Example of an Exponential Function
A biologist grows bacteria in a culture. If initially
there were three grams of bacteria, after one day there were six grams of
bacteria, and after two days, there were twelve grams, how many grams will there
be at the end of the week?
We draw a t chart
P(t) = 3(2t)
Hence after one week, we calculate
P(7) = 3(27) = 384 grams of bacteria.
We call P(t) an exponential
function with base 2.
Below is the graph of y = 2x. It turns out that for any b > 1 the graph of y = bt looks similar.
Shifting techniques can also be used to graph variations of this
y = 2-x
We see that the graph is reflected about the y-axis:
Three Properties of Exponents
We define negative exponents below
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.
A = 2000(1 + .04/12)12(2) = $2,166.29.
For continuously compounded interest, we have the formula:
A = 200e(.08)(45) = $7319 per month!