# Exponential Growth & Decay Equations & Examples

Recent questions in Exponential growth and decay
Felix Cohen 2022-09-11

## Understanding e and e to the power of imaginary numberHow did the fee of e come from compound hobby equation. What does the cost of e honestly meanCapacitors and inductors charge and discharge exponentially, radioactive elements decay exponentially and even bacterial growth follows exponential i.e., $\left(2.71{\right)}^{x}$ ,why can't it be ${2}^{x}$ or something.Also ${e}^{2}$ means $e\ast e$ ,${e}^{3}$ means $e\ast e\ast e$ But what exactly ${e}^{ix}$ meanI want to know how to visualise ${e}^{i\pi }=-1$ in graphs

tashiiexb0o5c 2022-09-10

## what happened to the other constant of integration? The standard equation for exponential growth and decay starts and is derived like this:$\frac{dP}{dt}=kP$$\frac{dP}{P}=kdt$$\int \frac{dP}{P}=\int kdt$$\mathrm{ln}|P|=kt+C$I don't understand the left hand side at this point, isn't $\int \frac{1}{x}dx=\mathrm{ln}|x|+C$? Where did the constant of integration from the left integral go?

Staffangz 2022-09-10

## Why do I have to double an exponential growth and half an exponential decay function to find out how fast/slow the process is?Let’s take the exponential growth function to be $y={A}_{0}{e}^{kt}$ and exponential decay function to be $y={A}_{0}{e}^{-kt}$.I’m just told that to find out How fast the process is, you have to double the exponential growth function and Half-time the exponential decay function. Why is this so? I never seem to understand it. Is there a way to prove it? Or is my tutor right by just telling me the fact and nothing else?

Sasha Hess 2022-09-10

## What do the "real" and "imaginary" parts of the Laplace and Z transform represent?I understand that the Fourier transform brings you from the time domain into frequency domain, and that the Fourier transform is just the Laplace transform but where $\sigma$, the real valued portion of $s=\sigma +j\omega$, is set to 0. So if the imaginary portion, $\omega$, is the frequency, what does the real $\sigma$ represent?Furthermore, why is it not like this between the DTFT and the Z transform? The DTFT is a specialized case not where $\sigma =0$, but where r in $z=r{e}^{j\omega }$is set to 0, i.e when $|z|=1$. Do the real and imaginary parts of the signal change what they represent in continuous and discrete signals?

ubwicanyil5 2022-09-09

## Under ideal conditions a certain bacteria population is know to double every three hours. Suppose that there are initially 100 bacteria. How would you go about formulating a function for this?

rustenig 2022-09-08

## Is all acclimatization modeled by Newton’s Law of Cooling? I have in mind things like language learning (native or foreign), and growth to adulthood.Is acclimatization the same thing as saturation? Can the logistic function be regarded as simply an upside-down version of Newton’s Law of Cooling?

Lina Neal 2022-09-07

## Size of Object = # of pixels falling on the objectThe object is of arbitrary dimensions, so if we take photos of object from 1 meter distance, then 2 meter distance and so on. The number of pixels falling on the object will decrease. I want to know how will they decrease linearly or exponentially.

Natalya Mayer 2022-09-05

## There's a cup of coffee made with boiling water standing at room where room temperature is 20ºC. If H(t) is the temperature of this cup of coffee at the time t, in minutes, explain what the differential equation says in everyday terms. What is the sign of k?$\frac{dh}{dt}=-k\left(H-20\right)$Then solve the differential equation for 90ºC in 2 minutes and how long it will take to cool to 60ºCObserving $\frac{dh}{dt}=0$ we find that H=20 this means that the function stops changing at the room temperature H=20. As t is implied to be $H=20+A{e}^{-kt}$ as t approaches infinity H=20.

Sasha Hess 2022-09-05

## In the equation$A=B{e}^{kt}$where B is the initial amount and t is the time taken what is k,I know it's a constant of proportionality ,but is it the same as the number of time a certain amount of money gets compounded every year?For example, if an amount of \$ 500 is getting compounded four times at the rate of 5 % per year ,here if they ask what is the amount if the money is compounded every instant ,the equation will the somewhat similar to exponential growth equation,is the number 4 here same as k?

batystowy2b 2022-09-05

## Exponential growth and decay question A city has a growing population at a rate proportional to the current population, that is:$\frac{dP}{dx}=kP.$Verify that $P\left(t\right)={P}_{0}{e}^{kt}$, $t>0$ is a solution of the equation.If the population on 1st January 2006 which is t=1 was 147,200 and on 1st January 2007 when t=2 was 154,800, find the initial population and the value of k. Round your answer down to the 3dpl.Find the population on 1st January 2012.Find the time it takes for the population to double.

Deacon House 2022-09-05

## This is the question: "If you want to have \$75,000 after 35 years in your account that pays 12% annual interest compounded quarterly, how much should you put in as your original investment?"The formula I'm using is $y=a\left(1+r{\right)}^{t}$, with a being the initial amount, r being the rate in decimal form, and t is time relative to the rate. Or $y=a\left(1+r/t{\right)}^{t}$Although my biggest problem is that I'm not sure whether to have the 1+r or 1−r.So after plugging in what I have either:$75000=a\left(1-.12{\right)}^{35}$$75000=a\left(1+.12{\right)}^{35}$or you can use this formula (preferably):$75000=a\left(1-.12/35{\right)}^{35}$$75000=a\left(1+.12/35{\right)}^{35}$

Nyasia Flowers 2022-09-05

## What is the formula for exponential growth with a decay rate?Exponential growth can be modeled as$b\left(1+r{\right)}^{N}$For b your starting quantity, (1+r) your rate of growth, and N the number of periods. But for $N\to \mathrm{\infty }$, this formula can get out of control.Is there a traditional way of controlling for this by factoring in some notion of a decay factor (so that for periods N past some threshold, you stop growing asymptotically)?

Kiana Arias 2022-09-05

## Inferring exponential decay from difference equations?I'm trying to justify, why the graph of the following system:$\begin{array}{r}v\left(n\right)=0.6\cdot v\left(n-1\right)\\ p\left(n\right)=0.13\cdot v\left(n\right)+0.87\cdot p\left(n-1\right)+25\end{array}$with initial values $v\left(0\right)\approx 1441.67$, p(0)=3000 and p(1)=3500,seems to show exponential decay for both v(n) and p(n) (after some initial growth).Justifying this for v(n) is easy: decays exponentiallybut what about p(n)?I can write for example p(3):$\begin{array}{r}p\left(3\right)=a\ast {b}^{3}\ast v\left(0\right)+c\ast \left[a\ast {b}^{2}\ast v\left(0\right)+c\ast \left[a\ast b\ast v\left(0\right)+c\ast p\left(0\right)+25\right]\right]\\ =a\ast {b}^{3}\ast v\left(0\right)+c\ast \left[a\ast {b}^{2}\ast v\left(0\right)+c\ast a\ast b\ast v\left(0\right)+{c}^{2}\ast p\left(0\right)+c\ast 25\right]\\ =a\ast {b}^{3}\ast v\left(0\right)+c\ast a\ast {b}^{2}\ast v\left(0\right)+{c}^{2}\ast a\ast {b}^{2}\ast v\left(0\right)+{c}^{3}\ast p\left(0\right)+{c}^{2}\ast 25\end{array}$where $a=13/100,b=60/100,c=87/100$and one can see exponential terms, but the whole expression is too complicated in order to infer whether the decaying is "clean" exponential decay or whether it would exhibit some sort of other more complex curves.

s2vunov 2022-09-03

## Let a>0 and A>0, I am looking for the decay rate of the integralThere is no closed form answer for the integral. I have tried on Matlab that it should converge to zero much faster than power growth. I think the growth should be exponential types. Do we have some literature discussing this kind of issue? Thanks!I have successfully obtained the growth rate ofbe expanding the denominator in power series.But I do not know to deal with the integral in [−M,M].

Algebra IOpen question
hannahb862r 2022-09-03

## What is the difference between exponential growth and decay?A colleague came across this terminology question.What are the definitions of exponential growth and exponential decay? In particular:1) Is $f\left(x\right)=-{e}^{x}$ exponential growth, decay, or neither?2) Is $g\left(x\right)=-{e}^{-x}$ exponential growth, decay, or neither?Consider $f\left(x\right)=A{e}^{kx}.$. I can't find any sources that specify A>0. My answer is that:f exhibits1. exponential growth for A>0,k>0, and2. exponential decay for A>0,k<0whereas $|f|$ exhibits3. exponential growth for A<0,k>0, and4. exponential decay for A<0,k<0.In case (3) we shouldn't call f an exponential growth function without noting that it is "negative growth". Also it wouldn't be called it an exponential decay function without specifying the "direction of decay", so it is neither.In case (4) it's neither as well. One should specify that it is the magnitude of f which decays exponentially although f is increasing in value. Although f is increasing in value, is it growing? It seems odd to say it is exponentially growing.It just doesn't sit right with me to refer to a function as growing if it is decreasing in value. Certainly, it's magnitude may be growing.Next consider a function with exponential asymptotic behavior (e.g. logistic) so that as $x\to \mathrm{\infty },$, $f\left(x\right)\approx A{e}^{-kx}+C$ for some k>0. I feel the best way to describe this would be "exponential decay towards C" with a qualification as being from as being from above or below depending on the sign of A.If someone is to just use the terminology "exponential growth (decay)", it implies $f\left(x\right)=A{e}^{kx}$ with positive A and k>0 (k<0) unless there is a specific context or further clarification as to what the actual nature of the function is.

sincsenekdq 2022-09-03

## Using the exponential growth and decay formula for compound interestI have what seems to be a rather simple question but one that is confusing me a lot.When looking at standard exponential growth/decay models(such as the decay of Carbon-14 etc),we can use the formula $A=P\left(1+r{\right)}^{t}$ in order to find things such as the half-life/rate.However, in these models, aren't the substances (such as Carbon-14) assumed to decay continuously?If so, why can we NOT use the formula $P{e}^{rt}$,when is this assumed to be the formula for continuous growth/decay?For instance, in a compound interest problem where interest is compoundedcontinuously, we would have to use the $P{e}^{rt}$ formula right?(We couldn't use the formula $A\left(1+r{\right)}^{t}$

Algebra IOpen question