6-4 The Number “e”

1. 6-4 6-4The Number “e”

2. 6-4 Compound Interest: • A(t) = Amount after time t • P = Principal/initial amount • r = interest rate • t = time (in years!) • n = number of times the $ is compounded per year

3. 6-4 Interest rate vs. Yield: Interest rate is the rate of change of your money. Yield is how much you actually get because with compound interest, you earn interest on your interest. So: If one invests $5000 with an interest rate of 2.75% and it is compounded monthly, how much did that person yield after one year? Percent yield vs. amount yield…

4. 6-4 Investigation: We are taking our money to a fantasy bank: We are only investing $1, but we are earning 100% interest. Let’s see how much money we earn after one year if the number of times we compound increases… Let’s compound quarterly: Now compound monthly: Daily? How about secondly?

5. 6-4 Continuous Change Model What we just saw was the occurrence of the natural number “e”. Let “e” mean that we are compounding “e”very possible measure of time. We call this continuous compounding. If a quantity P grows or decays continuously at an annual rate r, the amount A(t) is given by:

6. 6-4 Examples: If you invest $10,000 in bonds at 3.35% compounded continuously, • What is the value of the investment after 1 year? • What is the percent annual yield?

7. 6-4 Examples: If $10,000 is put into a 5 year CD paying 3.25% compounded continuously, • What is the balance at the end of the period? • How does this compare if the interest were compounded annually?

8. 6-4 The Natural Logarithmic Function: The Natural Log = base “e” • Written: ln x which implies loge x • All properties of other log functions still apply because we have only changed the base. • Graph is the same as other log graphs • Solving natural logs follow the same definition, and the same rules…

9. 6-4 Examples: Evaluate without a calculator: ln e ln e4 Evaluate with a calculator: ln 27 Solve: ex = 16