Transcript 8.3 The number e

7.3 The number e
p. 492
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.
Natural Base e
• Like Л 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.
Examples
• ·
7
•e
3
e
4
e
=
3
•10e =
5e2
3-2
=
•2e
•2e
-4x
2
•(3e )
(-4x)2
•9e
•9e-8x
• 9
e8x
More Examples!
•
8
24e =
5
8e
• 3e3
-5x
-2
•(2e ) =
•2-2e10x=
10x
•e
4
Using a calculator
2
• Evaluate e using
a graphing
calculator
• Locate the ex
button
• you need to use
the second button
7.389
Evaluate e-.06
with a calculator
Graphing
• f(x) =
rx
ae 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)
(1,2.7)
(0,1)
Graphing cont.
• Graph y=e-x
(0,1)
(1,.368)
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)
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 =
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
Assignment