Explanation Given information: The interval is [ 0 , 2 ] . Number of subintervals is 2. Calculation: Let consider a function f ( x ) = | x − 1 | with the interval 0 ≤ x ≤ 2 and draw the graph for the function. The sketch of the graph of the function f ( x ) = | x − 1 | is shown in Figure 1. The right endpoint approximation is shown below: R n = f ( x 1 ) Δ x + f ( x 2 ) Δ x + ... + f ( x n ) Δ x (1) Here, the right endpoint height of x 1 is f ( x 1 ) , the width is Δ x , the right endpoint height of x 2 is f ( x 2 ) , and the right endpoint height of x n is f ( x n ) . Find the width ( Δ x ) using the relation: Δ x = b − a n (2) Here, the upper limit is b , the lower limit is a , and the number of rectangles is n . Substitute 2 for b , 0 for a , and 2 for n in Equation (2). Δ x = 2 − 0 2 = 1 Refer Figure (1). Take the right endpoint height of f ( x 1 ) value as 0 and the right endpoint height of f ( x 2 ) value as 1. The area of ∫ 0 2 f ( x ) d x is equal to 1. Substitute 2 for n , 0 for f ( x 1 ) , 1 for Δ x , and 1 for f ( x 2 ) in Equation (1). R 2 = ( 0 × 1 ) + ( 1 × 1 ) = 0 + 1 = 1 Therefore, the right endpoint approximation for n of 2 is 1. Show the Simpson’s rule as follows: ∫ a b f ( x ) d x ≈ S n = Δ x 3 [ f ( x 0 ) + 4 f ( x 1 ) + 2 f ( x 2 ) + 4 f ( x 3 )

Bundle: Stewart, Essential Calculus: Early Transcendentals, 2nd (hardound) + WebAssign Printed Access Card for Stewart's Essential Calculus: Early ... Multi-Term + WebAssign - Start Smart Guide

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ISBN: 9781133425946

Author: James Stewart

Publisher: Cengage Learning

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Chapter 6.5, Problem 37E

To determine

To sketch:

The graph of a continuous function on [0, 2].

To verify:

For the continuous function, the right endpoint approximation is more accurate than the Simpson’s rule.

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Chapter 6.5, Problem 36E

Chapter 6.5, Problem 39E

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Chapter 6 Solutions

Bundle: Stewart, Essential Calculus: Early Transcendentals, 2nd (hardound) + WebAssign Printed Access Card for Stewart's Essential Calculus: Early ... Multi-Term + WebAssign - Start Smart Guide

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The graph of a continuous function on [ 0 , 2 ] . To verify: For the continuous function, the right endpoint approximation is more accurate than the Simpson’s rule.

Solution Summary: The author illustrates how the right endpoint approximation is more accurate than the Simpson's rule for a continuous function.

Concept explainers

Videos

To sketch:

The graph of a continuous function on [0, 2].

To verify:

For the continuous function, the right endpoint approximation is more accurate than the Simpson’s rule.

Want to see the full answer?

Chapter 6 Solutions

The graph of a continuous function on [ 0 , 2 ] . To verify: For the continuous function, the right endpoint approximation is more accurate than the Simpson’s rule.

Solution Summary: The author illustrates how the right endpoint approximation is more accurate than the Simpson's rule for a continuous function.

Concept explainers

Videos

To sketch: The graph of a continuous function on [0, 2]. To verify: For the continuous function, the right endpoint approximation is more accurate than the Simpson’s rule.

Want to see the full answer?

Chapter 6 Solutions

To sketch:

The graph of a continuous function on [0, 2].

To verify:

For the continuous function, the right endpoint approximation is more accurate than the Simpson’s rule.