Warm up• Graph 13 xy

Lesson 11-3 The Number e

Objective: To use the exponential function y = ex

Natural Base e• Like 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.

Using a calculator

• Evaluate e2 using a graphing calculator

• Locate the ex button • you need to use the

second button

7.389

Graphing examples

• Graph y=ex

• Remember the rules for graphing exponential functions!

• The graph goes thru (0,1) and (1,e)

(0,1)

(1,2.7)

Graphing cont.

• Graph y=e-x

(0,1) (1,.368)

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

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

Practice• An amount of $1,240.00 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

Exponential Decay• An artifact originally had 12 grams of carbon-14

present. The decay model A = 12e-0.000121t

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-0.000121t

A = 12e-0.000121(10,000)

A = 12e-1.21

A = 3.58

Sources

• myteacherpages.com/webpages/rrowe, 2/22/14.