Exponential versus Linear Growth
Sally is a new employee and is faced with a rather perplexing problem. Her new employer gave her a choice between two ways in which she could be paid each month. The first option has her being paid $100 on the first day of the month. Each day of the month after that, her pay will be $50 more than the previous day. The second option has her being paid $0.01 on the first day. Each day of the month after that, her pay will be twice as much as it was the day before.
Which option should she take?
The first salary option pays Sally $100 on the first day, pays her $150 on the second day, pays her $200 on the third day, and continues in such a way that on the last day of the month she will be paid $1,550. Her salary for the month will be $24,750. Very nice!
The second salary option pays her a penny on the first day, 2 cents on the second day, and 4 cents on the third day. This is not looking like a particularly good deal for the new employee. However, Sally is a patient person and continues to work out how much she will be paid for the month. The big smile on her face reveals that she will choose option 2. Sally will be paid $5,368,709.12 for the last day of the month and a total of $10,737,418.23 for the entire month!
When you examine the daily wage for each option, you see that the first option is linear, because the change in wage from day to day is always the same, $50. In other words, the daily change is found by adding. For the second option offered to Sally, however, the change in daily wage is found by multiplying, not adding. This is a classic example of an exponential function, and exponential functions are the subject of this chapter.