Exponential growth and growth rate

[anbj]

Hi all,

I'm reading about exponential growth, and finding it very interesting.

However, I'm a bit confused. I wanted to have some spreadsheed fun and do some calculation in practice.

Given n=e^kx, I want to calculate some values for n, with k=0.07. Alright no problem.

I than wanted to check the increase in % for each x. I thought to myself, 'duh, this is given by k! I already know this is 0.07!'. But I got surprised! When I calculate the %-growth I get a %-growth that is > k. In the case of K=0.07, I get an %-increase by 0.072508181254217 for each term. Why is this!? Why is it not 0.07?

I'm calculating the %-growth with the formula: [n(x)-n(x-1)]/n(x-1) (i.e. [current-previous]/previous).

Please see attached a picture of my spreadsheed.

Thank you for any guidance!

Best,
Andreas

 

[Harry_the_cat]

If you divide one value of n by the one before it, lets just say when x=5 and x=4 what you are doing is calculating
\(\displaystyle \frac{e^{(0.07*5)}}{e^{(0.07*4)}}\) which gives \(\displaystyle e^{0.07} = 1.072508.....\) giving a growth rate of 0.072508.. ie 7.2508% ... NOT 0.07 or 7%.

 

[Harry_the_cat]

The reason behind this is when expressed as \(\displaystyle n=e^{kx}\) the growth is continuous and not discrete.

 

[anbj]

I do not understand why, when somethings grows (continously) exponentially with 7% , the growth factor is > 7% and not 0.07! When would this be case (if any)?

Thanks for helping me grasp this Harry_the_cat.

 

[Harry_the_cat]

Have a read of https://en.wikipedia.org/wiki/E_(mathematical_constant) . Read the bit under the heading "Applications", particularly compound interest. It explains it quite well.

 

[Deleted member 4993]

Hi all,

I'm reading about exponential growth, and finding it very interesting.

However, I'm a bit confused. I wanted to have some spreadsheed fun and do some calculation in practice.

Given n=e^kx, I want to calculate some values for n, with k=0.07. Alright no problem.

I than wanted to check the increase in % for each x. I thought to myself, 'duh, this is given by k! I already know this is 0.07!'. But I got surprised! When I calculate the %-growth I get a %-growth that is > k. In the case of K=0.07, I get an %-increase by 0.072508181254217 for each term. Why is this!? Why is it not 0.07?

I'm calculating the %-growth with the formula: [n(x)-n(x-1)]/n(x-1) (i.e. [current-previous]/previous).

Please see attached a picture of my spreadsheet.

Thank you for any guidance!

Best,
Andreas

Please run similar experiments with

Linear growth: y = a + k*x

power (second order) growth: y = a + k * x²

Please tell us what you found.....

 

[anbj]

Alright, I hope I did it correct.

This is (I think) similar to what I espected. Linear growth, the growth rate will decrease for every term. It makes sense; the value added by each term will represent a decreasing percentage of what you have already.

Power growth looks in a way to be similar; the growth rate will decrease for every term, again, the addition of each term will represent a decreasing percentage of what you already have. What is special about this is that even though the difference in value added increases by every term, the percentage of what this represents of the total still decreases.

I appreciate all comments!