Explanation The statement is “If the area under the graph of f on [ a , b ] is equal to both the left sum L n and right sum R n for some positive integer n , then f is a constant on [ a , b ] .” In order to say the given statement is false, its enough to give one counter example. Counter example: Let f ( x ) = x 2 on [ − 3 , 3 ] , and a positive integer be n = 3 . Then, the left sum L n and right sum L n is calculated as follows. In order to check the equality of the area under the curve and left sum for arbitrary choice of a positive integer, first find the right and left sum. Divide the interval [ − 3 , 3 ] into 3 subintervals. Then, the equal length. Δ x = 3 + 3 3 = 6 3 = 2 Therefore, the subintervals are [ − 3 , − 1 ] , [ − 1 , 1 ] , and [ 1 , 3 ] , respectively. The value of left end points are − 3 , − 1 , and 1 , from the above subintervals. Calculate the value of f ( x ) = x 2 at x = − 3 . f ( − 3 ) = ( − 3 ) 2 = 9 Calculate the value of f ( x ) = x 2 at x = − 1 . f ( − 1 ) = ( − 1 ) 2 = 1 Calculate the value of f ( x ) = x 2 at x = 1 . f ( 1 ) = ( 1 ) 2 = 1 Use the formula mentioned below obtain the left sum. Left sum is L n = f ( c 1 ) Δ x + f ( c 2 ) Δ x + … + f ( c n − 1 ) Δ x , where c n ∈ [ x n − 1 , x n ]

Calculus for Business, Economics, Life Sciences, and Social Sciences (14th Edition)

14th Edition

ISBN: 9780134668574

Author: Raymond A. Barnett, Michael R. Ziegler, Karl E. Byleen, Christopher J. Stocker

Publisher: PEARSON

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Chapter 5.4, Problem 59E

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To find: Whether the given statement is true or false.

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Chapter 5.4, Problem 58E

Chapter 5.4, Problem 60E

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Chapter 5 Solutions

Calculus for Business, Economics, Life Sciences, and Social Sciences (14th Edition)

Ch. 5.1 - Matched Problem 1 Note that ddx(x3)=3x2 A. Find...Ch. 5.1 - Find each indefinite integral: (A) 2dx (B) 16etdt...Ch. 5.1 - Matched Problem 3 Find each indefinite integral:...Ch. 5.1 - Matched Problem 4 Find the equation of the curve...Ch. 5.1 - Matched Problem 5 Find the revenue function R(x)...Ch. 5.1 - Matched Problem 6 There are currently 64,000...Ch. 5.1 - Explore and Discuss 1 Formulas 1, 2, and 3 do not...Ch. 5.1 - In Problems 18, write each function as a sum of...Ch. 5.1 - In Problems 18, write each function as a sum of...Ch. 5.1 - In Problems 18, write each function as a sum of...

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Whether the given statement is true or false.

Solution Summary: The author explains that if the area under the graph of f on left[a,bright] is equal to both the left and right sums, the statement is false.

To find: Whether the given statement is true or false.

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Chapter 5 Solutions