Explanation Given Information: The provided formulae are S L = 14 3 − 6 n + 4 3 n 2 and S R = 14 n 2 + 18 n + 4 3 n 2 . Calculation: Consider the formulae, S L = 14 3 − 6 n + 4 3 n 2 and S R = 14 n 2 + 18 n + 4 3 n 2 Recall that to calculate the value of a function f ( x ) at x = a , substitute a for x in the formula. First, calculate S L ( 10 ) . Substitute 10 for n in S L = 14 3 − 6 n + 4 3 n 2 . S L ( 10 ) = 14 3 − 6 10 + 4 3 ( 10 ) 2 = 14 3 − 6 10 + 4 300 = 1224 300 = 4.08 Thus, the value of S L ( 10 ) is 4.08 . Next, calculate S R ( 10 ) . Substitute 10 for n in S R = 14 n 2 + 18 n + 4 3 n 2 . S R = 14 ( 10 ) 2 + 18 ( 10 ) + 4 3 ( 10 ) 2 = 1400 + 180 + 4 300 = 1584 300 = 5.28 Thus, the value of S R ( 10 ) is 5.28 . Hence, the value of S L ( 10 ) is 4.08 and the value of S R ( 10 ) is 5.28 . Since, the function f ( x ) = x 2 + x defined over the interval [ 0 , 2 ] is an increasing function. Thus, the area calculated at any point in each subinterval is between the values of S L ( 10 ) S R ( 10 ) . Let A be the area of the function f ( x ) = x 2 + x . Thus, S L ( 10 ) ≤ A ≤ S R ( 10 ) . Consider that n → ∞ . Recall that for functions f ( x ) and g ( x ) , lim n → ∞ [ f ( x ) ± g ( x ) ] = lim n → ∞ f ( x ) ± lim n → ∞ g ( x ) . First, calculate lim n → ∞ S L . Substitute S L = 14 3 − 6 n + 4 3 n 2 . lim n → ∞ S L = lim n → ∞ ( 14 3 − 6 n + 4 3 n 2 ) = lim n → ∞ 14 3 − lim n → ∞ 6 n + lim n → ∞ 4 3 n 2 = 14 3 − 6 lim n → ∞ 1 n + 4 3 lim n → ∞ ( 1 n ) 2 Use lim n → ∞ 1 n = 0 to simplify further

WebAssign Printed Access Card for Harshbarger/Reynolds' Mathematical Applications for the Management, Life, and Social Sciences, 12th Edition, Multi-Term

12th Edition

ISBN: 9781337630542

Author: Ronald J. Harshbarger, James J. Reynolds

Publisher: Cengage Learning

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Chapter 13.1, Problem 13E

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The comparison between the approximated area of the increasing function f(x)=x2+x over the interval from x=0 to x=2 using n=10 and any other point in each subinterval with SL(10) and SR(10). Also, calculate the area as n→∞ if any other point in each subinterval is used.

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Chapter 13.1, Problem 12E

Chapter 13.1, Problem 14E

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Ch. 13.1 - CHECKPOINT 1. For the interval [0, 2]. determine...Ch. 13.1 - Prob. 2CP Ch. 13.1 - Prob. 1E Ch. 13.1 - Prob. 2E Ch. 13.1 - Prob. 3E Ch. 13.1 - Prob. 4E Ch. 13.1 - Prob. 5E Ch. 13.1 - Prob. 6E Ch. 13.1 - Prob. 7E Ch. 13.1 - Prob. 8E

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The comparison between the approximated area of the increasing function f ( x ) = x 2 + x over the interval from x = 0 to x = 2 using n = 10 and any other point in each subinterval with S L ( 10 ) and S R ( 10 ) . Also, calculate the area as n → ∞ if any other point in each subinterval is used.

Solution Summary: The author explains how to calculate the approximated area of the increasing function f(x)=x

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The comparison between the approximated area of the increasing function f(x)=x2+x over the interval from x=0 to x=2 using n=10 and any other point in each subinterval with SL(10) and SR(10). Also, calculate the area as n→∞ if any other point in each subinterval is used.

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Chapter 13 Solutions