## 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 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:

### Population growth is typically exponential

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):

### 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?

More recently...EIA data 1860-present.

### A last exponential growth curve

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

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