Transcript 8.3 The number e

4.3 Use Functions Involving e
p. 244
What is the Euler number?
How is it defined?
Do laws of exponents apply to “e” number?
How do you use “e” on your calculator?
When graphing base e, how do you know if
you have growth or decay?
What is the formula for continuously
compounded interest?
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, π, and imaginary
numbers.
Natural Base e
• Like π 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….
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
Use a calculator to evaluate the expression.
Expression Keystrokes
a. e4
b.
e –0.09
[ex]
[ex ]
Display
54.59815003
4
0.09
0.9139311853
Simplify the expression.
3
3
–4x
3
–4x
)
4. (10e ) = 10 ( e
= 1000 e –12x
1000
= e12x
5. Use a calculator to evaluate e 3/4.
e 3/4 = 2.117
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)
y=0
Graphing cont.
• Graph y=e-x
(0,1)
y=0
(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)
y=0
Graph the function. State the domain and range.
b.
y = e –0.75(x – 2) + 1
SOLUTION
a = 1 is positive and r = –0.75
is negative, so the function
is an exponential decay
function. Translate the graph
of y = e –0.75x right 2 units and
up 1 unit.
The domain is all real numbers, and the range
is y > 1.
Using e in real life.
• In 4.1 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
Continuously Compounded Interest
A=
rt
Pe
“Shampoo”
Problems
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
FINANCE: You deposit $2500 in an account that
pays 5% annual interest compounded
continuously. Find the balance after each amount
of time?
a. 2 years
SOLUTION
Use the formula for continuously compounded interest.
A = Pert
Write formula.
= 2500e (0.05 •2)
= 2500 e0.10
= 2500 •1.105
≈2762.9
Substitute 2500 for P, 0.05 for r, and 2 for t.
ANSWER
The balance at the end of 2 years is $2762.90.
• What is the Euler number?
Natural base e
• How is it defined?
2.718 - - it is an irrational number like pi
• Do laws of exponents apply to “e” number?
Yes- - all of them.
• How do When graphing base e, how do you know if
you have growth or decay?
Growth rises on the right and decay rises on the
left.
• What is the formula for continuously compounded
interest?
Pert
4.3 Assignment
Page 247, 3-48
every third
problem