【MATLAB】数学小实验

sin(x)的泰勒展开式

做一个MATLAB小实验：

x = -2:0.001:2;

y = sin(x);

hold on;

plot(x,y,'r-','linewidth',1);

y1 = x;

plot(x,y1,'linewidth',1);

y2 = x - x.^3/3;

plot(x,y2,'linewidth',1);

y3 = x - x.^3/3 + x.^5/120;

plot(x,y3,'linewidth',1);

y4 = x - x.^3/3 + x.^5/120 + x.^7/5040;

plot(x,y4,'linewidth',1);

xlabel('x');

ylabel('y');

legend('真实值','1阶泰勒展开','2阶泰勒展开','3阶泰勒展开','4阶泰勒展开');

title('泰勒展开：y = sin(x)')

cos(x)的泰勒展开式：

做个小实验：

x = -2:0.001:2;

y = cos(x);

hold on;

plot(x,y,'r-','linewidth',1);

y1 = ones(size(x));

plot(x,y1,'linewidth',1);

y2 = 1 - x.^2/2;

plot(x,y2,'linewidth',1);

y3 = 1 - x.^2/2 + x.^4/24;

plot(x,y3,'linewidth',1);

y4 = 1 - x.^2/2 + x.^4/24 + x.^6/720;

plot(x,y4,'linewidth',1);

xlabel('x');

ylabel('y');

legend('真实值','1阶泰勒展开','2阶泰勒展开','3阶泰勒展开','4阶泰勒展开');

title('泰勒展开：y = cos(x)')

log(x)的泰勒展开式：

x = -0.999:0.001:1;

y = log(1+x);

hold on;

plot(x,y,'r-','linewidth',1);

y1 = x;

plot(x,y1,'linewidth',1);

y2 = x - x.^2/2;

plot(x,y2,'linewidth',1);

y3 = x - x.^2/2 + x.^3/3;

plot(x,y3,'linewidth',1);

y4 = x - x.^2/2 + x.^3/3 - x.^4/4;

plot(x,y4,'linewidth',1);

xlabel('x');

ylabel('y');

legend('真实值','1阶泰勒展开','2阶泰勒展开','3阶泰勒展开','4阶泰勒展开');

title('泰勒展开：y = ln(1+x)')

1/(1-x)的泰勒展开式：

数学小实验：

x = -0.5:0.001:0.5;

y = 1./(1-x);

hold on;

plot(x,y,'r-','linewidth',1);

y1 = 1 + x;

plot(x,y1,'linewidth',1);

y2 = 1 + x + x.^2;

plot(x,y2,'linewidth',1);

y3 = 1 + x + x.^2 + x.^3;

plot(x,y3,'linewidth',1);

y4 = 1 + x + x.^2 + x.^3 + x.^4;

plot(x,y4,'linewidth',1);

xlabel('x');

ylabel('y');

legend('真实值','1阶泰勒展开','2阶泰勒展开','3阶泰勒展开','4阶泰勒展开');

title('泰勒展开：y = 1/(1-x)')

exp(x)的泰勒展开式：

做个小实验：

x = -2:0.001:2;

y = exp(x);

hold on;

plot(x,y,'r-','linewidth',1);

y1 = 1 + x;

plot(x,y1,'linewidth',1);

y2 = 1 + x + x.^2/2;

plot(x,y2,'linewidth',1);

y3 = 1 + x + x.^2/2 + x.^3/6;

plot(x,y3,'linewidth',1);

y4 = 1 + x + x.^2/2 + x.^3/6 + x.^4/24;

plot(x,y4,'linewidth',1);

xlabel('x');

ylabel('y');

legend('真实值','1阶泰勒展开','2阶泰勒展开','3阶泰勒展开','4阶泰勒展开');

title('泰勒展开：y = e^x')