Exponentials

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?

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(2^t)

Hence after one week, we calculate

        P(7) = 3(2⁷) = 384 grams of bacteria.

We call P(t) an exponential function with base 2.

Graphing Exponentials 

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

Notice the following properties

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:

Three Properties of Exponents  

Below are three properties of exponents that will be useful.

1. b^x b^y  =  b^x+y
   

2.    b^x
             =  b^x-y  
      b^y 
   

3. (b^x)^y  =  b^xy
   

^{We define negative exponents below}

Example

Simplify  

              3⁴(-3)^-1
                                  
                 (3²)³

Solution

              3⁴(-3)^-1                 3⁴(-3)^-1 
                                  =                             
                 (3²)³                       3⁶

                     -3⁴                   3⁴ 
             =                  =    -                
                   3⁶ 3¹                  3⁷

                     1                  1 
             =  -           =    -                
                    3³                27

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! 

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