1. One lucky day, you find $8,000 on the street. At the Bank of Baker- that’s my bank, I am offering you an interest rate of 10% a year. Being the smart students you are, you invest your money at my bank. After the first year, your account collects 10% interest, so I would have to payout 8000+8000(.1)= $8,800 Or, 8,000(1 +.1) = $8,800 The second year, your $8,800 will collect even more interest and become 8,800(1 +.1) = 8,000(1 +.1)(1+.1)= $9,680

2. Complete the table below Year12345 Payout Amou nt 8,800 9,680 10,648 11, 71212,884 One lucky day, you find $8,000 on the street. At the Bank of Baker- that’s my bank. I am offering you an interest rate of 10% a year. Being the smart students you are, you invest your money at my bank.

3. Today’s objectives: 1.) Understand the real world situations and applications of logarithms and exponentials 2.) Learn the history of the number e and recognize its importance in mathematics

4. Don’t FORGET The decimal number of 10% =.1 5.8 % = =.058 Move the decimal point over 2 to the left.

5. Revisit warm-up problem If you make the initial investment(I) of $8,000 at Bank of Baker, and I offer an interest rate (r) of 10%, how much money will your investment grow to after 20 years? Write an equation of the payout with respect to years. 1 st year … 8000(1 +.1) = 8800 2 nd year… (8000(1+.1))(1+.1) = 9680 3 rd year… (8,000(1 +.1)(1+.1))(1 +.1)= $11, 712.80

7. Deal or No Deal? I will compound/apply your interest rate twice in one year. But I am going to cut your interest rate in half.

8. Deal or No Deal? You come to me with $5000. I have an interest rate of 4.1 %. You want to establish this amount in my bank for 20 years. What if I compound your investment quarterly. I will apply a compounded interest rate 4 times but I will divide the interest rate by 4.

9. Initial investment Interest rate in decimal form I will pay 4 times per year for 20 years, but as consequence I will divide interest rate by 4 11,168.24 11,305.21

10. Compound interest

11. Suppose Damon only has $3500 to invest but wants $4000 for a hot tub. He finds a bank offering 5.25% interest compounded quarterly. How long will he have to leave his money in the account to have it earn itself $4000. t = 2.56 years

12. WS 1, 3,5

13. Compound Interest: An account starts out with $1, and it pays an interest rate of 100% a year If the interest is “compounded” once a year, the value is 1(1+1) 1 = 2 If the interest is applied/compounded twice, I will apply interest twice that year, but I will half the interest rate. $1(1 + ) 2 = 2.25

14. Compound Interest: To compound the interest means I will apply the interest as many times as you want, but I will also divide your interest rate by as many times as I compound your investment. If the interest is “compounded” quarterly …? If the interest is compounded monthly $1(1 + ) 4 = $2.44 $1(1 + ) 12 = $2.61

15. What if you wanted to compound every minute, every second, every millisecond…?

16. In 1683, mathematician Jacob Bernoulli considered the value of as n approaches infinity. His study was the first approximation of e

17. e= 2.71828182845904523546028747135266249 7757246093699959574077078727723076630 3535475945713821785251664274663919320 0305992181741349662904357290033829880 7531952510190115728241879307….. Comparable to an irrational number like ∏

18. Continuously Compounded Interest Equation P = Ie rt I= Initial Investment Amount P= Final Amount/payout r = annual interest rate t= time in years

19. If Marie invests $2000 and it is compounded continuously for 30 years