Section 5-3:  Exponential Functions
 

Demo: Exponential Applet (Kennesaw State University)
 
Any function in the form f(x) = abx, where a > 0, b > 0 and b not equal to 1 is called an exponential function with base b. Let's take a look at a couple of simple exponential graphs.
f(x) = 2x
 
x f(x)
3 8
2 4
1 2
0 1
-1 1/2
-2 1/4
-3 1/8
 
 
Notice the domain is all real numbers and the range is y > 0.  As x gets larger (right), y gets very large.  As x gets smaller(left), y approaches zero asymptotically.  Notice also that the graph crosses the y-axis at (0, 1).  The above is the general shape of an exponential with b > 1.  This is an example of exponential growth.

 Now let's look at the graph of
f(x) = (1/2)x
 
x f(x)
3 1/8
2 1/4
1 1/2
0 1
-1 2
-2 4
-3 8
 
 
 Observe that this graph is the reflection about the y-axis of the first graph.  The domain is still all real numbers and the range is y > 0.  The y-intercept is (0, 1).  This is the general form of an exponential graph if 0 < b < 1.  It is an example of an exponential decay.

 
 Look at the following graphs that illustrate the general properties of exponentials.
Do you see the similarities of each graph?
How about this one?

Many of the functions associated with exponential growth or decay are functions of time.  We have already had one form:
A(t) = Ao(1 + r)t
A second form looks like:
A(t) = Aobt/k
where k = time needed to multiply Ao by b

 Rule of 72
 
If a quantity is growing at r% per year then the doubling time is approximately 72/r  years.

For example, if a quantity grows at 10% per year, then it will take 72/10 or 7.2 years to double in value.  In other words, it will take you 7.2 years to double your money if you put it into an account that pays 10% interest.  At the current bank rate or 2%, it will take you 72/2 or 36 years to double your money!!  Boy, jump all over that investment!!

Sample Problems
 
1)  Suppose you invest money so that it grows at A(t) = 1000(2)t/8 
                 a)  How much money did you invest?
                 b)  How long will it take to double your money?
 
                     Solutions:
                         a)  The original amount in the formula is $1000.
                         b)  This means what time will it take to get $2000.
2000 = 1000(2)t/8
2 = 2t/8
1 = t/8
8 = t
It will take 8 years to double your money!!
 
2)  Suppose that t hours from now the population of a bacteria colony is given by:  P(t) = 150(100)t/10
                     a)  What is the initial population?
                     b)  How long does it take for the population to be multiplied by
                          100?
                     c)  What is the population at t = 20?
 
                     Solutions:
                         a)  It is 150 from the original equation.
                         b)  It takes 10 hours.  That's the definition of the exponential
                              function.
                         c)  P(20) = 150(100)20/10 = 150(100)2 = 1,500,000
 
3)  The half life of a substance is 5 days.  We have 4 kg present now.
                     a)  Write a formula for this decay problem.
                     b)  How much is left after 10 days?  15 days?  20 days?
 
                         Solutions:
                             a)  A(t) = 4(1/2)t/5
                             b)  A(10) = 4(1/2)10/5 = 4(1/2)2 = 1 kg.
                                   A(15) = 4(1/2)15/5 = 4(1/2)3 = 1/2 kg.
                                   A(20) = 4(1/2)20/5 = 4(1/2)4 = 1/4 kg.
 
4)  The value of a car is given by the equation V(t) = 6000(.82)t
                     a)  What is the annual rate of depreciation?
                     b)  What is the current value?
                     c)  What will be the value in three years?
 
                     Solutions:
                         a)  It is 1 - .82 or .18 = 18%
                         b)  The current value is given in the formula, $6000.
                         c)  V(3) = 6000(.82)3 = 3308.21 Which is $3308.21

On to the number e!! 
 
Back up, I'm lost!!