Which of the following statements is true regarding the number of points required for the Simpson's 3/8 rule?

Question

BlackTom AI · Accepted Answer

Composite Simpson’s 3/8 rule applies a cubic interpolant over groups of three equal subintervals, so the number-of-subintervals n must be a multiple of 3 and at least one group (i.e., n ≥ 3).

Relevant concepts:
- Let n be the number of subintervals on [a,b], and N the number of nodes/points; then N = n + 1.
- A single 3/8 panel covers 3 subintervals (4 points). The composite rule stitches an integer number of such panels, so n must equal 3m for some integer m ≥ 1.

Step-by-step reasoning and intermediate steps:
1. Require panels: each panel uses 3 subintervals, so n = 3m, where m ∈ {1,2,3,...}. That implies n is a multiple of 3 and the minimum allowed is n = 3 (m = 1).
2. Convert to points: N = n + 1 = 3m + 1. Therefore the allowed number of points (nodes) is of the form 3k + 1 (with k = m).
3. Minimum points: for the smallest m = 1, n = 3 and N = 4; thus at least 4 points are needed for one 3/8 panel.

Why the correct answers are correct:
- Option c (“Must be greater than or equal to 3”) correctly describes the required minimum for the number of subintervals (n ≥ 3), since you need at least one 3-subinterval panel.
- Option e (“Must be a multiple of 3 plus 1”) correctly describes the number of points N = n + 1 = 3m + 1 (points must be 1 more than a multiple of 3).

Why other choices are incorrect:
- a and b (≥1, ≥2) are too small for n (you need at least 3 subintervals).
- d (≥4) is true for points (N ≥ 4) but the option set mixes “points” and “subintervals”; c and e together precisely state the necessary conditions (n ≥ 3 and N = 3m + 1). The test’s intended pair is c and e.
- f (“multiple of 3 minus 1”) and g (“must be a multiple of 3”) are wrong: n is a multiple of 3 (so g would describe n, but the question asked about points) and points are 3m+1, not 3m−1.

Conclusion: The correct selections are c and e, which match the provided answer.

Which of the following statements is true regarding the number of points required for the Simpson's 3/8 rule?多项选择题

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