A Double Exponential Equation

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Since and
, there are points on the graphs of
and
where
. These graphs are the special cases of
where
and
. All points with
can be found as intersections of the graph with the lines
with slope
. In this case, parametric equations in terms of
have simple formulas.
Contributed by: Roger B. Kirchner (March 2011)
Open content licensed under CC BY-NC-SA
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Assume . Conveniently,
iff
iff
iff
. Thus,
is on the graph of
where
iff
and
, where
.
Let and
. Then
and
,
and
are parametric equations for the graph of
. Since the graph of
is symmetric with respect to
,
and
.
Parametric equations for the graphs of and
are obtained by differentiating the identity
. They are
,
, and
,
.
Where does the graph of meet the line
? Two ways to determine it are 1) observe it is the point
such that
, where
, and 2) compute
using L'Hospital's rule. You will find
.
What are the domain and range of ? It is an interesting exercise in L'Hospital's rule to determine that
and
when
and
and
when
. By symmetry, the domain and range of
are
when
and
when
.
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