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Mean Value Theorem

# Mean Value Theorem - Average Value of a Function Suppose we...

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Unformatted text preview: Average Value of a Function Suppose we want to find the average temperature over a 24 hour period. We will take n measurements, so T (t1 ) + T (t 2 ) + ... + T (t n ) Average Temperature = n The more measurements we take, the better the approximation. Let’s rewrite this as a Riemann sum. 24 24 Δt = so n = n Δt Then, T (t1 ) + T (t 2 ) + ... + T (t n ) Average Temperature = 24 Δt Δt [T (t1 ) + T (t2 ) + ... + T (tn )] 24 1n = ∑ T (ti )Δt 24 i =1 = Take the limit as n → ∞ to get 1n ∑ T (ti )Δt n → ∞ 24 i =1 Average Temperature = lim 24 1 = T (t ) dt 24 ∫ 0 Theorem: The average value of a function f on an interval [ a, b ] is b 1 fav = f ( x ) dx b−a∫ a Example 1: Find the average value of f ( x ) = x 2 − 1 on [ 0, 3 ]. Example 2: Find the average value of f ( x ) = ⎛ 3 x⎞ cos ⎜ ⎝8⎟ ⎠ x 2 3 on [0, 8]. Mean Value Theorem: If f is continuous on [a, b], there exists a number c in [a, b], such that b 1 f (c ) = f ( x ) dx b − a ∫a = fav 1 Example 3: Use the MVT to find the value c in [1, 3] such that for f ( x ) = , x f (c) = fav . ...
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