Which of the following statements is true regarding the number of segments required for the trapezoidal rule?

Question

BlackTom AI · Accepted Answer

The trapezoidal rule approximates an integral by partitioning the interval [a,b] into n subintervals (segments) and summing the areas of trapezoids. Key facts: n is the number of subintervals (must be a positive integer), the subinterval width is h = (b − a)/n, and the composite trapezoidal formula is
T_n = h [ (f(a) + f(b))/2 + sum_{i=1}^{n-1} f(a + i h) ].

Step-by-step reasoning:
1. Definition of n: n counts how many equal subintervals the interval [a,b] is split into. By definition n ∈ {1,2,3,...}; it cannot be zero or fractional.
2. Example with n = 1: h = b − a, and T_1 = (b − a)[(f(a)+f(b))/2], which is the area of a single trapezoid spanning the whole interval. This demonstrates n = 1 is valid.
3. Parity: no factor in the trapezoidal rule requires n to be even or odd. The formula works for any positive integer n because the summation index and spacing are defined for any n ≥ 1.
4. Comparison with Simpson’s rule: Simpson’s rule requires an even number of subintervals (n even) because it pieces together parabolic arcs over pairs of subintervals. This parity restriction is specific to Simpson’s method, not the trapezoidal rule.

Why the correct answer is correct:
- Option c (“Must be greater than or equal to 1”) matches the requirement that n is a positive integer. The trapezoidal rule works for n = 1, 2, 3, ….

Why other options are incorrect:
- a (“Must be odd”) and b (“Must be even”): parity is not required for trapezoidal rule (only Simpson’s needs even).
- d, e, f (“Must be ≥ 2, ≥ 3, ≥ 4”): these are unnecessarily restrictive; n = 1 is valid as shown.

Conclusion: The correct choice is c. This matches the provided answer.

Which of the following statements is true regarding the number of segments required for the trapezoidal rule?多项选择题

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