11.1: The Constant e and Continuous Compound Interest. 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.
for a sufficiently large x
Use your calculator, e = 2.718 281 828 459 …
DEFINITION OF THE NUMBER e
with n = 1 (interest is compounded annually
– once per year)
semiannually(2), quarterly(4), monthly(12),
weekly(52), daily(365), hourly(8760)…
A = Pert
A: future value
r: interest rate
t: number of years
Your Grandma puts $1,000 in a bank for you, at 5% interest. Calculate the amount after 20 years.
A = 1000 (1 + 0.0520) = $2,000.00
A = 1000 (1 + .05)20 =$2,653.30
A = 1000 e(.05)(20) = $2,718.28
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?
A = Pert = 100e(.08*5)
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.Example 4
How long will it take an investment of $10000 that earns 5.61% compounded continuously, graph the amount in the account relative to time for a period of 4 years.
to grow to $15000 if it is invested at 9% compounded
Formula: A =P ert
15000 = 10000 e .09t
1.5 = 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 yearsExample 5
How long will it take money to triple if it is that earns 5.61% compounded continuously, graph the amount in the account relative to time for a period of 4 years.
invested at 5.5% compounded continuously?
Formula: A =P ert
3P = P e .055t
3 = e .055t
Ln 3 = ln (e .055t)
Ln 3 = .055t
So t = ln3 / .055
t = 19.97
It will take about 19.97 yearsExample 6