Example of an Exponential Function
A biologist grows bacteria in a culture. If initially there were three
grams of bacteria, after one day there are six grams of bacteria, and
after two days, there are twelve grams, how many grams will there be at the
end of the week?
We draw a t chart
||3 = 3(20)
||6 = 3(21)
||12 = 3(22)
We see that the general formula is
P(t) = 3(2t)
Hence after one week, we calculate
P(7) = 3(27) = 384 grams of bacteria.
We call P(t) and exponential function with base 2.
Money and Compound Interest
We have the formula for compound interest
where A corresponds to the amount in the account after
t years in a bank
that gives an annual interest rate r compounded n times per year.
Suppose we have $2,000 to put into a savings account at a 4% interest
rate compounded monthly. How much will be in the account after 2 years?
P = 2,000, r = .04, n = 12 and
t = 2
We want A.
A = 2000(1 + .04/12)12(2) = $2,166.29.
For continuously compounded interest, we have the formula:
With an 8% rate of inflation in the health industry, how much will health
insurance cost in 45 years if currently I pay $200 per month?
r = .08, P = 200, and
t = 45
A = 200e(.08)(45) = $7319 per month!