7.3 Use Functions Involving
e
Algebra II
7.3 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.
• 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.
Ex. 1 Simplify the expression.
a.)
• e3 · e4 =
7
•e
b.)
•10e3 =
2
5e
•2e3-2 =
•2e
c.)
-4x
2
•(3e )
•9e(-4x)2
9
e8x
Ex.2 Simplify the expression.
a.)
• 24e8 =
5
8e
3
• 3e
b.)
•2-2e10x=
10x
•e
4
Using a calculator
• Evaluate e2 using a
scientific calculator
• Locate the ex button
• you need to use the
second button
7.389
-.06
e
Evaluate
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
Ex. 3a
• 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 3b
• Graph y=e-x
(0,1)
(1,.368)
Graphing Ex. 4
• 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 7.1 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
Ex. 6) Continuously Compounded
Interest
• You deposit $1000.00 into an account
that pays 8% annual interest
compounded continuously. What is the
balance after 1 year?
Ex. 6) 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