Exponential growth

  • Jack is offered a weekly wage of $\$ $10,000, which will increase by $\$ $10,000 each week he works.
  • Jill is offered a weekly wage of $\$ $1.00. But her weekly wage will double every 2 weeks.

Which would you choose??

After 25 weeks

But 8 weeks later, a "J-curve" is apparent...

When you graph on "semi-log" paper...

Using semi-log paper

Our writing exercise involves using semi-log graph paper.

Modification... only graph your balance every other year!

Linear scale: numbers are evenly spaced.

Logarithmic scale: powers of 10 are evenly spaced.

Exponential 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
$$P(t)=P_0e^{kt}$$ 'J'-curve on regular plot

straight line on semi-log plot

Has a constant doubling time or half-life
$$y(t)=b+m*t$$ straight line on regular plot

flattening-out curve on semi-log plot

Has a constant slope or rate-of-change

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...

 

New technologies

New technologies typically "take off" (grow exponentially) for a while before market saturation kicks in (everyone's got one).

Renewable resources

Renewable resources follow the same shape for consumption vs time: An S-curve or logistic curve.

Hydroelectric power

U.S. Hydroelectric capacity (USGS)

More recently...Wikipedia graph of EIA data. [What do you think accounts for all that recent variation?]

Non-renewable resources

Resources that are not being replenished do not have a long-term "plateau" at which they can be eternally exploited.

Oil production

What have you heard about "peak oil"? What is that?

Oil rig

More recently...EIA data 1860-present.

A last exponential growth curve

PhotoVoltaic capacity

For comparison a large coal power plant might have a capacity of 1500 MW, and the Three Gorges Dam plans on 22,500 MW.

 

 

Image credits

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