A brief calculation shows that if 0 ≤ x ≤ 1, then the second derivative of
f(x) = √(1 + x⁴)
lies between 0 and 8.
Based on this, about how many subdivisions would you need to estimate the integral of f from 0 to 1
with an error no greater than 10⁻³ in absolute value using the Trapezoidal Rule?

검증된 단계별 안내

Recall the error bound formula for the Trapezoidal Rule: the absolute error \( E_T \) satisfies \( |E_T| \leq \frac{(b - a)^3}{12 n^2} \max_{a \leq x \leq b} |f''(x)| \), where \( n \) is the number of subdivisions.

Identify the interval \([a, b]\) for the integral, which is from 0 to 1, so \( a = 0 \) and \( b = 1 \).

Use the given information that \( 0 \leq f''(x) \leq 8 \) on \([0,1]\), so \( \max |f''(x)| = 8 \).

Set the error bound \( \frac{(1 - 0)^3}{12 n^2} \times 8 \leq 10^{-3} \) to ensure the error is no greater than \( 10^{-3} \).

Solve the inequality for \( n \) to find the minimum number of subdivisions needed: \( \frac{8}{12 n^2} \leq 10^{-3} \) which simplifies to \( n^2 \geq \frac{8}{12 \times 10^{-3}} \). Then take the square root to find \( n \).

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Second Derivative and Error Bound in the Trapezoidal Rule

The error bound for the Trapezoidal Rule depends on the maximum absolute value of the second derivative of the function over the interval. Specifically, the error is proportional to the square of the width of the subdivisions and the maximum second derivative. Knowing the range of the second derivative helps estimate the number of subdivisions needed for a desired accuracy.

Trapezoidal Rule for Numerical Integration

The Trapezoidal Rule approximates the integral of a function by dividing the interval into subintervals and summing the areas of trapezoids under the curve. The accuracy improves as the number of subdivisions increases, reducing the width of each trapezoid and thus the approximation error.

Error Tolerance and Subdivision Calculation

To achieve a specific error tolerance, such as 10⁻³, one uses the error bound formula involving the second derivative and interval length. By rearranging this formula, the minimum number of subdivisions needed can be calculated to ensure the approximation error does not exceed the given tolerance.