Math 2
B
Dr. C. Famiglietti
************************************************************************
Section 6.3:
The Exponential Function
The inverse of the natural logarithmic function (presented in Section 6.1) is the natural
exponential function
y
=
e
x
, where
e
≈
2.71828
…
Since the two are inverse functions, the graph of
y
=
e
x
can be obtained from the graph of
y
=
ln
x
by reflecting the graph of
y
=
ln
x
about the line
y
=
x
.
The graph of
y
=
e
x
is shown here
and has the following characteristics:
Domain:
−∞
,
∞
( )
Range:
0,
∞
( )
Increasing throughout its domain
Concave upward
Passes through
0,1
( )
since
e
0
=
1
lim
x
→∞
e
x
=
∞
lim
x
→−∞
e
x
=
0
.
The graph of
y
=
e
−
x
is a reflection of the graph of
y
=
e
x
about the yaxis and is
therefore, decreasing throughout its domain with
lim
x
→∞
e
−
x
=
0
and
lim
x
→−∞
e
−
x
=
∞
.
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View Full DocumentProperties of the natural exponential function,
y
=
e
x
1.
e
ln
x
=
x
, for
x
>
0
.
2.
ln
e
x
( )
=
x
, for
−∞ ≤
x
≤ ∞
.
Derivative of the natural exponential function
d
dx
e
x
( )
=
e
x
To derive the differentiation formula (without using the definition of the
derivative) for
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 Spring '08
 Famigleitti
 Calculus, Exponential Function, Derivative, Inverse Functions, dx

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