Let \(f\) satisfy \(|f''(x)|\le M\) on \([0,1]\). Show that the composite trapezoidal rule with \(n\) subintervals for \(\int_0^1 f\) has error bounded by \(\frac{M}{12n^2}\). Which statement is correct?
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B. The bound is \(\frac{M}{12n^2}\), because \(h=1/n\) and the error is \(\frac{(b-a)h^2}{12}|f''|\) with \(b-a=1\).
With \(b-a=1\) and \(h=1/n\), the composite error is bounded by \(\frac{M}{12n^2}\). The others change the power of \(h\), the constant factor, or restrict \(f\) without reason.
The trapezoidal rule is a quadrature method that approximates the definite integral of a function \(f\) on \([a,b]\) by replacing the curve with the straight line joining \((a,f(a))\) and \((b,f(b))\).