Explanation Given Information: The function is f ( x ) = cos ( 2 x ) . Calculation: Consider the provided function, f ( x ) = cos ( 2 x ) , To evaluate the average of the function f ( x ) = cos ( 2 x ) over the interval [ 0 , π 4 ] , apply the formula, f ¯ = 1 b − a ∫ a b f ( x ) d x , Substitute a = 0 , b = π 4 in the above formula, f ¯ = 1 π 4 − 0 ∫ 0 π / 4 cos ( 2 x ) d x = 4 π ∫ 0 π / 4 cos ( 2 x ) d x Now, evaluate the integral of 4 π ∫ 0 π / 4 cos ( 2 x ) d x . 4 π ∫ 0 π / 4 cos ( 2 x ) d x = 4 π [ 1 2 sin ( 2 x ) ] 0 π / 4 = 2 π [ sin ( 2 x ) ] 0 π / 4 Now evaluate the value of integral at upper limit π 4 , 2 π [ sin x ] 0 π / 4 = 2 π [ sin ( 2 × π 4 ) ] = 2 π [ sin ( π 2 ) ] Substitute sin ( π 2 ) = 1 . 2 π [ sin ( π 2 ) ] = 2 π × 1 = 2 π Now evaluate the value of integral at lower limit 0, 2 π [ sin x ] 0 π / 4 = 2 π [ sin ( 2 × 0 ) ] = 2 π [ sin ( 0 ) ] Substitute sin ( 0 ) = 0 . 2 π [ sin ( 0 ) ] = 2 π × 0 2 π [ sin ( 0 ) ] = 0 Subtract the value of lower limit 0 from upper limit 2 π . 4 π ∫ 0 π / 4 cos ( 2 x ) d x = 2 π − 0 = 2 π Thus, the average of the function is 2 π . Graph: Consider the function f ( x ) = cos ( 2 x ) and f ¯ = 2 π

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Chapter 16.3, Problem 54E

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To calculate: The average of the function f(x)=cos(2x) over the interval [0,π4] and sketch the graph of the function and its average.

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Chapter 16.3, Problem 53E

Chapter 16.3, Problem 55E

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Chapter 16 Solutions

EBK FINITE MATH AND APPLIED CALCULUS

Ch. 16.1 - Prob. 1E Ch. 16.1 - Prob. 2E Ch. 16.1 - Prob. 3E Ch. 16.1 - Prob. 4E Ch. 16.1 - Prob. 5E Ch. 16.1 - Prob. 6E Ch. 16.1 - Prob. 7E Ch. 16.1 - Prob. 8E Ch. 16.1 - Prob. 9E Ch. 16.1 - Prob. 10E

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The average of the function f ( x ) = cos ( 2 x ) over the interval [ 0 , π 4 ] and sketch the graph of the function and its average.

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To calculate: The average of the function f(x)=cos(2x) over the interval [0,π4] and sketch the graph of the function and its average.

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Chapter 16 Solutions