Differential Calculus

Taylor's Formula

• Theorem 1.1.
  Let \(f\): \(I=(c,d)->\mathbb{R}\) be a n-times differentiable function. If \(a, b\) are distinct points in \({I}\), then there exists a point \(\overline{x}\) strictly between a and b such that
  \[
  f(b)=f(a)+f'(a)(b-a)+\frac{f''(a)}{2}(b-a)^2+...+\frac{f^{(n-1)}(a)}{(n-1)!}(b-a)^{n-1}+ \frac{f^{(n)}(\overline{x})}{n!}(b-a)^{n}
  \]