Complete the table below

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)= $8,800Or, 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 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. 9,680 11, 712 12,884 10,648

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. 1st year … 8000(1 +.1) = 8800 2nd year… (8000(1+.1))(1+.1) = 9680 3rd 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. 11,168.24 Interest rate in decimal form Initial investment I will pay 4 times per year for 20 years, but as consequence I will divide interest rate by 4 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= 2.718281828459045235460287471352662497757246093699959574077078727723076630353547594571382178525166427466391932003059921817413496629043572900338298807531952510190115728241879307….. Comparable to an irrational number like ∏

18. Continuously Compounded Interest Equation P= Iert 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