1. 11.1: The Constant e and Continuous Compound Interest

2. Review (Mat 115)
Just like π, e is an irrational number which can not be represented exactly by any finite decimal fraction.
However, it can be approximated by
for a sufficiently large x e e e

3. The Constant e DEFINITION OF THE NUMBER e Reminder:
Use your calculator, e = …
DEFINITION OF THE NUMBER e

4. Check the limit using table and graph

5. Review Simple interest: A = P + Prt = P(1 + rt)
Compound Interest: A = P(1 + r)t or
with n = 1 (interest is compounded annually
– once per year)
Other compounding periods:
semiannually(2), quarterly(4), monthly(12),
weekly(52), daily(365), hourly(8760)…
Continuous Compounding: A = Pert
A: future value
P: principal
r: interest rate
t: number of years

6. Example 1: Generous Grandma
Your Grandma puts $1,000 in a bank for you, at 5% interest. Calculate the amount after 20 years.
Simple interest:
A = 1000 ( 20) = $2,000.00
Compounded annually:
A = 1000 ( )20 =$2,653.30
Compounded daily:
Compounded continuously:
A = 1000 e(.05*20) = $2,718.28

7. Example 2: IRA
After graduating from Barnett College, Sam Spartan landed a great job with Springettsbury Manufacturing, Inc. His first year he bought a $3,000 Roth IRA and invested it in a stock sensitive mutual fund that grows at 12% a year, compounded continuously. He plans to retire in 35 years.
What will be its value at the end of the time period?
A = Pert = 3000 e(0.12*35) =$200,058.99
The second year he repeated the purchase of an identical Roth IRA. What will be its value in 34 years?
A = Pert = 3000 e(0.12*34) =$177,436.41

8. Example 3 What amount (to the nearest cent) will an account have
after 5 years if $100 is invested at an annual nominal rate
of 8% compounded annually? Semiannually? continuously?
Compounded annually
Compounded semiannually
Compounded continuously
A = Pert = 100e(.08*5) =

9. Example 4
If $5000 is invested in a Union Savings Bank 4-year CD that earns 5.61% compounded continuously, graph the amount in the account relative to time for a period of 4 years.
Use your graphing calculator:
Press y=
Type in 5000e^(x*0.0561)
Press ZOOM, scroll down, then press ZoomFit
You will see the graph
To find out the amount after 4 years
Press 2ND, TRACE, 1:VALUE
Then type in 4, ENTER

10. Example 5 How long will it take an investment of $10000
to grow to $15000 if it is invested at 9% compounded
continuously?
Formula: A =P ert
15000 = e .09t
= e .09t
Ln (1.5) = ln (e .09t)
Ln (1.5) = .09 t
So t = ln(1.5) / .09
t = 4.51
It will take about 4.51 years

11. Example 6 How long will it take money to triple if it is
invested at 5.5% compounded continuously?
Formula: A =P ert
3P = P e .055t
3 = e .055t
Ln = ln (e .055t)
Ln = .055t
So t = ln3 / .055
t = 19.97
It will take about years

12. Review on how to solve exponential equations that involves e if needed (materials in MAT 115)