題目 7. (10 points) Find the Maclaurin series of order 444 for the function f(x)=e−x2cosxf(x)=e^{-x^2}\cos xf(x)=e−x2cosx, i.e., approximate f(x)f(x)f(x) by a polynomial a0+a1x+a2x2+a3x3+a4x4a_0+a_1x+a_2x^2+a_3x^3+a_4x^4a0+a1x+a2x2+a3x3+a4x4. 解答 解法一 只保留至四次項： exp(−x2)=1−x2+x42+更高次項,cosx=1−x22+x424+更高次項.\exp(-x^2)=1-x^2+\frac{x^4}{2}+\text{更高次項}, \qquad \cos x=1-\frac{x^2}{2}+\frac{x^4}{24}+\text{更高次項}.exp(−x2)=1−x2+2x4+更高次項,cosx=1−2x2+24x4+更高次項. 相乘後得到 exp(−x2)cosx=1−32x2+2524x4+更高次項.\boxed{\exp(-x^2)\cos x =1-\frac32x^2+\frac{25}{24}x^4+\text{更高次項}}.exp(−x2)cosx=1−23x2+2425x4+更高次項.
