2.4. Average Value of a Function (Mean Value Theorem) 2.4.1 ...
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For each problem, find the average value of the function over the given interval. Then, find the values of c that satisf
f(x)dx . 2.4.2. The Mean Value Theorem for Integrals. If f is con- tinuous on [a, b], then there exists a number c in [a
2.4. AVERAGE VALUE OF A FUNCTION (MEAN VALUE THEOREM)
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2.4. Average Value of a Function (Mean Value Theorem) 2.4.1. Average Value of a Function. The average value of finitely many numbers y1 , y2 , . . . , yn is defined as yave =
y1 + y2 + · · · + yn . n
The average value has the property that if each of the numbers y1 , y2 , . . . , yn is replaced by yave , their sum remains the same: (n times)
z }| { y1 + y2 + · · · + yn = yave + yave + · · · + yave Analogously, the average value of a function y = f (x) in the interval [a, b] can be defined as the value of a constant fave whose integral over [a, b] equals the integral of f (x): Z b Z b f (x) dx = fave dx = (b − a) fave . a
a
Hence: fave
1 = b−a
Z
b
f (x) dx .
a
2.4.2. The Mean Value Theorem for Integrals. If f is continuous on [a, b], then there exists a number c in [a, b] such that Z b 1 f (x) dx , f (c) = fave = b−a a i.e.,
Z
b a
f (x) dx = f (c)(b − a) .
Example: Assume that in a certain city the temperature (in ◦ F) t hours after 9 A.M. is represented by the function T (t) = 50 + 14 sin
πt . 12
Find the average temperature in that city during the period from 9 A.M. to 9 P.M.
2.4. AVERAGE VALUE OF A FUNCTION (MEAN VALUE THEOREM)
