In this section we will pose a number of short problems involving
compound interest and the Mathematica functions that you've learned
about so far. You are to use a Mathematica notbook to prepare the
answers to these questions. For each question, your report should
contain the text of the question as given below, your answer to the
question, and the output from Mathematica that shows how you arrived
at your answer.
When you are done with your report, you should save it to a file
called hw2.ma and submit if for grading.
(1) Between 1970 and 1990, the population of the United States
increased from approximately 200 million to approximately 250
million. What has been the average annual rate of population
increase during that period? What will the population of the United
States be in 2090 if this rate of increase continues?
(2) Derive a function that takes an interest rate R as its argument and
returns the number of years that it would take an investment at that rate to
double in value under continuous compounding.
(3) Derive a function triple that takes an interest rate R as its
argument and returns the number of years it would take an investment
to triple in value under interest that is compounded annually.
(4) Derive a good approximation to triple (called approx). Like
the approximation to double earlier in this lesson, it should involve
only a single division.
(5) Consider a bank account that pays interest R, compounded m times per
year. We say that this account is effectively paying interest at an
annual rate of S if an investment P in the account will be worth P(1+S)
at the end of the year. Define a function effective that takes an
interest rate and a compounding interval as arguments and returns the effective
annual interest rate.
(6) By taking the limit of your effective function as the number of
compounding intervals approaches infinity, determine the effective interest
rate of a continuously compounded investment.