16-week Lesson 30 (8-week Lesson 24) Interest Compounded Continuously
1
As shown in Lesson 29, one application of exponential functions is
compound interest, which is when interest is calculated on the total value
of a sum and not just on the principal like with simple interest. We saw in
Lesson 29 that one way interest can be compounded is ๐ times per year,
where ๐ represents some number of compounding periods (quarterly,
monthly, weekly, daily, etc.). The other way interest can be compounded
is continuously, where interest is compounded essentially every second of
every day for the entire term. This means ๐ is essentially infinite, and so
we will use a different formula which contains the natural number ๐ to
calculate the value of an investment. The formula for interest
compounded continuously is ๐ด = ๐๐๐๐ก.
Formula for Interest Compounded Continuously:
- when interest is compounded continuously, we use the formula
๐ด = ๐๐๐๐ก
o when interest is compounded continuously, there are essentially
an infinite number of compounding periods (๐ โ โ), so that is
why we use the natural number ๐
o ๐ด is the accumulated value of the investment
o ๐ is the principal (the original amount invested)
o ๐ is the annual interest rate
o ๐ก is the number of years the principal is invested (the term)
Example 1: If $17,000 is invested at a rate of 6.25% per year for
39 years, find value of the investment to the nearest penny if the interest is
compounded continuously. Use either ๐ด = ๐ (1 + ๐
๐)๐๐ก or ๐ด = ๐๐๐๐ก.
๐ด = ๐๐๐๐ก
๐ด = 17000๐(0.0625)(39)
๐ด = 17000๐2.4375
๐ด = 17000(11.44439396โฆ)
๐จ = $๐๐๐,๐๐๐.๐๐
