# hw3 - Homework 3 – Math 118B Winter 2010 Due on Thursday...

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Unformatted text preview: Homework 3 – Math 118B, Winter 2010 Due on Thursday, February 4, 2010 1. Let α be a ﬁxed increasing function on [a, b]. For u ∈ R(α), deﬁne 1/2 b u 2 2 = |u| dα . a Suppose f, g, h ∈ R(α), and prove the triangle inequality f −h 2 ≤ f −g 2 + g − h 2, as a consequence of the Schwarz inequality. 2. Let α be a ﬁxed increasing function on [a, b]. Suppose f ∈ R(α) and ǫ > 0. Prove that there exists a continuous function g on [a, b] such that f − g 2 < ǫ. 3. Deﬁne x+1 sin(t2 ) dt. f (x) = x (a) Prove that |f (x)| < 1/x for x > 0. (b) Prove that 2xf (x) = cos(x2 ) − cos[(x + 1)2 ] + r(x) where |r(x)| < c/x for some constant c. (c) Find the upper and lower limits of xf (x), as x → ∞. (d) Does ∞ sin(t2 ) dt 0 converge? 4. Suppose f is a real, continuously diﬀerentiable function on [a, b], f (a) = f (b) = 0, and b f 2 (x) dx = 1. a 1 2 Prove that b a and that 1 xf (x)f ′ (x) dx = − , 2 b b [f ′ (x)]2 dx · a a 1 x2 f 2 (x) dx ≥ . 4 5. Suppose f ∈ R on [a, b] for every b > a, where a is ﬁxed. Deﬁne b ∞ f (x) dx = lim b→∞ a f (x) dx a if the limit exists and is ﬁnite. In that case, we say that the integral on the left converges. If it also converges after f has been replaced by |f |, it is said to converge absolutely Assume that f (x) ≥ 0 and that f decreases monotonically on [1, ∞). Prove that ∞ f (x) dx 1 converges if and only if ∞ f (n) n=1 converges. ...
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hw3 - Homework 3 – Math 118B Winter 2010 Due on Thursday...

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