What is the minimum number of points required for a single application of the trapezoidal rule?

Question

BlackTom AI · Accepted Answer

The trapezoidal rule approximates an integral over a single subinterval by replacing the curve with the straight line segment joining the function values at the two endpoints. For an interval [a, b] you need the function values at x0 = a and x1 = b: f(x0) and f(x1). The single-application formula is
Integral from a to b f(x) dx ≈ (b − a)/2 · [f(a) + f(b)].
Relevant concepts: numerical quadrature, linear interpolation on one subinterval, and endpoint sampling. A single trapezoid is defined by two vertices on the curve (the endpoints), so only two sample points are necessary.

Step-by-step reasoning:
1. Choose the interval [a, b]. Label endpoints x0 = a and x1 = b.
2. Evaluate f at these two points: y0 = f(x0), y1 = f(x1).
3. The area of the trapezoid with parallel sides y0 and y1 and base (b − a) is (1/2)·(sum of parallel sides)·(base) = (b − a)/2 · (y0 + y1).
4. Therefore the approximation uses exactly the two function values.

Illustrative calculation: Let f(x) = x on [0,1]. Exact integral = 1/2. Trapezoidal approximation with the two endpoints: (1 − 0)/2 · [f(0) + f(1)] = 1/2 · [0 + 1] = 1/2, matching the exact result. Intermediate steps shown: base = 1, sum of heights = 1, multiply by 1/2 gives 1/2.

Why the correct answer is correct: the geometric trapezoid requires two vertices on the function; mathematically the formula involves only f(a) and f(b). Other options:
- b. 3: Three points are needed for a single application of Simpson’s rule (a quadratic fit), not the single trapezoid.
- c. 4: Four points are unnecessary; that would be for higher-degree composite formulas or multiple subintervals.
- d. None of the above: incorrect because the standard single trapezoid uses exactly two points.

Thus the minimum number of points for a single trapezoidal application is 2, matching the given answer (a. 2).

What is the minimum number of points required for a single application of the trapezoidal rule?单项选择题

类似问题

An approximation for the area under the graph [math: y=12x] and the x-axis over the interval x = 1 to x = 4 using the trapezoidal rule and interval widths of 1 unit is:

Integrate the following data using the composite trapezoidal rule. What is the output? [table] 𝑥x | -5 | -4.5 | -4 | -3.5 𝑓(𝑥)f(x) | 8 | 5 | 6 | 5 [/table]

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

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

What is the order of the polynomial that is applied for the trapezoidal rule?

An approximation for the area under the graph \( y= \frac{12}{x} \) and the x-axis over the interval x = 1 to x = 4 using the trapezoidal rule and interval widths of 1 unit is:

An approximation for the area between the curve and the x-axis over the interval indicated in the diagram using the combined area of the trapeziums is:

An approximation for the area between the curve and the x-axis over the interval indicated in the diagram using the combined area of the trapeziums is:

更多留学生实用工具

智能学习助手

风格化写作助手

论文查重助手

文献引用助手

课堂转译助手

课堂笔记助手

Quiz搜索助手

学校历年真题

智能刷题助手

智能匹配练习

希望你的学习变得更简单