5.1 5.3 Average Function Value

5.1 5.3 Average Function Value - x-c ) r on [ c,c + d ]....

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1 5.1, 5.3 Average Function Value Imagine that we measured the outside temperature for a period of time from t = a to t = b and recorded n measurements: T 1 , T 2 , ... T n . What is the average temperature? T av = We multiply and divide by ( b - a ) to transform the expression, so that it looks like a Riemann sum: T av = Temperature doesn’t change abruptly, it is a continuous function of time, thus our calculation is only an approximation. To get closer to the true average we need to increase the number of measurements n . The true average is a limit: T av = This motivates the following definition: Definition Average Function Value If f is integrable on [ a,b ] then its average value (also called mean value ) on [ a,b ] is: f av =
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2 Example 1 Find the average value of the function f ( x ) = x 2 - 1 on [0 , 1] Example 2 Find the average value of the function f ( x ) = x - 2 3 cos ± 3 x 8 ² on [0 , 8] Example 3 To do Find the average value of the function f ( x ) = (
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Unformatted text preview: x-c ) r on [ c,c + d ]. Here c, d, r are constants. Answer: d r r + 1 3 5.4 Mean Value Theorem for Denite Integrals If the function f is continuous on [ a,b ] the property of the denite integrals: 6 Z b a f ( x ) dx 6 guarantees that 6 f av 6 Recall that the Mean Value Theorem for continuous function states that f attains any value between its maximum and minimum. In other words: This proves the following theorem: Theorem Mean Value Theorem for Denite Integrals If f is on [ a,b ], then there exists at least one point x ? inside [ a,b ], such that: Example 4 Find the point(s) x ? that satisfy the conclusion of the Mean Value Theo-rem for the following functions: f ( x ) = x 2-1 on [-1 , 1] 4 f ( x ) = (2 x-3) 3 on [1 , 4] To do f ( x ) = 1 x , on [1 , 3]...
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5.1 5.3 Average Function Value - x-c ) r on [ c,c + d ]....

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