Unformatted text preview: , it must be the case that c = 2 . Example 2 (Exercise 6.5.13 in the text) If f is continuous and R 3 1 f ( x ) dx = 8 , show that f takes on the value 4 at least once on teh interval [1,3] . Solution. By the mean value theorem, there is a number c in [1,3] such that f ( c ) = f ave = 1 31 Z 3 1 f ( x ) dx = 1 2 · 8 = 4. Example 3 (Imitation of exercise 6.5.14 in the text) Find the numbers b such that the average value of f ( x ) = 6 x 2 + 10 x8 on the interval [0, b ] is equal to 4 , where b > . Solution. The average value of f on the interval [0, b ] is given by f ave = 1 bZ b (6 x 2 + 10 x8) dx = 2 b 2 + 5 b8. Thus f ave = 4 if and only if 2 b 2 + 5 b8 = 4 or 2 b 2 + 5 b12 = (2 b3)( b + 4) = . This implies that either b = 3 2 ....
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 Fall '09
 GELINE
 Calculus, Geometry, Mean Value Theorem

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