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Find the mean value of f(x) = x2 + 5 on [0, 5] |
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Answer» Answer: The average value is = 1
Explanation: The average value of a function f ( x ) over an interval [ a , ] is ¯ x = 1 b − a ∫ b a f ( x ) d x
Here, f ( x ) = ( x − 3 ) 2 = x 2 − 6 x + 9
and [ a , b ] = [ 2 , 5 ]
Therefore, ¯ x = 1 5 − 2 ∫ 5 2 ( x − 3 ) 2 d x
= 1 3 ∫ 5 2 ( x 2 − 6 x + 9 ) d x
= 1 3 [ x 3 3 − 6 x 2 2 + 9 x ] 5 2
= 1 3 ( ( 125 3 − 75 + 45 ) − ( 8 3 − 12 + 18 ) ) = 1 3 ( 117 3 − 36 )
= 1 3 ⋅ 3
= 1 |
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