Explanation Given: The function f ( x ) = 3 x + 1 , and the interval [ 0 , 4 ] . Concept Used: The average value of an integrable functionf on the interval [ a , b ] , is given by 1 b − a ∫ a b f ( x ) d x . The area that is bounded by the graph of f ( x ) above the x-axis is positive. The area that is bounded by the graph of f ( x ) below the x-axis is negative. Power rule of integration: ∫ x n d x = x n + 1 n + 1 Calculation: Observe that the average value of an integrable function f on the interval [ a , b ] , is given by 1 b − a ∫ a b f ( x ) d x , so using this concept the exact average value of f ( x ) = 3 x + 1 on the interval [ 0 , 4 ] is given by, f avg = 1 4 − 0 ∫ 0 4 ( 3 x + 1 ) d x = 1 4 [ 3 2 x 2 + x ] 0 <

14th Edition

ISBN: 9781464151088

Author: Associate Professor Laura Taalman, Peter Kohn

Publisher: W.H. Freeman & Company

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Chapter 4.6, Problem 57S

To determine

The exact average value of f(x)=3x+1 on the interval [0, 4] using definite integrals and the Fundamental Theorem of Calculus.

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Chapter 4.6, Problem 56S

Chapter 4.6, Problem 58S

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The exact average value of f ( x ) = 3 x + 1 on the interval [ 0 , 4 ] using definite integrals and the Fundamental Theorem of Calculus.

Solution Summary: The author calculates the exact average value of f(x)=3x+1 using definite integrals and the Fundamental Theorem of Calculus.

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The exact average value of f(x)=3x+1 on the interval [0, 4] using definite integrals and the Fundamental Theorem of Calculus.

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