Page 171 - How To Solve Word Problems In Calculus

n
r
r
It can be shown that lim 1 + = e , so the amount of
n→∞ n
rt
money after t years compounded continuously is A = Pe .
The growth of money compounded continuously is expo-
nential.

EXAMPLE 5
Compute the amount of money in the bank after 10 years
when $1000 is compounded quarterly, monthly, daily, and
continuously at an annual rate of 6 percent.

Solution
40
0.06
Quarterly : A = 1000 1 + = $1814.02
4
120
0.06
Monthly : A = 1000 1 + = $1819.40
12
3650
0.06
Daily : A = 1000 1 + = $1822.03
365
Continuously : A = 1000e 0.6 = $1822.12

EXAMPLE 6
How long will it take money to double if it is compounded
continuously at an annual rate of 5 percent?

Solution
The amount of money we start with is irrelevant. The
important thing is that we end up with twice as much as we
started with. If we start with P dollars, we must end up with
2P dollars.

A = Pe rt

2P = Pe 0.05 t
2 = e 0.05 t

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