Graphical comparison of exponential functions
This example discusses a graphical approach to determining the larger of two values: and .
To graphically compare the values of and , we plot the surface of the difference function of the exponential functions .
To do this, first of all, install and connect the library CairoMakie
.
import Pkg; Pkg.add("CairoMakie");
using CairoMakie;
Let's define arrays of surface coordinates.
x = 0:0.05:4; y = 0:0.05:4;
X = [i for i in x, j in 1:length(y)]; Y = [j for i in 1:length(x), j in y];
Z = X.^Y-Y.^X;
Build the surface by the given function.
using CairoMakie
# Создаем фигуру и 3D оси
fig = Figure(;size = (600, 600));
ax = Axis3(fig[1, 1],
title = "Показательные функции",
xlabel = "Ось X", ylabel = "Ось Y", zlabel = "Ось Z",
aspect=(1,1,1), azimuth=-pi*0.64);
# Создаём поверхность
CairoMakie.surface!(ax, x, y, Z,
colormap = :prism,
colorrange = (minimum(Z), maximum(Z)));
# Добавляем цветовую шкалу
Colorbar(fig[1, 2],
limits = (minimum(Z), maximum(Z)),
colormap = :prism, label = "Значения");
display(fig);
In the plane draw the contour on the constructed surface.
CairoMakie.contour3d!(ax, x, y, Z,
levels = [0],
color=:black, linewidth=2)
display(fig);
On the constructed contours there are several points of integer solutions of the equation . Let us plot these points on the contours.
# массивы целочисленных решений уравнения
ix=[0,1,2,2,3,4,4]
iy=[0,1,2,4,3,2,4]
iz=zeros(7)
CairoMakie.scatter!(ax, ix, iy, iz;
color=:black)
display(fig);
Now plot the function points for the sought and at the coordinates: and .
e=exp(1);
tx=[pi,e]
ty=[e,pi]
textt=["pi,e", "e,pi"]
tz=tx.^ty-ty.^tx
CairoMakie.scatter!(ax, tx, ty, tz;
color=:white)
CairoMakie.text!(ax, tx, ty, tz;
text = textt, align = (:right, :bottom),
color = :white)
display(fig);
From the construction we can see that the point with coordinates is located in the part of the surface that is above the plane , and the point - in the part of the surface below the secant plane. For convenience of viewing the location of points you can edit the azimuth of axes and code cell mask:
Азимут=-0.38 # @param {type:"slider",min:-6.28,max:6.28,step:0.1}
ax.azimuth[] = Азимут;
display(fig);
From this we can conclude that the value of is greater than . In confirmation let's calculate these values.
@show e^pi, pi^e;
Conclusion
In this example, we have looked at an illustrative way to graphically find the larger of two values.