Exponential Base-e - Intermediate Algebra - Lecture Slides, Slides of Algebra

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§9.1b

The Base e

Review §

 Any QUESTIONS About

• §9.1 → Exponential Functions, base a

 Any QUESTIONS About HomeWork

• §9.1 → HW-

9.1 MTH 55

Compound Interest  Terms

• TIME: Suppose P dollars is borrowed. The borrower agrees to pay back the initial P dollars, plus the interest amount, within a specified period. This period is called the time (or time-period) of the loan and is denoted by t.
• SIMPLE INTEREST ≡ The amount of interest computed only on the principal is called simple interest.

Compound Interest  Terms

• INTEREST RATE: The interest rate is the percent charged for the use of the principal for the given period. The interest rate is expressed as a decimal and denoted by r.
• Unless stated otherwise, it is assumed the time-base for the rate is one year ; that is, r is thus an annual interest rate.

Example  Calc Simple Interest

• Rosarita deposited $8000 in a bank for 5 years at a simple interest rate of 6% a) How much interest will she receive? b) How much money will she receive at the end of five years?
• SOLUTION a) Use the simple interest formula with: P = 8000, r = 0.06, and t = 5

Example  Calc Simple Interest

• SOLUTION a) Use Formula

I = Prt

I = $8000 0.06( )( ) 5

I = $

 SOLUTION b) The total amount, A , due her in five years is the sum of the original principal and the interest earned

A = P + I

A = $8000 + $

A = $10, 400

Compare Compounding Periods

• One hundred dollars is deposited in a bank that pays 5% annual interest. Find the future-value amount, A , after one year if the interest is compounded: a) Annually. b) SemiAnnually. c) Quarterly. d) Monthly. e) Daily.

Compare Compounding Periods

• SOLUTION

In each of the computations that follow, P = 100 and r = 0.05 and t = 1. Only n , the number of times interest is compounded each year, is changing. Since t = 1, nt = n ∙1 = n. a) Annual Amount:

A = P 1 +

r n

^

n

A = 100 1( + 0.05) = $105.

Compare Compounding Periods

d) Monthly Amount: A^ =^ P^^1 +^

r 12

^

12

A = 100 1 +

^

12 = $105.

A = P 1 +

r 365

^

365

A = 100 1 +

^

365 = $105.

e) Daily Amount:

The Value of the Natural Base e

• The number e , an irrational number, is sometimes called the Euler constant.
• Mathematically speaking, e is the fixed number that the expression (^1) +

1 h

 ^

 

h

approaches e as h gets larger & larger

 The value of e to 15 places:

e = 2.

Example  Continuous Interest

• Find the amount when a principal of $ is invested at a 7.5% annual rate of interest compounded continuously for eight years and three months.
• SOLUTION: Convert 8-yrs & 3-months to 8.25 years. P = $8300 and r = 0.075 then use Formula

A = Pe rt

A = $8300 e (0.075^ )(^ 8.25^ ) A = $15, 409.

Compare Continuous Compounding

• Italy's Banca Monte dei Paschi di Siena (MPS), the world's oldest bank founded in 1472 and is today one the top five banks in Italy
• If in 1797 Thomas Jefferson Placed a Deposit of $450k the MPS bank at an interest rate of 6%, then find the value $-Amount for the this Account Today, 213 years Later

Account $Value for $450k invested at 6% Interest for 213 Years

0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170

Continuous

Quarterly

Yearly

Simple

Interest Compounding

Account Value ($B) (^) M55_Sec9_1_Compare_Compounding_0810.xls

The NATURAL Exponential Fcn

• The exponential function

 with base e is so prevalent in the

sciences that it is often referred to

as THE exponential function or the

NATURAL exponential function.

f (^) ( ) x = e

x