设f(x)=x^4-2x^2+3x+4,写出它在x0=1处的四阶泰勒公式

题目

设f(x)=x^4-2x^2+3x+4,写出它在x0=1处的四阶泰勒公式

答案

f(x)=x^4-2x^2+3x+4
f(1)=6
f'(x) =4x^3-4x+3
f'(1) = 3
f''(x) = 12x^2-4
f''(1)=8
f'''(x) = 24x
f'''(1)=24
f''''(x) =24
f''''(1) =24
f(x) = f(1) + f'(x) (x-1) + f''(1)(x-1)^2/2 + f'''(1) (x-1)^3/6 + f''''(1)(x-1)^4/24
=6 + 3(x-1) + 4(x-1)^2+ 4(x-1)^3+ (x-1)^4