Differential Equations: Population Growth: Logistic Equation: Example 2
In this video I prove that the growth rate in a population, assuming it uses the logistic model, is fastest at half the carrying capacity. I prove this using the first derivative test but apply it to the 1st and 2nd derivatives of the population function. In my last video I graphically showed that in fact the population appears to be growing fastest at about the half way mark, but in this video I prove it definitely. This is a very good video to understand how population sizes increase so make sure to watch this video!
In this video I go over Question 2 of the Discovery Project: Patterns in Integrals section of my calculus book. In this question I go over the family of integrals of the form with integrand sin(ax)cos(bx) and show how we can derive the general integral formula by finding the pattern that arises from using a computer to quickly solve different examples within that integral family.
Download the notes in my video: https://1drv.ms/b/s!As32ynv0LoaIheQwjuYUOlKh5_pw-g
View Video Notes on Steemit: https://steemit.com/mathematics/@mes/discovery-project-patterns-in-integrals-question-2-sin-ax-cos-bx
Related Videos:
Discovery Project: Patterns in Integrals: Question 1: http://youtu.be/uQUu0XFMw8I
Integration using Tables of Integrals: Example 1: http://youtu.be/AOToFP-4Mfg
Can We Integrate All Continuous Functions?: http://youtu.be/OFEDLJYqYps
Integration by Partial Fractions: http://youtu.be/r07NnKf76og
Trigonometry Identities: sin(-x) = - sin(x) and cos(-x) = cos(x): http://youtu.be/tD5EA2SXyFQ
Trigonometry: Product Identities: http://youtu.be/os8XKWAj9P8 .
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...
https://www.youtube.com/watch?v=O0p1DGMOcfM
In #MESExperiments 28 I demonstrate the amazing ability of the GyroTop magnetic spinning top to always balance opposite the magnetic attractive force even when turning the apparatus upside down. The GyroTop, also called Space Top, is the invention of an obscure person named Owen Liang, and whom created it to test his hypothesis that spinning tops want to align themselves opposite of gravity rather than simply "maintaining angular momentum".
The first part of the video involves just a regular spinning top with the ring magnet removed, and is used as a comparison. The spinning top keeps upright opposite the force of gravity even when the apparatus is tilted. This demonstration is often used to illustrate "angular momentum".
In the second part of the video, I attach the ring magnet and repeat the same procedure of spinning it up with a drill. Note the GyroTop has a tube to manually blow air into to spin it up, but I used my drill instead. As I rotate the apparatus the GyroTop always moves to align against the magnetic attraction. This is counterintuitive since the angular momentum should keep the top stationary until the magnetic attractive force is strong enough to grab hold of the top. This GyroTop opens up lots of possibilities!
Further information, screenshots, and relevant links are included on the Hive blockchain: https://peakd.com/hive-128780/@mes/mesexperiments-28-magnetic-spinning-top-aligns-opposite-of-magnetic-attraction
The timestamps of the regular vs magnetic spinning tops are shown below:
- Regular Spinning Top: 0:00
- Magnetic Spinning Top: 0:59
Stay tuned for #MESExperiments 29...
Related Videos:
#MESExperiments video series: https://peakd.com/experiments/@mes/list
DRAFT #MESExperiments video series: https://mes.fm/experiments-draft
#MESScience video series: https://mes.fm/science-playlist
#AntiGravity video series: https://peakd.com/antigravity/@mes/series
#FreeEnergy video series: https://mes.fm/freeenergy-playlist
?#MESExperiments 27: Spin Friction Can Make Gyroscope Rise from Steep 70° Angle (No Casing Rotation)
?#MESExperiments 25: Increasing Spin Friction Can Make Gyroscopes Rise Much Faster: https://peakd.com/hive-128780/@mes/mesexperiments-25-increasing-spin-friction-can-make-gyroscopes-rise-much-faster
?#MESExperiments 13: Steepest Gyroscope Rising Experiment Ever! 74 Degrees from the Vertical: https://peakd.com/mesexperiments/@mes/mesexperiments-13-steepest-gyroscope-rising-experiment-ever-74-degrees-from-the-vertical-magic
?#MESExperiments 3: Gyroscopes Exert 'Inertial Forces' With ZERO Loss of Spin Rate: https://peakd.com/mesexperiments/@mes/mesexperiments-3-gyroscopes-exert-inertial-forces-with-zero-loss-of-spin-rate-freeenergy
?#MESScience 1: How Does a Powerball Gyroscope Work? + Gyros Are Inverted Pendulums: https://peakd.com/messcience/@mes/messcience-1-how-does-a-powerball-gyroscope-work-gyros-are-inverted-pendulums
☝#AntiGravity Part 6: Objects in Rotation Defy ‘Mainstream’ Physics + MES Duality Concept: https://peakd.com/antigravity/@mes/antigravity-part-6-video-1-objects-in-rotation-defy-mainstream-physics-mes-duality-concept .
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In this video I go over a really fascinating curve, and that is the catenary which is the shape formed by handing a heavy cable across two heights of equal heights. For example telephone cables and power lines hang according to this catenary shape. The word catenary stems from the Latin “catena” which means “chain”. While the curve “appears” to look like a parabola it is in fact not a parabolic shape, but rather a hyperbolic function, hyperbolic cosine (cosh) to be exact. The catenary formula is usually written as y = a*cosh(x/a) + c or y = cosh(a*x)/a + c, whereas my calculus book uses the former.
The derivation involves some basic static equilibrium physics by first breaking up a segment of the chain and using a Free-Body Diagram to illustrate all of the forces at play. Summing up the horizontal and vertical components of the forces to ensure they sum up to zero (i.e. that the chain is static and not moving with an acceleration), we are able to obtain a differential equation for the curve. This equation includes the arc length of the chain segment, and thus we can apply the formula arc length, and a whole lot of tedious algebra later to finally obtain the hyperbolic catenary function!
I also go over a brief history lesson on the catenary and show how the likes of Galileo, Huygens, Leibniz, and Bernoulli brothers Jacob and Johann were involved in a rigorous mathematical battle to try to determine the formula of the catenary, which shocked Galileo’s assumption of it being a parabola. The catenary has some very interesting properties, and the fact that it is formed by simply holding a heavy cable up by two heights helps explain how it has a very low center of gravity and small potential energy. I may investigate more into the catenary so stay tuned!
Download the notes in my video: https://1drv.ms/b/s!As32ynv0LoaIhvofvd_G4uxmjT0IjQ
View Video Notes on Steemit: https://steemit.com/mathematics/@mes/vcsmh-video-notes-hyperbolic-functions-catenary-formula-and-proof
Related Videos:
Hyperbolic Functions - tanh(x), sinh(x), cosh(x) - Introduction: http://youtu.be/EmJKuQBEdlc
Applications of Integrals: Arc Length Proof: https://youtu.be/2rb4H_rmgxg .
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...
https://www.youtube.com/watch?v=EYb1p9r1fnM
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/ .
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https://www.youtube.com/watch?v=tazw6CAcrKo
In this video I go over another example on improper integrals and this time evaluate the integral of the function 1/(x - 1) from x = 0 to x = 3. This is a very important example because it illustrates the need to always check the function being integrated to see if it is continuous or discontinuous in the interval being integrated. In this particular example there is a vertical asymptote at x = 1 so we must treat this as an improper integral of type 2 and evaluate it accordingly. In this example, I show how this integral diverges to negative infinite so the integral does not exist as a definite number. If I had not noticed that vertical asymptote and instead solved it like a regular integral the answer would have yielded the number ln(2) which is wrong! So make sure to always check if you are dealing with a proper or improper integral before you start evaluating the integral.
Download the notes in my video: http://1drv.ms/1U2vvqF
View Video Notes on Steemit: https://steemit.com/mathematics/@mes/improper-integrals-example-7-important-example
Related Videos:
Improper Integrals: Example 6: sec(x): https://youtu.be/yHnv2w9OVWk
Improper Integrals Example 5: 1/sqrt(x-2): https://youtu.be/C8X_TUdTo-A
Improper Integrals: Type 2: Discontinuous Integrands: https://youtu.be/livlGUBuM54
Improper Integrals: Introduction: https://youtu.be/ud44W-0jC-c
The Definite Integral - Brief Introduction: http://youtu.be/vhMP5SKbQjU
Integrals and Areas Between Curves: http://youtu.be/2F03KMLIzbk
Infinite Limits - Precise Definition: http://youtu.be/QyVTgSzn7dE
Infinite Limits - Vertical Asymptotes: http://youtu.be/H-5Txa8SOwQ
Integration by Parts: Example 2: ln(x): http://youtu.be/wVjra6DCZzE .
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https://www.youtube.com/watch?v=MbjghfT0IoM
In this video I go over an example on Euler's Method for numerically approximating solutions to a differential equation and this time look at the initial-value problem: y' = x + y with y(0) = 1. Initially I use the step size of 0.1 to roughly approximate the solution at x = 0.1, 0.2, ..., 1.0. But to get more accurate approximations we can decrease the step size even further. As shown in this video, when we decrease the step size further and further the solution approaches closer and closer to the exact value. This is a very useful video to understand Euler's method in detail, as well as how to develop spreadsheets to calculate it, so make sure to watch this video!
Download the notes in my video:
Notes: https://1drv.ms/b/s!As32ynv0LoaIhsQlsgrDZEb68Fl1LQ
Excel Notes: https://1drv.ms/x/s!As32ynv0LoaIhsQnSuqAQ7rhC3PsFg
View Video Notes on Steemit: https://steemit.com/mathematics/@mes/differential-equations-euler-s-method-example-1
Related Videos:
Differential Equations: Euler's Method: https://youtu.be/VlwVl-3oPDM
Differential Equations: Electric Circuit: Introduction: https://youtu.be/E6vij-RzQ-o
Differential Equations: Direction Fields: Example 1: https://youtu.be/mtbMQQZeMoQ
Differential Equations: Direction Fields: https://youtu.be/zWv1y8Xp1ac
Differential Equations: General Overview: https://youtu.be/jit59tIY4UI
Differential Equations: Spring Motion: Example 1: https://youtu.be/Twu30EJ93Wg
Differential Equations: Motion of a Spring: https://youtu.be/mk2TiR5dwVs
Differential Equations: Population Growth: https://youtu.be/Td8C_cTEGkA .
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https://www.youtube.com/watch?v=L_l5DLZsZLQ
In this video I go over more examples on infinite limits at infinity to better illustrate the concept. In one example I show how you can graph a function by knowing the x and y intercepts as well as the limits at infinity.
Download the notes in my video: https://www.dropbox.com/s/qsk3x1ll24epwkx/334%20-%20Infinite%20Limits%20at%20Infinity%20Examples%20Part%202.pdf
Related Videos:
Infinite Limits at Infinity + Examples Part 1: http://youtu.be/qXLYInFkW7Y
Limits at Infinity: Horizontal Asymptotes: http://youtu.be/6pdgb09wRvI
Infinite Limits: Horizontal and Vertical Asymptote Lines and VERY Useful Examples: http://youtu.be/__Y6IRcwdP8
Guidelines to Curve Sketching: http://youtu.be/tOn7ZSAntKs .
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https://www.youtube.com/watch?v=TSyFU3PlNTY
On Sunday, my soccer team, Club Ireland Celtic, was in the championship game of the President's Cup against the team Elegant Flooring. We were down 1-0 at the 70th minute but we came back and ending up winning 3-1! I scored the game-winning goal and it was a beauty so I just had to upload it here!
Watch the Full Game here: https://youtu.be/oQZeN6tas9I
Related Videos:
Probability that my friend Dmitry Scores Goals: http://youtu.be/zbAN04Kj3D8
Lebron James In depth Stats in the 2012 NBA Finals: http://youtu.be/75dd0Tii3WY
Analysis of Jack Taylor's 138 Points - Grinnell College vs Faith Baptist Bible: http://youtu.be/x5x0VF0xaSo
Stephen Curry Not Making the 2013 All Star Game was a Good Thing: http://youtu.be/WEVK2-9gd3I
Amazing NBA Streetball Crossovers and Blocks: http://youtu.be/90J7oE22I50 .
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https://www.youtube.com/watch?v=JsZN2uPuh-Q
In this video I go over the gyrocompass and show how it is used to find True North by aligning itself with the Earth's rotation, thus debunking Flat Earth. A gyrocompass is just a gyroscope that is forced to remain horizontal on the Earth's surface, such as by adding compensator weights to the sides. Since a gyroscope wants to maintain its orientation when spun up, then as the Earth rotates, the gyrocompass will also want to maintain orientation but to do so it will have to lift the compensator weights. Gravity acts upon the compensator weights thus inducing precession in the gyrocompass such that it will move until it aligns with Earth's rotation. Once it aligns with Earth's rotation, there will be no more induced torque, thus no induced precession. This also means that the gyrocompass will be pointing to the True North Pole. Gyrocompasses have applications in ships and planes, especially where there are areas of magnetic compass interference, such as nearby metal or magnetic mines as in during World War 2.
I demonstrate the principle of the gyrocompass with a handheld gimbled gyroscope as well as showing a WW2 gyrocompass tutorial as well as showing factory and testing footage. The gyrocompass is an incredible engineering feet with very delicate construction, design, and rigorous testing. It also has a "phantom ring" that electronically detects the gyrocompass rotation and gradually rotates with the string that is holding up the gyrocompass. This is to prevent the string from getting tangled up and thus inducing a reverse torque to the gyrocompass. In my opinion, the gyrocompass is irrefutable evidence for Earth's rotation and is one of the best pieces of evidence against a Flat Earth. I will be going over more debunking Flat Earth videos so stay tuned!
The links shown in the video are listed below:
- MES Links: https://mes.fm/links
- Earth's Curvature Calculator: https://mes.fm/earth-curvature-calculator/
- Debunking Flat Earth playlist: https://www.youtube.com/playlist?list=PLai3U8-WIK0HUyOvHJiyX74GmJ79exCkh
- #AntiGravity Part 6
- Video 2: https://youtu.be/dKwwJe0znpA?t=6874
- HIVE notes: https://peakd.com/antigravity/@mes/antigravity-part-6-video-1-objects-in-rotation-defy-mainstream-physics-mes-duality-concept
- Gimbaled gyroscope: https://youtu.be/1kOL6qQf6I0
- WW2 Gyrocompass: https://youtu.be/EM051IXJD9Q
- WW2 Gyrocompass factory and testing: https://youtu.be/CNcDz1MDPE0
- Gyrocompass Wiki: https://en.wikipedia.org/wiki/Gyrocompass
MES Experiments: https://www.youtube.com/playlist?list=PLai3U8-WIK0HUHoank-Lj9q6RGAS51QRh
Timestamps:
- Introduction and Links: 0:00
- MES Links: 1:11
- MES Earth Curvature Calculator: 1:36
- #AntiGravity Part 6: 2:07
- Gyroscope spin aligns with precession direction: 3:07
- Gyrocompass aligns with external torque: 4:14
- Gyrocompass wants to align with Earth's rotation: 8:55
- WW2 Gyrocompass: 9:35
- Gyrocompass vs Gyroscope comparison: 10:36
- WW2 Gyrocompass tutorial: 12:33
- WW2 Gyrocompass factory and testing: 19:52
- Gyrocompass Wikipedia: 23:17
- #MESExperiments playlist: 23:53
Related Videos:
- Anti-Gravity: https://www.youtube.com/playlist?list=PLai3U8-WIK0HkT4DmvQSg0XZfY5Bxkxq1
- Free Energy: https://www.youtube.com/playlist?list=PLai3U8-WIK0FKVpHL_onhaqBeVP-8qJXV
- MES Experiments: https://www.youtube.com/playlist?list=PLai3U8-WIK0HUHoank-Lj9q6RGAS51QRh .
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