SLIDE 2 3/6/2012 2
Key Facts about Exponential Growth
- Exponential growth leads to repeated doublings.
With each doubling, the amount of increase is approximately equal to the sum of all preceding doublings.
- Exponential growth cannot continue indefinitely.
After only a relatively small number of doublings, exponentially growing quantities reach impossible proportions.
Growth: Linear versus Exponential
Example 8.A.1: Recall simple interest versus compound
- interest. Simple interest is the same amount of interest
every time as where compound interest is the same percent of interest at each step of time. For example, let's say we invested $500 with an interest rate of 10%.
Year Sim ple Com pound 1 $1,000 $1,000 2 $1,000+100=$1,100 $1,000+100=$1,100 3 $1,100+100=$1,200 $1,100+110=$1,210 4 $1,200+100=$1,300 $1,210+121=$1,331 5 $1,300+100=$1,400 $1,331+133.10=$1,464.10 Linear: We add the same amount, $100, every time. Exponential: We add the same percent, 10%, every time.
Growth: Linear versus Exponential
Example 8.A.2: Bacteria in a Bottle: Suppose you put a single bacterium in a bottle at 11:00 a.m. It grows and at 11:01, it divides into two bacteria. These two bacteria grow and at 11:02 divide into four bacteria, which grow and at 11:03 divide into eight bacteria, and so on. Thus, the bacteria doubles every minute. If the bottle is half-full at 11:59, when will the bottle be completely full?
Growth: Linear versus Exponential
Example 8.A.3: You are given a choice, take $1000 each month for the rest of your life or be given a magic penny. The magic penny will turn into two pennies after one day. Then double again into four pennies the next day, and so
- n. Which option would you rather take?
After 30 Years!
- $1,000 option: You have $1000 12
30 $360,000
- Penny: You have $0.012 $10,737,418.24
