In this video I show how for some examples you can solve the integral of a function by simply summing up the area under the curve by calculating the area of a circle or a triangle if those functions happen to be in those shapes. This saves a lot of time and greatly simplifies calculating integrals.
Download the notes in my video: https://www.dropbox.com/s/av6n8kjw3t2kcwv/217%20-%20Evaluating%20Integrals%20Examples%20Part%202.pdf
Related Videos:
The Definite Integral - Brief Introduction: http://youtu.be/vhMP5SKbQjU
Area Under a Curve: Introduction to Integral Calculus: http://youtu.be/JbEbhv8ybmE
Area of a Triangle - A simple proof that A = b*h/2: http://youtu.be/wUT_KN5pshQ
Equation of a Circle and it's proof: http://youtu.be/xMXYJ9UeF4I
Equation of a Line - A Simple Explanation: http://youtu.be/a3ntO-xNN5M .
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https://www.youtube.com/watch?v=yRP_7umJmIo
Recently I was asked whether I could go over a visual proof of the Cauchy's Mean Value Theorem, as I had done for the Lagrange or simple version of the Mean Value Theorem (MFT). This was a very interesting question so I decided to go ahead and go over the graphical visualization of the theorem. In this video I show that the Cauchy or general mean value theorem can be graphically represented in the same way as for the simple MFT. The only difference is that the horizontal axis is not x, but a more general function of x, g(x). This difference causes the formulation to be more general, and the instantaneous slope to be formulated in the more general definition of the derivative, as shown in my last video. This is a very interesting topic to understand, but make sure to watch my earlier video on the general definition of derivative to get a better understanding of this!
Download the notes in my video: https://1drv.ms/b/s!As32ynv0LoaIhtgYCYyHYoceOBojYA
View Video Notes on Steemit: https://steemit.com/mathematics/@mes/cauchy-s-mean-value-theorem-visual-proof
Related Videos:
Definition of Derivative: More General Definition: https://youtu.be/OpH1D4JAOS8
Cauchy's Mean Value Theorem - A simple proof: http://youtu.be/sBW-LwoFWLY
Mean Value Theorem - A Simple Proof: http://youtu.be/x-2MiiG2E38
Mean Value Theorem: Corollary Theorems: http://youtu.be/Os2dO9XKmVE .
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https://www.youtube.com/watch?v=VxH89U2E1bQ
In this video I ask ChatGPT to give me the main paper on COVID "virus" isolation. The paper is listed below:
- A Novel Coronavirus from Patients with Pneumonia in China, 2019
- Authors: Na Zhu et al
- Published on January 24, 2020 in The New England Journal of Medicine (NEJM)
- https://www.nejm.org/doi/full/10.1056/NEJMoa2001017
ChatGPT uses artificial intelligence (AI) to synthesize data on the internet into human-like answers. Note that ChatGPT is a lot like Wikipedia so it is biased towards information already widely accepted by the mainstream public and institutions.
Note also that the ChatGPT response was from March 18, 2023 and since then the answer they have provided is different (although both papers are very similar and with many of the same authors). The following link is for the more recent June 11, 2023 response from ChatGPT: https://chat.openai.com/share/3be5aa52-b5c2-4b44-9ff6-aaebcae8fafb
The timestamps of key parts of the video are listed below:
- Disclaimer on ChatGPT: 0:00
- ChatGPT on COVID-19 "Virus" Isolation Paper: 0:59
- Paper: A Novel Coronavirus from Patients with Pneumonia in China, 2019: 2:47
This video is taken from my earlier video listed below:
- ?#MESScience 4: Review of COVID-19 "Virus" Isolation Paper: https://youtu.be/cvDO85Rw4d8
- Video notes: https://peakd.com/hive-128780/@mes/messcience-4-review-of-covid-19-virus-isolation-paper
- Playlist: https://www.youtube.com/playlist?list=PLai3U8-WIK0E4aQ_cq4ZDD2WGiDU5vVgx
Related Videos:
?#MESScience: MES Science Tutorials: https://mes.fm/science-playlist .
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https://www.youtube.com/watch?v=U3WKmIogG84
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 .
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https://www.youtube.com/watch?v=ss3ZClOuUk4
In this video I go over a quick tutorial on adding two matrices. Matrix addition is defined in a straight forward way, and just requires 2 matrices with the same number of rows and columns. Then each entry in the resulting matrix involves adding up the corresponding entries from the other 2 matrices.
This video was taken from my earlier video listed below:
- Complex Numbers as Rotation Matrices: https://youtu.be/Mgp2vrQeLEw
- Video notes: https://peakd.com/hive-128780/@mes/complex-numbers-as-rotation-matrices
- Playlist: https://www.youtube.com/playlist?list=PLai3U8-WIK0E6x0dZEAx77Kqv7LLxYdbx
Related Videos:
Vectors and the Geometry of Space Playlist: https://www.youtube.com/playlist?list=PLai3U8-WIK0FjJpwnxwdrOR7L8Ul8VZoZ .
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https://www.youtube.com/watch?v=hH5LQ9sSeBU
In this video I go further into determining the area of polar curves and this time do an example on evaluating the area of one loop of a 4 leaved rose given by the polar curve y = cos 2ϴ. I had graphed this curve in my Example 8 video, and I use the fact that one of the loops is bounded by the angles -π/4 and π/4. Thus we can apply the formula for area I derived in my last video, or remember the fact that the formula was derived by summing up an infinite amount of sectors of a circle, which we know the formula for as ½*r2ϴ. I also use symmetry to simplify the resulting integral and show that the area of one of the loops is π/8, thus the full curve is four times that, π/2. This is a very useful video in applying the area formula for polar curves so make sure to watch this video!
Download the notes in my video: https://1drv.ms/b/s!As32ynv0LoaIhvU1cfT6XTWOuHvAjw
View Video Notes On Steemit: https://steemit.com/mathematics/@mes/video-notes-polar-coordinates-area-example-1-r-cos-2
Related Videos:
Polar Coordinates: Area Formula: https://youtu.be/wd-EwiKzqa0
Polar Coordinates: Graphing With Polar Curves with Desmos Calculator: https://youtu.be/xY9Be9HHRsk
Polar Coordinates: Cartesian Connection: https://youtu.be/HcaTYrpmGaU
Polar Coordinates: https://youtu.be/-KAdZL-N4ok
Parametric Equations and Polar Coordinates: https://youtu.be/usSors49Gdw
Area of a Sector of a Circle: Proof: https://youtu.be/lglnySF8DyY
Equation of a Circle and it's proof: http://youtu.be/xMXYJ9UeF4I
Integration Overview: How are Riemann Sums, Antiderivatives, and Integrals Linked?: https://youtu.be/TGnnu1vnD_U
Polar Coordinates: Example 8: Four-Leaved Rose: https://youtu.be/mTk5j0jD3dE .
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https://www.youtube.com/watch?v=6kavQyGcODo
In this video I go over further into differential equations and expand on my earlier videos on modeling population growth. The same basic model used in modeling population growth occurs actually very often in nature: in bacteria growth, radioactive decay, and compound interest, to name only a few. This model stems from the idea that the rate of change, either growth or decay, of a substance is often proportional to the amount of that substance. In other words, doubling the amount of a substance often doubles the rate of change as well. The result is a simple separable differential equation and the solution is an exponential function. This is a very interesting and important video because of the many applications that used the resulting solution, so make sure to watch this video!
Download the notes in my video: https://1drv.ms/b/s!As32ynv0LoaIhtATE4Zy5KKR3_5Htw
View Video Notes on Steemit: https://steemit.com/mathematics/@mes/differential-equations-exponential-growth-and-decay
Related Videos:
Differential Equations: Population Growth: https://youtu.be/Td8C_cTEGkA
Differential Equations: Separable Equations: https://youtu.be/pBV-xT9ty94
Derivative of y = ln|x| or absolute value of x: http://youtu.be/Trzm6tagKws
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=DZtDUIZuxcg
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 further into Hyperbolas and this time determine the foci, asymptotes, and vertices of the hyperbola 9x^2 – 16y^2 = 144. This particular hyperbola is not written in “standard form”, which is the form x^2/a^2 – y^2/b^2 = 1 and which I derived in my earlier Conic Sections video. But we can get it in standard from by simply dividing both sides of the equation by 144 and then simplifying the resulting fractions. Doing so, I show that the hyperbola in standard form is x^2/4^2 – y^2/3^2 = 1. Thus, as shown also in my earlier derivation video, the foci are located at (+/- c, 0) = (+/- 5, 0); vertices at (+/- a, 0) = (+/- 4, 0), and asymptotes y = +/- (b/a)x = +/- (3/4)x. Graphing this using the amazing Desmos graphing calculator confirms that it a typical horizontal hyperbola, as expected. This is a very useful illustrating on walking through the steps in analyzing a hyperbola by first converting it into standard form and then graphing; so make sure to watch this video!
Download the notes in my video: https://1drv.ms/b/s!As32ynv0LoaIh5MfEMYv-svSfjLUCg
View video notes on the Hive blockchain: https://peakd.com/mathematics/@mes/conic-sections-hyperbolas-example-1
Related Videos:
Conic Sections: Hyperbola: Definition and Formula: https://youtu.be/UBIHovXNV9U
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
Hyperbola - Definition and derivation of the equation: x^2/a^2 - y^2/b^2 = 1: http://youtu.be/Y6iYC4VEAi0
Hyperbola Examples: http://youtu.be/LvccUMFrY3k .
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