Math Easy Solutions

In this video I explore why the shape of cans that you see in the supermarket are the way they are and use calculus to determine what ratios of can height to can radius are used in real life. This is an applied project question in my calculus text book and I thought it was really interesting and worth doing a math easy on it! Enjoy!

View video notes on the Hive blockchain: https://peakd.com/hive-128780/@mes/applied-project-shape-of-a-can-science-behind-can-manufacturing

Related Videos:

Area of a Regular Hexagon - Simple Proof and other Properties: http://youtu.be/RAReWAs_dEY
Power Functions and their Properties Part 1 - A Simple Explanation: http://youtu.be/2MKko4ZkSf0
Derivative of y = x^n - Part 2: General Power Rule: http://youtu.be/Ibvu33oh49o
Local and Absolute Max/Min and Critical Points/Numbers: http://youtu.be/aOvwQaEFbjw .

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In this video I go through a useful example on marginal cost vs average costs and show how to apply the principle which I proved in my previous video that the average cost is minimum when average cost equals marginal cost.

View video notes on the Hive blockchain: https://peakd.com/hive-128780/@mes/marginal-cost-vs-average-cost-example-economics-101

Related Videos:

Marginal Cost vs Average Cost - Economics 101: http://youtu.be/HiMaLvsTstc
Marginal Costs - Economics 101: http://youtu.be/XS-1L6Iq4Wk
Local and Absolute Max/Min and Critical Points/Numbers: http://youtu.be/aOvwQaEFbjw
First Derivative Test Introduction and Examples: http://youtu.be/60UMYG7Ppeg
Second Derivative Test - A brief Introduction: http://youtu.be/lOwK5rQh0Kk
Derivative of y = x^n - Part 2: General Power Rule: http://youtu.be/Ibvu33oh49o .

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In this video I look further into economics and show that the average cost of producing one item when x items are produced is minimum when the average cost equals the marginal cost. This is an important principle to understand and it is easily derived using the calculus principles that I have covered in my previous videos.

View video notes on the Hive blockchain: https://peakd.com/hive-128780/@mes/marginal-cost-vs-average-cost-economics-101

Related Videos:

Marginal Costs - Economics 101: http://youtu.be/XS-1L6Iq4Wk
Local and Absolute Max/Min and Critical Points/Numbers: http://youtu.be/aOvwQaEFbjw
First Derivative Test Introduction and Examples: http://youtu.be/60UMYG7Ppeg
Second Derivative Test - A brief Introduction: http://youtu.be/lOwK5rQh0Kk
Derivative of y = x^n - Part 2: General Power Rule: http://youtu.be/Ibvu33oh49oDerivative Rules - Proof of the Product Rule: http://youtu.be/EIjvGJhDAOk .

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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 expand on the area of a triangle proof and show that regardless of the shape of the triangle the Area of the triangle remains the same no matter which side is used as the base. The formula for all cases still remains at Area = base*height/2.

View video notes on the Hive blockchain: https://peakd.com/hive-128780/@mes/area-of-a-triangle-proof-part-2-base-is-not-the-longest-side

Related Videos:

Area of a Triangle - A simple proof that A = b*h/2: http://youtu.be/wUT_KN5pshQ
Similar Triangles - Part 1: http://youtu.be/gXcQEKVcesc .

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In this video I show a quick and very easy proof of why the area of any triangle is just the base*height/2.

View video notes on the Hive blockchain: https://peakd.com/hive-128780/@mes/area-of-a-triangle-proof

Related Videos:

Area of a Triangle Proof - Part 2: When the base is NOT the longest side of the triangle: http://youtu.be/xTVP-HHBP0U
Similar Triangles - Part 1: http://youtu.be/gXcQEKVcesc .

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In this video I go through a useful word problem and show how you can derive the demand and revenue functions for a simple consumer sales example. I also illustrate how you can increase revenue by offering rebates, or basically refunding a percentage of the product back to the buyer.

View video notes on the Hive blockchain: https://peakd.com/hive-128780/@mes/marginal-cost-vs-marginal-revenue-maximizing-revenue

Related Videos:

Marginal Cost vs Marginal Revenue: When is maximum profit realized??: http://youtu.be/ufMgmYbZPdU
Marginal Cost vs Marginal Revenue - Examples Part 1: Maximizing Profit: http://youtu.be/cQ83F5qynco
Marginal Cost vs Average Cost - Economics 101: http://youtu.be/HiMaLvsTstc
Marginal Costs - Economics 101: http://youtu.be/XS-1L6Iq4Wk
Marginal Cost vs Average Cost Example - Economics 101: http://youtu.be/W2xx0Wtl608
Local and Absolute Max/Min and Critical Points/Numbers: http://youtu.be/aOvwQaEFbjw
First Derivative Test Introduction and Examples: http://youtu.be/60UMYG7Ppeg
Derivative of y = x^n - Part 2: General Power Rule: http://youtu.be/Ibvu33oh49o .

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In this video I go through a useful example illustrating how to apply the principle that I showed in my earlier video: if the total profit is at a maximum than the marginal revenue has to equal the marginal cost.

View video notes on the Hive blockchain: https://peakd.com/hive-128780/@mes/marginal-cost-vs-marginal-revenue

Related Videos:

Marginal Cost vs Marginal Revenue: When is maximum profit realized??: http://youtu.be/ufMgmYbZPdU
Marginal Cost vs Average Cost - Economics 101: http://youtu.be/HiMaLvsTstc
Marginal Costs - Economics 101: http://youtu.be/XS-1L6Iq4Wk
Marginal Cost vs Average Cost Example - Economics 101: http://youtu.be/W2xx0Wtl608
Local and Absolute Max/Min and Critical Points/Numbers: http://youtu.be/aOvwQaEFbjw
First Derivative Test Introduction and Examples: http://youtu.be/60UMYG7Ppeg
Second Derivative Test - A brief Introduction: http://youtu.be/lOwK5rQh0Kk
Derivative of y = x^n - Part 2: General Power Rule: http://youtu.be/Ibvu33oh49o .

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In this video I show that when the total profit is at a maximum than the marginal revenue has to equal the marginal cost.

View video notes on the Hive blockchain: https://peakd.com/hive-128780/@mes/marginal-cost-vs-marginal-revenue

Related Videos:

Marginal Cost vs Average Cost - Economics 101: http://youtu.be/HiMaLvsTstc
Marginal Costs - Economics 101: http://youtu.be/XS-1L6Iq4Wk
Marginal Cost vs Average Cost Example - Economics 101: http://youtu.be/W2xx0Wtl608
Local and Absolute Max/Min and Critical Points/Numbers: http://youtu.be/aOvwQaEFbjw
First Derivative Test Introduction and Examples: http://youtu.be/60UMYG7Ppeg
Second Derivative Test - A brief Introduction: http://youtu.be/lOwK5rQh0Kk
Derivative of y = x^n - Part 2: General Power Rule: http://youtu.be/Ibvu33oh49o .

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Ever wonder how your calculator is able to graph or find the roots of complex functions?? Well chances are, that your calculator either uses Newton's Method or a variation of Newton's Method. Note also that this method is sometimes called the Newton-Raphson Method.

View video notes on the Hive blockchain: https://peakd.com/hive-128780/@mes/newtons-method-of-linear-approximation-introduction

Related Videos:

Simple Proof of the Quadratic Formula: http://youtu.be/9eQLZar3g6g
Linear Approximation - Introduction and Examples: http://youtu.be/bXEK8bkWTtM .

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In this video I show how you can use modern powerful spreadsheet capabilities to prove that pi is indeed 3.14... at least 2 decimal places. This is a pretty ridiculous method I just figured out which basically involves computing the area of many squares (or Microsoft Excel cells) inscribed inside a circle and using this area to compute what Pi should be.

View video notes on the Hive blockchain: https://peakd.com/hive-128780/@mes/proving-pi-314-with-microsoft-excel

Related Videos:

What is Pi and Proof that it equals 3.14159265....: http://youtu.be/mZc7Uvae4Fw .

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In this video I recap on the Newton's Method of Linear Approximation which I showed in my earlier video and go through some very useful examples to illustrate the cases in which this method works and converges fairly quickly. In the examples I utilized Microsoft's powerful spreadsheet capabilities and use Excel tables to automate Newton's Method. I also show some useful tips on how to speed up creating a table in Excel.

View video notes on the Hive blockchain: https://peakd.com/hive-128780/@mes/newtons-method-on-linear-approximation-examples-part-1

Related Videos:

Newton's Method of Linear Approximation - Introduction: http://youtu.be/aT4b_5l50RI
Linear Approximation - Introduction and Examples: http://youtu.be/bXEK8bkWTtM
Derivative of Trigonometry Functions: Derivative of cos(x): http://youtu.be/LMjVp-GsrCw
Derivative of Power Functions Part 1 - x^n where n is a positive integer: http://youtu.be/-Yv85MZNYgU .

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Is your video too loud or too quiet?? Well if it is than you need to download Handbrake, which is a free program, from http://handbrake.fr and use it to lower or increase the audio volume in your video. In this video I show how you can easily do it with Handbrake.

My earlier video which was about my cousin, Simone, doing one-handed push-ups was initially way too loud and I needed to figure out a way to decrease the volume before uploading to YouTube. You can watch that video on the link below:

My 11 year old Cousin Simone doing One Hand Pushups: http://youtu.be/P6HyWrpbDlM

Related videos:

Handbrake Tutorial: Compress/Convert Large Video Files!: http://youtu.be/qZPFfS_ZF8w
Cut / Split Video Files Easily with Virtual Dub!!: http://youtu.be/rRUKFP54LiQ .

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In this video I go over a very interesting concept and that is in interpreting complex or imaginary numbers as and equivalent rotational matrix form. This concept was brought to my attention by Lori Gardi from the Fractal Woman YouTube channel: https://www.youtube.com/user/FractalWoman/search?query=complex%20numbers

Rewriting complex numbers as rotation matrices allows for looking at complex numbers from a different perspective, and this case that perspective is in simply rotating vectors. The imaginary number "i" is thus viewed as simply a 90 degree rotation when viewed as a rotation matrix. I also rewrite the famous Euler's formula into a rotation matrix form and that turns out to just be a 180 degree rotation. Since complex numbers appear often in mathematics and physics, it may be insightful to reinterpret such complex equations as simply vector rotations!

Note that while I only covered 2D rotation matrices, a similar concept can be applied to 3D rotation matrices, and which I may cover in future videos so stay tuned.

The topics covered as well as their timestamps are listed below.

- Introduction: 0:00
- Topics to Cover: 0:41
1. Fractal Woman and Other References: 1:28
2. Rotating a Vector: 3:01
3. Matrix Multiplication: 12:47
5. Rotation Matrix: 27:13
6. Complex Numbers: 36:58
7. Relating the Complex Plane with the Rotation Matrix: 43:30
- Complex Numbers as Vector Rotations: 57:12
- Reinterpreting Euler's Formula: 1:10:02
- Applications to Other Fields: 1:20:51

View video notes on the Hive blockchain: https://peakd.com/hive-128780/@mes/complex-numbers-as-rotation-matrices

Related Videos:

Vector Space and Geometry Playlist: https://www.youtube.com/playlist?list=PLai3U8-WIK0FjJpwnxwdrOR7L8Ul8VZoZ

3D Coordinate Systems: https://peakd.com/hive-128780/@mes/vector-space-and-geometry-3d-coordinate-systems .

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In this video I recap on Newton's Method on Linear Approximation and go through 4 very useful examples illustrating when Newton's Method either fails to converge or converges very slowly depending on the initial starting approximation you select.

View video notes on the Hive blockchain: https://peakd.com/hive-128780/@mes/newtons-method-on-linear-approximation-examples-part-2

Related Videos:

Newton's Method on Linear Approximation - Examples Part 1: Where it Converges: http://youtu.be/u_Uo5aShAUs
Newton's Method of Linear Approximation - Introduction: http://youtu.be/aT4b_5l50RI
Linear Approximation - Introduction and Examples: http://youtu.be/bXEK8bkWTtM
Derivative of Power Functions Part 1 - x^n where n is a positive integer: http://youtu.be/-Yv85MZNYgU
Power Functions and their Properties Part 1 - A Simple Explanation: http://youtu.be/2MKko4ZkSf0 .

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In this video I explain what Antiderivatives are and allude to further videos where I will go through integrals.

View video notes on the Hive blockchain: https://peakd.com/hive-128780/@mes/antiderivatives-brief-introduction

Related Videos:

Mean Value Theorem - A Simple Proof: http://youtu.be/x-2MiiG2E38
Derivative of Power Functions Part 1 - x^n where n is a positive integer: http://youtu.be/-Yv85MZNYgU
Definition of Derivative Simple Explanation: http://youtu.be/0rjGMpM06Eg .

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In this video I recap what antiderivatives are which I showed in my earlier video and go through some very useful examples on antiderivatives. I also list a table of common functions and their antiderivatives.

View video notes on the Hive blockchain: https://peakd.com/hive-128780/@mes/antiderivatives-examples-part-1

Related Videos:

Antiderivatives - A brief Introduction: http://youtu.be/zDzUHqAUSEw
Derivative of Power Functions Part 1 - x^n where n is a positive integer: http://youtu.be/-Yv85MZNYgU
Derivative of y = Log(x) and y = Ln(x): http://youtu.be/5e6MisvvMPE
Definition of Derivative Simple Explanation: http://youtu.be/0rjGMpM06Eg
Derivative of Trigonometry Functions: Derivative of tan(x): http://youtu.be/Jf6p4crjrSc
Derivative of Trigonometry Functions: Derivative of cos(x): http://youtu.be/LMjVp-GsrCw
Derivative of Trigonometry Functions: Derivative of sin(x): http://youtu.be/elEvQ4Wu7Pk
Derivative of arcsine or Inverse sin(x): http://youtu.be/rj2nQmiekz4
Derivative of arccosine or Inverse cos(x): http://youtu.be/ShNAwpojuXA
Derivative of arctan(x) or Inverse tan(x): http://youtu.be/q9jdOiQ7pZY .

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In this video, I continue further on Antiderivatives and go over some further examples which include initial value problems that involve solving for the constants of the general antiderivative function. I also graph out the function and its antiderivative to show how the function can be used as a check of the antiderivative by looking at the inflection points and whether the antiderivative is increasing, decreasing, or concaving up or down.

View video notes on the Hive blockchain: https://peakd.com/hive-128780/@mes/antiderivatives-examples-part-2

Related Videos:

Antiderivatives - A brief Introduction: http://youtu.be/zDzUHqAUSEw
Antiderivatives - Examples, Part 1: http://youtu.be/Ug9QJT8Z6Bs
Derivative of Power Functions Part 1 - x^n where n is a positive integer: http://youtu.be/-Yv85MZNYgU
Pi is WRONG! Tau is much better!: http://youtu.be/3fVEQrEmo9o
Power Functions and their Properties Part 1 - A Simple Explanation: http://youtu.be/2MKko4ZkSf0
Inverse Trigonometric Functions: Inverse cosine, sine, and tan Functions
Derivative of y = Log(x) and y = Ln(x): http://youtu.be/5e6MisvvMPE
Definition of Derivative Simple Explanation: http://youtu.be/0rjGMpM06Eg .

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You probably have used the foil method in Elementary and High-school math class but you may not know quite why it works. Well the proof is quite simple and it involves just considering the multiplication as an Area of a rectangle. Also in the video, I show a very quick method to foil out very complex equations and it involves creating a simple table and is very easy to memorize. If you don't know the foil method inside and out, then this video is a must watch since you'll have to use the foil method in almost every math, engineering, physics class.

View video notes on the Hive blockchain: https://peakd.com/hive-128780/@mes/foil-method-proof-and-alternative-method

Related Videos:

Factoring: Difference of Squares: http://youtu.be/qFdohskRQB0
Rationalizing the Denominator: http://youtu.be/fRb2TQQxH_Y
Factoring Quadratic Polynomials by Guessing: http://youtu.be/biEfGwT5pn4
Factoring: Difference and Addition of Cubes: http://youtu.be/LAUSBMod4Ks .

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In this video I go further into the wonderful world of antiderivatives and explore how we can apply the calculus graphing methods I showed in my earlier videos to graph the antiderivative a function without actually determining what the antiderivative is. I also show how you can graph the antiderivative in a very accurate method using the Directional Fields drawing technique which basically draws the tangent lines of the antiderivative at many x values and then graphing can be done simply by following the direction of the tangents and hence the name directional field.

View video notes on the Hive blockchain: https://peakd.com/hive-128780/@mes/antiderivatives-examples-part-3-graphing-and-directional-fields

Related Videos:

Antiderivatives - A brief Introduction: http://youtu.be/zDzUHqAUSEw
Antiderivatives - Examples, Part 1: http://youtu.be/Ug9QJT8Z6Bs
Antiderivatives - Examples, Part 2: http://youtu.be/mV1RyLjUOns
First Derivative Test Introduction and Examples: http://youtu.be/60UMYG7Ppeg
Second Derivative Test - A brief Introduction: http://youtu.be/lOwK5rQh0Kk .

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In this video I go further into the examples series on Antiderivatives and basically show how they can be used to determine the position function when given the velocity or the acceleration functions and also initial position and velocity values. I also discuss gravity and how near the surface of the Earth the acceleration due to gravity can be taken as a constant which is equal to 9.8 m/s^2 or 32 ft/s^2.

View video notes on the Hive blockchain: https://peakd.com/hive-128780/@mes/antiderivatives-examples-part-4-position-velocity-acceleration

Related Videos:

Antiderivatives - A brief Introduction: http://youtu.be/zDzUHqAUSEw
Antiderivatives - Examples, Part 1: http://youtu.be/Ug9QJT8Z6Bs
Antiderivatives - Examples, Part 2: http://youtu.be/mV1RyLjUOns
Antiderivatives - Examples, Part 3 - Graphing and Directional Fields: http://youtu.be/Y0Zw4INaoxU
First Derivative Test Introduction and Examples: http://youtu.be/60UMYG7Ppeg
Second Derivative Test - A brief Introduction: http://youtu.be/lOwK5rQh0Kk .

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In this video I go over the second section in my Vectors and the Geometry of Space calculus book chapter and this time go over Vectors. Vectors are mathematical objects that describe both magnitude and direction. These are very useful in physics and science in general as it can simplify the mathematical description of 3D-dimensional space. I go over an introduction on Vectors as well as their properties and corresponding vector algebra.

The topics covered as well as their timestamps are listed below.

- Introduction: 0:00
- Sections in Calculus Book Chapter: 0:58
- Topics to Cover: 2:16
1. Vectors: 3:01
2. Combining Vectors: 5:46
- Definition of Vector Addition: 7:03
- Triangle Law: 7:54
- Parallelogram Law: 8:40
- Example 1: 11:21
- Definition of Scalar Multiplication: 14:50
- Example 2: 21:01
3. Components: 22:27
- Equation 1: 35:36
- Example 3: 36:11
- Magnitude or Length of a Vector: 37:48
- Vector Algebra: 39:42
- Example 4: 50:08
- Note on n-Dimensional Vectors: 56:31
- Properties of Vectors: 58:57
- Standard Basis Vectors: 1:08:20
- Example 5: 1:20:56
- Historical Note: 1:25:04
- Unit Vectors: 1:27:15
- Example 6: 1:31:44
4. Applications: 1:35:00
- Example 7: 1:35:45

View video notes on the Hive blockchain: https://peakd.com/hive-128780/@mes/vectors-and-the-geometry-of-space-vectors

Related Videos:

Vector Space and Geometry Playlist: https://www.youtube.com/playlist?list=PLai3U8-WIK0FjJpwnxwdrOR7L8Ul8VZoZ

Vectors and the Geometry of Space: 3D Coordinate Systems: https://peakd.com/hive-128780/@mes/vector-space-and-geometry-3d-coordinate-systems .

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My calculus book has some pretty challenging questions at the end of each chapter and I think it would be a good idea if I do some of those examples. In this video I go through a seemingly complex equation by showing that you can break it down into simpler terms and apply what we have learned in finding the local or absolute minimum or maximum to solve the problem.

View video notes on the Hive blockchain: https://peakd.com/hive-128780/@mes/problems-plus-example-1-simplifying-complex-equations

Related Videos:

First Derivative Test Introduction and Examples: http://youtu.be/60UMYG7Ppeg
Second Derivative Test - A brief Introduction: http://youtu.be/lOwK5rQh0Kk
Definition of Derivative Simple Explanation: http://youtu.be/0rjGMpM06Eg
Foil Method - Simple Proof and Quick Alternative Method: http://youtu.be/tmj_r94D6wQ .

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In this video I introduce Sigma Notation and show how it is a really convenient method of writing down sums. I will be using Sigma Notation a great deal in my later videos on integral calculus, so make sure you watch this video to learn more about Sigma Notation or at least get a refresher if you have seen it before.

View video notes on the Hive blockchain: https://peakd.com/hive-128780/@mes/sigma-notation-a-introduction

Related Video:

Sum of "n" Consecutive Integers - Simple Proof : http://youtu.be/tpkzn2e5mtI .

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In this video I go through Karl Gauss's ingenious proof for the formula of a sum of the first n positive and consecutive integers. Gauss derived this when he was only 10 years old!

View video notes on the Hive blockchain: https://peakd.com/hive-128780/@mes/sum-of-n-consecutive-integers-proof

Related Videos:

Sigma Notation - A brief Introduction: http://youtu.be/Gew7y73NY30
Types of Numbers: Natural, Integers, Rational, Irrational, and Real Numbers: http://youtu.be/U22Z1q_Ibqg .

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