1. Warm up
Graph

2. Objective: To use the exponential function y = ex
Lesson 11-3 The Number e
Objective: To use the exponential function y = ex

3. Natural Base e Like and ‘i’, ‘e’ denotes a number.
Called The Euler Number after Leonhard Euler ( )
It can be defined by:
e= …
0! 1! 2! 3! 4! 5!
= ½ + 1/6 + 1/24 + 1/
≈ ….

4. 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 …..
Which is the value of e.

5. 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

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

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

8. A = Pert Using e in real life.
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

9. 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 ≈ $

10. Practice
An amount of $1, is deposited in a bank paying an annual interest rate of 2.85 %, compounded continuously. Find the balance after 2½ years.
A = 1240e(.0285)(2.5)
= $1,331.57
A = 1240e(.0285)(2.5)
= $1,331.57

11. Exponential Decay
An artifact originally had 12 grams of carbon-14 present.  The decay model A = 12e t describes the amount of carbon-14 present after t years.  How many grams of carbon-14 will be present in this artifact after 10,000 years?
A = 12e t
A = 12e (10,000)
A = 12e-1.21
A = 3.58

12. Sources
myteacherpages.com/webpages/rrowe, 2/22/14.