What is the minimum number of points required for a single application of the Simpson's 3/8 rule?

Question

BlackTom AI · Accepted Answer

Simpson’s 3/8 rule uses a cubic interpolating polynomial over three equal subintervals, so it requires four function values (four points).

1. Relevant concepts
- Newton–Cotes closed formulas: a closed formula over n subintervals uses n+1 equally spaced nodes.  
- Simpson’s 3/8 rule is the closed Newton–Cotes rule of degree 3 (cubic), i.e., it approximates the integral on [a,b] by fitting a cubic through four points.  
- The standard 3/8 composite formula for a single application (one panel covering [a,b]) is: ∫_a^b f(x) dx ≈ (3h/8)[f(x0) + 3f(x1) + 3f(x2) + f(x3)], where h = (b−a)/3.

2. Step-by-step reasoning
- To fit a cubic polynomial you need 4 coefficients, so you need 4 data points (nodes) to determine it uniquely.  
- The 3/8 rule divides [a,b] into 3 equal subintervals, so the nodes are x0 = a, x1 = a + h, x2 = a + 2h, x3 = b with h = (b−a)/3. That is four points total (n+1 = 3+1 = 4).  
- Plugging these into the Newton–Cotes derivation yields the weight pattern [1, 3, 3, 1] and the factor 3h/8, confirming the single-panel formula above.

3. Intermediate calculation example
- Let h = (b−a)/3. Then the approximation explicitly uses f(a), f(a+h), f(a+2h), f(b). There are four function evaluations.

4. Why the correct answer is correct
- Because the method fits a cubic (degree 3) across three subintervals, it needs four equally spaced nodes; thus the minimum number of points for one application is 4.

5. Why other choices are incorrect
- a. 2: two points correspond to the trapezoidal rule (one subinterval), not 3/8.  
- b. 3: three points correspond to Simpson’s 1/3 rule (two subintervals, quadratic fit), not the 3/8 rule.  
- d. None of the above: incorrect because 4 is a listed option and is correct.

Conclusion: The minimum number of points required for a single application of Simpson’s 3/8 rule is 4 (choice c).

What is the minimum number of points required for a single application of the Simpson's 3/8 rule?单项选择题

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