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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
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 = − ,
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 If it also converges after f has been replaced by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, ∞).
f (x) dx
1 converges if and only if
∞ f (n)
n=1 converges. ...
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- Fall '09