高等数学——泰勒公式

泰勒公式

\(\Delta y \approx f'(x)\Delta x\)

\(fy = f'(x)dx\)

\(f(x)-f(x_{0}) \approx f'(x_{0})(x-x_{0})\)

\(f(x)\approx f(x_{0})+f'(x_{0})(x-x_{0})\)

\[P_{n}(x)=a_{0}+a_{1}(x-x_{0})+a_{2}(x-x_{0})^2+\dots+a_{n}(x-x_{0})^n \]

\[a_{0}=f(x_{0}), a_{1} = f'(x_{0}), 2!a_{2}f''(x_{0})\dots n!a_{n}=f^{n}(x_{0}) \]

\[a_{0} = f(x_{0}),a_{1} = \frac{f'(x0)}{1!},\dots,a_{n}=\frac{f^{(n)}(x_{0})}{n!} \]

泰勒中值定理：

\(f(x)\) 在 \(x_{0}\) 处 \(n\) 阶导，存在 \(x_{0}\) 的一个领域，对于该领域内的任一 \(x\)，有：

\[f(x)=f(x_{0})+f'(x_{0})(x-x_{0})+\frac{f''(x_{0})}{2!}(x-x_{0})^2+\dots+\frac{f^{(n)}(x_{0})}{n!}(x-x_{0})^n+R_{n}(x) \]

\[R_{n}(x)=\frac{f^{(n + 1)}(\xi)}{(n + 1)!}(x-x_{0})^{n+1},\xi \text{介于}x_{0},x \]