泰勒公式笔记(一)-泰勒公式推导数学的学习

一些函数很难求值,比如 cos2,e x e^x ex 等等,为了解决这个问题.泰勒想到了用仿造函数 g(x) 去逼近这个函数, 然后对 g(x) 求值即可.

1. 一个可以无限求导的函数g(x)用来仿造f(x): ?? g ( x ) = a 0 + a 1 x 1 + a 2 x 2 . . . + a n x n g(x) = a_0+a_1x^1+a_2x^2...+a_nx^n g(x)=a0?+a1?x1+a2?x2...+an?xn
2. 仿造过程:

以下操作会使f(x)与g(x)函数图像越来越趋近

g ( x 0 ) = f ( x 0 ) g(x_0) = f(x_0) g(x0?)=f(x0?)
g ′ ( x 0 ) = f ′ ( x 0 ) g'(x_0) = f'(x_0) g′(x0?)=f′(x0?)
…
g n ( x 0 ) = f n ( x 0 ) g^n(x_0) = f^n(x_0) gn(x0?)=fn(x0?)
3. 推论 3.11 非常数项推论 g n ( x 0 ) = n ! a n g^n(x_0) = n!a_n gn(x0?)=n!an?
=>a n = g n ( x 0 ) n ! a_n=\frac{g^n(x0)}{n!} an?=n!gn(x0)?
=>a n = f n ( x 0 ) n ! a_n=\frac{f^n(x0)}{n!} an?=n!fn(x0)?(此推论仅仅只用于非常数项)
3.12 常数项推论 g ( x 0 ) = f ( x 0 ) g(x_0) = f(x_0) g(x0?)=f(x0?)
=>g ( x 0 ) = a 0 + a 1 x 0 1 + a 2 x 0 2 . . . + a n x 0 n g(x_0)=a_0+a_1x_0^1+a_2x_0^2...+a_nx_0^n g(x0?)=a0?+a1?x01?+a2?x02?...+an?x0n?
=>a 0 = g ( x 0 ) ? a 1 x 0 1 ? a 2 x 0 2 . . . ? a n x 0 n a_0 = g(x_0) -a_1x_0^1-a_2x_0^2...-a_nx_0^n a0?=g(x0?)?a1?x01??a2?x02?...?an?x0n?
=>a 0 = f ( x 0 ) ? a 1 x 0 1 ? a 2 x 0 2 . . . ? a n x 0 n a_0 = f(x_0) -a_1x_0^1-a_2x_0^2...-a_nx_0^n a0?=f(x0?)?a1?x01??a2?x02?...?an?x0n?
结合非常数项推论替换a i a_i ai?
=>a 0 = f ( x 0 ) ? f 1 ( x 0 ) 1 ! x 0 1 ? f 2 ( x 0 ) 2 ! x 0 2 . . . ? f n ( x 0 ) n ! x 0 n a_0 = f(x_0) -\frac{f^1(x0)}{1!}x_0^1-\frac{f^2(x0)}{2!}x_0^2...-\frac{f^n(x0)}{n!}x_0^n a0?=f(x0?)?1!f1(x0)?x01??2!f2(x0)?x02?...?n!fn(x0)?x0n?
4. 结论

综上所述, 用推论替换原始的 g(x):
4.1 非常数项推论替换如下:
?? g ( x ) = a 0 + f 1 ( x 0 ) 1 ! x 1 + f 2 ( x 0 ) 2 ! x 2 + . . . + f n ( x 0 ) n ! x n g(x) = a_0+\frac{f^1(x0)}{1!}x^1+\frac{f^2(x0)}{2!}x^2+...+\frac{f^n(x0)}{n!}x^n g(x)=a0?+1!f1(x0)?x1+2!f2(x0)?x2+...+n!fn(x0)?xn
4.2 常数项推论将 a 0 a_0 a0?替换如下:
g ( x ) = g(x) = g(x)=
? f ( x 0 ) ? f 1 ( x 0 ) 1 ! x 0 1 ? f 2 ( x 0 ) 2 ! x 0 2 . . . ? f n ( x 0 ) n ! x 0 n f(x_0) -\frac{f^1(x0)}{1!}x_0^1-\frac{f^2(x0)}{2!}x_0^2...-\frac{f^n(x0)}{n!}x_0^n f(x0?)?1!f1(x0)?x01??2!f2(x0)?x02?...?n!fn(x0)?x0n?
??? + f 1 ( x 0 ) 1 ! x 1 + f 2 ( x 0 ) 2 ! x 2 + . . . + f n ( x 0 ) n ! x n +\frac{f^1(x0)}{1!}x^1+\frac{f^2(x0)}{2!}x^2+...+\frac{f^n(x0)}{n!}x^n +1!f1(x0)?x1+2!f2(x0)?x2+...+n!fn(x0)?xn

【泰勒公式笔记(一)-泰勒公式推导】化简得泰勒公式如下:
g ( x ) = f ( x 0 ) + f 1 ( x 0 ) 1 ! ( x ? x 0 ) 1 + f 2 ( x 0 ) 2 ! ( x ? x 0 ) 2 . . . + f n ( x 0 ) n ! ( x ? x 0 ) n g(x) = f(x_0) +\frac{f^1(x0)}{1!}(x-x_0)^1+\frac{f^2(x0)}{2!}(x-x_0)^2...+\frac{f^n(x0)}{n!}(x-x_0)^n g(x)=f(x0?)+1!f1(x0)?(x?x0?)1+2!f2(x0)?(x?x0?)2...+n!fn(x0)?(x?x0?)n