 evitagimm9h

2022-11-06

${e}^{pt}$or $\left(1+p{\right)}^{t}$ What is the difference in modeling exponential growth and decay?
I would really like to better recognize, whilst the feature ${e}^{pt}$ is the "higher" desire and while (if at all) $\left(1+p{\right)}^{t}$ have to be used.
to provide a conventional example: Say we need to version radioactive decay of some detail A. allow t be in units of one half-life of A. Then $p=-\frac{1}{2}$
Now I'd say the standard approach to modeling this is via the function
${A}_{1}\left(t\right)={A}_{0}\cdot {\left(1-\frac{1}{2}\right)}^{t}.$
On the other hand, if we approach this problem as an ODE, we can say that at any point t we want A(t) to decrease at a rate of half of its momentary amount:

${A}_{2}\left(t\right)={A}_{0}\cdot {e}^{-\frac{1}{2}t}.$
But which approach would be "better" here? I think ${A}_{1}$ is much more commonly (if not exceptionally) used when it comes to modeling atomic decay. On the other hand, I know that
$e=\underset{n\to \mathrm{\infty }}{lim}{\left(1+\frac{1}{n}\right)}^{n}$
which essentially means that the rate of change of ${A}_{2}$ is continously updated, while ${A}_{1}$ is updated discretely, right? That's the best way I can phrase it at the moment.
So to conclude: Does this mean that ${A}_{2}$ is always the "better", more accurate choice or are there situations where ${A}_{1}$ is actually "correct"? dilettato5t1

"It depends how large the decay period is. Let´s say the length of the decay period is $\frac{1}{n}$. Then in n periods the amount of units of the atom after t periods is
${A}_{t}={A}_{0}\cdot {\left(1-\frac{\frac{1}{2}}{n}\right)}^{n\cdot t}$, where ${A}_{0}$ is the initial amount. Important: $\frac{1}{2}$ is the decay rate in a period of the legnth $\frac{1}{n}$.
The period can get smaller and smaller, which means that $n\to \mathrm{\infty }$.
${A}_{t}={A}_{0}\cdot \underset{n\to \mathrm{\infty }}{lim}{\left(1-\frac{\frac{1}{2}}{n}\right)}^{n\cdot t}={A}_{0}\cdot {e}^{-\frac{1}{2}\cdot t}$
Conclusion: The smaller the decay/growth period gets the more it is reasonable to apply the continuous growth/decay model."

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