Lagrange Error Bound calculates the maximum error when approximating a function with a Taylor polynomial. The formula ∣Rn(x)∣≤M(n+1)!∣x−a∣n+1|R_n(x)| \le \frac{M}{(n+1)!}|x-a|^{n+1}∣Rn(x)∣≤(n+1)!M∣x−a∣n+1 uses MMM, the largest (n+1)(n+1)(n+1)-th derivative value between aaa and xxx. It ensures...