### Understanding the Trapezoidal Rule Error EstimationRecall that the error in using the Trapezoidal rule to estimate the definite integral \(\int_{a}^{b} f(x) \, dx\) is given by:\[ |E_T| \leq \frac{M(b - a)^3}{12n^2} \]where:- \( n \) is the number of subintervals,- \( M \) is the maximum value of the second derivative \( f''(x) \) over the interval \([a, b]\).### Problem StatementDetermine the smallest number of subintervals \( n \) so that the error in estimating the integral:\[ \int_{0}^{1} \left( \frac{1}{3} x^4 + \frac{2}{3} x^3 + x + \frac{4}{3} \right) \, dx \]is less than \( 10^{-3} \).### Detailed ExplanationThe given integral is:\[ \int_{0}^{1} \left( \frac{1}{3} x^4 + \frac{2}{3} x^3 + x + \frac{4}{3} \right) \, dx \]To apply the trapezoidal rule, we need to consider the second derivative of the integrand:\[ f(x) = \frac{1}{3} x^4 + \frac{2}{3} x^3 + x + \frac{4}{3} \]Calculating the first and second derivatives:\[ f'(x) = \frac{4}{3} x^3 + 2 x^2 + 1 \]\[ f''(x) = 4 x^2 + 4 x \]To find \( M \), which is the maximum value of \( f''(x) \) on the interval \([0, 1]\):\[ f''(x) = 4 x^2 + 4 x \]Since \( f''(x) \) is a quadratic function that increases on \([0, 1]\), it reaches its maximum at \( x = 1 \).\[ f''(1) = 4(1)^2 + 4(1) = 8 \]Thus, \( M = 8 \).Substituting back into the

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Recall that the error in using the Trapezoidal rule to estimate f(x) dr is M (b – a)3 |Er| < 12n2 where n is the number of subintervals and M is the maximum of f" on [a, b]. Determine the smallest number of subintervals n so that the error in estimating 2 4 4 dr 3 3 is less that 10

Recall that the error in using the Trapezoidal rule to estimate f(x) dr is M (b – a)3 |Er| < 12n2 where n is the number of subintervals and M is the maximum of f" on [a, b]. Determine the smallest number of subintervals n so that the error in estimating 2 4 4 dr 3 3 is less that 10

Calculus: Early Transcendentals

8th Edition

ISBN:9781285741550

Author:James Stewart

Publisher:James Stewart

Chapter1: Functions And Models

Section: Chapter Questions

Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...

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