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%1求多项式在【-2，5】区间的数值
a = [9,-5,3,7]; x = -2:0.01:5;
f = polyval(a,x); 
plot(x,f,'LineWidth', 2);
xlabel('x'); ylabel('f(x)');
set(gca, 'FontSize', 14)

%%
%2 多项式微分
p=[5 0 -2 0 1];
polyder(p)
%求f(x)在7的数组
polyval(polyder(p),7)

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%3多项式积分
p=[5 0 -2 0 1];
polyint(p, 3)
polyval(polyint(p, 3),7)

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%4两数之间差异
x = [1 2 5 2 1];
diff(x)

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%5两点之间的斜率
x = [1 2]; y = [5 7];
slope = diff(y)./diff(x)

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%6 x0点的斜率
x0 = pi/2; h = 0.1;
x = [x0 x0+h];
y = [sin(x0) sin(x0+h)]; 
m = diff(y)./diff(x)

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%7 求区间内微分数值
h = 0.5; x = 0:h:2*pi;
y = sin(x); m = diff(y)./diff(x);

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%积分sin(x)差值主键变小的图像
g = colormap(lines); hold on;
for i=1:4
x = 0:power(10, -i):pi;
y = sin(x); m = diff(y)./diff(x);
plot(x(1:end-1), m, 'Color', g(i,:)); 
end
hold off;
set(gca, 'XLim', [0, pi/2]); 
set(gca, 'YLim', [0, 1.2]);
set(gca, 'FontSize', 18); 
set(gca, 'FontName', 'symbol');
set(gca, 'XTick', 0:pi/4:pi/2);
set(gca, 'XTickLabel', {'0', 'p/4', 'p/2'});
h = legend('h=0.1','h=0.01','h=0.001','h=0.0001');
set(h,'FontName', 'Times New Roman'); box on;

%%
%exercise
a=exp(1);
h=0.1;
g = colormap(lines);
hold on;
for i=1:3
x = 0:h:2*pi;
y = a.^(-x).*sin(x.^2/2);
m = diff(y)./diff(x);
plot(x(1:end-1), m, 'Color', g(i,:)); 
h=0.1*h;
end;
set(gca, 'XLim', [0, 2*pi]); 
set(gca, 'YLim', [-0.3,0.3]);
set(gca, 'FontSize', 18); 
set(gca, 'FontName', 'symbol');
set(gca, 'XTick', 0:pi/2:2*pi);
h = legend('h=0.1','h=0.01','h=0.001');
set(h, 'fontname', 'Times New Roman');
box on;

%%
%二次跟三次微分
x = -2:0.005:2; y = x.^3;
m = diff(y)./diff(x);
m2 = diff(m)./diff(x(1:end-1));
plot(x,y,x(1:end-1),m,x(1:end-2),m2);
xlabel('x', 'FontSize', 18); 
ylabel('y', 'FontSize', 18);
legend('f(x) = x^3','f''(x)','f''''(x)', 4);
set(gca, 'FontSize', 18);

%%
%求函数内的面积大小
h = 0.001; x = 0:h:2;
midpoint = (x(1:end-1)+x(2:end))./2;
y = 4*midpoint.^3;
s = sum(h*y)

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%梯形方法求函数在某区间内的范围（trapz方法）
h = 0.0001; x = 0:h:2; y = 4*x.^3;
s = h*trapz(y)

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%simpson方法
h = 0.05; x = 0:h:2; y = 4*x.^3;
s = h/3*(y(1)+2*sum(y(3:2:end-2))+...
4*sum(y(2:2:end))+y(end));

%%
%function方法（路径得对）
xy_plot(@tan,0:0.01:2*pi);

%%
%简单方法求数值积分
y = @(x) 1./(x.^3-2*x-5);
integral(y,0,2)

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%二重积分和三重积分
%二重积分
f = @(x,y) y.*sin(x)+x.*cos(y);
integral2(f,pi,2*pi,0,pi)
%三重积分
f = @(x,y,z) y.*sin(x)+z.*cos(y);
integral3(f,0,pi,0,1,-1,1)