Exponential Applet (Kennesaw State University)
Any function in the
f(x) = abx,
a > 0, b > 0 and b not equal to 1 is called an exponential
function with base b. Let's take a look
a couple of simple exponential graphs.
f(x) = 2x
Notice the domain
all real numbers and the range is y > 0. As x gets larger
y gets very large. As x gets smaller(left), y approaches zero
Notice also that the graph crosses the y-axis at (0, 1). The
is the general shape
of an exponential with b > 1.
This is an example of exponential growth.
at the graph of
f(x) = (1/2)x
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
is (0, 1). This is the general form of an exponential graph if 0
< b < 1. It is an example of an exponential
Look at the
graphs that illustrate the general properties of exponentials.
Do you see the
of each graph?
How about this one?
Many of the functions
with exponential growth or decay are functions of time. We have
had one form:
A(t) = Ao(1 +
A second form looks like:
A(t) = Aobt/k
where k = time needed to
Ao by b
Rule of 72
If a quantity is
at r% per year then the doubling time is approximately 72/r years.
For example, if a
grows at 10% per year, then it will take 72/10 or 7.2 years to double
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
current bank rate or 2%, it will take you 72/2 or 36 years to double
money!! Boy, jump all over that investment!!
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?
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
to double your money!!
2) Suppose that t
from now the population of a bacteria colony is given by: P(t) =
a) What is the initial population?
b) How long does it take for the population to be multiplied by
c) What is the population at t = 20?
a) It is 150 from the original equation.
b) It takes 10 hours. That's the definition of the
c) P(20) = 150(100)20/10 = 150(100)2 =
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?
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?
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
Back up, I'm