Unformatted text preview: . a If f (x) â‰¥ 0 on [a, b], then F is nondecreasing on [a, b] and F (b) âˆ’ F (a) â‰¥ 0.
If f (x) > 0 on [a, b], then F is increasing on [a, b] and F (b) âˆ’ F (a) > 0.
31. Since m â‰¤ f (x) â‰¤ M for all x âˆˆ [a, b], it follows from (5.8.3) that
b b m dx â‰¤ a 33. b f (x) dx â‰¤ 2x a and so m(b âˆ’ a) â‰¤ x dt
âˆš =x
1+ t x 3 âˆ’4
2x dt
âˆš
1+ t M dx a
b x 3 âˆ’4 H (x ) = H (x ) = x f (x) dx â‰¤ M (b âˆ’ a) a H (2) = 2 3x 2
2
âˆ’
+
âˆš
âˆš
1 + 2x
1 + x3 âˆ’ 4
12
2
âˆ’
+
3
3 4
4 x 3 âˆ’4
2x dt
â...
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 Fall '13
 1 L, Tier One, Scaled Composites, Vavg, Scaled Composites White Knight

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