In this video I go over an example on average value and show that the average velocity when defined as the change in position over the change in time is equal to the average of the average velocities or the average velocity function.
In this video I go over the Discovery Project titled The Geometry of a Tetrahedron. Discovery Projects are advanced math problems in each chapter of my calculus book, and this one is part of the Vectors and Geometry of Space chapter. This project involves 3 Questions regarding the tetrahedron, which is a triangular pyramid. The first question involves finding the areas of each triangular face using the Cross Product. The second question involves finding the volume of a tetrahedron when written in coordinate vector form. And the third question involves deriving the 3D version of the Pythagorean Theorem, which is: the 3 triangular areas squared at a vertex sum up to equal the square of the opposite triangular area!
The topics covered as well as their timestamps are listed below.
- Introduction: 0:00
- Calculus Book Reference: 0:56
- Sections in Calculus Book Chapter: 1:12
- Topics to Cover: 2:03
1. Discovery Project: The Geometry of a Tetrahedron: 2:25
2. 3 Questions: 3:05
- Question 1: 5:00
- Question 2: 34:59
- Solution to Part (a): 36:10
- The Equation of a Plane: 41:20
- The Distance from a Point to a Plane: 45:31
- Solution to Part (b): 52:43
- Question 3: 1:05:35
Download video notes: https://1drv.ms/b/s!As32ynv0LoaIicNsAiLSiBI1ivS39w?e=V33CkF
View video notes on the Hive blockchain: https://peakd.com/hive-128780/@mes/discovery-project-geometry-of-a-tetrahedron
View video sections playlist: https://www.youtube.com/playlist?list=PLai3U8-WIK0GoEi9wxl8nTFcfw1ay-_1T
View the full Vectors and the Geometry of Space series: https://www.youtube.com/playlist?list=PLai3U8-WIK0FjJpwnxwdrOR7L8Ul8VZoZ
Related Videos:
The Cross Product: https://peakd.com/hive-128780/@mes/vectors-and-the-geometry-of-space-the-cross-product
The Dot Product: https://peakd.com/hive-128780/@mes/vectors-and-the-geometry-of-space-the-dot-product
Complex Numbers as Rotation Matrices: https://peakd.com/hive-128780/@mes/complex-numbers-as-rotation-matrices
Vectors: https://peakd.com/hive-128780/@mes/vectors-and-the-geometry-of-space-vectors
3D Coordinate Systems: https://peakd.com/hive-128780/@mes/vector-space-and-geometry-3d-coordinate-systems .
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...
https://www.youtube.com/watch?v=yRws7Jk2iHU
My bros Ivan and Mike go at it in an intense NBA Streetball basketball Game. Mike gets the better of Ivan though with two amazing blocks and some sweet handles!
I also made a sweet GIF of Mike crossover Ivan and you can view it here: https://lh6.googleusercontent.com/-8x3Gj1c40Xc/UblH5p3lE6I/AAAAAAAAUSM/UpxlJw0_IpY/w467-h278-no/2013-06-12
Related Videos:
And1 Remix Baby: http://youtu.be/ya0FGZtLr2w
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
Lebron James In depth Stats in the 2012 NBA Finals: http://youtu.be/75dd0Tii3WY .
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https://www.youtube.com/watch?v=90J7oE22I50
In this video I go over an example graphing parametric equations x = t^2 - 2t and y = t + 1. Plotting several points from these equations shows a shape that looks like a parabola, but we can in fact confirm this is actually this case. If we eliminate the t variable from both x and y functions we do in fact get a parabola that opens up horizontally.
Also in this video I show that sometimes we restrict the t variable to be within a given interval, in which the resulting curve has a initial point, (f(a), g(a)), and a terminal point, (f(b), g(b)). Overall this is a very interesting and useful video to show how common functions such as the parabola can be written in parametric equation form; so make sure to watch this video!
Download the notes in my video:
Written Notes: https://1drv.ms/b/s!As32ynv0LoaIht5QH8nHZYJM2c58gA
Excel File: https://1drv.ms/x/s!As32ynv0LoaIht5SWYnOE01B-zwGpg
View Video Notes on Steemit: https://steemit.com/mathematics/@mes/parametric-curves-example-1-parabola
Related Videos:
Parametric Equations and Curves: https://youtu.be/Kd3XF4LZoFE
Parametric Equations and Polar Coordinates: https://youtu.be/usSors49Gdw
Equation of a Circle and it's proof: http://youtu.be/xMXYJ9UeF4I
Trigonometry: Sine, Cosine and Tan Functions: http://youtu.be/WKTIlF2oWw8
Trigonometry Identities: Proof that sin^2(x) + cos^2(x) = 1: http://youtu.be/o-fAx_96lgw .
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https://www.youtube.com/watch?v=0k7QyWa9-HA
In this video I go over the derivative of hyperbolic cotangent or coth(x) and show that it is equal to -csch^2(x)
Download the notes in my video: https://www.dropbox.com/s/8snbqgy02cfl29z/443%20-%20Derivative%20Hyperbolic%20Functions%20Proof%20-%20coth%28x%29.pdf
Related Videos:
Derivatives of Hyperbolic Trigonometry: sech(x): http://youtu.be/ozsEwXpHXZs
Derivatives of Hyperbolic Trigonometry: csch(x): http://youtu.be/K2mqMP-gVTw
Derivatives of Hyperbolic Trigonometry: tanh(x): http://youtu.be/BAPcdco7zBk
Derivatives of Hyperbolic Trigonometry: cosh(x): http://youtu.be/4AGkxTGxASQ
Derivatives of Hyperbolic Trigonometry: sinh(x): http://youtu.be/9QpIDqYg52c
Hyperbola - Definition and derivation of the equation: x^2/a^2 - y^2/b^2 = 1: http://youtu.be/Y6iYC4VEAi0
Hyperbolic Functions - tanh(x), sinh(x), cosh(x) - Introduction: http://youtu.be/EmJKuQBEdlc
Derivative of Hyperbolic functions - y = sinh(x), tanh(x), cosh(x): http://youtu.be/tdlbLP5pzis
Derivative of Inverse Hyperbolic Functions: inverse sinh(x), cosh(x), tanh(x): http://youtu.be/ubYbYbJOsNs
Derivative Rules: Proof of Chain Rule: http://youtu.be/tYDDpKzP-VU
Derivative of y = x^n: Power Rule Part 1: n is positive integer: http://youtu.be/-Yv85MZNYgU .
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https://www.youtube.com/watch?v=4EkTEYFMOqI
In this video I show that if the limit as n approaches infinity of the sequence a_n approaches L than so does the limit as n approaches infinity of the sequence a_(n+1). This is because if n approaches infinity, then so does n + 1 (since infinity + 1 is still infinity). Thus the limits are the same and the question is true.
Timestamps:
- Question 3: 0:00
- Solution: True: 0:13
- Both limits approach L as n approaches infinity: 1:00
Full video 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:
Infinite Sequences and Series playlist: https://www.youtube.com/playlist?list=PLai3U8-WIK0FjJpwnxwdrOR7L8Ul8VZoZ .
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In this video I go over further into proving the parametric equations for the Cycloid curve and this time show that these equations do in fact work for all angles that the circle rotates to. In my earlier videos I derived the proof for the cases where θ is between 0 and π/2, as well as between π/2 and π. In this video I first show that the equations still work for the values θ = 0, π/2, and pi. From there I show that it still works for all other values of θ because of the cyclical nature of the trigonometric functions in the cycloid equations. Thus as the values of θ change, the derivation for the parametric equations will mirror all our previous derivations and thus give the exact same result. This is a great example to see how we can spot patterns in the derivation of parametric equations that involve trig functions, so make sure to watch this video!
Download the notes in my video: https://1drv.ms/b/s!As32ynv0LoaIhuAu4o8BZJsHTf-FQg
View Video Notes on Steemit: https://steemit.com/mathematics/@mes/parametric-curves-example-9-the-cycloid-proof-part-3
Related Videos:
Parametric Curves: Example 8: The Cycloid: Proof Part 2: https://youtu.be/_LAmOMbXXts
Parametric Curves: Example 7: The Cycloid: Proof Part 1: https://youtu.be/XmHpIZqeRKk
Parametric Curves: Example 6: Graphing Devices: https://youtu.be/cwpEasc9mhQ
Parametric Curves: Example 5: Lissajous Figure: https://youtu.be/0OT1OnnAQ9I
Parametric Curves: Example 4: https://youtu.be/WZYIS1My-g4
Parametric Curves: Example 3: https://youtu.be/VpiJdEviBbk
Parametric Curves: Example 2: https://youtu.be/SBkzZDlqTTE
Parametric Curves: Example 1: https://youtu.be/0k7QyWa9-HA
Parametric Equations and Curves: https://youtu.be/Kd3XF4LZoFE
Parametric Equations and Polar Coordinates: https://youtu.be/usSors49Gdw
Trigonometry: Sine, Cosine and Tan Functions: http://youtu.be/WKTIlF2oWw8
Angles - Degrees vs Radians: What are Radians??: http://youtu.be/bOdrVvQWRdE
Trigonometry Identities: cos(x +/- y) = cos(x)cos(y) -/+ sin(x)sin(y) : http://youtu.be/VuQczhk7HOs
Trigonometry Identities: sin(x +/- y) = sin(x)cos(y) +/- cos(x)sin(y): http://youtu.be/edtk9thfwbM .
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...
https://www.youtube.com/watch?v=4VuYJRFAI_g
In this video I go over an example on using the mean value theorem for integrals to first show that there are 2 x-values that correspond to the average value of the function and then I solve for the x-values.
Download the notes in my video: http://1drv.ms/1uwwEGs
View Video Notes on Steemit: https://steemit.com/mathematics/@mes/mean-value-theorem-for-integrals-example-1
Related Videos:
Mean Value Theorem for Integrals: Proof: http://youtu.be/HIhDyXM87pI
Integrals and Average Value: Mean Value Theorem: http://youtu.be/rjI7ropMwuI
Mean Value Theorem - A Simple Proof: http://youtu.be/x-2MiiG2E38
Integrals and Average Value: Example 1: http://youtu.be/Ps1IPP4O_Lo
Integrals and Average Value of a Function: http://youtu.be/Xo2SSQOki1g
Integrals and Work: Introduction: http://youtu.be/dNmfGmWyMqc
Integrals and Volumes by Cylindrical Shells: http://youtu.be/LbywV9X154E
Integrals and Volumes: http://youtu.be/-evdvkDwBuQ
Integrals and Areas Between Curves: Absolute Value: http://youtu.be/w7HlDFcDNUs
Integrals and Areas Between Curves: http://youtu.be/2F03KMLIzbk
Area Under a Curve: Introduction to Integral Calculus: http://youtu.be/JbEbhv8ybmE
The Definite Integral - Brief Introduction: http://youtu.be/vhMP5SKbQjU
Fundamental Theorem of Calculus - Intro and Proof of Part 2 of the Theorem: http://youtu.be/yuIl-BPQHss
Fundamental Theorem of Calculus - Introduction and Part 1 of the Theorem: http://youtu.be/3o8Q6UJzJyk
Fundamental Theorem of Calculus - Proof of Part 1 of the Theorem: http://youtu.be/CAqTwiPxYwU .
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https://www.youtube.com/watch?v=1rkqf6qVBXQ
In this video I go over the sum of an infinite series that involves taking multiple derivatives of the Geometric Series. Starting off with the Geometric Series for x^n, we can apply the derivative to it for its radius of convergence, as derived from my earlier video on Power Series. When taking derivatives, the Radius of Convergence remains the same but we still have to determine the endpoints separately. The series with the terms (n^3)x^n has a radius of convergence equal to 1 and diverges at the endpoints, that is it diverges at x = 1 and x = -1.
The timestamps of key parts of the video are listed below:
- Problem 9: 0:00
- Solution: Start with a Geometric Series: 0:20
- Differentiating with Quotient Rule: 6:42
- Radius of Convergence is 1: 16:40
- Checking for convergence at the endpoints: 18:18
- Endpoints diverge: 21:09
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 .
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https://www.youtube.com/watch?v=CCFcOCBfgAs
In this video I share a few of my thoughts as to what all of these Donald Trump birthday coincidences. My current take is that much of the coincidences and patterns involving Trump are foreshadowing his life path and which appear to be part of major global events (aka psychological operations). This is similar to how many coincidences foreshadowed 9/11, such as Neo from the Matrix having a passport that expires on September 11, 2001. Is this a case of "predictive programming"? Or just the way the world works, as defined by the term "synchronicity"? Or is this all just random chance? Only time will tell! But if you have coincidences I missed, comment below, as well as your take on what this all means!
Lastly, I also mention another strange coincidence involving Donald Trump's June 14, 1946 blood moon birthday, and that is the Sun was in between the horns of the Taurus the bull Zodiac constellation sign. And this of course has a whole series of historical and movie usage, which I may cover in another video so stay tuned!
The timestamps of key parts of the video are listed below:
- Conclusions: Trump Predictive Programming: 0:00
- Comment Below for Coincidences I Missed!: 1:07
- Another Trump Birthday Coincidence: Sun between Taurus' Horns: 1:21
This video was taken from my earlier video listed below:
- ? Strange Coincidences Involving Donald Trump's Birthday: https://youtu.be/Yd0Ah2Komco
- HIVE video notes: https://peakd.com/hive-106474/@mes/coincidences-involving-donald-trumps-birthday
- Video sections playlist: https://www.youtube.com/playlist?list=PLai3U8-WIK0ElXMhiVmHKyq9IiI0SFYBk
Related Videos:
Full MES Occult video series: https://www.youtube.com/playlist?list=PLai3U8-WIK0EX_SJASwVeZMA-CMgu6-Ox .
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https://www.youtube.com/watch?v=hvCRWszufhE