b. Choose the correct answer below. A. The statement is false because the error in the Midpoint Rule is proportional to ( \( \Delta x)^{2} \). B. The statement is false because the error in the Midpoint Rule is proportional to \( \Delta x \). C. The statement is false because the error in the Midpoint Rule is proportional to ( \( \Delta x)^{4} \). D. The statement is true because the error in the Midpoint Rule is proportional to \( \frac{\Delta x}{24} \).

The Midpoint Rule is a numerical method used to approximate definite integrals, and the error associated with it can often be understood through Taylor series expansion. Specifically, the error term involves the second derivative of the function being integrated, implying that the error is typically proportional to the square of the width of the subintervals, \( \Delta x^2 \), when the function is smooth. In practice, the Midpoint Rule can offer a surprisingly accurate approximation of the integral even with relatively large \( \Delta x \). Common mistakes occur when using it on functions that are not well-behaved—such as those with sharp corners or discontinuities—where the error can grow significantly. Always assess your function’s behavior to avoid surprises!