1. Section 5-4 The Number e and the Function

2. The number e You have already seen many exponential functions. In advanced mathematics, the most important base is the irrational number e, defined as This is read “the limit of as n approaches infinity.”

3. Activity Complete activity on p. 187.

4. Activity You can see that as n increases, appears to get closer and closer to 2.718… The Swiss mathematician Leonhard Euler proved this idea and the limit is called e in his honor.

5. The function Values of can be obtained using a calculator or table (p. 821). The function is called the natural exponential function. The number e is extremely important in advanced mathematics. Look at the graphs on p. 187

6. Compound Interest and the Number e The formula for compound interest is Where P 0 is the initial amount (or principal), r is the interest rate, n is the number of times the interested is compounded, and t is the amount of time the money is in the bank.

7. Continuous Compounding In general if you invest P (the principal) dollars at an annual rate r (expressed as a decimal), compounded continuously, then t year later your investment will be worth dollars.

8. Applications of Continuous Compounding The same principle applies to any quantity, such as population, where compounding takes place “all the time.” If is the initial amount, then the amount at any future time is

9. Additional Example: 1. Which plan yields the most interest? Plan A: A 7.5% annual rate compounded monthly for 4 years. Plan B: A 7.2% annual rate compounded daily for 4 years. Plan C: A 7% annual rate compounded continuously for 4 years.

10. Example: 2. Use the graph of y = to graph (a)(b)