8.3 The number e

p. 480

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

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

Homework

•P. 483 (17-73) odd