Exponential growth

Read Chapter 7

[spreadsheet]

If we think of a bank account, identify as many ways as you can of how

  • money comes in
  • money leaves

Which of those depends on how much money you have in the bank? Which are *independent* of how much money you have in the bank?

Doubling times for money in Bank L? Bank H?

Doubling times for boring money in checking account?

Semi-log plots

Linear scale: numbers are evenly spaced.

Logarithmic scale: powers of 10 are evenly spaced.

Semi-log plot: $y$-axis is logarithmic, $x$-axis is linear.

In Excel:

  • [PC] Double-click the $y$-axis or [Mac] Right-click on $y$-axis
  • Choose "Format axis"
  • Choose "Scale"
  • Check the "Logarithmic scale" box

Our writing exercise involves using semi-log graph paper.

railroad track data


Exponential vs linear growth

When a quantity grows...

  • by a fixed amount in each [month, or other time-interval], its growth is linear.

  • by a fixed percentage in each time-interval, its growth is exponential.
exponential growthlinear growth
  • 'J'-curve on linear plot:

  • straight line on semi-log plot
  • Has a constant doubling time or half-life
  • Growth depends on "balance" in account
  • $$P(t)=P_0e^{kt}$$
  • straight line on linear plot:

  • flattening-out curve on semi-log plot
  • Has a constant slope or rate-of-change
  • Growth does not depend on "balance" in account.
  • $$y(t)=b+m*t$$

Comparing exponentials

Jill was offered a weekly wage starting at $10 and doubling every 2 weeks.

Molly is offered a weekly wage of 10 cents, but it will triple every 2 weeks.

Comparing Jill and Molly

Comparing Jill and Molly: semi-log plot

Comparing doubling times

Moore's "Law" is the observation that the number of transistors in the Central Processing Units (CPUs) of computers has been doubling approximately every two years since ~1970. Note the semi-log y-axis:

Moore's Law

Population growth is typically exponential

bacteria

Bacteria grow by dividing. So each successive generation will have twice as many 'children' as the previous generation.

Sample problem

The number of lily pads on a pond doubles every 5 days.

Counting backwards from the time at which the pond is fully covered (100%), how many days before was 1/4 of the pond covered with lily pads?

Here is a similar graph for the population of humans on Earth...

What's the human population "doubling time"?

Semi-log plot, by region (from U.N. data):

Exponential growth can't go on forever...

 

Image credits

S. Foucher, Al, Mike Baird, W.G.Simon