## 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 growth | linear 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:

### 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):

### 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).

- Mobile phones
- Lots of new gadgets, all on one graph (NYTimes)

### 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.