Exponential Growth & Decay Equations & Examples

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

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

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?

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 $\mathrm{\sigma <!--\; \sigma \; -->}$, the real valued portion of $s=\mathrm{\sigma <!--\; \sigma \; -->}+j\mathrm{\omega <!--\; \omega \; -->}$, is set to 0. So if the imaginary portion, $\mathrm{\omega <!--\; \omega \; -->}$, is the frequency, what does the real $\mathrm{\sigma <!--\; \sigma \; -->}$ represent?
Furthermore, why is it not like this between the DTFT and the Z transform? The DTFT is a specialized case not where $\mathrm{\sigma <!--\; \sigma \; -->}=0$, but where r in $z=r{e}^{j\mathrm{\omega <!--\; \omega \; -->}}$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?

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?

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?

Size of Object = # of pixels falling on the object
The 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.

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?
$\frac{dh}{dt}=-<!--\; -\; -->k(H-<!--\; -\; -->20)$
Then solve the differential equation for 90ºC in 2 minutes and how long it will take to cool to 60ºC
Observing $\frac{dh}{dt}=0$ 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.

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?

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 $P(t)=P_0{e}^{kt}$, $t>0$ 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.

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 $y=a(1+r{)}^t$, with a being the initial amount, r being the rate in decimal form, and t is time relative to the rate. Or $y=a(1+r/t{)}^t$
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:
$75000=a(1-<!--\; -\; -->.12{)}^{35}$
$75000=a(1+.12{)}^{35}$
or you can use this formula (preferably):
$75000=a(1-<!--\; -\; -->.12/35{)}^{35}$
$75000=a(1+.12/35{)}^{35}$

What is the formula for exponential growth with a decay rate?
Exponential growth can be modeled as
$$b(1+r{)}^N$$
For b your starting quantity, (1+r) your rate of growth, and N the number of periods. But for $N\to <!--\; \to \; -->\mathrm{\infty <!--\; \infty \; -->}$, 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)?

Inferring exponential decay from difference equations?
I'm trying to justify, why the graph of the following system:
$$\begin{array}{r}v(n)=0.6\cdot <!--\; \cdot \; -->v(n-<!--\; -\; -->1)\\ p(n)=0.13\cdot <!--\; \cdot \; -->v(n)+0.87\cdot <!--\; \cdot \; -->p(n-<!--\; -\; -->1)+25\end{array}$$
with initial values $v(0)\approx <!--\; \approx \; -->1441.67$, 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:
$$v(n)={0.6}^{n}v(0)⟹<!--\; ⟹\; -->v(n)\text{ decays exponentially}$$ decays exponentially
but what about p(n)?
I can write for example p(3):
$$\begin{array}{r}p(3)=a\ast <!--\; \ast \; -->b^3\ast <!--\; \ast \; -->v(0)+c\ast <!--\; \ast \; -->[a\ast <!--\; \ast \; -->b^2\ast <!--\; \ast \; -->v(0)+c\ast <!--\; \ast \; -->[a\ast <!--\; \ast \; -->b\ast <!--\; \ast \; -->v(0)+c\ast <!--\; \ast \; -->p(0)+25]]\\ =a\ast <!--\; \ast \; -->b^3\ast <!--\; \ast \; -->v(0)+c\ast <!--\; \ast \; -->[a\ast <!--\; \ast \; -->b^2\ast <!--\; \ast \; -->v(0)+c\ast <!--\; \ast \; -->a\ast <!--\; \ast \; -->b\ast <!--\; \ast \; -->v(0)+c^2\ast <!--\; \ast \; -->p(0)+c\ast <!--\; \ast \; -->25]\\ =a\ast <!--\; \ast \; -->b^3\ast <!--\; \ast \; -->v(0)+c\ast <!--\; \ast \; -->a\ast <!--\; \ast \; -->b^2\ast <!--\; \ast \; -->v(0)+c^2\ast <!--\; \ast \; -->a\ast <!--\; \ast \; -->b^2\ast <!--\; \ast \; -->v(0)+c^3\ast <!--\; \ast \; -->p(0)+c^2\ast <!--\; \ast \; -->25\end{array}$$
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.

Let a>0 and A>0, I am looking for the decay rate of the integral
$${\int <!--\; \int \; -->}_M^{\mathrm{\infty <!--\; \infty \; -->}}+{\int <!--\; \int \; -->}_{-<!--\; -\; -->\mathrm{\infty <!--\; \infty \; -->}}^{-<!--\; -\; -->M}\frac{x^2\exp <!--\; \; -->(-<!--\; -\; -->(x-<!--\; -\; -->a{)}^2)}{1+A{a}^{-<!--\; -\; -->1}\exp <!--\; \; -->(-<!--\; -\; -->x^2/(1+a^2))}dx\text{ as }a\to <!--\; \to \; -->0$$
There 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 of
$${\int <!--\; \int \; -->}_M^{\mathrm{\infty <!--\; \infty \; -->}}+{\int <!--\; \int \; -->}_{-<!--\; -\; -->\mathrm{\infty <!--\; \infty \; -->}}^{-<!--\; -\; -->M}\frac{x^2\exp <!--\; \; -->(-<!--\; -\; -->(x-<!--\; -\; -->a{)}^2)}{1+A{a}^{-<!--\; -\; -->1}\exp <!--\; \; -->(-<!--\; -\; -->x^2/(1+a^2))}dx\text{ as }a\to <!--\; \to \; -->0$$
be expanding the denominator in power series.
But I do not know to deal with the integral in [−M,M].

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 $f(x)=-<!--\; -\; -->e^x$ exponential growth, decay, or neither?
2) Is $g(x)=-<!--\; -\; -->e^{-<!--\; -\; -->x}$ exponential growth, decay, or neither?
Consider $f(x)=A{e}^{kx}.$. I can't find any sources that specify A>0. My answer is that:
f exhibits
1. exponential growth for A>0,k>0, and
2. exponential decay for A>0,k<0
whereas $|f|$ exhibits
3. exponential growth for A<0,k>0, and
4. 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 $x\to <!--\; \to \; -->\mathrm{\infty <!--\; \infty \; -->},$, $f(x)\approx <!--\; \approx \; -->A{e}^{-<!--\; -\; -->kx}+C$ 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 $f(x)=A{e}^{kx}$ 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.

Using the exponential growth and decay formula for compound interest
I 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 $A=P(1+r{)}^t$ 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 compounded
continuously, we would have to use the $P{e}^{rt}$ formula right?
(We couldn't use the formula $A(1+r{)}^t$

First time poster, and, as my post will intimate, not a mathematician, just someone searching for answers. My question has two parts:
1) In a genealogical chart for a single individual starting with yourself and working backwards, you'll find a simple exponential trait to your preceding/antecedent group of ancestors, e.g.
you have one set of parents (2 people)
you have two sets of grandparents (4 people)
you have four sets of great-grandparents (8 people)
and so on...I'm only counting genetically-linked lineage (no step/half) for simplicity and using "sets" of ancestors rather than individuals.
However, whether you believe in Adam and Eve or in Darwin and Haldane, at a certain point all of this must converge back to an original set of antecedents (your common, original male/female ancestors, and logically the common human ancestors for all--the question I'll leave to philosophers and Richard Dawkins is how you get to a single ancestor not a single set of ancestors). Again, for simplicity, I'm only counting homo homo sapiens and not trying to take this back to the first unicellular organisms.
The question I'm trying to answer is, as one moves back in time, away from yourself (x=1) on a genealogical chart, your ancestors increase exponentially, but at some point they must start decreasing again to get back to a single set of common original ancestors (y=1)--for argument's sake, let's assume the decrease is perfectly proportional to the rate of increase and the time-series is based on finite generations not years--though if someone wants to try and model out interbreeding have at it.
When would this conversion/inflection across the generations need to occur--put another way, what is the maximum number of grand-nth parent sets you'd need to have before we started to see a need for this decrease--one would imagine it's about half-way back? In highly simple form it would go 1:2:4:2:1, but on a much grander scale.
There's an excellent article here from the BBC that talks about this issue as well as one known as the "genealogical paradox" (i.e. that most genealogy models show one to have more potential ancestors than human beings to have ever lived), and it also provides an important parameter for the time series: human history back to a single set of common ancestors for all humans is only about 3000 years or 100 generations. It also points out the need to assume inbreeding, consanguity, and incest as part of any genealogy, but for reasons both moral and mathematical let us keep things pure and simple.
2) The second part of my question pertains to the first: formulaically, how would one model out the math for the specific question above using the parameters described (e.g. 100 generations)? And, more generally, how would one write the formula for an exponential growth time series that starts at 1 and that must then suddenly inflect, and start to decay in proportion to its original exponential growth to ensure the final result is 1 at the end of the sequence? Put another way, what is the general formula for expressing a pattern that both increases and decreases across a time series such as 1:2:4:2:1 and could this be expressed in a single formula?
For bonus points: what fields of math are we discussing in this question and what would the graph for the specific ancestor and general formula equation look like? I believe in graph theory this is something called a directed acrylic graph?

The population P of bacteria in an experiment grows according to the equation $\frac{dP}{dt}=kP$, where k is a constant and t is measured in hours. If the population of bacteria doubles every 24 hours, what is the value of k?
I was given this problem and I'm not sure what to do with it. I know the formula for this kind of equation is $c{e}^{kx}$. But, how do you plug in the values given?

Exponential Growth and Decay
Find the solution to $\frac{dy}{dt}=4y$ satisfying y(4)=9. I am not sure where to begin. If I can get help with that I am sure I can finish the problem.

Is exponential growth and decay faster than polynomial growth and decay?
I know the answer is yes for growth conditions, but I don't see how it's obvious that exponential decay is faster than polynomial decay, say for a polynomial $x^2$.