Real Exponents

11-3
OBJECTIVES
• Use the
exponential
function y e x .
The Number e
Real World
MEDICINE Swiss entomologist
Dr. Paul Mueller was awarded the
Nobel Prize in medicine in 1948
for his work with the pesticide DDT. Dr. Mueller
discovered that DDT is effective against insects that
destroy agricultural crops, mosquitoes that transmit
malaria and yellow fever, as well as lice that carry
typhus.
It was later discovered that DDT presented a risk
to humans. Effective January 1, 1973, the United
States Environmental Protection Agency banned all
uses of DDT. More than 1.0 10 10 kilograms of
DDT had been used in the U.S. before the ban. How much will remain in the
environment in 2005? This problem will be solved in Example 1.
A p plic atio n
DDT degrades into harmless materials over time. To find the amount of a
substance that decays exponentially remaining after a certain amount of time,
you can use the following formula for exponential growth or decay, which
involves the number e.
Exponential
Growth or
Decay
(in terms of e)
N N 0
e kt , where N is the final amount, N 0
is the initial amount, k is a
constant and t is time.
The number e in the formula is not a variable. It is a special irrational
number. This number is the sum of the infinite series shown below.
e 1 1 1 1
1 2 1
1
… 1
1 2 3 1 2 3 4 1 2 3 … …
n
The following computation for e is correct to three decimal places.
e 1 1 1 1
1 2 1
1
1
1 2 3 1 2 3 4 1 2 3 4 5
1
1
1 2 3 4 5 6 1 2 3 4 5 6 7
1 1 1 2 1 6 1 1
24
1 20 1
720 1
5040
1 1 0.5 0.16667 0.04167 0.00833
0.00139 0.000198
2.718
y
y e x
The function y e x is one of the most important
exponential functions. The graph of y e x is shown
at the right.
O
x
712 Chapter 11 Exponential and Logarithmic Functions