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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 '06
 EDeSturler
 Approximation, Mean Value Theorem, Trigraph, ΔT, 24 hour, average temperature

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