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:
The ISS’s orbit is slowly decaying ... Over the 25-year lifespan of the station, hundreds of tons of hydrazine rocket fuel have been carried to it to enable rocket-propelled orbital maneuvers to keep its orbit from decaying.
For those who haven’t played, “HorizonZeroDawn,” is an open world action adventure game set in the far future, where humanity is divided into strange, primitive tribes who live amid the decaying ruins of 21st century civilization.
'The problem is the constant sugar exposure in the mouth creates a situation where decay can affect the front teeth, possibly requiring fillings and/or cosmetic work later on in the new year.
The constant heat from the long-lived, decaying plutonium-238 isotope (88.7-year half-life) in Perseverance’s RTG is converted into electricity through thermocouples ... They quickly decay to promethium-147, which lasts more than two years.
The problem is the constant sugar exposure in the mouth creates a situation where decay can affect the front teeth, possibly requiring fillings and/or cosmetic work later on in the new year, once the last cork has long been popped.
By Michael Segalov / The Observer... She’d have been constantly cold and hungry. Emaciated, unable to move or grow ... “Reminders that my brain and body are constantly decaying,” he says, “can become a bit much from time to time; it can feel heightened ... .
Tragically, her life expectancy was just a matter of months ... She’d have been constantly cold and hungry ... Unchangeable ... “Reminders that my brain and body are constantly decaying,” he says, “can become a bit much from time to time; it can feel heightened.
... let the property decay to a “dangerous” state that has left them fearful for their lives with constant leaks, floods, collapsed ceilings, heat outages — and for six months this year, no water.
However, the academic admits he 'can only speculate' exactly what the writing says ... The rate of decay of carbon-14, a carbon isotope, is constant and easily measured, making it ideal for providing age estimates for anything over 300 years old ... .
Just as the shell of air around our planet protects us from cosmic rays and the meteors that constantly rain down on us, it also burns up satellites whose orbits have decayed – provided they haven't been equipped with special protection ...Source. ORNL. .
Moreover, the mean recurrence time of QPEs, calculated to be 2.35 hours, was found to be constant between 2022 and 2024, with a hint of a decay of approximately six minutes between August 2020 and June 2022.
Temperatures plummeted, and ice sheets that may have been several miles thick crept over every inch of Earth's surface... Because uranium atoms decay into lead at a constant rate, the team could use them as a sort of timekeeper for the planet's rocks.
... in HFIR decay to plutonium-238 nuclei, she said ... The constant heat of the long-lived decaying plutonium-238 isotope in the RTG powers the PerseveranceRover, which landed on Mars after a 2020 launch.
“As soon as we started introducing refined sugars into the diet, decay exploded ... “If you are constantly bathing your teeth in food, then you are constantly feeding the bacteria, which causes decay.”Wait for 30 minutes after eating to brush.