# Math Easy Solutions

In this video I go through part 3 of the examples on the Substitution Rule for integrals video series and solve the integral of the function sqrt(1+x^2)·x^5.

View video notes on the Hive blockchain: https://peakd.com/hive-128780/@mes/the-substitution-rule-for-integrals-examples-part-3

Related Videos:

The Substitution Rule for Integrals: Examples Part 2: http://youtu.be/s2BHKUo738Y
The Substitution Rule for Integrals: Examples Part 1: http://youtu.be/ootUKdimk9g
The Substitution Rule for Integrals: http://youtu.be/VsLC-0g6hVg
Differentials Notation in Linear Approximation: http://youtu.be/s0adatWiZg4
Linear Approximation: Differentials Notation: http://youtu.be/NBXH3P_qekU
Indefinite Integrals - Introduction: http://youtu.be/_jl9Zy2sBwk
Fundamental Theorem of Calculus - Recap and Summary: http://youtu.be/CD7-PFg7omc
Antiderivatives - A brief Introduction: http://youtu.be/zDzUHqAUSEw
Foil Method - Simple Proof and Quick Alternative Method: http://youtu.be/tmj_r94D6wQ .

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In this video I go over proving the trigonometry Identity of the integral of tan(x) = ln|sec(x)| + C. In the proof I use the Substitution Rule for integrals to change the variable from x to u.

View video notes on the Hive blockchain: https://peakd.com/hive-128780/@mes/trigonometry-identity-integral-of-tan-x

Related Videos:

The Substitution Rule for Integrals: http://youtu.be/VsLC-0g6hVg
The Substitution Rule for Integrals: Examples Part 1: http://youtu.be/ootUKdimk9g
Trigonometry: Sine, Cosine and Tan Functions: http://youtu.be/WKTIlF2oWw8
Trigonometry: Derivative of sec(x): Proof: http://youtu.be/0sO0TehqMOw
Derivative of Trigonometry Functions: Derivative of cos(x): http://youtu.be/LMjVp-GsrCw
Derivative of y = Log(x) and y = Ln(x): http://youtu.be/5e6MisvvMPE .

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In this video I go over how to remove the background colour of multiple images at a time by converting any given colour into a transparent or alpha layer. To do this you would first need to download Image Magick from http://imagemagick.org. The program relies on the Command Line so you will need to watch the video carefully to see how I input code into the command line. This can save you a lot of time and effort if you have 100s of images to convert so make sure you watch this video!

For converting Single photos I used the code:

convert "photo (1).png" -transparent #374633 "output\photo (1).png"

For multiple photos I used the code:

FOR %G IN ("*.png") DO convert "%G" -transparent #374633 "output\%G"

View video notes on the Hive blockchain: https://peakd.com/photoshop/@mes/remove-background-from-multiple-images

Related Videos:

GIMP Photoshop: Compressing Images: http://youtu.be/srmjXGl8ywQ
GIMP Photoshop Tutorial: http://youtu.be/dSYjD4ov6ao
Edit Multiple Layers in GIMP Photoshop: http://youtu.be/ifQOah2IA7A
GIMP Photoshop Tutorial - Adding a Color Border to Text and Objects: http://youtu.be/fTyLewIVB-8
GIMP: Rounding Text Edges with the Gaussian Blurr Function: http://youtu.be/lEAO6Ajr1AY
GIF Animations with GIMP 2.8 - A very simple step-by-step tutorial: http://youtu.be/vg8IFbHsvng
Whiten Teeth with GIMP 2.8 Photoshop - Simple Tutorial: http://youtu.be/2ENln3t6J94
GIMP Tutorial: Curve Text and Objects with Cage Transform: http://youtu.be/frZj7DmObhY .

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In this video I go over the Substitution Rule for definite integrals and show how converting the limits of integration into the new variable makes integration much easier. I also go over a simple proof for this theorem.

View video notes on the Hive blockchain: https://peakd.com/hive-128780/@mes/the-substitution-rule-for-definite-integrals

Related Videos:

The Substitution Rule for Integrals: http://youtu.be/VsLC-0g6hVg
The Substitution Rule for Integrals: Examples Part 1: http://youtu.be/ootUKdimk9g
The Definite Integral - Brief Introduction: http://youtu.be/vhMP5SKbQjU
Properties of Definite Integrals - Part 1: http://youtu.be/XAel-Zti_Hs
Properties of Definite Integrals - Part 2: http://youtu.be/lxkQmB11qoU .

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In this video I go further into the Substitution Rule for definite integrals by solving a couple of very useful examples.

View video notes on the Hive blockchain: https://peakd.com/hive-128780/@mes/the-substitution-rule-for-definite-integrals-examples

Related Videos:

The Substiution Rule for Definite Integrals: http://youtu.be/AzmYfV1vsbU
The Substitution Rule for Integrals: http://youtu.be/VsLC-0g6hVg
The Substitution Rule for Integrals: Examples Part 1: http://youtu.be/ootUKdimk9g
Antiderivatives - A brief Introduction: http://youtu.be/zDzUHqAUSEw
Antiderivatives - Examples, Part 1: http://youtu.be/Ug9QJT8Z6Bs
Antiderivatives - Examples, Part 2: http://youtu.be/mV1RyLjUOns
Derivative of y = Log(x) and y = Ln(x): http://youtu.be/5e6MisvvMPE
Derivative of y = x^n - Part 2: General Power Rule: http://youtu.be/Ibvu33oh49o .

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In this video I go over the theorem on integrating symmetric functions which greatly simplifies integration. For even functions the integral from -a to a is just two times the integral from 0 to a. For odd functions, the integral from -a to a is simply zero. In this video I also provide a simple proof of this theorem while utilizing the substitution rule for integrals and properties of definite integrals.

View video notes on the Hive blockchain: https://peakd.com/hive-128780/@mes/symmetry-in-integrals-even-vs-odd-integrals

Related Videos:

The Substitution Rule for Definite Integrals: http://youtu.be/AzmYfV1vsbU
The Substitution Rule for Integrals: http://youtu.be/VsLC-0g6hVg
The Definite Integral - Brief Introduction: http://youtu.be/vhMP5SKbQjU
Properties of Definite Integrals - Part 1: http://youtu.be/XAel-Zti_Hs
Properties of Definite Integrals - Part 2: http://youtu.be/lxkQmB11qoU .

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In this video I prove the trigonometry identity tan(-x) = -tan(x) and show how it is considered an odd function with the property f(-x) = -f(x).

View video notes on the Hive blockchain: https://peakd.com/hive-128780/@mes/trigonometry-identity-tan-x-tan-x

Related Videos:

Trigonometric Identity: sin(2x) = 2sin(x)cos(x): http://youtu.be/IaZFDmVtyyU
Trigonometric Identity: cos(2x) = 1 - 2sin^2(x): http://youtu.be/XGk2CpbEl8I
Trigonometry Identity: Integral of tan(x): http://youtu.be/DuBB3xl0bdc
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
Trigonometry Graphing: Sin, Cos, Tan Functions: http://youtu.be/l_3cgeq-YXE
Trigonometry Identities: sin(-x) = - sin(x) and cos(-x) = cos(x): http://youtu.be/tD5EA2SXyFQ
Trigonometry Identities: cos(x +/- y) = cos(x)cos(y) -/+ sin(x)sin(y) : http://youtu.be/VuQczhk7HOs
Trigonometry Identities: cos(2π + x), sin(2π + x), cos(π/2 -x), sin(π/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 go further into symmetry in integrals by going over 2 useful examples. One of the examples is integrating an even function and the other one is integrating an odd function. Using the concept of symmetry we can greatly simplify the integration so make sure to watch this video!

View video notes on the Hive blockchain: https://peakd.com/hive-128780/@mes/symmetry-in-integrals-examples

Related Videos:

Symmetry in Integrals: Even vs. Odd Integrals: http://youtu.be/NbwJOQnT1eM
Trigonometry Identity: tan(-x) = tan(x): http://youtu.be/nvaP_091Ar8
Antiderivatives - A brief Introduction: http://youtu.be/zDzUHqAUSEw
Antiderivatives - Examples, Part 1: http://youtu.be/Ug9QJT8Z6Bs .

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In this video I go over logarithms but this define natural logarithms using integrals through the Fundamental Theorem of Calculus. In my previous videos I went over the definition of the logarithm as an inverse to the exponential function but the definition I go over in this video is more concrete and better suited when dealing with derivatives and integrals.

View video notes on the Hive blockchain: https://peakd.com/hive-128780/@mes/natural-logarithms-defined-as-integrals-introduction

Related Videos:

Inverse Functions - f-1(x) - An Introduction: http://youtu.be/qIqj3oKwFi8
Logarithms and their Properties - An Introduction: http://youtu.be/AZ6KKym19gI
Natural Logarithms, Log base 10, and Some Examples Using Logs: http://youtu.be/XRSkMk5L3pk
Inverse Functions Part 2: One to One Functions and the Horizontal Line Test: http://youtu.be/iSNuplUWrbA
Derivative of y = Log(x) and y = Ln(x): http://youtu.be/5e6MisvvMPE
Exponential Functions and Graphing: http://youtu.be/FFNwh8KR9-Q
Fundamental Theorem of Calculus - Introduction and Part 1 of the Theorem: http://youtu.be/3o8Q6UJzJyk
Fundamental Theorem of Calculus - Intro and Proof of Part 2 of the Theorem: http://youtu.be/yuIl-BPQHss .

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In this video I go over some laws of logarithms and prove the law ln(xy) = ln(x) + ln(y). In my earlier videos on logs and their properties I proved this same law but using the definition of logarithms as the inverse of the exponential function. In this video, however, I show how to prove logarithm laws using integrals and derivatives like I explained in my last video.

View video notes on the Hive blockchain: https://peakd.com/hive-128780/@mes/laws-of-logarithms-ln-xy-ln-x-ln-y

Related Videos:

Natural Logarithmns Defined as Integrals: Introduction: http://youtu.be/M-N2PQ5UZns
Inverse Functions - f-1(x) - An Introduction: http://youtu.be/qIqj3oKwFi8
Logarithms and their Properties - An Introduction: http://youtu.be/AZ6KKym19gI
Natural Logarithms, Log base 10, and Some Examples Using Logs: http://youtu.be/XRSkMk5L3pk
Fundamental Theorem of Calculus - Introduction and Part 1 of the Theorem: http://youtu.be/3o8Q6UJzJyk

In this video I go over another logarithm law and prove that ln(x^r) = r·ln(x) where r is a rational number. In this proof I use derivatives and integrals as opposed to my earlier video which I proved using the definition of ln(x) as the inverse of the exponential function, e^x.

View video notes on the Hive blockchain: https://peakd.com/hive-128780/@mes/laws-of-logarithms-ln-x-r-r-ln-x

Related Videos:

Natural Logarithmns Defined as Integrals: Introduction: http://youtu.be/M-N2PQ5UZns
Laws of Logarithms: ln(x*y) = ln(x) + ln(y): http://youtu.be/2SCZzFy2b2s
Laws of Logarithms: ln(x/y) = ln(x) - ln(y): http://youtu.be/GXkyY9EmDGM
Fundamental Theorem of Calculus - Introduction and Part 1 of the Theorem: http://youtu.be/3o8Q6UJzJyk
Inverse Functions - f-1(x) - An Introduction: http://youtu.be/qIqj3oKwFi8
Logarithms and their Properties - An Introduction: http://youtu.be/AZ6KKym19gI
Natural Logarithms, Log base 10, and Some Examples Using Logs: http://youtu.be/XRSkMk5L3pk

In this video I go over step by step in graphing the natural logarithm function, y = ln(x) by first finding the limits at infinity and zero from the right side.

View video notes on the Hive blockchain: https://peakd.com/hive-128780/@mes/graph-of-natural-logarithm-y-ln-x

Related Videos:

Natural Logarithmns Defined as Integrals: Introduction: http://youtu.be/M-N2PQ5UZns
Laws of Logarithms: ln(x*y) = ln(x) + ln(y): http://youtu.be/2SCZzFy2b2s
Laws of Logarithms: ln(x/y) = ln(x) - ln(y): http://youtu.be/GXkyY9EmDGM
Laws of Logarithms: ln(x^r) = r*ln(x): http://youtu.be/mvJK8lQcdTY
Derivative of y = Log(x) and y = Ln(x): http://youtu.be/5e6MisvvMPE
Logarithms and their Properties - An Introduction: http://youtu.be/AZ6KKym19gI
Natural Logarithms, Log base 10, and Some Examples Using Logs: http://youtu.be/XRSkMk5L3pk

In this video I go over the natural exponential function y = e^x and define it strictly from looking at the inverse of the natural logarithmic function y = ln(x). The natural exponential function is one of the most widely used and important function in all of mathematics so it is important to understand the definition of it.

View video notes on the Hive blockchain: https://peakd.com/hive-128780/@mes/natural-exponential-function-y-e-x

Related Videos:

Natural Logarithmns Defined as Integrals: Introduction: http://youtu.be/M-N2PQ5UZns
Laws of Logarithms: ln(x*y) = ln(x) + ln(y): http://youtu.be/2SCZzFy2b2s
Laws of Logarithms: ln(x/y) = ln(x) - ln(y): http://youtu.be/GXkyY9EmDGM
Laws of Logarithms: ln(x^r) = r*ln(x): http://youtu.be/mvJK8lQcdTY
Graph of Natural Logarithm y = ln(x): http://youtu.be/GZkwYVdbluc
Exponential Functions and Graphing: http://youtu.be/FFNwh8KR9-Q
The Number e - A Brief Introduction and it's Derivative: http://youtu.be/o_s_YYD6v3g
A Simple Proof that the number e = 2.71828183...: http://youtu.be/NDb80lP3Tr4
Inverse Functions - f-1(x) - An Introduction: http://youtu.be/qIqj3oKwFi8
Logarithms and their Properties - An Introduction: http://youtu.be/AZ6KKym19gI

In this video I go over how to graph the natural exponential function or y = e^x in a step by step fashion. This is one of the most important functions in all of mathematics so it is important to understand the graph of it!

View video notes on the Hive blockchain: https://peakd.com/hive-128780/@mes/graphing-the-natural-exponential-function-y-e-x

Related Videos:

Natural Exponential Function: y = e^x: http://youtu.be/vGsOA2eqkig
e^π is greater than π^e - A "Simple" Proof for Math Enthusiasts ;): http://youtu.be/9NW30GlCUlg
Derivative of y = a^x and e^x: Using the Chain Rule: http://youtu.be/uXwwgMzikao .

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In this video I go over the exponential function y = e^x and write it as a limit. Earlier I proved the number e as a limit which can be simply considered as e^1.

View video notes on the Hive blockchain: https://hive.blog/hive-128780/@mes/e-x-as-a-limit

Related Videos:

The Number e - A Brief Introduction and it's Derivative: http://youtu.be/o_s_YYD6v3g
A Simple Proof that the number e = 2.71828183...: http://youtu.be/NDb80lP3Tr4
Limit Laws - Part 1: Brief Overview: http://youtu.be/qfk5c-43dLg
Melinda and Tania amazing Band!: http://youtu.be/N5OhLvZ4WXM
Limit Laws - Part 1 Examples: http://youtu.be/tL_c7ZKkUBQ
Limit Laws - Part 2 + Brief History of Newton and Limits: http://youtu.be/K5n7y2BvnOs
What are Limits? A Simple Explanation: http://youtu.be/FbV7TzlkTZk
Natural Exponential Function: y = e^x: http://youtu.be/vGsOA2eqkig
e^π is greater than π^e - A "Simple" Proof for Math Enthusiasts ;): http://youtu.be/9NW30GlCUlg .

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In this video I go over laws of exponents and prove that e^(x+y) = e^x·e^y. In this prove I first consider the natural logarithm function, ln(x), and use log properties and laws to simplify and breakdown the function in proofing the exponent law.

View video notes on the Steem blockchain: https://steemit.com/hive-128780/@mes/5eulbh-laws-of-exponents-e-x-y-e-x-e-y

Related Videos:

Laws of Logarithms: ln(x*y) = ln(x) + ln(y): http://youtu.be/2SCZzFy2b2s
Laws of Logarithms: ln(x/y) = ln(x) - ln(y): http://youtu.be/GXkyY9EmDGM
Laws of Logarithms: ln(x^r) = r*ln(x): http://youtu.be/mvJK8lQcdTY
Logarithms and their Properties - An Introduction: http://youtu.be/AZ6KKym19gI
Natural Logarithmns Defined as Integrals: Introduction: http://youtu.be/M-N2PQ5UZns
Inverse Functions Part 2: One to One Functions and the Horizontal Line Test: http://youtu.be/iSNuplUWrbA .

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In this video I go over further into the laws of exponents and prove that e^(x-y) = e^x/e^y by using the natural logarithm function and using the log rules and properties to simplify and breakdown the function to obtain the exponent law.

View video notes on the Steem blockchain: https://steemit.com/hive-128780/@mes/laws-of-exponents-e-x-y-e-x-e-y

Related Videos:

Laws of Exponents: e^(x+y) = e^x*e^y: http://youtu.be/8z4zpIjsVjk
Laws of Logarithms: ln(x*y) = ln(x) + ln(y): http://youtu.be/2SCZzFy2b2s
Laws of Logarithms: ln(x/y) = ln(x) - ln(y): http://youtu.be/GXkyY9EmDGM
Laws of Logarithms: ln(x^r) = r*ln(x): http://youtu.be/mvJK8lQcdTY
Logarithms and their Properties - An Introduction: http://youtu.be/AZ6KKym19gI
Natural Logarithmns Defined as Integrals: Introduction: http://youtu.be/M-N2PQ5UZns
Inverse Functions Part 2: One to One Functions and the Horizontal Line Test: http://youtu.be/iSNuplUWrbA .

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In this video I go over another law of exponent and prove that (e^x)^r = e^(rx) where r is a rational number. In this proof I use a similar method as in my previous videos on the laws of exponents and utilize the natural logarithm function, ln(x), in simplifying and breaking down the function to obtain the exponent law.

View video notes on the Steem blockchain: https://steemit.com/hive-128780/@mes/laws-of-exponents-e-x-r-e-rx

Related Videos:

Laws of Exponents: e^(x+y) = e^x*e^y: http://youtu.be/8z4zpIjsVjk
Laws of Exponents: e^(x-y) = e^x/e^y: http://youtu.be/DpGjtkyQVPI
Laws of Logarithms: ln(x*y) = ln(x) + ln(y): http://youtu.be/2SCZzFy2b2s
Laws of Logarithms: ln(x/y) = ln(x) - ln(y): http://youtu.be/GXkyY9EmDGM
Laws of Logarithms: ln(x^r) = r*ln(x): http://youtu.be/mvJK8lQcdTY
Logarithms and their Properties - An Introduction: http://youtu.be/AZ6KKym19gI
Natural Logarithmns Defined as Integrals: Introduction: http://youtu.be/M-N2PQ5UZns
Inverse Functions Part 2: One to One Functions and the Horizontal Line Test: http://youtu.be/iSNuplUWrbA .

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In #MESExperiments 26 I continue further in experimenting with attaching tape to a gyroscope’s rotor to increase spin friction and this time show that doing so makes it possible for a gyroscope to rise from a very steep 72° angle from the vertical. For this particular gyroscope, the previous steepest angle I had been able to make rise, until now, was about 61° from the vertical (see Experiment 12) but was done so without increasing spin friction. Overall, the steepest angle I had made a gyroscope rise from was 74°, see Experiment 13, and it was done so with a different gyroscope which interestingly had naturally more friction in the bearings than that of the precession gyro in this particular Experiment 26. Thus, in both cases it appears that increasing spin friction, even by applying a tiny amount of tape to the rotor, can increase both the rising rate and the maximum angle at which it is possible to get to a fully risen position.

Interestingly, the casing rotation in this experiment was very rapid, and near the risen position the gyroscope becomes very chaotic as it tilts and rises in multiple oscillating sequences. I have included a slow motion footage of this chaotic rising at the end of this video. Had the casing not rotated, I suspect it would just make the rising more smooth since in Experiment 13 the gyro rose from 74° without rotating the casing.

The original unedited experiment can be view here: https://youtu.be/37ewZYMGGFE

The gyroscope weighs 150.69 g and the added tape friction is 0.08 g, which is about 0.05% of the gyroscope’s weight.

The full experiment results along with the screenshots of the experiment progression can be viewed here: https://steemit.com/hive-128780/@mes/mesexperiments-26-increasing-spin-friction-can-make-a-gyroscope-rise-from-a-very-steep-72-angle

Stay Tuned for #MESExperiments 27…

Related Videos:

🔬#MESExperiments 25: Increasing Spin Friction Can Make Gyroscopes Rise Much Faster: https://steemit.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 #MAGIC: https://steemit.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! #FreeEnergy: https://steemit.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://steemit.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://steemit.com/antigravity/@mes/antigravity-part-6-video-1-objects-in-rotation-defy-mainstream-physics-mes-duality-concept

#MESExperiments Video Series: https://steemit.com/mesexperiments/@mes/list
DRAFT #MESExperiments Video Series: https://mes.fm/experiments-draft
#MESScience Video Series: https://steemit.com/science/@mes/tutorials
#AntiGravity Video Series: https://steemit.com/antigravity/@mes/series
#FreeEnergy Video Series: https://mes.fm/freeenergy-playlist .

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In this video I go over how to use the inverse function method to easily find the derivative of e^x. Using the fact that e^x is the inverse of the ln(x) or natural log function, it can be proved very easily using implicit differentiation and the chain rule that the derivative of e^x is simply itself, e^x.

View video notes on the Steem blockchain: https://steemit.com/hive-128780/@mes/derivative-of-e-x-using-the-inverse-function-method

Related Videos:

Derivative of y = a^x and e^x: Using the Chain Rule: http://youtu.be/uXwwgMzikao
Derivative Rules: Proof of Chain Rule: http://youtu.be/tYDDpKzP-VU
Definition of Derivative Simple Explanation: http://youtu.be/0rjGMpM06Eg
Implicit Differentiation - A Brief Introduction: http://youtu.be/12OY1b3DYHQ
Derivative of y = Log(x) and y = Ln(x): http://youtu.be/5e6MisvvMPE
Inverse Functions - f-1(x) - An Introduction: http://youtu.be/qIqj3oKwFi8 .

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In this video I go over general exponential functions and look at exponential functions with base a, such as a^x and derive some exponential and logarithmic laws that can be used as the basis for exponent laws that I will prove in later videos. So stay tuned!

View video notes on the Steem blockchain: https://steemit.com/hive-128780/@mes/general-exponential-functions

Related Videos:

Derivative of e^x using the Inverse Function Method: http://youtu.be/vjEnL9Ppmsg
Laws of Exponents: e^(x+y) = e^x*e^y: http://youtu.be/8z4zpIjsVjk
Laws of Exponents: e^(x-y) = e^x/e^y: http://youtu.be/DpGjtkyQVPI
Laws of Exponents: (e^x)^r = e^(rx): http://youtu.be/0pIdZdg3YTg
Natural Logarithmns Defined as Integrals: Introduction: http://youtu.be/M-N2PQ5UZns
Laws of Logarithms: ln(x*y) = ln(x) + ln(y): http://youtu.be/2SCZzFy2b2s
Laws of Logarithms: ln(x/y) = ln(x) - ln(y): http://youtu.be/GXkyY9EmDGM
Laws of Logarithms: ln(x^r) = r*ln(x): http://youtu.be/mvJK8lQcdTY .

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In this video I go over a simple proof of the general law of exponent with base a: a^(x+y) = (a^x)(a^y). In the proof I use the definition a^x = e^(xlna) and the laws of exponents with base e which I covered in my earlier videos.

View video notes on the Steem blockchain: https://steemit.com/hive-128780/@mes/laws-of-exponents-a-x-y-a-x-a-y

Related Videos:

General Exponential Functions: http://youtu.be/c4R2urO1GCU
Laws of Exponents: e^(x+y) = e^x*e^y: http://youtu.be/8z4zpIjsVjk
Laws of Exponents: e^(x-y) = e^x/e^y: http://youtu.be/DpGjtkyQVPI
Laws of Exponents: (e^x)^r = e^(rx): http://youtu.be/0pIdZdg3YTg .

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In this video I go over a simple proof of the general exponent law with base a: a^(x-y) = (a^x)/(a^y). In the proof I utilize exponent laws with base e as well the definition which I covered in my earlier video, a^x = e^(x ln a).

View video notes on the Steem blockchain: https://steemit.com/hive-128780/@mes/law-of-exponents-a-x-y-a-x-a-y

Related Videos:

Laws of Exponents: a^(x+y) = (a^x)(a^y): http://youtu.be/qgDPsFD_vSA
General Exponential Functions: http://youtu.be/c4R2urO1GCU
Laws of Exponents: e^(x+y) = e^x*e^y: http://youtu.be/8z4zpIjsVjk
Laws of Exponents: e^(x-y) = e^x/e^y: http://youtu.be/DpGjtkyQVPI
Laws of Exponents: (e^x)^r = e^(rx): http://youtu.be/0pIdZdg3YTg .

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In this video I go over the general law of exponents with base a: (a^x)^y = a^(xy). In the proof I utilize the laws which I covered earlier: a^x = e^(x ln a) and ln(a^x) = x ln a.

View video notes on the Steem blockchain: https://steemit.com/hive-128780/@mes/law-of-exponents-a-x-y-a-xy

Related Videos:

Law of Exponents a^(x-y) = (a^x)/(a^y): http://youtu.be/djwpuNtmYJY
Laws of Exponents: a^(x+y) = (a^x)(a^y): http://youtu.be/qgDPsFD_vSA
General Exponential Functions: http://youtu.be/c4R2urO1GCU
Laws of Exponents: e^(x+y) = e^x*e^y: http://youtu.be/8z4zpIjsVjk
Laws of Exponents: e^(x-y) = e^x/e^y: http://youtu.be/DpGjtkyQVPI
Laws of Exponents: (e^x)^r = e^(rx): http://youtu.be/0pIdZdg3YTg .

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In #MESExperiments 25 I continue experimenting with different ways of making a gyroscope rise faster, and this time I have discovered that simply attaching tape to the rotor axis to increase spin friction can dramatically increase the rising rate! And even increase the rising rate by over 5.5X+ but at the expense of greatly cutting short the total balancing, sleep, and spin time. Also of note is that with the tape friction the gyroscope’s outer casing also rotated throughout the experiment, whereas the low friction gyro comparison test had a much smoother but longer rising time. It appears that the rate at which the gyro spin speed lowers due to friction may in fact be the key driver, along with minimizing any precessional friction as in Experiment 1, to enabling gyroscopes to rise upwards as well as governing the rate of rising.

Although increasing spin friction can increase the rate of rising, it is important to keep in mind my earlier Experiment 3 which shows that a gyroscope precesses, rises, and overall exerts "inertial forces" but with no additional loss of spin speed. In other words, there is no direct conversion from the loss of spin speed due to friction to a gyroscope rising; or no direct conversion from the conventional view of "kinetic energy" to "potential energy". Thus, had the gyro rose or not, the overall spin time should be the same for the same friction environment.

The original unedited experiments are shown in the links below:

- No tape friction added: https://youtu.be/YvkGfYcISZo

The gyroscope weighs 150.69 g and the added tape friction is 0.08 g, which is about 0.05% of the gyroscope’s weight.

The full experiment results along with the screenshots of the experiment progression can be viewed here: https://steemit.com/hive-128780/@mes/mesexperiments-25-increasing-spin-friction-can-make-gyroscopes-rise-much-faster

Stay Tuned for #MESExperiments 26…

Related Videos:

🔬#MESExperiments 23: Added Weight Doesn’t Always Make Gyroscopes Rise Faster #Nuance: https://steemit.com/mesexperiments/@mes/mesexperiments-23-added-weight-doesn-t-always-make-gyroscopes-rise-faster-nuance

🔬#MESExperiments 22: Added Weight Can Make a Gyroscope Rise Faster (No Casing Rotation): https://steemit.com/mesexperiments/@mes/mesexperiments-22-added-weight-can-make-a-gyroscope-rise-faster-no-casing-rotation

🔬#MESExperiments 21: Added Weight Can Make a Gyroscope Rise Faster #Magic: https://steemit.com/mesexperiments/@mes/mesexperiments-21-added-weight-can-make-a-gyroscope-rise-faster-magic

🔬#MESExperiments 3: Gyroscopes Exert 'Inertial Forces' With ZERO Loss of Spin Rate! #FreeEnergy: https://steemit.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://steemit.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://steemit.com/antigravity/@mes/antigravity-part-6-video-1-objects-in-rotation-defy-mainstream-physics-mes-duality-concept

#MESExperiments Video Series: https://steemit.com/mesexperiments/@mes/list
DRAFT #MESExperiments Video Series: https://mes.fm/experiments-draft
#MESScience Video Series: https://steemit.com/science/@mes/tutorials
#AntiGravity Video Series: https://steemit.com/antigravity/@mes/series
#FreeEnergy Video Series: https://mes.fm/freeenergy-playlist .

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