a. Choose the correct answer below. A. The statement is false because the Trapezoid Rule is just an approximation of a definite integral. B. The statement is false because the error in the Trapezoid Rule is proportional to \( (\Delta x)^{2} \), so it cannot be zero unless \( \Delta x \) is zero. C. The statement is false because the slope of the linear function may not exactly match the slope of the top of the approximating trapezoid in every subinterval. D. The statement is true because the area of the trapezoid in each subinterval is always exactly the same as the area under a line.

The Trapezoid Rule is a numerical method for estimating the area under a curve, which means it has its limitations. In fact, when applying this technique, the approximation relies heavily on the behavior of the function in question. If the function has high curvature or changes rapidly, the trapezoidal approximation can diverge from the true integral, leading to larger errors. This characteristic highlights the importance of understanding when and how to apply this rule effectively for accurate results. When implementing the Trapezoid Rule, a common mistake is to assume that the approximation will always yield a high degree of accuracy regardless of the function's nature. It's crucial to consider factors such as the number of subintervals and the shape of the function. If you're dealing with complex shapes, increasing the number of partitions can enhance accuracy. However, it's essential to balance between the computational effort and the desired precision to avoid unnecessary complexity.