A quantity is subject to exponential decay if it decreases at a rate proportional to its current value. Symbolically, this process can be expressed by the following differential equation, where N is the quantity and λ (lambda) is a positive rate called the exponential decay constant:
The solution to this equation (see derivation below) is:
Here N(t) is the quantity at time t, and N0 = N(0) is the initial quantity, i.e. the quantity at time t = 0.
Measuring rates of decay
Mean lifetime
If the decaying quantity, N(t), is the number of discrete elements in a certain set, it is possible to compute the average length of time that an element remains in the set. This is called the mean lifetime (or simply the lifetime or the exponential time constant), τ, and it can be shown that it relates to the decay rate, λ, in the following way:
The mean lifetime can be looked at as a "scaling time", because we can write the exponential decay equation in terms of the mean lifetime, τ, instead of the decay constant, λ:
Exponential Growth and Decay Word Problems & Functions - Algebra & Precalculus
This algebra and precalculus video tutorial explains how to solve exponential growth and decay word problems. It provides the formulas and equations / functions that you need to solve it.
Algebra For Beginners: https://www.youtube.com/watch?v=MHeirBPOI6w
Logarithms - The Easy Way!
https://www.youtube.com/watch?v=kqVpPSzkTYA
Solving Exponential Equations:
https://www.youtube.com/watch?v=9tutJ5xrRwg
Solving Logarithmic Equations:
https://www.youtube.com/watch?v=fnhFneOz6n8
Logarithmic Equations - Different Bases:
https://www.youtube.com/watch?v=XvwPB21Gm9A
__________________________________
Compound Interest Word Problems:
https://www.yout...
published: 06 Dec 2016
Exponential Decay
Exponential Decay as a measure of growth
published: 05 Sep 2022
Ex: Exponential Decay Function - Half Life
This video explains how to determine an exponential decay function from given information. Then it explains how to determine when a certain level of decay will be reached and how to determine half-life.
Library: http://mathispower4u.com
Search by Topic: http://mathispower4u.wordpress.com
published: 30 Dec 2011
Exponential Growth and Decay Calculus, Relative Growth Rate, Differential Equations, Word Problems
This calculus video tutorial focuses on exponential growth and decay. it shows you how to derive a general equation / formula for population growth starting with a differential equation.
Introduction to Limits: https://www.youtube.com/watch?v=YNstP0ESndU
Continuity & Differentiability: https://www.youtube.com/watch?v=fml0-ELYLaE&
Calculus 1 - Derivatives: https://www.youtube.com/watch?v=5yfh5cf4-0w
Introduction to Related Rates: https://www.youtube.com/watch?v=I9mVUo-bhM8
Local Maximum & Minimum: https://www.youtube.com/watch?v=WCq3sRzsJfs
L'Hopital's Rule: https://www.youtube.com/watch?v=Gh48aOvWcxw
Curve Sketching With Derivatives: https://www.youtube.com/watch?v=JTVN...
published: 04 Feb 2017
Exponential Growth: a Commonsense Explanation.
published: 18 Sep 2016
Exponential Decay / Finding Half Life
Thanks to all of you who support me on Patreon. You da real mvps! $1 per month helps!! :) https://www.patreon.com/patrickjmt !! Exponential Decay / Finding Half Life - In this video, I find the half life of a substance that is decreasing annually by 4%.
For more free math videos, visit http://PatrickJMT.com
published: 26 Mar 2009
Find the Laplace transform of the sine function (Laplace transform of sin(t)).
To calculate the Laplace transform of sin(t), we take the integral on 0 to infinity of e^-st*sin(t), where s is a constant with respect to the t integral.
This Laplace transform combines improper integrals with the integration by parts looping trick, which made it a perfect bonus problem for my last Calculus II exam!
To proceed with the integral, we apply integration by parts, letting u=e^-st and dv=sin(t)dt. This results in a second integral with a cosine instead of a sine, and we let u=e^-st and dv=cos(t)dt. After cleaning things up, we find a copy of the original integral on the right hand side of our work.
We evaluate the leftover terms from 0 to infinity, noting that e^(-infinity) is unambiguously zero, so the integral converges if s is greater than 0. These leftover terms yiel...
published: 27 Oct 2024
Ex: Exponential Decay Function with Logarithms
This video explains how to determine an exponential decay function from given information. Then it explains how to determine when a certain level of decay will be reached using logarithms.
Library: http://mathispower4u.com
Search by Topic: http://mathispower4u.wordpress.com
This algebra and precalculus video tutorial explains how to solve exponential growth and decay word problems. It provides the formulas and equations / function...
This algebra and precalculus video tutorial explains how to solve exponential growth and decay word problems. It provides the formulas and equations / functions that you need to solve it.
Algebra For Beginners: https://www.youtube.com/watch?v=MHeirBPOI6w
Logarithms - The Easy Way!
https://www.youtube.com/watch?v=kqVpPSzkTYA
Solving Exponential Equations:
https://www.youtube.com/watch?v=9tutJ5xrRwg
Solving Logarithmic Equations:
https://www.youtube.com/watch?v=fnhFneOz6n8
Logarithmic Equations - Different Bases:
https://www.youtube.com/watch?v=XvwPB21Gm9A
__________________________________
Compound Interest Word Problems:
https://www.youtube.com/watch?v=Hn0eLcOSQGw
Interest Compounded Continuously:
https://www.youtube.com/watch?v=Ln97Hd7AiDc
Population Growth Word Problems:
https://www.youtube.com/watch?v=k4LLdFFLRmQ
Logarithms Practice Problems:
https://www.youtube.com/watch?v=7DVbQKI600k
Where does "e" come from?
https://www.youtube.com/watch?v=pDFcu_wLOzo
____________________________________
Logistic Growth Function:
https://www.youtube.com/watch?v=JgMvB22XQs0
Newton's Law of Cooling:
https://www.youtube.com/watch?v=ejEXSjdMpck
Final Exams and Video Playlists:
https://www.video-tutor.net/
Full-Length Videos and Worksheets:
https://www.patreon.com/MathScienceTutor/collections
This algebra and precalculus video tutorial explains how to solve exponential growth and decay word problems. It provides the formulas and equations / functions that you need to solve it.
Algebra For Beginners: https://www.youtube.com/watch?v=MHeirBPOI6w
Logarithms - The Easy Way!
https://www.youtube.com/watch?v=kqVpPSzkTYA
Solving Exponential Equations:
https://www.youtube.com/watch?v=9tutJ5xrRwg
Solving Logarithmic Equations:
https://www.youtube.com/watch?v=fnhFneOz6n8
Logarithmic Equations - Different Bases:
https://www.youtube.com/watch?v=XvwPB21Gm9A
__________________________________
Compound Interest Word Problems:
https://www.youtube.com/watch?v=Hn0eLcOSQGw
Interest Compounded Continuously:
https://www.youtube.com/watch?v=Ln97Hd7AiDc
Population Growth Word Problems:
https://www.youtube.com/watch?v=k4LLdFFLRmQ
Logarithms Practice Problems:
https://www.youtube.com/watch?v=7DVbQKI600k
Where does "e" come from?
https://www.youtube.com/watch?v=pDFcu_wLOzo
____________________________________
Logistic Growth Function:
https://www.youtube.com/watch?v=JgMvB22XQs0
Newton's Law of Cooling:
https://www.youtube.com/watch?v=ejEXSjdMpck
Final Exams and Video Playlists:
https://www.video-tutor.net/
Full-Length Videos and Worksheets:
https://www.patreon.com/MathScienceTutor/collections
This video explains how to determine an exponential decay function from given information. Then it explains how to determine when a certain level of decay will...
This video explains how to determine an exponential decay function from given information. Then it explains how to determine when a certain level of decay will be reached and how to determine half-life.
Library: http://mathispower4u.com
Search by Topic: http://mathispower4u.wordpress.com
This video explains how to determine an exponential decay function from given information. Then it explains how to determine when a certain level of decay will be reached and how to determine half-life.
Library: http://mathispower4u.com
Search by Topic: http://mathispower4u.wordpress.com
This calculus video tutorial focuses on exponential growth and decay. it shows you how to derive a general equation / formula for population growth starting wi...
This calculus video tutorial focuses on exponential growth and decay. it shows you how to derive a general equation / formula for population growth starting with a differential equation.
Introduction to Limits: https://www.youtube.com/watch?v=YNstP0ESndU
Continuity & Differentiability: https://www.youtube.com/watch?v=fml0-ELYLaE&
Calculus 1 - Derivatives: https://www.youtube.com/watch?v=5yfh5cf4-0w
Introduction to Related Rates: https://www.youtube.com/watch?v=I9mVUo-bhM8
Local Maximum & Minimum: https://www.youtube.com/watch?v=WCq3sRzsJfs
L'Hopital's Rule: https://www.youtube.com/watch?v=Gh48aOvWcxw
Curve Sketching With Derivatives: https://www.youtube.com/watch?v=JTVNUdL7sWs
Newton's Method: https://www.youtube.com/watch?v=-5e2cULI3H8
Optimization Problems: https://www.youtube.com/watch?v=lx8RcYcYVuU
____________________________________________________________________________________
Antiderivatives: https://www.youtube.com/watch?v=xaCPDMEkbig
Basic Integration Problems: https://www.youtube.com/watch?v=zOxaUlRkFG0
Indefinite Integral: https://www.youtube.com/watch?v=JTFMeSCxgcA
Definite Integral: https://www.youtube.com/watch?v=Gc3QvUB0PkI
Differential Equations: https://www.youtube.com/watch?v=H5tD_NtPDuU
Properties of Definite Integrals: https://www.youtube.com/watch?v=QcHz3h81U-s
Rectilinear Motion Problems: https://www.youtube.com/watch?v=LBmET4sH460
Sigma Notation - Calculus: https://www.youtube.com/watch?v=XJkIaw2e1Pw
Riemann Sums - Area: https://www.youtube.com/watch?v=YTKQswb60Pw
The Midpoint Rule: https://www.youtube.com/watch?v=5XreKMJDJsg
____________________________________________________________________________________
Finding Area - Limit Definition: https://www.youtube.com/watch?v=ctEpKZyxqFU
Definite Integrals - Geometry: https://www.youtube.com/watch?v=ghxEOz9rmwE
Fundamental Theorem - Part 1: https://www.youtube.com/watch?v=aeB5BWY0RlE
Fundamental Theorem - Part 2: https://www.youtube.com/watch?v=ns8N1UuXl4w
Net Change Theorem: https://www.youtube.com/watch?v=df1Qr8pepx0
Mean Value Theorem - Integrals: https://www.youtube.com/watch?v=bLeglo-c5Tw
Average Value of a Function: https://www.youtube.com/watch?v=MB1xDNKimNc
U-Substitution - Indefinite Integrals: https://www.youtube.com/watch?v=sdYdnpYn-1o
U-Substitution - Definite Integrals: https://www.youtube.com/watch?v=tM4RWc9ryx0
1st Order Differential Equations: https://www.youtube.com/watch?v=C7nuJcJriWM
_____________________________________________________________________________________
Initial Value Problem: https://www.youtube.com/watch?v=kwGukY_2qWQ
Area Between Two Curves: https://www.youtube.com/watch?v=kgg5Rspf1Js
Disk and Washer Method: https://www.youtube.com/watch?v=SAHSVg7Jw_A
Volume By The Shell Method: https://www.youtube.com/watch?v=D5sT1br9soI
Volume By Cross Sections: https://www.youtube.com/watch?v=qMXPnfx2MQM
Arc Length Calculus Problems: https://www.youtube.com/watch?v=DNDAwWIL5FY
Surface Area of Revolution: https://www.youtube.com/watch?v=lQM-0Nqs9Pg
Work Problems - Calculus: https://www.youtube.com/watch?v=TLw8xbmnY3c
Integration By Parts: https://www.youtube.com/watch?v=sWSLLO3DS1I
Trigonometric Integrals: https://www.youtube.com/watch?v=3pXALn2ovIE
_____________________________________________________________________________________
GPA Calculator: https://www.youtube.com/watch?v=qYHsThZWydY
Save Money In College: https://www.youtube.com/watch?v=yNO02qfMSwI
SAT Test Prep: https://www.youtube.com/watch?v=fTGuTEQCsZY
ACT Test Prep: https://www.youtube.com/watch?v=SsA7rZ8kczM
GRE Math Test Prep: https://www.youtube.com/watch?v=z6lbrzaCbdk
Calculus 1 Final Exam Review: https://www.youtube.com/watch?v=WmBzmHru78w
Full Length Exams + Worksheets: https://bit.ly/4990rzU
This calculus video tutorial focuses on exponential growth and decay. it shows you how to derive a general equation / formula for population growth starting with a differential equation.
Introduction to Limits: https://www.youtube.com/watch?v=YNstP0ESndU
Continuity & Differentiability: https://www.youtube.com/watch?v=fml0-ELYLaE&
Calculus 1 - Derivatives: https://www.youtube.com/watch?v=5yfh5cf4-0w
Introduction to Related Rates: https://www.youtube.com/watch?v=I9mVUo-bhM8
Local Maximum & Minimum: https://www.youtube.com/watch?v=WCq3sRzsJfs
L'Hopital's Rule: https://www.youtube.com/watch?v=Gh48aOvWcxw
Curve Sketching With Derivatives: https://www.youtube.com/watch?v=JTVNUdL7sWs
Newton's Method: https://www.youtube.com/watch?v=-5e2cULI3H8
Optimization Problems: https://www.youtube.com/watch?v=lx8RcYcYVuU
____________________________________________________________________________________
Antiderivatives: https://www.youtube.com/watch?v=xaCPDMEkbig
Basic Integration Problems: https://www.youtube.com/watch?v=zOxaUlRkFG0
Indefinite Integral: https://www.youtube.com/watch?v=JTFMeSCxgcA
Definite Integral: https://www.youtube.com/watch?v=Gc3QvUB0PkI
Differential Equations: https://www.youtube.com/watch?v=H5tD_NtPDuU
Properties of Definite Integrals: https://www.youtube.com/watch?v=QcHz3h81U-s
Rectilinear Motion Problems: https://www.youtube.com/watch?v=LBmET4sH460
Sigma Notation - Calculus: https://www.youtube.com/watch?v=XJkIaw2e1Pw
Riemann Sums - Area: https://www.youtube.com/watch?v=YTKQswb60Pw
The Midpoint Rule: https://www.youtube.com/watch?v=5XreKMJDJsg
____________________________________________________________________________________
Finding Area - Limit Definition: https://www.youtube.com/watch?v=ctEpKZyxqFU
Definite Integrals - Geometry: https://www.youtube.com/watch?v=ghxEOz9rmwE
Fundamental Theorem - Part 1: https://www.youtube.com/watch?v=aeB5BWY0RlE
Fundamental Theorem - Part 2: https://www.youtube.com/watch?v=ns8N1UuXl4w
Net Change Theorem: https://www.youtube.com/watch?v=df1Qr8pepx0
Mean Value Theorem - Integrals: https://www.youtube.com/watch?v=bLeglo-c5Tw
Average Value of a Function: https://www.youtube.com/watch?v=MB1xDNKimNc
U-Substitution - Indefinite Integrals: https://www.youtube.com/watch?v=sdYdnpYn-1o
U-Substitution - Definite Integrals: https://www.youtube.com/watch?v=tM4RWc9ryx0
1st Order Differential Equations: https://www.youtube.com/watch?v=C7nuJcJriWM
_____________________________________________________________________________________
Initial Value Problem: https://www.youtube.com/watch?v=kwGukY_2qWQ
Area Between Two Curves: https://www.youtube.com/watch?v=kgg5Rspf1Js
Disk and Washer Method: https://www.youtube.com/watch?v=SAHSVg7Jw_A
Volume By The Shell Method: https://www.youtube.com/watch?v=D5sT1br9soI
Volume By Cross Sections: https://www.youtube.com/watch?v=qMXPnfx2MQM
Arc Length Calculus Problems: https://www.youtube.com/watch?v=DNDAwWIL5FY
Surface Area of Revolution: https://www.youtube.com/watch?v=lQM-0Nqs9Pg
Work Problems - Calculus: https://www.youtube.com/watch?v=TLw8xbmnY3c
Integration By Parts: https://www.youtube.com/watch?v=sWSLLO3DS1I
Trigonometric Integrals: https://www.youtube.com/watch?v=3pXALn2ovIE
_____________________________________________________________________________________
GPA Calculator: https://www.youtube.com/watch?v=qYHsThZWydY
Save Money In College: https://www.youtube.com/watch?v=yNO02qfMSwI
SAT Test Prep: https://www.youtube.com/watch?v=fTGuTEQCsZY
ACT Test Prep: https://www.youtube.com/watch?v=SsA7rZ8kczM
GRE Math Test Prep: https://www.youtube.com/watch?v=z6lbrzaCbdk
Calculus 1 Final Exam Review: https://www.youtube.com/watch?v=WmBzmHru78w
Full Length Exams + Worksheets: https://bit.ly/4990rzU
Thanks to all of you who support me on Patreon. You da real mvps! $1 per month helps!! :) https://www.patreon.com/patrickjmt !! Exponential Decay / Finding H...
Thanks to all of you who support me on Patreon. You da real mvps! $1 per month helps!! :) https://www.patreon.com/patrickjmt !! Exponential Decay / Finding Half Life - In this video, I find the half life of a substance that is decreasing annually by 4%.
For more free math videos, visit http://PatrickJMT.com
Thanks to all of you who support me on Patreon. You da real mvps! $1 per month helps!! :) https://www.patreon.com/patrickjmt !! Exponential Decay / Finding Half Life - In this video, I find the half life of a substance that is decreasing annually by 4%.
For more free math videos, visit http://PatrickJMT.com
To calculate the Laplace transform of sin(t), we take the integral on 0 to infinity of e^-st*sin(t), where s is a constant with respect to the t integral.
Thi...
To calculate the Laplace transform of sin(t), we take the integral on 0 to infinity of e^-st*sin(t), where s is a constant with respect to the t integral.
This Laplace transform combines improper integrals with the integration by parts looping trick, which made it a perfect bonus problem for my last Calculus II exam!
To proceed with the integral, we apply integration by parts, letting u=e^-st and dv=sin(t)dt. This results in a second integral with a cosine instead of a sine, and we let u=e^-st and dv=cos(t)dt. After cleaning things up, we find a copy of the original integral on the right hand side of our work.
We evaluate the leftover terms from 0 to infinity, noting that e^(-infinity) is unambiguously zero, so the integral converges if s is greater than 0. These leftover terms yield a constant 1 as the result.
Now we close the deal on the looping trick: we gather both copies of the original integral on the left hand side, and factor out the integral. Using the name F(s) for the integral, we find F(s)(1+s^2)=1 or F(s)=1/(1+s^2), and that's the Laplace transform of sin(t).
We comment at the end on the fact that Laplace transforms are very useful in solving differential equations, where we transform an entire differential equation to Laplace transform s-space and find the solution using simple algebra. Then we transform the solution back to t space! This is a great prototype for mathematical physics in general: the idea that a shift in perspective to an abstract space can render all the mathematics trivial, then you solve the problem in the abstract space, then you take the solution back to the real world. That's a beautiful mathematical idea that we see all the time in mathematical physics!
To calculate the Laplace transform of sin(t), we take the integral on 0 to infinity of e^-st*sin(t), where s is a constant with respect to the t integral.
This Laplace transform combines improper integrals with the integration by parts looping trick, which made it a perfect bonus problem for my last Calculus II exam!
To proceed with the integral, we apply integration by parts, letting u=e^-st and dv=sin(t)dt. This results in a second integral with a cosine instead of a sine, and we let u=e^-st and dv=cos(t)dt. After cleaning things up, we find a copy of the original integral on the right hand side of our work.
We evaluate the leftover terms from 0 to infinity, noting that e^(-infinity) is unambiguously zero, so the integral converges if s is greater than 0. These leftover terms yield a constant 1 as the result.
Now we close the deal on the looping trick: we gather both copies of the original integral on the left hand side, and factor out the integral. Using the name F(s) for the integral, we find F(s)(1+s^2)=1 or F(s)=1/(1+s^2), and that's the Laplace transform of sin(t).
We comment at the end on the fact that Laplace transforms are very useful in solving differential equations, where we transform an entire differential equation to Laplace transform s-space and find the solution using simple algebra. Then we transform the solution back to t space! This is a great prototype for mathematical physics in general: the idea that a shift in perspective to an abstract space can render all the mathematics trivial, then you solve the problem in the abstract space, then you take the solution back to the real world. That's a beautiful mathematical idea that we see all the time in mathematical physics!
This video explains how to determine an exponential decay function from given information. Then it explains how to determine when a certain level of decay will...
This video explains how to determine an exponential decay function from given information. Then it explains how to determine when a certain level of decay will be reached using logarithms.
Library: http://mathispower4u.com
Search by Topic: http://mathispower4u.wordpress.com
This video explains how to determine an exponential decay function from given information. Then it explains how to determine when a certain level of decay will be reached using logarithms.
Library: http://mathispower4u.com
Search by Topic: http://mathispower4u.wordpress.com
This algebra and precalculus video tutorial explains how to solve exponential growth and decay word problems. It provides the formulas and equations / functions that you need to solve it.
Algebra For Beginners: https://www.youtube.com/watch?v=MHeirBPOI6w
Logarithms - The Easy Way!
https://www.youtube.com/watch?v=kqVpPSzkTYA
Solving Exponential Equations:
https://www.youtube.com/watch?v=9tutJ5xrRwg
Solving Logarithmic Equations:
https://www.youtube.com/watch?v=fnhFneOz6n8
Logarithmic Equations - Different Bases:
https://www.youtube.com/watch?v=XvwPB21Gm9A
__________________________________
Compound Interest Word Problems:
https://www.youtube.com/watch?v=Hn0eLcOSQGw
Interest Compounded Continuously:
https://www.youtube.com/watch?v=Ln97Hd7AiDc
Population Growth Word Problems:
https://www.youtube.com/watch?v=k4LLdFFLRmQ
Logarithms Practice Problems:
https://www.youtube.com/watch?v=7DVbQKI600k
Where does "e" come from?
https://www.youtube.com/watch?v=pDFcu_wLOzo
____________________________________
Logistic Growth Function:
https://www.youtube.com/watch?v=JgMvB22XQs0
Newton's Law of Cooling:
https://www.youtube.com/watch?v=ejEXSjdMpck
Final Exams and Video Playlists:
https://www.video-tutor.net/
Full-Length Videos and Worksheets:
https://www.patreon.com/MathScienceTutor/collections
This video explains how to determine an exponential decay function from given information. Then it explains how to determine when a certain level of decay will be reached and how to determine half-life.
Library: http://mathispower4u.com
Search by Topic: http://mathispower4u.wordpress.com
This calculus video tutorial focuses on exponential growth and decay. it shows you how to derive a general equation / formula for population growth starting with a differential equation.
Introduction to Limits: https://www.youtube.com/watch?v=YNstP0ESndU
Continuity & Differentiability: https://www.youtube.com/watch?v=fml0-ELYLaE&
Calculus 1 - Derivatives: https://www.youtube.com/watch?v=5yfh5cf4-0w
Introduction to Related Rates: https://www.youtube.com/watch?v=I9mVUo-bhM8
Local Maximum & Minimum: https://www.youtube.com/watch?v=WCq3sRzsJfs
L'Hopital's Rule: https://www.youtube.com/watch?v=Gh48aOvWcxw
Curve Sketching With Derivatives: https://www.youtube.com/watch?v=JTVNUdL7sWs
Newton's Method: https://www.youtube.com/watch?v=-5e2cULI3H8
Optimization Problems: https://www.youtube.com/watch?v=lx8RcYcYVuU
____________________________________________________________________________________
Antiderivatives: https://www.youtube.com/watch?v=xaCPDMEkbig
Basic Integration Problems: https://www.youtube.com/watch?v=zOxaUlRkFG0
Indefinite Integral: https://www.youtube.com/watch?v=JTFMeSCxgcA
Definite Integral: https://www.youtube.com/watch?v=Gc3QvUB0PkI
Differential Equations: https://www.youtube.com/watch?v=H5tD_NtPDuU
Properties of Definite Integrals: https://www.youtube.com/watch?v=QcHz3h81U-s
Rectilinear Motion Problems: https://www.youtube.com/watch?v=LBmET4sH460
Sigma Notation - Calculus: https://www.youtube.com/watch?v=XJkIaw2e1Pw
Riemann Sums - Area: https://www.youtube.com/watch?v=YTKQswb60Pw
The Midpoint Rule: https://www.youtube.com/watch?v=5XreKMJDJsg
____________________________________________________________________________________
Finding Area - Limit Definition: https://www.youtube.com/watch?v=ctEpKZyxqFU
Definite Integrals - Geometry: https://www.youtube.com/watch?v=ghxEOz9rmwE
Fundamental Theorem - Part 1: https://www.youtube.com/watch?v=aeB5BWY0RlE
Fundamental Theorem - Part 2: https://www.youtube.com/watch?v=ns8N1UuXl4w
Net Change Theorem: https://www.youtube.com/watch?v=df1Qr8pepx0
Mean Value Theorem - Integrals: https://www.youtube.com/watch?v=bLeglo-c5Tw
Average Value of a Function: https://www.youtube.com/watch?v=MB1xDNKimNc
U-Substitution - Indefinite Integrals: https://www.youtube.com/watch?v=sdYdnpYn-1o
U-Substitution - Definite Integrals: https://www.youtube.com/watch?v=tM4RWc9ryx0
1st Order Differential Equations: https://www.youtube.com/watch?v=C7nuJcJriWM
_____________________________________________________________________________________
Initial Value Problem: https://www.youtube.com/watch?v=kwGukY_2qWQ
Area Between Two Curves: https://www.youtube.com/watch?v=kgg5Rspf1Js
Disk and Washer Method: https://www.youtube.com/watch?v=SAHSVg7Jw_A
Volume By The Shell Method: https://www.youtube.com/watch?v=D5sT1br9soI
Volume By Cross Sections: https://www.youtube.com/watch?v=qMXPnfx2MQM
Arc Length Calculus Problems: https://www.youtube.com/watch?v=DNDAwWIL5FY
Surface Area of Revolution: https://www.youtube.com/watch?v=lQM-0Nqs9Pg
Work Problems - Calculus: https://www.youtube.com/watch?v=TLw8xbmnY3c
Integration By Parts: https://www.youtube.com/watch?v=sWSLLO3DS1I
Trigonometric Integrals: https://www.youtube.com/watch?v=3pXALn2ovIE
_____________________________________________________________________________________
GPA Calculator: https://www.youtube.com/watch?v=qYHsThZWydY
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Thanks to all of you who support me on Patreon. You da real mvps! $1 per month helps!! :) https://www.patreon.com/patrickjmt !! Exponential Decay / Finding Half Life - In this video, I find the half life of a substance that is decreasing annually by 4%.
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To calculate the Laplace transform of sin(t), we take the integral on 0 to infinity of e^-st*sin(t), where s is a constant with respect to the t integral.
This Laplace transform combines improper integrals with the integration by parts looping trick, which made it a perfect bonus problem for my last Calculus II exam!
To proceed with the integral, we apply integration by parts, letting u=e^-st and dv=sin(t)dt. This results in a second integral with a cosine instead of a sine, and we let u=e^-st and dv=cos(t)dt. After cleaning things up, we find a copy of the original integral on the right hand side of our work.
We evaluate the leftover terms from 0 to infinity, noting that e^(-infinity) is unambiguously zero, so the integral converges if s is greater than 0. These leftover terms yield a constant 1 as the result.
Now we close the deal on the looping trick: we gather both copies of the original integral on the left hand side, and factor out the integral. Using the name F(s) for the integral, we find F(s)(1+s^2)=1 or F(s)=1/(1+s^2), and that's the Laplace transform of sin(t).
We comment at the end on the fact that Laplace transforms are very useful in solving differential equations, where we transform an entire differential equation to Laplace transform s-space and find the solution using simple algebra. Then we transform the solution back to t space! This is a great prototype for mathematical physics in general: the idea that a shift in perspective to an abstract space can render all the mathematics trivial, then you solve the problem in the abstract space, then you take the solution back to the real world. That's a beautiful mathematical idea that we see all the time in mathematical physics!
This video explains how to determine an exponential decay function from given information. Then it explains how to determine when a certain level of decay will be reached using logarithms.
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A quantity is subject to exponential decay if it decreases at a rate proportional to its current value. Symbolically, this process can be expressed by the following differential equation, where N is the quantity and λ (lambda) is a positive rate called the exponential decay constant:
The solution to this equation (see derivation below) is:
Here N(t) is the quantity at time t, and N0 = N(0) is the initial quantity, i.e. the quantity at time t = 0.
Measuring rates of decay
Mean lifetime
If the decaying quantity, N(t), is the number of discrete elements in a certain set, it is possible to compute the average length of time that an element remains in the set. This is called the mean lifetime (or simply the lifetime or the exponential time constant), τ, and it can be shown that it relates to the decay rate, λ, in the following way:
The mean lifetime can be looked at as a "scaling time", because we can write the exponential decay equation in terms of the mean lifetime, τ, instead of the decay constant, λ:
Taking to X, Brandt stated ... Should Bitcoin fail to make a decisive new all-time high and decline below $55,000, he would raise the probability of an “Exponential Decay.” ... Astronomer remarked ... The terminal price does that very well ... .
Shiba Inu price has seen a significant increase since its inception, and investors are speculating about the probability of the meme coin surging to $1 ... No ... Using a simplified exponential decay model, we arrive at the probability of 0.000000000537 ... .
The observations found that the 2022 superflare on HD 251108 had a peak flux of around 10 decillion erg/s in the 0.5–4.0 keV band and an exponential decay time of 2.2 days in the early decay> phase.
There are four fundamental forces in the Universe... The resulting force was similar to the inverse-square forces of gravity and electromagnetism but had an exponential decay aspect, so it only affected nucleons ... Credit. Tsai, et al ... Reference ... 311. .
Bitcoin Power Law And Exponential Decay Theory... This trend uses a power law equation and an exponential decay to help pinpoint how high the Bitcoin price will go in each cycle and how low it could possibly drop after.
The Bitcoin price decay model aims at a more conservative prediction curve ... Unlike other models, the decay chart does not plot the upper bound of BTC as an exponential move, avoiding the over-optimistic scenario ... The Bitcoin Decay Channel™.
CryptoMarket Stumbles. Binance Report... Market Downturn Triggered by Major Bitcoin Movements ... Mt ... Source ... The analyst shared that this exponential decay model, based on historical data, indicates that significant price movements could be on the horizon.
Based on historical post-halving trends and the assumption that current Miner Capitulation is set to end soon, Cryptonary has stated that an exponential decay model suggests that BTC’s price, at ...
The discovery could increase the efficiency of photocatalysis for eliminating micropollutants in wastewater ... "That band bending has an exponential decay profile that reaches microns away and gave rise to this long-range adsorption enhancement." ... DOI.
Exponential functions for bacterial growth and decay. If you ever studied biology in high school, you may remember exponential functions being mentioned ... Exponential functions are often used in biological studies to measure bacterial growth and decay.
A new study led by Rice University's Peter Wolynes offers new insights into the evolution of foldable proteins ... The findings indicated deviations in the exon size distribution from exponential decay, suggesting there was evolutionary selection ... DOI.
Currently, MEG requires a magnetically shielded room for operation ... "Additionally, for MEGs, a short measurement distance is required as the decay of magnetic fields increases exponentially with distance ... More information. N ... DOI ... On arXiv. DOI ... Citation.
“Should Bitcoin fail to make a decisive new ATH (all-time high) and decline below $55,000 I will raise the probability of the Exponential Decay.”. The trader defines exponential decay as ...