Math 131	Grinnell College	Fall, 2009

Section 1: MWF 8:00-8:50, T 11:00-11:50 Section 2: MTWF: 10:00-10:50	Calculus I

Calculus Group Assignment: Exponential Growth and Decay

For each of the following problems, write a careful solution. That is,

When calculations are needed, you should show your work.
When statements are needed, you should use complete sentences with careful wording.
When a questions indicates "Justify your answer", you should write a thorough statement — usually a short paragraph.

Continuous Compounding of Interest

Suppose a savings account or CD at a bank has an initial balance of P, called the principal. Suppose further that the account pays r percent interest compounded n times per year. For example,

if interest is compounded annually, then n = 1,
if interest is compounded semi-annually, then n = 2,
if interest is compounded quarterly, then n = 4, and
if interest is compounded monthly, then n = 12.

To clarify, if interest is compounded once per year, the interest earned for the year is rP, and the balance at the end of the year is the initial principal plus the interest: P + rP = (1+r)P.

If the interest is compounded twice per year, then the interest for the first 6-month period is r/2 and the interest for the second 6-month period is the remaining r/2. The computation of interest for the first year is shown in the following table:

Compounding Period	Principal at Start of Period	Interest Earned during Period	Balance at End of Period (interest plus principal)
1	P	(r/2)P	(1 + (r/2))P
2	(1 + (r/2))P	(r/2)(1 + (r/2))P	(1 + (r/2))²P

If the interest is compounded four times per year, then the interest for the first 3-month period is r/4 and the interest for each subsequent 3-month period is also r/4. The computation of interest for the first year is shown in the following table:

Compounding Period	Principal at Start of Period	Interest Earned during Period	Balance at End of Period (interest plus principal)
1	P	(r/4)P	(1 + (r/4))P
2	(1 + (r/4))P	(r/4)(1 + (r/4))P	(1 + (r/4))²P
3	(1 + (r/4))²P	(r/4)(1 + (r/4))²P	(1 + (r/4))³P
4	(1 + (r/4))³P	(r/4)(1 + (r/4))³P	(1 + (r/4))⁴P

These computations illustrate the general formulae for compounding n times per year:

The balance at the end of 1 year: (1 + (r/n))ⁿP
The balance at the end of t years: (1 + (r/n))^ntP

As the number of compounding periods (n) increases, the actual amount of interest earned increases somewhat, since the interest earned in later periods is based on interest plus principal earned earlier in the year. As a limit, we could consider the result of compounding more and more times per year (i.e., taking the limit as n approaches infinity. This is called continuous compounding. The balance for an interest rate of r and continuous compounding for t years is:

The balance at the end of 1 year: B(1) = lim _{{n —> ∞}}(1 + (r/n))ⁿP
The balance at the end of t years: B(t) = lim _{{n —> ∞}}(1 + (r/n))^ntP

The first part of this lab as you to supply details for computing this limit.

Pages 418-419 of Stewart's Calculus shows that
e = lim _{{x —> 0}}(1 + x)^1/x
Using this as a starting point and setting x = r/n, show that
e = lim _{{n —> ∞}}(1 + r/n)^n/r
and
B(1) = e^rP
Do additional algebra to show that
B(t) = e^rtP
Compute the balance in a savings account after 1 year, starting with $1,000, paying 5% annual interest,
1. compounded annually
2. compounded quarterly
3. compounded monthly
4. compounded continuously
Suppose a savings account pays 5% annual interest compounded annually. How long does it take for the initial principal to double?
Suppose a savings account pays 5% annual interest compounded annually. If the account has $1,000 as its initial balance, how long does it take for the initial principal to grow to $1,500?

Exponential Growth

Refer to Section 7.5, pages 447-450, of Stewart's Calculus to answer these "Real" problems:

"Real" problem III.(75)
"Real" problem III.(81)
"Real" problem III.(83)

Work to Turn In

You must work in groups of 2 or 3 for this group assignment.

Solutions to each of the 7 problems specified.
Solutions should be organized into problem order.

This document is available on the World Wide Web as

http://www.cs.grinnell.edu/~walker/courses/131.fa09/groups/exponentials.shtml

created 16 August 2009 last revised 1 December 2009

For more information, please contact Henry M. Walker at walker@cs.grinnell.edu.