Differential Equations: Population Growth: Logistic Equation: Example 5
In this video I go over another example on the logistic differential equation for modeling population growth and this time compare the results with that of the natural growth exponential model. The example involves modeling the growth of the bacteria protozoan Paramecium, which were part of a series of experiments by the famous Russian biologist Georgy Gause in the 1930s. In modeling the growth of bacteria, it is clear that the logistic equation fits the observed data far better than that of the simple exponential model.
Also in this video I go over some brief definitions on biology and ecology, as well as a brief history on Georgy Gause, who is best known for his paper on the competitive exclusion principle. This a great video to learn more about biology and its history so make sure to watch it!
In #MESExperiments Part 8, I have uploaded the late Francis J. McCabe’s amazing Gyro Wheel Torque “Over-Unity” demonstration, which he had up on his YouTube Channel in June 27, 2012. Francis showed that a large 50 pound, 32 inch diameter gyro, precessing from about 25 inches from the pivot point, can still be “torqued-up” by a very small and visibly weak 6 volt battery with a “Torque Rating” of 2.5 inch-pounds at 12 volts.
Crunching in the numbers, we get roughly:
- Output Torque = 50 pounds * 25 inches = 50 * 25 = 1,250 inch-pounds
- Input Torque = 2.5 inch-pounds * (6 volts input) / (12 volts rating) = 1.25 inch-pounds.
- Output Torque / Input Torque = 1250/1.25 = 1000 TIMES OVER-UNITY!
Recall that the mainstream definition of “Torque” is essentially force multiplied by the distance perpendicular from the fulcrum or point of rotation. For a fixed horizontal wheel, the torque would be the “Moment of Inertia” multiplied by the Acceleration. But while precessing, this torque is also equal to the weight multiplied by the distance to the pivot. More on this *mainstream* torque and precession can be found here: http://hyperphysics.phy-astr.gsu.edu/hbase/rotv2.html
Essentially what this means is that although a very weak motor won’t be able to spin a fixed horizontal heavy wheel, but if spun initially through the vertical near zero torque position and then allowed to precess, the motor magically is able to increase spin speed and maintain precession!
This is literally as if the very magical act of precession is causing the gyroscope to be “lifted” in a horizontal spiral plane. Later Francis turns off the power and the gyro-wheel begins precessing in a downwards spiral.
This is an absolutely mind-boggling demonstration by Francis J. McCabe, and just adds to amazing properties of gyroscopes shown in my earlier experiments.
To see the original video on Francis’ YouTube Channel, you can watch here:
https://youtu.be/QmqIg00xSFo
To learn more about Francis J. McCabe, such as his previous work for both NASA and Boeing as a Rocket Scientist before realizing that Gyro Science is where the real science is at, then make sure to read up his life on the following links:
- https://web.archive.org/web/20190405163327/https://en.wikipedia.org/wiki/Francis_J._McCabe
- https://www.chestnuthilllocal.com/2013/01/08/mt-airy-inventor-finds-new-uses-for-the-gyroscope/
- https://www.chestnuthilllocal.com/2014/03/26/francis-j-mccabe-inventor-business-owner/
- http://www.franmccabe.com/
- https://www.youtube.com/user/Mccabefj/videos
Of note is that Francis viewed the Earth and in fact everything as a gyroscope; which I tend to agree with him on this! And it is no surprise that in his lecture, he mentioned he worked with THE Eric Laithwaite:
https://youtu.be/TVoXdgkSOfY?t=54
The world is definitely a small place, and nothing smaller than the number of people awakened to the magical and game-changing secrets (in plain sight) of gyroscopes….
Stay Tuned for #MESExperiments Number 9!
View full experiment results on the Hive blockchain: https://peakd.com/mesexperiments/@mes/mesexperiments-8-large-gyro-wheel-precesses-at-1000x-torque-over-unity-freeenergy
Related Videos:
#MESExperiments video series: https://peakd.com/mesexperiments/@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 .
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In this video I go over a very useful example on logarithmic differentiation and find the the derivative of y = x^sqrt(x). I solve the derivative using two ways, the first using logarithmic differentiation and the second a very useful trick to avoid using logarithmic differentiation by utilizing the property that e^lnx = x. This is a very useful example so make sure to watch it!
Download the notes in my video: https://www.dropbox.com/s/qw3qrmtz8lh0mup/431%20-%20Logarithmic%20Differentiation%20Example.pdf
Related Videos:
Logarithmic Differentiation: Easy way to simplify derivatives: http://youtu.be/_hfZdRztUjU
Derivative of y = Log(x) and y = Ln(x): http://youtu.be/5e6MisvvMPE
Derivatives of Logarithmic Functions: Examples: http://youtu.be/bAU_BhSEpnE
Derivatives of Logarithmic Functions: Example on using Log Rules: http://youtu.be/X2mv1_xHOIk
Logarithmic Differentiation: Useful Example: http://youtu.be/kTfGBvgxOes
The Number e - A Brief Introduction and it's Derivative: http://youtu.be/o_s_YYD6v3g
A Simple Proof that the number e = 2.71828183...: http://youtu.be/NDb80lP3Tr4
Logarithms and their Properties - An Introduction: http://youtu.be/AZ6KKym19gI
Natural Logarithms, Log base 10, and Some Examples Using Logs: http://youtu.be/XRSkMk5L3pk .
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https://www.youtube.com/watch?v=kTfGBvgxOes
In this video I go over the same “cow grazing area” problem in which a cow (or goat or other farmyard animal) is tied to a circular silo and we are asked to determine the total area in which the cow can graze around. But this time I go over another method in solving this problem, Method 2, and show break the area under the involute curve of the circle into infinite sectors of a circle, which we can sum up as an integral. This method is actually much easier than in the parametric equations used in Method 1, and saves plenty of time too. This integration method is based on the area formula that I derived in my earlier video for the area of a polar curve. The only difference is that instead of having a direct formula for the involute curve, we can simply project the angle of a sector of a circle perpendicularly towards the involute because we know the length of the rope. This is actually an ingenious method in solving the grazing area problem and shows that complex problems can be solved easily with a thorough understanding of mathematics. The difference in the amount of work required in both Method 1 and Method 2 also show how certain techniques are better suited than others in certain situations, which illustrates the remarkable flexibility of calculus, so make sure to watch both methods!
Download the notes in my video: https://1drv.ms/b/s!As32ynv0LoaIhvV-EUB2O3mB0v8-SA
View Video Notes on Steemit: https://steemit.com/mathematics/@mes/video-notes-polar-coordinates-area-example-3-cow-grazing-area-method-2
Related Videos:
Parametric Calculus: Areas: Example 2: Cow Grazing Area: Method 1: https://youtu.be/pYf_c_CK5kE
Polar Coordinates: Area Formula: https://youtu.be/wd-EwiKzqa0
Polar Coordinates: Area: Example 1: r = cos 2ϴ: https://youtu.be/6kavQyGcODo
Polar Coordinates: Area: Example 2: Cardioid and Circle: https://youtu.be/vSzxzNn3EiA
Parametric Equations and Polar Coordinates: https://youtu.be/usSors49Gdw
Area of a Sector of a Circle: Proof: https://youtu.be/lglnySF8DyY
Angles - Degrees vs Radians: What are Radians??: http://youtu.be/bOdrVvQWRdE
Parametric Calculus: Example 1: Involute of a Circle: https://youtu.be/NIGw_dlEzQ4 .
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https://www.youtube.com/watch?v=oyoLlcVZWVE
In #PizzaGate Part 25 I revisit the widely publicized Madeleine McCann “missing” child case after learning of the peculiar case of Brenda Leyland and her alleged suicide. I was shocked at first to realize that I never heard about Brenda but then realized that the information that the ruling elites and fakestream mainstream media don’t want you to here usually aren’t covered much if at all. I started researching into Brenda Leyland by reading the mainstream version of events, i.e. Wikipedia, and the only mention of Brenda was in a disgusting paragraph demonizing people on social media as committing “trial by media”, when as I later uncovered it was actually the evil mainstream media whom is the judge and likely executioner… Brenda was mentioned in only half a sentence, and without even mentioning her name!! This is more than just under-reporting, this is demonizing a person so that people think less about their strange death…
Brenda Leyland is alleged to have tweeted critically about the McCann case, and in October 1, 2014, the pathetic reporter Martin Brunt at Sky News ‘doorstepped’ to demand questions to very vague questions about “attacking” the McCanns. He also warned her that she maybe part of a police investigation on “hateful” trolls. Brenda was clearly distraught and drove away. Initially only her Twitter username was known, “sweepyface”, but soon after her name, video, and image was plastered all over the media as an evil “troll” attacking the parents of Madeleine, Gerry and Kate McCann. This led to Gerry calling for an “example” to be made of “trolls”, much like John ‘Skippy’ Podesta had said in his email regarding whistle-blowers. Days later Brenda is found dead in a hotel with the cause of death initially unknown, ruled “unsuspicious”, and an inquest months later “revealed” she died by suicide through “helium overdose”…. #YouCantMakeThisStuff
What prompted this doorstepping was a mysterious “dossier” obtained by Martin and the UK Police from an anonymous source that allegedly compiled 80 or so pages “evil” tweets. But what the media doesn’t want you to know is that there is no evidence that Brenda was named in the dossier, in fact witnesses to the inquest trial state she was not, and Buzzfeed reported that Brenda had tweeted a year earlier that if she were die of “odd circumstances” for people to not believe the initial report… #WOW
Also the calls by the McCanns for online censorship, combined with the horrible reporting by Sky News, and extremely bizarre circumstances, raises concerns that this is in fact a Psychological Operation (PsyOp). And to top this off, I show a bizarre twist at the end of the video and reveal that the anonymous source is ALMOST CERTAI
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https://www.youtube.com/watch?v=b2mKpQ3hoDI
In this video I derive an expression for the derivative of any power function f(x) = x^n where n is a positive integer. In later videos I show that this is applicable to all values of n.
Download the notes in my video: https://1drv.ms/b/s!As32ynv0LoaIichipz1n0PXTRvn0sw?e=iTIKlp
View video notes on the Hive blockchain: https://peakd.com/hive-128780/@mes/derivative-power-rule-part-1-n-positive-integer
Related Videos:
Definition of Derivative Simple Explanation: http://youtu.be/0rjGMpM06Eg
Derivative of y = x^n - Part 2: General Power Rule: http://youtu.be/Ibvu33oh49o
Derivative of y = x^n - Part 3: General Power Rule when x = 0: http://youtu.be/c2Vfuk4Xcv4 .
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In this video I quickly go over a list of common Maclaurin series and their radius of convergence. The first series is for the function 1/(1-x), then e^x, sin x, cos x, arctan x, and finally ln(1+x). Note that in my earlier video I had used an earlier version of my Stewart calculus textbook, which went over the Maclaurin series for ln(1-x) instead of ln(1+x). I quickly recap on that derivation and show that we can simply input -x into ln(1-x) to obtain the Maclaurin series for ln(1+x).
The timestamps of key parts of the video are listed below:
- Question 11: 0:00
- Solution: Recap on Taylor and Maclaurin series: 0:53
- (a) to (f) Table of Maclaurin series: 1:33
- Note on the Maclaurin series for ln(1+x): 5:00
- Maclaurin series for ln(1-x): 5:18
- Replacing x with -x to get ln(1+x): 6:00
- Maclaurin series for ln(1+x): 8:08
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 .
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https://www.youtube.com/watch?v=Ze2-PIvip0E
In this video I go over the linear equation of a plane, which involves rewriting and simplifying the scalar equation of a plane. I do this by starting with the scalar equation of a plane and then collecting all like terms and replacing the constants addition with a single letter "d", thus obtaining the linear equation of a plane in the form: ax + by + cz + d = 0.
I also do 2 examples on planes to illustrate this concept. The first example involves finding the equation of a plane when given 3 points on the plane. The second example involves finding the point of intersection of a line and plane, whose equations are both given. And as always, I graph out the results using the amazing GeoGebra 3D graphing calculator as a double check of my work. You can play around with the calculator via these links: https://www.geogebra.org/calculator/mm6t7n4q and https://www.geogebra.org/calculator/raqsvnad.
The timestamps of key parts of the video are listed below:
- Linear Equation of a Plane: 0:00
- Example 5: Determine Equation of a Plane 3:18
- Graphing Plane with GeoGebra 3D Graphing Calculator: 11:28
- Example 6: Find Point At Intersection of Line and Plane 12:46
- Graphing with GeoGebra: 17:20
This video was taken from my earlier video listed below:
- Equations of Lines and Planes: https://youtu.be/qWQz6qPhXR8
- Video notes: https://peakd.com/hive-128780/@mes/equations-of-lines-and-planes
- Playlist: https://www.youtube.com/playlist?list=PLai3U8-WIK0FO3u0IupqllNNkZTsffpIV
Related videos:
Vectors and the Geometry of Space video series: https://www.youtube.com/playlist?list=PLai3U8-WIK0FjJpwnxwdrOR7L8Ul8VZoZ .
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https://www.youtube.com/watch?v=dhhlgzGIfp0
In this video I go over Isaac Newton's Law of Cooling (and Warming) and formulate it as a differential equation. The law states that the rate of cooling of an object is proportional to the temperature difference between the object and its surroundings, provided that this difference is not too large; this also applies to warming as well. The resulting differential equation is similar to the separable equations that I have covered in my earlier videos and thus the solution is an exponential function, again similar to the other applications that I covered in my earlier videos. I will go over an example on cooling in my next video so stay tuned for that!
Download the notes in my video: https://1drv.ms/b/s!As32ynv0LoaIhtFXWUtbs-c0_XACog
View Video Notes on Steemit: https://steemit.com/mathematics/@mes/differential-equations-newton-s-law-of-cooling
Related Videos:
Differential Equations: Radioactive Decay: Example 1: https://youtu.be/O7XHjBYuscA
Differential Equations: Radioactive Decay: https://youtu.be/JWVC6cIyasw
Differential Equations: Population Growth: Proportionality Constant: https://youtu.be/y4cJX0rcXqw
Differential Equations: Exponential Growth and Decay: https://youtu.be/DZtDUIZuxcg
Power Functions and their Properties Part 1 - A Simple Explanation: http://youtu.be/2MKko4ZkSf0
Logarithms and their Properties - An Introduction: http://youtu.be/AZ6KKym19gI .
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https://www.youtube.com/watch?v=9jw9xvM3krg
In this video I go over another example on trig substitution for integrals and this time show that sometimes we need to further simplify the integrand to enable trig substitution to be used. In this example I solve the integral of the function x^3/(4*x^2+9)^(3/2) and first complete the square inside the denominator before using trig substitution to solve the integral.
Download the notes in my video: http://1drv.ms/1JI96Ha
View Video Notes on Steemit: https://steemit.com/mathematics/@mes/trigonometric-substitution-for-integrals-example-6
Related Videos:
Trigonometric Substitution for Integrals: Example 6: http://youtu.be/wh2dlrFbxrQ
Trigonometric Substitution for Integrals: Example 5: http://youtu.be/X-35GgXPEHQ
Trigonometric Substitution for Integrals: One to One Function: http://youtu.be/QO26MkCC3eU
Trigonometric Substitution for Integrals: Example 4: http://youtu.be/v3tHdU63Vio
Trigonometric Substitution for Integrals: Example 3: http://youtu.be/Tfl90TqW3jA
The Area of an Ellipse (and Circle): http://youtu.be/7hSkKPQA71s
Trigonometric Substitution for Integrals: Example 1: http://youtu.be/lpp6YLWB2GM
Trigonometric Substitution for Integrals: http://youtu.be/2pWvGXwtVJo
Completing the Square: http://youtu.be/5uxV6-f_qc0
Exact Trigonometry Ratios Part 1: 0, 30, 45, 60, and 90 Degrees: http://youtu.be/ln03a5KvQAY .
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https://www.youtube.com/watch?v=wh2dlrFbxrQ