A Double Exponential Equation
A Double Exponential Equation
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.
1/2
(1/2)
1/4
(1/4)
1/2
2
1/4
4
x
x
y
y
1/x
x
1/y
y
x≠y
p
x
x
p
y
y
p=1
p=-1
x≠y
y=mx
m≠1
m
The graph of is black. The graph of interest, = where , is blue for and red for , and is the graph of a function . The intersection points with and , for , and corresponding points on , are plotted.
y=
p
x
x
p
x
x
p
y
y
x≠y
0<m<1
1<m
F
y=mx
y=(1/m)x
0<m<1
y=
p
x
x
It is interesting to see that when is varied between 0 and 2, the graph of bows from concave up to concave down, and appears to be a line segment from to for some . The graphs of and are shown to help you decide whether the graph of for this really is straight. The special satisfies =1/2.
p
F
(0,1)
(1,0)
p
F'
F''
F
p
p=1.4427
-1/p
e
The case is especially interesting because then the equation is equivalent to =, which has a solution and . (The slider for can take values from -2 to 5.)
p=-1
y
x
x
y
x=4
y=2
p