1. 8.3 The Number e

2. Objectives/Assignment
Use the number e as the base of exponential functions
Use the natural base e
Assignment: odd

3. 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.

4. Natural Base e Like and ‘i’, ‘e’ denotes a number.
Called The Euler Number after Leonard Euler ( )
The number e is irrational – its’ decimal representation does not terminate or follow a repeating pattern.
As n gets larger (n→∞), (1+1/n)n gets closer and closer to …..
Which is the value of e.

5. e3 · e4 = e7 (3e-4x)2 9e(-4x)2 9e-8x 9 e8x 10e3 = 5e2 2e3-2 = 2e
Examples
e3 · e4 = e7
(3e-4x)2
9e(-4x)2
9e-8x 9
e8x
10e3 =
5e2
2e3-2 = 2e

6. More Examples!
(2e-5x)-2=
2-2e10x=
e10x 4
24e8 =
8e5
3e3

7. Using a calculator Evaluate e2 using a graphing calculator
7.389
Evaluate e2 using a graphing calculator
Locate the ex button
you need to use the second button

8. Evaluate e-.06 with a calculator

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

10. Graphing examples Graph y=ex
Remember the rules for graphing exponential functions!
The graph goes thru (0,a) and (1,e)
(1,2.7)
(0,1)

11. Graphing cont.
Graph y=e-x
(1,.368)
(0,1)

12. 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.
(1,4.23)
(0,2)

13. A = Pert 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

14. Example of continuously compounded interest
You deposit $ 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 ≈ $