4.3 use functions involving e

4.3 Use Functions Involving e

p. 244What 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, π, and imaginary numbers.

Natural Base e• Like π 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….

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

Use a calculator to evaluate the expression.

a. e4

b. e –0.09

Expression Keystrokes Display

54.59815003

0.9139311853

[ ] ex 4

[ ]ex 0.09

Simplify the expression.4. (10e –4x )3 = 103(e –4x ) 3

= 1000e12x

Use a calculator to evaluate5. e 3/4 .

2.117e 3/4 =

= 1000e –12x

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 & RangeBecause 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

Graph the function. State the domain and range.

SOLUTION

The domain is all real numbers, and the range is y > 1.

b. y = e –0.75(x – 2) + 1

a = 1 is positive and r = –0.75 is negative, so the function is an exponential decay function. Translate the graph of y = right 2 units and up 1 unit.

e –0.75x

Using e in real life.

• In 4.1 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

A = Pert

SOLUTION

FINANCE: You deposit $2500 in an account that pays 5% annual interest compounded continuously. Find the balance after each amount of time?

Use the formula for continuously compounded interest.

Write formula.

Substitute 2500 for P, 0.05 for r, and 2 for t.

≈2762.9

The balance at the end of 2 years is $2762.90.ANSWER

a. 2 years

= 2500e0.10

= 2500e (0.05 •2)

= 2500 •1.105

• 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

4.3 Assignment

Page 247, 3-48 every third problem