In this problem, we will approximate the value of the definite integral I = = [₁²ea²0the Trapezoid Rule.M(2)(b − a)³12n²Here are the error bound for the Trapezoid Rule approximation and the second derivative of thefunction f(x) = e³.andf"(x) = 3.xex³6x3(2+3x³)dx usingIn - I| ≤where M(2) is an upper bound forf"(x)| on [0,1].(a) Find a practical upper bound for f"(x)], 0≤ x ≤ 1.(b) How many slices n should we use to guarantee that the error of approximating I using theTrapezoid Rule is no bigger than 10-4?

Answered: = In this problem, we will approximate…

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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Question

In this problem, we will approximate the value of the definite integral I = = [₁²ea²
0
the Trapezoid Rule.
M(2)(b − a)³
12n²
Here are the error bound for the Trapezoid Rule approximation and the second derivative of the
function f(x) = e³.
and
f"(x) = 3.xex³
6x3
(2+3x³)
dx using
In - I| ≤
where M(2) is an upper bound for
f"(x)| on [0,1].
(a) Find a practical upper bound for f"(x)], 0≤ x ≤ 1.
(b) How many slices n should we use to guarantee that the error of approximating I using the
Trapezoid Rule is no bigger than 10-4?

Expert Solution

Step 1

Concept:

One of the significant integration rules is the trapezoidal rule. Because little trapezoids rather than rectangles are used to divide the overall area when the area under the curve is calculated, this shape is known as a trapezoid. This rule uses the linear approximations of the functions to approximate the definite integrals.

The technique of numerical analysis mostly employs the trapezoidal rule. We can also utilise Riemann sums, which calculate the area under the curve using tiny rectangles, to evaluate the definite integrals.