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 For the last problem, figure out the growth factors for 12% compounded annually, semi- annually, 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 We get 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) = e t ? –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) = ae kt –Then the growth rate (or decay rate) is e k So from y = ab t, b = e k –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