# Math Easy Solutions

In this video, I discuss finite limits in terms of horizontal and vertical asymptote lines and also go through some very useful examples and tricks in solving limits in which you approach infinity.

View video notes on the Hive blockchain: https://peakd.com/hive-128780/@mes/infinite-limits-horizontal-and-vertical-asymptote-lines

Related Videos:

The Limit of a Function: http://youtu.be/0RX1o7KeZ28
What are Limits? A Simple Explanation: http://youtu.be/FbV7TzlkTZk
One Sided Limits - Plus Examples: http://youtu.be/QA2v_NCej10
Limit Laws - Brief Overview: http://youtu.be/qfk5c-43dLg
Squeeze Theorem - A Brief Explanation: http://youtu.be/nRrzgneyEb4 .

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In this video I go over two examples involving spheres. The first example involves completing the square in simplifying an equation to that of the standard equation of a sphere. The second example involves applying inequalities to a sphere, which thus results in a semi-spherical shell solid surface being formed.

The timestamps of key parts of the video are listed below:

- Example 6: Equation of a Sphere: 0:00
- Example 7: Spherical Inequalities: 7:39

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

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In this video I show how you can add borders of any color or pattern around text and objects using the free and powerful Photoshop Program GIMP 2.0. I show how to use the Stroke, Add Alpha to Selection and Fuzzy Select functions in GIMP to select text and the Border with any color to text or objects. Using these methods you can have text or objects that are white but with a black border on white background.

Related Videos:

GIF Animations with GIMP 2.8 - A very simple step-by-step tutorial: http://youtu.be/vg8IFbHsvng
GIMP 2.0: Rounding Text Edges with the Gaussian Blurr Function: http://youtu.be/lEAO6Ajr1AY
GIMP 2.0 Photoshop Tutorial - Adding a Color Border to Text and Objects: http://youtu.be/fTyLewIVB-8
GIMP Photoshop Tutorial: http://youtu.be/dSYjD4ov6ao .

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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

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In this video I discuss the difference between Tangent and Secant Lines and show how the definition of derivative idea came about from the idea of tangents.

View video notes on the Hive blockchain: https://peakd.com/hive-128780/@mes/tangent-and-secant-lines

Related Videos:

Equation of a Line - A Simple Explanation: http://youtu.be/a3ntO-xNN5M
Definition of Derivative Simple Explanation: http://youtu.be/0rjGMpM06Eg .

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In this tutorial I show how you can round text edges with the Gaussian Blur Function in GIMP 2.0 and also with a little trick of duplicating and merging layers. This is a must learn technique for anyone even photo-shopping for fun.

Related Videos:

GIF Animations with GIMP 2.8 - A very simple step-by-step tutorial: http://youtu.be/vg8IFbHsvng
GIMP Photoshop Tutorial - Adding a Color Border to Text and Objects: http://youtu.be/fTyLewIVB-8
GIMP Photoshop Tutorial: http://youtu.be/dSYjD4ov6ao .

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In this video I prove the distance formula between 2 points in both two and three dimensions. The distance formula in 2D is readily derived using the Pythagorean theorem. In 3D we can extend the 2D distance formula by drawing a box and then applying the Pythagorean Theorem twice. Both distance formulas appear identical but with the 3D formula having the term in the z-axis. I also go over a quick example on calculating the distance between 2 points in 3D space.

The timestamps of key parts of the video are listed below:

- Distance Formula in 2D: 0:00
- Distance Formula in 3D: 1:49
- Example 4: 12:28

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

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In this video I show various ways of writing the derivative as well as showing how you can easily graph the derivative of a function if you are given its graph. I then show the 2 cases where a function is NOT differentiable. Being differentiable means that the derivative is defined at every point of the graph.

View video notes on the Hive blockchain: https://peakd.com/hive-128780/@mes/is-the-function-differentiable

Related Videos:

Definition of Derivative Simple Explanation: http://youtu.be/0rjGMpM06Eg
Continuity and Open and Closed Intervals: http://youtu.be/u29fd3WYT-g .

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In this video I go over a short example on graphing the equation y = x in three dimensions. The surface is just a vertical plane that is split directly between the x- and y-axes.

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

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In this video I show that the derivative of a constant function is just 0.

View video notes on the Hive blockchain: https://peakd.com/hive-128780/@mes/derivative-of-a-constant-function-using-the-definition-of-the-derivative

Related Videos:

Definition of Derivative Simple Explanation: http://youtu.be/0rjGMpM06Eg
Derivative Rules - Proof of the Quotient Rule: http://youtu.be/fJcgnLKkISE
Derivative Rules - Proof of the Product Rule: http://youtu.be/EIjvGJhDAOk
Derivative Rules - Constant Multiple, Sum, and Difference Rules: http://youtu.be/_y8Y3Us3q9c .

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In this video I derive differentiation rules for taking the derivative of functions multiplied by a constant or functions added or subtracted to other functions.

View video notes on the Hive blockchain: https://peakd.com/hive-128780/@mes/derivative-rules-constant-multiple-sum-difference

Related Videos:

Definition of Derivative Simple Explanation: http://youtu.be/0rjGMpM06Eg
Derivative of Power Functions Part 1 - x^n where n is a positive integer: http://youtu.be/-Yv85MZNYgU
Derivative Rules - Proof of the Quotient Rule: http://youtu.be/fJcgnLKkISE
Derivative Rules - Proof of the Product Rule: http://youtu.be/EIjvGJhDAOk .

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In this video I go over an example on graphing a circle and a cylinder in 3D coordinates. The circle that I graph has the equation x^2 + y^2 = 1 and z = 3. This means that the circle is centered about the xy-axis origin but at a height of z = 3. The cylinder that I graph just has the equation x^2 + y^2 = 1. In the context of 3D coordinates this means it is a vertical cylinder with an indefinite height. Note that in the context of 2D coordinates, this would just be a circle centered about the xy-axis origin.

The timestamps of key parts of the video are listed below:

- Example 2a: Circle: 0:00
- Example 2b: Cylinder: 2:05

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

Related videos:

Vectors and Geometry of Space video series: https://www.youtube.com/playlist?list=PLai3U8-WIK0FjJpwnxwdrOR7L8Ul8VZoZ .

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In this video I prove the very useful product rule which says that the derivative of a product of the 2 functions is the first function multiplied the derivative of the second function added to the derivative of the first function multiplied by the second function.

View video notes on the Hive blockchain: https://peakd.com/hive-128780/@mes/derivative-rules-proof-of-the-product-rule

Related Videos:

Definition of Derivative Simple Explanation: http://youtu.be/0rjGMpM06Eg
Derivative Rules - Constant Multiple, Sum, and Difference Rules: http://youtu.be/_y8Y3Us3q9c .

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In this video I prove the Quotient rule for derivatives in which you have a function divided by another function. Later I will prove the chain rule and from there you would not have to memorize this equation and just calculate all derivatives using the product and chain rules.

View video notes on the Hive blockchain: https://peakd.com/hive-128780/@mes/derivative-rules-proof-of-the-quotient-rule

Related Videos:

Derivative Rules - Proof of the Product Rule: http://youtu.be/EIjvGJhDAOk
Definition of Derivative Simple Explanation: http://youtu.be/0rjGMpM06Eg
Derivative Rules: Proof of Chain Rule: http://youtu.be/tYDDpKzP-VU
Derivative of Power Functions Part 1 - x^n where n is a positive integer: http://youtu.be/-Yv85MZNYgU .

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In this video I go over graphing vertical and horizontal planes in 3D space. I start off with an example on plotting the equations z = 3 and y = 5. The first equation, z = 3, is just a horizontal plane in three-dimensional space and represents all the points on a surface where the z-coordinate is equal to 3. Similarly, the equation y = 5 is the set of all points where the y-coordinate is equal to 5, and which represents a vertical plane. Note that it is important to consider the context in which we are asked to plot these equations. Because if the context was graphing in 2D space, then the equation y = 5 is just a horizontal line. In general: the equation x = k is a vertical plane in the yz-plane, y = k is a vertical plane in the xz-plane, and z = k is a horizontal plane in the xy-plane.

The timestamps of key parts of the video are listed below:

- Example 1a: Horizontal Plane: 0:00
- Example 1b: Vertical Plane: 2:29
- Note on Context: 4:29
- Equations of Vertical or Horizontal Planes: 5:30

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

Related videos:

Vectors and Geometry of Space video series: https://www.youtube.com/playlist?list=PLai3U8-WIK0FjJpwnxwdrOR7L8Ul8VZoZ .

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In this video I go through an example of using the derivative rules I showed in my earlier videos. I look at determining the velocity and distance traveled of a random particle with a distance function. The slope or instantaneous change in distance is the velocity and I use the derivative to derive the velocity equation from the distance function. This is a very useful example and shows how useful calculus is in real world problems.

View video notes on the Hive blockchain: https://peakd.com/hive-128780/@mes/derivative-application-velocity-and-distance

Related Videos:

Completing the Square: http://youtu.be/5uxV6-f_qc0
Definition of Derivative Simple Explanation: http://youtu.be/0rjGMpM06Eg
Derivative of a Constant Function using the Definition of the Derivative: http://youtu.be/YdxeLS2qvp4
Derivative Rules - Constant Multiple, Sum, and Difference Rules: http://youtu.be/_y8Y3Us3q9c
Derivative of y = x^n: Power Rule Part 1: n is positive integer: http//youtu.be/-Yv85MZNYgU
Derivative Rules - Constant Multiple, Sum, and Difference Rules: http://youtu.be/_y8Y3Us3q9c
Derivative Rules - Proof of the Product Rule: http://youtu.be/EIjvGJhDAOk
Derivative Rules - Proof of the Quotient Rule: http://youtu.be/fJcgnLKkISE .

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In this video I show a simple geometric proof of the cosine law. This is very useful in real world applications. Also from the proof of the cosine law you can easily proof the Pythagorean theorem as well by setting the angle equal to 90 degrees.

View video notes on the Hive blockchain: https://peakd.com/hive-128780/@mes/trigonometry-proof-of-law-of-cosines-and-pythagorean-theorem

Related Videos:

Simple Proof of the Pythagorean Theorem: http://youtu.be/yt-EJlbJQp8
Trigonometry Identities: cos(x +/- y) = cos(x)cos(y) -/+ sin(x)sin(y) : http://youtu.be/VuQczhk7HOs
Trigonometry Identities: sin(-x) = - sin(x) and cos(-x) = cos(x): http://youtu.be/tD5EA2SXyFQ
Trigonometry Identities: cos and sin of angles (2p + x) and (p/2 -x): http://youtu.be/EynSefQs308
Trigonometry Identities: sin(x +/- y) = sin(x)cos(y) +/- cos(x)sin(y): http://youtu.be/edtk9thfwbM .

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In this video I provide an introduction to 3D or Three-Dimensional Coordinate Systems. This section is the first part of the chapter on Vectors and the Geometry of Space from my calculus book. A 3D coordinate system is typically defined with the x-, y-, and z-axes in which the direction of the z-axis is determined by the Right-Hand Rule. Visualizing 3D geometry is often easy to use by considering the analogy of a house with 8 rooms (4 on top and 4 on the bottom) and all being in contact with the center point. In the context of 3D- coordinate systems, the 8 rooms are called octants, and the center point is the origin. I illustrate this by plotting several points in 3D. And finally I describe mathematical terms for all real numbers in a rectangular 3D coordinate system, and which is referred to as a Dimensional Rectangular Coordinate System.

The timestamps of key parts of the video are listed below:

- Introduction to Vectors and the Geometry of Space: 0:00
- Three-Dimensional Coordinate Systems: 1:30
- Right-Hand Rule: 3:08
- Octants: 5:00
- Point P in 3D Space: 7:16
- Plotting Points Examples: 13:22
- Dimensional Rectangular Coordinate System: 17:57

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

Related videos:

Vectors and Geometry of Space video series: https://www.youtube.com/playlist?list=PLai3U8-WIK0FjJpwnxwdrOR7L8Ul8VZoZ .

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In this video I prove the two identities, sin(-x) = - sin(x) and cos(-x) = cos(x), which show that cosine is an even function while sine is an odd function.

View video notes on the Hive blockchain: https://peakd.com/hive-128780/@mes/trigonometry-identities-sine-cosine-even-odd-functions

Related Videos:

Trigonometry Identities: cos(x +/- y) = cos(x)cos(y) -/+ sin(x)sin(y) : http://youtu.be/VuQczhk7HOs
Trigonometry: Proof of Law of Cosines (and Pythagorean Theorem): http://youtu.be/wOCYb3Ku6yw
Trigonometry Identities: cos and sin of angles (2p + x) and (p/2 -x): http://youtu.be/EynSefQs308
Trigonometry Identities: sin(x +/- y) = sin(x)cos(y) +/- cos(x)sin(y): http://youtu.be/edtk9thfwbM
Trigonometry: Sine, Cosine and Tan Functions: http://youtu.be/WKTIlF2oWw8 .

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In this video I have uploaded NASA's RS-25 space shuttle engine test which quite literally creates giant clouds! The test is from October 19, 2017 and took place at NASA's Stennis Space Center in Mississippi, USA. The RS-25 engine is produced by the American company Aerojet Rocketdyne and is a liquid-fuel cryogenic rocket engine. The engine burns liquid hydrogen and liquid oxygen, which together form H20 or water, and hence the giant clouds of water vapor being exhausted outwards into the sky. The RS-25 is the latest engine being developed by NASA with hopes of using it in the new Artemis Moon missions program as well as possibly for future Mars missions.

Note that I have sped up the majority of the video by 10X speed to better grasp just how much water vapor is being exhausted outwards.

#RS25 #NASA #Artemis #clouds #Moon

The original video as well as links to more information on the RS-25 and Artemis are shown below:

- Original video: NASA Tests 2nd RS-25 Flight Engine for Space Launch System: https://images.nasa.gov/details/SSC_2017-10-19%20-%20RS-25%20Test and https://youtu.be/lFqfCDEp6iw
- RS-25 engine Wikipedia: https://en.wikipedia.org/wiki/RS-25
- How the RS-25 works: https://archive.ph/wip/4UWWP
- NASA Artemis Moon missions: https://www.nasa.gov/specials/artemis/
- Artemis program Wikipedia: https://en.wikipedia.org/wiki/Artemis_program
- April 5, 2023 RS-25 test: https://images.nasa.gov/details/SSC_2023-04-05_RS-25_Engine_Test and https://youtu.be/I4ldyG9NSeU
- Top Gear films NASA engine tests on October 29, 2010: https://youtu.be/BIpeNs5OWbo

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In this video I derive addition and subtraction identities, cos(x+y) and cos(x-y), using the Law of Cosines and the Pythagorean Theorem. This identity is very useful in all of mathematics and is used in the derivations of many trigonometric derivatives and integrals.

View video notes on the Hive blockchain: https://peakd.com/hive-128780/@mes/trigonometry-identities-cosine-addition-subtraction

Related Videos:

Trigonometry Identities: sin(-x) = - sin(x) and cos(-x) = cos(x): http://youtu.be/tD5EA2SXyFQ
Trigonometry: Proof of Law of Cosines (and Pythagorean Theorem): http://youtu.be/wOCYb3Ku6yw
Trigonometry: Sine, Cosine and Tan Functions: http://youtu.be/WKTIlF2oWw8
Trigonometry Identities: cos and sin of angles (2p + x) and (p/2 -x): http://youtu.be/EynSefQs308
Trigonometry Identities: sin(x +/- y) = sin(x)cos(y) +/- cos(x)sin(y): http://youtu.be/edtk9thfwbM .

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In this video I show some very useful trigonometric identities which basically involve shifting the sine or cosine functions by either 2π or 360 degrees and π/2 or 90 degrees. Make sure you understand these and keep them in mind because I will use these many times in later videos.

View video notes on the Hive blockchain: https://peakd.com/hive-128780/@mes/trigonometry-identities-cos-sin-of-2pi-x-and-pi-2-x

Related Videos:

Trigonometry Identities: sin(-x) = - sin(x) and cos(-x) = cos(x): http://youtu.be/tD5EA2SXyFQ
Trigonometry: Proof of Law of Cosines (and Pythagorean Theorem): http://youtu.be/wOCYb3Ku6yw
Trigonometry: Sine, Cosine and Tan Functions: http://youtu.be/WKTIlF2oWw8
Trigonometry Identities: sin(x +/- y) = sin(x)cos(y) +/- cos(x)sin(y): http://youtu.be/edtk9thfwbM
Exact Trigonometry Ratios Part 1: 0, 30, 45, 60, and 90 Degrees: http://youtu.be/ln03a5KvQAY .

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In this video my brother Mark aka MFA is back and this time with an amazing phone hack for loose headphone jacks. The hack involves taping a small piece of paper behind the phone and then have it overlap into the headphone jack slot. This makes the headphone jack more firm when inserted inside. And when placing your phone case on top, the paper is barely visible. This is a pretty epic phone hack and let me know if this works for you!

Related Videos:

How to Reverse an App Update on your iPhone: https://youtu.be/U7KKb8H4r2E
Replace Samsung S Voice with Google Voice Search on Your Galaxy S4: https://youtu.be/Gcv88jlyQOs
How to Replace the Hard Disk on your Laptop: https://youtu.be/U7qeacrvvbc .

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In this video I use the cos(x - y) identity which I proved in my earlier video to derive a very useful trigonometric identity for sin(x + y) and sin(x -y).

View video notes on the Hive blockchain: https://peakd.com/hive-128780/@mes/trigonometry-identities-sinx-y-sinxcosy-cosxsiny

Related Videos:

Trigonometry Identities: Addition and Subtraction Identities - cos(x + y) and cos(x - y): http://youtu.be/VuQczhk7HOs
Trigonometry Identities: sin(-x) = - sin(x) and cos(-x) = cos(x): http://youtu.be/tD5EA2SXyFQ
Trigonometry: Proof of Law of Cosines (and Pythagorean Theorem): http://youtu.be/wOCYb3Ku6yw
Trigonometry: Sine, Cosine and Tan Functions: http://youtu.be/WKTIlF2oWw8
Trigonometry Identities: cos(2π + x), sin(2π + x), cos(π/2 -x), sin(π/2 -x): http://youtu.be/EynSefQs308 .

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In this video I show a geometric proof of this very useful limit of sin(x)/x as x goes to 0. I will use this in later videos to prove many trig derivatives and integrals.

View video notes on the Hive blockchain: https://peakd.com/hive-128780/@mes/trigonometry-and-limits-limit-of-sinxx-as-x-goes-to-0

Related Videos:

What are Limits? A Simple Explanation: http://youtu.be/FbV7TzlkTZk
One Sided Limits - Plus Examples: http://youtu.be/QA2v_NCej10
Limit Laws - Part 1: Brief Overview: http://youtu.be/qfk5c-43dLg
Limit Laws - Part 1 Examples: http://youtu.be/tL_c7ZKkUBQ
Limit Laws - Part 2 + Brief History of Newton and Limits: http://youtu.be/K5n7y2BvnOs
Limit Laws Part 2 - Examples + Intro to Direct Substitution Property: http://youtu.be/cIjWhV7YdOE
Limits - where Direct Substitution doesn't work: http://youtu.be/L9jrwf-IKeU .

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