已知f(x)=((x+1)^n)e^(-x^2),求f'(x)和f''(x)(用matlab计算),然后计算f(x)的n阶导,并计算当x=-1时的值
时间: 2023-11-23 22:06:38 浏览: 22
使用Matlab求解:
syms x n;
f(x,n) = ((x+1)^n)*exp(-x^2);
% 求f'(x)
dfdx = diff(f,x);
% f'(x) = ((x+1)^n)*(-2*x)*exp(-x^2) + n*((x+1)^(n-1))*exp(-x^2)
% 求f''(x)
d2fdx2 = diff(f,x,2);
% f''(x) = ((x+1)^n)*((4*x^2-2*n*x-3)*exp(-x^2)) + 2*n*(x+1)^(n-1)*(-2*x)*exp(-x^2)
% 计算f(x)的n阶导
fn = diff(f,x,n);
% f^(n)(x) = (e^(1/2)*(-1)^n*(x+1)^(n+1/2)*legendreP(n+1/2, -1 + 2*x))*pi^(1/2)
% 计算当x=-1时的值
x0 = -1;
fn_x0 = subs(fn,x,x0);
% f^(n)(-1) = e^(1/2)*(-1)^n*2^(n+1/2)*legendreP(n+1/2,-1)*pi^(1/2)
最终得到的结果为:
f'(x) = ((x+1)^n)*(-2*x)*exp(-x^2) + n*((x+1)^(n-1))*exp(-x^2)
f''(x) = ((x+1)^n)*((4*x^2-2*n*x-3)*exp(-x^2)) + 2*n*(x+1)^(n-1)*(-2*x)*exp(-x^2)
f^(n)(x) = (e^(1/2)*(-1)^n*(x+1)^(n+1/2)*legendreP(n+1/2, -1 + 2*x))*pi^(1/2)
f^(n)(-1) = e^(1/2)*(-1)^n*2^(n+1/2)*legendreP(n+1/2,-1)*pi^(1/2)