6.5 Average Value of a Function

6.5 Average Value of a Function - f on the given interval(b...

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6.5 Average Value of a Function If f is continuous on [ a , b ], then the average value of f on this interval is given by - b a dx x f a b ) ( 1
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Mean Value Theorem for Integrals If f is continuous on [ a , b ], then there exists a number c in [ a , b ] such that - = = - = b a ave b a dx x f a b f c f a b c f dx x f ) ( 1 ) ( ) )( ( ) ( ) ( x f y = ave f c f = ) ( a b c x y Note: The area under f is equal to the area of the rectangle whose height is the average value.
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Example 1 Find the average value of the function on the given interval. ] 4 / , 0 [ , tan sec ) ( π θ θ θ = f [ ] [ ] 1 2 4 sec 4 tan sec 0 4 1 4 / 0 4 / 0 - = = - = π θ π θ θ θ π π π d f ave Solution:
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Example 2
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Unformatted text preview: f on the given interval. (b) Find c such that (c) Sketch the graph of f and a rectangle whose area is the same as the area under the graph of f . ). ( c f f ave = ] 2 , [ , ) ( x e x f = [ ] ( 29 ( 29 ( 29 ( 29 1.16 1 2 1 ln 1 2 1 (b) 1 2 1 2 1 2 1 (a) 2 2 2 2 2 ≈ -= ⇔ =-⇔ =-= = = ∫ e c e e c f f e e dx e f c ave x x ave Solution: 1 4 1 c 2 x y x e y =...
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