文档介绍：Section
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Models of Growth and Decay
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In this section we will look at several applications of the exponential and logarithm func-
tions to problems involving growth and decay, pound interest, radioactive
decay, and population growth.
Compound interest
Suppose a principal of P dollars is deposited in a bank which pays 100i% -
pounded n times a year. That is, each year is divided into n units and after each unit of
100i
time the bank pays n % interest on all money currently in the account, including money
that was earned as interest at an earlier time. Thus if xm represents the amount of money
in the account after m units of time, xm must satisfy the difference equation
i
x − x = x , ()
m+1 m n m
m = 0, 1, 2, . . ., with initial condition x0 = P . Hence the sequence {xm} satisfies the linear
difference equation
i
xm+1 = 1 + xm, ()
n
and so, from our work in Section , we know that
i m i m
xm = 1 + x0 = 1 + P ()
n n
for m = 0, 1, 2, . . .. If we let A(t) be the amount in the account after t years, then, since
there are pounding periods in t years,
i nt
A(t) = xnt = 1 + P ()
n
Example Suppose $1,000 is deposited at 5% interest which pounded quarterly. If
A(t) is the amount in the account after t years, then, for example,
20
A(5) = 1000 1 + = 1, ,
4
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2 Models of Growth and Decay Section
rounded to the nearest cent. If the interest pounded monthly instead, then we
would have
60
A(5) = 1000 1 + = 1, .
12
Of course, the more frequent pounding, the faster the amount in the account
will grow. At the same time, there is no limit to how often the bank pound.
However, is there some limit to how fast the account can grow? That is, for a fixed value
of t, is A(t) bounded as n grows? To answer this question, we need to consider
i nt
lim 1 + .
n→∞ n
To evalu