No Calculator 4 Me

Calculus

When you have no calculator, calculus is sure handy.

Derivatives of functions are employed in Taylor's Theorem:

\begin{matrix} f (x + δ) = f (x) + δ f' (x) + \frac{δ^{2}}{2!} f'' (x) + \frac{δ^{3}}{3!} f''' (x) + ... + \frac{δ^{n}}{n!} f^{(n)} (x) + R_{n} (x) \\ where R_{n} (x) = \frac{δ^{n + 1}}{(n + 1)!} f^{(n + 1)} (c) for some x \leq c \leq x + δ \end{matrix}

especially when δ/x << 1. In this case, it is handy to rewrite Taylor's Theorem to emphasize this. Let ϵ = δ/x << 1.

\begin{matrix} f (x + δ) = f (x (1 + \frac{δ}{x})) = f (x (1 + ϵ)) = g (1 + ϵ) \\ = g (1) + ϵ g' (1) + \frac{ϵ^{2}}{2!} g'' (1) + \frac{ϵ^{3}}{3!} g''' (1) + ... + \frac{ϵ^{n}}{n!} g^{(n)} (1) + R_{n} (1) \\ where R_{n} (1) = \frac{ϵ^{n + 1}}{(n + 1)!} g^{(n + 1)} (c) for some 1 \leq c \leq 1 + ϵ \end{matrix}