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Homework 14 Calculus 1
1. Given f, f1, f2 ∈ R([a, b]), prove that
• If f1(x) ≤ f2(x) ∀x ∈ [a, b], then Rb
a f1(x)dx ≤Rb
a f2(x)dx.
• Rb
a −f (x)dx = −Rb
a f (x)dx.
2. • Given f ∈ R([a, b]) and c ∈ [a, b], prove that Z b
a
f (x)dx = Z c
a
f (x)dx + Z b
c
f (x)dx.
• Prove that if |f (x)| ≤ M ∀x ∈ [a, b], then |Rb
af (x)dx| ≤ M (b − a).
3. Rudin Chapter 6, Problem 8.
4. Suppose that
f (x) = Z x
0
f (t)dt.
• Prove that f (x) = Cex for some C ∈ R.
• Prove that f = 0.
5. Prove the Cauchy-Schwartz inequality for integrals:
Z b a
f (x)g(x)dx
2
≤ Z b
a
[f (x)]2dx Z b
a
[g(x)]2dx.
(Hint: start by showing that Rb
a(f − tg)2dx ≥ 0 for ALL t ∈ R.) 6. Let f (x) be Lipschitz continuous on [0, 1]. That is, ∃M such that
|f (x) − f (y)| < M |x − y| ∀x, y ∈ [0, 1].
Prove that for all n ∈ N,
Z 1 0
f (x)dx − 1 n
n
X
j=1
f j n
< M 2n,
7. Given differentiable functions G, H and continuous function f , prove that d
dx Z G(x)
H(x)
f (t)dt = f (G(x))G0(x) − f (H(x))H0(x).
8. Salas 5.3: 2, 10, 12, 29, 33.
9. Salas 5.4: 8, 17, 27, 31, 46, 62, 64.
