvrvphfkr6L Related Videos: Sequences and Series playlist: https://www.youtube.com/playlist?list=PLai3U8-WIK0FjJpwnxwdrOR7L8Ul8VZoZ . ------------------------------------------------------ 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

Join me live as I answer questions regarding 9/11 and debunk the mainstream pancakes and alt-media explosive nonsense; both of which cover up the use of suppressed DEW technology that turned the towers to dust! Related Videos: MES Livestreams: https://www.youtube.com/@mes/streams . ------------------------------------------------------ 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 derive the formula for projecting 3D coordinates onto a 2D screen using similar triangles. First I plot out the coordinates in 3D and then draw a straight line from the point where we are projecting the coordinates and onto the x = 0 plane. The point of projection can be thought of as the point at which our camera or eyes are located at, thus the resulting 2D projection maintains 3D perspective from that specific point. Drawing a second line passing through the z coordinates of the 3D line, we can see two similar triangles. Either of the similar triangles can be used to obtain identical formulas for the y and z projection coordinates. Note that in this projection derivation, I project the x coordinates to x = 0 and the camera or eyes location is at x = 1000. Epic stuff! The timestamps of key parts of the video are listed below: - Projecting 3D coordinates to 2D coordinates: 0:00 - Two Similar Triangles: 2:45 - Determining the formula for the y and z projection: 4:41 - 3D to 2D projection formula: 8:53 This video was taken from my earlier video listed below: - Laboratory Project: Putting 3D in Perspective: https://youtu.be/3txedAqdtkQ - HIVE video notes: https://peakd.com/hive-128780/@mes/laboratory-project-putting-3d-in-perspective - Video sections playlist: https://www.youtube.com/playlist?list=PLai3U8-WIK0ElsrMs_IBprUoHocIwyAmK Related Videos: Sequences and Series playlist: https://www.youtube.com/playlist?list=PLai3U8-WIK0FjJpwnxwdrOR7L8Ul8VZoZ . ------------------------------------------------------ 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 project a 3D rectangle onto a 2D screen, while also removing a segment of a line that is behind the rectangle so that it maintains its 3D perspective. I first project the given 3D rectangle coordinates using the 3D to 2D projection equation I derived earlier. Then, I determine the point of intersection of the 3D rectangle and 3D line. The next point that is needed is the point of intersection of the projected 2D line and the bottom portion of the projected 2D rectangle. Once I solve for these, I use GeoGebra to plot the projected line while removing the portion that is behind the rectangle. The resulting 3D chart is truly amazing! https://www.geogebra.org/calculator/twezjbu5 The timestamps of key parts of the video are listed below: - Question 4: Projecting a 3D rectangle to 2D: 0:00 - Solution: Graphing out the rectangle and its projection in GeoGebra: 1:04 - Remove the portion of the line behind the rectangle: 6:12 - Find point of intersection of the 3D line and rectangle: 7:15 - Determining the equation of the plane of the rectangle: 7:47 - Cross product to get the plane normal vector: 15:48 - Calculating the equation of the plane: 21:12 - Plugging in the line equation into the rectangle equation: 27:03 - Graphing the intersection point in GeoGebra: 32:18 - Finding the point of intersection of the projected line and rectangle: 35:03 - Vector equation of bottom right portion of projected rectangle: 38:08 - Equating the projected line equation with the bottom right rectangle line equation: 41:07 - Calculating the point of intersection of projected line and rectangle: 47:10 - Graphing it all out in GeoGebra: 50:30 This video was taken from my earlier video listed below: p - Laboratory Project: Putting 3D in Perspective: https://youtu.be/3txedAqdtkQ - HIVE video notes: https://peakd.com/hive-128780/@mes/laboratory-project-putting-3d-in-perspective - Video sections playlist: https://www.youtube.com/playlist?list=PLai3U8-WIK0ElsrMs_IBprUoHocIwyAmK Related Videos: Sequences and Series playlist: https://www.youtube.com/playlist?list=PLai3U8-WIK0FjJpwnxwdrOR7L8Ul8VZoZ . ------------------------------------------------------ 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 quickly go over how to verify if our 3D to 2D projection is correct, which I do so by adding sight lines in GeoGebra. These "sight lines" are just lines drawn connecting the coordinates of the camera and the projected points on the screen. If these sight lines intersect our clipped points along the 3D line, then it means our projection is indeed correct. This is because the clipped points are simply projected along a straight line from the camera to the screen. In the amazing GeoGebra 3D graphing calculator, we can quickly create sight lines by using the line segment function: https://www.geogebra.org/calculator/twezjbu5 The timestamps of key parts of the video are listed below: - Question 3: Verifying the Projection with Sight Lines: 0:00 - Solution: 0:18 - Adding sight lines with GeoGebra: 1:31 This video was taken from my earlier video listed below: - Laboratory Project: Putting 3D in Perspective: https://youtu.be/3txedAqdtkQ - HIVE video notes: https://peakd.com/hive-128780/@mes/laboratory-project-putting-3d-in-perspective - Video sections playlist: https://www.youtube.com/playlist?list=PLai3U8-WIK0ElsrMs_IBprUoHocIwyAmK Related Videos: Sequences and Series playlist: https://www.youtube.com/playlist?list=PLai3U8-WIK0FjJpwnxwdrOR7L8Ul8VZoZ . ------------------------------------------------------ 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 first derive the formula for projecting a 3D point to a 2D screen and then apply that formula to the 3D line from the previous question. To project a 3D image onto a 2D screen, we just have to draw a line extending from the camera or viewpoint that intersects our desired point and extends to the screen. Then applying similar triangles, we can derive the projected coordinates for the y and z coordinates; note that the x-coordinates in this case all get projected to x = 0. Next we apply the derived formula to the clipped 3D line points from Question 1 to obtain the projected coordinates on the 2D screen. And as always, I graph this all out using the amazing GeoGebra 3D graphing calculator: https://www.geogebra.org/calculator/twezjbu5 The timestamps of key parts of the video are listed below: - Question 1: Projecting Clipped 3D Line to 2D: 0:00 - Solution: Deriving 3D to 2D projection equation: 0:12 - Applying similar triangles: 7:18 - Projecting the clipped points of the 3D line: 10:51 - Graphing with GeoGebra: 14:31 This video was taken from my earlier video listed below: - Laboratory Project: Putting 3D in Perspective: https://youtu.be/3txedAqdtkQ - HIVE video notes: https://peakd.com/hive-128780/@mes/laboratory-project-putting-3d-in-perspective - Video sections playlist: https://www.youtube.com/playlist?list=PLai3U8-WIK0ElsrMs_IBprUoHocIwyAmK Related Videos: Sequences and Series playlist: https://www.youtube.com/playlist?list=PLai3U8-WIK0FjJpwnxwdrOR7L8Ul8VZoZ . ------------------------------------------------------ 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 Question 1 of the Laboratory Project: Putting 3D in Perspective and this time determine the points at which a 3D line needs to be clipped before it is projected onto a 2D screen. To do this, I first obtain the equations of the 4 clipping planes which project from the point of view of the camera or eyes and unto the vertices of the projected screen. Then we can plug in the equation of the 3D line into each linear equation of the planes and obtain points at which the line and planes intersect. Graphing this out in the amazing GeoGebra 3D graphing calculator, it becomes clear that the line only needs to be clipped at the left and top clipping planes. This is an important exercise in the mathematics of projecting 3D images onto 2D screens, which are used throughout computer graphics programming. Here is a link to GeoGebra if you want to play around with the projection: https://www.geogebra.org/calculator/twezjbu5 The timestamps of key parts of the video are listed below: - Laboratory Project: Putting 3D in Perspective: 0:00 - Question 1: Clipping a 3D Line: 2:08 - Solution: Graphing out the question in GeoGebra: 2:38 - We need 4 clipping planes: 4:35 - Recap on the equations of lines and planes: 6:58 - Vector and Parametric Equations of the 3D line: 8:17 - Equation of the right clipping plane: 12:21 - Note on simplified vectors: 21:49 - Linear equation of the right clipping plane: 22:49 - Plugging in equation of a line into the equation of a plane: 24:30 - Plotting our right clipping plane in GeoGebra: 30:49 - Equation of the top clipping plane: 32:47 - Plugging in equation of a line into the equation of a plane: 40:46 - Equation of the left clipping plane: 47:13 - Plugging in L into the plane equation: 50:42 - Equation of the bottom clipping plane: 58:52 - Plugging in L into the plane equation: 1:01:04 - Summary: Line is clipped at left and top clipping planes: 1:03:08 This video was taken from my earlier video listed below: - Laboratory Project: Putting 3D in Perspective: https://youtu.be/3txedAqdtkQ - HIVE video notes: https://peakd.com/hive-128780/@mes/laboratory-project-putting-3d-in-perspective - Video sections playlist: https://www.youtube.com/playlist?list=PLai3U8-WIK0ElsrMs_IBprUoHocIwyAmK Related Videos: Sequences and Series playlist: https://www.youtube.com/playlist?list=PLai3U8-WIK0FjJpwnxwdrOR7L8Ul8VZoZ . ------------------------------------------------------ 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 the mathematics involved in computer graphics programming in order to clip and project a 3D image onto a 2D screen while still maintaining 3D perspective. This was taken from my Stewart Calculus book in the Laboratory Project titled Putting 3D in Perspective. Computer graphics programmers have to somehow display 3D images stored in memory onto a 2D screen for users to observe. To do this, the 3D images need to be projected from a point at which the camera or eye is viewing and then onto a 2D screen behind the object, while maintaining perspective. To do this, the equations of 4 clipping planes are first needed to be obtained to clip off the portions of image that will be beyond the projection. In this Project, I derive the clipping plane equations, as well as equations of a 3D line and 3D rectangle, and their their projected formulas. I also go over the concept of "hidden line rendering" to remove objects that are behind other 3D objects, thus saving computing power by not needing to project them onto the screen. And as always, I use the amazing GeoGebra 3D graphing calculator to plot out all the projections, and which you can play around with in this link: https://www.geogebra.org/calculator/twezjbu5 The topics covered as well as their timestamps are listed below. - Introduction: 0:00 - Calculus Book Reference: 1:00 - Sections in Calculus Book Chapter: 1:14 - Topics to Cover: 2:02 - Laboratory Project: Putting 3D in Perspective: 2:54 - 4 Questions: 5:03 - Question 1: 8:10 - Equation of the 3D line: 13:00 - Equation of the Right Clipping Plane: 18:22 - Equation of the Top Clipping Plane: 38:49 - Equation of the Left Clipping Plane: 53:15 - Equation Bottom Clipping Plane: 1:04:55 - Summary: 1:09:10 - Question 2: 1:10:54 - Projecting 3D on to 2D: 1:12:13 - Question 3: 1:27:31 - Question 4: 1:29:39 - Point of Intersection of the Line and Rectangle: 1:36:56 - Point of Intersection of the Projected Line and Projected Rectangle: 2:04:43 - Graphing it All Out in GeoGebra 3D Graphing Calculator: 2:20:10 - Outro: 2:23:31 - PDF video notes: https://1drv.ms/b/s!As32ynv0LoaIirBjY_SU7z2CXBOG1A - HIVE video notes: https://peakd.com/hive-128780/@mes/laboratory-project-putting-3d-in-perspective - Video sections playlist: https://www.youtube.com/playlist?list=PLai3U8-WIK0ElsrMs_IBprUoHocIwyAmK - Full Vectors and the Geometry of Space series: https://www.youtube.com/playlist?list=PLai3U8-WIK0FjJpwnxwdrOR7L8Ul8VZoZ Related Videos: - Vectors and the Geometry of Space: Equations of Lines and Planes: https://peakd.com/hive-128780/@mes/equations-of-lines-and-planes . ------------------------------------------------------ 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 very unique problem on showing that the n-th partial sum of the harmonic series is NOT an integer. The harmonic series is the sum of the terms 1/i where i is a positive integer. To show that the partial sum is not an integer, we require some truly outside-of-the-box thinking, which fortunately for us is made easier with the given hint. To prove this, we first assume that the partial sum IS an integer and then observe what happens to 2 similar equations. The first is the multiplication of (the product of all odd integers less than or equal to n) * (the largest power of 2 that is less than or equal to n) * (partial sum). The second equation is the same as the first but with the assumption that the partial sum is an integer. After going over some very unique mathematical reasoning, I show that the left side of the equation is always odd while the right side is always even. This is a contradiction and thus the partial sum can not be an integer! Pretty epic brain twister! The timestamps of key parts of the video are listed below: - Problem 26: Partial Sum of Harmonic Series is NOT an Integer: 0:00 - Solution: 2:27 - Right side of hint equation is even: 4:16 - Left side of hint equation is odd: 7:23 - Checking when r is less than k: 13:41 - Checking when r = k: 16:56 - Summary: Partial Sum of Harmonic Series is NOT an integer: 20:45 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 have uploaded an epic edit made by 9/11 Revisionist, which I further corrected, showing a 3D model of the WTC overlaid on top the very tiny rubble pile after the Towers turned to dust in mid-air. The 3D models show the shear scale of the size of the Twin Towers and the rest of the World Trade Center (WTC) complex, which comprised of 7 buildings in total. The WTC was constructed in a shallow slurry wall known as the Bathtub which stretched 70 feet from ground level to the bedrock and was used to protect from the Hudson River. There were 6 floors below ground level. The 3D model shown of the underground floors is not to scale, so I have added a correction in the video. Each Twin Tower was 208 feet wide so the 70 feet to bedrock is relatively very shallow, and much more shallow than depicted in the 3D model. Besides this error, the 3D model does a good job illustrating the construction features and size of the WTC. Later in the video, parts of a presentation by Dr. Judy Wood are shown and voiced over another 3D model of the WTC. This time the 3D model is overlaid ontop of the tiny leftover rubble pile, and clearly demonstrate the astonishing lack of debris and overall disappearance of the Twin Towers. The Twin Towers were made with 110 floors of thick structural steel columns and Dr. Judy Wood correctly points out that their near instant disintegration could not have been done by any conventional gravity collapse or even explosive demolition. The question reiterates throughout the end of the video: Where did the towers go? Original video by 9/11 Revisionist: https://odysee.com/@911revisited:7/Where-did-the-towers-go:4 The timestamps of key parts of the video are listed below: - 3D model of the WTC: 0:00 - Underground Bathtub, NOT TO SCALE: 0:57 - Actual depth of Twin Towers foundation: 1:30 - Cross Section view of the Twin Towers: 1:54 - September 11, 2001: Presentation by Dr. Judy Wood: 2:21 - 3D model of the WTC overlaid above WTC 3: 3:21 - 3D model of the WTC overlaid above the tiny rubble pile: 4:30 - Top view of the WTC tiny rubble pile: 5:02 - Where did the towers go?: 5:43 Related Videos: Related videos: - 9/11 Observable Evidence video sections playlist: https://www.youtube.com/playlist?list=PLai3U8-WIK0G_HHWt33moIqEeUBP3cgCh - #911Truth video series: https://www.youtube.com/playlist?list=PLai3U8-WIK0EzqTamtIXtgX8QudQSxuxh and https://peakd.com/truth/@mes/911 . ------------------------------------------------------ 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 an interesting math problem which involves differentiating 3 power series to find relationships among them in order to show the given equation is true. In order to differentiate power series, we first need to determine if the series are convergent, which I do so using the Ratio Test. Next we can differentiate the given equation and using our obtained relationships, can show that the equation equals a constant. Plugging in x = 0 into the equation, we can thus simply solve for this constant, and thus prove our given equation. The timestamps of key parts of the video are listed below: - Problem 25: 0:00 - Solution: Using the Ratio Test to determine convergence: 1:33 - Term by Term Differentiation: 8:35 - Differentiating the desired equation: 14:42 - Determining the constant that equals to the desired equation: 20:09 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