You are planning to use Simpson’s Rule to estimate the value of the integral Estimate ∫ from 1 to 2 of f(x) dx with an error magnitude less than 10⁻⁵ using Simpson’s Rule.
You have determined that |f⁽⁴⁾(x)| ≤ 3 throughout the interval of integration. How many subintervals should you use to ensure the required accuracy?
(Remember that for Simpson’s Rule the number of subintervals must be even.)

Guida verificata passo dopo passo

Recall the error bound formula for Simpson's Rule: the error \( E_S \) satisfies \( |E_S| \leq \frac{K(b - a)^5}{180 n^4} \), where \( K \) is an upper bound on \( |f^{(4)}(x)| \) over \([a, b]\), and \( n \) is the number of subintervals (which must be even).

Identify the given values: \( a = 1 \), \( b = 2 \), \( K = 3 \), and the desired error bound \( |E_S| < 10^{-5} \).

Set up the inequality for the error bound: \( \frac{3 (2 - 1)^5}{180 n^4} < 10^{-5} \).

Simplify the inequality to solve for \( n^4 \): \( \frac{3}{180 n^4} < 10^{-5} \) which leads to \( n^4 > \frac{3}{180 \times 10^{-5}} \).

Calculate the right side and then take the fourth root to find \( n \). Finally, choose the smallest even integer \( n \) that satisfies this inequality to ensure the error is less than \( 10^{-5} \).

Risposta video verificata per un problema simile:

Questa soluzione video è stata consigliata dai nostri tutor come utile per risolvere questo problema.

Durata del video:

Concetti chiave

Ecco i concetti essenziali che devi comprendere per rispondere correttamente alla domanda.

Simpson’s Rule

Simpson’s Rule is a numerical method for approximating definite integrals by fitting parabolas through subintervals of the function. It requires an even number of subintervals and generally provides more accurate results than the trapezoidal rule for smooth functions.

Video consigliato:

5:50

Power Rules

Error Bound for Simpson’s Rule

The error bound for Simpson’s Rule depends on the fourth derivative of the function. Specifically, the error magnitude is bounded by (K(b−a)^5) / (180 n^4), where K is the maximum absolute value of the fourth derivative on [a, b], and n is the number of subintervals.

Video consigliato:

Percorso guidato

04:57

Determining Error and Relative Error

Choosing Number of Subintervals for Desired Accuracy

To ensure the approximation error is below a specified tolerance, solve the error bound inequality for n, the number of subintervals. Since Simpson’s Rule requires n to be even, round up to the nearest even integer after calculating the minimum n needed.

Video consigliato:

Percorso guidato