Section 5.2a The Natural Exponential Function
Objective 1: Understanding the Characteristics of the Natural Exponential Function
The number e is an irrational number that is defined as the value of the expression
1
1n
n
as n approaches
infinity. The table below shows the values of the expression
1
1n
n
for increasingly large values of n.
As the values of n get large, the value e (rounded to 6 decimal places) is
2.718281
. The function
( ) x
f x e
is
called the natural exponential function.
The graph of the natural
exponential function
( ) x
f x e
Characteristics of the Natural Exponential Function
The Natural Exponential Function is the exponential function with base e and is defined as
( ) x
f x e
.
The domain of
( ) x
f x e
is
,
and the range is
0,
.
The graph of
( ) x
f x e
intersects the y-axis at
0 1,
.
as
x
e x
0 as
x
e x
The line
0y
is a horizontal asymptote.
The function
( ) x
f x e
is one-to-one.
5.2.1 and 4
Use a calculator to approximate the exponential expression to 6 decimal places.
n
1
1n
n
1 2
2 2.25
10 2.5937424601
100 2.7048138294
1000 2.7169239322
10,000 2.7181459268
100,000 2.7182682372
1,000,000 2.7182804693
10,000,000 2.7182816925
100,000,000 2.7182818149
2
x
y
3
x
y
( )
x
f x e
( )
x
f x e
