Question

如何用matlab绘制z=x^y-y^x的图形，来比较e^pi和pi^e的大小？

Answer 1

点展开看, 蛋疼的百度.
%两个思路

%其一, z(x,y)写成 g(x)=z(x,pi)=x^pi-pi^x; 画图
x=linspace(2,3.5,100);
g=x.^pi-pi.^x;
plot(x,g); hold on
plot(x,zeros(size(x)),'k'); % 横轴
% 再画出x=e的直线.
plot(exp(ones(1,10)), linspace(-4,2,10),'r');
text(exp(1), exp(pi)-pi^(exp(1))+0.5, '交点g(e)=z(e,\pi)');
% 注意交点处大于0
% 所以g(e)=z(e,pi)>0, 因此 e^pi > pi^e

%其二，绘制3d图
figure
g=linspace(2,3.5,100);
[x,y]=meshgrid(g,g);
z=x.^y-y.^x;
surf(x,y,z,'edgecolor','none','facealpha',0.5);
hold on
z0=zeros(1,10);
plot3(z0+exp(1), z0+pi, linspace(-3,3,10),'k');
plot3(exp(1), pi, exp(pi)-pi^(exp(1)), 'ko','markerfacecolor','k')
xlabel('X');ylabel('Y');zlabel('Z')
mesh(x,y,zeros(size(x)),'edgecolor','none','facealpha',0.3,'facecolor','k');
view([130,18])
% 注意过(e, pi)的直线与曲面交点在黑色平面之上, 说明z(e,pi)>0