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 times per year,
where 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 is essentially infinite, and so
we will use a different formula which contains the natural number to
calculate the value of an investment. The formula for interest
compounded continuously is
Formula for Interest Compounded Continuously:
- when interest is compounded continuously, we use the formula
o when interest is compounded continuously, there are essentially
an infinite number of compounding periods
, so that is
why we use the natural number
o is the accumulated value of the investment
o is the principal (the original amount invested)
o is the annual interest rate
o is the number of years the principal is invested (the term)
Example 1: If is invested at a rate of per year for
years, find value of the investment to the nearest penny if the interest is
compounded continuously. Use either
or
.
16-week Lesson 30 (8-week Lesson 24) Interest Compounded Continuously
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When working with compound interest formulas, remember to keep
in mind order of operation (PEMA):
1. simplify parentheses
2. simplify exponents
3. simplify multiplication/division, working from left to right
4. simplify addition/subtraction, working from left to right
Example 2: If is invested at a rate of per year compounded
continuously, find value of the investment at each given time and round to
the nearest cent. Use either
or
.
a. months b. months c. years d. years
For each of these problems you will use the formula
since
interest is compounded continuously. The principal will be for
each problem part
and the interest rate will be
. However the term will vary from part to part:
Once again do your best to leave all calculated values in your calculator.
For instance when calculating
from Example 2 part a,
do not calculate
and then try to write that down on paper to 5 or 6
decimal places. Leave calculated values in your calcul ator to avoid
approximating.
Once again for help with entering expressions such as
in your calculator, take a look at the
Calculator Tips document in Brightspace or stop by my office
hours. Also, be sure to use the same calculator on homework
(handheld or computer calculator) as you will on the exam.
16-week Lesson 30 (8-week Lesson 24) Interest Compounded Continuously
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Example 3: A recent college graduate decides to open a credit card in
order to pay for their upcoming trip across Europe. In order to get a card
with a large enough credit limit to pay for their trip
, the student
agrees to an interest rate of compounded continuously. If no
payments are made for an entire year, what will be the balance on the card
rounded to the nearest penny? Use either
or
.
Example 4: Parents of a newborn baby are given a gift of and
will choose between two options to invest for their child’s college fund.
Option 1 is to invest the gift in a fund that pays an average annual interest
rate of compounded semiannually; option 2 is to invest the gift in a
fund that pays an average annual interest rate of compounded
continuously. Assuming each investment has a term of years, calculate
the value of each investment and round your answer to the nearest penny.
Use either
or
.
If the rates are the same, which is the better option for the parents?
16-week Lesson 30 (8-week Lesson 24) Interest Compounded Continuously
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Answers to Examples:
1. ; 2a. ; 2b. ; 2c.
; 2d. ; 3. ;
4.
;
