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• Graph
Warm up
y  3 1
x
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)
(1,2.7)
(0,1)
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 =
rt
Pe
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 carbon14 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.