Explanation Given: The given function is f ( x ) = ln ( x 2 + 1 ) ; [ a , b ] = [ 0 , 1 ] Calculation: Since, the function is continuous and differentiable at c ∈ [ 0 , 1 ] Therefore, using Mean Value Theorem; f ' ( c ) = f ( 1 ) − f ( 0 ) 1 − 0 f ' ( c ) = ln ( 1 2 + 1 ) − ln ( 1 2 + 1 ) 1 f ' ( c ) = ln 2 − ln 0 f ' ( c ) = 0.6931 Now, the differentiation of the function is given by f ' ( x ) = d d x ln ( x 2 + 1 ) f ' ( x ) = 1 x 2 + 1 2 x f ' ( x ) = 2 x x 2 + 1 Now, for finding the value of c; f ' ( c ) = 0

5th Edition

ISBN: 9781429241861

Author: Laura Taalman, Peter Kohn

Publisher: W. H. Freeman

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Chapter 3.1, Problem 57S

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The reason why the given function ‘f’ satisfies the hypothesis of the Mean Value Theorem on the given interval [a, b] and the exact values c∈(a,b) which satisfies the conclusion of the Mean Value Theorem

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Chapter 3.1, Problem 56S

Chapter 3.1, Problem 58S

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The reason why the given function ‘f’ satisfies the hypothesis of the Mean Value Theorem on the given interval [a, b] and the exact values c ∈ ( a , b ) which satisfies the conclusion of the Mean Value Theorem

Solution Summary: The author explains that the given function satisfies the hypothesis of the Mean Value Theorem and the exact values of cin (a,b).

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The reason why the given function ‘f’ satisfies the hypothesis of the Mean Value Theorem on the given interval [a, b] and the exact values c∈(a,b) which satisfies the conclusion of the Mean Value Theorem

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Chapter 3 Solutions