# average - Averageofafunc on Deni&on.Letfbeafunc...

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Defini&on. Let f be a func+on which is con+nuous on the closed interval [a, b]. The average value of f from x = a to x = b is the integral 1 b a f ( x ) dx a b Average of a func+on (this is a number!)

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f ave = 1 b a f ( x ) dx = a b = 1 2 ( 1) ( x 2 + 1) dx = 1 2 = 1 3 x + x 3 3 1 2 = 2 Find the average of the func+on f(x)=x 2 +1 over [‐1,2] An example of average
Find the average of over [‐1,5/2] Another example of average b‐a f(x)

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Suppose that f is a con+nuous func+on on [a,b], and that f is non‐nega+ve. f ave = 1 b a f ( x ) dx a b vcnvvcvxv f ave ( b a ) = f ( x ) dx a b fdfdfdffffdd BvhvghM f ave b‐a f ave Geometric meaning of average (for f≥0)
f ave f ave Geometric meaning of average (for f≥0) The green and the pink region must compensate each other

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Geometric meaning of average for f≥0 b‐a ( b a ) f ave f ( x ) dx a b Example : f(x)=1+ x 2 over [‐1,2]
Geometric meaning of average for a (possibly non‐posi+ve) f Example: f ave =‐9/5 f ave =

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• Fall '09
• FAMIGLIETTI,C
• Calculus, Topology, dx, Geometric meaning, Mean Value Theorem

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