Derivation of the Trapezoidal Rule Error Estimate

and |g′′(x)| = M. We will show that |f(x)| ≤ g(x) on [0, 1]. The desired conclusion follows. Suppose however that this is false, that there is a number q ∈ [0, 1] for which f(q) > g(q). (The other case, f(q) < −g(q), is similar.) Our strategy will be to show that there are real numbers s and t with s < t such that f ′(s) > g′(s) and f ′(t) < g′(t). See figure. We will then apply the Mean Value Theorem to f ′ and g′ and use facts about g to force |f ′′(x)| > M , for some value of x. This, of course, contradicts a hypothesis of the Theorem. Thus, if the hypotheses are true then, |f(x)| ≤ g(x) on [0, 1], is also true. We proceed. By the Fundamental Theorem of Calculus ∫ q