8.3 The number ep. 480

What is the Euler number?How is it defined?

Do laws of exponents apply to “e” number?

How do you use “e” on your calculator?When graphing base e, how do you know

if you have growth or decay?What is the formula for continuously

compounded interest?

The Natural base e

• Much of the history of mathematics is marked by the discovery of special types of numbers like counting numbers, zero, negative numbers, Л, and imaginary numbers.

Natural Base e• Like Л and ‘i’, ‘e’ denotes a number.

• Called The Euler Number after Leonhard Euler (1707-1783)

• It can be defined by:

e= 1 + 1 + 1 + 1 + 1 + 1 +…

0! 1! 2! 3! 4! 5!

= 1 + 1 + ½ + 1/6 + 1/24 +1/120+...

≈ 2.718281828459….

• The number e is irrational – its’ decimal representation does not terminate or follow a repeating pattern.

• The previous sequence of e can also be represented:

• As n gets larger (n→∞), (1+1/n)n gets closer and closer to 2.71828…..

• Which is the value of e.

Examples

• e3 · e4 =

• e7

•10e3 = 5e2

•2e3-2 =

•2e

•(3e-4x)2

•9e(-4x)2

•9e-8x

• 9 e8x

More Examples!

• 24e8 =

8e5

• 3e3

•(2e-5x)-2=•2-2e10x=• e10x

4

Using a calculator

• Evaluate e2 using a graphing calculator

• Locate the ex button

• you need to use the second button

7.389

Evaluate e-.06 with a calculator

Graphing

• f(x) = aerx is a natural base exponential function

• If a>0 & r>0 it is a growth function

• If a>0 & r<0 it is a decay function

Graphing examples

• Graph y=ex

• Remember the rules for graphing exponential functions!

• The graph goes thru (0,a) and (1,e)

(0,1)

(1,2.7)

y=0

Graphing cont.

• Graph y=e-x

(0,1) (1,.368)

y=0

Graphing Example

• Graph y=2e0.75x

• State the Domain & Range

• Because a=2 is positive and r=0.75, the function is exponential growth.

• Plot (0,2)&(1,4.23) and draw the curve.

(0,2)

(1,4.23)

y=0

Using e in real life.

• In 8.1 we learned the formula for compounding interest n times a year.

• In that equation, as n approaches infinity, the compound interest formula approaches the formula for continuously compounded interest:

•A = Pert

Continuously Compounded Interest

A = Pert

Shampoo Problems

Example of continuously compounded interest

• You deposit $1000.00 into an account that pays 8% annual interest compounded continuously. What is the balance after 1 year?

• P = 1000, r = .08, and t = 1

•A=Pert = 1000e.08*1 ≈ $1083.29

• What is the Euler number?Natural base e• How is it defined?2.718 - - it is an irrational number like pi• Do laws of exponents apply to “e” number?Yes- - all of them.• How do When graphing base e, how do you

know if you have growth or decay?Growth rises on the right and decay rises on the

left.• What is the formula for continuously

compounded interest?Pert

Assignment

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