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

Complete the table below

Year 1 2 3 4 5Payout Amou

nt

8,800 9,680 10,648 11, 712 12,884

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.

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

Don’t FORGET

The decimal number of 10% = .1

5.8 % = = .058

Move the decimal point over 2 to the left.

100

8.5

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) = 88002nd year… (8000(1+.1))(1+.1) = 96803rd year… (8,000(1 +.1)(1+.1))(1 +.1)= $11, 712.80

trIP 1

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.

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.

trIP 1

20041.15000 P

20*4

4

041.15000

P

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

Compound interest

nt

n

rIP

1

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

WS 1, 3,5

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 = 2.252

1

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

4

1$1(1 + )4 = $2.44

12

1$1(1 + )12 = $2.61

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

In 1683, mathematician Jacob Bernoulli considered the value of

as n approaches infinity. His study was the first approximation of e

n

n

11

e= 2.718281828459045235460287471352662497757246093699959574077078727723076630353547594571382178525166427466391932003059921817413496629043572900338298807531952510190115728241879307…..

Comparable to an irrational number like ∏

Continuously Compounded Interest Equation

P = Iert

I= Initial Investment AmountP= Final Amount/payoutr = annual interest ratet= time in years

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