Exponentials

1. 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?

Solution:

We draw a t chart

 t P(t) 0 3 = 3(20) 1 6 = 3(21) 2 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.

2. Graphing Exponentials

Below is the graph of y = 2x.    It turns out that for any b > 1 the graph of y = bt looks similar.

Notice that

1.  the left horizontal asymptote at 0

2. The y-intercept is 1

3. The graph is always increasing.

Shifting techniques can also be used to graph variations of this curve.

Example

Graph y = 2-x

Solution:

We see that the graph is reflected about the y-axis:

3. Three Properties of Exponents

1. bx by = bx+y

2. bx / by = bx-y

3. (bx)y = bxy

 Definition         b-x  = 1 / bx

Example

Simplify

34(-3)-1/[(32)3]

Solution

34(-3)-1/[(32)3] = 34(-3)-1/ 3

= -34 /(3136)  =  -34 / 37

=  -1/33 = -1/27

4. Applications

Money and Compound Interest

We have the formula for compound interest

 A = P(1 + r/n)nt

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.

Example

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?

We have

P = 2,000, r = .04, n = 12 and t = 2

We want A.

A = 2000(1 + .04/12)12(2)  = \$2,166.29.

Continuous Interest

For continuously compounded interest, we have the formula:

 A = Pert

Inflation Example

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?

Solution

We have

r = .08, P = 200, and t = 45

So that

A = 200e(.08)(45) = \$7319 per month!