Explanation Let us consider a function h ( x ) = f ( x ) . ( b − a ) − x . ( f ( b ) − f ( a ) ) ……………… equation (1) Since f ( x ) is continuous on the interval [ a , b ] , h ( x ) is also continuous on the same interval. Taking the derivative of h ( x ) , we obtain, h ′ ( x ) = f ′ ( x ) . ( b − a ) − ( f ( b ) − f ( a ) ) h ′ ( x ) is differentiable on the interval [ a , b ] since f ( x ) is differentiable in the same interval. By substituting x = a in h ( x ) , i.e., equation (1), we obtain, h ( a ) = f ( a ) . ( b − a ) − a . ( f ( b ) − f ( a ) ) = b . f ( a ) − a . f ( a ) − a . f ( b ) + a . f ( a ) h ( a ) = b . f ( a ) − a . f ( b ) ………………………………………………...

3rd Edition

ISBN: 9781464125263

Author: Jon Rogawski, Colin Adams

Publisher: W.H. Freeman & Co

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Question

Chapter 6.2, Problem 68E

To determine

To prove:

To prove the Mean Value Theorem for Integrals.

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Chapter 6.2, Problem 67E

Chapter 6.3, Problem 1PQ

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Chapter 6 Solutions

Calculus - Standalone book

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To prove: To prove the Mean Value Theorem for Integrals.

Solution Summary: The author explains how the Mean Value Theorem can be proved for a continuous and differentiable function.