Section 3.4
Continuous Growth and the Number e
• Let’s say you just won $1000 that you would
like to invest. You have the choice of three
different accounts:
– Account 1 pays 12% interest each year
– Account 2 pays 6% interest every months (this is
called 12% compounded semi-annually)
– Account 3 pays out 1% interest every month (this
is called 12% compounded monthly)
• Do all the accounts give you the same return
after one year? What about after t years?
• If not, which one should you choose?
• If an annual interest r is compounded n times
per year, then the balance, B, on an initial
deposit P after t years is
nt
 r
B  P 1  
 n
• For the last problem, figure out the growth
factors for 12% compounded annually, semiannually, monthly, daily, and hourly
– We’ll put them up on the board
– Also note the nominal rate versus the effective rate
• The nominal rate for each is 12%
• Now let’s look at continuously compounded
rt
• We get B  Pe
• Find the growth rate for 12%
– How does it compare to our previous growth rates?
• Find the formula for our $1000 compounded
continuously for t years
• Now 2 < e < 3 so what do you think we can
say about the graph of Q(t) = et?
– What about the graph of f(t) = e-t
• It turns out that the number e is called the
natural base
– It is an irrational number introduced by Lheonard
Euler in 1727
– It makes many formulas in calculus simpler which
is why it is so often used
• Consider the exponential function Q(t) = aekt
– Then the growth rate (or decay rate) is ek
• So from y = abt, b = ek
– If k is positive then Q(t) is increasing and k is
called the continuous growth rate
– If k is negative then Q(t) is decreasing and k is
called the continuous decay rate
• Note: for the above cases we are assuming a > 0
Example
• Suppose a lake is evaporating at a continuous
rate of 3.5% per month.
– Find a formula that gives the amount of water
remaining after t months
– What is the decay factor?
– By what percentage does the amount of water
decrease each month?
Example
• Suppose that $500 is invested in an account
that pays 8%, find the amount after t years if it
is compounded
–
–
–
–
Annually
Semi-annually
Monthly
Continuously
• Find the effective rate for 8% compounded
annually
• In your groups try problems 3, 11, and 16