In this video I go over another example on improper integrals and this time solve the integral of the function 1/sqrt(x-2) from x =2 to x = 5. This integral is a Type 2 improper integral because the integrand has an infinite discontinuity or vertical asymptote at x = 2. But when we write this integral as a limit of a definite integral, the limit ends up converging to the value 2*sqrt(3) and thus represents the area under the curve.
In this video I derive a similar formula as in Problem 7 but this time use arccot or inverse cotangent. I then use that formula, as well as the useful method of the Telescoping Sum, to determine the sum of a series involving arccot and show that it equals to π/2.
The timestamps of key parts of the video are listed below:
- Problem 8: 0:00
- Solution to Part (a): 0:30
- Solution to Part (b): 18:11
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 .
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https://www.youtube.com/watch?v=rTmVAlyc8BA
In this video I review the expression for the derivative of exponential functions, i.e. f(x) = a^x where a is a constant, which I derived in my earlier video using the definition of the derivative. That expression was complicated and hard to remember. But with the chain rule and making use of the number e and natural logs, I derive a much simpler expression for the derivative of exponential functions.
Download the notes in my video: https://1drv.ms/b/s!As32ynv0LoaIidhbHivOlej6v69NWA?e=SBKPEZ
View video notes on the Hive blockchain: https://peakd.com/hive-128780/@mes/derivative-of-y-ax-and-ex-using-the-chain-rule
Related Videos:
Derivative Rules: Proof of Chain Rule: http://youtu.be/tYDDpKzP-VU
Derivative Examples using the Chain Rule: http://youtu.be/KB9_lBxTrwI
Definition of Derivative Simple Explanation: http://youtu.be/0rjGMpM06Eg
Logarithms and their Properties - An Introduction: http://youtu.be/AZ6KKym19gI
Natural Logarithms, Log base 10, and Some Examples Using Logs: http://youtu.be/XRSkMk5L3pk
Power Functions and their Properties Part 1 - A Simple Explanation: http://youtu.be/2MKko4ZkSf0
What are Limits? A Simple Explanation: http://youtu.be/FbV7TzlkTZk .
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If you have athlete's foot or want to know how to avoid getting it, make sure you watch this video! In this video I discuss how I got athlete's foot and show you the very effective treatment that I am currently on and also how to avoid getting athlete's foot!
Download the notes in my video: https://www.dropbox.com/s/h1e5st3tyq4yfuo/173%20-%20Athlete%27s%20Foot.pdf
Related Videos:
Athlete's Foot Part 2 - New Simple and Effective Treatment: http://youtu.be/piw3zUQnaaY
Why you shouldn't remove ear wax: http://youtu.be/V-_2LLYC68U
Complete Proteins - Vegetarian Diet and Myths about Plant Proteins: http://youtu.be/LHzzutoupY0 .
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https://www.youtube.com/watch?v=4yMD0_SSw1c
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 this video I go over Question 5 of the Discovery Project: Rotating on a Slant video series. In this question, I derive the surface area formula of the shape generated by rotating the curve about a slanted line. The derivation is fairly easy because the majority of the work was in Question 1 when the variables on the slanted line were rotated to write in terms of x.
Download the notes in my video: https://onedrive.live.com/redir?resid=88862EF47BCAF6CD!104392&authkey=!AM7ZI568EbxWse0&ithint=file%2cpdf
View Video Notes on Steemit: https://steemit.com/mathematics/@mes/discovery-project-rotating-on-a-slant-question-5
Related Videos:
Discovery Project: Rotating on a Slant: Question 4: https://youtu.be/w3eR3d-dJRI
Discovery Project: Rotating on a Slant: Question 3: https://youtu.be/zwodt1OsFjE
Discovery Project: Rotating on a Slant: Question 3: https://youtu.be/zwodt1OsFjE
Discovery Project: Rotating on a Slant: Question 1: https://youtu.be/dXlhpDOBDe0
Discovery Project: Patterns in Integrals: Question 1: 1/((x+a)(x+b)): http://youtu.be/uQUu0XFMw8I
Discovery Project: Patterns in Integrals: Question 2: sin(ax)cos(bx): http://youtu.be/O0p1DGMOcfM
Discovery Project: Patterns in Integrals: Question 3: x^n * ln(x): http://youtu.be/PA9-Me1ko10
Discovery Project: Patterns in Integrals: Question 4: x^n * e^6: http://youtu.be/XgE75Pa0n-c
Applications of Integrals: Surface Area: https://youtu.be/JkDPmAD37qk
Applications of Integrals: Arc Length Proof: https://youtu.be/2rb4H_rmgxg
Applications of Integrals: Arc Length Function: https://youtu.be/MWKK3qLvSwU .
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https://www.youtube.com/watch?v=6c_4v9JeFsI
In this video I go over a quick introduction on a brand new video series on presenting some of the amazing MES Science Experiments that I have been performing in my quest for Suppressed #FreeEnergy and #AntiGravity Science. After spending months of countless hours researching for my #AntiGravity Part 6 video, it appears that I may have a few more countless of research to adequately present my bombshell findings! That being said, I have gone on too long of a hiatus of not uploading videos so this definitely has to change! Thus uploading some of the experiments that I am doing behind the scenes is a great way to show you some of the research I am working on and also to get your feedback and thoughts of your own as to exactly what is happening. Here at MES, simple regurgitation of mainstream “science” talking points is not good enough, but feel free to provide your very own explanation… regardless of how far-out it may be. I don’t pretend like I have a clue what reality is so I am open to all suggestions.
My #AntiGravity Part 6 research is on all things spinning, and this has led me to explore ALL avenues of life: from physics, electromagnetism, gravity, to religion, philosophy, and even real time sports. Simply trying to discover the true mechanism and causation of how a simple spinning top stays upright (and no, simply saying “angular momentum” is not an answer) involves diving into the world of duality: objects at rest & objects in motion, light & darkness, life & death, good & evil, conscious & subconscious, Humanity & God, etc.; and with Rotation representing the merger of seemingly opposing forces. This interconnectedness of all things has led me to believe that simply understanding the true fundamental behavior of anything, regardless of how specific, involves understanding everything: If you know anything, you know everything.
With those MES words of wisdom being said, stay tuned as I will be uploading some truly amazing science experiments! Also, while I continue to research for my epic #AntiGravity Part 6 video, I will also be throwing in some mathematical challenging videos throughout, since Mathematics is the name of the game here at MES.
And instead of just waiting to see what experiments I perform, get started by buying a gyroscope and see what magical experiments you can discover!
Download the notes in my video: https://1drv.ms/b/s!As32ynv0LoaIh6Rqg2ythWPKFpBaLg
View video notes on the Hive blockchain: https://peakd.com/mesexperiments/@mes/mesexperiments-introduction-to-mes-science-experiments
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 the equation of a sphere by utilizing the distance formula in 3D I solved earlier. A sphere is by definition just the set of all points whose distance from a particular point (the center) is equal to a constant r or radius. I go over the equation of the sphere for a center at an arbitrary point in space and also when the center is at the origin.
The timestamps of key parts of the video are listed below:
- Example 5: 0:00
- Equation of a Sphere 3:07
This video was taken from my earlier video listed below:
- Vectors and the Geometry of Space: 3D Coordinate Systems: https://youtu.be/xJ-qPJFf2gE
- Video notes: https://peakd.com/hive-128780/@mes/vector-space-and-geometry-3d-coordinate-systems
- Playlist: https://www.youtube.com/playlist?list=PLai3U8-WIK0GiAJeLMzhOsrMliibfce5m
Related videos:
Vectors and Geometry of Space video series: https://www.youtube.com/playlist?list=PLai3U8-WIK0FjJpwnxwdrOR7L8Ul8VZoZ .
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https://www.youtube.com/watch?v=49qZEuhNhbI
In this video I go further into the properties of the cross product and this time derive the vector triple product. Note that this is Property 6 of Theorem 3 from my earlier video, and I solve it as part of Exercise 5. The vector triple product involves the cross product of a cross product and it produces a vector with a pair of dot products. This has applications in astronomy such as in Kepler's First Law of planetary motion, so it is important to learn how the Property is derived.
This video was taken from my earlier video listed below:
- Vectors and the Geometry of Space: The Cross Product: https://youtu.be/k8GRt95i-Gc
- Video notes: https://peakd.com/hive-128780/@mes/vectors-and-the-geometry-of-space-the-cross-product
- Playlist: https://www.youtube.com/playlist?list=PLai3U8-WIK0FuRJ8rg-YVQvfPoPOwhuRW .
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https://www.youtube.com/watch?v=KKRMlFaNtes
In this video I go through some important examples in understanding how to sketch and obtain the equation of a hyperbola.
Download the notes in my video: https://1drv.ms/u/s!As32ynv0LoaIicg9gFI6sv6jFFHZBw?e=Geqcdf
View video notes on the Hive blockchain: https://peakd.com/hive-128780/@mes/hyperbola-examples
Related Videos:
Hyperbola - Definition and derivation of the equation: x^2/a^2 - y^2/b^2 = 1: http://youtu.be/Y6iYC4VEAi0
Completing the Square: http://youtu.be/5uxV6-f_qc0
Transformation of Functions Part 1: Vertical and Horizontal Shifts: http://youtu.be/CBZd19GFOTE
Transformations of Functions Part 2: Horizontal Stretching and Reflecting: http://youtu.be/8sUgreHjTt8 .
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